Design Guide on the ACI 318 Building Code Requirements for Structural Concrete 1943961514, 9781943961511

Table of contents :
Cover
Contents
Foreword
Acknowledgements
Photo Credits
Chapter 1
Introduction
1.1 Overview
1.2 Example Buildings
1.2.1 Building #1 – 5-Story Office Building
1.2.2 Building #2 – 12-Story Emergency Operations Center
1.2.3 Building #3 – 16-Story Residential Building
1.2.4 Building #4 – 30-Story Office Building
1.3 Organization of This Design Guide
Chapter 2 Material Requirements and Strength Reduction Factors
2.1 Overview
2.2 Material Requirements
2.2.1 Concrete Design Properties
Specified Compressive Strength
Modulus of Elasticity
Modulus of Rupture
Lightweight Concrete Modification Factor
2.2.2 Nonprestressed Steel Reinforcement
Material Properties
Design Properties
2.2.3 Headed Shear Stud Reinforcement
2.2.4 Durability of Steel Reinforcement
Specified Concrete Cover
Nonprestressed Coated Reinforcement
2.3 Strength Reduction Factors
2.3.1 Overview
2.3.2 Strength Reduction Factors Based On Action or Structural Element
2.3.3 Strength Reduction Factors For Moment, Axial Force, Or Combined Moment and Axial Force
2.3.4 Strength Reduction Factors For Shear In Structures Relying On Special Moment Frames and SpecialStructural Walls
Chapter 3
Design Loads and Load Combinations
3.1 Overview
3.2 Design Loads
3.3 Seismic Design Category
3.4 Live Load Reduction
3.5 Load Factors and Combinations
3.6 Determination of Wind Forces
3.7 Determination of Seismic Forces
3.7.1 Seismic Forces on the SFRS
3.7.2 Seismic Forces on Diaphragms, Chords, and Collectors
3.8 Examples
3.8.1 Example 3.1 – Determination of Wind Forces: Building #1
3.8.2 Example 3.2 – Determination of Wind Forces: Building #2
3.8.3 Example 3.3 – Determination of Wind Forces: Building #3
3.8.4 Example 3.4 – Determination of Wind Forces: Building #4
3.8.5 Example 3.5 – Determination of the Seismic Design Category: Building #1
3.8.6 Example 3.6 – Determination of the Seismic Design Category: Building #2
3.8.7 Example 3.7 – Determination of the Seismic Design Category: Building #3
3.8.8 Example 3.8 – Determination of the Seismic Design Category: Building #4
3.8.9 Example 3.9 – Determination of Seismic Forces: SFRS of Building #1 (Framing Option A)
3.8.10 Example 3.10 – Determination of Seismic Forces: SFRS of Building #2
3.8.11 Example 3.11 – Determination of Seismic Forces: SFRS of Building #3
3.8.12 Example 3.12 – Determination of Seismic Forces: SFRS of Building #4
3.8.13 Example 3.13 – Determination of Seismic Forces: Diaphragms of Building #1 (Framing Option A)
3.8.14 Example 3.14 – Determination of Seismic Forces: Diaphragms of Building #2
3.8.15 Example 3.15 – Determination of Seismic Forces: Diaphragms of Building #3
3.8.16 Example 3.16 – Determination of Seismic Forces: Diaphragms of Building #4
Chapter 4
One-way Slabs
4.1 Overview
4.2 Minimum Slab Thickness
4.3 Required Strength
4.3.1 Analysis Methods
4.3.2 Critical Sections for Flexure and Shear
4.4 Design Strength
4.4.1 General
4.4.2 Nominal Flexural Strength
4.4.3 Nominal Shear Strength
4.5 Determination of Required Reinforcement
4.5.1 Required Flexural Reinforcement
4.5.2 Minimum Shrinkage and Temperature Reinforcement
4.6 Reinforcement Detailing
4.6.1 Concrete Cover
4.6.2 Minimum Spacing of Flexural Reinforcing Bars
4.6.3 Maximum Spacing of Flexural Reinforcing Bars
4.6.4 Selection of Flexural Reinforcement
4.6.5 Development of Flexural Reinforcement
Development of Deformed Bars in Tension
Development of Standard Hooks in Tension
Development of Headed Deformed Bars in Tension
Development of Mechanically Anchored Deformed Bars in Tension
Development of Positive and Negative Flexural Reinforcement
Negative Flexural Reinforcement
Positive Flexural Reinforcement
4.6.6 Splices of Reinforcement
Overview
Lap Splices
Mechanical Splices
Welded Splices
4.6.7 Structural Integrity Reinforcement
4.6.8 Recommended Flexural Reinforcement Details
4.7 Design Procedure
4.8 Examples
4.8.1 Example 4.1 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2,Normalweight Concrete
4.8.2 Example 4.2 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2,Lightweight Concrete
4.8.3 Example 4.3 – Determination of Required Reinforcement: One-way Slab System, Building #2
4.8.4 Example 4.4 – Determination of Lap Splice Lengths: One-way Slab System, Building #2
4.8.5 Example 4.5 – Determination of Reinforcement Details: One-way Slab System, Building #2
Chapter 5
Two-way Slabs
5.1 Overview
5.2 Minimum Slab Thickness
5.2.1 Overview
5.2.2 Flat Plates
5.2.3 Flat Slabs
5.2.4 Two-way Beam-Supported Slabs
5.2.5 Two-way Joists
5.2.6 Flat Plate Voided Concrete Slabs
5.3 Required Strength
5.3.1 Analysis Methods
5.3.2 Critical Sections for Flexure
5.3.3 Critical Sections for Shear
One-way Shear
Two-way Shear
Section Properties of Critical Sections
5.3.4 Direct Design Method
Overview
Determination of Factored Bending Moments in a Design Strip
Determination of Factored Moments in Columns and Walls
Determination of Factored Shear in Slabs with Beams
5.3.5 Lateral Loads
5.4 Design Strength
5.4.1 General
5.4.2 Nominal Flexural Strength
5.4.3 Nominal One-way Shear Strength
5.4.4 Nominal Two-way Shear Strength
Overview
Two-way Shear Strength Provided by Concrete in Slabs without Shear Reinforcement
Two-way Shear Strength Provided by Concrete in Slabs with Shear Reinforcement
Two-way Shear Strength Provided by Single- or Multiple-leg Stirrups
Two-way Shear Strength Provided by Headed Shear Stud Reinforcement
Summary of Nominal Two-way Shear Strength Requirements
5.4.5 Openings in Two-way Slab Systems
5.5 Determination of Required Reinforcement
5.5.1 Required Flexural Reinforcement
5.5.2 Required Shear Reinforcement
Stirrups
Headed Shear Stud Reinforcement
5.6 Reinforcement Detailing
5.6.1 Concrete Cover
5.6.2 Minimum Spacing of Flexural Reinforcing Bars
5.6.3 Maximum Spacing of Flexural Reinforcing Bars
5.6.4 Selection of Flexural Reinforcement
5.6.5 Corner Restraint in Slabs
5.6.6 Termination of Flexural Reinforcement
5.6.7 Splices of Reinforcement
5.6.8 Structural Integrity Reinforcement
5.6.9 Shear Reinforcement Details
5.7 Design Procedure
5.8 Examples
5.8.1 Example 5.1 – Determination of Minimum Slab Thickness: Flat Plate System, Building #1(Framing Option A)
5.8.2 Example 5.2 – Determination of Minimum Slab Thickness: Flat Plate System with Edge Beams,Building #1 (Framing Option B)
5.8.3 Example 5.3 – Determination of Minimum Slab Thickness: Two-way Beam-Supported Slab System,Building #1 (Framing Option C)
5.8.4 Example 5.4 – Determination of Minimum Slab Thickness: Flat Slab System With Edge Beams,Building #1 (Framing Option D)
5.8.5 Example 5.5 – Determination of Minimum Thickness: Two-way Joist System,Building #1 (Framing Option E)
5.8.6 Example 5.6 – Determination of Minimum Slab Thickness: Flat Plate System, Building #3
5.8.7 Example 5.7 – Determination of Required Flexural Reinforcement: Flat Plate System,Building #1 (Framing Option A), SDC A
5.8.8 Example 5.8 – Determination of Required Flexural Reinforcement: Flat Plate System With EdgeBeams, Building #1 (Framing
5.8.9 Example 5.9 – Determination of Required Flexural Reinforcement: Two-way Beam-Supported SlabSystem, Building #1 (Framing Option C), SDC A
5.8.10 Example 5.10 – Determination of Required Flexural Reinforcement: Flat Slab System With EdgeBeams, Building #1 (Framing Option D), SDC A
5.8.11 Example 5.11 – Determination of Required Flexural Reinforcement: Two-way Joist System,Building #1 (Framing Option E), SDC A
5.8.12 Example 5.12 – Determination of Required Flexural Reinforcement: Flat Plate System, Building #1(Framing Option A), SDC B
5.8.13 Example 5.13 – Check of Shear Strength Requirements: Flat Plate System, Building #1(Framing Option A), SDC B
5.8.14 Example 5.14 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (FramingOption A), SDC B, Shear Cap
5.8.15 Example 5.15 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (FramingOption A), SDC B, Slab Opening
5.8.16 Example 5.16 – Check of Shear Strength Requirements: Flat Slab System With Edge Beams,Building #1 (Framing Option D), SDC A, Circular Columns
5.8.17 Example 5.17 – Determination of Shear Reinforcement: Flat Plate System, Building #1 (FramingOption A), SDC B, Stirrups
5.8.18 Example 5.18 – Determination of Shear Reinforcement: Flat Plate System, Building #1(Framing Option A), SDC B, Headed Shear Studs
Chapter 6
Beams
6.1 Overview
6.2 Sizing the Cross-Section
6.2.1 Determining the Beam Depth
6.2.2 Determining the Beam Width
6.2.3 General Guidelines for Sizing Beams for Economy
6.3 Required Strength
6.3.1 Analysis Methods
Overview
Bending Moments and Shear Forces
Torsional Moments
6.3.2 Critical Sections for Flexure, Shear, and Torsion
6.3.3 Redistribution of Moments in Continuous Flexural Members
6.4 Design Strength
6.4.1 General
6.4.2 Nominal Flexural Strength
Rectangular Sections with Tension Reinforcement Only
Rectangular Sections with Tension and Compression Reinforcement
T-Beams and Inverted L-Beams with Tension Reinforcement
6.4.3 Nominal Shear Strength
Overview
Nominal Shear Strength Provided by Concrete
Nominal Shear Strength Provided by Shear Reinforcement
6.4.4 Nominal Torsional Strength
6.5 Determination of Required Reinforcement
6.5.1 Required Flexural Reinforcement
Rectangular Sections with Tension Reinforcement Only
Rectangular Sections with Tension and Compression Reinforcement
T-Beams and Inverted L-Beams with Tension Reinforcement
6.5.3 Required Torsion Reinforcement
Transverse Reinforcement
Longitudinal Reinforcement
6.5.4 Reinforcement Requirements for Combined Flexure, Shear, and Torsion
6.6 Reinforcement Detailing
6.6.1 Concrete Cover
6.6.2 Flexural Reinforcement Spacing
Minimum Spacing of Flexural Reinforcing Bars
Maximum Spacing of Flexural Reinforcing Bars for Crack Control
Distribution of Tension Reinforcement in Flanges of T-Beams
Crack Control Reinforcement in Deep Flexural Members
6.6.3 Selection of Flexural Reinforcement
6.6.4 Development of Flexural Reinforcement
Overview
Development of Deformed Bars in Tension
Development of Standard Hooks in Tension
Development of Headed Deformed Bars in Tension
Development of Mechanically Anchored Deformed Bars in Tension
Development of Positive and Negative Flexural Reinforcement
Negative Flexural Reinforcement
Positive Flexural Reinforcement
6.6.5 Splices of Deformed Reinforcement
Overview
Lap Splices
Mechanical Splices
Welded Splices
6.6.6 Longitudinal Torsional Reinforcement
6.6.7 Transverse Reinforcement
Overview
Shear Reinforcement
Torsion Reinforcement
6.6.8 Structural Integrity Reinforcement
6.6.9 Flexural Reinforcement Requirements for SDC B
6.6.10 Recommended Flexural Reinforcement Details
6.7 Deflections
6.7.1 Overview
6.7.2 Immediate Deflections
Uncracked Sections
Cracked Sections
Effective Moment of Inertia
Approximate Immediate Deflections
6.7.3 Time-Dependent Deflections
6.7.4 Maximum Permissible Calculated Deflections
6.8 Design Procedure
6.9 Examples
6.9.1 Example 6.1 – Determination of Beam Size: Building #1 (Framing Option C), Beam is Not Part of theLFRS, SDC A
6.9.2 Example 6.2 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C),Beam is Not Part of the LFRS, SDC A, Single Layer of Tension Reinforcement
6.9.3 Example 6.3 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C),Beam is Not Part of the LFRS, SDC A
6.9.4 Example 6.4 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C),Beam is Not Part of the LFRS, SDC A
6.9.5 Example 6.5 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is NotPart of the LFRS, SDC A
6.9.6 Example 6.6 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C),Beam is Part of the LFRS, SDC A, Multiple Layers of Tension Reinforcement
6.9.7 Example 6.7 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C),Beam is Part of the LFRS, SDC A
6.9.8 Example 6.8 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C),Beam is Part of the LFRS, SDC A
6.9.9 Example 6.9 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is Part ofthe LFRS, SDC A, Includes Compression Reinforcement
6.9.10 Example 6.10 – Determination of Joist Size: Joist in Building #2, Joist is Not Part of the LFRS, SDC C
6.9.11 Example 6.11 – Determination of Flexural Reinforcement: Joist in Building #2, Joist Not Part ofthe LFRS, SDC C
6.9.12 Example 6.12 – Determination of Shear Reinforcement: Joist in Building #2, Joist is Not Part ofthe LFRS, SDC C
6.9.13 Example 6.13 – Determination of Reinforcement Details: Joist in Building #2, Joist is Not Part ofthe LFRS, SDC C
6.9.14 Example 6.14 – Determination of Deflections: Joist in Building #2, Typical Floor, Joist is Not Part ofthe LFRS, SDC C
6.9.15 Example 6.15 – Determination of Beam Size: Edge Beam in Building #2, Beam is Not Part of theLFRS, SDC C
6.9.16 Example 6.16 – Determination of Flexural Reinforcement: Edge Beam in Building #2, Beam is NotPart of the LFRS, SDC C
6.9.17 Example 6.17 – Determination of Shear Reinforcement: Edge Beam in Building #2, Beam is Not Partof the LFRS, SDC C
6.9.18 Example 6.18 – Determination of Torsion Reinforcement: Edge Beam in Building #2, Second-FloorLevel, Beam is Not Part of the LFRS, SDC C
6.9.19 Example 6.19 – Design for Combined Flexure, Shear, and Torsion: Edge Beam in Building #2, Beam isNot Part of the LFRS, SDC C
6.9.20 Example 6.20 – Determination of Reinforcement Details: Edge Beam in Building #2, Beam is Not Partof the LFRS, SDC C
Chapter 7
Columns
7.1 Overview
7.2 Dimensional Limits
7.3 Required Strength
7.3.1 Analysis Methods
Overview
Linear Elastic First-Order Analysis
Linear Elastic Second-Order Analysis
Inelastic Analysis
Finite Element Analysis
Section Properties
7.3.2 Factored Axial Force and Moment
7.3.3 Slenderness Effects
Overview
Columns in Nonsway and Sway Frames
Consideration of Slenderness Effects
Moment Magnification Method
Overview
Nonsway Frames
Sway Frames
7.3.4 Required Shear Strength for Columns in Buildings Assigned to Seismic Design Category B
7.4 Design Strength
7.4.1 General
7.4.2 Nominal Axial Strength
Nominal Axial Compressive Strength
Nominal Axial Tensile Strength
7.4.3 Nominal Strength of Columns Subjected to Moment and Axial Forces
Overview
Rectangular Sections
Circular Sections
Interaction Diagrams
Biaxial Loading
Overview
Reciprocal Load Method
Load Contour Method
7.4.4 Nominal Shear Strength
Overview
Nominal Shear Strength Provided by Concrete
Nominal Shear Strength Provided by Shear Reinforcement
Biaxial Shear Strength
7.4.5 Nominal Torsional Strength
7.5 Reinforcement Limits
7.5.1 Longitudinal Reinforcement
7.5.2 Shear Reinforcement
7.6 Sizing the Cross-Section
7.6.1 Axial Compression
7.6.2 Combined Moment and Axial Force
7.6.3 Slenderness Effects
7.7 Determination of Required Reinforcement
7.7.1 Required Longitudinal Reinforcement
Axial Compression
Combined Moment and Axial Force
7.7.2 Required Shear Reinforcement
7.8 Reinforcement Detailing
7.8.1 Concrete Cover
7.8.2 Minimum Number of Longitudinal Bars
7.8.3 Spacing of Longitudinal Bars
7.8.4 Offset Bent Longitudinal Reinforcement
7.8.5 Splices of Longitudinal Reinforcement
Overview
Lap Splices
End-Bearing Splices
Mechanical and Welded Splices
7.8.6 Transverse Reinforcement
Overview
Tie Reinforcement
Spiral Reinforcement
7.9 Connections to Foundations
7.9.1 Overview
7.9.2 Vertical Transfer
Compression
Tension
7.9.3 Horizontal Transfer
7.10 Design Procedure
7.11 Examples
7.11.1 Example 7.1 – Determination of Preliminary Column Size: Building #1 (Framing Option B),Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.2 Example 7.2 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.3 Example 7.3 – Determination of Transverse Reinforcement: Building #1 (Framing Option B),Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.4 Example 7.4 – Determination of Dowel Reinforcement at the Foundation: Building #1 (FramingOption B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to PrimarilyAxial Forces
7.11.5 Example 7.5 – Determination of Preliminary Column Size: Building #1 (Framing Option B),Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.6 Example 7.6 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.7 Example 7.7 – Determination of Preliminary Column Size: Building #1 (Framing Option B), Circular,Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.8 Example 7.8 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.9 Example 7.9 – Determination of Transverse Reinforcement: Building #1 (Framing Option B), Circular,Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
7.11.10 Example 7.10 – Determination of Dowel Reinforcement at the Foundation: Building #1 (FramingOption B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to PrimarilyAxial Forces
7.11.11 Example 7.11 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1(Framing Option B), Rectangular, Tied Column, Grade 60 Longitudinal Reinforcement
7.11.12 Example 7.12 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1(Framing Option B), Rectangular, Tied Column, Grade 100 Longitudinal Reinforcement
7.11.13 Example 7.13 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1(Framing Option B), Circular, Tied Column, Grade 60 Longitudinal Reinforcement
7.11.14 Example 7.14 – Determination of Preliminary Column Size: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Column Subjected to Uniaxial Bendingand Axial Forces
7.11.15 Example 7.15 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A
7.11.16 Example 7.16 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame
7.11.17 Example 7.17 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected toUniaxial Bending and Axial Forces
7.11.18 Example 7.18 – Determination of Transverse Reinforcement: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected toUniaxial Bending and Axial Forces
7.11.19 Example 7.19 – Determination of Dowel Reinforcement at the Foundation: Building #1 (FramingOption C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjectedto Uniaxial Bending and Axial Forces
7.11.20 Example 7.20 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to BiaxialBending and Axial Forces
7.11.21 Example 7.21 – Determination of Transverse Reinforcement: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected toBiaxial Bending and Axial Forces
7.11.22 Example 7.22 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected toUniaxial Bending and Axial Forces, Slenderness Effects
7.11.23 Example 7.23 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option B),Rectangular, Tied Column is Part of the LFRS, SDC A
7.11.24 Example 7.24 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option B),Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame
7.11.25 Example 7.25 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to UniaxialBending and Axial Forces, Slenderness Effects
7.11.26 Example 7.26 – Determination of Transverse Reinforcement: Building #1 (Framing Option B),Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to UniaxialBending and Axial Forces, Slenderness Effects
7.11.27 Example 7.27 – Determination of Dowel Reinforcement at the Foundation: Building #1 (FramingOption B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjectedto Uniaxial Bending and Axial Forces, Slenderness Effects
Chapter 8
Walls
8.1 Overview
8.2 Design Limits
8.2.1 Minimum Wall Thickness
8.2.2 Intersecting Elements
8.3 Required Strength
8.3.1 Analysis Methods
8.3.2 Factored Axial Force, Moment, and Shear
8.3.3 Slenderness Effects
Overview
Moment Magnification Method
Alternative Method for Out-of-Plane Slender Wall Analysis
8.4 Design Strength
8.4.1 General
8.4.2 Nominal Axial Strength
Nominal Axial Compressive Strength
Nominal Axial Tensile Strength
8.4.3 Nominal Strength of Walls Subjected to Moment and Axial Forces
Overview
Rectangular Sections
I-, T-, and L-Shaped Sections
Simplified Design Method
8.4.4 Nominal Shear Strength
In-Plane Shear
Out-of-Plane Shear
8.5 Reinforcement Limits
8.6 Determining the Wall Thickness
8.7 Determination of Required Reinforcement
8.7.1 Longitudinal Reinforcement
8.7.2 Transverse Reinforcement
8.8 Reinforcement Detailing
8.8.1 Concrete Cover
8.8.2 Splices of Reinforcement
Overview
Lap Splices
Mechanical and Welded Splices
8.8.3 Spacing of Longitudinal Reinforcement
8.8.4 Spacing of Transverse Reinforcement
8.8.5 Lateral Support of Longitudinal Reinforcement
8.8.6 Reinforcement Around Openings
8.9 Connections to Foundations
8.9.1 Overview
8.9.2 Vertical Transfer
Compression
Tension
8.9.3 Horizontal Transfer
8.10 Design Procedure
8.11 Examples
8.11.1 Example 8.1 – Design of Reinforced Concrete Wall: Building #2, Interior Wall is Not Part of the SFRS,Simplified Design Method
8.11.2 Example 8.2 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS,Moment Magnification Method, Out-of-Plane Forces
8.11.3 Example 8.3 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS,Alternative Method for Out-of-Plane Forces
8.11.4 Example 8.4 – Determination of Trial Wall Thickness of Reinforced Concrete Wall: Building #2, SDC C,Interior Wall is Part of the SFRS
8.11.5 Example 8.5 – Design of Reinforced Concrete Wall for Combined Flexure and Axal Forces: Building#2, SDC C, Interior Wall is Part of the SFRS
8.11.6 Example 8.6 – Design of Reinforced Concrete Wall for Shear Forces: Building #2, SDC C, Interior Wallis Part of the SFRS
8.11.7 Example 8.7 – Determination of Dowel Reinforcement at the Foundation of a Reinforced ConcreteWall: Building #2, SDC C, Interior Wall is Part of the SFRS
Chapter 9
Diaphragms
9.1 Overview
9.2 Minimum Diaphragm Thickness
9.3 Required Strength
9.3.1 General
9.3.2 Diaphragm Design Forces
Overview
In-Plane Forces due to Lateral Loads
In-Plane Forces due to Transfer Forces
Connection Forces Between the Diaphragm and Vertical Framing or Nonstructural Elements
Forces Resulting from Bracing Vertical or Sloped Building Elements
Out-of-Plane Forces
Collector Design Forces
9.3.3 Diaphragm Modeling and Analysis
Overview
In-Plane Stiffness Modeling
Analysis Methods
Equivalent Beam Model with Rigid Supports
Diaphragm Forces
Collector Forces
Corrected Equivalent Beam Method with Spring Supports
Diaphragms with Openings
9.4 Design Strength
9.4.1 General
9.4.2 Nominal Moment and Axial Force Strength
9.4.3 Nominal Shear Strength
Nominal Shear Strength of Diaphragms
Shear Transfer
9.4.4 Collectors
9.5 Reinforcement Limits
9.6 Determination of Required Reinforcement
9.6.1 Chord Reinforcement
9.6.2 Diaphragm Shear Reinforcement
9.6.3 Shear Transfer Reinforcement
9.6.4 Reinforcement Due to Eccentricity of Collector Forces
9.6.5 Collector Reinforcement
Overview
Slabs
Beams
9.7 Reinforcement Detailing
9.8 Design Procedure
9.9 Examples
9.9.1 Example 9.1 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A,Collectors Not Required
9.9.2 Example 9.2 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A,Collectors Not Required
9.9.3 Example 9.3 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A,Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
9.9.4 Example 9.4 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A,Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
9.9.5 Example 9.5 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option C), SDC A,Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
9.9.6 Example 9.6 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option C), SDC A,Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
9.9.7 Example 9.7 – Determination of Diaphragm In-Plane Forces: Building #2, SDC C, Collectors Required,Collector Width the Same as the Width of the Vertical Elements of the SFRS
9.9.8 Example 9.8 – Determination of Diaphragm Reinforcement: Building #2, SDC C, Collectors Required,
Collector Width the Same as the Width of the Vertical Elements of the SFRS
9.9.9 Example 9.9 – Determination of Diaphragm In-Plane Forces: Building #2, SDC C, Collectors Required,Collector Width Wider than the Width of the Vertical Elements of the SFRS
9.9.10 Example 9.10 – Determination of Diaphragm Reinforcement: Building #2, SDC C, Collectors Required,Collector Width Wider than the Width of the Vertical Elements of the SFRS
Chapter 10
Foundations
10.1 Overview
10.2 Design Criteria
10.3 Footings
10.3.1 Overview
10.3.2 Determining the Base Area of a Footing
Overview
Isolated Spread Footing
Combined Footings
10.3.3 Determining the Thickness of a Footing
Overview
Minimum Thickness Based on Flexural Requirements
Minimum Thickness Based on Shear Requirements
10.3.4 Determining the Flexural Reinforcement
10.3.5 Detailing the Flexural Reinforcement
10.3.6 Development of Flexural Reinforcement
10.3.7 Force Transfer at the Base of Supported Members
Overview
Vertical Transfer – Compression
Vertical Transfer – Tension
Horizontal Transfer
10.3.8 Design Procedure
10.4 Drilled Piers
10.4.1 Overview
10.4.2 Design Methods
Overview
Allowable Axial Strength
Strength Design
10.4.3 Determining the Pier Size
Allowable Axial Strength
Strength Design
10.4.4 Determining the Bell Diameter
10.4.5 Reinforcement Details
10.5 Examples
10.5.1 Example 10.1 – Design of a Wall Footing Subjected to Axial Compression: Building #2
10.5.2 Example 10.2 – Design of a Square Isolated Spread Footing Subjected to Axial Compression:Building #1 (Framing Option B), SDC A
10.5.3 Example 10.3 – Design of a Rectangular Isolated Spread Footing Subjected to Axial Compression:Building #1 (Framing Option B), SDC A
10.5.4 Example 10.4 – Design of a Square Isolated Spread Footing Subjected to Axial Compression andFlexure: Building #1 (Framing Option C), SDC A
10.5.5 Example 10.5 – Design of a Square Isolated Spread Footing Subjected to Axial Compression andFlexure: Building #1 (Framing Option B), SDC A
10.5.6 Example 10.6 – Design of a Combined Rectangular Spread Footing Subjected to Axial Compression:Building #1 (Framing Option B), SDC A
10.5.7 Example 10.7 – Design of a Drilled Pier Subjected to Axial Compression: Building #1 (FramingOption B), SDC A
Chapter 11
Beam-Column and Slab-Column Joints
11.1 Overview
11.2 Design Criteria
11.3 Detailing of Joints
11.3.1 Beam-Column Joint Transverse Reinforcement
11.3.2 Slab-Column Joint Transverse Reinforcement
11.3.3 Longitudinal Reinforcement
11.4 Strength Requirements for Beam-Column Joints
11.4.1 Required Shear Strength
Overview
Joints in Moment Frames Subjected to Gravity Loads Only
Joints in Moment Frames Subjected to Gravity and Lateral Loads
11.4.2 Design Shear Strength
11.5 Transfer of Column Axial Force Through the Floor System
11.6 Examples
11.6.1 Example 11.1 – Check of Joint Shear Strength, Edge Column is Not Part of the LFRS: Building #1(Framing Option C), SDC A
11.6.2 Example 11.2 – Check of Joint Shear Strength, Edge Column is Part of the LFRS: Building #1(Framing Option C), SDC A
11.6.3 Example 11.3 – Check of Joint Shear Strength, Corner Column is Part of the LFRS: Building #1(Framing Option B), SDC A
11.6.4 Example 11.4 – Check of Joint Shear Strength, Edge Column is Part of the SFRS: Building #1(Framing Option B), SDC B
11.6.5 Example 11.5 – Adequacy of Transfer of Column Axial Force, Interior Column: Building #1 (FramingOption B), SDC A
Chapter 12
Earthquake-Resistant Structures – Overview
12.1 Overview
12.2 Seismic Design Category
12.3 Design and Detailing Requirements
12.4 Structural Systems
12.4.1 Overview
12.4.2 Bearing Wall Systems
Overview
SDC B
SDC C
SDC D, E, or F
12.4.3 Building Frame Systems
Overview
SDC B
SDC C
SDC D, E, or F
12.4.4 Moment-Resisting Frame Systems
Overview
SDC B
SDC C
SDC D, E, or F
12.4.5 Dual Systems
Overview
SDC B
SDC C
SDC D, E, or F
12.4.6 Shear Wall-Frame Interactive Systems
Chapter 13
Earthquake-Resistant Structures – SDC B and C
13.1 Overview
13.2 Ordinary Moment Frames (SDC B)
13.2.1 Overview
13.2.2 Beams
13.2.3 Columns
13.2.4 Beam-Column Joints
13.3 Intermediate Moment Frames (SDC C)
13.3.1 Overview
13.3.2 Beams
Overview
Flexural Strength Requirements
Shear Strength Requirements
13.3.3 Columns
Overview
Shear Strength Requirements
Columns Supporting Reactions from Discontinuous Stiff Members
13.3.4 Joints
Beam-Column Joints
Slab-Column Joints
Shear Strength Requirements for Beam-Column Joints
13.3.5 Two-way Slabs Without Beams
Overview
Analysis Methods
Required Flexural Reinforcement
Detailing the Flexural Reinforcement
Shear Strength Requirements
13.4 Foundations
13.5 Examples
13.5.1 Example 13.1 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option B),Beam is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.2 Example 13.2 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option B),Beam is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.3 Example 13.3 – Determination of Torsion Reinforcement: Beam in Building #1 (Framing Option B),Beam is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.4 Example 13.4 – Design for Combined Flexure, Shear, and Torsion: Beam in Building #1 (FramingOption B), Beam is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.5 Example 13.5 – Determination of Longitudinal Reinforcement: Column in Building #1 (FramingOption B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.6 Example 13.6 – Determination of Transverse Reinforcement: Column in Building #1 (FramingOption B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.7 Example 13.7 – Determination of Lap Splice Length: Column in Building #1 (Framing Option B),Column is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.8 Example 13.8 – Check of Joint Shear Strength: Column in Building #1 (Framing Option B), Columnis Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.9 Example 13.9 – Determination of Flexural Reinforcement: Two-way Slab in Building #1 (FramingOption A), Two-way Slab is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.10 Example 13.10 – Check of Two-way Shear Strength Requirements: Two-way Slab in Building #1(Framing Option A), Two-way Slab is Part of the SFRS (Intermediate Moment Frame), SDC C
13.5.11 Example 13.11 – Design of Foundation Seismic Tie: Column in Building #1 (Framing Option B),Column is Part of the SFRS (Intermediate Moment Frame), SDC C
Chapter 14
Earthquake-Resistant Structures – SDC D, E and F
14.1 Overview
14.2 Beams of Special Moment Frames
14.2.1 Overview
14.2.2 Dimensional Limits
14.2.3 Longitudinal Reinforcement
Determining the Required Flexural Reinforcement
Detailing the Flexural Reinforcement
14.2.4 Transverse Reinforcement
Determining the Required Transverse Reinforcement
Detailing the Transverse Reinforcement
14.3 Columns of Special Moment Frames
14.3.1 Overview
14.3.2 Dimensional Limits
14.3.3 Minimum Flexural Strength of Columns
14.3.4 Longitudinal Reinforcement
Determining the Required Longitudinal Reinforcement
Detailing the Longitudinal Reinforcement
14.3.5 Transverse Reinforcement
Determining the Required Transverse Reinforcement
Detailing the Transverse Reinforcement
14.4 Joints of Special Moment Frames
14.4.1 Overview
14.4.2 Transverse Reinforcement
14.4.3 Shear Strength
14.4.4 Development Length of Bars in Tension
14.5 Special Structural Walls
14.5.1 Overview
14.5.2 Reinforcement
Minimum Reinforcement Requirements
Tension Development and Splice Requirements
14.5.3 Design Shear Force
14.5.4 Shear Strength
14.5.5 Design for Flexure and Axial Force
14.5.6 Boundary Elements of Special Structural Walls
Overview
Displacement-Based Approach (ACI 18.10.6.2)
Compressive Stress Approach (ACI 18.10.6.3)
Design and Detailing Requirements for Special Boundary Elements
Design and Detailing Requirements Where Special Boundary Elements Are Not Required
Summary of Boundary Element Requirements for Special Structural Walls
14.5.7 Coupling Beams
Overview
Design and Detailing Requirements
14.5.8 Wall Piers
14.5.9 Ductile Coupled Structural Walls
14.5.10 Construction Joints
14.5.11 Discontinuous Walls
14.6 Diaphragms
14.6.1 Overview
14.6.2 Minimum Thickness
14.6.3 Reinforcement
Minimum Reinforcement
Development and Splices
Collectors
14.6.4 Flexural Strength
14.6.5 Shear Strength
14.6.6 Construction Joints
14.7 Foundations
14.7.1 Overview
14.7.2 Footings, Foundation Mats, and Pile Caps
14.7.3 Grade Beams and Slabs-on-ground
14.7.4 Foundation Seismic Ties
Overview
Design and Detailing Requirements
14.7.5 Deep Foundations
14.8 Members Not Designated as Part of the SFRS
14.8.1 Overview
14.8.2 Beams
14.8.3 Columns
14.8.4 Joints
14.8.5 Slab-Column Connections
14.8.6 Wall Piers
14.9 Examples
14.9.1 Example 14.1 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C),Beam is Part of the SFRS (Special Moment Frame), SDC D
14.9.2 Example 14.2 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C),Beam is Part of the SFRS (Special Moment Frame), SDC D
14.9.3 Example 14.3 – Determination of Cutoff Points of Flexural Reinforcement: Beam in Building #1(Framing Option C), Beam is Part of the SFRS (Special Moment Frame), SDC D
14.9.4 Example 14.4 – Determination of Longitudinal Reinforcement: Interior Column in Building #1(Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.5 Example 14.5 – Determination of Transverse Reinforcement: Interior Column in Building #1 (FramingOption C), Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.6 Example 14.6 – Check of Joint Shear Strength: Interior Column in Building #1 (Framing Option C),Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.7 Example 14.7 – Determination of Longitudinal Reinforcement: Corner Column in Building #1(Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.8 Example 14.8 – Determination of Transverse Reinforcement: Corner Column in Building #1 (FramingOption C), Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.9 Example 14.9 – Check of Joint Shear Strength: Corner Column in Building #1 (Framing Option C),Column is Part of the SFRS (Special Moment Frame), SDC D
14.9.10 Example 14.10 – Design of Special Structural Wall: Building #3, Wall is Part of the SFRS (BuildingFrame System), SDC D, Displacement-Based Approach
14.9.11 Example 14.11 – Design of Special Structural Wall: Building #4, Wall is Part of the SFRS (DualSystem), SDC D, Compressive Stress Approach
14.9.12 Example 14.12 – Design of a Coupling Beam (Dual System): Building #4, SDC D
14.9.13 Example 14.13 – Determination of Diaphragm Reinforcement: Building #4, SDC D
14.9.14 Example 14.14 – Design of Foundation Seismic Tie: Building #1 (Framing Option C), SDC D
14.9.15 Example 14.15 – Determination of Required Reinforcement: Beam in Building #4, Beam is Not Partof the SFRS, SDC D
14.9.16 Example 14.16 – Determination of Required Reinforcement: Column in Building #4, Column is NotPart of the SFRS, SDC D
14.9.17 Example 14.17 – Check of Slab-Column Connection: Column in Building #3, Column is Not Part ofthe SFRS, SDC D
Appendix A
References
Appendix B
Reinforcing Bar Data
Table B.1 ASTM Standard Reinforcing Bars
Table B.2 Overall Reinforcing Bar Diameters
Appendix C
Section Index

Citation preview

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A comprehensive guide to assist design professionals on the design and detailing of reinforced concrete buildings. Based on ACI 318-19

2020

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Publication No: DG-ACI318-19 ISBN: 978-1-943961-52-8 Copyright © 2020 By Concrete Reinforcing Steel Institute First Edition 2020, First Printing Contains errata material, December 2020

All rights reserved. This guide or any part thereof may not be reproduced in any form without the written permission of the Concrete Reinforcing Steel Institute.

Printed in the U.S.A

This publication is intended for the use of professionals competent to evaluate the significance and limitations of its contents and who will accept responsibility for the application of the material it contains. The Concrete Reinforcing Steel Institute reports the foregoing material as a matter of information and, therefore, disclaims any and all responsibility for application of the stated principles or for the accuracy of the sources other than material developed by the Institute.

Founded in 1924, the Concrete Reinforcing Steel Institute (CRSI) is a technical institute and an ANSI-accredited Standards Developing Organization (SDO) that stands as the authoritative resource for information related to steel reinforced concrete construction. Serving the needs of engineers, architects and construction professionals, CRSI offers many industry-trusted technical publications, standards documents, design aids, reference materials and educational opportunities. CRSI Industry members include manufacturers, fabricators, material suppliers and placers of steel reinforcing bars and related products. Our Professional members are involved in the research, design, and construction of steel reinforced concrete. CRSI also has a broad Region Manager network that supports both members and industry professionals and creates awareness among the design/construction community through outreach activities. Together, they form a complete network of industry information and support.

i

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Foreword The purpose of this Design Guide is to assist in the proper application of the design and detailing requirements in Building Code Requirements for Structural Concrete (ACI 318-19) for cast-in-place concrete buildings with nonprestressed steel reinforcement. Many design aids and worked-out examples are provided to make designing and detailing reinforced concrete members simpler and faster. The goal is to acquire an understanding of the code UHTXLUHPHQWVDQGWRDSSO\WKHPSURSHUO\DQGHI¿FLHQWO\ In addition to structural engineers, this Design Guide is a valuable resource for educators, students, individuals VWXG\LQJIRUOLFHQVLQJH[DPVDQGEXLOGLQJRI¿FLDOV 6LQFHWKH¿UVW&56,'HVLJQ+DQGERRNLQXVHUVRI&56,SXEOLFDWLRQVKDYHEHHQFRRSHUDWLYHLQVXJJHVWLQJWR WKH'HVLJQ$LGV&RPPLWWHHDQG&56,6WDIIPDQ\LPSURYHPHQWVFODUL¿FDWLRQVDQGDGGLWLRQDOGHVLJQVKRUWFXWV This professional assistance is very helpful, and is appreciated. Comments on this design guide are welcome so that future editions can be further improved. Please direct all comments to Amy Trygestad, P.E., F.ACI, CRSI Vice President of Engineering.

Acknowledgements Special recognition is due to the following individuals who provided a review of the publication and made insightful comments and suggestions for improvement: David A. Fanella, S.E., P.E., F.ACI, F.ASCE, F.SEI, Senior Director of Engineering, is the author of this Design Guide. Amy Trygestad, P.E., F.ACI, Vice President of Engineering Nathan Westin, P.E., Technical Director Special recognition is also due to Dave Mounce, Director of Communications, ZKRZDVLQFKDUJHRISURGXFWLRQDQGOD\RXWRIWKLV'HVLJQ*XLGH+LVGHGLFDWLRQ and expertise is greatly appreciated.

Photo Credits Front cover image courtesy of Skidmore, Owings & Merrill, SEOR 3URMHFW1DPH0LVVLRQ Client: Kilroy Realty General Contractor: Webcor Builders Back cover image courtesy of DeSimone Consulting Engineers Project Name: Panorama Tower Client: Florida East Coast Realty (FECR) General Contractor: Florida East Coast Realty (FECR)

ii

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Contents Foreword

Chapter 3

Acknowledgements

Design Loads and Load Combinations 3.1 Overview

3-1

3.2 Design Loads

3-1

Chapter 1

3.3 Seismic Design Category

3-1

Introduction

3.4 Live Load Reduction

3-3

Photo Credits

1.1 Overview

1-1

3.5 Load Factors and Combinations

3-4

1.2 Example Buildings

1-2

3.6 Determination of Wind Forces

3-8

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3.7 Determination of Seismic Forces

3-8

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Chapter 2 Material Requirements and Strength Reduction Factors 2.1 Overview

2-1

2.2 Material Requirements

2-1

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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4-22

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4-25

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4-1

4.2 Minimum Slab Thickness

4-1

4.3 Required Strength

4-2

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4-7

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   5-36

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5-53

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5-56

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 6 Beams 6.1 Overview

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1RPLQDO6KHDU6WUHQJWK Overview Nominal Shear Strength Provided by Concrete Nominal Shear Strength Provided by 6KHDU5HLQIRUFHPHQW

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6-69

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6.9 Examples

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7-1

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vii

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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7.8 Reinforcement Detailing

7-33

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7.9 Connections to Foundations

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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8-1

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8-1

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8-1

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8.4 Design Strength *HQHUDO

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8.5 Reinforcement Limits

8-17

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8-17

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8-21

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9-1

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9-1

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9-2

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9.3.3 Diaphragm Modeling and Analysis 9-6 Overview 9-6 In-Plane Stiffness Modeling 9-6 Analysis Methods 9-8 Equivalent Beam Model with Rigid Supports 9-9 Corrected Equivalent Beam Method with Spring Supports 9-13 Diaphragms with Openings 9-16 9.4 Design Strength *HQHUDO

9-19 

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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9-22

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Chapter 10

9.7 Reinforcement Detailing

9-28

10.1 Overview

10-1

9.8 Design Procedure

9-29

10.2 Design Criteria

10-1

9.9 Examples

9-29

10.3 Footings

10-1

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9-36

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9-39

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Foundations

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   xi

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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10.5 Examples

10-28

11.5 Transfer of Column Axial Force Through the Floor System

11-15

11.6 Examples

11-16

([DPSOH±&KHFNRI-RLQW6KHDU Strength, Edge Column is Not Part RIWKH/)56%XLOGLQJ)UDPLQJ Option C), SDC A

11-16

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11-19

([DPSOH±'HVLJQRID6TXDUH Isolated Spread Footing Subjected to Axial Compression and Flexure: %XLOGLQJ)UDPLQJ2SWLRQ&  6'&$

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1-33





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Chapter 12 Earthquake-Resistant Structures – Overview 12.1 Overview

12-1

12.2 Seismic Design Category

12-1

12.3 Design and Detailing Requirements

12-1

12.4 Structural Systems

12-2

Chapter 11

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Beam-Column and Slab-Column Joints

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11.1 Overview

11-1

11.2 Design Criteria

11-1

11.3 Detailing of Joints

11-2

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6ODE&ROXPQ-RLQW7UDQVYHUVH Reinforcement

11-3

11.3.3 Longitudinal Reinforcement

11-3

11.4 Strength Requirements for Beam-Column Joints

xii

11-7

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Chapter 13 Earthquake-Resistant Structures – SDC B and C 13.1 Overview

13-1

13.2 Ordinary Moment Frames (SDC B)

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13.4 Foundations

13-25

13.5 Examples

13-29

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Chapter 14 Earthquake-Resistant Structures – SDC D, E and F 14.1 Overview

14-1

14.2 Beams of Special Moment Frames

14-2

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xiii

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7UDQVYHUVH5HLQIRUFHPHQW  Determining the Required Transverse 5HLQIRUFHPHQW  'HWDLOLQJWKH7UDQVYHUVH5HLQIRUFHPHQW  14.3 Columns of Special Moment Frames

14-15

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14-29

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14-36 

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14-60

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14-69

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14-76



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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

([DPSOH±'HWHUPLQDWLRQRI Transverse Reinforcement: Interior &ROXPQLQ%XLOGLQJ)UDPLQJ Option C), Column is Part of the SFRS 6SHFLDO0RPHQW)UDPH 6'&'

Appendix A References 

([DPSOH±&KHFNRI-RLQW6KHDU 6WUHQJWK,QWHULRU&ROXPQLQ%XLOGLQJ (Framing Option C), Column is Part of the SFRS (Special Moment Frame), 6'&'  ([DPSOH±'HWHUPLQDWLRQRI Longitudinal Reinforcement: Corner &ROXPQLQ%XLOGLQJ)UDPLQJ Option C), Column is Part of the SFRS 6SHFLDO0RPHQW)UDPH 6'&'

A-1

Appendix B Reinforcing Bar Data Table B.1 ASTM Standard Reinforcing Bars

B-1

Table B.2 Overall Reinforcing Bar Diameters

B-2

Appendix C 

Section Index

C-1

([DPSOH±'HWHUPLQDWLRQRI Transverse Reinforcement: Corner &ROXPQLQ%XLOGLQJ)UDPLQJ Option C), Column is Part of the SFRS 6SHFLDO0RPHQW)UDPH 6'&'  ([DPSOH±&KHFNRI-RLQW6KHDU 6WUHQJWK&RUQHU&ROXPQLQ%XLOGLQJ (Framing Option C), Column is Part of the SFRS (Special Moment Frame), 6'&'  ([DPSOH±'HVLJQRI6SHFLDO 6WUXFWXUDO:DOO%XLOGLQJ:DOOLV Part of the SFRS (Building Frame System), SDC D, Displacement%DVHG$SSURDFK  ([DPSOH±'HVLJQRI6SHFLDO 6WUXFWXUDO:DOO%XLOGLQJ:DOO is Part of the SFRS (Dual System), SDC D, Compressive Stress $SSURDFK



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xv

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

xvi

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 1 INTRODUCTION 1.1 Overview 7KHSXUSRVHRIWKLVGHVLJQJXLGHLVWRDVVLVWLQWKHSURSHUDSSOLFDWLRQRIWKHSURYLVLRQVLQWKHHGLWLRQRIBuilding Code Requirements for Structural Concrete (ACI 318-19) and Commentary (ACI 318R-19) for cast-in-place concrete buildings with nonprestressed steel reinforcing bars (Reference 1; see Appendix A for a list of references in this design guide). The main goals of this publication are to provide: •

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step-by-step design procedures and design aids that make designing and detailing reinforced concrete buildings simpler and faster.

7KHLQIRUPDWLRQFRQWDLQHGLQWKLVSXEOLFDWLRQFDQEHXVHGIRUUHLQIRUFHGFRQFUHWHEXLOGLQJVRIDQ\VL]HDVVLJQHGWR Seismic Design Categories A through F. Throughout this publication, emphasis is placed on how to correctly apply the ACI 318 requirements, and explanatory material is provided to supplement the requirements. Readers who are interested in the origins of the provisions FDQ¿QGDZHDOWKRILQIRUPDWLRQLQWKH$&,&RPPHQWDU\DQGDFFRPSDQ\LQJUHIHUHQFHV 2YHUZRUNHGRXWSUDFWLFDOGHVLJQH[DPSOHVDUHSURYLGHGWKDWLOOXVWUDWHWKHSURSHUDSSOLFDWLRQRIWKH$&, UHTXLUHPHQWV7KHH[DPSOHVDUHGHULYHGIURPWKHIRXUEXLOGLQJVLQ6HFWLRQRIWKLVSXEOLFDWLRQZKLFKUDQJHLQ KHLJKWIURPWRVWRULHV7KHVHEXLOGLQJVLQFOXGHDZLGHDUUD\RIJUDYLW\DQGODWHUDOIRUFHUHVLVWLQJV\VWHPVDQG examples are given throughout the chapters of this design guide that systematically show the design and detailing of WKHVWUXFWXUDOPHPEHUVLQWKHVHV\VWHPVXVLQJWKHH[SODQDWRU\PDWHULDODQGGHVLJQDLGVLQWKH¿UVWSDUWRIWKHFKDSter. A number of the examples address certain requirements in ACI 318 not covered in any other reinforced concrete resources. The titles for the examples are purposely very descriptive so users can quickly locate what they are looking for in the table of contents of this publication. 7KHVHFWLRQLQGH[LQ$SSHQGL[&FDQEHXVHGWRTXLFNO\ORFDWHLQIRUPDWLRQLQWKLVGHVLJQJXLGHRQDVSHFL¿F$&, 318 provision: ACI 318 section numbers are given in the left column of the index and the corresponding chapters and page numbers in this design guide are given in the right column of the index. This method of cross-referencing between ACI 318 and this design guide is a very useful feature for users who are interested in a particular requirement. Throughout this publication, section numbers from ACI 318-19 are referenced as illustrated by the following: SecWLRQRI$&,LVGHQRWHGDV$&,6LPLODUO\VHFWLRQQXPEHUVIURPWKH,QWHUQDWLRQDO%XLOGLQJ&RGH ,%& >5HIHUHQFH@DQG$6&(6(,5HIHUHQFH DUHUHIHUHQFHGDVIROORZV6HFWLRQIURPWKH,%& DQG6HFWLRQRI$6&(6(,DUHGHQRWHGDV,%&DQG$6&(6(,UHVSHFWLYHO\1RWH7KH HGLWLRQRI$&,LVUHIHUHQFHGLQWKH,%&EXWWKHLQIRUPDWLRQUHIHUHQFHGLQWKLVGHVLJQJXLGHLVDSSOLFDEOH regardless of the edition of the IBC). 7KHQRWDWLRQXVHGWKURXJKRXWWKLVGHVLJQJXLGHFRQIRUPVWRWKHQRWDWLRQJLYHQLQ6HFWLRQRI$&,:KHUH DGGLWLRQDOQRWDWLRQLVQHHGHGLWLVFOHDUO\GH¿QHGZKHUHLWLV¿UVWLQWURGXFHG In addition to practicing engineers, this design guide is a valuable resource for educators, students, individuals VWXG\LQJIRUOLFHQVLQJH[DPVDQGEXLOGLQJRI¿FLDOV

1-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.2 Example Buildings 1.2.1 Building #1 – 5-Story Office Building

1A

2A

3A

4A ᇱ

ᇳ

ᇱ

ᇳ

5A

6A

7A

5A

6A

7A

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1A

2A

3A

4A

A

25 -0 ሺtyp.ሻ

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E

23ᇱ -6ᇳ ሺtyp.ሻ

D

A

N

A

C

A

B

A

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Figure 1.1 Building #1 – 5-story office building.

1-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A

6A

7A

A

25ᇱ -0ᇳ ሺtyp.ሻ

A

E

23ᇱ -6ᇳ ሺtyp.ሻ

D

A

N

A

C

A

B

A

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1A

2A

3A

4A

5A

6A

7A

A

25ᇱ -0ᇳ ሺtyp.ሻ

A

E

23ᇱ -6ᇳ ሺtyp.ሻ

D

A

N

A

C

A

B

A

۴‫ ܖܗܑܜܘ۽ ܏ܖܑܕ܉ܚ‬۱

Figure 1.1 Building #1 – 5-story office building (cont.).

1-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

8ᇱ -6ᇳ ሺtyp.ሻ

A

8ᇱ -6ᇳ ሺtyp.ሻ

E

23ᇱ -6ᇳ ሺtyp.ሻ

D

A

N

C

A

5ᇱ -3ᇳ ሺtyp.ሻ

A

B

A

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1A

2A

3A

4A

A

50ᇱ -0ᇳ ሺtyp.ሻ

A

31ᇱ -4ᇳ

D

C

A

31ᇱ -4ᇳ

N

A

31ᇱ -4ᇳ

B

A

۴‫ ܖܗܑܜܘ۽ ܏ܖܑܕ܉ܚ‬۳

Figure 1.1 Building #1 – 5-story office building (cont.).

1-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 1.1 Design Information for Building #1 Design Data

Location

Latitude

Occupancy

, Longitude Business Group B

Topography

Not situated on a hill, ridge, or escarpment

Exposure category

C

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Superimposed dead load Live load

Roof Floor

Live load

Superimposed dead load (includes

for partitions)

Cladding Structural Systems

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Lateral

Moment-resisting frames in both directions

1-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.2.2 Building #2 – 12-Story Emergency Operations Center 1A

2A

3A

4A

35ᇱ -6ᇳ

42ᇱ -6ᇳ

A

42ᇱ -6ᇳ

A

G

A

F

E

ᇱ

30 -0 ሺtyp.ሻ

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Wide-module joist system ሺtyp.ሻ

A

N

A

D

A

C

A

B

Pan depth

Slab Thickness

A

Rib width

12 1

Pan width

Section through wide-module joist system

Figure 1.2 Building #2 – 12-story emergency operations center.

1-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳA

ʹA

͵A ͵ͷᇱ Ǧ͸ᇳ

ͶA Ͷʹᇱ Ǧ͸ᇳ

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Figure 1.2 Building #2 – 12-story emergency operations center (cont.).

Table 1.2 Design Information for Building #2 Design Data

Location

Latitude

, Longitude

Occupancy

Essential facility

Topography

Not situated on a hill, ridge, or escarpment

Exposure category

B

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Roof

Superimposed dead load Live load

Floor

Superimposed dead load Live load

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Wide-module joist system

Lateral

Building frame system in both directions

1-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.2.3 Building #3 – 16-Story Residential Building 1A

2A

3A

4A

5A

A

22ᇱ -0ᇳ ሺtyp.ሻ

A

ᇱ

19 -0

ᇳ

D

C

A

ᇱ

20 -6

ᇳ

N

A

ᇱ

19 -0

ᇳ

B

A

Story heights ൌ 9ᇱ -6ᇳ

Figure 1.3 Building #3 – 16-story residential building.

Table 1.3 Design Information for Building #3 Design Data

Location

Latitude

, Longitude

Occupancy

Residential Group R

Topography

Not situated on a hill, ridge, or escarpment

Exposure category

B

6RLOFODVVL¿FDWLRQ

Site Class D (stiff soil; determined) Load Data

Roof

Superimposed dead load Live load

Floor

Superimposed dead load Live load

Cladding Structural Systems

1-8

)ORRUURRI

Flat plate system

Lateral

Building frame system in both directions

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.2.4 Building #4 – 30-Story Office Building 1



2 40ᇱ -0ԣ

3 30ᇱ -0ԣ

4 40ᇱ -0ԣ

G

F

Wide-module joists ሺtyp.ሻ 18ᇱ -0ᇳ 

ሺtyp.ሻ

25ᇱ -0ᇳ 

E

N

9ᇱ -0ᇳ  8ᇱ -0ᇳ 

D

C

B

A Story heights ൌ 11ᇱ -0ᇳ 

Figure 1.4 Building #4 – 30-story office building.

1-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 1.4 Design Information for Building #4 Design Data

Location

Latitude

, Longitude

Occupancy

Business Group B

Topography

Not situated on a hill, ridge, or escarpment

Exposure category

B

6RLOFODVVL¿FDWLRQ

Site Class D (default) Load Data

Superimposed dead load Live load

Roof Floor

Live load

Superimposed dead load (includes

for partitions)

Cladding Structural Systems

)ORRUURRI

Wide-module joist system

Lateral

Dual system in both directions

1.3 Organization of This Design Guide Properties of concrete and nonprestressed steel reinforcing bars permitted to be used in the design of reinforced conFUHWHPHPEHUVDUHFRYHUHGLQ&KDSWHU$&,VWUHQJWKUHGXFWLRQIDFWRUVDUHDOVRFRYHUHGLQWKLVFKDSWHU Presented in Chapter 3 are typical loads and required load combinations applicable in the design of reinforced concrete buildings. Methods are given to determine the Seismic Design Category (SDC) of a building and live load reduction on structural members. Step-by-step procedures on how to determine wind forces on the main wind force resisting system (MWFRS), seismic forces on the seismic-force-resisting system (SFRS), and seismic design forces on diaphragms (including collectors) are also presented along with examples illustrating the load calculations for each of the example buildings. 'HVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRURQHZD\VODEVDUHFRYHUHGLQ&KDSWHU,QFOXGHGDUHPHWKRGVWRGHWHUPLQH PLQLPXPVODEWKLFNQHVVDQGWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQW7\SLFDOUHLQIRUFHPHQWGHWDLOVDUHDOVRJLYHQDQG examples are provided for a one-way slab that is part of a wide-module joist system. &KDSWHUFRQWDLQVWKHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWZRZD\VODEVLQEXLOGLQJVDVVLJQHGWR6'&$DQG% Design aids are provided on how to determine the slab thickness based on serviceability and two-way shear strength UHTXLUHPHQWVIRUÀDWSODWHVÀDWVODEVWZRZD\EHDPVXSSRUWHGVODEVDQGWZRZD\MRLVWVZDIÀHVODEV $JHQHUDO method to determine the section properties of any critical section for two-way shear is also given, including properties for polygon-shaped sections, which are required where shear reinforcement is provided. Examples include analysis and design for two-way slabs systems subjected to both gravity and lateral load effects, two-way shear design for slabs with headed shear studs and stirrups, and analysis for two-way shear with a slab opening adjacent to a column. The design and detailing requirements for beams are covered in Chapter 6 and are applicable to buildings assigned WR6'&$DQG%,QDGGLWLRQWRGHVLJQDLGVIRUVL]LQJWKHVHFWLRQDQGVHOHFWLQJWKHUHLQIRUFHPHQWIRUÀH[XUHVKHDU DQGWRUVLRQPHWKRGVWRFDOFXODWHERWKVKRUWDQGORQJWHUPGHÀHFWLRQVDUHSURYLGHG,QFOXGHGDUHJXLGHOLQHVIRU economical detailing and design examples for wide-module joist systems and beams subjected to combined gravity and lateral load effects.

1-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

&KDSWHUFRYHUVWKHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUFROXPQVLQEXLOGLQJVDVVLJQHGWR6'&$DQG%,QFOXGHG DUHPHWKRGVWRVL]HWKHFURVVVHFWLRQDQGWRGHWHUPLQHWKHUHTXLUHGORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWWLHV and spirals) for both nonslender and slender columns. Examples include the determination of nominal and design strength interaction diagrams for rectangular and circular columns, columns subjected to combined gravity and latHUDOORDGHIIHFWVDQGWKHGHVLJQDQGGHWDLOLQJRIDFRUQHUFROXPQVXEMHFWHGWRELD[LDOÀH[XUHDQGVKHDU Structural walls in buildings assigned to SDC A, B, or C are covered in Chapter 8. Information is given on the moPHQWPDJQL¿FDWLRQPHWKRGWKHDOWHUQDWLYHPHWKRGIRURXWRISODQHVOHQGHUZDOODQDO\VLVDQGWKHVLPSOL¿HGGHVLJQ method. Typical reinforcement details are provided, including details for walls with openings, and examples are given for nonslender and slender walls subjected to axial forces and in-plane and out-of-plane lateral forces. Design and detailing requirements for diaphragms are given in Chapter 9 for buildings assigned to SDC A, B, or C. Diaphragm modeling and analysis, in-plane stiffness modeling, and analysis methods for diaphragms with and without openings are the primary topics covered. Methods are presented on how to determine the center of rigidity, how to distribute the in-plane forces to the vertical elements of the lateral force-resisting system, and how to determine the required diaphragm and collector reinforcement. &KDSWHUFRQWDLQVWKHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRULVRODWHGVSUHDGIRRWLQJVZDOOIRRWLQJVFRPELQHG spread footings, and drilled piers (caissons) supporting buildings assigned to SDC A and B. Procedures are presentHGRQKRZWRVL]HWKHIRXQGDWLRQPHPEHUDQGKRZWRGHWHUPLQHDQGGHWDLOWKHUHTXLUHGUHLQIRUFHPHQWIRUPHPEHUV subjected to axial forces and combined bending and axial forces. Design and detailing requirements for beam-column and slab-column joints in buildings assigned to SDC A and B are covered in Chapter 11. Step-by-step procedures are given on how to determine joint forces in moment frames subjected to gravity loads only and to gravity and lateral loads. Examples are provided illustrating application of the requirements. An overview of the design and detailing requirements for structures assigned to SDC B through F is given in ChapWHU7KHUHTXLUHPHQWVEDVHGRQ6'&DUHSURYLGHGDORQJZLWKWKHDSSOLFDEOH$&,VHFWLRQQXPEHUVWKDWPXVW EHVDWLV¿HG,QFOXGHGLVDQRYHUYLHZRIVHLVPLFIRUFHUHVLVWLQJV\VWHPVIRUUHLQIRUFHGFRQFUHWHEXLOGLQJVWKHFRUUHVSRQGLQJGHVLJQFRHI¿FLHQWVDQGIDFWRUVDQGWKHVWUXFWXUDOV\VWHPOLPLWDWLRQVIRUWKHVHV\VWHPV The design and detailing requirements for frame members and foundations in buildings assigned to SDC B and C DUHJLYHQLQ&KDSWHU$FRPSLODWLRQRIXVHIXOGHWDLOVLVSURYLGHGWKDWFOHDUO\VXPPDUL]HWKHSURYLVLRQVIRUEHDPV columns, and beam-column joints in intermediate moment frames and for drilled piers (caissons). Included are examples for the design of the structural members in an intermediate moment frame and a foundation seismic tie. &KDSWHUFRQWDLQVGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWKHIROORZLQJVWUXFWXUDOPHPEHUVLQEXLOGLQJVDVVLJQHGWR SDC D, E, or F: • • • • • • •

beams, columns, and beam-column joints in special moment frames; special structural walls; coupling beams; wall piers; diaphragms; shallow and deep foundations (including foundation seismic ties); and members not designated as part of the SFRS.

1XPHURXVGHVLJQDLGVDQGDFDWDORJXHRIXVHIXOGHWDLOVDUHSURYLGHGWKDWVXPPDUL]HWKH$&,UHTXLUHPHQWV along with examples illustrating the proper application of the provisions. References in this design guide are given in Appendix A. Appendix B contains properties of currently available steel reinforcing bars. The section index described in Section 1.1 above is in Appendix C.

1-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 2 MATERIAL REQUIREMENTS AND STRENGTH REDUCTION FACTORS 2.1 Overview Properties of concrete and nonprestressed reinforcement permitted to be used in the design of reinforced concrete PHPEHUVDUHFRYHUHGLQWKLVFKDSWHU$FFRUGLQJWR$&,GHVLJQSURSHUWLHVRIFRQFUHWHPXVWEHVHOHFWHGLQ accordance with the requirements in ACI Chapter 19. Similarly, the design properties of reinforcement must be VHOHFWHGLQDFFRUGDQFHZLWKWKHUHTXLUHPHQWVLQ$&,&KDSWHU$&,  play a key role in the Strength reduction factors, which are commonly referred to as resistance factors or determination of the design strength of a reinforced concrete member. ACI strength reduction factors are given in $&,&KDSWHUDQGDUHDOVRFRYHUHGLQWKLVFKDSWHU

2.2 Material Requirements 2.2.1 Concrete Design Properties Specified Compressive Strength

Stress-strain curves for compression tests on concrete mixtures of varying compressive strengths are given in FigXUH ͳͶǡͲͲͲ ͳʹǡͲͲͲ

–”‡••ሺ’•‹ሻ

ͳͲǡͲͲͲ ͺǡͲͲͲ ͸ǡͲͲͲ ͶǡͲͲͲ ʹǡͲͲͲ Ͳ ͲǤͲͲͲ

ͲǤͲͲͳ

ͲǤͲͲʹ

ͲǤͲͲ͵

ͲǤͲͲͶ

–”ƒ‹ሺ‹ǤȀ‹Ǥሻ Figure 2.1 Stress-strain curves for compression tests on concrete mixtures of varying compressive strengths.

/LPLWVRQWKHVSHFL¿HGFRPSUHVVLYHVWUHQJWKRIFRQFUHWH DUHJLYHQLQ$&,7DEOHZKLFKDSSO\WRERWK QRUPDOZHLJKWDQGOLJKWZHLJKWFRQFUHWH>$&,D @7KHOLPLWVRQ for cast-in-place structural members based on Seismic Design Category (SDC), including foundations supporting concrete structures with other than UHVLGHQWLDODQGXWLOLW\XVHDUHJLYHQLQ7DEOH

2-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 2.1 Limits for Concrete Compressive Strength Application

Minimum f 'c (psi)

General



Foundations for structures assigned to SDC A, B, or C



Foundations for structures assigned to SDC D, E, or F



Special moment frames 6SHFLDOVWUXFWXUDOZDOOVZLWK*UDGHRUUHLQIRUFHPHQW



6SHFLDOVWUXFWXUDOZDOOVZLWK*UDGHUHLQIRUFHPHQW



A minimum RISVLIRUVSHFLDOVWUXFWXUDOZDOOVZLWK*UDGHUHLQIRUFHPHQWLVVSHFL¿HGEHFDXVHWHVWGDWD are not available for lower concrete strengths. In general, minimum concrete strengths are increased for special VHLVPLFV\VWHPVVSHFLDOPRPHQWIUDPHVDQGVSHFLDOVWUXFWXUDOZDOOV ZLWK*UDGHDQG*UDGHUHLQIRUFHPHQW to enhance reinforcing bar anchorage and to improve structural performance by reducing the neutral axis depth. 7KHGXUDELOLW\UHTXLUHPHQWVLQ$&,PD\KDYHDQLPSDFWRQWKHPLQLPXPFRQFUHWHFRPSUHVVLYHVWUHQJWK WKDWPXVWEHVSHFL¿HG>$&,E @([SRVXUHFDWHJRULHVDQGFODVVHVDUHGH¿QHGLQ$&,7DEOHDQG structural members must be assigned an exposure class for the applicable exposure category. Based on the severity based on durability requirements may be higher than that required for strucof the anticipated exposure class, tural purposes. Structural strength requirements may also have an effect on the minimum >$&,F @7KHRQO\ZD\WR satisfy certain strength requirements for a structural member is to increase the compressive strength of the concrete. No upper limits on DUHVSHFL¿HGH[FHSWIRUOLJKWZHLJKWFRQFUHWHLQVSHFLDOPRPHQWIUDPHVVSHFLDOVWUXFWXUDO walls, and their foundations; in such cases, PXVWEHOHVVWKDQRUHTXDOWRSVL>$&,G @7KLVOLPLW LVLPSRVHGEHFDXVHRIWKHYHU\OLPLWHGDPRXQWRIH[SHULPHQWDODQG¿HOGGDWDDYDLODEOHRQWKHEHKDYLRUDQGSHUIRUPDQFHRIPHPEHUVPDGHZLWKOLJKWZHLJKWFRQFUHWH+RZHYHUYDOXHVRI JUHDWHUWKDQSVLDUHSHUPLWWHG if it can be shown that members made with lightweight concrete provide strength and toughness levels equal to or exceeding those of comparable members made with normalweight concrete of the same compressive strength. $FFRUGLQJWR$&,WKHVSHFL¿HGFRPSUHVVLYHVWUHQJWKPXVWEHXVHGIRUSURSRUWLRQLQJFRQFUHWHPL[WXUHVLQ $&,DQGIRUWHVWLQJDQGDFFHSWDQFHRIFRQFUHWHLQ$&, Concrete compressive strength is typically determined for a concrete mixture on the basis of compression tests of F\OLQGHUVPROGHGDQGFXUHGIRUDVSHFL¿HGQXPEHURIGD\VLQDFFRUGDQFHZLWK$670VSHFL¿FDWLRQV6WDQGDUGF\OLQGHUVDUHXVXDOO\WHVWHGDWGD\V,I LVGHWHUPLQHGRWKHUWKDQDWGD\VWKHWHVWDJHPXVWEHLQGLFDWHGLQWKH FRQVWUXFWLRQGRFXPHQWV$&,  Modulus of Elasticity

The modulus of elasticity of concrete, ZKLFKLVXVHGLQWKHGHVLJQRIFRQFUHWHPHPEHUVLQFOXGLQJGHÀHFWLRQV DQGVOHQGHUQHVVHIIHFWVLQFROXPQV LVSHUPLWWHGWREHGHWHUPLQHGE\$&,(TXDWLRQD RUE >$&, @ For concrete with

EHWZHHQDQGOEIW3: (2.1)

For normalweight concrete (2.2)

2-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,Q(TXDWLRQ  is the density (unit weight) of normalweight concrete or the equilibrium density of lightweight has the units of pounds per square inch (psi). concrete. In both equations, It is permitted to specify EDVHGRQWHVWVRIFRQFUHWHPL[WXUHV$&, 7HVWLQJPD\EHZDUUDQWHGZKHUH IRUH[DPSOHZKHQFDOFXODWLQJWKHGHÀHFWLRQRIDEHDPVXSSRUWdesign conditions are sensitive to the value of LQJDQRQVWUXFWXUDOPHPEHUVHQVLWLYHWRGHÀHFWLRQVRUYLEUDWLRQRUZKHQFDOFXODWLQJWKHODWHUDOVWLIIQHVVRIDWDOO EXLOGLQJVHH$&,5  Modulus of Rupture

LVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@

The modulus of rupture for concrete,

(2.3)

In this equation, has the units of pounds per square inch (psi) and LVDPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,VHHWKH following section). 7KHPRGXOXVRIUXSWXUHLVXVHGLQGHÀHFWLRQFDOFXODWLRQVRIEHDPVDQGVODEV Lightweight Concrete Modification Factor

The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODtive to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium density, RIWKHFRQFUHWHPL[WXUHRU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[WXUHVHH$&,  9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOHD @DQGYDOXHVRI based on composiWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOHE @1RWHWKDW LVSHUPLWWHGWREHWDNHQDVIRU OLJKWZHLJKWFRQFUHWH$&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH$&,  Table 2.2 Values of Ȝ Based on Equilibrium Density, wc

Ȝ

Equilibrium Density, wc



 Table 2.3 Values of Ȝ Based on Composition of Aggregates Concrete

All-lightweight /LJKWZHLJKW¿QHEOHQG Sand-lightweight Sand-lightweight, coarse blend

Composition of Aggregates

)LQH$670& &RDUVH$670& )LQH&RPELQDWLRQRI$670&DQG$670& &RDUVH$670& Fine: ASTM C33 &RDUVH$670& Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670&DQG$670&

Ȝ  WR(1)  WR

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

2-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

2.2.2 Nonprestressed Steel Reinforcement Material Properties

Yield Strength 7KHWHQVLOHVWUHVVVWUDLQFXUYHZLWKDZHOOGH¿QHG\LHOGVWUHQJWK for a steel reinforcing bar is given in Figure 7KLVJHQHULFVWUHVVVWUDLQFXUYHKDVWKUHHGLVWLQFWVHJPHQWVZKLFKLVFKDUDFWHULVWLFRIUHLQIRUFLQJEDUVZLWKD \LHOGVWUHQJWKRISVLRUOHVV • $QLQLWLDOOLQHDUHODVWLFVHJPHQWXSWRDZHOOGH¿QHG\LHOGVWUHQJWK • $UHODWLYHO\ÀDW\LHOGSODWHDXXSWRWKHRQVHWRIVWUDLQKDUGHQLQJWKHVWUDLQDWZKLFKLVGHVLJQDWHG • A rounded strain-hardening segment  ͳͲͲ

‫ܧ‬௦ ൌ ‘†—Ž—•‘ˆ‡Žƒ•–‹…‹–› ‫ܧ‬௦௛ ൌ –”ƒ‹ǦŠƒ”†‡‹‰‘†—Ž—•

݂௨ 

ͺͲ ‫ܛܛ܍ܚܜ܁‬ሺ‫ܛܘܑܓ‬Ȁܑ‫ܖ‬Ǥ૛ ሻ

‫ܧ‬௦௛  ݂௨௬ 

͸Ͳ

݂௙ 

݂௨ ൌ ‡•‹Ž‡•–”‡‰–Š ߝ௨ ൌ –”ƒ‹ƒ––‡•‹Ž‡•–”‡‰–Š

݂௬ 

ͶͲ ‫ܧ‬௦  ʹͲ

ߝ௬  Ͳ ͲǤͲͲͲ

݂௬ ൌ ‹‡Ž†•–”‡‰–Š ݂௨௬ ൌ ’’‡”›‹‡Ž†’‘‹– ߝ௬ ൌ –”ƒ‹ƒ–›‹‡Ž†•–”‡‰–Š ߝ௦௛ ൌ –”ƒ‹ƒ–•–”ƒ‹ǦŠƒ”†‡‹‰

݂௙ ൌ ”ƒ…–—”‡•–”‡‰–Š ߝ௙ ൌ –”ƒ‹ƒ–ˆ”ƒ…–—”‡’‘‹–

ߝ௨ 

ߝ௦௛  ͲǤͲʹͲ

ͲǤͲͶͲ

ͲǤͲ͸Ͳ

ߝ௙  ͲǤͲͺͲ

ͲǤͳͲͲ

‫ܖܑ܉ܚܜ܁‬ሺܑ‫ܖ‬ǤȀܑ‫ܖ‬Ǥ ሻ Figure 2.2 Stress-strain curve of a steel reinforcing bar with a well-defined yield strength.

:KHUHWKHVWUHVVVWUDLQGLDJUDPLVFKDUDFWHUL]HGE\DVKDUSNQHHGRUZHOOGH¿QHG\LHOGSRLQWVXFKDVWKHRQHLOOXVWUDWHGLQ)LJXUHWKHKDOWRIIRUFHPHWKRGFDQEHXVHGWRGHWHUPLQHWKH\LHOGVWUHQJWK>$&,E @,QWKLV PHWKRGDQLQFUHDVLQJIRUFHLVDSSOLHGWRDWHQVLOHWHVWVSHFLPHQDWDVSHFL¿HGXQLIRUPUDWH7KHORDGDWZKLFKWKH force hesitates corresponds to the yield strength of the reinforcing bar. Test values of The slope of the initial linear segment of the stress-strain curve is the modulus of elasticity, PD\EHRIWKHRUGHURIWRSVLEHFDXVHWKHUHLQIRUFLQJEDUDUHDLVQRWFRQVWDQWGXHWRWUDQVYHUVHGHIRUPDWLRQVRQWKHEDUV+RZHYHUIRUGHVLJQSXUSRVHV LVSHUPLWWHGWREHWDNHQDVSVLVHH $&,  is the slope tangent to the initial portion of the strain-hardening segment of the The strain-hardening modulus, VWUHVVVWUDLQFXUYH,WLVYDULDEOHDQGQRWVSHFL¿HGLQDQ\RIWKH$670VSHFL¿FDWLRQVQRUGHVFULEHGLQ$&, The stress-strain curves for some types of higher strength reinforcing bars can be more rounded in shape than the RQHVKRZQLQ)LJXUHWKHVHDUHFRPPRQO\UHIHUUHGWRDVURXQGKRXVHRUFRQWLQXRXVO\\LHOGLQJFXUYHVVHH)LJXUH IRUDJHQHULFVWUHVVVWUDLQFXUYHIRU*UDGHEDUV $IWHUDQLQLWLDOOLQHDUHODVWLFVHJPHQWDJUDGXDOUHGXFWLRQ

2-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

LQVWLIIQHVVRFFXUVEHKDYLRUEHFRPHVQRQOLQHDUEHIRUHUHDFKLQJD\LHOGVWUHQJWKGH¿QHGE\WKHSHUFHQWRIIVHW method, which must be used to determine ZKHUHDVWUHVVVWUDLQFXUYHGRHVQRWKDYHDZHOOGH¿QHG\LHOGSRLQW is reached. >$&,D @7KLVLVIROORZHGE\JUDGXDOVRIWHQLQJXQWLOWKHWHQVLOHVWUHQJWK

”ƒ†‡ͳͲͲ ݂௨ 

–”‡••ሺ‹’•Ȁ‹Ǥʹሻ

ͳͷͲ

ͳͲͲ

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ͷͲ

Ͳ ͲǤʹ’‡”…‡–‘ˆˆ•‡–

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–”ƒ‹ሺ‹ǤȀ‹Ǥሻ Figure 2.3 Rounded stress-strain curve for Grade 100 steel reinforcing bars.

,QWKHSHUFHQWRIIVHWPHWKRGDVWUDLQLVORFDWHGRQWKHVWUDLQD[LVDGLVWDQFHRILQLQIURPWKHRULJLQVHH )LJXUH $OLQHHPDQDWLQJIURPWKLVSRLQWLVWKHQGUDZQSDUDOOHOWRWKHLQLWLDOOLQHDUSRUWLRQRIWKHVWUHVVVWUDLQ FXUYH7KHSRLQWZKHUHWKLVOLQHLQWHUVHFWVWKHVWUHVVVWUDLQFXUYHLVGH¿QHGDV Tensile Property and Uniform Elongation Requirements 3URSHUWLHVRIGHIRUPHGUHLQIRUFLQJEDUVPXVWFRQIRUPWR$&,7KHDSSOLFDEOH$670VSHFL¿FDWLRQVDQGDGGLWLRQDOUHTXLUHPHQWVWKDWPXVWEHVDWLV¿HGIRUWKHDYDLODEOHEDUW\SHVDUHJLYHQLQ7DEOH

2-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 2.4 ASTM Specifications and Additional Requirements for Deformed Reinforcing Bars Steel Type

ASTM Specification/Additional Requirements

Carbon steel

ASTM A615/ACI Table 20.2.1.3(a)

Low-alloy steel

ASTM A706/ACI 20.2.1.3(b)(i)-(iii)

Axle steel, rail steel, and bars from rail steel

ASTM A996/Bars from rail steel must be Type R

Stainless steel

ASTM A955

Low-carbon chromium steel

ASTM A1035

The additional requirements in ACI Table 20.2.1.3(a) for ASTM A615 reinforcement are given in Table 2.5. Table 2.5 Modified Tensile Strength and Additional Tensile Strength Property Requirements for ASTM A615 Reinforcement

Minimum tensile strength (psi) Minimum ratio of actual tensile strength to actual yield strength

Grade 40

Grade 60

Grade 80

Grade 100

60,000

80,000

100,000

115,000

1.10

1.10

1.10

1.10

7KHDGGLWLRQDOUHTXLUHPHQWVWKDWPXVWEHVDWLV¿HGIRU$670$UHLQIRUFHPHQWDUHWKHIROORZLQJ 7HQVLOHSURSHUW\UHTXLUHPHQWVIRU$670$*UDGHUHLQIRUFHPHQWPXVWEHDVVSHFL¿HGLQ$&, Table 20.2.1.3(b) [see Table 2.6]. Bend test requirements for ASTM A706 Grade 100 reinforcement must be the same as those for ASTM A706 Grade 80 reinforcement. ii. 8QLIRUPHORQJDWLRQUHTXLUHPHQWVIRUDOOJUDGHVRI$670$UHLQIRUFHPHQWPXVWEHDVVSHFL¿HGLQ ACI Table 20.2.1.3(c) [see Table 2.7] where the uniform elongation is determined as the elongation at the maximum force sustained by the reinforcing bar test piece. iii. For all grades of ASTM A706 reinforcement, the radius at the base of each bar deformation must be at least 1.5 times the height of the deformation. This requirement applies to all deformations, including transverse lugs, longitudinal ribs, grade ribs, grade marks, and intersections between deformations. Conformance must be assessed by measurements taken on newly-machined rolls used to manufacture reinforcing bars instead of measurements taken on reinforcing bar samples. i.

Table 2.6 Tensile Property Requirements for ASTM A706 Grade 100 Reinforcement

Minimum tensile strength, psi

117,000

Minimum ratio of actual tensile strength to actual yield strength

1.17

Minimum yield strength, psi

100,000

Maximum yield strength, psi

118,000

Minimum fracture elongation in 8 in., percent

10

Table 2.7 Uniform Elongation Requirements for ASTM A706 Reinforcement Bar Size

2-6

Minimum Uniform Elongation, Percent Grade 60

Grade 80

Grade 100

#3 through #10

9

7

6

#11, #14, #18

6

6

6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3ODLQEDUVZKLFKDUHRQO\SHUPLWWHGIRUVSLUDOUHLQIRUFHPHQWPXVWFRQIRUPWR$670$$$RU$ $&,  +HDGHGGHIRUPHGEDUVPXVWFRQIRUPWR$670$LQFOXGLQJWKHUHTXLUHPHQWVLQ$QQH[$IRU&ODVV+$KHDG GLPHQVLRQV$&, 7KHOLPLWDWLRQWR&ODVV+$KHDGGLPHQVLRQVLVGXHWRWKHODFNRIWHVWGDWDIRUKHDGHG GHIRUPHGEDUVWKDWGRQRWPHHW&ODVV+$GLPHQVLRQDOUHTXLUHPHQWV Design Properties

Stress-Strain Relationships For deformed reinforcing bars, it is permitted to assume the stress in the reinforcing bars, is proportional to that is, $&, )RUVWUDLQVJUHDWHUWKDQWKDWFRUUHstrain in cases where the strain is below the increase in strength due to strain hardening is neglected for nominal strength calculasponding to tions. Modulus of Elasticity As noted above, the modulus of elasticity of the reinforcing steel, UHJDUGOHVVRIUHLQIRUFHPHQWJUDGH$&, 

LVSHUPLWWHGWREHWDNHQDVSVL

3HUPLWWHG9DOXHVRI6SHFL¿HGVHH7DEOH@6LPLODULQIRUPDWLRQ LVJLYHQLQ$&,7DEOHE IRUSODLQUHLQIRUFHPHQWVHH7DEOH  Table 2.8 Nonprestressed Deformed Reinforcing Bars Usage

• Flexure • Axial force • Shrinkage and temperature • Lateral support of longitudinal bars • Concrete FRQ¿QHPHQW

Shear

Torsion Anchor reinforcement

Application

Special seismic systems

Maximum Value of f y or f yt Permitted for Design Calculations (psi)

Special moment frames



Special structural walls(1)



Other

 

Special seismic systems



Spirals



Other Special Special moment frames(6) seismic  systems Special structural walls Spirals

  

ASTM Specification

$ A$$$ $ A$$$ $ $$$$ $ $$$$ $$$$



$$$$

Shear friction



Stirrups, ties, hoops



Longitudinal and transverse



$$$$ $$$$ $ $$$$

Special seismic systems



$

Other



$$$$ (table continued on next page)

2-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 2.8 Nonprestressed Deformed Reinforcing Bars (cont.) Usage

Application

Maximum Value of f y or f yt Permitted for Design Calculations (psi)

Regions designed using the strut-andtie method

Longitudinal ties



Other



(1) (2) (3) (4) (5) (6) (7)

ASTM Specification

$$$$

All components of special structural walls, including coupling beams and wall piers. ASTM A615 Grade 60 bars are permitted provided the requirements of ACI 20.2.2.5(b) are satisfied. In slabs and beams that are not part of a special seismic system, reinforcing bars passing through or extending from special structural walls must satisfy ACI 20.2.2.5. Longitudinal reinforcement with fy>80,000 psi is not permitted for intermediate moment frames and ordinary moment frames resisting earthquake demands, The maximum design value of fy or fyt is 80,000 psi for use in design calculations when used for shear reinforcement in diaphragms and foundations for load combinations including earthquake forces where the diaphragms or foundations are part of a building with a special seismic system. Shear reinforcement in this application includes stirrups, ties, hoops, and spirals in special moment frames. Shear reinforcement in this application includes all transverse reinforcement in special structural walls, coupling beams, and wall piers. Diagonal bars in coupling beams must comply with ASTM A706 or Footnote (2).

Table 2.9 Nonprestressed Plain Spiral Reinforcing Bars Application

Maximum Value of f y or f yt Permitted for Design Calculations (psi)

ASTM Specification

Spirals in special seismic systems



$$$ $

Spirals



$$$ $

Shear

Spirals



$$$ $

Torsion in nonprestressed beams

Spirals



$$$ $

Usage

• Lateral support of longitudinal bars • &RQFUHWHFRQ¿QHPHQW

Special Seismic Systems and Anchor Reinforcement $FFRUGLQJWR$&,GHIRUPHGQRQSUHVWUHVVHGORQJLWXGLQDOUHLQIRUFHPHQWUHVLVWLQJHDUWKTXDNHLQGXFHGEHQGing moments, axial forces, or both, in special seismic systems and anchor reinforcement in SDCs C, D, E, and F must be in accordance with (a) or (b): a. $  670$*UDGHRUIRUVSHFLDOVWUXFWXUDOZDOOVDQG*UDGHDQGIRUVSHFLDOPRPHQW frames. b. $670$*UDGHLIL WKURXJKLY DUHVDWLV¿HG$670$*UDGHDQG*UDGHDUHQRWSHUPLWWHG in special seismic systems because of concern associated with low-cycle fatigue behavior). i. Actual yield strength based on mill tests does not exceed E\PRUHWKDQSVL ii. 5DWLRRIWKHDFWXDOWHQVLOHVWUHQJWKWRWKHDFWXDO\LHOGVWUHQJWKLVDWOHDVW iii. 0LQLPXPIUDFWXUHHORQJDWLRQLQLQPXVWEHDWOHDVWSHUFHQWIRUEDUVL]HVWKURXJKDWOHDVW SHUFHQWIRUEDUVL]HVWKURXJKDQGDWOHDVWSHUFHQWIRUEDUVL]HVDQG iv. 0LQLPXPXQLIRUPHORQJDWLRQPXVWEHDWOHDVWSHUFHQWIRUEDUVL]HVWKURXJKDQGDWOHDVW SHUFHQWIRUEDUVL]HVDQG occurs at the tensile strength is commonly referred to as uniform elongation, which is the largest The strain HORQJDWLRQLQWKHEDUIRUZKLFKWKHWHQVLOHVWUDLQVDUHXQLIRUPWKURXJKRXWWKHOHQJWKRIWKHEDUVHH)LJXUH 7KLV generally occurs right before the onset of necking. This is the maximum strain relied upon at a location where yielding of the reinforcing bar may occur (that is, in an anticipated plastic hinge region in the member).

2-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The strain at the fracture point is the total elongation over a prescribed gauge length extending across the fracture RIDUHLQIRUFLQJEDU$FFRUGLQJWR$670$DUHLQIRUFLQJEDULVPDUNHGZLWKDQLQLWLDOLQJDXJHOHQJWKDQG LVSXOOHGWRIUDFWXUH7KHHQGVRIWKHIUDFWXUHGUHLQIRUFLQJEDUDUH¿WWRJHWKHUDQGWKHGLVWDQFHEHWZHHQWKHJDXJH marks is re-measured. The total elongation is calculated as the percent increase in length relative to the original gauge length: (2.4) 2.2.3 Headed Shear Stud Reinforcement

+HDGHGVKHDUVWXGUHLQIRUFHPHQWDQGVWXGDVVHPEOLHVZKLFKDUHW\SLFDOO\XVHGDVVKHDUUHLQIRUFHPHQWDURXQG FROXPQVLQWZRZD\UHLQIRUFHGFRQFUHWHVODEVPXVWFRQIRUPWR$670$$&, 5HTXLUHGVWXGKHDG FRQ¿JXUDWLRQVDUHJLYHQLQ$&,)LJXUH5 2.2.4 Durability of Steel Reinforcement Specified Concrete Cover

Overview Concrete cover, which is provided to reinforcing bars to protect them from weather and other effects, is measured IURPWKHFRQFUHWHVXUIDFHWRWKHRXWHUPRVWVXUIDFHRIWKHUHLQIRUFHPHQW0LQLPXPVSHFL¿HGFRQFUHWHFRYHULV JLYHQLQ$&,WKURXJK$&, $JUHDWHUFRYHUWKDQWKDWSUHVFULEHGLQWKHVHVHFWLRQVPD\ EHUHTXLUHGE\WKHJHQHUDOEXLOGLQJFRGHIRU¿UHSURWHFWLRQ&RQFUHWHÀRRU¿QLVKHVDUHSHUPLWWHGWREHSDUWRIWKH UHTXLUHGFRYHUIRUQRQVWUXFWXUDOSXUSRVHV$&,  Nonprestressed Cast-in-place Members 0LQLPXPFRQFUHWHFRYHUIRUQRQSUHVWUHVVHGFDVWLQSODFHFRQFUHWHPHPEHUVLVJLYHQLQ$&,7DEOH$&, VHH7DEOH  Table 2.10 Minimum Concrete Cover for Cast-in-place Nonprestressed Concrete Members Concrete Exposure

Member

Reinforcement

Specified Cover (in.)

Cast against and permanently in contact with the ground

All

All



Exposed to weather or in contact with the ground

WKURXJKEDUV



All EDUVDQGVPDOOHU



DQGEDUV



EDUVDQGVPDOOHU



Slabs, joists, and walls Not exposed to weather or in contact with the ground

Beams, columns, pedestals, Primary reinforcement, stirrups, and tension ties ties, spirals, and hoops



Deep Foundation Members 6SHFL¿HGFRQFUHWHFRYHUIRUFDVWLQSODFHGHHSIRXQGDWLRQPHPEHUVIRUH[DPSOHSLHUVDQGFDLVVRQV LVJLYHQLQ $&,7DEOH$&,VHH7DEOH 

2-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 2.11 Minimum Concrete Cover for Cast-in-place Deep Foundation Members Concrete Exposure

Reinforcement

Specified Cover (in.)

Cast against and permanently in contact with the ground and not enclosed by steel pipe, tube, permanent casing, or stable rock socket

All



Enclosed by steel pipe, tube, permanent casing, or stable rock socket

All



%XQGOHG5HLQIRUFLQJ%DUV )RUEXQGOHGUHLQIRUFLQJEDUVWKHVSHFL¿HGFRQFUHWHFRYHULVHTXDOWRWKHVPDOOHURIWKHHTXLYDOHQWGLDPHWHURIWKH EXQGOHRULQ$&,  For bundled bars in members where the concrete is cast against and is permanently in contact with the ground, the VSHFL¿HGFRYHULVLQ Headed Shear Stud Reinforcement 7KHVSHFL¿HGFRQFUHWHFRYHUIRUKHDGVDQGEDVHUDLOVLQKHDGHGVKHDUVWXGDVVHPEOLHVPXVWEHDWOHDVWHTXDOWRWKDW UHTXLUHGIRUWKHUHLQIRUFHPHQWLQWKHPHPEHU$&,VHH$&,)LJXUH5 )RUH[DPSOHWKHPLQLPXPFRQFUHWHFRYHUWRWKHKHDGVDQGUDLOVLVLQIRUDKHDGHGVKHDUVWXGDVVHPEO\XVHGDVVKHDUUHLQIRUFHPHQW in a two-way reinforced concrete slab not exposed to the elements and reinforced with longitudinal reinforcement VPDOOHUWKDQEDUVVHH7DEOH  Corrosive Environments For members in corrosive environments where it is expected the concrete will be exposed to external sources of chlorides (such as deicing salts, brackish water, seawater, or spray from these sources) or other severe exposure FRQGLWLRQVVSHFL¿HGFRQFUHWHFRYHUVKRXOGEHLQFUHDVHGDERYHWKHPLQLPXPYDOXHVVSHFL¿HGLQ$&,DV GHHPHGQHFHVVDU\$&, ,QVXFKFDVHVWKHFRQFUHWHVKRXOGEHSURSRUWLRQHGWRVDWLVI\WKHUHTXLUHPHQWV IRUWKHDSSOLFDEOHH[SRVXUHFODVVVSHFL¿HGLQ$&,&KDSWHU $VSHFL¿HGFRQFUHWHFRYHURIQRWOHVVWKDQLQLVUHFRPPHQGHGLQ$&,5IRUUHLQIRUFHPHQWLQZDOOV DQGVODEVLQFRUURVLYHHQYLURQPHQWV)RURWKHUPHPEHUVDLQFRQFUHWHFRYHULVUHFRPPHQGHG Nonprestressed Coated Reinforcement

Coated reinforcing bars are used in applications where corrosion resistance is required (such as parking structures or other buildings and structures where the members are exposed to the elements). Nonprestressed coated reinforcing EDUVPXVWFRQIRUPWR$&,7DEOH$&,  =LQFFRDWHGKRWGLSSHGJDOYDQL]HG $670$ (SR[\FRDWHG$670$RU$ =LQFDQGHSR[\GXDOFRDWHG$670$ 7KHDIRUHPHQWLRQHGFRDWHGEDUW\SHVPXVWFRQIRUPWRWKHUHTXLUHPHQWVLQ$&,D E RUF 

2.3 Strength Reduction Factors 2.3.1 Overview

Strength reduction factors, used in the determination of the design strength of a reinforced concrete member are JLYHQLQ$&,&KDSWHU7KHSXUSRVHVRIVWUHQJWKUHGXFWLRQIDFWRUVDUHWKHIROORZLQJ$&,5 

2-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1. To account for the probability of under-strength members due to variations in material strengths and dimensions. The strength of concrete can vary because concrete is a composite material made of constituent materials whose properties vary. The strength of reinforcing steel can also vary but usually to a much lesser degree than concrete.  To account for inaccuracies in design equations. $QXPEHURIDVVXPSWLRQVDQGVLPSOL¿FDWLRQVDUHPDGHLQWKHHTXDWLRQVWRGHWHUPLQHQRPLQDOVWUHQJWKV$V such, inaccuracies are introduced that must be accounted for when determining the design strength of a reinforced concrete member. 3. 7RUHÀHFWWKHDYDLODEOHGXFWLOLW\DQGUHTXLUHGUHOLDELOLW\RIWKHUHLQIRUFHGFRQFUHWHPHPEHUXQGHUWKHORDGHIfects being considered. Reinforced concrete members that are more ductile, such as beams or slabs, are less sensitive to variations in concrete strength compared to members that are less ductile, such as columns. Spiral reinforcement con¿QHVWKHFRQFUHWHLQDFROXPQEHWWHUWKDQWLHGUHLQIRUFHPHQW7KXVVSLUDOFROXPQVDUHPRUHGXFWLOHDQGKDYH greater toughness than tied columns.  7RUHÀHFWWKHLPSRUWDQFHRIWKHUHLQIRUFHGFRQFUHWHPHPEHULQWKHVWUXFWXUH The failure of a vertical member, such as a column or wall, in a structure is usually considered more detrimenWDOWKDQIDLOXUHRIDKRUL]RQWDOPHPEHUVXFKDVDEHDP 2.3.2 Strength Reduction Factors Based On Action or Structural Element

Strength reduction factors based on different types of actions and nonprestressed structural elements are given in $&,7DEOHVHH$&,DQG7DEOHIRUWKHVWUHQJWKUHGXFWLRQIDFWRUVDSSOLFDEOHLQWKLVSXEOLFDWLRQ  The strength reduction factor for shear in this table is not applicable to members in special moment frames (including beam-column joints), special structural walls (including diagonally reinforced coupling beams), and the other PHPEHUVQRWHGLQ$&,VHH6HFWLRQRIWKLVSXEOLFDWLRQ  Table 2.12 Strength Reduction Factors, Ҁ Action

Strength Reduction Factor, Ҁ

Moment, axial force, or combined moment and axial force Shear

WRLQDFFRUGDQFHZLWK$&, VHH6HFWLRQRIWKLVSXEOLFDWLRQ 

Torsion



Bearing



2.3.3 Strength Reduction Factors For Moment, Axial Force, Or Combined Moment and Axial Force

Strength reduction factors for reinforced concrete members subjected to bending moments, axial forces, or a combiQDWLRQRIERWKDUHJLYHQLQ$&,7DEOHVHH$&,7DEOHDQG)LJXUH  Table 2.13 Strength Reduction Factors, Ҁ, For Moment, Axal Force, Or Combined Moment and Axial Force Strength Reduction Factor, Ҁ Net Tensile Strain, İt

Classification

Compression-controlled

Type of Transverse Reinforcement Spirals Conforming to ACI 25.7.3

Other

0.75

0.65





Transition* Tension-controlled *For sections classified as transition, it is permitted to use

corresponding to compression-controlled sections.

2-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Figure 2.4 Variation of

with net tensile strain

The nominal strength of a member subjected to bending moments or combined bending moments and axial forces is GHWHUPLQHGDVVXPLQJWKHVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHURIWKHVHFWLRQLVHTXDOWRVHH$&, is the tensile strain calculated in the extreme tension reinforcement at nominal  7KHQHWWHQVLOHVWUDLQ VWUHQJWKVHH)LJXUHIRUWKHFDVHRIDUHLQIRUFHGFRQFUHWHEHDPZLWKWZROD\HUVRIWHQVLRQUHLQIRUFHPHQWDWWKH section under consideration). 2-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

&RPSUHVVLRQFRQWUROOHGVHFWLRQVDUHGH¿QHGDVVHFWLRQVZKHUH The term is the value of the net tensile VWUDLQLQWKHH[WUHPHOD\HURIORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQ $&,  (2.5)

)RU*UDGHUHLQIRUFHPHQW LVSHUPLWWHGWREHWDNHQHTXDOWR$&, 0HPEHUVVXEMHFWHGWRRQO\ axial compression, like columns, are compression-controlled. In such cases, a brittle compression failure is expected with little or no warning prior to failure. As such, lower strength reduction factors are assigned to compressioncontrolled sections. Additionally, columns typically support areas much greater than beams, which are tensionFRQWUROOHGDQGDVQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHFRQVHTXHQFHVRIFROXPQIDLOXUHDUHJHQHUDOO\PRUH severe than those attributed to beam failure. 7KLVOLPLWSURYLGHVVXI¿FLHQWGXFWLOLW\IRUPRVWDS$VHFWLRQLVGH¿QHGDVWHQVLRQFRQWUROOHGZKHUH SOLFDWLRQVDQGHVVHQWLDOO\SURYLGHVDZDUQLQJWRIDLOXUHWKDWLVH[FHVVLYHFUDFNLQJDQGGHÀHFWLRQDUHH[SHFWHGSULRU to failure). Beams and slabs must be tension-controlled. A linear transition in the strength reduction factor is permitted between the limits for compression-controlled and WHQVLRQFRQWUROOHGVHFWLRQVVHH)LJXUH ,WLVSHUPLWWHGWRXVHVWUHQJWKUHGXFWLRQIDFWRUVFRUUHVSRQGLQJWR FRPSUHVVLRQFRQWUROOHGVHFWLRQVIRUVHFWLRQVFODVVL¿HGDVWUDQVLWLRQ $FFRUGLQJWR$&,GHYHORSPHQWOHQJWKVGRQRWUHTXLUHDVWUHQJWKUHGXFWLRQIDFWRU%HFDXVHODSVSOLFHOHQJWKV are based on development lengths, a strength reduction factor is not required for lap splices as well. An allowance for VWUHQJWKUHGXFWLRQLVLQFOXGHGLQWKHH[SUHVVLRQVLQ$&,&KDSWHUIRUGHWHUPLQLQJGHYHORSPHQWDQGVSOLFHOHQJWKV 2.3.4 Strength Reduction Factors For Shear In Structures Relying On Special Moment Frames and Special Structural Walls

)RUVWUXFWXUHVWKDWXWLOL]HVSHFLDOPRPHQWIUDPHVDQGVSHFLDOVWUXFWXUDOZDOOVWRUHVLVWHDUWKTXDNHHIIHFWV the value of the strength reduction factor, IRUVKHDULVWREHPRGL¿HGLQDFFRUGDQFHZLWK$&,WKURXJK $&,  For any member designed to resist where the nominal shear strength of the member is less than the shear corresponding to the development of the nominal moment strength of the member, IRUVKHDUPXVWEHWDNHQDV $&, 7KHQRPLQDOPRPHQWVWUHQJWKLQVXFKFDVHVLVWREHFDOFXODWHGXVLQJWKHIDFWRUHGD[LDOIRUFHVIURP ORDGFRPELQDWLRQVLQ$&,7DEOHWKDWLQFOXGH This provision is applicable to shear-controlled members, such as low-rise structural walls, portions of walls between openings, or diaphragms where nominal shear strength is less than the shear corresponding to the development of nominal moment strength. In most cases, it would be impractical to design the member otherwise. For diaphragms, for shear must not exceed the value of for shear required for the vertical components of the SULPDU\VHLVPLFIRUFHUHVLVWLQJV\VWHP6)56 >$&,@7KLVSURYLVLRQLVLQWHQGHGWRLQFUHDVHWKHVWUHQJWK for the walls assigned to resist the earthquake of the diaphragm and its connections in buildings where forces. This requirement is also applicable to foundation elements supporting the primary vertical elements of the 6)56$&,  For beam-column joints of special moment frames and diagonally reinforced coupling beams, WR$&, 

for shear is equal

2-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

2-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 3 DESIGN LOADS AND LOAD COMBINATIONS 3.1 Overview $FFRUGLQJWR$&,WKHORDGVDQGORDGFRPELQDWLRQVFRQVLGHUHGLQGHVLJQPXVWEHLQDFFRUGDQFHZLWK$&,&KDSWHU3UHVHQWHGLQWKLVFKDSWHUDUHWKHW\SLFDOORDGVDQGUHTXLUHGORDGFRPELQDWLRQVDSSOLFDEOHLQWKHGHVLJQRIUHinforced concrete buildings. Methods are given to determine the Seismic Design Category (SDC) of a building and live load reduction on structural members. Step-by-step procedures on how to determine wind forces on the main wind force resisting system (MWFRS), seismic forces on the seismic-force-resisting system (SFRS), and seismic design forces on diaphragms (including collectors) are also presented.

3.2 Design Loads In the design of reinforced concrete members, the loads that must be included are self-weight, applied loads, and the HIIHFWVIURPZLQGHDUWKTXDNHVUHVWUDLQWRIYROXPHFKDQJHDQGGLIIHUHQWLDOVHWWOHPHQW$&, /RDGVDUHWREH GHWHUPLQHGLQDFFRUGDQFHZLWKWKHJHQHUDOEXLOGLQJFRGHRUE\WKHEXLOGLQJRI¿FLDO$&,  7KHORDGHIIHFWVLQFOXGHGLQWKH,%&DQG$6&(6(,DUHJLYHQLQ7DEOH Table 3.1 Summary of Load Effects Section/Chapter No. Notation

Load Effect IBC

ASCE/SEI 7

Dead loads



3.1

Weight of ice





Seismic load effects

1613



Seismic load effects including overstrength

1613



/RDGVGXHWRÀXLGVZLWKZHOOGH¿QHGSUHVVXUHVDQGPD[LPXPKHLJKWV

—

—

Flood loads Loads due to lateral earth pressures, ground water pressure, or pressure of bulk materials 5RRIOLYHORDGVJUHDWHUWKDQOEIWDQGÀRRUOLYHORDG













5RRIOLYHORDGVRIOEIW or less





Rain loads

1611

8

Snow loads





—



Loads due to wind pressure



±

Wind-on-ice loads





Cumulative effects of self-straining forces and effects

3.3 Seismic Design Category The SDC of a building is determined in accordance with the general building code or determined by the building RI¿FLDO$&, $OOEXLOGLQJVDQGVWUXFWXUHVPXVWEHDVVLJQHGWRDQ6'&LQDFFRUGDQFHZLWK,%&RU $6&(6(,$&, )LJXUHFDQEHXVHGWRGHWHUPLQHWKH6'&EDVHGRQWKHUHTXLUHPHQWVLQWKHVH and are the VHFWLRQV7KHVHFWLRQ¿JXUHDQGWDEOHQXPEHUVLQWKH¿JXUHDUHIURP$6&(6(,7KHWHUPV GHVLJQSHUFHQWGDPSHGVSHFWUDOUHVSRQVHDFFHOHUDWLRQSDUDPHWHUVDWVKRUWSHULRGVDQGDWDSHULRGRIVHFRQG UHVSHFWLYHO\ZKLFKDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$6&(6(,

3-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine ܵܵ and ܵ1 from  Figures 22-1 through 22-8 ሺ11.4.2ሻ

EŽ

Determine the Risk Category from Table 1.5-1

EŽ

EŽ

zĞƐ

Is ܵܵ ൑ 0.15 and ܵ1 ൑ 0.04? 

Structure is permitted to be assigned to SDC A and must only comply with 11.7

Is the Risk Category I, II, or III?

Is ܵ1 ൒ 0.75? 

zĞƐ

zĞƐ

Is ܵ1 ൒ 0.75? 

EŽ

zĞƐ

Structure is assigned to SDC E

Structure is assigned to SDC F

Determineܵ‫ ܵܦ‬and ܵ‫ܦ‬1 E\

EŽ

Are all 4 conditions in 11.6 satisfied?

'HWHUPLQHWKH6'&DVWKHPRUHVHYHUHRIWKH WZR6'&VLQ7DEOHVDQGEDVHG RQWKH5LVN&DWHJRU\DQGܵ‫ ܵܦ‬and ܵ‫ܦ‬1 , UHVSHFWLYHO\

zĞƐ

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Figure 3.1 Determination of Seismic Design Category (SDC).

Structural members in buildings assigned to SDC A must satisfy all the applicable requirements in ACI 318 except IRUWKHUHTXLUHPHQWVLQ$&,&KDSWHUZKLFKDUHQRWDSSOLFDEOH$&, )RUEXLOGLQJVDVVLJQHGWR6'&% WKURXJK)WKHSURYLVLRQVRI&KDSWHUPXVWEHVDWLV¿HGLQDGGLWLRQWRWKHDSSOLFDEOHUHTXLUHPHQWVRIRWKHUFKDSWHUV $&, 7KHUHTXLUHPHQWVRI$&,SHUWDLQWRVWUXFWXUDOPHPEHUVGHVLJQDWHGQRWWREHSDUWRIWKH6)56 in buildings assigned to SDC B through F. 3-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3.4 Live Load Reduction $FFRUGLQJWR$&,OLYHORDGUHGXFWLRQLVSHUPLWWHGLQDFFRUGDQFHZLWKWKHJHQHUDOEXLOGLQJFRGH,QWKHDEVHQFHRIDJHQHUDOEXLOGLQJFRGHWKHOLYHORDGUHGXFWLRQSURYLVLRQVLQ$6&(6(,DQGIRUXQLIRUPOLYHORDGV and uniform roof live loads, respectively, may be used.

Determine ‫ ݋ܮ‬from Table 4.3-1

zĞƐ

Is ‫ ݋ܮ‬൐ 100 psf? 

EŽ

Live load reduction is not permitted*

EŽ

Is the member a one-way slab?

zĞƐ

‫ ܶ ܣ‬൑ 1.5ሺslab spanሻ2 

EŽ

Is the structure a passenger vehicle garage?

zĞƐ

Live load reduction is not permitted*

Is the area classified as an assembly occupancy?

EŽ

zĞƐ

Live load reduction is not permitted

EŽ

Is‫ ܶ ܣ ܮܮܭ‬൏ 400 sq ft?

zĞƐ

Live load reduction is not permitted

‫ ܮ‬ൌ ‫ ݋ܮ‬ቆ0.25 ൅

 

15 ඥ‫ܶܣ ܮܮܭ‬

ቇ ሾASCE/SEI Eq. ሺ4.7-1ሻሿ

൒ 0.50‫ ݋ܮ‬for members supporting one floor

* Live loads for members supporting two or more floors are permitted to be reduced by a maximum of 20 percent, but the reduced live load must not be less than L determined in accordance with ASCE/SEI 4.7.2.

൒ 0.40‫ ݋ܮ‬for members supporting two or more floors

 Figure 3.2 Determination of live load reduction in accordance with ASCE/SEI 4.7.

3-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Reduced uniform live load, LQDFFRUGDQFHZLWK$6&(6(,FDQEHGHWHUPLQHGXVLQJWKHÀRZFKDUWLQ)LJXUH is the live load element ,QWKLV¿JXUH LVWKHXQLIRUPO\GLVWULEXWHGOLYHORDGLQ$6&(6(,7DEOH is the tributary area in square IDFWRULQ$6&(6(,7DEOHVHH$6&(6(,)LJXUH&DQG7DEOH DQG FDQEHGHWHUPLQHGE\WKHÀRZFKDUWLQ)LJXUHZKHUH and are feet. Similarly, reduced roof live load, and the factor respectively. For pitched roofs, is equal to reduction factors based on the tributary area, WKHQXPEHURILQFKHVRIULVHSHUIRRWDQGIRUDUFKHVRUGRPHVLWLVWKHULVHWRVSDQUDWLRPXOWLSOLHGE\ Table 3.2 Live Load Element Factor, KLL Element

KLL

Interior columns



Exterior columns without cantilever slabs



Edge columns with cantilever slabs

3

Corner columns with cantilever slabs



Edge beams without cantilever slabs



Interior beams



$OORWKHUPHPEHUVQRWLGHQWL¿HGLQFOXGLQJ • Edge beams with cantilever slabs • Cantilever beams • One-way slabs • Two-way slabs • Members without provisions for continuous shear transfer normal to their span

1

3.5 Load Factors and Combinations The load combinations needed to determine the required strength, of a structural member are given in ACI Table VHH7DEOH 7KHORDGFRPELQDWLRQVLQ,%&DUHWKHVDPHDVWKRVHLQ7DEOHZLWKWKHIROORZLQJ exceptions: 1. The variable LQ,%&(TXDWLRQV  DQG DUHQRWLQ$&,(TXDWLRQVF G DQG H ,QVWHDGWKHORDGIDFWRURQWKHOLYHORDG LQWKH$&,ORDGFRPELQDWLRQVLVHTXDOWRZLWKWKHH[ception that the load factor on LVSHUPLWWHGWRHTXDOIRUDOORFFXSDQFLHVH[FHSWIRUSDUNLQJJDUDJHVDUHDV occupied as places of public assembly, and areas where LVJUHDWHUWKDQOEIW$&, 7KLVH[FHStion makes these load combinations the same in the IBC and ACI 318.  The variable LQ,%&(TXDWLRQ LVQRWLQ$&,(TXDWLRQH ,QVWHDGDORDGIDFWRURILVDSSOLHG to $FFRUGLQJWRWKHVHFRQGH[FHSWLRQLQ$6&(6(,$6&(6(,ORDGFRPELQDWLRQ>ZKLFKLVWKH or the sloped roof snow VDPHDV$&,(TXDWLRQH @ PXVWEHWDNHQDVHLWKHUWKHÀDWURRIVQRZORDG 7KLVPHDQVWKHEDODQFHGVQRZORDGGH¿QHGLQ$6&(6(,IRUÀDWURRIVDQGLQ$6&(6(,IRU load, VORSHGURRIVFDQEHXVHGLQ$6&(6(,ORDGFRPELQDWLRQ1RWHWKDW LQ$6&(6(,ORDGFRPELQDWLRQVDQG LVGH¿QHGLQWKHVDPHZD\VHHH[FHSWLRQLQ$6&(6(, 'ULIWORDGVDQGXQEDODQFHGVQRZORDGVDUH FRYHUHGE\$6&(6(,ORDGFRPELQDWLRQ

3-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

zĞƐ

EŽ

EŽ

zĞƐ

EŽ

zĞƐ

EŽ

zĞƐ



Figure 3.3 Determination of roof live load reduction in accordance with ASCE/SEI 4.8.

3-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.3 ACI and ASCE/SEI Strength Design Load Combinations ACI Equation Number

ASCE/SEI 7 Load Combination

D

1

E



F

3

G



H

6

I



J



$FFRUGLQJWR$&, • • • • • •

Load Combination

must include the applicable effects from the following:

&RQFHQWUDWHGOLYHORDGV$6&(6(,7DEOH 9HKLFXODUORDGV$6&(6(, &UDQHORDGV$6&(6(, /RDGVRQKDQGUDLOVJXDUGUDLOVDQGYHKLFXODUEDUULHUV\VWHPV$6&(6(, ,PSDFWHIIHFWV$6&(6(, Vibration effects

is provided at service-level loads, Where wind load, must be used in place of G DQGI DQG Seismic load effect,

must be used in place of in ACI Equations LQ$&,(TXDWLRQF >$&,@

LVGH¿QHGLQ$6&(6(,DVIROORZV

• )RUXVHLQ$&,(TXDWLRQH RUORDGFRPELQDWLRQLQ$6&(6(, (3.1)

• )RUXVHLQ$&,(TXDWLRQJ RUORDGFRPELQDWLRQLQ$6&(6(, (3.2)

where

LVSHUPLWWHGWREHWDNHQDVIRUEXLOGLQJV assigned to SDC B or C)

6HLVPLFORDGFRPELQDWLRQVEDVHGRQ6'&XVLQJWKHSURYLVLRQVRI$6&(6(,DQGDUHJLYHQLQ7DEOH

3-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.4 Seismic Load Combinations SDC

ACI Equation Number

ASCE/SEI 7 Load Combination

H

6

J



H

6

J



H

6

J



B C D, E, and F Seismic load effect,

Load Combination

LVGH¿QHGLQ$6&(6(,DVIROORZV

• )RUXVHLQ$&,(TXDWLRQH RUORDGFRPELQDWLRQLQ$6&(6(, (3.3)

• )RUXVHLQ$&,(TXDWLRQJ RUORDGFRPELQDWLRQLQ$6&(6(, (3.4)

where

Seismic load combinations based on SDC where seismic load effects including overstrength are required are given LQ7DEOH Table 3.5 Seismic Load Combinations Where Seismic Load Effects Including Overstrength are Required SDC

B C, D, E, and F

ACI Equation Number

ASCE/SEI 7 Load Combination

H

6

J



H

6

J



Load Combination

The following load effects must be considered, where applicable, in combination with other load effects: • Cumulative service effects of temperature, creep, shrinkage, and differential settlement, ZLWK$&, • (IIHFWVRIVHUYLFHODWHUDOORDGVGXHWRÀXLGV LQDFFRUGDQFHZLWK$&, LQDFFRUGDQFHZLWK$&, • Effects of service load due to lateral earth pressure, • (IIHFWVGXHWRÀRRGORDGVLQDFFRUGDQFHZLWK$&, • (IIHFWVGXHWRDWPRVSKHULFLFHORDGVLQDFFRUGDQFHZLWK$&, /RDGFRPELQDWLRQVZKHUHÀRRGORDGV

in accordance

and atmospheric ice loads must be considered are given in Table 3.6.

3-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.6 Strength Design Load Combination for Flood Loads and Atmospheric Ice Loads Load Effect

Flood load,

ASCE/SEI load combination

V Zones or Coastal A Zones



Noncoastal A Zones



*

Load Combination



  

Atmospheric ice load   * Definitions of V Zones and Coastal A Zones are given in ASCE/SEI 5.2.

$FFRUGLQJWR$&,ORDGFRPELQDWLRQVPXVWEHLQYHVWLJDWHGZLWKRQHRUPRUHRIWKHYDULDEOHORDGVVHWHTXDO WR]HUR,WLVSRVVLEOHWKDWWKHPRVWFULWLFDOORDGHIIHFWVRQDPHPEHURFFXUZKHQRQHRUPRUHYDULDEOHORDGVDUHQRW present. /RDGFRPELQDWLRQVIRUH[WUDRUGLQDU\HYHQWVDQGIRUJHQHUDOVWUXFWXUDOLQWHJULW\ORDGVDUHJLYHQLQ$6&(6(,DQG UHVSHFWLYHO\

3.6 Determination of Wind Forces 7KHÀRZFKDUWLQ)LJXUHFDQEHXVHGWRGHWHUPLQHWKHWRWDOGHVLJQZLQGSUHVVXUHVDWDKHLJKW above ground OHYHORQWKH0:)56RIDQHQFORVHGRUSDUWLDOO\HQFORVHGULJLGRUÀH[LEOHUHLQIRUFHGFRQFUHWHEXLOGLQJZLWKRXWD SDUDSHWDQGZLWKDQHVVHQWLDOO\ÀDWURRIXVLQJ3DUWLQ&KDSWHURI$6&(6(,7KHVHFWLRQ¿JXUHWDEOHDQG HTXDWLRQQXPEHUVLQ)LJXUHDUHIURP$6&(6(,+RUL]RQWDOLQWHUQDOSUHVVXUHVFDQFHORXWZKHQGHWHUPLQLQJ total wind pressures on the MWFRS of a building. )RUEXLOGLQJVWUXFWXUHVLWLVFXVWRPDU\WRFDOFXODWHZLQGSUHVVXUHVDWWKHURRIDQGÀRRUOHYHOVDQGWRDVVXPHWKDWWKH SUHVVXUHVDUHXQLIRUPO\GLVWULEXWHGRYHUWKHWULEXWDU\DUHDDWWKDWOHYHO7KXVWKHWRWDOZLQGIRUFHDWDURRIRUÀRRU level is determined by multiplying the total wind pressure at that level (windward plus leeward pressure) by the tributary area, which is equal to the tributary story height times the width of the windward face of the building (see )LJXUH  ,QOLHXRIXVLQJWKH¿JXUHVLQ$6&(6(,&KDSWHUWKHEDVLFZLQGVSHHG

FDQEHREWDLQHGIURP5HIHUHQFHRU

3.7 Determination of Seismic Forces 3.7.1 Seismic Forces on the SFRS

7KHÀRZFKDUWLQ)LJXUHFDQEHXVHGWRGHWHUPLQHVHLVPLFIRUFHVRYHUWKHKHLJKWRIDUHLQIRUFHGFRQFUHWH6)56 LQDFFRUGDQFHZLWKWKH(TXLYDOHQW/DWHUDO)RUFH(/) 3URFHGXUHRI$6&(6(,LWLVDSSOLFDEOHWRVLWHVZKHUHD VLWHUHVSRQVHDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(,QHHGQRWEHSHUIRUPHGDQGWRVWUXFWXUHVZLWKRXWVHLVPLF LVRODWLRQRUGDPSLQJV\VWHPV7KHVHFWLRQ¿JXUHWDEOHDQGHTXDWLRQQXPEHUVLQ)LJXUHDUHIURP$6&(6(, 7KHW\SHRI6)56SHUPLWWHGWREHXVHGLVGHVLJQDWHGE\WKHJHQHUDOEXLOGLQJFRGHRUE\WKHEXLOGLQJRI¿FLDOLQDUHDV ZLWKRXWDOHJDOO\DGRSWHGEXLOGLQJFRGH$&, 

3-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine the Risk Category of the building in accordance with Table 1.5-1

Determine the basic wind speed, V, for the applicable Risk Category from Figs. 26.5-1A through 26.5-1D or from Figs. 26.5-2A through 26.5-2D

Determine the wind directionality factor, Kd, from Table 26.6-1 ሺ26.6ሻ

Determine the exposure category ሺ26.7ሻ

EŽ

Are all 5 conditions of 26.8.1 met?

zĞƐ

Determine ground elevation factor, Ke, from Table 26.9-1 ሺ26.9ሻ

Determine velocity pressure exposure coefficients, Kz and Kh, from Table 26.10-1 ሺ26.10.1ሻ

Determine pressure coefficients, Cp, for the windward and leeward walls from Fig. 27.3-1

A

Figure 3.4 Determination of total wind pressure on the MWFRS of a building in accordance with Part 1 in Chapter 27 of ASCE/SEI 7.

3-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

Determine whether the building is rigid or flexible in accordance with 26.11.2

EŽ

Determine Gf from 26.11.5

Is the building rigid?

zĞƐ

G ൌ 0.85

Determine G from 26.11.4

Figure 3.4 (cont.) Determination of total wind pressure on the MWFRS of a building in accordance with Part 1 in Chapter 27 of ASCE/SEI 7.

3-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



‫ܮ‬

‫݌‬௛ 

݄

݄௭ାଵ 

‫ܤ‬

‡˜‡Ž‫ݖ‬

‫ݖ‬

݄௭ିଵ 

‫݌‬௭ 

‘–ƒŽ™‹†ˆ‘”…‡ƒ–Ž‡˜‡Ž‫ ݖ‬ൌ ሺ‫݌‬௭ ൅ ‫݌‬௛ ሻ ൬

݄௭ାଵ ൅ ݄௭ିଵ ൰ ‫ܤ‬ ʹ

Figure 3.5 Determination of the total wind force at a level of the MWFRS.

3.7.2 Seismic Forces on Diaphragms, Chords, and Collectors

Floor and roof diaphragms must be designed to resist the effects due to out-of-plane gravity loads and in-plane latHUDOIRUFHVXVLQJWKHORDGFRPELQDWLRQVLQ$&,$&,  )RUEXLOGLQJVDVVLJQHGWR6'&%WKURXJK)GLDSKUDJPVDUHWREHGHVLJQHGIRUWKHODUJHURIWKHIRUFHVLQ DQG  EHORZ$6&(6(,  determined from the structural analysis (see Figure 3.6). 1. Design seismic force, GHWHUPLQHGE\$6&(6(,(TXDWLRQ   Diaphragm design force,

where

weight tributary to the diaphragm at level portion of the seismic base shear,

induced at level

(see Figure 3.6)

weight tributary to level Minimum and maximum values of DUHGHWHUPLQHGE\$6&(6(,(TXDWLRQV DQG UHVSHFtively, which are independent of the design base shear, Minimum Maximum

3-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

zĞƐ

EŽ

Determine the Site Class in accordance with 11.4.3 and Chapter 20

Structure is permitted to be assigned to SDC A and must only comply with 11.7

Determine the SDC from Figure 3.1 of this publication

Determine the response modification coefficient, R, from Table 12.2-1 for the appropriate structural system based on the SDC

Determine the importance factor, Ie, from Table 1.5-2 based on the Risk Category

A

Figure 3.6 Determination of seismic forces on a SFRS in accordance with the ELF Procedure in ASCE/SEI 12.8.

3-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

Determine approximate fundamental period of the building, Ta, in accordance 12.8.2.1 and set T ൌ Ta

Determine fundamental period of the building, T, in accordance 12.8.2

Determine TL from Figs. 22-14 through 22-17

zĞƐ

EŽ

B

EŽ

EŽ

zĞƐ

zĞƐ

EŽ

zĞƐ

C

* Value of CS is permitted to be calculated using a value of 1.0 for SDS, but not less than 0.7SDS, provided all criteria of 12.8.1.3 are met

Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance with the ELF Procedure in ASCE/SEI 12.8.

3-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

B

EŽ

EŽ

zĞƐ

zĞƐ

EŽ

C

Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance with the ELF Procedure in ASCE/SEI 12.8.

3-14

zĞƐ

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

C

Determine the effective seismic weight, W, in accordance with 12.7.2

Determine base shear, V, by Eq. ሺ12.8-1ሻ: V ൌ CsW

EŽ

EŽ

zĞƐ

zĞƐ

Exponent related to structure period k ൌ 2 ሺ12.8.3ሻ

Exponent related to structure period k ൌ 2 ሺ12.8.3ሻ

Exponent related to structure period k ൌ 1 ሺ12.8.3ሻ

Exponent related to structure period k ൌ 0.75 ൅ 0.5T ሺ12.8.3ሻ

Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance with the ELF Procedure in ASCE/SEI 12.8.

3-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In addition to

GLDSKUDJPVPXVWEHGHVLJQHGIRUDSSOLFDEOHWUDQVIHUIRUFHVVHH$6&(6(, 

Collectors must be provided where required to transmit forces between diaphragms and the vertical elements of the /)56$&, )RUEXLOGLQJVDVVLJQHGWR6'&&WKURXJK)FROOHFWRUVDQGWKHLUFRQQHFWLRQVWRWKHYHUWLFDO HOHPHQWVRIWKH6)56PXVWEHGHVLJQHGWRUHVLVWWKHHIIHFWVIURPWKHPD[LPXPRIWKHIROORZLQJIRUFHV$6&(6(,   1. )RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKVHLVPLF IRUFHVGHWHUPLQHGE\WKH(/)3URFHGXUHRI$6&(6(,RUWKHPRGDOUHVSRQVHVSHFWUXPDQDO\VLVSURFHGXUHRI$6&(6(, where ,Q$&,(TXDWLRQVH DQGJ RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH is determined using  )RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKVHLVPLF IRUFHVGHWHUPLQHGE\$6&(6(,(TXDWLRQ IRUGLDSKUDJPV where ,Q$&,(TXDWLRQVH DQGJ RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH is determined using 3. )RUFHVFDOFXODWHGXVLQJWKHORDGFRPELQDWLRQVRI$6&(6(,ZLWKVHLVPLFIRUFHVGHWHUPLQHGE\$6&( 6(,(TXDWLRQ ZKLFKLVWKHORZHUOLPLWGLDSKUDJPIRUFH where is ,Q$&,(TXDWLRQVH DQGJ RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH determined using Diaphragms and collectors in buildings assigned to SDC D, E, and F are designed in accordance with ACI Chapter $&, 

3.8 Examples 3.8.1 Example 3.1 – Determination of Wind Forces: Building #1

Determine the wind forces on the MWFRS in the north-south and east-west directions (see Figure 1.1). Assume the RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 2 – Determine the basic wind speed

For the given location of this Risk Category II building, Step 3 – Determine the wind directionality factor

ASCE/SEI Figure 26.5-1B

ZKLFKFDQEHREWDLQHGIURP5HIRU  ASCE/SEI Table 26.6-1

For the MWFRS of a building, Step 4 – Determine the exposure category

ASCE/SEI 26.7

The exposure category is given as C in the design data. Step 5 – Determine the topographic factor

The building is not located on a hill or escarpment, so

3-16

ASCE/SEI 26.8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 6 – Determine the ground elevation factor

The ground elevation factor,

ASCE/SEI Table 26.9-1

LVSHUPLWWHGWREHLQDOOFDVHVVHH1RWHLQ$6&(6(,7DEOH 

Step 7 – Determine the velocity pressure exposure coefficients

ASCE/SEI Table 26.10-1

7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH LQ$6&(6(,7DEOH

For Exposure C, Values of

and

and

IURP$6&(6(,7DEOH

DUHJLYHQLQ7DEOH

Table 3.7 Velocity Pressure Exposure Coefficients, Kz, Building #1 Height Above Ground Level, z (ft)

Kz





















Step 8 – Determine the velocity pressures

Values of

ASCE/SEI Eq. (26.10-1)

are given in Table 3.8.

Table 3.8 Velocity Pressures, qz, Building #1 Height Above Ground Level, z (ft)

Kz

qz (lb/ft2)































Step 9 – Determine the external pressure coefficients

ASCE/SEI Figure 27.3-1

For wind in the north-south direction: • Windward wall: • Leeward wall:

3-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For wind in the east-west direction: • Windward wall: • Leeward wall: Step 10 – Determine whether the building is rigid or not

ASCE/SEI 26.11.2

&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(, 1. Building height  Building height

(building length in each direction is constant

over the height, which means

is equal to the building length parallel to the wind direction)

7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\EHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\ )RUFRQFUHWHPRPHQWUHVLVWLQJIUDPHEXLOGLQJV>$6&(6(,(T @

Therefore, the building is rigid in both directions. Step 11 – Determine the gust-effect factor

For rigid buildings,

LVSHUPLWWHGWREHWDNHQDV

Step 12 – Determine the design wind pressures

For wind in the north-south direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS For wind in the east-west direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS Total design wind pressures in the north-south and east-west directions are given in Table 3.9.

3-18

ASCE/SEI 26.11.1

ASCE/SEI Eq. (27.3-1)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.9 Total Design Wind Pressures, p (lb/ft2), Building #1 Height Above Ground Level, (ft)

North-South

East-West































7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK LVHTXDOWROEIW for walls. Step 13 – Determine the total design wind forces

Total design wind forces are obtained at each level by multiplying the total design wind pressures in Table 3.9 by WKHWULEXWDU\DUHDIRUHDFKOHYHOVHH7DEOH  Table 3.10 Total Design Wind Forces, Building 1 North-South

East-West

Height Above Ground Level, z (ft)

Tributary Area (sq ft)

Wind Force (kips)

Tributary Area (sq ft)

Wind Force (kips)











































1,933.8







Ȉ





3.8.2 Example 3.2 – Determination of Wind Forces: Building #2

'HWHUPLQHWKHZLQGIRUFHVRQWKH0:)56LQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVVHH)LJXUH $VVXPHWKH RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For an essential facility, the Risk Category is IV. Step 2 – Determine the basic wind speed

For the given location of this Risk Category IV building, Step 3 – Determine the wind directionality factor

ASCE/SEI Figure 26.5-1D

ZKLFKFDQEHREWDLQHGIURP5HIRU  ASCE/SEI Table 26.6-1

For the MWFRS of a building, Step 4 – Determine the exposure category

ASCE/SEI 26.7

The exposure category is given as B in the design data. 3-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the topographic factor

ASCE/SEI 26.8

The building is not located on a hill or escarpment, so ASCE/SEI Table 26.9-1

Step 6 – Determine the ground elevation factor

The ground elevation factor,

LVSHUPLWWHGWREHLQDOOFDVHVVHH1RWHLQ$6&(6(,7DEOH 

Step 7 – Determine the velocity pressure exposure coefficients

ASCE/SEI Table 26.10-1

7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH LQ$6&(6(,7DEOH

For Exposure B, Values of

and

and

IURP$6&(6(,7DEOH

are given in Table 3.11.

Table 3.11 Velocity Pressure Exposure Coefficients, Kz, Building #2 Height Above Ground Level, z (ft)

Kz

















































Step 8 – Determine the velocity pressures

Values of

3-20

DUHJLYHQLQ7DEOH

ASCE/SEI Eq. (26.10-1)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.12 Velocity Pressures, qz, Building #2 Height Above Ground Level, z (ft)

Kz

qz (lb/ft2)























36.6























31.1

























Step 9 – Determine the external pressure coefficients

ASCE/SEI Figure 27.3-1

For wind in the north-south direction: Windward wall: Leeward wall: For wind in the east-west direction: Windward wall: Leeward wall: Step 10 – Determine whether the building is rigid or not

ASCE/SEI 26.11.2

&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(, 1. Building height  Building height over the height, which means

(building length in each direction is constant is equal to the building length parallel to the wind direction)

7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\EHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\ )RUDFRQFUHWHEXLOGLQJZLWKD0:)56RWKHUWKDQPRPHQWUHVLVWLQJIUDPHV>$6&(6(,(T @

7KHUHIRUHWKHEXLOGLQJLVÀH[LEOHLQERWKGLUHFWLRQV ,WFDQEHGHWHUPLQHGWKHEXLOGLQJLVDOVRÀH[LEOHXVLQJ$6&(6(,(T ZKLFKLVDSSOLFDEOHIRUFRQFUHWH shear wall buildings.

3-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUFRPSDULVRQSXUSRVHVWKHIXQGDPHQWDOIUHTXHQFLHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHHTXDOWR +]DQG+]UHVSHFWLYHO\IURPDG\QDPLFDQDO\VLVRIWKHVWUXFWXUHZKLFKVLJQL¿HVWKHEXLOGLQJLVÀH[LEOHLQWKH is used in subsenorth-south direction and rigid in the east-west direction. The approximate natural frequency, quent calculations. Step 11 – Determine the gust-effect factor

)RUÀH[LEOHEXLOGLQJV Calculations for

ASCE/SEI 26.11.5

LVGHWHUPLQHGE\$6&(6(,(T 

in both the north-south and east-west directions are given in Table 3.13.

Table 3.13 Determination of Gust-Effect Factor, Gf , Building #2 Item

ASCE/SEI Reference

 (T 7DEOH  7DEOH 7DEOH 7DEOH (T (T

(T

Damping ratio

for concrete buildings

& 7DEOH 7DEOH (T (T (T  (table continued on next page)

3-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.13 Determination of Gust-Effect Factor, Gf , Building #2 (cont.) Item

ASCE/SEI Reference

(TD



(TD



(TD

(T

(T

Step 12 – Determine the design wind pressures

ASCE/SEI Eq. (27.3-1)

For wind in the north-south direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS For wind in the east-west direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS 7RWDOGHVLJQZLQGSUHVVXUHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH

3-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.14 Total Design Wind Pressures, p (lb/ft2), Building #2 Height Above Ground Level, z (ft)

North-South

East-West









































































7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK LVHTXDOWROEIW for walls. Step 13 – Determine the total design wind forces

7RWDOGHVLJQZLQGIRUFHVDUHREWDLQHGDWHDFKOHYHOE\PXOWLSO\LQJWKHWRWDOGHVLJQZLQGSUHVVXUHVLQ7DEOHE\ WKHWULEXWDU\DUHDIRUHDFKOHYHOVHH7DEOH  Table 3.15 Total Design Wind Forces, Building 2 North-South

East-West

Height Above Ground Level, z (ft)

Tributary Area (sq ft)

Wind Force (kips)

Tributary Area (sq ft)

Wind Force (kips)



















91.1



















88.3









86.3





































































88.9

Ȉ

3-24



963.8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3.8.3 Example 3.3 – Determination of Wind Forces: Building #3

Determine the wind forces on the MWFRS in the north-south and east-west directions (see Figure 1.3). Assume the RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Residential Group R occupancy, the Risk Category is II. Step 2 – Determine the basic wind speed

For the given location of this Risk Category II building,

ASCE/SEI Figure 26.5-1B

ZKLFKFDQEHREWDLQHGIURP5HIRU 

Step 3 – Determine the wind directionality factor

ASCE/SEI Table 26.6-1

For the MWFRS of a building, Step 4 – Determine the exposure category

ASCE/SEI 26.7

The exposure category is given as B in the design data. Step 5 – Determine the topographic factor

ASCE/SEI 26.8

The building is not located on a hill or escarpment, so Step 6 – Determine the ground elevation factor

The ground elevation factor,

ASCE/SEI Table 26.9-1

LVSHUPLWWHGWREHLQDOOFDVHVVHH1RWHLQ$6&(6(,7DEOH 

Step 7 – Determine the velocity pressure exposure coefficients

ASCE/SEI Table 26.10-1

7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH LQ$6&(6(,7DEOH

For Exposure B, Values of

and

and

IURP$6&(6(,7DEOH

are given in Table 3.16.

Table 3.16 Velocity Pressure Exposure Coefficients, Kz, Building #3 Height Above Ground Level, z (ft)

Kz



1.11











 (table continued on next page)

3-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.16 Velocity Pressure Exposure Coefficients, Kz, Building #3 (cont.) Height Above Ground Level, z (ft)

Kz

















































Step 8 – Determine the velocity pressures

Values of

ASCE/SEI Eq. (26.10-1)

DUHJLYHQLQ7DEOH

Table 3.17 Velocity Pressures, qz, Building #3 Height Above Ground Level, z (ft)

Kz

qz (lb/ft2)



1.11











































19.9















































11.9

3-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 9 – Determine the external pressure coefficients

ASCE/SEI Figure 27.3-1

For wind in the north-south direction: Windward wall: Leeward wall: For wind in the east-west direction: Windward wall: Leeward wall: Step 10 – Determine whether the building is rigid or not

ASCE/SEI 26.11.2

&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(, 1. Building height

 Building height

(building length in each direction is constant

over the height, which means

is equal to the building length parallel to the wind direction)

7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\EHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\ )RUDFRQFUHWHEXLOGLQJZLWKD0:)56RWKHUWKDQPRPHQWUHVLVWLQJIUDPHV>$6&(6(,(T @

7KHUHIRUHWKHEXLOGLQJLVÀH[LEOHLQERWKGLUHFWLRQV ,WFDQEHGHWHUPLQHGWKHEXLOGLQJLVDOVRÀH[LEOHXVLQJ$6&(6(,(T ZKLFKLVDSSOLFDEOHIRUFRQFUHWH shear wall buildings. )RUFRPSDULVRQSXUSRVHVWKHIXQGDPHQWDOIUHTXHQFLHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHHTXDOWR +]DQG+]UHVSHFWLYHO\IURPDG\QDPLFDQDO\VLVRIWKHVWUXFWXUHZKLFKDOVRVLJQL¿HVWKHEXLOGLQJLVÀH[LEOH is used in subsequent calculations. in both directions. The approximate natural frequency, Step 11 – Determine the gust-effect factor

)RUÀH[LEOHEXLOGLQJV Calculations for

ASCE/SEI 26.11.5

LVGHWHUPLQHGE\$6&(6(,(T 

in both the north-south and east-west directions are given in Table 3.18.

Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3 Item

ASCE/SEI Reference

 (T 7DEOH (table continued on next page)

3-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3 (cont.) Item

ASCE/SEI Reference

 7DEOH 7DEOH 7DEOH (T (T

(T

Damping ratio

for concrete buildings

& 7DEOH 7DEOH (T (T (T  (TD



(TD

 (table continued on next page)

3-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3 (cont.) Item

ASCE/SEI Reference

(TD

(T

(T

Step 12 – Determine the design wind pressures, pz

ASCE/SEI Eq. (27.3-1)

For wind in the north-south direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS For wind in the east-west direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS Total design wind pressures in the north-south and east-west directions are given in Table 3.19. Table 3.19 Total Design Wind Pressures, p (lb/ft2), Building #3 Height Above Ground Level, z (ft)

North-South

East-West



























































 (table continued on next page)

3-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.19 Total Design Wind Pressures, p (lb/ft2), Building #3 (cont.) Height Above Ground Level, z (ft)

North-South

East-West

































19.6



7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK LVHTXDOWROEIW for walls. Step 13 – Determine the total design wind forces

Total design wind forces are obtained at each level by multiplying the total design wind pressures in Table 3.19 by WKHWULEXWDU\DUHDIRUHDFKOHYHOVHH7DEOH  Table 3.20 Total Design Wind Forces, Building #3 North-South

East-West

Height Above Ground Level, z (ft)

Tributary Area (sq ft)

Wind Force (kips)

Tributary Area (sq ft)

Wind Force (kips)















































































13.9























































18.8

























Ȉ

3-30





Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3.8.4 Example 3.4 – Determination of Wind Forces: Building #4

'HWHUPLQHWKHZLQGIRUFHVRQWKH0:)56LQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVVHH)LJXUH $VVXPHWKH RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 2 – Determine the basic wind speed

For the given location of this Risk Category II building,

ASCE/SEI Figure 26.5-1B

ZKLFKFDQEHREWDLQHGIURP5HIRU 

Step 3 – Determine the wind directionality factor

ASCE/SEI Table 26.6-1

For the MWFRS of a building, Step 4 – Determine the exposure category

ASCE/SEI 26.7

The exposure category is given as B in the design data. Step 5 – Determine the topographic factor

ASCE/SEI 26.8

The building is not located on a hill or escarpment, so Step 6 – Determine the ground elevation factor

The ground elevation factor,

ASCE/SEI Table 26.9-1

LVSHUPLWWHGWREHLQDOOFDVHVVHH1RWHLQ$6&(6(,7DEOH 

Step 7 – Determine the velocity pressure exposure coefficients

ASCE/SEI Table 26.10-1

7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH LQ$6&(6(,7DEOH

For Exposure B, Values of

and

and

IURP$6&(6(,7DEOH

DUHJLYHQLQ7DEOH

Table 3.21 Velocity Pressure Exposure Coefficients, Kz, Building #4 Height A Height Above Ground Level, z (ft)und

Kz



1.39



1.38



1.36 (table continued on next page)

3-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.21 Velocity Pressure Exposure Coefficients, Kz, Building #4 (cont.) Height A Height Above Ground Level, z (ft)und

Kz







1.33



































1.18



1.16





























































Step 8 – Determine the velocity pressures

Values of

3-32

DUHJLYHQLQ7DEOH

ASCE/SEI Eq. (26.10-1)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.22 Velocity Pressures, qz, Building #4 Height Above Ground Level, z (ft)

Kz

qz (lb/ft2)



1.39





1.38





1.36











1.33





















































1.18





1.16

















































19.9











18.3











16.3



















Step 9 – Determine the external pressure coefficients

ASCE/SEI Figure 27.3-1

For wind in the north-south direction: Windward wall: Leeward wall:

3-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For wind in the east-west direction: Windward wall: Leeward wall: Step 10 – Determine whether the building is rigid or not

ASCE/SEI 26.11.2

&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(, Building height 7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\QRWEHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\ )URPDG\QDPLFDQDO\VLVRIWKHVWUXFWXUHWKHEXLOGLQJLVIRXQGWREHÀH[LEOHLQERWKGLUHFWLRQV in the east-west direction). north-south direction and Step 11 – Determine the gust-effect factor

)RUÀH[LEOHEXLOGLQJV Calculations for

in the

ASCE/SEI 26.11.5

LVGHWHUPLQHGE\$6&(6(,(T 

LQERWKWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH

Table 3.23 Determination of Gust-Effect Factor, Gf , Building #4 Item

ASCE/SEI Reference 26.11.5

Eq. (26.11-11)

Table 26.11-1

26.11.4

Table 26.11-1 Table 26.11-1 Table 26.11-1 Eq. (26.11-7) Eq. (26.11-9)

Eq. (26.11-8)

Damping ratio

for concrete buildings

C26.11 Table 26.11-1 (table continued on next page)

3-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.23 Determination of Gust-Effect Factor, Gf , Building #4 (cont.) Item

ASCE/SEI Reference Table 26.11-1 Eq. (26.11-16)

Eq. (26.11-14)

Eq. (26.11-13)

26.11.5

Eq. (26.11-15a)

26.11.5

Eq. (26.11-15a)

26.11.5

Eq. (26.11-15a)

Eq. (26.11-12)

Eq. (26.11-10)

3-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 12 – Determine the design wind pressures, pz

ASCE/SEI Eq. (27.3-1)

For wind in the north-south direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS For wind in the east-west direction: • Windward wall: • Leeward wall: • Total pressure on MWFRS 7RWDOGHVLJQZLQGSUHVVXUHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH Table 3.24 Total Design Wind Pressures, p (lb/ft2), Building #4 Height Above Ground Level, z (ft)

North-South

East-West

330.0

33.6

33.2

319.0

33.4

33.0

308.0

33.2

32.8

297.0

33.0

32.5

286.0

32.7

32.3

275.0

32.5

32.1

264.0

32.3

31.9

253.0

32.0

31.6

242.0

31.8

31.4

231.0

31.5

31.1

220.0

31.2

30.9

209.0

30.9

30.7

198.0

30.6

30.4

187.0

30.3

30.1

176.0

30.0

29.8

165.0

29.7

29.5

154.0

29.4

29.2

143.0

29.0

28.8

132.0

28.6

28.5

121.0

28.2

28.1

110.0

27.7

27.6

99.0

27.3

27.2 (table continued on next page)

3-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.24 Total Design Wind Pressures, p (lb/ft2), Building #4 (cont.) Height Above Ground Level, z (ft)

North-South

East-West

88.0

26.7

26.7

77.0

26.2

26.2

66.0

25.5

25.6

55.0

24.9

25.0

44.0

24.0

24.2

33.0

23.1

23.4

22.0

21.9

22.2

11.0

20.8

21.2

7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK LVHTXDOWROEIW for walls. Step 13 – Determine the total design wind forces

7RWDOGHVLJQZLQGIRUFHVDUHREWDLQHGDWHDFKOHYHOE\PXOWLSO\LQJWKHWRWDOGHVLJQZLQGSUHVVXUHVLQ7DEOHE\ WKHWULEXWDU\DUHDIRUHDFKOHYHOVHH7DEOH  Table 3.25 Total Design Wind Forces, Building #4 North-South

East-West

Height Above Ground Level, z (ft)

Tributary Area (sq ft)

Wind Force (kips)

Tributary Area (sq ft)

Wind Force (kips)











































































39.8





























38.8

















































36.9













 (table continued on next page)

3-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.25 Total Design Wind Forces, Building #4 (cont.) North-South

East-West

Height Above Ground Level, z (ft)

Tributary Area (sq ft)

Wind Force (kips)

Tributary Area (sq ft)

Wind Force (kips)





36.1







































33.9





















































































Ȉ





3.8.5 Example 3.5 – Determination of the Seismic Design Category: Building #1

Determine the SDC for (a) Site Class C, (b) Site Class D (default), and (c) Site Class E (see Figure 1.1). 'HVLJQGDWDDUHJLYHQLQ6HFW Part (a): Site Class C Step 1 – Determine the mapped acceleration parameters

ASCE/SEI 11.4.2

In lieu of determining the mapped acceleration parameters and IURP$6&(6(,)LJXUHVDQGUHand for the latitude and longitude of the site. VSHFWLYHO\XVH5HIRU Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because SDC A.

and

ASCE/SEI 11.4.2

, the building is not permitted to be automatically assigned to

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 4 – Determine if the SDC is E or F

Because the Risk Category is II and

ASCE/SEI 11.6

the SDC is not E or F.

Step 5 – Determine the design acceleration parameters

For Site Class C and

3-38

IURP$6&(6(,7DEOH

ASCE/SEI 11.4.4 and 11.4.5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

IURP$6&(6(,7DEOH

For Site Class C and

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

For concrete moment-resisting frames in both directions: ASCE/SEI Eq. (12.8-7)

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is A.

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is A.

Therefore, the SDC is A for this building. Part (b): Site Class D (default) Step 1 – Determine the mapped acceleration parameters

From Step 1 in Part (a):

ASCE/SEI 11.4.2

and

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because SDC A.

and

ASCE/SEI 11.4.2

the building is not permitted to be automatically assigned to

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 4 – Determine if the SDC is E or F

Because the Risk Category is II and

ASCE/SEI 11.6

the SDC is not E or F.

Step 5 – Determine the design acceleration parameters

For Site Class D (default) and

ASCE/SEI 11.4.4 and 11.4.5

IURP$6&(6(,7DEOH

3-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

IURP$6&(6(,7DEOH

For Site Class D (default) and

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

From Step 6 in Part (a),

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is A.

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is B.

Therefore, the SDC is B for this building. Part (c): Site Class E Step 1 – Determine the mapped acceleration parameters

From Step 1 in Part (a):

ASCE/SEI 11.4.2

and

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because SDC A.

and

ASCE/SEI 11.4.2

the building is not permitted to be automatically assigned to

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 4 – Determine if the SDC is E or F

Because the Risk Category is II and

ASCE/SEI 11.6

the SDC is not E or F.

Step 5 – Determine the design acceleration parameters

For Site Class E and

IURP$6&(6(,7DEOH

For Site Class E and

IURP$6&(6(,7DEOH

3-40

ASCE/SEI 11.4.4 and 11.4.5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

From Step 6 in Part (a),

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is B.

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is C.

Therefore, the SDC is C for this building. 3.8.6 Example 3.6 – Determination of the Seismic Design Category: Building #2

'HWHUPLQHWKH6'&IRU6LWH&ODVV'VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

ASCE/SEI 11.4.2

In lieu of determining the mapped acceleration parameters and IURP$6&(6(,)LJXUHVDQGUHand for the latitude and longitude of the site. VSHFWLYHO\XVH5HIRU Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because SDC A.

and

ASCE/SEI 11.4.2

the building is not permitted to be automatically assigned to

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For this essential facility, the Risk Category is IV. Step 4 – Determine if the SDC is E or F

Because the Risk Category is IV and

ASCE/SEI 11.6

the SDC is not E or F.

Step 5 – Determine the design acceleration parameters

For Site Class D (stiff soil) and  For Site Class D (stiff soil) and

ASCE/SEI 11.4.4 and 11.4.5

IURPOLQHDULQWHUSRODWLRQLQ$6&(6(,7DEOH IURP$6&(6(,7DEOH 3-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

For all other structural systems in both directions: ASCE/SEI Eq. (12.8-7)

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category IV, the SDC is C.

)URP$6&(6(,7DEOHZLWK

and Risk Category IV, the SDC is C.

Therefore, the SDC is C for this building. 3.8.7 Example 3.7 – Determination of the Seismic Design Category: Building #3

'HWHUPLQHWKH6'&IRU6LWH&ODVV'VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

ASCE/SEI 11.4.2

In lieu of determining the mapped acceleration parameters and IURP$6&(6(,)LJXUHVDQGUHand for the latitude and longitude of the site. VSHFWLYHO\XVH5HIRU Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because to SDC A.

and

ASCE/SEI 11.4.2

the building is not permitted to be automatically be assigned

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Residential Group R occupancy, the Risk Category is II. Step 4 – Determine if the SDC is E or F

Because the Risk Category is II and

3-42

ASCE/SEI 11.6

the SDC is not E or F.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the design acceleration parameters

ASCE/SEI 11.4.4, 11.4.5, and 11.4.8

Because the Site Class is D and DJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(, PD\EHUHTXLUHG$VVXPHWKHVHFRQGH[FHSWLRQLQ$6&(6(,LVDSSOLFDEOHWKXVVXFKDQDQDO\VLVQHHG not be performed. For Site Class D (stiff soil) and

IURP$6&(6(,7DEOH

For Site Class D (stiff soil) and 

IURPOLQHDULQWHUSRODWLRQLQ$6&(6(,7DEOH

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

For all other structural systems in both directions: ASCE/SEI Eq. (12.8-7)

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is D.

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is D.

Therefore, the SDC is D for this building. 3.8.8 Example 3.8 – Determination of the Seismic Design Category: Building #4

'HWHUPLQHWKH6'&IRU6LWH&ODVV'GHIDXOW >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

ASCE/SEI 11.4.2

In lieu of determining the mapped acceleration parameters and IURP$6&(6(,)LJXUHVDQGUHand for the latitude and longitude of the site. VSHFWLYHO\XVH5HIRU Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

Because to SDC A.

and

ASCE/SEI 11.4.2

the building is not permitted to be automatically be assigned

3-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the Risk Category

ASCE/SEI Table 1.5-1

For the Business Group B occupancy, the Risk Category is II. Step 4 – Determine if the SDC is E or F

ASCE/SEI 11.6

Because the Risk Category is II and

the SDC is not E or F.

Step 5 – Determine the design acceleration parameters

Because the Site Class is D and QHHGQRWEHSHUIRUPHG

ASCE/SEI 11.4.4, 11.4.5, and 11.4.8

DJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(,

For Site Class D (default) and

IURP7DEOH$6&(6(,E\OLQHDULQWHUSRODWLRQ

For Site Class D (default) and

IURP$6&(6(,7DEOHE\OLQHDULQWHUSRODWLRQ

Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone

ASCE/SEI 11.6

&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG Determine the approximate fundamental period,

$6&(6(, 

For all other structural systems in both directions: ASCE/SEI Eq. (12.8-7)

Because VDWLV¿HG 

WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQHWKDWLVFRQGLWLRQLVQRW

Step 7 – Determine the SDC

ASCE/SEI Tables 11.6-1 and 11.6-2

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is D.

)URP$6&(6(,7DEOHZLWK

and Risk Category II, the SDC is D.

Therefore, the SDC is D for this building. 3.8.9 Example 3.9 – Determination of Seismic Forces: SFRS of Building #1 (Framing Option A)

'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV'GHIDXOW >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

)URP6WHSLQ3DUWE RI([DPSOH

3-44

and

ASCE/SEI 11.4.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

ASCE/SEI 11.4.2

)URP6WHSLQ3DUWE RI([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$ Step 3 – Determine the Site Class

ASCE/SEI 11.4.3

From the design data, the Site Class is given as D (default). Step 4 – Determine the acceleration parameters adjusted for site class effects

)URP6WHSRI([DPSOH

and

Step 5 – Determine the design acceleration parameters

)URP6WHSRI([DPSOH

ASCE/SEI 11.4.4

ASCE/SEI 11.4.5

and

Step 6 – Determine the SDC

ASCE/SEI 11.6

)URP6WHSRI([DPSOHWKH6'&LV% Step 7 – Determine the response modification coefficient

ASCE/SEI Table 12.2-1

Because the building is assigned to SDC B, ordinary reinforced concrete moment frames may be used in both direcWLRQVZLWKQROLPLWDWLRQV6)56&  For this SFRS, Step 8 – Determine the importance factor

ASCE/SEI Table 1.5-2

For Risk Category II buildings, Step 9 – Determine the approximate period

ASCE/SEI 12.8.2.1

)URP6WHSRI([DPSOH Step 10 – Determine the long-period transition period

)URP$6&(6(,)LJXUH

WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU 

Step 11 – Determine the seismic response coefficient

Because

ASCE/SEI Figures 22-14 through 22-17

ASCE/SEI 12.8.1.1

is determined as follows: ASCE/SEI Eq. (12.8-3)

need not exceed the following: ASCE/SEI Eq. (12.8-2)

3-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Minimum

ASCE/SEI Eq. (12.8-5)

Therefore, Step 12 – Determine the effective seismic weight

ASCE/SEI 12.7.2

7KHHIIHFWLYHVHLVPLFZHLJKWIRUWKLVEXLOGLQJLQFOXGHVWKHGHDGORDGRIWKHVWUXFWXUHDVVXPLQJDLQWKLFNVODE DQGLQE\LQFROXPQV WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHOEIW partition weight RQWKHÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ 7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKW

are given in Table 3.

Table 3.26 Seismic Forces and Story Shears, Building #1 Level

Story Weight, wx (kips)

Height, hx (ft)

wx hxk

Lateral Force, Fx (kips)

Story Shear, Vx (kips)

R



































3



























Ȉ

W 

Step 13 – Determine the seismic base shear

Step 14 – Determine the exponent related to the structure period

Step 15 – Determine the seismic forces at each level

Values of

DUHJLYHQLQ7DEOH

)RUH[DPSOHDWOHYHO

3-46

ASCE/SEI Eq. (12.8-1)

ASCE/SEI 12.8.3

ASCE/SEI Eqs. (12.8-11) and (12.8-12)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3.8.10 Example 3.10 – Determination of Seismic Forces: SFRS of Building #2

'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV'VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

From Step 1 in Example 3.6,

ASCE/SEI 11.4.2

and

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

ASCE/SEI 11.4.2

)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$ Step 3 – Determine the Site Class

ASCE/SEI 11.4.3

From the design data, the Site Class is given as D (stiff soil). Step 4 – Determine the acceleration parameters adjusted for site class effects

)URP6WHSLQ([DPSOH

ASCE/SEI 11.4.4

and

Step 5 – Determine the design acceleration parameters

)URP6WHSLQ([DPSOH

ASCE/SEI 11.4.5

and

Step 6 – Determine the SDC

ASCE/SEI 11.6

)URP6WHSLQ([DPSOHWKH6'&LV& Step 7 – Determine the response modification coefficient

ASCE/SEI Table 12.2-1

Because the building is assigned to SDC C, a building frame system with ordinary reinforced concrete shear (strucWXUDO ZDOOVPD\EHXVHGLQERWKGLUHFWLRQVZLWKQROLPLWDWLRQV6)56%  For this SFRS, Step 8 – Determine the importance factor

ASCE/SEI Table 1.5-2

For Risk Category IV buildings, Step 9 – Determine the approximate period

ASCE/SEI 12.8.2.1

From Step 6 in Example 3.6, Step 10 – Determine the long-period transition period

)URP$6&(6(,)LJXUH

ASCE/SEI Figures 22-14 through 22-17

WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU 

3-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 11 – Determine the seismic response coefficient

Because

ASCE/SEI 12.8.1.1

is determined as follows: ASCE/SEI Eq. (12.8-3)

need not exceed the following: ASCE/SEI Eq. (12.8-2)

Minimum ASCE/SEI Eq. (12.8-5)

Therefore, Step 12 – Determine the effective seismic weight

ASCE/SEI 12.7.2

The effective seismic weight for this building includes the dead load of the structure (assuming a ZLGHPRGXOHMRLVWV\VWHPLQE\LQLQWHULRUEHDPVLQE\LQHGJHEHDPVLQE\LQDQG LQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQDQGLQE\LQFROXPQVLQVWRULHV WKURXJKLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHVWKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHOEIWSDUWLWLRQZHLJKWRQWKHÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ 7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKW

are given in Table 3.

Table 3.27 Seismic Forces and Story Shears, Building #2 Level

Story Weight, wx (kips)

Height, hx (ft)

wx hxk

Lateral Force, Fx (kips)

Story Shear, Vx (kips)

R





















11

 





















9











8























6





99.8

1,369.8



 





















3



























Ȉ 3-48

W 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 13 – Determine the seismic base shear

ASCE/SEI Eq. (12.8-1)

Step 14 – Determine the exponent related to the structure period

Step 15 – Determine the seismic forces at each level

Values of

ASCE/SEI 12.8.3

ASCE/SEI Eqs. (12.8-11) and (12.8-12)

are given in Table 3.

)RUH[DPSOHDWOHYHO

3.8.11 Example 3.11 – Determination of Seismic Forces: SFRS of Building #3

'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV'VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

)URP6WHSLQ([DPSOH

ASCE/SEI 11.4.2

and

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

ASCE/SEI 11.4.2

)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$ Step 3 – Determine the Site Class

ASCE/SEI 11.4.3

From the design data, the Site Class is given as D (stiff soil). Step 4 – Determine the acceleration parameters adjusted for site class effects

)URP6WHSLQ([DPSOH

and

Step 5 – Determine the design acceleration parameters

)URP6WHSLQ([DPSOH

ASCE/SEI 11.4.4

ASCE/SEI 11.4.5

and

Step 6 – Determine the SDC

ASCE/SEI 11.6

)URP6WHSLQ([DPSOHWKH6'&LV'

3-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 7 – Determine the response modification coefficient

ASCE/SEI Table 12.2-1

7KHKHLJKWRIWKLVEXLOGLQJLVHTXDOWRIW%HFDXVHWKHEXLOGLQJLVDVVLJQHGWR6'&'DEXLOGLQJIUDPHV\VWHP ZLWKVSHFLDOUHLQIRUFHGFRQFUHWHVKHDUVWUXFWXUDO ZDOOVPD\EHXVHGLQERWKGLUHFWLRQVZLWKDKHLJKWOLPLWRI IW6)56%  For this SFRS, Step 8 – Determine the importance factor

ASCE/SEI Table 1.5-2

For Risk Category II buildings, Step 9 – Determine the approximate period

ASCE/SEI 12.8.2.1

)URP6WHSLQ([DPSOH From a dynamic analysis of the structure, the periods in the north-south and east-west directions are determined to EHVDQGVUHVSHFWLYHO\$FFRUGLQJWR$6&(6(,WKHIXQGDPHQWDOSHULRG must not exceed the IURP$6&(6(,7DEOHDQGWKHDSSUR[LSURGXFWRIWKHFRHI¿FLHQWIRUXSSHUOLPLWRQFDOFXODWHGSHULRG mate fundamental period, In the north-south direction: In the east-west direction: where Therefore,

IURP$6&(6(,7DEOHIRU is permitted to be used to determine the base shear,

Step 10 – Determine the long-period transition period

)URP$6&(6(,)LJXUH

ASCE/SEI Figures 22-14 through 22-17

WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU 

Step 11 – Determine the seismic response coefficient

Because

in both directions.

ASCE/SEI 12.8.1.1

is determined as follows: ASCE/SEI Eq. (12.8-3)

need not exceed the following: ASCE/SEI Eq. (12.8-2)

Minimum ASCE/SEI Eq. (12.8-5)

3-50

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,WZDVDVVXPHGLQ6WHSRI([DPSOHWKDWDJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(, ZDVQRWUHTXLUHGEHFDXVHWKHVHFRQGH[FHSWLRQLQ$6&(6(,LVDSSOLFDEOH7KHUHIRUHGHWHUPLQH based on that exception:

Thus,

PXVWEHWDNHQDVWLPHVWKHYDOXHFRPSXWHGLQDFFRUGDQFHZLWK$6&(6(,(T 

Step 12 – Determine the effective seismic weight

ASCE/SEI 12.7.2

7KHHIIHFWLYHVHLVPLFZHLJKWIRUWKLVEXLOGLQJLQFOXGHVWKHGHDGORDGRIWKHVWUXFWXUHDVVXPLQJDQLQWKLFNVODE LQE\LQDQGLQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQDQGLQE\ LQFROXPQVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHV WKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ 7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKWDUHJLYHQLQ7DEOH Table 3.28 Seismic Forces and Story Shears, Building #3 Level

Story Weight, wx (kips)

Height, hx (ft)

wx hxk

Lateral Force, Fx (kips)

Story Shear, Vx (kips)

R











16









398.3





















168.3



13























11







118.1















9











8























6



































3



























Ȉ

W = 13,319

Step 13 – Determine the seismic base shear

Step 14 – Determine the exponent related to the structure period

ASCE/SEI Eq. (12.8-1)

ASCE/SEI 12.8.3

3-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 15 – Determine the seismic forces at each level

Values of

ASCE/SEI Eqs. (12.8-11) and (12.8-12)

DUHJLYHQLQ7DEOH

)RUH[DPSOHDWOHYHO

3.8.12 Example 3.12 – Determination of Seismic Forces: SFRS of Building #4

'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV'GHIDXOW >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the mapped acceleration parameters

From Step 1 in Example 3.8,

ASCE/SEI 11.4.2

and

Step 2 – Determine if the building is permitted to be automatically assigned to SDC A

ASCE/SEI 11.4.2

)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$ Step 3 – Determine the Site Class

ASCE/SEI 11.4.3

From the design data, the Site Class is given as D (default). Step 4 – Determine the acceleration parameters adjusted for site class effects

)URP6WHSLQ([DPSOH

and

Step 5 – Determine the design acceleration parameters

)URP6WHSLQ([DPSOH

ASCE/SEI 11.4.4

ASCE/SEI 11.4.5

and

Step 6 – Determine the SDC

ASCE/SEI 11.6

)URP6WHSLQ([DPSOHWKH6'&LV' Step 7 – Determine the response modification coefficient

ASCE/SEI Table 12.2-1

Because the building is assigned to SDC D, a dual system with special reinforced concrete shear (structural) walls DQGVSHFLDOPRPHQWIUDPHVRIUHLQIRUFHGFRQFUHWHFDSDEOHRIUHVLVWLQJDWOHDVWSHUFHQWRIWKHSUHVFULEHGVHLVPLF forces may be used with no limits in both directions (SFRS D3). For this SFRS, A building frame system with special reinforced concrete shear (structural) walls is not permitted because the buildLQJKHLJKWZKLFKLVHTXDOWRIWH[FHHGVWKHKHLJKWOLPLWRIIWIRUWKLVV\VWHPLQ6'&'

3-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 8 – Determine the importance factor

ASCE/SEI Table 1.5-2

For Risk Category II buildings, Step 9 – Determine the approximate period

ASCE/SEI 12.8.2.1

From Step 6 in Example 3.8, From a dynamic analysis of the structure, the periods in the north-south and east-west directions are determined to EHVDQGVUHVSHFWLYHO\$FFRUGLQJWR$6&(6(,WKHIXQGDPHQWDOSHULRG must not exceed the IURP$6&(6(,7DEOHDQGWKHDSSUR[LSURGXFWRIWKHFRHI¿FLHQWIRUXSSHUOLPLWRQFDOFXODWHGSHULRG mate fundamental period, In the north-south direction: In the east-west direction: where Therefore, directions.

IURP$6&(6(,7DEOHE\OLQHDULQWHUSRODWLRQIRU is permitted to be used to determine the base shear,

in both the north-south and east-west

,QRUGHUWRXVHWKH(/)3URFHGXUHVWUXFWXUHVH[FHHGLQJIWLQKHLJKWZLWKQRVWUXFWXUDOLUUHJXODULWLHVPXVWVDWLVI\WKHIROORZLQJ$6&(6(,7DEOH 

Therefore, use

in both directions.

Step 10 – Determine the long-period transition period

)URP$6&(6(,)LJXUH

WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU 

Step 11 – Determine the seismic response coefficient

Because

ASCE/SEI Figures 22-14 through 22-17

ASCE/SEI 12.8.1.1

is determined as follows: ASCE/SEI Eq. (12.8-3)

need not exceed the following: ASCE/SEI Eq. (12.8-2)

Minimum ASCE/SEI Eq. (12.8-5)

Thus,

3-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 12 – Determine the effective seismic weight

ASCE/SEI 12.7.2

The effective seismic weight for this building includes the dead load of the structure (assuming wide-module roof LQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQ DQGÀRRUV\VWHPVZLWK FROXPQVLQVWRULHVWKURXJKLQE\LQFROXPQVLQVWRULHVWKURXJKEHDPVLQFOXGLQJFROOHFWRU EHDPV DUHDVZLGHDVWKHFROXPQVZLWKDGHSWKRILQFRXSOLQJEHDPVDUHZLGHDVWKHZDOOVZLWKDGHSWKRI LQLQWKLFNZDOOVLQVWRULHVWKURXJKLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFN ZDOOVLQVWRULHVWKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHZHLJKWRIWKHSDUWLWLRQVDQG the weight of the cladding. 7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKWDUHJLYHQLQ7DEOH Table 3.29 Seismic Forces and Story Shears, Building #4 Level

Story Weight, wx (kips)

Height, hx (ft)

wx hxk

Lateral Force, Fx (kips)

Story Shear, Vx (kips)

R



















196.1



































163.1























































































19







98.3



18























16



































13























11























9











8



















16.9



6









 (table continued on next page)

3-54

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.29 Seismic Forces and Story Shears, Building #4 (cont.) Level

Story Weight, wx (kips)

Height, hx (ft)

wx hxk

Lateral Force, Fx (kips)

Story Shear, Vx (kips)

























3



























Ȉ

W 

Step 13 – Determine the seismic base shear

ASCE/SEI Eq. (12.8-1)

Step 14 – Determine the exponent related to the structure period

Step 15 – Determine the seismic forces at each level

Values of

ASCE/SEI 12.8.3

ASCE/SEI Eqs. (12.8-11) and (12.8-12)

DUHJLYHQLQ7DEOH

For example, at the roof level:

3.8.13 Example 3.13 – Determination of Seismic Forces: Diaphragms of Building #1 (Framing Option A)

'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(, IRU6LWH&ODVV'GHIDXOW >VHH)LJXUH@ 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the seismic forces on the SFRS

ASCE/SEI 12.8.3

Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOHVHH7DEOH DQGDUH JLYHQLQ7DEOH Table 3.30 Seismic Diaphragm Design Forces, Building #1 Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)

R











Ȉwi

ȈFi

Design Force (kips)**

(kips)

wpx

(kips)

ȈFi  Ȉwi

(kips)

























Fpx

(table continued on next page)

3-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.30 Seismic Diaphragm Design Forces, Building #1 (cont.) Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)







3







Ȉwi

ȈFi

Design Force (kips)**

(kips)

wpx

(kips)

ȈFi  Ȉwi

(kips)









































Fpx

* Max. ** Max. of

Step 2 – Determine the seismic diaphragm forces

ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)

Diaphragm forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHJLYHQLQ7DEOH9DOXHVRIWKHVWRU\ZHLJKW and the seismic lateral force, DUHJLYHQLQ7DEOHRI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO is set equal to ,WLVHYLGHQWIURP7DEOHWKDWWKHPD[LPXP the three top levels of the building. For example, at the roof level:

governs at

Also, minimum Step 3 – Determine the seismic diaphragm design forces

Seismic diaphragm design forces are the larger of JLYHQLQ7DEOH

and

ASCE/SEI 12.10.1.1

The design forces over the height of the building are

3.8.14 Example 3.14 – Determination of Seismic Forces: Diaphragms of Building #2

'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(, VHH)LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the seismic forces on the SFRS

ASCE/SEI 12.8.3

Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOHVHH7DEOH DQG are given in Table 3.31.

3-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.31 Seismic Diaphragm Design Forces, Building #2 Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)

R









11

Ȉwi

ȈFi

Fpx

Design Force (kips)**

wpx

(kips)

(kips)

ȈFi Ȉwi

(kips)





























































9

















8



































6



99.8



1,369.8





331.8

331.8















331.8

331.8















331.8

331.8

3













331.8

331.8



















* Min. ** Max. of

Step 2 – Determine the seismic diaphragm forces

ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)

Diaphragm forces over the height of the building are given in Table 3.31. Values of the story weight, and the are given in Table 3.RI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO seismic lateral force, is set equal to It is evident from Table 3.31 that the minimum DOOOHYHOVRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO

governs at

Also, maximum Step 3 – Determine the seismic diaphragm design forces

Seismic diaphragm design forces are the larger of given in Table 3.31.

and

ASCE/SEI 12.10.1.1

The design forces over the height of the building are

3-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3.8.15 Example 3.15 – Determination of Seismic Forces: Diaphragms of Building #3

'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(, VHH)LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the seismic forces on the SFRS

ASCE/SEI 12.8.3

Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOHVHH7DEOH DQG DUHJLYHQLQ7DEOH Table 3.32 Seismic Diaphragm Design Forces, Building #3 Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)

R





16





Ȉwi

ȈFi

Fpx

Design Force (kips)**

(kips)

wpx

(kips)

ȈFi  Ȉwi

(kips)

















398.3





























168.3













13





3,936





























11



118.1









168.3

168.3



















9

















8



































6





















































3























13,319











* Min. ** Max. of

Step 2 – Determine the seismic diaphragm forces

3-58

ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Diaphragm forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHJLYHQLQ7DEOH9DOXHVRIWKHVWRU\ZHLJKW and the are given in Table 3.RI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO seismic lateral force, is set equal to ,WLVHYLGHQWIURP7DEOHWKDWWKHPLQLPXP OHYHOVWKURXJKRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO

governs at

Also, maximum Step 3 – Determine the seismic diaphragm design forces

Seismic diaphragm design forces are the larger of JLYHQLQ7DEOH

and

ASCE/SEI 12.10.1.1

The design forces over the height of the building are

3.8.16 Example 3.16 – Determination of Seismic Forces: Diaphragms of Building #4

'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(, VHH)LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the seismic forces on the SFRS

ASCE/SEI 12.8.3

Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOHLQWKHQRUWKVRXWKDQG HDVWZHVWGLUHFWLRQVVHH7DEOH DQGDUHJLYHQLQ7DEOH Table 3.33 Seismic Diaphragm Design Forces, Building #4 Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)

R











Ȉwi

ȈFi

Fpx

Design Force (kips)**

(kips)

wpx

(kips)

ȈFi  Ȉwi

(kips)













196.1



















































163.1





















































































































368.6

368.6

(table continued on next page)

3-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 3.33 Seismic Diaphragm Design Forces, Building #4 (cont.) Level

Story Weight, wi (kips)

Lateral Force, Fi (kips)

(kips)







19



18

Ȉwi

ȈFi

Fpx

Design Force (kips)**

wpx

(kips)

(kips)

ȈFi Ȉwi

(kips)













98.3















































16





















































13



































11



































9

















8





















16.9













6





















































3



































* Min. ** Max. of

Step 2 – Determine the seismic diaphragm forces

ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)

Diaphragm forces over the height of the building are given in Table 3.33. Values of the story weight, and the seismic lateral force, DUHJLYHQLQ7DEOHRI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO is set equal to It is evident from Table 3.33 that the minimum DOOOHYHOVRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO

3-60

governs at

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Also, maximum Step 3 – Determine the seismic diaphragm design forces

Seismic diaphragm design forces are the larger of given in Table 3.33.

and

ASCE/SEI 12.10.1.1

The design forces over the height of the building are

3-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 4 ONE-WAY SLABS 4.1 Overview One-way slabs are elements in one-way construction where the members are designed to support loads through EHQGLQJLQDVLQJOHGLUHFWLRQ:KHUHWKHUDWLRRIWKHORQJWRWKHVKRUWVLGHRIDVODESDQHOLVJUHDWHUWKDQORDG WUDQVIHULVSUHGRPLQDWHO\E\EHQGLQJLQWKHVKRUWGLUHFWLRQDQGWKHSDQHOLVGH¿QHGDVDRQHZD\VODEVHH)LJXUH  7KHPDLQÀH[XUDOUHLQIRUFHPHQWLQDRQHZD\VODEV\VWHPUXQVSDUDOOHOWRWKHGLUHFWLRQRIORDGWUDQVIHU 2

3

κଵ  E

Beam ሺtyp. ሻ

κଶ 

One-way slab κଶ Τκଵ ൐ 2 A

A

D

Section A-A

Figure 4.1 A one-way slab system.

The design and detailing of solid, cast-in-place one-way slabs with nonprestressed reinforcement are covered in this FKDSWHU3URYLVLRQVIRURQHZD\VODEVDUHJLYHQLQ$&,&KDSWHU

4.2 Minimum Slab Thickness 2QHZD\VODEVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRWKDWDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG,QOLHXRIFDOFXODWLQJGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,DQGVXEVHTXHQWO\FKHFNLQJWKDWWKHGHÀHFWLRQV GRQRWH[FHHGWKHOLPLWVLQ$&,$&, WKHPLQLPXPRYHUDOOVODEWKLFNQHVV for solid, nonprestressed VODEVQRWVXSSRUWLQJRUDWWDFKHGWRSDUWLWLRQVRURWKHUW\SHVRIFRQVWUXFWLRQOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV FDQEHGHWHUPLQHGXVLQJWKHOLPLWVLQ$&,7DEOHWKHVHOLPLWVDUHDSSOLFDEOHWRRQHZD\VODEVZLWKUHLQIRUFHPHQWWKDWKDVDVSHFL¿HG\LHOGVWUHQJWK HTXDOWRSVLDQGQRUPDOZHLJKWFRQFUHWH$&, 0RGL¿FDWLRQIDFWRUVIRUVODEVZLWKUHLQIRUFHPHQWRWKHUWKDQSVLDQGOLJKWZHLJKWFRQFUHWHDUHJLYHQLQ$&, DQGUHVSHFWLYHO\

4-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

0LQLPXPVODEWKLFNQHVVHVEDVHGRQYDULRXVVXSSRUWFRQGLWLRQVDUHJLYHQLQ7DEOH,QWKHH[SUHVVLRQVIRUPLQLis the span length of the one-way slab in inches; for cantilevers, is the clear projection of the cantilever mum in inches. Also, and DUHWKHPRGL¿FDWLRQIDFWRUVIRUUHLQIRUFHPHQWJUDGHDQGOLJKWZHLJKWFRQFUHWHUHVSHFtively. Table 4.1 Minimum Thickness of Solid, Nonprestressed One-way Slabs Lightweight Concrete(1)

Normalweight Concrete Support Condition

ƒy = 60 ksi

ƒy other than 60 ksi(2)

ƒy = 60 ksi(3)

ƒy other than 60 ksi(2),(3)

Simply supported One end continuous Both ends continuous Cantilever (1) Applicable where equilibrium density, (2) (3)

is in the range of 90 to 115 lb/ft3

For the usual case of continuous construction, the thickness of the slab, should be the same for all spans and it should be determined on the basis of the span yielding the largest minimum depth; this results in economical IRUPZRUNVHH5HIHUHQFH )RUWKHRQHZD\VODEV\VWHPLQ)LJXUHZLWKQRUPDOZHLJKWFRQFUHWHDQG*UDGH UHLQIRUFHPHQWWKHPLQLPXPVODEWKLFNQHVVWREHXVHGIRUDOOWKHVSDQVLVHTXDOWRLQZKLFKLVURXQGHGXSWR LQ6ODEWKLFNQHVVLVW\SLFDOO\VSHFL¿HGLQZKROHLQFKLQFUHPHQWVDOWKRXJKKDOILQFKLQFUHPHQWVDUHFRPPRQO\XVHG in wide-module joist construction.



ͳͷᇱ ǦͲᇳ 

ͷᇱ ǦͲᇳ 

κΤͳͲ ൌ ͸ǤͲᇳ 

κΤʹͶ ൌ ͹Ǥͷᇳ 

ͳͺᇱ ǦͲᇳ 

κΤʹͺ ൌ ͹Ǥ͹ᇳ 

ͳͶᇱ Ǧ͸ᇳ 

κΤʹͶ ൌ ͹Ǥ͵ᇳ 

Figure 4.2 Calculation of minimum slab thickness for a one-way slab system.

Fire-resistance requirements of the general building code must also be considered when specifying a slab thickQHVV$&, 7KLVLVHVSHFLDOO\LPSRUWDQWIRURQHZD\VODEVWKDWDUHSDUWRIDMRLVWV\VWHPZKHUHWKHMRLVWVDUH spaced relatively closely together, which means the required slab thickness for serviceability is relatively small. In FHUWDLQVLWXDWLRQVWKHUHTXLUHGVODEWKLFNQHVVEDVHGRQ¿UHUHVLVWDQFHUHTXLUHPHQWVIRXQGLQ,%&7DEOH LV JUHDWHUWKDQWKDWUHTXLUHGE\$&,IRUVHUYLFHDELOLW\$&, 

4.3 Required Strength 4.3.1 Analysis Methods

7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKVHH$&,DQGUHVSHFWLYHO\ )RUWKHFDVHRIJUDYLW\ORDGVWKH VLPSOL¿HGPHWKRGIRUDQDO\VLVRIFRQWLQXRXVEHDPVDQGRQHZD\VODEVLQ$&,LVSHUPLWWHGWREHXVHGSURYLGHG WKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG>$&,D @ 1. Members are prismatic  Gravity loads are uniformly distributed 4-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3. The service live load, is less than or equal to 3 times the service dead load,  There are at least two spans  7KHORQJHURIWZRDGMDFHQWVSDQVGRHVQRWH[FHHGWKHVKRUWHUVSDQE\PRUHWKDQSHUFHQW and factored shear forces, LQ$&,DQGUHVSHFWLYHThe approximate factored bending moments, is the factored uniformly distributed gravity load on the one-way slab. O\DUHJLYHQLQ)LJXUHZKHUH

Uniformly distributed gravity load, ‫ݓ‬௨ , with ‫ ܮ‬൑ 3‫ܦ‬ Two or more spans Integral with support

Prismatic members

Simple support

Positive moment

Spandrel support Column support

Negative moment

**

Shear

* For two-span condition, first interior negative moment ൌ ‫ݓ‬௨ κଶ௡ /9 κ௡,௔௩௚ ൌ ሺκ௡,ଵ ൅ κ௡,ଶ ሻ/2  ** Applicable to slabs with spans equal to or less than 10 ft and beams where the ratio of the sum of column stiffness to beam stiffness is greater than 8 at each end of the span. A  Interior face of exterior support B  Exterior face of first interior support C  Other faces of interior supports

Figure 4.3 Simplified analysis method for continuous beams and one-way slabs.

4.3.2 Critical Sections for Flexure and Shear

,QDFFRUGDQFHZLWK$&,DQG and are permitted to be calculated at the faces of the supports IRURQHZD\VODEVEXLOWLQWHJUDOO\ZLWKWKHVXSSRUWVVHH)LJXUH )RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVRFcur at the faces of the supports where negative moments are maximum and in the span where positive moments are maximum. One-way slabs are permitted to be designed for the shear force at a critical section located a distance IURPWKHIDFHRIWKHVXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVHH)LJXUH  • Support reaction, in direction of applied shear, introduces compression into the end region of the slab • Loads are applied at or near the top surface of the slab • No concentrated load occurs between the face of the support and the critical section

4-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ݓ‬௨ ƒ’’Ž‹‡†ƒ–‘”‡ƒ”–‘’ •—”ˆƒ…‡‘ˆ•Žƒ„

ܸ௨ ̷ˆƒ…‡

‘…‡–”ƒ–‡† Ž‘ƒ†ǡܲ௨ 

ܸ௨ ̷݀

”‹–‹…ƒŽ•‡…–‹‘ ݀ ൐ ݀

Figure 4.4 Critical section for shear in a one-way slab satisfying the conditions of ACI 7.4.3.2.

The critical section for shear must be taken at the face of the support where one or more of the three conditions in $&,DUHQRWPHW)RUH[DPSOHWKHFULWLFDOVHFWLRQPXVWEHDWWKHIDFHRIWKHVXSSRUWIRUWKHIUDPLQJFRQ¿JXUDWLRQLQ)LJXUHEHFDXVHWKHVXSSRUWLQJPHPEHULVLQWHQVLRQ

‡•‹‘ˆ‘”…‡ǡܶ௨ 

‫ݓ‬௨ 

ܸ௨ ̷ˆƒ…‡

”‹–‹…ƒŽ•‡…–‹‘ Figure 4.5 Critical section for shear in a one-way slab supported by a member in tension.

0RPHQWUHGLVWULEXWLRQLVQRWSHUPLWWHGZKHQEHQGLQJPRPHQWVDUHFDOFXODWHGXVLQJWKHVLPSOL¿HGPHWKRG$&,  :KHUHRQHRUPRUHRIWKHFRQGLWLRQVLQ$&,LVQRWVDWLV¿HGRQHRIWKHRWKHUIRXUPHWKRGVRIDQDO\VLV and at the critical sections. JLYHQLQ$&,PXVWEHXVHGWRGHWHUPLQH

4.4 Design Strength 4.4.1 General

7KHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\VHFWLRQLQDRQHZD\VODEIRUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOH$&, 

4-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(4.1) (4.2)

Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGRQHZD\VODEVPXVW EHGHVLJQHGDVWHQVLRQFRQWUROOHGLQDFFRUGDQFHZLWK$&,7DEOH$&, WKXVWKHQHWWHQVLOHVWUDLQLQ must be greater than or equal the extreme layer of the longitudinal tension reinforcement at nominal strength, and $&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPHOD\HURIORQJLWXGLQDOWHQVLRQ to is equal to $&, 7KHPRGXreinforcement corresponding to compression-controlled sections, LVSHUPLWWHGWREHWDNHQDVSVLUHJDUGOHVVRIWKHJUDGHRIWKH lus of elasticity of the reinforcement, $&,7DEOH  UHLQIRUFHPHQW$&, )RUVKHDU Methods to determine the nominal strengths

and

DUHJLYHQLQ6HFWLRQVDQGUHVSHFWLYHO\

4.4.2 Nominal Flexural Strength

7KHQRPLQDOÀH[XUDOVWUHQJWK of a rectangular section with one layer of tension reinforcement is determined in DFFRUGDQFHZLWK$&,DQGLVEDVHGRQPRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQVHH$&,DQG)LJXUH   (4.3)

ߝ௧ 

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ ݀ െ ሺܽȀʹሻ

ܿ ݀ ൌ ݀௧ 

݄

‫ܣ‬௦ 

ܽ ൌ ߚଵ ܿ

ܽ

ͲǤͺͷ݂௖ᇱ  ͲǤͲͲ͵

ܶ ൌ ‫ܣ‬௦ ݂௬ 

ܾ

Figure 4.6 Strain and stress distributions at a positive moment section in a one-way slab.

In this equation, LVWKHWRWDODUHDRIWKHÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUH are at a positive moment section and are equally applicable at negative moment sections. )RURQHZD\VODEVWKHDYHUDJHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQUsing an approximate SURYLGHVVXI¿FLHQWDFFXUDF\ sion reinforcement, can be approximated as ZKHQGHWHUPLQLQJWKHUHTXLUHGDPRXQWRIÀH[XUDOUHLQIRUFHPHQWLQW\SLFDORQHZD\VODEV\VWHPV The depth of the equivalent stress block, is determined from force equilibrium, that is, it is determined by setequal to the tension force in the reinforcement, ting the resultant compressive force in the concrete, and solving for VHH)LJXUH  (4.4)

,WLVDVVXPHGWKDWDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK is distributed over where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV$&,  the depth

4-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPD[LPXPVWUDLQLQWKHFRQFUHWHLVDVVXPHGWREH$&, DQGWKHWHUP SUHVVLRQIDFHRIWKHPHPEHUZKLFKLVW\SLFDOO\WDNHQDVLQIRUDRQHZD\VODE

is the width of the com-

is the factor relating the depth of the equivalent rectangular compression block, The term LVGH¿QHGLQ$&,7DEOHDVIROORZV$&,  neutral axis,

to the depth of the

• For • For • For 4.4.3 Nominal Shear Strength

The nominal one-way shear strength,

DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  (4.5)

In this equation, is the nominal shear strength provided by concrete and is the nominal shear strength provided by shear reinforcement. Shear reinforcement is typically not used in one-way slabs because of the issues associated with cost, fabrication, and placement of such reinforcement in a relatively thin slab; where additional shear strength is required, the slab thickness is often increased. For nonprestressed members, LVGHWHUPLQHGE\$&,7DEOH$&, (TXDWLRQF LQ$&,7DEOH LVDSSOLFDEOHZKHUHQRVKHDUUHLQIRUFHPHQWLVXVHGLQDVHFWLRQ7KLVHTXDWLRQUHGXFHVWRWKHIROORZLQJIRU LVHTXDOWR]HUR the case where the axial force, (4.6)

In this equation, is the width of the web of the section; for one-way slabs, must not be taken greater than

$FFRUGLQJWR$&,

7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU accounts for the phenomenon indicated in test results that the shear strength attributed to concrete in members without shear reinforcement does not increase in direct proportion with member GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (4.7)

,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK IRUVODEVZLWKDQRYHUDOOWKLFNQHVVOHVVWKDQDERXWLQ means that

For one-way slabs, this

The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[VHH$&,DQG density, $&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOHD @DQGYDOXHVRI based on FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOHE @1RWHWKDW is permitted to be taken as IRUOLJKWZHLJKWFRQFUHWH$&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH$&,  Table 4.2 Values of Ȝ Based on Equilibrium Density, wc Equilibrium Density, wc

Ȝ 



4-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 4.3 Values of Ȝ Based on Composition of Aggregates Concrete

Composition of Aggregates

)LQH$670& &RDUVH$670&

All-lightweight

)LQH&RPELQDWLRQRI$670& and ASTM C33 &RDUVH$670&

/LJKWZHLJKW¿QHEOHQG

Fine: ASTM C33 &RDUVH$670&

Sand-lightweight

Sand-lightweight, coarse blend

Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670& and ASTM C33

Ȝ 

WR(1)



WR

1. Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. 2. Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

The term LQ(TXDWLRQ LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW at the section divided by $FFRUGLQJWR$&,5 PD\EHWDNHQDVWKHVXPRIWKHDUHDVRIWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW ORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUV with one layer of tension reinforcement, like one-way slabs, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDWVHFtion. used to calculate DUHOLPLWHGWRSVLH[FHSWDVDOORZHGLQ$&,ZKLFKLVQRWDSSOLFDEOH Values of WRRQHZD\VODEV$&, 7KLVOLPLWDWLRQRQ is primarily due to the fact that there is a lack of test data and SUDFWLFDOH[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\ for oneZD\VODEVXVXDOO\GRHVQRWH[FHHGSVL5HIHUHQFH  In situations where increased, although increasing

WKHWKLFNQHVVRIWKHVODEDQGRUWKHFRQFUHWHFRPSUHVVLYHVWUHQJWKVKRXOGEH is less effective than increasing (or, equivalently, ).

7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@ (4.8)

4.5 Determination of Required Reinforcement 4.5.1 Required Flexural Reinforcement

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW , at a critical section of a one-way slab is determined based on the reand , for tension-controlled sections. Required can be determined TXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV E\WKHIROORZLQJHTXDWLRQZKLFKLVEDVHGRQ(TXDWLRQV DQG  (4.9)

7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH

LVGH¿QHGDVIROORZV (4.10)

where

for tension-controlled sections. 4-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is important to check that FDOFXODWHGE\(TXDWLRQ LVDWOHDVWHTXDOWRWKHUHTXLUHGPLQLPXPDUHDRI , which is equal to where the gross area of the slab, is equal to (ACI ÀH[XUDOUHLQIRUFHPHQW   It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear for tension-controlled sections (see strain distribution where is the depth of the neutral axis when $&,DQG)LJXUH  (4.11)

Substituting LQWR(TXDWLRQ ZLWK HTXDOWRWKDWIURP(TXDWLRQ WKHIROORZLQJHTXDWLRQFDQEH corresponding to tension-controlled sections for one-way XVHGWRFDOFXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW slabs with (4.12)

)RU*UDGHUHLQIRUFHPHQW duces to the following for these material properties:

and for

(TXDWLRQ UH(4.13)

If FDOFXODWHGE\(TXDWLRQ LVJUHDWHUWKDQ the latter of which is essentially the maximum amount of and must be inÀH[XUDOUHLQIRUFHPHQWSHUPLWWHGDWDQ\VHFWLRQWKHVHFWLRQLVQRWWHQVLRQFRQWUROOHG creased accordingly to attain a tension-controlled section. 4.5.2 Minimum Shrinkage and Temperature Reinforcement

A minimum amount of shrinkage and temperature reinforcement must be provided in a one-way slab perpendicular WRWKHÀH[XUDOUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,$&, 7KHPLQLPXPDUHDRIVXFKUHLQIRUFHPHQW $&,  is equal to The maximum center-to-center spacing of the shrinkage and temperature reinforcement is equal to the lesser of DQGLQ$&, DQGWKHVHUHLQIRUFLQJEDUVPXVWEHGHVLJQHGWRGHYHORS in tension at all sections where UHTXLUHG$&, 

4.6 Reinforcement Detailing 4.6.1 Concrete Cover

5HLQIRUFLQJEDUVDUHSODFHGLQRQHZD\VODEVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UH DQGRWKHUHIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&,$&, :KHQJHQHUDOEXLOGLQJFRGH PLQLPXPFRYHUUHTXLUHPHQWVIRU¿UHSURWHFWLRQDUHJUHDWHUWKDQ$&,$&,UHTXLUHVWKHJUHDWHUFRQFUHWHFRYHUWKLFNQHVVWRJRYHUQ,%&7DEOH SURYLGHVPLQLPXPFRQFUHWHFRYHUUHTXLUHPHQWVIRU¿UHUDWLQJVUDQJLQJIURPKUWRKU)RUUHLQIRUFLQJEDUVLQRQHZD\VODEVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKH JURXQGDQGDPD[LPXPKU¿UHUDWLQJWKHPLQLPXPFRYHULVHTXDOWRLQIRUDQGVPDOOHUEDUV$&,7DEOH  )RURQHZD\VODEVZLWKRXWWUDQVYHUVHUHLQIRUFHPHQWFRQFUHWHFRYHULVPHDVXUHGIURPWKHVXUIDFHRIWKH FRQFUHWHWRWKHRXWPRVWOD\HURIUHLQIRUFHPHQWVHH)LJXUH  4.6.2 Minimum Spacing of Flexural Reinforcing Bars

0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&,$&,  7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRWKDWFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJ bars. 4-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‘˜‡” Figure 4.7 Concrete cover for one-way slabs.

Ž‡š—”ƒŽ”‡‹ˆ‘”…‡‡–ሺ–›’Ǥሻ

݀௕  Š”‹ƒ‰‡ƒ†–‡’‡”ƒ–—”‡ ”‡‹ˆ‘”…‡‡– ‫ۓ‬ ۖ

ͳ‹Ǥ

‹‹—…Ž‡ƒ”•’ƒ…‡൒ ‰”‡ƒ–‡”‘ˆ ݀௕   ‫۔‬  ۖ ‫ە‬ሺͶȀ͵ሻ݀௔௚௚

Figure 4.8 Minimum clear spacing requirements for reinforcing bars.

7KHVSDFLQJUHTXLUHPHQWVDUHJLYHQLQ)LJXUHIRUDRQHZD\VODEZKHUH VL]HLQWKHFRQFUHWHPL[

is the nominal maximum aggregate

4.6.3 Maximum Spacing of Flexural Reinforcing Bars

0D[LPXPFHQWHUWRFHQWHUVSDFLQJRIUHLQIRUFLQJEDUVLVJLYHQLQ$&,$&, 7KHLQWHQWRIWKHVH UHTXLUHPHQWVLVWRFRQWUROÀH[XUDOFUDFNLQJ,QJHQHUDODODUJHUQXPEHURI¿QHUFUDFNVLVSUHIHUDEOHWRDIHZZLGH cracks mainly for reasons of durability and appearance. The maximum center-to-center bar spacing, LVGHWHUPLQHGE\WKHHTXDWLRQVLQ$&,7DEOH$&,  The following equation is applicable to deformed reinforcing bars:

(4.14)

In this equation, LVWKHOHDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKHVHFtion and LVWKHFDOFXODWHGVWUHVVLQWKHÀH[XUDOUHLQIRUFHPHQWFORVHWWRWKHWHQVLRQIDFHDWVHUYLFHORDGVLQSVL  In lieu of calculating using the unfactored bending moment at that section, is permitted to be taken as and maximum center-to-center $&, )RU*UDGHUHLQIRUFHPHQW IURP(TXDWLRQ  bar spacing $FFRUGLQJWR$&,WKHPD[LPXPVSDFLQJRIGHIRUPHGÀH[XUDOUHLQIRUFHPHQWLQDRQHZD\VODELVHTXDO and 18 in. For typical one-way slab thicknesses, the maximum spacing based on crack control to the lesser of requirements usually governs.

4-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

4.6.4 Selection of Flexural Reinforcement

7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ DQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV JLYHQLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\ In one-way slabs, LVW\SLFDOO\FDOFXODWHGSHUIRRWZLGWKRIVODE$EDUVL]HDQGVSDFLQJVKRXOGEHVHOHFWHGVRWKDW is equal to or slightly greater than the required considering minimum reinforcement requirethe provided PHQWVDQGPLQLPXPDQGPD[LPXPEDUVSDFLQJUHTXLUHPHQWV7KHLQIRUPDWLRQLQ7DEOHFDQEHXVHGWRIDFLOLWDWH VHOHFWLRQRIEDUVL]HDQGVSDFLQJDWDFULWLFDOVHFWLRQIRUÀH[XUHLQDRQHZD\VODE Table 4.4 Area of Reinforcement (in.2) Based on Center-to-Center Spacing of the Bars Spacing (in.) Bar Size 4

5

6

7

8

9

10

11

12

















































































4.6.5 Development of Flexural Reinforcement Development of Deformed Bars in Tension

'HYHORSPHQWOHQJWKVRIGHIRUPHGUHLQIRUFHPHQWDUHJLYHQLQ$&,$&, 3URYLVLRQVIRUWKHGHYHOis RSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHORSPHQWOHQJWK GHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RUDORQJZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KH UHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHUUHTXLUHPHQWVDUHFRYHUHG¿UVW Method 1 – ACI 25.4.2.4 The development length in tension of a deformed reinforcing bar,

LVGHWHUPLQHGE\$&,(TXDWLRQD  (4.15)

7KHWHUPVLQ(TXDWLRQ DUHDVIROORZV$&,7DEOH  0RGL¿FDWLRQIDFWRUIRUOLJKWZHLJKWFRQFUHWH crete:

7KLVIDFWRUUHÀHFWVWKHORZHUWHQVLOHVWUHQJWKRIOLJKWZHLJKWFRQ-

(4.16)

5HLQIRUFHPHQWJUDGHIDFWRU development length:

This factor accounts for the effect of reinforcement yield strength on the required

(4.17)

4-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

*UDGHÀH[XUDOUHLQIRUFHPHQWLVW\SLFDOO\XVHGLQRQHZD\VODEVVR

.

5HLQIRUFHPHQWFRDWLQJIDFWRU This factor accounts for the reduced bond strength between the concrete and HSR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFLQJEDUV

(4.18)

5HLQIRUFHPHQWVL]HIDFWRU

7KLVIDFWRUUHÀHFWVWKHPRUHIDYRUDEOHSHUIRUPDQFHRIVPDOOHUGLDPHWHUUHLQIRUFLQJEDUV (4.19)

&DVWLQJSRVLWLRQIDFWRU 7KLVIDFWRUUHÀHFWVWKHDGYHUVHHIIHFWVWKDWFDQRFFXUWRWKHWRSUHLQIRUFHPHQWLQDPHPber due to vertical migration of water and mortar, which collect on the underside of the bars during placement of the concrete: (4.20)

7KLVIDFWRULVXVXDOO\IRURQHZD\VODEV$OVRDFFRUGLQJWRWKHIRRWQRWHLQ$&,7DEOHWKHSURGXFW QHHGQRWH[FHHG Spacing or cover dimension,

7KLVWHUPLVGH¿QHGDVIROORZVVHH)LJXUH  (4.21)

ܿଵ 

For one-way slabs, the distance from the center of the bar to the nearest concrete surface typically governs.

ܿଶ 

‫ݏ‬ ‫ۓ‬ ۖ

ܿଵ 

ܿ௕ ൌ Ž‡••‡”‘ˆ ܿଶ  ‫۔‬ ۖ ‫ݏە‬Τʹ Figure 4.9 Spacing or cover dimension, cb.

4-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7UDQVYHUVHUHLQIRUFHPHQWLQGH[ 7KHWUDQVYHUVHUHLQIRUFHPHQWLQGH[UHSUHVHQWVWKHHIIHFWRIFRQ¿QLQJUHLQIRUFHPHQWDFURVVWKHSRWHQWLDOVSOLWWLQJSODQHVODUJHUDPRXQWVRIFRQ¿QLQJUHLQIRUFHPHQWUHGXFHWKHSRWHQWLDOIRU splitting failure, thereby reducing the overall required development length. This index is determined by ACI EquaWLRQE  (4.22)

where

total cross-sectional area of all transverse reinforcement within a spacing potential plane of splitting through the reinforcement being developed

that crosses the

center-to-center spacing of the transverse reinforcement number of bars being developed across the plane of splitting It is permitted to conservatively use LIWUDQVYHUVHUHLQIRUFHPHQWLVSUHVHQWRUUHTXLUHG$&, )RU . Also, for reinforcing bars one-way slabs, where it is typical that transverse reinforcement is not used, VSDFHGFORVHUWKDQLQRQFHQWHUWUDQVYHUVHUHLQIRUFHPHQWPXVWEHSURYLGHGVXFKWKDW with $&,  7KHFRQ¿QLQJWHUP PXVWEHWDNHQOHVVWKDQRUHTXDOWRLQ(TXDWLRQ >$&,@7HVWV KDYHVKRZQWKDWZKHQWKLVWHUPLVOHVVWKDQVSOLWWLQJIDLOXUHVDUHOLNHO\WRRFFXU$SXOORXWIDLOXUHRIWKHUHLQIRUFHPHQWLVPRUHOLNHO\ZKHQWKLVWHUPLVJUHDWHUWKDQVRDQLQFUHDVHLQWKHDQFKRUDJHFDSDFLW\GXHWRDQ LQFUHDVHLQFRYHURUDPRXQWRIFRQ¿QLQJUHLQIRUFHPHQWLVQRWOLNHO\ used in calculating PXVWEHOHVVWKDQRUHTXDOWRSVL$&, 7KLVUHTXLUHPHQWLVDSValues of SOLFDEOHWRDOOWKHUHLQIRUFHPHQWGHYHORSPHQWSURYLVLRQVLQ$&, The tension development length LVSHUPLWWHGWREHUHGXFHGLQFDVHVZKHUHWKHÀH[XUDOUHLQIRUFHPHQWLVJUHDWHU WKDQWKDWUHTXLUHGIURPDQDO\VLVH[FHSWIRUWKHVL[FDVHVLQ$&,$&, :KHUHSHUPLWWHGWKH reduction factor applied to LVHTXDOWRWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWGLYLGHGE\WKHSURYLGHGDUHDRI reinforcement. for deformed bars in one-way slabs based on normalweight concrete with , uncoated Values of , , and DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI *UDGHUHLQIRUFHPHQW are rounded up to the next whole number. Values of for one-way slabs with coated bars can be obtained by multiplying the tabulated values by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI for oneZD\VODEVPDGHRIOLJKWZHLJKWFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFHZLWK (TXDWLRQ  Table 4.5 Tension Development Length, ℓd, for Deformed Bars in One-way Slabs in Accordance with ACI 25.4.2.4*

* Normalweight concrete with Uncoated Grade 60 reinforcement

4-12

Bar Size

ℓd (in.)

















Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Method 2 – ACI 25.4.2.3 7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUH. Two sets of spacing and cover cases are given in ACI Table VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP and )RURQHZD\VODEVLWLVFRPPRQIRUWKHFOHDUVSDFLQJRIWKHEDUVEHLQJGHYHORSHGWREHDWOHDVW 7KHUHIRUHWKHIROORZLQJHTXDWLRQIURP$&,7DEOHIRUWKHFDVHRIDQG the clear cover to be at least for reinforcing bars in one-way slabs: smaller longitudinal bars can be used to determine (4.23)

Values of for deformed bars in one-way slabs based on normalweight concrete with , uncoated DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI are rounded up to the next *UDGHUHLQIRUFHPHQWDQG for one-way slabs with coated bars can be obtained by multiplying the tabulated values whole number. Values of by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI for one-way slabs made of lightweight DJJUHJDWHFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFHZLWK(TXDWLRQ  Table 4.6 Tension Development Length, ℓd, for Deformed Bars in One-way Slabs in Accordance with ACI 25.4.2.3* Bar Size

ℓd (in.)







19









* Normalweight concrete with Uncoated Grade 60 reinforcement

Development of Standard Hooks in Tension

+RRNVDUHSURYLGHGDWWKHHQGVRIUHLQIRUFLQJEDUVWRSURYLGHDGGLWLRQDODQFKRUDJHZKHUHGHYHORSPHQWOHQJWKFDQƒ”‘Ǥ ͻͲǦ†‡‰”‡‡‘‘ QRWEHDWWDLQHGZLWKVWUDLJKWEDUV6WDQGDUGKRRNVDUHGH¿QHGLQ$&,VHH)LJXUH  ͓ͻǡ͓ͳͲǡ

κௗ௛  ݀௕ 

κ௘௫௧ ൌ ͳʹ݀௕ 

‫ܦ‬

ͻͲǦ†‡‰”‡‡‘‘

ͻͲǦ†‡‰”‡‡‘‘

͓ͳͶǡ͓ͳ

κௗ௛  ݀௕  ”‹–‹…ƒŽ•‡…–‹‘

‫ܦ‬

ƒ”‘Ǥ

ࡰ

͓͵െ͓ͺ

͸݀௕ 

͓ͻǡ͓ͳͲǡ͓ͳͳ ͺ݀௕  ƒƒ”‘Ǥ ͳͲ݀  ͓ͳͶǡ͓ͳͺ ௕

͓͵

”‹–‹…ƒŽ•‡…–‹‘

κ௘௫௧ ൌ Ͷ݀௕ ൒ ʹΦ̶



ͳͺͲǦ†‡‰”‡‡‘‘

Figure 4.10 Standard hooks in accordance with ACI 25.3.1.

4-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The development length of a deformed reinforcing bar in tension with a standard hook,

LVJLYHQLQ$&,

(4.24)

7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRNVHH)LJXUH 7KH PRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  Table 4.7 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Lightweight,

Epoxy,

&RQ¿QLQJ reinforcement,

Location,

Condition

Lightweight concrete



Normalweight concrete



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG reinforcement



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK or



Other

1.6

)RUDQGVPDOOHUGLDPHWHUKRRNHGEDUV WHUPLnating inside a column core with side cover normal RU ZLWKVLGH to the plane of the hook cover normal to the plane of the hook



Other Concrete strength,

Value of Factor





The term LVWKHWRWDOFURVVVHFWLRQDODUHDRIWLHVRUVWLUUXSVFRQ¿QLQJKRRNHGEDUVDQG is the total cross-sectional area of hooked bars being developed at the same critical section. For one-way slabs, it is typical for (that is, ties or stirrups are not present). The term is the center-to-center spacing of the hooked bars, which have a nominal diameter 7KHUHTXLUHPHQWVLQ$&,DSSO\WRKRRNVDWGLVFRQWLQXRXVHQGVRIPHPEHUVWKDWLVDWHQGVRIVLPSO\ supported members, at the free ends of cantilevers, and at exterior joints where members do not extend beyond the MRLQW ZKHUHERWKVLGHFRYHUDQGWRSRUERWWRP FRYHUWRWKHKRRNLVOHVVWKDQLQ7KHVHSURYLVLRQVDUHW\SLFDOO\ applicable at beam-column joints, and thus, do not apply to one-way slabs in continuous construction where con¿QHPHQWLVSURYLGHGE\WKHVODERQERWKVLGHVSHUSHQGLFXODUWRWKHSODQHRIWKHKRRN for hooked deformed bars in one-way slabs based on normalweight concrete with Values of and side cover normal to the plane XQFRDWHG*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKRRNHGEDUVSDFLQJ DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI are rounded up to the next whole number. of the hook

4-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 4.8 Tension Development Length, ℓdh, for Hooked Deformed Bars in One-way Slabs* Bar Size

ℓdh (in.)

   

6 6 8 

* Normalweight concrete with Uncoated Grade 60 reinforcement

Side cover normal to the plane of the hook

Development of Headed Deformed Bars in Tension

3URYLVLRQVIRUWKHGHYHORSPHQWRIKHDGHGGHIRUPHGEDUVLQWHQVLRQDUHJLYHQLQ$&,([DPSOHVRIKHDGHG UHLQIRUFLQJEDUVDUHJLYHQLQ)LJXUH

Figure 4.11 Headed reinforcing bars.

+HDGHGGHIRUPHGEDUVDUHSHUPLWWHGWREHXVHGRQO\ZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG (a) (b) (c) (d) (e) (f)

%DUPXVWFRQIRUPWR$&, %DUVL]HPXVWEHRUVPDOOHU must be at least where is the area of the bar Net bearing area of head, Concrete must be normalweight where is the nominal diameter of the bar Clear cover to the bar must be greater than or equal to Center-to-center spacing between bars must be greater than or equal to

The development length of a headed deformed reinforcing bar in tension,

LVJLYHQLQ$&,

(4.25)

This development length is measured from the critical section to the bearing face of the head (see ACI Figure 5D 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  4-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 4.9 Modification Factors for Development of Headed Bars in Tension Modification Factor

Condition

Epoxy,

Parallel tie reinforcement,

Value of Factor

(SR[\FRDWHGRU]LQFDQG epoxy dual-coated reinforcement



8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK or



Other

1.6

For headed bars (1) terminating inside a column core with side cover to bar RU ZLWKVLGHFRYHUWREDU

Location,

 

Other Concrete strength,

 The term is the total cross-sectional area of ties or stirrups acting as parallel tie reinforcement for headed bars is the area of nonprestressed reinforcement in a tie or stirrup. For anchorages other than in beam-column and PXVWQRWEHFRQVLGHUHG$&, 7KHWHUP is the centerjoints (which is applicable to one-way slabs), to-center spacing of the headed bars, which have a nominal diameter 7KHUHTXLUHPHQWVRI$&,DQGDUHQRWDSSOLFDEOHWRRQHZD\VODEV Values of for headed deformed bars in one-way slabs based on normalweight concrete with , and side cover to the bar XQFRDWHG*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKHDGHGEDUVSDFLQJ are rounded up to the next whole number. JLYHQLQ7DEOH&DOFXODWHGYDOXHVRI

are

Table 4.10 Tension Development Length, ℓdt, for Headed Deformed Bars in One-way Slabs* Bar Size

ℓdt (in.)

#3

6

#4

6

#5

6

#6

8

* Normalweight concrete with Uncoated Grade 60 reinforcement Side cover normal to the bar

Development of Mechanically Anchored Deformed Bars in Tension

It is permitted to use any mechanical attachment or device capable of developing of the deformed bars provided LWLVDSSURYHGE\WKHEXLOGLQJRI¿FLDOLQDFFRUGDQFHZLWK$&,$&, 7KHXVHRIPHFKDQLFDOGHYLFHV WKDWGRQRWPHHWWKHUHTXLUHPHQWVLQ$&,RUDUHQRWGHYHORSHGLQDFFRUGDQFHZLWK$&,PD\EHXVHG provided test results are available that demonstrate the ability of the head and bar system to develop or anchor the required force in the bar.

4-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Development of Positive and Negative Flexural Reinforcement

Flexural reinforcement must be properly developed or anchored in a reinforced concrete one-way slab for the slab WRSHUIRUPDVLQWHQGHGLQDFFRUGDQFHZLWKWKHVWUHQJWKGHVLJQPHWKRG&ULWLFDOVHFWLRQVIRUGHYHORSPHQWRIÀH[XUDO UHLQIRUFHPHQWRFFXUDWWKHIROORZLQJ$&,  1. Points of maximum stress (that is, sections of maximum bending moment)  Points along the span where adjacent reinforcement is terminated because it is no longer required to UHVLVWÀH[XUH In continuous one-way slabs subjected to uniform gravity loads, the maximum positive and negative bending moPHQWVW\SLFDOO\RFFXUQHDUPLGVSDQDQGDWWKHIDFHVRIWKHVXSSRUWVUHVSHFWLYHO\3RVLWLYHDQGQHJDWLYHÀH[XUDO UHLQIRUFLQJEDUVPXVWEHGHYHORSHGRUDQFKRUHGRQERWKVLGHVRIWKHVHFULWLFDOVHFWLRQV$&,  7KHUHTXLUHGDUHDRIUHLQIRUFHPHQWDWDFULWLFDOVHFWLRQFDQEHGHWHUPLQHGXVLQJWKHPHWKRGVJLYHQLQ6HFWLRQRI this publication. )RUWKHRQHZD\VODEV\VWHPLQ)LJXUHWKHWRWDOUHTXLUHGDUHDRIQHJDWLYHUHLQIRUFHPHQWIRUWKHPD[LPXP at critical section A is equal to and the total number of reinforcing negative factored bending moment, bars at this location is equal to ZKHUHDOOWKHEDUVDUHWKHVDPHVL]H6LPLODUO\WKHWRWDOUHTXLUHGDUHDRISRVLWLYH at critical section C is equal to and reinforcement for the maximum positive factored bending moment, the total number of reinforcing bars at this location is equal to Requirements for the development of these two sets of reinforcing bars are discussed next. ݊ ൌ ݊ଵ ൅ ݊ଶ  ݊ଵ ൒ ݊Τ3

‫ିܣ‬ ௦ ሺ݊ barsሻ

‫ିܣ‬ ௦ଵ ሺ݊ଵ barsሻ

‫ܣ‬ା ௦ ሺ‫ ݌‬barsሻ ‫ܣ‬ା ௦ଶ ሺ‫݌‬ଶ barsሻ

‫ିܣ‬ ௦ଶ ሺ݊ଶ barsሻ

‫ ݌‬ൌ ‫݌‬ଵ ൅ ‫݌‬ଶ  ‫݌‬ଵ ൒ ‫݌‬Τ4 ൒ 6"

A

B

C

D

‫ܣ‬ା ௦ଵ ሺ‫݌‬ଵ barsሻ

൒ ݀ or 12݀௕  ൒ ݀ or 12݀௕  ݊ଶ bars

൒ κௗ 

൒ κௗ  ‫ݔ‬஺஻ 

݊ଵ bars

‫ݔ‬஼஽ 

൒ κௗ 

൒ ݀, 12݀௕ , or κ௡ /16

ሺ‫ܯ‬௨ା ሻ஽ 

‫݌‬ଶ bars ሺ‫ܯ‬௨ା ሻ஼ 

‫ݔ‬஻ா  E Point of inflection ሺ‫ܯ‬௨ି ሻ஻ 

ሺ‫ܯ‬௨ି ሻ஺  Figure 4.12 Development of flexural reinforcement in a one-way slab.

4-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Negative Flexural Reinforcement Assume a portion of LVFXWRIIDWVHFWLRQ%ZKHUHLWLVQRORQJHUUHTXLUHGIRUÀH[XUDOVWUHQJWK7KLVPDNHV and the number of section B a critical section. At this location, the cutoff reinforcement has an area equal to The area of the remaining portion of reinforcing bars is equal to and reinforcing bars is equal to This reinforcement must be able to resist the the corresponding number of reinforcing bars is equal to at section B. negative factored bending moment Because section B is a critical section, bars must be adequately developed to the right of this section. This is achieved by extending the bars a minimum distance of SDVWVHFWLRQ%DVVKRZQLQ)LJXUHZKHUH is the GHYHORSPHQWOHQJWKLQWHQVLRQRIDGHIRUPHGEDUZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  $FFRUGLQJWR$&,DWOHDVWRQHWKLUGRIWKHWRWDOQHJDWLYHUHLQIRUFHPHQWSURYLGHGDWDVXSSRUWPXVWKDYH , and SDVWWKHSRLQWRILQÀHFWLRQZKHUH is the clear an embedment length equal to the larger of , span length measured face-to-face of supports (which in this case are beams). Therefore, to satisfy this requirement, ZKLFKDFFRXQWVIRUSRVVLEOHVKLIWLQJRIWKHEHQGLQJPRPHQWGLDJUDPDWWKHSRLQWRILQÀHFWLRQGXHWRWKHDSSUR[L. The minimum length of bars to the right mate bending moment diagram customarily used in design, RIFULWLFDOVHFWLRQ$LVHTXDOWRWKHIROORZLQJ$&,DQG 

(4.26)

In these equations, is the distance from section A to the theoretical cutoff point at section B and GLVWDQFHIURPVHFWLRQ%WRWKHSRLQWRILQÀHFWLRQDWVHFWLRQ(

is the

Because section A is a critical section, the bars cutoff at section B must be developed a distance equal to at least EH\RQGWKDWVHFWLRQ$GGLWLRQDOO\$&,VWLSXODWHVWKDWWKHVHEDUVPXVWH[WHQGEH\RQGWKHSRLQWZKHUHWKH\DUH The minimum length of bars to the right no longer required a distance equal to at least the greater of and of critical section A is equal to the following:

(4.27)

5HLQIRUFLQJEDUVDUHQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRIWKHIROORZLQJFRQGLWLRQVDUH VDWLV¿HG$&,  where is the design shear strength of the section at that point (see Sec1. At the cutoff point, WLRQRIWKLVSXEOLFDWLRQ  )RUDQGVPDOOHUEDUVFRQWLQXLQJUHLQIRUFHPHQWSURYLGHVDWOHDVWGRXEOHWKHDUHDUHTXLUHGIRUÀH[XUHDWWKH cutoff point and 7KHWKLUGFRQGLWLRQLQ$&,ZKLFKLVUHODWHGWRVWLUUXSDUHDLQWKHVHFWLRQLVW\SLFDOO\QRWDSSOLFDEOHWRRQH way slabs.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The development of the negative reinforcing bars to the left of section A depends on its location within the framing V\VWHP$WLQWHULRUMRLQWVOLNHWKHRQHGHSLFWHGLQ)LJXUHGHYHORSPHQWLVDFKLHYHGE\FRQWLQXLQJWKHQHJDtive reinforcing bars into the span to the left of the beam. In an end span where the slab terminates at edge beams, a standard hook is provided at the ends of the negative reinforcing bars. Positive Flexural Reinforcement is cut off at section D. This makes section D a critical section. At this location, the reinAssume a portion of and the number of reinforcing bars is equal to The area of the remaining forcement has an area equal to and the corresponding number of reinforcing bars is equal to portion of reinforcing bars is equal to This reinforcement must be able to resist the negative factored bending moment at section D. The bars cut off at section D must be developed to a distance greater than or equal to beyond the critical section C. Like in the case of the negative reinforcement, these bars must extend beyond the point they are no longer $&, 7KHPLQLPXPOHQJWKRI bars to the left required by a distance equal to the greater of and of critical section C is equal to the following:

(4.28)

In this equation,

is the distance between critical section C and the theoretical cutoff point located at section D.

7KHUHTXLUHPHQWVRI$&,SHUWDLQLQJWRUHLQIRUFHPHQWWHUPLQDWHGLQDWHQVLRQ]RQHDUHDOVRDSSOLFDEOHLQWKLV case. $FFRUGLQJWR$&,DWOHDVWRQHIRXUWKRIWKHSRVLWLYHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQLQWRWKHVXSAlthough not common in typical reinforced concrete construction, at least one-third of the port. Thus, maximum positive reinforcement must extend along the slab bottom into simple supports. 7KHGLDPHWHURIWKHSRVLWLYHUHLQIRUFHPHQWLVOLPLWHGDWSRLQWVRILQÀHFWLRQLQDFFRUGDQFHZLWK$&,WKLV requirement helps to ensure the bars are developed in a length short enough such that the moment capacity is greater than the applied moment over the entire length of the slab:

(4.29)

In this equation, LVWKHQRPLQDOÀH[XUDOVWUHQJWKRIWKHVODEVHFWLRQDVVXPLQJDOOWKHUHLQIRUFHPHQWDWWKDWVHFis the factored shear force at the section, and is the embedment length of the reinforcetion is stressed to Additional information on this topic PHQWEH\RQGWKHLQÀHFWLRQSRLQWZKLFKLVOLPLWHGWRWKHJUHDWHURI and FDQEHIRXQGLQ$&,5IRUEHDPV 7KHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,DOVRKDYHDQLPSDFWRQGHYHORSPHQWOHQJWKVVHH6HFWLRQRI this publication).

4-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

4.6.6 Splices of Reinforcement Overview

6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQRQHZD\VODEVPXVWEHLQDFFRUGDQFHZLWK$&,$&, 7KHSULmary reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to WKHFRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG /DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQWDQGHDFK type of splice is examined next. Lap Splices

/DSVSOLFHVDUHIUHTXHQWO\VSHFL¿HGDQGDUHXVXDOO\WKHPRVWHFRQRPLFDOW\SHRIVSOLFH7KHUHDUHEDVLFDOO\WZR types of lap splices: contact and noncontact. In a contact lap splice, the bars are generally in contact over a speci¿HGOHQJWKDQGDUHWLHGWRJHWKHU>VHH)LJXUHD @,QDQRQFRQWDFWODSVSOLFHWKHEDUVDUHQRWLQFRQWDFWDQGWKH FHQWHUWRFHQWHUVSDFLQJRIWKHEDUVEHLQJVSOLFHGPXVWQRWH[FHHGWKHOLPLWVLQGLFDWHGLQ)LJXUHE VHH$&,  &RQWDFWODSVSOLFHVDUHXVXDOO\SUHIHUUHGEHFDXVHWKHEDUVDUHWLHGWRJHWKHUDQGDUHOHVVOLNHO\WRGLVSODFH during construction. The minimum clear spacing between a contact lap splice and adjacent splices or bars is the VDPHDVWKDWIRULQGLYLGXDOEDUV$&,VHH)LJXUH 

(a)

” lap length/5, 6"

(b)

Figure 4.13 Lap splices (a) Contact lap splice (b) Noncontact lap splice.

Lap splices should be provided at locations away from maximum stress (that is, away from sections of maximum bending moment) and should be staggered wherever possible. Reinforcing bars provided for negative bending moments (top bars) should be spliced near the midspan of a member, and reinforcing bars provided for positive bending moments (bottom bars) should be spliced over the supports. depends on the tension development length of the bars, the area of The required tension lap splice length, reinforcement provided over the length of the splice, and the percentage of reinforcement spliced at any one locaWLRQ/DSVSOLFHVDUHFODVVL¿HGDV&ODVV$RU&ODVV%%HFDXVHH[SHULPHQWDOGDWDRQODSVSOLFHVZLWKDQG EDUVDUHVSDUVHWKHXVHRIWHQVLRQODSVSOLFHVIRUWKHVHEDUVDUHSURKLELWHGH[FHSWDVSHUPLWWHGLQ$&,IRU FRPSUHVVLRQODSVSOLFHV$&,  The length of a tension lap splice is given as a multiple of

$&,  (4.30) (4.31)

When determining to be used in determining WKHLQPLQLPXPOHQJWKVSHFL¿HGLQ$&,E DQGWKH H[FHVVUHLQIRUFHPHQWPRGL¿FDWLRQIDFWRURI$&,DUHQRWDSSOLFDEOH$&, $OVRIRUUHLQIRUFLQJ spaced closer than 6 in. on center, transverse reinforcement must be provided such that bars with $&, 

4-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHGHIDXOWVSOLFHFODVVL¿FDWLRQIRUDWHQVLRQODSVSOLFHLV&ODVV%$&ODVV$VSOLFHLVSHUPLWWHGZKHUHERWKRIWKH IROORZLQJFRQGLWLRQVDUHVDWLV¿HG$&,7DEOH  over the entire splice length where and are the area of reinforcement provided at the splice location and the area of reinforcement required by analysis at the splice location, respectively • /HVVWKDQRUHTXDOWRSHUFHQWRIWKHUHLQIRUFHPHQWLVVSOLFHGZLWKLQWKHUHTXLUHGODSVSOLFHOHQJWK •

:KHUHEDUVRIGLIIHUHQWVL]HDUHODSVSOLFHGLQWHQVLRQ WKHVPDOOHUEDU$&, 

is equal to the greater of

of the larger bar and

of

Mechanical Splices

Mechanical splices are, in general, a complete assembly of a coupler, a coupling sleeve, or an end-bearing sleeve, including any additional material or components, required to accomplish the splicing of the reinforcing bars. Such RIWKHEDU$&, 7KLVLVWRHQVXUHWKDW\LHOGsplices must develop in tension or compression at least ing occurs in the reinforcing bar adjacent to the mechanical splice prior to the failure of the splice. ,OOXVWUDWHGLQ)LJXUHDUHWKUHHRIWKHPRUHSRSXODUW\SHVRIPHFKDQLFDOVSOLFHV7KHFROGVZDJHGFRXSOLQJ VOHHYHLQ)LJXUHD XVHVDK\GUDXOLFVZDJLQJSUHVVZLWKVSHFLDOGLHVWRGHIRUPWKHVOHHYHDURXQGWKHHQGVRI the spliced bars to produce positive mechanical interlock with the reinforcing bars. The shear screw coupling sleeve LQ)LJXUHE FRQVLVWVRIDFRXSOLQJVOHHYHZLWKVKHDUKHDGVFUHZVZKLFKDUHGHVLJQHGWRVKHDURIIDWDVSHFL¿HGWRUTXH7KHUHLQIRUFLQJEDUVDUHLQVHUWHGWRPHHWDWWKHFHQWHURIWKHFRXSOLQJVOHHYHDQGWKHQWKHVFUHZVDUH tightened. The tightening process embeds the pointed screws into the bars. The non-upset straight thread coupler LQ)LJXUHF FRQVLVWVRIDFRXSOHUZLWKLQWHUQDOVWUDLJKWWKUHDGVDWHDFKHQGWKDWMRLQVWKHWZRUHLQIRUFLQJEDUV with matching external threads. Additional information on these and other mechanical splices can be found in Reference 9.

(a)

(b)

(c)

Figure 4.14 Mechanical splices (a) Cold-swaged coupling sleeve (b) Shear screw coupling sleeve (c) Non-upset straight thread coupler.

Mechanical splices can be advantageous in a number of situations, including the following: • Where long lap splices are needed • Where lap splices cause reinforcement congestion • :KHUHVSDFLQJRIWKHÀH[XUDOUHLQIRUFHPHQWLVLQVXI¿FLHQWWRSHUPLWODSVSOLFHV

4-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Welded Splices

$&,SHUPLWVWKHXVHRIZHOGHGVSOLFHVZKLFKPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&,/LNHPHFKDQLFDOVSOLFHVDIXOOZHOGHGVSOLFHZKLFKLVJHQHUDOO\LQWHQGHGIRUDQGODUJHUEDUVPXVWEHDEOHWRGHYHORSDWOHDVW of the bar. 4.6.7 Structural Integrity Reinforcement

5HTXLUHPHQWVIRUVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWLQFDVWLQSODFHRQHZD\VODEVDUHJLYHQLQ$&,7KHSXUpose of this reinforcement is to improve the redundancy and ductility in the structure so that in the event of damage WRDPDMRUVXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFture will have a higher probability of maintaining overall stability. 7KUHHUHTXLUHPHQWVDUHJLYHQLQ$&, 1. Longitudinal structural integrity reinforcement must consist of at least one-quarter of the maximum positive moment reinforcement and it must be continuous.  Longitudinal structural integrity reinforcement at noncontinuous supports must be anchored to develop at the face of the support. 3. Where splices are needed in the continuous structural integrity reinforcement, the reinforcement must be VSOLFHGQHDUWKHVXSSRUWV0HFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQ ODSVSOLFHVLQDFFRUGDQFHZLWK$&,PXVWEHSURYLGHG 4.6.8 Recommended Flexural Reinforcement Details

5HFRPPHQGHGÀH[XUDOUHLQIRUFHPHQWGHWDLOVIRURQHZD\VODEVEDVHGRQWKHUHTXLUHPHQWVFRYHUHGDERYHLQFOXGLQJWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&,DUHJLYHQLQ)LJXUH7KHEDUOHQJWKVJLYHQLQWKH¿JXUH are based on a one-way slab subjected to uniformly distributed gravity loads; these lengths can be used for one-way VODEVWKDWKDYHEHHQGHVLJQHGXVLQJWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,VHH6HFWLRQRIWKLVSXEOLFDtion). For one-way slabs subjected to the effects from other types of loads, required bar lengths must be determined by calculations. ‘–‡ͳ

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‘–‡ʹ

‘–‡͵

–ƒ†ƒ”† Š‘‘ሺ–›’Ǥሻ ൒ ‫ܣ‬ା௦ ȀͶ ൒ ͸̶ ሺ–›’Ǥሻ

‫ܣ‬ା௦ 

‫ܣ‬ା௦ 

൑ ͲǤͳʹͷκ௡ଵ  κ௡ଵ 

൑ ͲǤͳʹͷκ௡ଶ 

‫ܣ‬ା௦  ‘–‡ͷ

κ௡ଶ 

൑ ͲǤͳʹͷκ௡ଷ  κ௡ଷ 

‘–‡•ǣ ሺͳሻ൒ κ௡ଵ ΤͶ ሺʹሻ൒ ‰”‡ƒ–‡”‘ˆሺκ௡ଵ Τ͵ ǡ κ௡ଶ Τ͵ሻ ሺ͵ሻ൒ ‰”‡ƒ–‡”‘ˆሺκ௡ଶ Τ͵ ǡ κ௡ଷ Τ͵ሻ ሺͶሻŠ”‹ƒ‰‡ƒ†–‡’‡”ƒ–—”‡”‡‹ˆ‘”…‡‡–ሺ–›’Ǥሻ ሺͷሻŽƒ••Žƒ’•’Ž‹…‡‘”‡“—‹˜ƒŽ‡–

Figure 4.15 Recommended flexural reinforcement details for one-way slabs.

4.7 Design Procedure 7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIRQHZD\VODEV,QFOXGHGLQWKH¿JXUH DUHWKHVHFWLRQQXPEHUVWDEOHQXPEHUVDQG¿JXUHQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFLQWKLVFKDSWHU can be found.

4-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1

EŽ

zĞƐ

EŽ

zĞƐ

1 A

Figure 4.16 Design procedure for one-way slabs.

4-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

EŽ

zĞƐ

1

zĞƐ

EŽ

1

Detail the flexural reinforcement in accordance with Sects. 4.6.5 – 4.6.8

Figure 4.16 (cont.) Design procedure for one-way slabs.

4-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

4.8 Examples 4.8.1

Example 4.1 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2, Normalweight Concrete

'HWHUPLQHWKHWKLFNQHVVRIWKHRQHZD\VODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDO ÀRRUDVVXPLQJWKHMRLVWVDUHVSDFHGIWRQFHQWHUDQGWKHULEZLGWKLVHTXDOWRLQVHH)LJXUH $OVRDVVXPH DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum slab thickness based on serviceability requirements

ACI 7.3.1.1

Because all span lengths are equal, the governing minimum slab thickness is based on a support condition of one end continuous: Table 4.1

7U\DLQWKLFNVODE Step 2 – Check shear strength requirements

ACI 7.5.1.1

• Step 2a – Determine the maximum factored distributed load For a one-foot design strip:

Maximum

LVGHWHUPLQHGE\$&,(TE 

Table 3.3

• 6WHSE±'HWHUPLQHLIWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFH (1) Members are prismatic   Gravity loads are uniformly distributed (3)   All spans are equal in length

ACI 6.5.1

7KHUHIRUHWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHG • Step 2c – Determine the maximum factored shear force Maximum RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUEHDP

ACI 6.5.4 Figure 4.3

where is conservatively taken as the clear distance between the bottom instead of the top of the tapered ribs VHH)LJXUH  Although the critical section for shear is permitted to be taken a distance FDVHLWLVFRQVHUYDWLYHO\WDNHQDWWKHIDFHRIWKHVXSSRUW$&, 

from the face of the support in this

4-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Step 2d – Check the shear strength requirements

ACI 7.5.1.1 Eq. (4.6)

Assuming Eq. (4.7)

for normalweight concrete

ACI 19.2.4.3

$VVXPLQJPLQLPXPÀH[XUDOUHLQIRUFHPHQW

For shear,

Also, it is evident that

ACI Table 21.2.1

Eq. (4.8)

7KHUHIRUHDLQWKLFNVODEVDWLV¿HVVHUYLFHDELOLW\DQGVKHDUVWUHQJWKUHTXLUHPHQWV 1RWH7KHLQWKLFNVODEPXVWDOVREHFKHFNHGIRU¿UHUHVLVWDQFHUHTXLUHPHQWVEDVHGRQWKHUHTXLUHG¿UHUHVLVtance rating prescribed in the governing building code. 4.8.2

Example 4.2 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2, Lightweight Concrete

'HWHUPLQHLIDLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJLVDGHTXDWHDWWKHURRI HTXDOWRSFI DQG*UDGH level assuming lightweight concrete with an equilibrium density, UHLQIRUFHPHQWVHH)LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the minimum slab thickness based on serviceability requirements

ACI 7.3.1.1

Because all span lengths are equal, minimum thickness based on a support condition of one end continuous governs: Table 4.1

7KHUHIRUHDLQWKLFNVODEVDWLV¿HVVHUYLFHDELOLW\UHTXLUHPHQWV

4-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Check shear strength requirements

ACI 7.5.1.1

• Step 2a – Determine the maximum factored distributed load For a one-foot design strip:

Maximum

LVGHWHUPLQHGE\$&,(TF 

Table 3.3

• 6WHSE±'HWHUPLQHLIWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFH ACI 6.5.1 (1) Members are prismatic   Gravity loads are uniformly distributed (3)   All spans are equal in length 7KHUHIRUHWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHG • Step 2c – Determine the maximum factored shear force Maximum RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUEHDP

ACI 6.5.4 Figure 4.3

where is conservatively taken as the clear distance between the bottom instead of the top of the tapered ribs VHH)LJXUH  Although the critical section for shear is permitted to be taken a distance FDVHLWLVFRQVHUYDWLYHO\WDNHQDWWKHIDFHRIWKHVXSSRUW$&,  • Step 2d – Check the shear strength requirements

from the face of the support in this ACI 7.5.1.1 Eq. (4.6)

Assuming Eq. (4.7)

Table 4.2

$VVXPLQJPLQLPXPÀH[XUDOUHLQIRUFHPHQW

4-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For shear,

ACI Table 21.2.1

Also, it is evident that

Eq. (4.8)

7KHUHIRUHWKHLQWKLFNVODEVDWLV¿HVVKHDUVWUHQJWKUHTXLUHPHQWV 1RWH7KHLQWKLFNVODEPXVWDOVREHFKHFNHGIRU¿UHUHVLVWDQFHUHTXLUHPHQWVEDVHGRQWKHUHTXLUHG¿UHUHVLVtance rating prescribed in the governing building code. 4.8.3

Example 4.3 – Determination of Required Reinforcement: One-way Slab System, Building #2

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWIRUWKHLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGDQG*UDGHUHLQIRUFHPHQWVHH LQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWHZLWK )LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the factored bending moments along the span

ACI 6.5

,WZDVGHWHUPLQHGLQ6WHSERI([DPSOHWKDWWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHIDFWRUHG bending moments and shear forces. %HFDXVHWKHVODEVKDYHVSDQVOHVVWKDQIWWKHQHJDWLYHEHQGLQJPRPHQWVDWWKHFULWLFDOVHFWLRQVIDFHRIVXSSRUWV  LVFDOFXODWHGLQ6WHSDRI([DPSOH are equal to the following where Figure 4.3

Similarly, the maximum positive bending moment occurs near midspan in an end span; this moment is used for all spans:

Step 2 – Determine the required flexural reinforcement

ACI 7.4.1.2 Eq. (4.10)

Eq. (4.9)

For tension-controlled sections, A summary of the required

4-28

LVJLYHQLQ7DEOHZKHUH

ACI Table 21.2.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 4.11 Summary of Required Flexural Reinforcement for the One-way Slab in Example 4.3 Location

Mu (ft-kips/ft)

Rn (psi)

As (in.2)

Face of support







Midspan







0LQLPXPÀH[XUDOUHLQIRUFHPHQW

Because

ACI 7.6.1.1

at all critical sections, use

Check if the sections are tension-controlled. )RU*UDGHUHLQIRUFHPHQWDQG Eq. (4.13)

Because

the sections are tension-controlled.

Step 3 – Select the reinforcing bars

Sect. 4.6.4

7U\EDUVVSDFHGDWLQRQFHQWHUIRUWKHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQW Provided

required

Table 4.4

Check spacing requirements. The minimum center-to-center spacing of the bars, PD[LPXPDJJUHJDWHVL]H

LVHTXDOWRWKHJUHDWHURIWKHIROORZLQJDVVXPLQJDLQ

Figure 4.8

The maximum center-to-center spacing of the bars, is equal to the minimum of LQDQGWKHPLQLPXPRIWKHIROORZLQJDVVXPLQJDLQFRQFUHWHFRYHUDQG



Eq. (4.14)

7KHLQVSDFLQJRIWKHEDUVLVJUHDWHUWKDQWKHPLQLPXPVSDFLQJRILQDQGLVHTXDOWRWKHPD[LPXP VSDFLQJRILQ 8VHEDUVVSDFHGDWLQRQFHQWHUIRUWKHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQW

4-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the shrinkage and temperature reinforcement

ACI 7.6.4.1

Minimum shrinkage and temperature reinforcement is equal to the following: ACI 24.4.3.2

The maximum center-to-center spacing of the bars,

is equal to the following: ACI 24.4.3.3

8VHEDUVVSDFHGDWLQRQFHQWHUIRUWKHVKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQWSURYLGHG 7KLVUHLQIRUFHPHQWLVSODFHGSHUSHQGLFXODUWRWKHÀH[XUDOUHLQIRUFHPHQWVHH)LJXUH required   4.8.4

Example 4.4 – Determination of Lap Splice Lengths: One-way Slab System, Building #2

'HWHUPLQHWKHUHTXLUHGODSVSOLFHOHQJWKIRUWKHSRVLWLYHUHLQIRUFHPHQWLQWKHLQWKLFNVODEWKDWLVSDUWRIWKH and ZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQWVHH)LJXUH  'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the tension development length

ACI 25.4.2.4

Eq. (4.15)

For normalweight concrete,

Eq. (4.16)

)RU*UDGHUHLQIRUFHPHQW

Eq. (4.17)

For uncoated reinforcing bars,

Eq. (4.18)

)RUUHLQIRUFLQJEDUV

Eq. (4.19)

)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Eq. (4.20)

Eq. (4.21)

There is no transverse reinforcement in this one-way slab, so

4-30

Eq. (4.22)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

PDWFKHVWKHYDOXHLQ7DEOH

The calculated value of

Step 2 – Determine the required lap splice length

ACI 25.5.2.1

$FFRUGLQJWRWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,DWOHDVWRQHTXDUWHURIWKHPD[LPXPSRVLWLYH moment reinforcement must be used as structural integrity reinforcement and it must be continuous. Also, where splices are needed, Class B tension lap splices must be provided for this reinforcement near the supports (joists). Class B lap splice,

Eq. (4.31)

8VHDIWLQODSVSOLFHOHQJWKIRUWKHSRVLWLYHUHLQIRUFHPHQWLQWKLVRQHZD\VODE 4.8.5

Example 4.5 – Determination of Reinforcement Details: One-way Slab System, Building #2

'HWHUPLQHWKHUHLQIRUFHPHQWGHWDLOVIRUWKHLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWH 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG

ͶΦᇳ 

5HLQIRUFHPHQWGHWDLOVIRUWKLVRQHZD\VODEV\VWHPDUHJLYHQLQ)LJXUHZKLFKDUHEDVHGRQWKHLQIRUPDWLRQ JLYHQLQ)LJXUHWKHGHWDLOVLQ)LJXUHFDQEHXVHGEHFDXVHWKHRQHZD\VODELQWKLVH[DPSOHLVVXEMHFWHGWR uniformly distributed gravity loads.

͓͵̷ͳʹᇳ ሺ–›’Ǥ ሻ

Ͷᇱ ǦͲᇳ 

ͳᇱ Ǧʹᇳ 

ͳᇱ ǦͲᇳ 

ʹᇱ ǦͲᇳ 

–ƒ†ƒ”† Š‘‘ሺ–›’Ǥሻ

͹ᇳ 

Ͷᇱ Ǧͷᇳ 

͹ᇳ 

Ͷᇱ Ǧͷᇳ 

͹ᇳ 

–Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 4.17 Reinforcement details for the one-way slab in Example 4.5.

Continuous positive reinforcement is provided because the span lengths are relatively small; this automatically satis¿HVRQHRIWKHVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVRI$&,VHH)LJXUH $OVRGHJUHHKRRNV are provided at the ends of the positive reinforcement in the end span at the noncontinuous supports (joists), which DOVRVDWLV¿HVRQHRIWKHVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWV$OORWKHUUHLQIRUFHPHQWLQWKHV\VWHPLVQRW shown for clarity.

4-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

4-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 5 TWO-WAY SLABS 5.1 Overview Two-way slabs are elements in two-way construction where the slabs are designed to support loads through bending LQWZRGLUHFWLRQV:KHUHWKHUDWLRRIWKHORQJWRWKHVKRUWVLGHRIDVODESDQHOLVRUOHVVORDGWUDQVIHULVE\EHQGLQJ SUHGRPLQDWHO\LQWZRGLUHFWLRQVDQGWKHSDQHOLVGH¿QHGDVDWZRZD\VODE7KHPDLQÀH[XUDOUHLQIRUFHPHQWLQD two-way slab system is parallel to the two orthogonal directions of load transfer. 'HVFULSWLRQVRIFRPPRQWZRZD\VODEV\VWHPVDUHJLYHQLQ7DEOHVHHDOVR)LJXUH  Table 5.1 Two-way Slab Systems System

Description

Flat plate

A two-way concrete slab supported directly on columns.

Flat slab

A two-way concrete slab supported directly on columns where the slab is thickened around the columns; the thickened portions of the slab are called drop panels.

Two-way beam-supported slab

A two-way concrete slab with column-line beams on all four sides of a panel.

7ZRZD\MRLVWZDIÀHVODE

Evenly spaced concrete joists spanning in both directions and a reinforced concrete slab cast integrally with the joists; solid slab sections are provided around the columns mainly to provide two-way shear resistance.

Flat plate voided concrete slab

A two-way concrete slab of uniform thickness containing regularly-spaced, KROORZSODVWLFEDOOVPDGHRIKLJKGHQVLW\UHF\FOHGSRO\HWK\OHQH+'3(  inside the concrete.

(a)

(b)

(d)

(c)

(e)

Figure 5.1 Two-way slab systems. (a) Flat plate. (b) Flat slab. (c) Two-way beam-supported slab. (d) Two-way joist. (e) Flat plate voided concrete slab.

5-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The design and detailing of solid, cast-in-place two-way slabs with nonprestressed reinforcement in buildings assigned to Seismic Design Category (SDC) A and B are covered in this chapter. Two-way slabs that are part of the lateral force-resisting system (LFRS) in buildings assigned to SDC C are covered in Chapter 13 of this publication. Provisions for two-way slabs are given in ACI Chapter 8.

5.2 Minimum Slab Thickness 5.2.1 Overview

7ZRZD\VODEVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG'HÀHFWLRQVDUHSHUPLWWHGWREHFDOFXODWHGLQDFFRUGDQFHZLWK$&,DQGVXEVHTXHQWO\FKHFNHGDJDLQVWWKH OLPLWVLQ$&,$&, )RUVODEVZLWKRXWLQWHULRUEHDPVVSDQQLQJEHWZHHQVXSSRUWVRQDOOVLGHVZKHUH in WKHÀH[XUDOUHLQIRUFHPHQWKDVDVSHFL¿HG\LHOGVWUHQJWK JUHDWHUWKDQSVLWKHPRGXOXVRIUXSWXUH instead of VHH$&,DQG )RUHFRQRP\*UDGHUHLQIRUFHFDOFXODWLQJGHÀHFWLRQVLV PHQWLVFRPPRQO\XVHGLQWZRZD\VODEV5HIHUHQFH  ,QOLHXRISHUIRUPLQJGHÀHFWLRQFRPSXWDWLRQVPLQLPXPVODEWKLFNQHVV can be obtained using the provisions in $&,IRUQRQSUHVWUHVVHGVODEVZLWKRXWLQWHULRUEHDPVRULQ$&,IRUQRQSUHVWUHVVHGVODEVZLWKEHDPV spanning between supports on all sides. Methods to determine minimum slab thicknesses for the two-way systems LOOXVWUDWHGLQ)LJXUHDUHJLYHQEHORZ 7KHWKLFNQHVVRIDFRQFUHWHÀRRU¿QLVKLVSHUPLWWHGWREHLQFOXGHGLQ ZKHUHLWLVSODFHGPRQROLWKLFDOO\ZLWKWKHÀRRU VODERUZKHUHWKHÀRRU¿QLVKLVGHVLJQHGWREHFRPSRVLWHZLWKWKHÀRRUVODELQDFFRUGDQFHZLWK$&,$&,  Where single- or multiple-leg stirrups are used as part of the two-way shear resistance at critical sections in a twoZD\VODEWKHPLQLPXPVODEWKLFNQHVVUHTXLUHPHQWVLQ$&,DQGPXVWEHVDWLV¿HG,QSDUWLFXODUWKH GLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQW must be at least equal to the greater of the following: (5.1)

In this equation, is the nominal diameter of the stirrups. For two-way slabs, an average can be approximated which means the minimum WRVDWLVI\WKHUHTXLUHPHQWVRI$&,PXVWEHHTXDOWRWKH as This requirement essentially ensures the slab is thick enough so the stirrups JUHDWHURILQDQG considering the minimum inside bend diameters at the corners of the FDQGHYHORSWKHVSHFL¿HG\LHOGVWUHQJWK VWLUUXSVVHH$&,  5.2.2 Flat Plates

0LQLPXPVODEWKLFNQHVVIRUÀDWSODWHVLVSHUPLWWHGWREHGHWHUPLQHGXVLQJWKHSURYLVLRQVLQ$&,IRUH[WHULRU DQGLQWHULRUSDQHOVZLWKRXWGURSSDQHOVVHH$&,7DEOHDQG7DEOH $VQRWHGSUHYLRXVO\*UDGHUHLQIRUFHPHQWLVRIWHQXVHGLQÀDWSODWHV\VWHPV Table 5.2 Minimum Thickness of Flat Plates Exterior Panels

fy (psi)

Interior Panels Without Edge Beams

With Edge Beams*

   * Slabs with beams between columns along exterior edges. Exterior panels are considered to be without edge beams where

5-2

.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In the expressions for minimum is the clear span length in the long direction of the panel measured face-toIDFHRIVXSSRUWVLQLQFKHV0LQLPXPVODEWKLFNQHVVHVGHWHUPLQHGE\7DEOHPXVWEHJUHDWHUWKDQRUHTXDOWR LQ>$&,D @ LQWKHIRRWQRWHRI7DEOHLVGH¿QHGDVWKHUDWLRRIWKHÀH[XUDOVWLIIQHVVRIWKHEHDPVHFWLRQWRWKH The term ÀH[XUDOVWLIIQHVVRIDZLGWKRIVODEERXQGHGODWHUDOO\E\FHQWHUOLQHVRIDGMDFHQWSDQHOVLIDQ\RQHDFKVLGHRIWKH beam: (5.2)

In this equation, and are the moduli of elasticity of the beam and slab concrete, respectively, which can be FDOFXODWHGE\$&,(TXDWLRQD ZKHUHWKHGHQVLW\XQLWZHLJKW RIQRUPDOZHLJKWFRQFUHWHRUWKHHTXLOLELVEHWZHHQDQGOEIW3LQFOXVLYH>$&,D @ rium density of lightweight concrete, (5.3)

FDQEHGHWHUPLQHGE\$&,(TXDWLRQE >$&,

For normalweight concrete E @

(5.4)

,Q(TXDWLRQV DQG  construction,

and

have the units of pounds per square inch. For the typical case of monolithic

,QOLHXRIXVLQJ(TXDWLRQV RU LWLVSHUPLWWHGWRVSHFLI\ FRUGDQFHZLWK$&,

based upon testing of concrete mixtures in ac-

The terms and LQ(TXDWLRQ DUHWKHPRPHQWVRILQHUWLDRIWKHJURVVVHFWLRQVRIWKHEHDPDQGVODEDERXW the centroidal axes, respectively. 

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ܾ௘ ൌ ܾ௪ ൅ ݄௕ ൑ ܾ௪ ൅ Ͷ݄ Figure 5.2 Effective beam and slab sections for computation of stiffness ratio, Įf , for edge beams.

5-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A portion of the slab on the sides of a beam is permitted to be included when determining the section properties of WKHVHFWLRQ$&, 7KHHIIHFWLYHVODEZLGWK permitted to be included in the determination of is the OHVVHURIWKHIROORZLQJIRUDQHGJHEHDPVHH)LJXUH  (5.5)

7KHWHUPVLQ(TXDWLRQ DUHGH¿QHGLQ)LJXUHIRUWKHFDVHZKHUHWKHEHDPSURMHFWVEHORZWKHVODE,QJHQeral, LVWKHJUHDWHURIWKHEHDPSURMHFWLRQDERYHRUEHORZWKHVODEVHH$&,)LJXUH5  7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQH and for an edge beam [Note: These equations are also valid for interior beams where LVGHWHUPLQHGE\(TXDWLRQ @7KHWHUP is the distance from the bottom of equal to that deterthe combined beam section [which is the beam plus the slab section with an effective width, PLQHGE\(TXDWLRQ RU @WRWKHFHQWURLGRIWKHVHFWLRQ



TXOEIW  Figure 5.3 Preliminary slab thickness for flat plate systems.

5-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.3 Equations for Ib and Is for Edge and Interior Beams

yb

Ib

Is

)RUÀDWSODWHV\VWHPVZLWKHGJHEHDPVDSUHOLPLQDU\VODEWKLFNQHVVLVGHWHUPLQHGXVLQJWKHH[SUHVVLRQVLQ7DEOH because the slab thickness, is not known at this stage; this assumption is checked after assuming has been calculated based on that slab thickness. 7KHWKLFNQHVVRIDÀDWSODWHLVXVXDOO\FRQWUROOHGE\WKHVHUYLFHDELOLW\UHTXLUHPHQWVLQ7DEOHIRUVSDQOHQJWKVLQ WKHUDQJHRIWRIWDQGOLYHORDGVOHVVWKDQRUHTXDOWROEIW. Otherwise, two-way shear requirements typically govern the thickness. Shear stresses at edge and corner columns are particularly critical because of relatively large bending moments transferred from the slab to the columns at these locations. Because two-way shear requirePHQWVDUHUHODWHGWRÀH[XUDOUHTXLUHPHQWVDFORVHGIRUPVROXWLRQIRUVODEWKLFNQHVVWKDWVDWLV¿HVERWKVHWVRIUHTXLUHPHQWVFDQQRWEHREWDLQHGXQOHVVVRPHVLPSOLI\LQJDVVXPSWLRQVDUHPDGH)LJXUHFDQEHXVHGWRGHWHUPLQH DSUHOLPLQDU\VODEWKLFNQHVVIRUÀDWSODWHVFRQVLGHULQJWZRZD\VKHDUVWUHVVHV7KHLQIRUPDWLRQLQWKH¿JXUHLVEDVHG on the following assumptions: • 6TXDUHHGJHFROXPQVRIVL]H bending perpendicular to the edge with a three-sided critical section • Column supports a tributary area • Concrete is normalweight with ,QWKH¿JXUH is the total factored uniformly distributed load on the slab, which must include the factored selfZHLJKWRIWKHVODEWKLVZHLJKWFDQEHHVWLPDWHGXVLQJDVODEWKLFNQHVVEDVHGRQWKHH[SUHVVLRQVLQ7DEOH$ preliminary slab thickness, LVREWDLQHGE\DGGLQJLQWRWKHYDOXHRI DFTXLUHGIURP)LJXUHIRUDJLYHQ and area ratio For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUÀDWSODWHVZLWKHGJHEHDPVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRIWKLV publication. Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness $&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHDELOLW\DQGRUVKHDUVWUHQJWKUHTXLUHPHQWV :KHUHLWLVQRWYLDEOHWRLQFUHDVHWKHVODEWKLFNQHVVDQGRUWKHFROXPQVL]HWRVDWLVI\WZRZD\VKHDUUHTXLUHPHQWV shear reinforcement or shear caps may be used. Information on common types of shear reinforcement is given in 6HFWLRQRIWKLVSXEOLFDWLRQ 5.2.3 Flat Slabs

0LQLPXPVODEWKLFNQHVVIRUÀDWVODEVLVSHUPLWWHGWREHGHWHUPLQHGXVLQJ$&,IRUH[WHULRUDQGLQWHULRU SDQHOVZLWKGURSSDQHOVVHH$&,7DEOHDQG7DEOH 7KHH[SUHVVLRQVLQ7DEOHFDQQRWEHXVHGXQOHVV WKHGLPHQVLRQVRIWKHGURSSDQHOVFRQIRUPWRWKHUHTXLUHPHQWVLQ$&,ZKLFKDUHLOOXVWUDWHGLQ)LJXUH and are the adjoining center-to-center span lengths (Note: The dimensional limitations of the ,QWKH¿JXUH GURSSDQHOPXVWDOVREHVDWLV¿HGLQWKHGLUHFWLRQSHUSHQGLFXODUWRWKDWVKRZQLQWKH¿JXUH )RUHFRQRP\*UDGH UHLQIRUFHPHQWLVRIWHQXVHGLQÀDWVODEV\VWHPV

5-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.4 Minimum Thickness of Flat Slabs Exterior Panels

fy (psi)

Interior Panels Without Edge Beams

With Edge Beams*

   * Slabs with beams between columns along exterior edges. Exterior panels are considered to be without edge beams where

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Figure 5.4 Dimensional requirements for drop panels.

/LNHLQWKHFDVHIRUÀDWSODWHV is the clear span length in the long direction of the panel measured face-to-face RIVXSSRUWVLQLQFKHV0LQLPXPVODEWKLFNQHVVHVGHWHUPLQHGE\WKHH[SUHVVLRQVLQ7DEOHPXVWEHJUHDWHUWKDQRU HTXDOWRLQ>$&,E @ 7RDFKLHYHHFRQRPLFDOIRUPZRUNVWDQGDUGOXPEHUGLPHQVLRQVVKRXOGEHXVHGWRIRUPGURSSDQHOV5HIHUHQFHV DQG 'URSSDQHOKHLJKWVWKDWLVWKHGHSWKVRIWKHSURMHFWLRQVEHORZWKHVODE EDVHGRQDFWXDOOXPEHUGLPHQVLRQV DQGLQWKLFNIRUPZRUNVKHDWKLQJDUHJLYHQLQ7DEOH)RUPZRUNFRVWVXQQHFHVVDULO\LQFUHDVHLIGURSSDQHO KHLJKWVRWKHUWKDQWKRVHLQ7DEOHDUHVSHFL¿HG         

Nominal

[ [ 6x 8x

5-6

݄

Table 5.5 Drop Panel Height for Formwork Economy

Drop panel height Actual lumber dimension ¾ԣ formwork sheathing ሺtyp.ሻ Lumber Size Lumber Size Actual Nominal Actual [ òƎ  [ òƎ [ òƎ  [ óƎ

 

Drop Panel Height óƎ óƎ óƎ Ǝ

Drop Panel Height

   

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUÀDWVODEVZLWKHGJHEHDPVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRIWKLV publication. Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness $&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHDELOLW\DQGRUVKHDUUHTXLUHPHQWV 5.2.4 Two-way Beam-Supported Slabs

0LQLPXPWKLFNQHVVUHTXLUHPHQWVIRUVODEVLQWZRZD\EHDPVXSSRUWHGVODEV\VWHPVDUHJLYHQLQ$&,DQG is the average value of for all beams on the DUHVXPPDUL]HGLQ7DEOHVHH$&,7DEOH 7KHWHUP HGJHVRIDSDQHO7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHVHFWLRQSURSHUWLHVRIWKHFRPELQHGEHDP of the slab is determined section and the slab for edge and interior beams. For edge beams, the effective width, E\(TXDWLRQ DQGIRULQWHULRUEHDPV LVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQVHH$&,DQG)LJXUH for the case of an interior beam projecting below the slab): (5.6)

In general,

LVWKHJUHDWHURIWKHEHDPSURMHFWLRQDERYHRUEHORZWKHVODEVHH$&,)LJXUH5 

Table 5.6 Minimum Thickness of Two-way Slabs with Beams Spanning Between Supports on All Sides

Įfm

Minimum Thickness, h



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ACI 8.3.1.1 applies VHH7DEOHVDQG

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ܾ௘ ൌ ܾ௪ ൅ ʹ݄௕ ൑ ܾ௪ ൅ ͺ݄ Figure 5.5 Effective beam and slab sections for computation of stiffness ratio, Įf , for interior beams.

5-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QWKHH[SUHVVLRQVLQ7DEOH is the clear span in the long direction measured face-to-face of beams in inches and is the ratio of the long-to-short dimensions of the clear spans in a panel. Also, has the units of pounds per square inch. the minimum In edge panels with edge beams where the stiffness ratio 7DEOHPXVWEHLQFUHDVHGE\DWOHDVWSHUFHQW$&, 

determined by the equations in

For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUWZRZD\EHDPVXSSRUWHGVODEVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRI this publication. Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness $&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHability requirements. 5.2.5 Two-way Joists



‘‡†‡’–Šǡ ݄௥ 

Žƒ„–Š‹…‡••ǡ ݄

General requirements for nonprestressed two-way joist systems are given in ACI 8.8. This type of system is formed E\DQGLQZLGHGRPHVUHVXOWLQJLQDQGIWPRGXOHVUHVSHFWLYHO\VHH)LJXUH 6ODEWKLFNness, LVW\SLFDOO\FRQWUROOHGE\¿UHUHVLVWDQFHUHTXLUHPHQWVDQGWKHMRLVWULE ZLGWK is set by the dome width, WKDWLVVSHFL¿HG7KHRQO\GHSWKWREHGHWHUPLQHGWRVDWLVI\VHUYLFHDELOLW\UHTXLUHPHQWVLVWKHGRPHGHSWK 6WDQGDUGIRUPGLPHQVLRQVIRUWZRZD\MRLVWFRQVWUXFWLRQDUHJLYHQLQ7DEOH

‘‹•–™‹†–Šǡ ܾ௥ 

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Figure 5.6 Cross-section of a two-way joist system.

5-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.7 Standard Form Dimensions for Two-way Joist Construction Dome Width, bd (in.)

Dome Depth, hr (in.)

Joist Width, br (in.)





6











8

)RUGHVLJQSXUSRVHVDWZRZD\MRLVWV\VWHPLVFRQVLGHUHGWREHDÀDWVODEZLWKWKHVROLGKHDGVDWWKHFROXPQVDFWLQJ as drop panels. The cross-section of the two-way joist system is transformed into an equivalent section of uniform thickness, with the same width as the actual section, which is equal to VHH)LJXUH 7KHHTXLYDOHQW is determined by setting the moment of inertia of the equivalent section equal to the uniform slab thickness, gross moment of inertia of the actual two-way joist section, (5.7)

Solving for (5.8)

݄

Equations to determine IRUDWZRZD\MRLVWV\VWHPDUHJLYHQLQ7DEOHIRUWKHFDVHZKHUHWKHVLGHIDFHVRIWKH GRPHVDUHVORSHGVHH)LJXUH 

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Figure 5.7 Determination of the gross moment of inertia, Ig for a two-way joist system.

5-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.8 Equations to Determine Ig for Two-way Joist Systems

yb

Ig

(TXLYDOHQWVODEWKLFNQHVVHVIRUDWZRZD\MRLVWV\VWHPZLWKDIWPRGXOHDLQZLGHMRLVWDQGDLQWKLFNVODE DUHJLYHQLQ7DEOH Table 5.9 Equivalent Slab Thickness, te, for a Two-way Joist System with a 3-ft Module Dome Depth, hd (in.)

Equivalent Slab Thickness, te (in.)

8

8.8











13.1

16











To satisfy serviceability requirements, an overall depth must be provided such that the corresponding equivalent thickness, ZKLFKLVGHWHUPLQHGE\(TXDWLRQ >RUZKHUHDSSOLFDEOHE\7DEOH@LVJUHDWHUWKDQ RUHTXDOWRWKHPLQLPXPUHTXLUHGRYHUDOOGHSWKGHWHUPLQHGE\WKHDSSOLFDEOHH[SUHVVLRQLQ7DEOHXVLQJWKHORQFor economy in formwork, the largest required overall depth from all the spans should be gest clear span length, XVHGZKHUHYHUSRVVLEOH5HIHUHQFH  7ZRZD\MRLVWV\VWHPVQRWVDWLVI\LQJWKHOLPLWDWLRQVRI$&,WKURXJKPXVWEHGHVLJQHGDVVODEVDQG beams (ACI 8.8.1.8). 5.2.6 Flat Plate Voided Concrete Slabs

(PSLULFDOVHUYLFHDELOLW\UHTXLUHPHQWVIRUÀDWSODWHYRLGHGFRQFUHWHVODEVV\VWHPVDUHQRWVSHFL¿FDOO\SURYLGHGLQ$&, %HFDXVHÀDWSODWHYRLGHGFRQFUHWHVODEV\VWHPVDUHYHU\VLPLODUWRWZRZD\VODEDQGWZRZD\MRLVWV\VWHPVWKH IRUVODEVZLWK*UDGHUHLQIRUFHPHQWVHH7DEOH  overall thickness can be initially estimated by ,QOLHXRIXVLQJDQHVWLPDWHGRYHUDOOVODEWKLFNQHVVLWLVUHFRPPHQGHGGHÀHFWLRQFDOFXODWLRQVEHSHUIRUPHGLQ DFFRUGDQFHZLWK$&,,QDGGLWLRQWRIDFWRUVDFFRXQWLQJIRUFRQFUHWHFUDFNLQJWKHDQDO\VLVPXVWLQFOXGHWKH stiffness reduction factor, which accounts for the presence of the voids in the slab; such factors can be found in the PDQXIDFWXUHUV¶OLWHUDWXUHRQDYHUDJHWKLVIDFWRULVHTXDOWRDERXW &DOFXODWHGGHÀHFWLRQVPXVWEHOHVVWKDQRU HTXDOWRWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQVLQ$&,7DEOH$&,  The largest required slab thickness from all the panels should be used wherever possible for overall economy. AdGLWLRQDOO\WKHVDPHVL]HYRLGIRUPHUVVKRXOGEHXVHGDVRIWHQDVSUDFWLFDOWKURXJKRXWWKHSURMHFWWRIDFLOLWDWHSODFHPHQWLQWKH¿HOG 'HVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUÀDWSODWHYRLGHGFRQFUHWHVODEV\VWHPVFDQEHIRXQGLQ5HIHUHQFH

5-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.3 Required Strength 5.3.1 Analysis Methods

7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUPXVW EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKVHH$&,DQGUHVSHFWLYHO\ 7KHIROORZLQJDQDO\VLVPHWKRGV DUHUHOHYDQWWRWZRZD\VODEV$&,DQG  (1) /LQHDUHODVWLF¿UVWRUGHUDQDO\VLV$&,   /LQHDUHODVWLFVHFRQGRUGHUDQDO\VLV$&, (3) Inelastic analysis (ACI 6.8)   Finite element analysis (ACI 6.9)   'LUHFWGHVLJQPHWKRG''0 >$&,D @ (6) (TXLYDOHQWIUDPHPHWKRG()0 >$&,E @ For purposes of analysis, two-way slab panels, which are bounded by columns, beams, or wall centerlines on all VLGHV$&, DUHSHUPLWWHGWREHGLYLGHGLQWRFROXPQVVWULSVDQGPLGGOHVWULSVWKHGH¿QLWLRQVRIZKLFKDUH JLYHQLQ$&,DQGUHVSHFWLYHO\7KHZLGWKVRIWKHVHVWULSVGHSHQGRQWKHVSDQOHQJWKV and is the length of the span in the direction moments are being determined, measured center-to-center of supwhere is the length of the span in the direction perpendicular to measured center-to-center of supports ports, and VHH)LJXUH $Q\FROXPQOLQHEHDPVSUHVHQWDUHWREHLQFOXGHGLQWKHFROXPQVWULS$&,  ሺκଶ ሻ஺ 

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‫ܘܑܚܜܛ‬

‫ܘܑܚܜܛܖ܏ܑܛ܍܌‬

‫ݔ‬ଵ ൌ Ž‡••‡”‘ˆ κଵ ΤͶ ‘”ሺκଶ ሻ஺ ȀͶ ‫ݔ‬ଶ ൌ Ž‡••‡”‘ˆ κଵ ΤͶ ‘”ሺκଶ ሻ஻ ȀͶ Figure 5.8 Design strips, column strips, and middle strips in a two-way slab system.

For slabs that are part of the LFRS, the results from a gravity load analysis are permitted to be combined with those IURPDODWHUDOORDGDQDO\VLVXVLQJWKHDSSOLFDEOHORDGFRPELQDWLRQVLQ$&,7DEOH$&,  7KH''0DQGWKH()0DUHQRWLQFOXGHGLQ$&,3URYLVLRQVIRUWKHVHPHWKRGVDUHJLYHQLQWKHWKURXJK HGLWLRQVRI$&,7KH''0LVFRYHUHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHLQIRUPDWLRQLQWKDWVHFWLRQ LVSULPDULO\IURP$&,LQ$&, 5-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.3.2 Critical Sections for Flexure

Factored bending moments, are permitted to be calculated at the faces of the supports for two-way slabs built LQWHJUDOO\ZLWKWKHVXSSRUWV$&, )RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVLQWKHGHVLJQVWULSVRFFXUDWWKH faces of the supports where negative moments are maximum and in the span where positive moments are maximum. Studies of moment transfer between slabs and columns have shown that the factored slab moment resisted by the GXHWRJUDYLW\ODWHUDODQGRURWKHUORDGHIIHFWVLVWUDQVIHUUHGE\DFRPELQDWLRQRIÀH[XUH column at a joint, DQGHFFHQWULFLW\RIVKHDU5HTXLUHPHQWVIRUWUDQVIHUE\ÀH[XUHDQGWUDQVIHUE\HFFHQWULFLW\RIVKHDUDUHJLYHQLQ$&, DQGUHVSHFWLYHO\ 7KHIUDFWLRQRIWKHIDFWRUHGVODEPRPHQWUHVLVWHGE\WKHFROXPQWUDQVIHUUHGE\ÀH[XUHLVHTXDOWR factor LVFDOFXODWHGE\$&,(TXDWLRQ >$&,@

where the

(5.9)

In this equation, and are the dimensions of the critical section for two-way shear measured parallel and perpendicular to the direction of analysis, respectively. The critical section for two-way shear is located so its perimis a minimum where the perimeter need not approach closer than to (1) edges or corners of columns, eter, FRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVRU FKDQJHVLQVODEWKLFNQHVVVXFKDVHGJHVRIFROXPQFDSLWDOVGURSSDQHOV RUVKHDUFDSV$&, )RUVTXDUHRUUHFWDQJXODUFROXPQVFRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVWKHFULWLFDO VHFWLRQLVSHUPLWWHGWREHGH¿QHGXVLQJVWUDLJKWVLGHV$&, )RUFLUFXODURUSRO\JRQVKDSHGFROXPQV WKHFULWLFDOVHFWLRQLVSHUPLWWHGWREHGH¿QHGE\DVTXDUHFROXPQWKDWKDVWKHVDPHDUHDDVWKHDFWXDOFROXPQ$&,  3URSHUWLHVRIFULWLFDOVHFWLRQVDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ to be used to resist $FFRUGLQJWR$&,WKHHIIHFWLYHVODEZLGWK two-way slab systems without drop panels or shear caps:

is equal to the following for

(5.10)

ܿଶ 

ܿଵ 

݀ Τʹሺ–›’Ǥሻ

ܾଵ 

ܾଶ 

ܾ௦௟௔௕ ൌ ܿଶ ൅ ͵݄ ߛ௙ ‫ܯ‬௦௖ 

‘Ž—•–”‹’™‹†–Š

݄ ܾଵ ൌ ܿଵ ൅ ݀ Τʹ ܾଶ ൌ ܿଶ ൅ ݀ ܾ௢ ൌ ʹܾଵ ൅ ܾଶ 

Figure 5.9 Effective slab width for transfer of moment at an edge column in a flat plate system.

5-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For two-way systems with drop panels or shear caps,

is equal to the following:

(5.11)

In these equations, is the dimension of the column in the direction perpendicular to the direction of analysis, is the thickness of the slab, and is the thickness of the drop panel or shear cap projection below the slab. For columns with capitals, the width of the capital is to be used in these equations instead of 7KHHIIHFWLYHVODEZLGWKVIRUDQHGJHFROXPQLQDÀDWSODWHV\VWHPDQGIRUDQLQWHULRUFROXPQLQDÀDWSODWHV\VWHP ZLWKDVKHDUFDSDUHLOOXVWUDWHGLQ)LJXUHVDQGUHVSHFWLYHO\

κ

ܿଵ 

κ

ܿଶ 

ܿଶ ൅ ͵ሺ݄ ൅ ݄ଵ ሻ  κ ൅ ͵݄

݄ଵ  ݄

ܾ௦௟௔௕ ൌ Ž‡••‡”‘ˆ ൝

Š‡ƒ”…ƒ’

ሺκ െ ܿଶ ͿȀʹ

Figure 5.10 Effective slab width for transfer of moment at an interior column in a flat plate system with a shear cap.

The factor LVSHUPLWWHGWREHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHEDVHGRQWKHOLPLWDWLRQVJLYHQLQWKDW in the table are the nomiWDEOHIRUGLIIHUHQWFROXPQORFDWLRQV$&,VHH7DEOH 7KHWHUPV and QDOVKHDUVWUHQJWKSURYLGHGE\WKHFRQFUHWH$&, DQGWKHIDFWRUHGVKHDUVWUHVVRQWKHVODEFULWLFDOVHFWLRQIRU two-way action, from the controlling load combination, without moment transfer, respectively. Information on how WRFDOFXODWHWKHVHVWUHVVHVLVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ

5-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.10 Maximum Modified Values of Ȗf for Nonprestressed Two-way Slabs

vuv

İt within bslab

Maximum Modified Ȗf

Column Location

Span Direction

Corner column

Either direction



Perpendicular to the edge



Edge column Parallel to the edge Interior column

Either direction

,QRUGHUWREHDEOHWRXVHWKHPD[LPXPPRGL¿HGYDOXHVRI LQ7DEOHWKHQHWWHQVLOHVWUDLQLQWKHH[WUHPH IRUWKHÀH[XUDOUHLQIRUFHPHQWLQ must be layer of longitudinal tension reinforcement at nominal strength, is the minimum strain corresponding greater than or equal to the strains indicated in the table. The strain is the value of the net tensile strain in the extreme layer of longito a tension-controlled section where WXGLQDOWHQVLRQUHLQIRUFHPHQWXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQVHH$&,7DEOH  results in a decrease in which is the factor used to determine the fraction of the factored slab Increasing >VHH$&,(TXDWLRQ @,W moment resisted by the column transferred by eccentricity of shear: results in a decrease in the moment and a corresponding decrease in factored is evident that a decrease in shear stresses, which may be advantageous in certain situations. KDVEHHQGHWHUPLQHGWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWZLWKLQ can be determined using Once (TXDWLRQV DQG LQ6HFWLRQRIWKLVSXEOLFDWLRQ)OH[XUDOUHLQIRUFHPHQWPD\QHHGWREHFRQFHQWUDWHG to resist $&, 7KLVFDQEHDFKLHYHGE\SURYLGLQJDSRUWLRQRIWKHQHJDWLYHFROXPQ within or by adding more reinforcement within strip reinforcement at a closer spacing within A reversal of moment can occur at slab-column joints that are part of the LFRS. In such cases, both top and bot$&,5UHFRPPHQGVDUDWLRRIWRSWRERWWRP WRPÀH[XUDOUHLQIRUFHPHQWPXVWEHFRQFHQWUDWHGZLWKLQ UHLQIRUFHPHQWRIDSSUR[LPDWHO\LQVXFKFDVHV 5.3.3 Critical Sections for Shear One-way Shear

Factored shear forces, are permitted to be calculated at the faces of the supports for two-way slabs built inteJUDOO\ZLWKWKHVXSSRUWV$&,  Two-way slabs are permitted to be designed for the factored shear force at a critical section located a distance IURPWKHIDFHRIWKHVXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVHH)LJXUH  (1) Support reaction, in direction of applied shear, introduces compression into the end region of the slab   Loads are applied at or near the top surface of the slab (3) No concentrated load occurs between the face of the support and the critical section The critical section for one-way shear must be taken at the face of the support where one or more of the three condiWLRQVLQ$&,DUHQRWPHW Two-way Shear

The critical section for two-way shear is located so its perimeter, is a minimum where the perimeter need not apWR HGJHVRUFRUQHUVRIFROXPQVFRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVRU FKDQJHVLQ proach closer than VODEWKLFNQHVVVXFKDVHGJHVRIFROXPQFDSLWDOVGURSSDQHOVRUVKHDUFDSV$&,DQG 

5-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κଵ 

”‹–‹…ƒŽ•‡…–‹‘

ܿଶ 

κଶ 

ܿଵ 

݀

”‹„—–ƒ”›ƒ”‡ƒ

Figure 5.11 Critical section for one-way shear in a two-way slab that satisfies the conditions of ACI 8.4.3.2.

κଵ 

ܿଵ ൅ ݀ ”‹–‹…ƒŽ•‡…–‹‘

ܿଶ 

ܿଶ ൅ ݀

κଶ 

ܿଵ 

݀Ȁʹሺ–›’Ǥሻ

”‹„—–ƒ”›ƒ”‡ƒ

Figure 5.12 Critical section for two-way shear in a flat plate.

$FFRUGLQJWR$&,DFULWLFDOVHFWLRQZLWKVWUDLJKWVLGHVLVSHUPLWWHGIRUVTXDUHRUUHFWDQJXODUFROXPQV concentrated loads, or reaction areas. The locations of the critical sections for two-way shear at an interior column LQÀDWSODWHDQGÀDWVODEV\VWHPVDUHJLYHQLQ)LJXUHVDQGUHVSHFWLYHO\)RUÀDWVODEVVKHDUUHTXLUHPHQWV from the face of the column and from the face of the drop panel beneed to be checked at a distance and are the average distances from the extreme comcause shear failure can occur at either location where SUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWGH¿QHGLQ)LJXUH A shear cap is a projection below the slab similar to a drop panel; unlike a drop panel, it is used exclusively to inFUHDVHWKHVODEVKHDUVWUHQJWKRIWKHVODE$FFRUGLQJWR$&,VKHDUFDSVPXVWH[WHQGKRUL]RQWDOO\IURPWKHIDFH RIWKHFROXPQDGLVWDQFHHTXDOWRDWOHDVWWKHWKLFNQHVVRIWKHSURMHFWLRQEHORZWKHVODEVHH)LJXUHIRUWKHFDVH of an interior column). Like slabs with drop panels, shear requirements must be checked at two locations.

5-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ሺκȀ͵ሻ ൅ ݀ଵ  κȀ͵

ܿଶ 

ܿଶ ൅ ݀ଶ 

κȀ͵

ሺκȀ͵ሻ ൅ ݀ଵ 

ܿଵ ൅ ݀ଶ  ܿଵ 

”‹–‹…ƒŽ •‡…–‹‘•

݀ଶ 

݀ଵ 

”‘’’ƒ‡Ž



Figure 5.13 Critical sections for two-way shear in a flat slab.

ܿଵ ൅ ʹܾ௖௔௣ ൅ ݀ଵ  ܿଵ ൅ ݀ଶ 

ܿଶ 

ܿଶ ൅ ݀ଶ 

ܿଶ ൅ ʹܾ௖௔௣ ൅ ݀ଵ 

ܿଵ 

”‹–‹…ƒŽ•‡…–‹‘•

݀ଵ 

݄ଵ  ݄

݀ଶ 

Š‡ƒ”…ƒ’

ܾ௖௔௣ ൒ ݄ଵ 

Figure 5.14 Critical sections for two-way shear at a shear cap.

5-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ ܦ‬൅ ݀ ‫ܦ‬

”‹–‹…ƒŽ•‡…–‹‘

݀

‘Ž—…ƒ’‹–ƒŽ

Figure 5.15 Critical section for two-way shear at a column capital.

ሺκȀ͵ሻ ൅ ݀ଵ  κȀ͵ ‫ ܦ‬൅ ݀ଶ 

κȀ͵

ሺκȀ͵ሻ ൅ ݀ଵ 

‫ܦ‬

”‹–‹…ƒŽ•‡…–‹‘•

݀ଶ 

݀ଵ 

”‘’’ƒ‡Ž ‘Ž—…ƒ’‹–ƒŽ

Figure 5.16 Critical sections for two-way shear at a column capital and drop panel.

5-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHFULWLFDOVHFWLRQIRUDQLQWHULRUFROXPQZLWKDFLUFXODUFDSLWDOLVLOOXVWUDWHGLQ)LJXUH%HFDXVHWKHFDSLWDOLV part of the column and not part of the slab, shear requirements need only be checked at the critical section located IURPWKHIDFHRIWKHFDSLWDO,QFDVHVZKHUHDFROXPQFDSLWDODQGGURSSDQHODUHERWKXWLOL]HGWKH a distance from the face of the capital and at the shear strength must be checked at the critical section located a distance IURPWKHIDFHRIWKHGURSSDQHOVHH)LJXUH  critical section located a distance &ULWLFDOVKHDUSHULPHWHUVDUHQRWDVFOHDUO\GH¿QHGLQFDVHVZKHUHWKHVODEHGJHVFDQWLOHYHUEH\RQGWKHIDFHV RIDQ HGJHFROXPQ7KHIROORZLQJJHQHUDOJXLGHOLQHVFDQEHXVHGEDVHGRQWKHSURYLVLRQVRI$&,ZKLFKUHTXLUHV be a minimum. the perimeter of the critical section, &RQVLGHUWKHHGJHFROXPQLQ)LJXUH7KHFULWLFDOVKHDUSHULPHWHUIRUWKLVFROXPQLVHLWKHUWKUHHVLGHGRUIRXUVLGed depending on the length of the cantilever (overhang), The following equation can be used to obtain minimum

(5.12)

Similar equations can be derived for corner columns.

ܿଵ 

ܿଶ 

ܿଶ ൅ ݀

ܿଵ ൅ ݀

ܿଵ 

ܿଶ 

ܿଵ ൅ ݀Ȁʹ

ܿଶ ൅ ݀

‫ ݔ‬൑ ܿଶ Ȁʹ ൅ ݀

‫ ݔ‬൐ ܿଶ Ȁʹ ൅ ݀ Žƒ„‡†‰‡

Š”‡‡Ǧ•‹†‡†…”‹–‹…ƒŽ’‡”‹‡–‡”

Žƒ„‡†‰‡

‘—”Ǧ•‹†‡†…”‹–‹…ƒŽ’‡”‹‡–‡”

Figure 5.17 Critical section perimeters for edge columns with slab cantilevers.

 ܿଵ ൅ ݀

ܿଵ ൌ ξߨ ‫ܦ‬Τʹ

‫ܦ‬

݀Ȁʹ

݀Ȁʹ

ܿଵ 

ܿଵ ൅ ݀

”‹–‹…ƒŽ •‡…–‹‘

Figure 5.18 Critical section for two-way shear in slabs with circular columns.

5-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUFLUFXODURUUHJXODUSRO\JRQVKDSHGFROXPQVWKHFULWLFDOVHFWLRQVIRUWZRZD\VKHDUDUHSHUPLWWHGWREHGH¿QHG DVVXPLQJDVTXDUHFROXPQWKDWKDVWKHVDPHDUHDDVWKHFLUFXODURUSRO\JRQVKDSHGFROXPQ$&, )RUD WKHVL]HRIWKHVTXDUHFROXPQ is equal to the following: circular column of diameter (5.13)

The dimensions of the critical section in this case are VHH)LJXUHIRUWKHFDVHRIDQLQWHULRU FROXPQ 1RWHWKDWWKLVSURYLVLRQVSHFL¿FDOO\UHIHUVWRFLUFXODUFROXPQVDQGQRWFLUFXODUFROXPQFDSLWDOVWKHFULWLFDO VHFWLRQIRUWKHODWWHULVFLUFXODUDVVKRZQLQ)LJXUHVDQG

Ž‘•‡†•–‹””—’ሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘

൑ ݀ Τʹ

݀ Τʹ

݀Ȁʹሺ–›’Ǥሻ

‫ ݏ‬൑ ݀ Τʹ ሺ–›’Ǥ ሻ

‡”‹’Š‡”ƒŽŽ‹‡‘ˆ •–‹””—’•ሺ–›’Ǥሻ

൑ ʹ݀ –‡”‹‘”‘Ž—

”‹–‹…ƒŽ•‡…–‹‘

݀Ȁʹ

݀Ȁʹ

൑ ݀ Τʹ

൑ ݀ Τʹ ݀Ȁʹሺ–›’Ǥሻ

‫ ݏ‬൑ ݀ Τʹ ሺ–›’Ǥ ሻ

݀Ȁʹሺ–›’Ǥሻ

‫ ݏ‬൑ ݀ Τʹ ”‹–‹…ƒŽ•‡…–‹‘ ሺ–›’Ǥ ሻ

”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘

Ž‘•‡†•–‹””—’ሺ–›’Ǥሻ ൑ ʹ݀ †‰‡‘Ž—

Ž‘•‡†•–‹””—’ሺ–›’Ǥሻ ൑ ʹ݀ ‘”‡”‘Ž—

Figure 5.19 Critical sections for two-way shear in slabs reinforced with stirrups.

5-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

     

‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘ ൑ ݀ Τʹ ݀Ȁʹ ݀Ȁʹሺ–›’Ǥሻ

‫ݏ‬

‡”‹’Š‡”ƒŽŽ‹‡‘ˆŠ‡ƒ†‡† •Š‡ƒ”•–—†•ሺ–›’Ǥሻ ൑ ʹ݀ –‡”‹‘”‘Ž— ‫ݏ‬൑ቊ

͵݀ ΤͶ™Š‡”‡‫ݒ‬௨ ൑ ߶͸ඥ݂௖ᇱ ݀ Τʹ ™Š‡”‡‫ݒ‬௨ ൐ ߶͸ඥ݂௖ᇱ 



‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ

൑ ݀ Τʹ

൑ ݀ Τʹ

݀Ȁʹ ‫ݏ‬

݀Ȁʹ ‫ݏ‬

݀Ȁʹሺ–›’Ǥሻ

݀Ȁʹሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ൑ ʹ݀

†‰‡‘Ž—

”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘ ൑ ʹ݀ ‘”‡”‘Ž—

Figure 5.20 Critical sections for two-way shear in slabs reinforced with headed shear stud reinforcement.

&ULWLFDOVHFWLRQVIRUWZRZD\VKHDULQVODEVUHLQIRUFHGZLWKVWLUUXSVRUKHDGHGVKHDUVWXGUHLQIRUFHPHQWDUHGH¿QHG from the edges or corners LQ$&,$&, ,QDGGLWLRQWRWKHFULWLFDOVHFWLRQORFDWHGDGLVWDQFH beyond the of the columns, concentrated loads, or reaction areas, a critical section with a perimeter located RXWHUPRVWSHULSKHUDOOLQHRIVKHDUUHLQIRUFHPHQWPXVWDOVREHFRQVLGHUHG,QRUGHUWRPLQLPL]H the shape of this critical section must be a polygon. The locations of the critical sections at interior, edge, and corner columns for VODEVUHLQIRUFHGZLWKVWLUUXSVDUHJLYHQLQ)LJXUH>DOVRVHH$&,)LJXUHV5D F @6LPLODUO\WKHORFDWLRQVRIWKHFULWLFDOVHFWLRQVIRUVODEVUHLQIRUFHGZLWKKHDGHGVKHDUVWXGUHLQIRUFHPHQWDUHJLYHQLQ)LJXUHDOVR VHH$&,)LJXUH5  is the fraction of the factored slab moment resisted by $VGLVFXVVHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ >$&,(TXDWLRQ @DQGWKHIDFWRU is the column transferred by eccentricity of shear where FDOFXODWHGE\(TXDWLRQ ZKLFKGHSHQGVRQWKHGLPHQVLRQVRIWKHFULWLFDOVHFWLRQIRUVKHDU7KHPRPHQW LVDSSOLHGWRWKHFHQWURLGRIWKHFULWLFDOVHFWLRQ$&, DQGSURGXFHVIDFWRUHGVKHDUVWUHVVHVDVVXPHGWRYDU\ OLQHDUO\DERXWWKHFHQWURLGRIWKHFULWLFDOVHFWLRQLQDFFRUGDQFHZLWK$&,$&, 

5-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾଵ 

‫ݒ‬௨|஺஻

ܿଵ 

‫ݒ‬௨|஼஽

Critical section

ܾଶ 

A

ܿଶ 

D

C ܿ  ܿ  B ஺஻ ஼஽

ܿ஼஽ 

ܿ஺஻ 

Centroid Interior Column

ܾଵ  ܿଵ 

ܿ஼஽ 

‫ݒ‬௨|஼஽

‫ݒ‬௨|஺஻

ܾଶ 

ܿଶ  C

Critical section

A

D

B

ܿ஼஽ 

ܿ஺஻ 

ܿ஺஻ 

Centroid Edge Column

Figure 5.21 Assumed distribution of shear stresses due to direct factored shear and eccentricity of shear.

The assumed distributions of shear stresses due to the combination of direct factored shear, and eccentricity of on the faces of the critical section for an interior column and an edge column bending perpendicular shear, WRWKHHGJHDUHLOOXVWUDWHGLQ)LJXUH The total factored shear stresses on faces AB and CD of the critical section can be determined by the following equations:

(5.14)

where

area of concrete section resisting shear transfer perimeter of critical section for two-way shear distance from centroid of critical section to face AB distance from centroid of critical section to face CD property of critical section analogous to the polar moment of inertia

Section Properties of Critical Sections

Section properties of the critical section for two-way shear for rectangular columns, which are applicable to critical IURPWKHIDFHVRIWKHFROXPQDUHJLYHQLQ7DEOH7KHFULWLFDOVKHDUSHULPHWHUVIRU sections located a distance WKHHGJHDQGFRUQHUFROXPQVDUHEDVHGRQWKHFROXPQVEHLQJÀXVKZLWKWKHVODEHGJHVWKDWLVQRVODEFDQWLOHYHUV  $SSHQGL['LQ5HIHUHQFHFRQWDLQVWDEXODWHGYDOXHVRIWKHFULWLFDOVHFWLRQSURSHUWLHVIRUFRPPRQVODEWKLFNQHVVHV DQGVTXDUHFROXPQVL]HVIRUHDFKRIWKHIRXUFDVHVLQ7DEOH 5-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.11 Section Properties of the Critical Section for Rectangular Columns ܾଶ ൌ ܿଶ ൅ ሺ݀ Τʹሻ

ܾଶ ൌ ܿଶ ൅ ݀ ܿଶ 

ܿଶ  ‫ܦ‬

ܿ஼஽  ܿ஺஻ 

ܿ஺஻  ‫ܣ‬

ߛ௩ ‫ܯ‬௦௖ 

‫ܦ‬

ܿଵ 

ܾଵ ൌ ܿଵ ൅ ݀

ܿ஼஽ 

‫ܥ‬

ܿଵ 

ܾଵ ൌ ܿଵ ൅ ݀

‫ܥ‬

‫ܣ‬

‫ܤ‬

ߛ௩ ‫ܯ‬௦௖ 

‫ܤ‬

†‰‡

Case

1

Section Property

2

Case 1: Interior rectangular column Case 2: Edge rectangular column bending parallel to the edge

Table 5.11 Section Properties of the Critical Section for Rectangular Columns (cont.) ܾଶ ൌ ܿଶ ൅ ሺ݀ Τʹሻ

ܾଶ ൌ ܿଶ ൅ ݀

ܿଶ 

ܿ஺஻ 

‫ܣ‬

‫ܤ‬

ܿ஼஽ 

‫ܦ‬

ܿ஺஻ 

‫ܤ‬

ߛ௩ ‫ܯ‬௦௖  †‰‡

Case Section Property

3

Case 3: Edge rectangular column bending perpendicular to the edge Case 4: Corner rectangular column bending perpendicular to the edge

5-22

†‰‡

‫ܥ‬

ܿଵ 

‫ܣ‬

ܿ஼஽ 

‫ܦ‬ ܾଵ ൌ ܿଵ ൅ ሺ݀Ȁʹሻ

†‰‡

‫ܥ‬

ܿଵ 

ܾଵ ൌ ܿଵ ൅ ሺ݀Ȁʹሻ

ܿଶ 

4

ߛ௩ ‫ܯ‬௦௖ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6HFWLRQSURSHUWLHVRISRO\JRQVKDSHGFULWLFDOVHFWLRQVOLNHWKRVHLOOXVWUDWHGLQ)LJXUHVDQGIRUWZRZD\ VODEVZLWKFORVHGVWLUUXSVDQGKHDGHGVKHDUVWXGUHLQIRUFHPHQWUHVSHFWLYHO\DUHJLYHQLQ7DEOH Table 5.12 Section Properties of Polygon-Shaped Critical Sections κଶ  ߛ௩ ‫ܯ‬௦௖  ‫ܥ‬

‫ܦ‬

ܿ஼஽  κଷ 

ܿ஼஽ 

†‰‡ ܿ஺஻ 

ܿଵ Τʹܿଵ Τʹ ‫ܣ‬

‫ܥ‬

ܿଵ Ȁʹ

ܾଶ 

κଵ 

ܾଵ 

κଶ 

ܿଵ Ȁʹ

ܿଶ 

ܿଶ 

‫ܦ‬

‫ܤ‬

ߛ௩ ‫ܯ‬௦௖ 

ܿ஺஻ 

κଷ 

κଵ  ‫ܣ‬

‫ܤ‬ ܾଶ 

ܾଵ 

Case Section Property

1

2

Case 1: Interior critical section with a rectangular column Case 2: Edge critical section with a rectangular column bending parallel to the edge (table continued on next page)

7KHHTXDWLRQVLQ7DEOHDUHGHWHUPLQHGXVLQJWKHIROORZLQJJHQHUDOHTXDWLRQVZKLFKDUHVXLWDEOHIRUFULWLFDO sections with irregular geometries, such as polygon-shaped critical sections or critical sections affected by slab RSHQLQJV5HIHUHQFH  (5.15)

(5.16)

5-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.12 Section Properties of Polygon-Shaped Critical Sections (cont.) ܾଵ  ܿ஼஽ 

†‰‡

ܿ஺஻ 

ܿଵ 

ܾଶ 

†‰‡

‫ܦ‬

κଵ 

ܿ஼஽ 

ܿଵ 

ܿ஼஽ 

‫ܥ‬

κଵ 

ܿଵ 

ܿଶ 

ܾଵ 

ܾଵ 

൫ξʹΤʹ൯ܿଵ 

κଷ Ȁ—

‫ݕ‬

‫ݔ‬௣ Ͷͷι

κଶ  ሺܾଶ ሻ௣ 

ܿ஺஻ 

ܿ஺஻ 

ߛ௩ ‫ܯ‬௦௖  κଷ 

‫ݔ‬

‫ݕ‬௣ 

ξʹሺܾଵ െ ܿଵ െ ܿ஺஻ ሻ

‫ܤ‬

‫ܣ‬

κଵ 

κଶ  †‰‡ ሺܾଵ ሻ௣ 

Case Section Property

3

4

Case 3: Edge critical section with a rectangular column bending perpendicular to the edge Case 4: Corner critical section with a square column bending perpendicular to the edge

The summation occurs over the number of straight segments in the critical section where each segment has a length are the coordinates of the ends of each straight segment along the and equal to The terms axes, respectively, with respect to the centroid of the critical section. The section properties LQ7DEOHDUH FRQFHSWXDOO\GLIIHUHQWWKDQWKRVHLQ7DEOHGHWHUPLQLQJ for irregular critical sections using the methodology to determine IRUUHJXODUO\VKDSHGFULWLFDOVHFWLRQVLVYHU\GLI¿FXOW7KHVHFWLRQSURSHUWLHVGHWHUPLQHGE\(TXDWLRQV DQG IRUDQ\VKDSHGFULWLFDOVHFWLRQDUHFRQVHUYDWLYHWKDWLVWKH\DUHVPDOOHUWKDQWKRVHGHWHUPLQHGE\WKHPHWKRGRORJ\XVHGIRUWKHUHJXODUO\VKDSHGFULWLFDOVHFWLRQVLQ7DEOH &DOFXODWLRQVIRUSRO\JRQ shaped critical sections for two-way slabs with closed stirrups and headed shear stud reinforcement are given in ([DPSOHVDQGUHVSHFWLYHO\ Information on how to determine the critical section properties for rectangular corner columns is given in Reference $OVRLQFOXGHGLVDPHWKRGWRFDOFXODWHWKHFULWLFDOVHFWLRQSURSHUWLHVDERXWWKHSULQFLSDOD[HVRIWKHVHFWLRQ7KH SULQFLSDOD[HVIRUWKHVTXDUHFROXPQLQ7DEOHDUHGHQRWHGE\DVXEVFULSW³p”. For square columns, the principal D[HVRFFXUDORQJWKHGLDJRQDORIWKHFROXPQDQGDWDQDQJOHRULHQWHGGHJUHHIURPWKHGLDJRQDO

5-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QOLHXRIXVLQJDQHTXLYDOHQWVTXDUHFROXPQLQDFFRUGDQFHZLWK$&,VHFWLRQSURSHUWLHVRIWKHFULWLFDO VHFWLRQIRUWZRZD\VKHDUIRUFLUFXODUFROXPQVFDQEHGHWHUPLQHGXVLQJWKHHTXDWLRQVLQ7DEOH)RUHGJHDQG corner circular columns, the critical section consists of a partial circular section plus two straight tangential segments perpendicular to and extending to the edge(s) of the slab. It is assumed the column is set back a distance equal from the edge(s), which provides conservative section properties for columns with larger setbacks. Similar to section properties can be derived for circular columns at the edge(s) of the slab. Table 5.13 Section Properties of the Critical Section for Circular Columns ߛ௩ ‫ܯ‬௦௖ 

݀Ȁʹ

݀Ȁʹ

ܿ஺ 

‫ ܦ‬൅ ݀

‫ܦ‬

‫ ܦ‬൅ ݀

ߛ௩ ‫ܯ‬௦௖ 

ܿ஺ 

‫ܦ‬

ܿ஻ 

ܿ஻ 

†‰‡

Case Section Property

1

2

Case 1: Interior circular column Case 2: Edge circular column bending parallel to the edge (table continued on next page)

5-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.13 Section Properties of the Critical Section for Circular Columns (cont.) ߛ௩ ‫ܯ‬௦௖ 

ߛ௩ ‫ܯ‬௦௖ 

†‰‡

‫ ܦ‬൅ ݀

‫ܦ‬

‫ ܦ‬൅ ݀

ܿ஻ 

ܿ஺ 

݀Ȁʹ

ܿ஻ 

ܿ஺ 

݀Ȁʹ

‫ܦ‬

†‰‡

Case Section Property

3

4

Case 3: Edge circular column bending perpendicular to the edge Case 4: Corner circular column bending perpendicular to the edge

5.3.4 Direct Design Method Overview

7KH'LUHFW'HVLJQ0HWKRG''0 LVDQDSSUR[LPDWHDQDO\VLVPHWKRGLQFOXGHGLQWKHWKURXJKHGLWLRQVRI $&,7KHLQIRUPDWLRQJLYHQEHORZLVSULPDULO\IURP6HFWLRQRI$&,$FFRUGLQJWR$&,D  WKLVPHWKRGLVSHUPLWWHGWREHXVHGWRDQDO\]HWZRZD\VODEV\VWHPVIRUJUDYLW\ORDGV The DDM method gives reasonably conservative values of bending moments at the critical sections for two-way VODEVVXEMHFWHGWRJUDYLW\ORDGVPHHWLQJWKHOLPLWDWLRQVRIWKHPHWKRGZKLFKDUHVXPPDUL]HGLQ)LJXUH 7KHIXQGDPHQWDOFRQFHSWRIWKH''0LVWKDWWKHVODELVDQDO\]HGE\GLYLGLQJLWLQWRGHVLJQVWULSVVHH)LJXUH  $QDO\VLVRIWKHVODEV\VWHPLVSHUIRUPHGLQHDFKGLUHFWLRQLQGHSHQGHQWO\DQGSHUFHQWRIWKHWRWDOJUDYLW\ORDG must be carried in each direction. 5-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κଵ 

൒ ʹκଵ Ȁ͵

κଵ 

Š”‡‡‘”‘”‡ •’ƒ•

κଶ 

‹ˆ‘”Ž›†‹•–”‹„—–‡†Ž‘ƒ† ‫ܮ‬Τ‫ ܦ‬൑ ʹ ‘Ž—‘ˆˆ•‡–൑ κଶ ȀͳͲ

‡…–ƒ‰—Žƒ”•Žƒ„’ƒ‡Ž• ሺʹ‘”Ž‡••ǣͳሻ Figure 5.22 Limitations of the Direct Design Method.

Determination of Factored Bending Moments in a Design Strip

The three fundamental steps in the DDM used to determine factored bending moments in a design strip are as follows: in each span. • Step 1: Determine the total factored static moment, in a span is determined by the following equation: The total factored static moment, (5.17)

where

total factored uniformly distributed gravity load on the slab clear span in the direction of analysis measured from face to face of columns, capitals, brackets, VHH)LJXUHIRUWKHGH¿QLWLRQRIFOHDU or walls, which must be taken greater than or equal to nonrectangular span for rectangular and nonrectangular columns; for the purpose of determining columns are converted to square columns of the same cross-sectional area) length of the span in the direction perpendicular to )LJXUH

measured center-to-center of supports (see

is different on each side of the design strip, an average of these transverse span lengths must be used Where LQ(TXDWLRQ ,QWKHFDVHRIVSDQVDGMDFHQWDQGSDUDOOHOWRDVODEHGJH WREHXVHGLQ(TXDWLRQ LV equal to the distance from the edge of the slab to the centerline of the panel. is determined for each clear span length; however, For varying span lengths in the direction of analysis, can be calculated based on the longest clear span and that value can conservatively be used for all spans in that design strip.

5-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄

݄ ݄ ݄ ͲǤͻ͵݄

݄

ͲǤͺͻ݄

݄

κ௡ ൒ ͲǤ͸ͷκଵ  κଵ  Figure 5.23 Definition of clear span length, ℓn

• 6WHS'LVWULEXWH into negative and positive bending moments in each span. into positive and negative bending moments at the critical sections within &RHI¿FLHQWVDUHXVHGWRGLVWULEXWH each span (that is, near or at midspan and the faces of the supports, respectively). $VXPPDU\RIWKHEHQGLQJPRPHQWFRHI¿FLHQWVIRUDQHQGVSDQDUHJLYHQLQ7DEOHIRUWKHW\SLFDOFDVHRI monolithic construction. A fully restrained exterior edge corresponds to a slab integrally constructed with a UHLQIRUFHGFRQFUHWHZDOOWKDWKDVDÀH[XUDOVWLIIQHVVPXFKJUHDWHUWKDQWKDWRIWKHVODEWKDWLVDUHODWLYHO\VPDOO amount of rotation occurs at the slab-wall connection). Table 5.14 Bending Moment Coefficients for an End Span Flat Plate and Flat Slab

Two-way beam-supported slab

Without Edge Beams

With Edge Beams

Exterior Edge Fully Restrained

Exterior negative









Positive









Interior negative









Location

)RULQWHULRUVSDQVWKHFRHI¿FLHQWVDUHHTXDOWRDQGDWWKHQHJDWLYHDQGSRVLWLYHORFDWLRQVLQWKHVSDQ respectively.

5-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The total factored bending moments in the design strips at the critical locations are obtained by multiplying E\WKHFRHI¿FLHQWVLQ7DEOHIRUDQHQGVSDQDQGE\WKHDSSOLFDEOHFRHI¿FLHQWVQRWHGDERYHIRULQWHULRU VSDQV)RUH[DPSOHWKHWRWDOIDFWRUHGGHVLJQVWULSEHQGLQJPRPHQWDWWKHH[WHULRUVXSSRUWRIDÀDWSODWHRUÀDW slab without edge beams is equal to • Step 3: Distribute the total negative and positive bending moments in the design strip to the column strips and middle strips. 2QFHWKHWRWDOIDFWRUHGGHVLJQSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVKDYHEHHQGHWHUPLQHGLQ6WHSWKHVH moments are distributed to the column strips and middle strips using percentages based on studies of linearly elastic slabs with different beam stiffnesses. The percentages depend on the aspect ratio of the panel and the presence of column-line beams in one or both directions. Distribution to Column Strips (TXDWLRQVWRGHWHUPLQHWKHSHUFHQWDJHVDWWKHFULWLFDOVHFWLRQVLQFROXPQVWULSVDUHJLYHQLQ7DEOH Table 5.15 Percentages of Factored Bending Moments in Column Strips Location

Percentage*

Exterior support

Positive

Interior support *Where

where

In these equations, LVWKHUDWLRRIWKHÀH[XUDOVWLIIQHVVRIWKHEHDPVHFWLRQWRWKHÀH[XUDOVWLIIQHVVRIDZLGWKRI slab bounded laterally by centerlines of adjacent panels for the beams in the direction of analysis, and is calculated E\(TXDWLRQ  The term LVWKHUDWLRRIWKHWRUVLRQDOVWLIIQHVVRIDQHGJHEHDPVHFWLRQWRWKHÀH[XUDOVWLIIQHVVRIDVODEZLGWK equal to the span length of the beam, measured center-to center of supports: (5.18)

In this equation, and are the moduli of elasticity of the beam and slab concrete, respectively, which are typically equal for monolithic construction; is the moment of inertia of the slab section using the full width of the and is the cross-sectional constant determined by dividing the slab tributary to the beam, that is, beam section into its component rectangles, each having a smaller dimension and a larger dimension and by summing the contribution of each rectangle: (5.19)

7KHVXEGLYLVLRQLVWREHGRQHLQVXFKDZD\DVWRPD[LPL]H Equations for the calculation of for an edge and LVWREHXVHGLQ(TXDWLRQ 1RWHWKDWWKH VSDQGUHO EHDPDUHJLYHQLQ)LJXUH7KHODUJHURI HIIHFWLYHZLGWKRIVODESHUPLWWHGWREHLQFOXGHGLQWKHHGJHEHDPLVJLYHQLQ)LJXUH

5-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ݕ‬ଵ 

‫ݔ‬ଶ 

‫ݕ‬ଶ 

‫ܥ‬஺ ൌ ൬1 െ 0.63

‫ݔ‬ଵ ‫ݔ‬ଵଷ ‫ݕ‬ଵ ‫ݔ‬ଶ ‫ݔ‬ଶଷ ‫ݕ‬ଶ ൰ ൅ ൬1 െ 0.63 ൰  ‫ݕ‬ଵ 3 ‫ݕ‬ଶ 3

‫ܥ‬஻ ൌ ൬1 െ 0.63

‫ݔ‬ଵ ‫ݔ‬ଵଷ ‫ݕ‬ଵ ‫ݔ‬ଶ ‫ݔ‬ଶଷ ‫ݕ‬ଶ ൰ ൅ ൬1 െ 0.63 ൰  ‫ݕ‬ଵ 3 ‫ݕ‬ଶ 3

‫ݔ‬ଵ  Case A

‫ݔ‬ଵ 

‫ݔ‬ଶ 

‫ݕ‬ଶ 

‫ݕ‬ଵ  Case B Figure 5.24 Calculation of cross-sectional constant

Values of

.

IRUFRPPRQVODEWKLFNQHVVHVDQGHGJHEHDPVL]HVDUHJLYHQLQ$SSHQGL[&RI5HIHUHQFH

In cases where FDOFXODWHGE\(TXDWLRQ LVJUHDWHUWKDQ PXVWEHWDNHQDVZKHQGHWHUPLQLQJWKH SHUFHQWDJHRIWKHIDFWRUHGEHQGLQJPRPHQWDWWKHH[WHULRUVXSSRUWE\WKHHTXDWLRQLQ7DEOH and without edge beams the percentages to use for For slabs without beams between supports GLVWULEXWLRQRIWRWDOQHJDWLYHPRPHQWVWRFROXPQVWULSVDUHDQGIRUH[WHULRUDQGLQWHULRUVXSSRUWVUHVSHFWLYHly. It is evident that all the exterior negative factored moment is assigned to the column strip unless an edge beam LVSURYLGHGZLWKDUHODWLYHO\ODUJHWRUVLRQDOVWLIIQHVVFRPSDUHGWRWKHÀH[XUDOVWLIIQHVVRIWKHVODE6LPLODUO\WKH GLVWULEXWLRQRIWRWDOSRVLWLYHPRPHQWWRFROXPQVWULSVLVSHUFHQW which Walls along column lines in the direction of analysis can be regarded as stiff beams with PXVWEHXVHGLQWKHHTXDWLRQVLQ7DEOH,IDQH[WHULRUVXSSRUWLVDUHLQIRUFHGFRQFUHWH means wall perpendicular to the direction of analysis and monolithic with the slab, it can be assumed the wall provides VLJQL¿FDQWWRUVLRQDOUHVLVWDQFHWKDWLV FDQEHWDNHQDV $QH[FHSWLRQWRWKHSHUFHQWDJHVLQ7DEOHRFFXUVZKHUHFROXPQVRUZDOOVKDYHDUHODWLYHO\ODUJHZLGWKSHUSHQdicular to the direction of analysis. In particular, where the transverse width of a column or wall extends a distance the negative factored bending moments are to JUHDWHUWKDQRUHTXDOWRSHUFHQWWKHZLGWKRIWKHGHVLJQVWULS be uniformly distributed across Distribution to Column-line Beams For relatively stiff column-line beams LWLVDVVXPHGWKHEHDPVDWWUDFWDVLJQL¿FDQWSRUWLRQRIWKH EHQGLQJPRPHQWVDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULS,QVXFKFDVHVWKHEHDPVPXVWEHGHVLJQHGWRUHVLVW percent of the factored column strip moments based on the applicable percentages determined by the equations in 7DEOH

5-30

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Where WKHSHUFHQWDJHUHVLVWHGE\WKHEHDPFDQEHGHWHUPLQHGE\OLQHDULQWHUSRODWLRQEHWZHHQ and respectively. DQG]HURSHUFHQWZKLFKFRUUHVSRQGWR In addition to the factored bending moments from the column strip, column-line beams must also be designed to resist the effects from any loads applied directly to the beam. Loads located on the slab outside of the beam width must be distributed accordingly between the slab and beam. Distribution to Middle Strips The portions of the total factored design negative and positive bending moments that are not assigned to the column strip are assigned to the half middle strips in the design strip. A middle strip adjacent and parallel to a panel edge and supported by a wall must be designed for two times the factored bending moment assigned to the half middle VWULSFRUUHVSRQGLQJWRWKH¿UVWURZRILQWHULRUVXSSRUWV 'HVLJQPRPHQWFRHI¿FLHQWVIRUWRWDOPRPHQWVFROXPQVWULSPRPHQWVDQGPLGGOHVWULSPRPHQWVDWWKHFULWLFDOVHFWLRQVLQ ÀDWSODWHVRUÀDWVODEVZLWKRXWHGJHEHDPV ÀDWSODWHVRUÀDWVODEVZLWKHGJHEHDPV ÀDWSODWHVRU ÀDWVODEVZLWKDQHQGVSDQLQWHJUDOZLWKDUHLQIRUFHGFRQFUHWHZDOODQG WZRZD\EHDPVXSSRUWHGVODEVDUHJLYHQ LQ7DEOHVDQGUHVSHFWLYHO\7KHVHFRHI¿FLHQWVDUHGHWHUPLQHGXVLQJWKHDSSOLFDEOHSHUFHQWDJHV discussed previously. Table 5.16 Design Moment Coefficients for Flat Plates or Flat Slabs Without Edge Beams

༃

༄

༅

†’ƒ

༆

ᬉ

–‡”‹‘”’ƒ

End Span Slab Moments

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Total Moment Column Strip Middle Strip



5-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.17 Design Moment Coefficients for Flat Plates or Flat Slabs With Edge Beams*

༃

༄

༅

†’ƒ

༆

ᬉ

–‡”‹‘”’ƒ

End Span Slab Moments

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Total Moment Column Strip M o

Middle Strip *Applicable to edge beams with

For edge beams with

exterior negative column strip moment is equal to

Table 5.18 Design Moment Coefficients for Flat Plates or Flat Slabs With End Span Integral with a Reinforced Concrete Wall

༃

༄

༅

†’ƒ

༆

ᬉ

–‡”‹‘”’ƒ

End Span Slab Moments

Total Moment Column Strip Middle Strip

5-32

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.19 Design Moment Coefficients for Two-Way Beam-Supported Slabs

༃

༄

༅

†’ƒ

༆

ᬉ

–‡”‹‘”’ƒ End Span

Span Ratio, ℓ2 / ℓ1

Slab and Beam Moments

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Total Moment



Column Strip

Beam* Slab

Middle Strip



Column Strip

Beam* Slab

Middle Strip



Column Strip

Beam* Slab

Middle Strip *Applicable to (1) edge beams with

and (2) column-line beams with

Determination of Factored Moments in Columns and Walls

Columns and walls supporting a two-way slab system must be designed to resist the appropriate negative factored bending moments transferred from the slab. In lieu of a more exact analysis, the following equation can be used transferred at an interior support due to factored gravity loads for the case of two to determine the moment, DGMRLQLQJVSDQVZLWKRQHVSDQORQJHUWKDQWKHRWKHUVHH)LJXUH  (5.20)

In this equation, and are the uniformly distributed factored dead and live loads, respectively, on the longer is the uniformly distributed factored dead load on the shorter span; and are the longer and shorter span; and are the span lengths perpendicular to and respectively. clear span lengths, respectively; and, Where

and

(TXDWLRQ UHGXFHVWRWKHIROORZLQJ (5.21)

5-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄ଶ 

ሺܿଶ ሻ஺ 

ሺܿଵ ሻ஺ 

ሺ‫ܯ‬௨ ሻ஺ 

‫ݍ‬஽௨ ൅ 0.5‫ݍ‬௅௨  ᇱ ‫ݍ‬஽௨ 

‫ܯ‬௦௖ 

݄ଵ 

Joint A

ሺܿଶ ሻ஻ 

ሺ‫ܯ‬௨ ሻ஻ 

ሺܿଵ ሻ஻  κᇱ௡ ൏ κ௡ 

κ௡ 

κଵᇱ

κଵ 

൏ κଵ 

Joint A

ᇱ ‫ܯ‬௦௖ ൌ 0.07ሾሺ‫ݍ‬஽௨ ൅ 0.5‫ݍ‬௅௨ ሻκଶ κଶ௡ െ ‫ݍ‬஽௨ κᇱଶ ሺκᇱ௡ ሻଶ ሿ  ሺܿଶ ሻ஺ ሺܿଵ ሻଷ஺ ሺ݇ሻ஺ ൌ  12݄ଶ  ሺܿଶ ሻ஻ ሺܿଵ ሻଷ஻ ሺ݇ሻ஻ ൌ  12݄ଵ  ሺ݇ሻ஺ ሺ‫ܯ‬௨ ሻ஺ ൌ ‫ ܯ‬ ሺ݇ሻ஺ ൅ ሺ݇ሻ஻ ௦௖  ሺ݇ሻ஻ ሺ‫ܯ‬௨ ሻ஻ ൌ ‫ ܯ‬ ሺ݇ሻ஺ ൅ ሺ݇ሻ஻ ௦௖ 

Figure 5.25 Determination of factored moments at interior supporting members.

The moment GHWHUPLQHGE\HLWKHU(TXDWLRQ RU LVGLVWULEXWHGWRWKHLQWHULRUVXSSRUWLQJPHPEHUV above and below the slab in direct proportion to the members’ stiffnesses, which, in general, can be determined by dividing the cross-sectional moment of inertia of the member in the direction of analysis by the length of the PHPEHUWKLVLVEDVHGRQWKHDVVXPSWLRQWKHSUHGRPLQDQWFRQWULEXWLRQWRWKHWRWDOGLVSODFHPHQWLVGXHWRÀH[XUH VHH)LJXUH :KHUHWKHFURVVVHFWLRQDOGLPHQVLRQVRIWKHVXSSRUWLQJPHPEHUVDERYHDQGEHORZWKHVODEDUH is transferred on the basis of the inverse of the member lengths (that is, the longer member resists a the same, than the shorter member). lesser amount of At an exterior support, the total negative bending moment from the slab is transferred directly to the support. Like interior supports, this moment is transferred to the members above and below the slab in direct proportion to their stiffnesses. For two-way shear design at edge columns, the factored gravity bending moment to be transferred bemust be when the DDM is used to tween the slab and edge column bending perpendicular to the edge, determine design strip bending moments. 5-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determination of Factored Shear in Slabs with Beams

,QDGGLWLRQWRWKHDSSOLFDEOHEHQGLQJPRPHQWVLQ7DEOHEHDPVPXVWEHGHVLJQHGWRUHVLVWDSSOLFDEOHWULEXWDU\ factored shear forces due to the factored loads. The tributary areas for beams in a two-way beam-supported system DUHJLYHQLQ)LJXUHWKHJUD\WULEXWDU\DUHDDSSOLHVWRWKHLQWHULRUEHDPWKDWVSDQVEHWZHHQFROXPQV$DQG% EHDPVPXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHIDFWRUHGVKHDUIRUFHVFDXVHGE\WKH Where IDFWRUHGORDGVRQWKHWULEXWDU\DUHDWKLVPHDQVWKHVKHDUIRUFHVLQWKHVODEDURXQGWKHFROXPQDUHHTXDOWR]HUR

A Tributary area

45 deg ሺtyp.ሻ B

Figure 5.26 Tributary area for calculation of factored shear forces on an interior beam in a two-way beam-supported slab.

Where the shear forces resisted by the beam can be obtained by linear interpolation assuming the (as noted above) and carries no load at In EHDPFDUULHVSHUFHQWRIWKHORDGZKHUH FDVHVZKHUHWKHEHDPFDUULHVOHVVWKDQSHUFHQWRIWKHORDGWKHVKHDUIRUFHVQRWUHVLVWHGE\WKHEHDPVPXVWEH resisted by the slab around the column. These shear forces and the applicable moments determined by the expresVLRQVLQ7DEOHFDXVHVKHDUVWUHVVHVLQWKHVODEZKLFKDUHFDOFXODWHGE\(TXDWLRQ  In addition to the shear forces noted above, column-line beams must also be designed to resist the factored shear forces due to any factored loads applied directly to the beam. 5.3.5 Lateral Loads

Effects from lateral loads on two-way systems can be determined using the analysis methods in ACI Chapter 6. Regardless of the analysis method, the effects of axial loads, cracking, and effects of load duration must be taken LQWRFRQVLGHUDWLRQVRGULIWFDXVHGE\ODWHUDOORDGVLVQRWXQGHUHVWLPDWHG:KHUHDOLQHDUHODVWLF¿UVWRUGHUDQDO\VLVLQ DFFRUGDQFHZLWK$&,LVSHUIRUPHGXVLQJIDFWRUHGORDGVWKHVHFWLRQSURSHUWLHVLQ$&,DQGDUH 5-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

permitted to be used in lieu of performing a more rigorous analysis. In cases where service-level loads are applied to WKHVWUXFWXUHWKHPRPHQWVRILQHUWLDJLYHQLQ$&,DUHDSSOLFDEOH6LPLODUO\ZKHUHDOLQHDUHODVWLFVHFRQG RUGHUDQDO\VLVLVSHUIRUPHGLQDFFRUGDQFHZLWK$&,XVLQJIDFWRUHGORDGVLWLVSHUPLWWHGWRXVHWKHVHFWLRQSURSHUWLHVFDOFXODWHGLQDFFRUGDQFHZLWK$&,$&, ZKHUHVHUYLFHOHYHOORDGVDUHXVHGWKHSURYLVLRQV RI$&,DUHDSSOLFDEOH In two-way slab systems with column-line beams, the beams usually resist most or all of the lateral load effects because they are typically much stiffer than the slab. )RUÀDWSODWHV\VWHPVWKDWDUHSDUWRIWKHVHLVPLFIRUFHUHVLVWLQJV\VWHP6)56 WKHPRPHQWRILQHUWLDIRUWKHVODE members must be determined by a model in substantial agreement with results of comprehensive tests and analysis; WKHPRPHQWVRILQHUWLDRIWKHRWKHUIUDPHPHPEHUVPXVWEHLQDFFRUGDQFHZLWK$&,DQG$&, 6.6.3.1.3). Several models satisfying this requirement are referenced in ACI R6.6.3.1.3. To account for cracking, the slab moment of inertia is typically reduced between one-half and one-quarter of the uncracked moment of inertia. When determining drifts or second-order effects in columns, a lower-bound slab stiffness should be used in the analysis. In structures where slab-column frames interact with shear walls, a range of slab stiffnesses should be inYHVWLJDWHGLQRUGHUWRDVVHVVWKHLPSRUWDQFHRIWKHLQWHUDFWLRQ1RWHWKDWÀDWSODWHVDUHQRWSHUPLWWHGWREHSDUWRIWKH SFRS in buildings assigned to Seismic Design Category (SDC) D, E, or F (see ACI Chapter 18).

5.4 Design Strength 5.4.1 General

7KHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGLQDWZRZD\VODEV\VWHPIRUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQ LQ$&,7DEOH$&,  (5.22) (5.23) (5.24) (5.25)

Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGWZRZD\VODEVPXVWEH GHVLJQHGDVWHQVLRQFRQWUROOHGLQDFFRUGDQFHZLWK$&,7DEOH$&, WKXVWKHQHWWHQVLOHVWUDLQLQWKH must be greater than or equal to extreme layer of the longitudinal tension reinforcement at nominal strength, and $&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPHOD\HURIGHIRUPHGORQJLWXGLQDO is equal to $&,  tension reinforcement corresponding to compression-controlled sections, LVSHUPLWWHGWREHWDNHQDVSVLUHJDUGOHVVRI where the modulus of elasticity of the reinforcement, $&,7DEOH  WKHJUDGHRIWKHUHLQIRUFHPHQW$&, )RUVKHDU 'HWHUPLQDWLRQRIQRPLQDOÀH[XUDODQGVKHDUVWUHQJWKVDUHJLYHQLQ6HFWLRQVDQGUHVSHFWLYHO\ 5.4.2 Nominal Flexural Strength

7KHQRPLQDOÀH[XUDOVWUHQJWK of a rectangular section with one layer of tension reinforcement is determined in DFFRUGDQFHZLWK$&,DQGLVEDVHGRQPRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQVHH$&,DQG)LJXUH   (5.26)

In this equation, LVWKHWRWDODUHDRIWKHÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUH are at a positive moment section and are equally applicable at negative moment sections.

5-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ ݀ െ ሺܽȀʹሻ

ܿ ݀ ൌ ݀௧ 

݄

‫ܣ‬௦ 

ܽ ൌ ߚଵ ܿ

ܽ

ͲǤͺͷ݂௖ᇱ  ͲǤͲͲ͵

ܶ ൌ ‫ܣ‬௦ ݂௬ 

ߝ௧ 

ܾ

Figure 5.27 Strain and stress distributions at a positive moment section in a two-way slab.

)RUWZRZD\VODEVWKHDYHUDJHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDO )RUÀDWVODEVVDWLVI\LQJWKHGLPHQVLRQDOUHtension reinforcement, can be approximated as TXLUHPHQWVLQ$&,IRUWKHGURSSDQHOVWKH XVHGLQGHWHUPLQLQJWKHQHJDWLYHÀH[XUDOVWUHQJWKDWWKHFULWLFDO LVFDOFXODWHGEDVHGRQWKHOLPLWDWLRQLQ$&,WKHWKLFNQHVVRIWKHGURSSDQHOEHORZWKHVODE section, must be less than or equal to one-fourth the distance from the edge of the drop panel to the face of the column or VHH)LJXUHIRUWKH column capital. This requirement essentially limits the value of used in determining case of an interior column with no capital).

݀

݄

൒ ͳǤʹͷ݄

      

”‘’’ƒ‡Ž ܾௗ௥௢௣  ൒ κ஺ Ȁ͸

൒ κ஻ Ȁ͸

݀ ൌ ͳǤʹͷ݄ െ ݀௕ െ …‘˜‡”  ൑ ݄ ൅ ൫ܾௗ௥௢௣ ΤͶ൯ െ ݀௕ െ …‘˜‡”  Figure 5.28 Determination of d for slabs with drop panels.

The depth of the equivalent stress block, is determined from force equilibrium, that is, it is determined by setequal to the tension force in the reinforcement, ting the resultant compressive force in the concrete, and solving for VHH)LJXUH  (5.27)

,WLVDVVXPHGDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK is distributed over the , where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV$&, 7KH depth PD[LPXPVWUDLQLQWKHFRQFUHWHLVDVVXPHGWREH$&, DQGWKHWHUP is the width of the compression face of the member, which is typically the width of the column strip or middle strip.

5-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term is the factor relating the depth of the equivalent rectangular compression block, LVGH¿QHGLQ$&,7DEOHDVIROORZV$&,  neutral axis,

to the depth of the

• For • For • For 5.4.3 Nominal One-way Shear Strength

The nominal one-way shear strength,

DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  (5.28)

In this equation, is the nominal shear strength provided by concrete and is the nominal shear strength provided by shear reinforcement. For nonprestressed members, LVGHWHUPLQHGE\$&,7DEOH$&,  $VVXPLQJVKHDUUHLQIRUFHPHQWLVQRWXVHGLQWZRZD\VODEVIRURQHZD\VKHDU(TXDWLRQF LQ$&,7DEOHLV LVHTXDOWR]HUR applicable. This equation reduces to the following where the axial force, (5.29)

In this equation, LVWKHZLGWKRIWKHSODQHH[WHQGLQJDFURVVWKHHQWLUHVODEZLGWK$&,VHH)LJXUH $FFRUGLQJWR$&, must not be taken greater than where 7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU accounts for the phenomenon indicated in test results that the shear strength attributed to concrete in members without shear reinforcement does not increase in direct proportion with member GHSWKWKLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (5.30)

,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK ZD\VODEVZLWKDQRYHUDOOGHSWKOHVVWKDQDERXWLQ

This means

for two-

The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH to normalweight concrete of the same compressive strength; it is determined based on either (1) the equilibrium RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[VHH$&,DQG density, $&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>VHH$&,7DEOHD @DQGYDOXHVRI based on FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>VHH$&,7DEOHE @1RWHWKDW is permitted to be taken DVIRUOLJKWZHLJKWFRQFUHWH$&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH$&,  Table 5.20 Values of Ȝ Based on Equilibrium Density, wc Equilibrium Density, wc

Ȝ 



5-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.21 Values of Ȝ Based on Composition of Aggregates Concrete

All-lightweight /LJKWZHLJKW¿QHEOHQG Sand-lightweight Sand-lightweight, coarse blend

Composition of Aggregates

)LQH$670& &RDUVH$670& )LQH&RPELQDWLRQRI$670&DQG& &RDUVH$670& Fine: ASTM C33 &RDUVH$670& Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670&DQG$670&

Ȝ  WR(1)  WR

(1) /LQHDULQWHUSRODWLRQIURPWRLVSHUPLWWHGEDVHGRQWKHDEVROXWHYROXPHRIQRUPDOZHLJKW¿QHDJJUHJDWHDVDIUDFWLRQRIWKHWRWDODEVROXWHYROXPH RI¿QHDJJUHJDWH (2) /LQHDULQWHUSRODWLRQIURPWRLVSHUPLWWHGEDVHGRQWKHDEVROXWHYROXPHRIQRUPDOZHLJKWFRDUVHDJJUHJDWHDVDIUDFWLRQRIWKHWRWDODEVROXWH volume of aggregate.

The term LQ(TXDWLRQ LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW at the section divided by $FFRUGLQJWR$&,5 PD\EHWDNHQDVWKHVXPRIWKHDUHDVRIWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW ORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUV with one layer of tension reinforcement, like two-way slabs, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDW section, which has a width equal to Values of used to calculate DUHOLPLWHGWRSVLH[FHSWDVDOORZHGLQ$&,$&, 7KLV is primarily due to a lack of test data and practical experience with concrete having compressive limitation on VWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\LWLVW\SLFDOIRU LQWZRZD\VODEVWREHOHVVWKDQRUHTXDOWR SVL5HIHUHQFH  One-way shear rarely governs in a two-way slab system, so shear reinforcement for one-way shear is not required; thus, 7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@ (5.31) 5.4.4 Nominal Two-way Shear Strength Overview

For two-way shear, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&, 3URYLVLRQVWRGHWHUPLQHQRPLQDO two-way shear strengths for slabs without and with shear reinforcement are given below. Two-way Shear Strength Provided by Concrete in Slabs without Shear Reinforcement

)RUWZRZD\VODEV\VWHPVZLWKRXWVKHDUUHLQIRUFHPHQW(TXDWLRQ PXVWEHVDWLV¿HGDWWKHFULWLFDOVHFWLRQV The stress corresponding to the nominal two-way shear GH¿QHGLQ6HFWLRQRIWKLVSXEOLFDWLRQZKHUH LVWKHOHDVWRIWKHYDOXHVFDOFXODWHGE\WKHHTXDWLRQVLQ$&,7DEOH strength provided by the concrete, $&, 

(5.32)

5-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU LVGHWHUPLQHGE\(TXDWLRQ >$&,@DQG LVWKHPRGL¿FDWLRQ IDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOZHLJKWFRQFUHWHRIWKH VDPHFRPSUHVVLYHVWUHQJWKVHH7DEOHVDQG  The term is the ratio of the long to short dimensions of a rectangular column, concentrated load, or reaction area. The constant LVHTXDOWRIRULQWHULRUFROXPQVIRUHGJHFROXPQVDQGIRUFRUQHUFROXPQV5HIHUHQFHWR references the these types of columns does not suggest the actual location of a column in a building; instead, number of critical sections available to resist shear around a column. For example, LVHTXDOWRIRUWKHLQWHULRUFROXPQLQ)LJXUHDGMDFHQWWRDUHODWLYHO\ODUJHRSHQLQJLQWKLVFDVHRQO\VLGHVRIWKHFULWLFDOVHFWLRQDUH available to resist the shear stress. Additional information on the effects of openings in two-way slab systems is JLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ

’‡‹‰

”‹–‹…ƒŽ •‡…–‹‘

Ƚ௦ ൌ ͵Ͳ

–‡”‹‘” …‘Ž—

Figure 5.29 Effect of an opening on the critical section of an interior column.

Two-way Shear Strength Provided by Concrete in Slabs with Shear Reinforcement

For two-way slab systems with shear reinforcement (stirrups or headed shear stud reinforcement), is determined LQDFFRUGDQFHZLWK$&,(TXDWLRQVWRGHWHUPLQH at the various critical section locations are given in Table VHH$&,7DEOH  Table 5.22 Nominal Two-way Shear Strength Provided by Concrete in Slabs with Shear Reinforcement Shear Reinforcement

Critical Sections

Stirrups

$&,DQG )LJXUH

+HDGHGVKHDUVWXGUHLQIRUFHPHQW

$&, )LJXUH

$&, )LJXUH

5-40

vc

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU GHWHUPLQHGE\(TXDWLRQ LVSHUPLWWHGWREHWDNHQDVZKHUHRQHRI WKHIROORZLQJWZRFRQGLWLRQVLVVDWLV¿HG$&,  1. 6WLUUXSVGHVLJQHGDQGGHWDLOHGLQDFFRUGDQFHZLWK$&,DUHSURYLGHGZKHUH  6PRRWKKHDGHGVKHDUVWXGUHLQIRUFHPHQWZLWKDVWXGVKDIWOHQJWKOHVVWKDQRUHTXDOWRLQGHVLJQHGDQG GHWLOHGLQDFFRUGDQFHZLWK$&,DUHSURYLGHGZKHUH In these equations, is the total area of stirrup legs or headed shear stud reinforcement on one peripheral line geometrically similar to the perimeter of the column section within the spacing Information on shear reinforcement in two-way slabs is given below. Two-way Shear Strength Provided by Single- or Multiple-leg Stirrups

Two-way shear strength of two-way slabs can be increased by shear reinforcement consisting of properly anchored VLQJOHRUPXOWLSOHOHJVWLUUXSVVHH$&,DQG)LJXUH 6WLUUXSVDUHSHUPLWWHGWREHXVHGSURYLGHG of where is the diameter of the stirrups (ACI the two-way slab is greater than or equal to the larger of 6 in. or VHH)LJXUH 

ηͳ2db

κext ζ2 d

db (typ.)

κext

κext

κext

(a)

(b) 6d For #3–#5 stirrups, κext = greater of ൜ b 3"

(c)

Figure 5.30 Stirrup shear reinforcement for use in two-way slabs. (a) Single-leg stirrup. (b) Multiple-leg stirrup. (c) Closed stirrups.

 ƒ…‡‘ˆ…‘Ž—

‘˜‡”

݀௕ 

͸ԣ ݀ ൒ ‰”‡ƒ–‡”‘ˆ ൝

 ͳ͸݀௕

൑ ʹ݀

‘˜‡” –Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

൑ ݀ Τʹ

‫ ݏ‬൑ ݀ Τʹ

Figure 5.31 Cover, spacing, and depth requirements for closed stirrups.

5-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The nominal shear strength,

LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, (5.33)

In this equation, is the area of all stirrup legs on one peripheral line geometrically similar to the perimeter of the FROXPQVHFWLRQ)RUH[DPSOHDWWKHLQWHULRUFROXPQGHSLFWHGLQ)LJXUHVWLUUXSVDUHSURYLGHGSDUDOOHOWRHDFK column face. Along one peripheral line of stirrups, there are two vertical stirrup legs on each face, so for four faces, where is equal to the area of one stirrup leg. The term is the spacing of the peripheral lines of stirrups in the direction perpendicular to the column face. The ORFDWLRQDQGVSDFLQJRIWKHVWLUUXSVPXVWFRQIRUPZLWK$&,7DEOHVHH$&,DQG)LJXUH 0HWKods to determine DUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ Two-way Shear Strength Provided by Headed Shear Stud Reinforcement

+HDGHGVKHDUVWXGUHLQIRUFHPHQWFRQIRUPLQJWR$&,FDQEHXVHGDORQHRULQFRQMXQFWLRQZLWKVWLUUXSVGURS SDQHOVVKHDUFDSVRUFROXPQFDSLWDOVWRLQFUHDVHWKHQRPLQDOWZRZD\VKHDUVWUHQJWKRIWZRZD\VODEV$&,  $W\SLFDOKHDGHGVKHDUVWXGDUUDQJHPHQWLVVKRZQLQ)LJXUH/DUJHKHDGHGVWXGVDUHZHOGHGWRÀDWUDLOVDQG WKHDVVHPEO\LVSRVLWLRQHGRQFKDLUVDURXQGWKHFROXPQVDQGVXEVHTXHQWO\QDLOHGWRWKHIRUPZRUN7KHVL]HDQG spacing of the studs and the length of the base rail depend on the shear requirements.

‡ƒ†‡†•Š‡ƒ”•–—†

–‡‡Ž„ƒ•‡”ƒ‹Ž Figure 5.32 Headed shear stud reinforcement.



/LNHUHLQIRUFLQJEDUVVXI¿FLHQWFRYHUPXVWEHSURYLGHGWRSURWHFWWKHDVVHPEO\IURPZHDWKHU¿UHDQGRWKHUHIIHFWV VHH)LJXUH 0LQLPXPFRYHUUHTXLUHPHQWVIRUKHDGHGVKHDUVWXGUHLQIRUFHPHQWDUHJLYHQLQ$&, WKHPLQLPXPFRYHUWRWKHKHDGVDQGEDVHUDLOVLVWKHVDPHDVWKDWUHTXLUHGIRUWKHÀH[XUDOUHLQIRUFHPHQWLQDWZR ZD\VODEZKLFKLVJLYHQLQ$&,7DEOHIRUFDVWLQSODFHQRQSUHVWUHVVHGPHPEHUV)RUWZRZD\VODEVQRW H[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQGWKHPLQLPXPFRYHULVLQIRUDQGVPDOOHUEDUV7KHRYHUDOOKHLJKWRIWKHKHDGHGVKHDUVWXGDVVHPEO\PXVWFRQIRUPWRWKHUHTXLUHPHQWVLQ$&,

5-42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ƒ…‡‘ˆ…‘Ž—

–—†ሺ•Šƒሻ †‹ƒ‡–‡”

‘˜‡”

˜‡”ƒŽŽƒ••‡„Ž› Š‡‹‰Š–

‡ƒ††‹ƒ‡–‡”

ƒ‹ŽŠ‘Ž‡

ƒ‹Ž–Š‹…‡••

‘˜‡”

Žƒ„”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

൑ ݀ Τʹ ͵݀ ΤͶ™Š‡”‡‫ݒ‬௨ ൑ ߶͸ඥ݂௖ᇱ  ‫ݏ‬൑ቐ ݀ Τʹ ™Š‡”‡‫ݒ‬௨ ൐ ߶͸ඥ݂௖ᇱ  Figure 5.33 Cover and spacing requirements for headed shear stud reinforcement.

The nominal shear strength, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,(TXDWLRQ LVXVHG WKLVHTXDWLRQLVWKHVDPHDV$&,(TXDWLRQ IRUVWLUUXSV>VHH(TXDWLRQ @7KHPDLQ to determine is the cross-sectional area of all headed shear studs on one difference is that for the case of headed shear studs, SHULSKHUDOOLQHJHRPHWULFDOO\VLPLODUWRWKHSHULPHWHURIWKHFROXPQVHFWLRQ)RUWKHLQWHULRUFROXPQLQ)LJXUH where is the area of one headed shear stud. The term is the spacing of the peripheral lines of headed shear studs in the direction perpendicular to the column IDFH7KHORFDWLRQDQGVSDFLQJRIWKHKHDGHGVKHDUVWXGVPXVWFRQIRUPZLWK$&,7DEOHVHH$&, DQG)LJXUHVDQG 0HWKRGVWRGHWHUPLQH DUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ $&,(TXDWLRQ PXVWDOVREHVDWLV¿HGZKHUHKHDGHGVKHDUVWXGUHLQIRUFHPHQWLVSURYLGHG$&,  (5.34)

Summary of Nominal Two-way Shear Strength Requirements

7KHLQIRUPDWLRQLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHQRPLQDOWZRZD\VKHDUVWUHQJWK for two-way slab for systems without and with shear reinforcement. Included in this table are the maximum permitted values of two-way slabs with shear reinforcement, which are equal to the values of LQ$&,7DEOHGLYLGHGE\WKH strength reduction factor, $&, 

5-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.23 Nominal Two-way Shear Strength Requirements for Two-way Slabs Shear Reinforcement

Critical Sections

None

$&, )LJXUHV±

Nominal Strengths



$&, )LJXUH Stirrups

$&, )LJXUH

+HDGHG6KHDU6WXG Reinforcement

$&, )LJXUH

$&, )LJXUH

5-44





Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.4.5 Openings in Two-way Slab Systems

Openings for mechanical ductwork, electrical and plumbing risers, stairs, elevators, and other elements are comPRQO\UHTXLUHGLQWZRZD\VODEV\VWHPV7KHHIIHFWVRIVXFKRSHQLQJVPXVWEHFRQVLGHUHGLQGHVLJQ$&,  $FFRUGLQJWR$&,RSHQLQJVRIDQ\VL]HDUHSHUPLWWHGLQDWZRZD\VODEV\VWHPSURYLGHGDQDQDO\VLVRIWKH V\VWHPZLWKWKHRSHQLQJVLVSHUIRUPHGWKDWVKRZVDOOVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG)RUWZR ZD\VODEV\VWHPVZLWKRXWEHDPVVXFKDQDQDO\VLVQHHGQRWEHSHUIRUPHGZKHQWKHSURYLVLRQVRI$&,DUH VDWLV¿HGWKH¿UVWWKUHHRIWKHVHSURYLVLRQVDUHJLYHQLQ7DEOHVHH)LJXUH  Table 5.24 Requirements for Openings in Two-way Slab Systems in Accordance with ACI 8.5.4.2 Location

Maximum Opening Size

Additional Requirements

2SHQLQJVRIDQ\VL]HDUHSHUPLWWHG

Total quantity of reinforcement in the panel must be at least that required for the panel without the opening

Area common to intersecting columns strips

One-eighth the width of the column strip in either span

A quantity of reinforcement at least equal to that interrupted by an opening must be added to the sides of the opening

Area common to one column strip and one middle strip

Not more than one-fourth of the reinforcement in either strip must be interrupted by openings

A quantity of reinforcement at least equal to that interrupted by an opening must be added to the sides of the opening

Area common to intersecting middle strips

κଶ 

CS

κଵ 

MS

CS

κଵ 

MS

CS

κଶ 

CS

MS

CS

MS

CS

Any size opening permitted ሾACI 8.5.4.2ሺaሻሿ Maximum opening width ൌ ሺCS widthሻ/8 ሾACI 8.5.4.2ሺbሻሿ See ACI 8.5.4.2ሺcሻ

Figure 5.34 Permitted openings in two-way slab systems without beams in accordance with ACI 8.5.4.2.

5-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The closer an opening is to a column, the greater effect it has on the critical shear perimeter available to resist shear from the periphery of a column, concentrated load, forces. In particular, where an opening is located closer than RUUHDFWLRQDUHDWKHSURYLVLRQVRI$&,PXVWEHVDWLV¿HG>$&,G @7KHSHULPHWHURIWKHFULWLFDO section must be reduced by a length equal to the projection of the opening formed by two straight lines that extend from the centroid of the column, concentrated load, or reaction area and are tangent to the boundaries of the opening. The ineffective portions of DUHLOOXVWUDWHGLQ)LJXUHIRUDFROXPQLQDÀDWSODWHV\VWHPZLWKRSHQLQJVIRU systems without and with shear reinforcement.

൏ Ͷ݄ ሺ–›’Ǥ ሻ

‡ˆˆ‡…–‹˜‡’‘”–‹‘• ‘ˆܾ௢ 

‡ˆˆ‡…–‹˜‡’‘”–‹‘•‘ˆܾ௢ 

ሺƒሻ



ሺ„ሻ

Figure 5.35 Effect of openings on critical shear perimeter in two-way slabs. (a) Without shear reinforcement. (b) With shear reinforcement.

The section properties of the remaining effective segments of the critical section can be determined using Equations  DQG LQ6HFWLRQRIWKLVSXEOLFDWLRQ7KHVKHDUVWUHVVHVGXHWRWKHGLUHFWVKHDUIRUFHDQGWKHSRUWLRQ RIWKHPRPHQWWUDQVIHUUHGWRWKHFROXPQE\HFFHQWULFLW\RIVKHDUDUHFDOFXODWHGE\(TXDWLRQ XVLQJWKHUHGXFHG section properties of the critical section. The maximum shear stresses must be less than or equal to applicable design LQ7DEOH two-way shear strengths,

5.5 Determination of Required Reinforcement 5.5.1 Required Flexural Reinforcement

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW at a critical section in a column strip or middle strip in a two-way and for tension-controlled VODEV\VWHPLVGHWHUPLQHGEDVHGRQWKHUHTXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV VHFWLRQV8VLQJ(TXDWLRQV DQG WKHUHTXLUHG can be determined by the following equation: (5.35)

7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH

LVGH¿QHGDVIROORZV (5.36)

and for tension-controlled sections. Factored bending moments, LQ(TXDWLRQ FDQEHREWDLQHGDW WKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSVDQGPLGGOHVWULSVXVLQJWKH''0ZKHUHDSSOLFDEOHVHH7DEOHVWKURXJK  

5-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHDUHDRIÀH[XUDOUHLQIRUFHPHQWDWDQ\FULWLFDOVHFWLRQLQDWZRZD\VODEPXVWQRWEHOHVVWKDQ is equal to the thickness of the slab, times the width of the column strip or middle strip where the gross area (ACI 8.6.1.1). FDQDOVREHGHWHUPLQHGE\(TXDWLRQ ZKHUH 7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWLQWKHVODEZLGWK LQ(TXDWLRQ LVFDOFXODWHGXVLQJ >VHH(TXDWLRQ @,QWKLVFDVH is equal to the folORZLQJVHH$&,DQG6HFWLRQRIWKLVSXEOLFDWLRQ 

(5.37)

In this equation, for shear and is the factored shear stress on the slab critical section for two-way action from the controlling load combination without moment transfer: (5.38)

It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear for tension-controlled sections (see strain distribution where is the depth of the neutral axis when )LJXUH  (5.39)

Substituting LQWR(TXDWLRQ ZLWK IURP(TXDWLRQ WKHIROORZLQJHTXDWLRQFDQEHXVHGWRFDOcorresponding to tension-controlled sections for two-way slabs with FXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW

(5.40)

)RU*UDGHUHLQIRUFHPHQW SURSHUWLHV(TXDWLRQ UHGXFHVWRWKHIROORZLQJ

and for

For these material (5.41)

In these equations, is the width of the column strip or middle strip. If FDOFXODWHGE\(TXDWLRQ LVJUHDWHU the section is not tension-controlled , and must be increased accordingly to attain a tensionthan FRQWUROOHGVHFWLRQ$VLPLODUFKHFNPXVWEHSHUIRUPHGIRUWKHÀH[XUDOUHLQIRUFHPHQWZLWKLQ 5.5.2 Required Shear Reinforcement

:KHUHUHTXLUHGWKHVL]HDQGVSDFLQJRIVKHDUUHLQIRUFHPHQWIRUWZRZD\VODEVFDQEHGHWHUPLQHGXVLQJWKHLQIRUPDWLRQSURYLGHGLQ7DEOH

5-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Stirrups

3ULRUWRGHWHUPLQLQJWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHVWLUUXSVLWVKRXOGEHYHUL¿HGWKDWWKHPD[LPXPVKHDU in accordance with ACI Table stress, DWWKHFULWLFDOVHFWLRQGH¿QHGE\$&,LVOHVVWKDQRUHTXDOWR DQGRU must be increased accordingly to satisfy this requirement. :KHUH The total number of stirrup legs, provided on one peripheral line geometrically similar to the perimeter of the FROXPQGHSHQGRQWKHQXPEHURIVLGHVRIWKHFULWLFDOVHFWLRQVHH)LJXUH $VVXPLQJDVWLUUXSEDUVL]HZLWKDQ the following equation can be used to determine the required spacing, VHH7DEOH  area equal to

(5.42)

Stirrups can be terminated where the design strength of the concrete can resist the factored shear stress without the VHH)LJXUHDQG7DEOH  contribution of the stirrups; this occurs where Headed Shear Stud Reinforcement

3ULRUWRGHWHUPLQLQJWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHKHDGHGVKHDUVWXGUHLQIRUFHPHQWLWVKRXOGEHYHUL¿HG in that the maximum shear stress, DWWKHFULWLFDOVHFWLRQGH¿QHGE\$&,LVOHVVWKDQRUHTXDOWR DQGRU must be increased accordingly to satisfy this DFFRUGDQFHZLWK$&,7DEOH:KHUH requirement. provided on one peripheral line geometrically similar to the perimeter of The number of headed shear studs, WKHFROXPQGHSHQGRQ WKHQXPEHURIVLGHVRIWKHFULWLFDOVHFWLRQDQG WKHVSDFLQJUHTXLUHPHQWSDUDOOHOWRWKH the FROXPQIDFHLQ$&,7DEOHVHH)LJXUH $VVXPLQJDKHDGHGVKHDUVWXGZLWKDQDUHDHTXDOWR following equation can be used to determine the required spacing, VHH7DEOH 

(5.43)

+HDGHGVKHDUVWXGVFDQEHWHUPLQDWHGZKHUHWKHGHVLJQVWUHQJWKRIWKHFRQFUHWHFDQUHVLVWWKHIDFWRUHGVKHDUVWUHVV VHH)LJXUHDQG7DEOH without the contribution of the headed shear studs; this occurs where  

5-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VQRWHGSUHYLRXVO\$&,UHTXLUHVKHDGHGVKHDUVWXGUHLQIRUFHPHQWDQGVWXGDVVHPEOLHVFRQIRUPWR$670 is equal to $5HIHUHQFH $FFRUGLQJWRWKDWGRFXPHQWWKHPLQLPXP\LHOGVWUHQJWKRIWKHVWXGPDWHULDO SVL$OVRWKHDUHDRIDKHDGHGVKHDUVWXGVKRXOGEHREWDLQHGIURPPDQXIDFWXUHUV¶OLWHUDWXUH

5.6 Reinforcement Detailing 5.6.1 Concrete Cover

Reinforcing bars, stirrups, and headed shear stud assemblies are placed in two-way slabs with a minimum concrete FRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHUHIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&, $&, )RUUHLQIRUFLQJEDUVLQWZRZD\VODEVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQGWKHPLQLPXPFRYHULVHTXDOWRLQIRUDQGVPDOOHUEDUV$&,7DEOH &RQFUHWHFRYHULVPHDVXUHGIURP the surface of the concrete to the following: • 2XWHUPRVWOD\HURIÀH[XUDOUHLQIRUFHPHQWIRUWZRZD\VODEVZLWKRXWVKHDUUHLQIRUFHPHQW)LJXUH • Outermost surface of the stirrup bars for two-way slabs with shear reinforcement consisting of stirrups (Figure  • Underside of the base rail and outermost surface of the head for two-way slabs with shear reinforcement conVLVWLQJRIKHDGHGVKHDUVWXGV)LJXUH

Ž‡š—”ƒŽ”‡‹ˆ‘”…‡‡–ሺ–›’Ǥሻ

‘˜‡” ݀௕  ‫ۓ‬ ۖ

ͳ‹Ǥ

‹‹—…Ž‡ƒ”•’ƒ…‡൒ ‰”‡ƒ–‡”‘ˆ ݀௕   ‫۔‬ ۖ ‫ە‬ሺͶȀ͵ሻ݀௔௚௚

Figure 5.36 Minimum clear cover and spacing requirements for reinforcing bars in two-way slabs.

5.6.2 Minimum Spacing of Flexural Reinforcing Bars

0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&,$&,  7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJEDUV 7KHVSDFLQJUHTXLUHPHQWVDUHVXPPDUL]HGLQ)LJXUHIRUDWZRZD\VODEV\VWHPZKHUH PD[LPXPDJJUHJDWHVL]HLQWKHFRQFUHWHPL[

is the nominal

5.6.3 Maximum Spacing of Flexural Reinforcing Bars

([FHSWIRUWZRZD\MRLVWV\VWHPVPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIÀH[XUDOUHLQIRUFLQJEDUVLVHTXDOWRWKH and 18 in. at critical sections and DQGLQDWRWKHUVHFWLRQV$&, ,QDGGLWLRQWRFRQWUROlesser of ling crack widths, this requirement takes into consideration the effect that could be caused by loads concentrated on small areas of the slab. 5.6.4 Selection of Flexural Reinforcement

7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ DQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV JLYHQLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\

5-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In two-way slabs, LVFDOFXODWHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSVDQGPLGGOHVWULSV$EDUVL]HDQG is equal to or slightly greater than the required considering spacing should be selected so that the provided PLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVDQGPLQLPXPDQGPD[LPXPEDUVSDFLQJUHTXLUHPHQWV7KHÀH[XUDOUHLQIRUFLQJEDUVDUHXQLIRUPO\GLVWULEXWHGLQWKHGHVLJQVWULSV(FRQRP\LVJHQHUDOO\DFKLHYHGE\XVLQJWKHODUJHVWVL]H UHLQIRUFLQJEDUVVDWLVI\LQJVWUHQJWKDQGPD[LPXPVSDFLQJUHTXLUHPHQWVDQGWRUHSHDWEDUVL]HDQGVSDFLQJDVRIWHQ as possible. may need ,QRUGHUWRVDWLVI\(TXDWLRQ WKHSURYLGHGDUHDRIUHLQIRUFHPHQWLQWKHFROXPQVWULSZLWKLQ WREHLQFUHDVHGWKLVLVW\SLFDOO\DFKLHYHGE\DGGLQJUHLQIRUFLQJEDUVRIWKHVDPHVL]HDVWKRVHLQWKHFROXPQVWULS considering the requirements for bar spacing. Where additional reinforcing bars are required, the bars should be and a special detail must be provided that clearly illustrates the extent and location of uniformly spaced within the reinforcing bars. 5.6.5 Corner Restraint in Slabs

At exterior corners of slabs supported by stiff elements such as walls and edge beams with corner reinIRUFHPHQWPXVWEHSURYLGHGWKDWVDWLV¿HVWKHUHTXLUHPHQWVRI$&,7KLVUHLQIRUFHPHQWLVUHTXLUHGEHFDXVHWKH stiff elements restrain the slab and cause additional bending moments at these locations. Corner reinforcement must be provided in both the top and bottom of the slab, and the reinforcement in each layer equal to the largest positive bending moment per in each direction must be designed for a factored moment, unit width in the slab panel. The top and bottom reinforcement must be placed parallel and perpendicular to the GLDJRQDOUHVSHFWLYHO\IRUDGLVWDQFHRIDWOHDVWRQH¿IWKRIWKHORQJHURIWKHWZRVSDQOHQJWKVLQWKHFRUQHUSDQHO >VHH)LJXUHD @,QOLHXRIWKHGLDJRQDOUHLQIRUFHPHQWOD\RXWUHLQIRUFHPHQWSODFHGSDUDOOHOWRWKHHGJHVLV SHUPLWWHGVHH$&,DQG)LJXUHE @WKLVOD\RXWLVSUHIHUUHGFRPSDUHGWRWKHGLDJRQDOOD\RXWEHFDXVH WKHUHLQIRUFLQJEDUVDUHVLPSOHUWRSODFHLQWKH¿HOG,QHLWKHURSWLRQWKHPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIWKH corner reinforcement is equal to κ஺ 

κ஺ 

ሺaሻ

κ஺ /5 κ஻ 

Top bars

Top and bottom bars in both directions κ஻ 

Bottom bars

κ஺ /5 κ஺ /5

κ஺ /5

ሺbሻ

Notes 1.Applies to walls and edge beams where one or both edge beams has ߙ௙ ൐ 1.0 Ϯ͘κ஺ ൐ κ஻  3.Maximum ‫ ݏ‬ൌ 2݄ Figure 5.37 Required reinforcement at corners of slabs supported by stiff edge members. (a) Reinforcement parallel and perpendicular to the diagonal. (b) Reinforcement parallel to the slab edges.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.6.6 Termination of Flexural Reinforcement

5HTXLUHPHQWVIRUWHUPLQDWLRQORFDWLRQVRISRVLWLYHDQGQHJDWLYHÀH[XUDOUHLQIRUFHPHQWSHUSHQGLFXODUWRDGLVFRQWLQXRXVHGJHLQFROXPQDQGPLGGOHVWULSVDUHJLYHQLQ$&,)RUVODEVVXSSRUWHGGLUHFWO\RQHGJHVSDQGUHO  EHDPVFROXPQVDQGZDOOVWKHSRVLWLYHÀH[XUDOUHLQIRUFHPHQWPXVWH[WHQGLQWRWKHVXSSRUWLQJPHPEHUDWOHDVW LQIURPLWVIDFHVHH$&,DQG)LJXUH 6WUDLJKWKRRNHGRUKHDGHGEDUVPD\EHXVHGLQVXFKFDVHV +RRNHGRUKHDGHGUHLQIRUFLQJEDUVPXVWEHSURYLGHGDWWKHHQGVRIQHJDWLYHUHLQIRUFHPHQWDQGWKHUHLQIRUFHPHQW must be developed in tension at the face of the supporting member with development lengths determined in accorGDQFHZLWK$&,$&, ,QIRUPDWLRQRQKRZWRGHWHUPLQHWHQVLRQGHYHORSPHQWOHQJWKVIRUKRRNHGEDUV DQGIRUKHDGHGEDUVFDQEHIRXQGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHLQIRUPDWLRQLQWKDWVHFWLRQIRURQHZD\ slabs is also applicable to two-way slabs.



൒ κௗ௛ 

–ƒ†ƒ”†ͻͲǦ†‡‰ Š‘‘ሺ‘”Š‡ƒ†‡†„ƒ”ሻ

–”ƒ‹‰Š–ǡŠ‘‘‡†ǡ‘”Š‡ƒ†‡†„ƒ”• ƒ…‡‘ˆ•—’’‘”–‹‰‡„‡” ሺ•’ƒ†”‡Ž„‡ƒǡ…‘Ž—ǡ‘”™ƒŽŽሻ ൒ ͸ԣ ‡ƒ”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 5.38 Termination of flexural reinforcement in a two-way slab supported by an edge (spandrel) beam, column, or wall.

Anchorage of reinforcement is permitted within a slab where the slab is not supported by a spandrel beam or a wall DWDGLVFRQWLQXRXVHGJHRUZKHUHDVODEFDQWLOHYHUVEH\RQGWKHVXSSRUW$&,  )RUWZRZD\VODEV\VWHPVZLWKRXWEHDPVWKHÀH[XUDOUHLQIRUFHPHQWPXVWKDYHWKHPLQLPXPH[WHQVLRQVJLYHQLQ $&,)LJXUH$&,VHH)LJXUH WKHVHH[WHQVLRQVDUHDSSOLFDEOHWRV\VWHPVVXSSRUWLQJRQO\ for the negative reinforcing bars, which governs gravity loads. The purpose of the minimum extension length of LVOHVVWKDQDERXWLVWRKHOSLQWHUFHSWSRWHQWLDOSXQFKLQJVKHDUFUDFNVWKDWFDQIRUPLQUHODWLYHO\ where WKLFNWZRZD\VODEVIRUH[DPSOHLQWUDQVIHUVODEVDQGSRGLXPVODEVVHH$&,)LJXUH5 ,QRWKHUZRUGV PD\QRWEHVXI¿FLHQWWRLQWHUFHSWSRWHQWLDOSXQFKLQJVKHDU the prescribed minimum extension length of cracks in such cases. Where a two-way slab resists the effects from combined gravity and lateral forces, the bar lengths must be based on DQDO\VLVEXWPXVWQRWEHOHVVWKDQWKRVHJLYHQLQ$&,)LJXUH3UHFLVHORFDWLRQVRILQÀHFWLRQSRLQWVFDQQRW be explicitly found using approximate methods, such as the DDM Method, because they depend on the ratio of the panel dimensions, the ratio of live to dead load, and the continuity conditions at the edges of the panels. Also, other WKDQSHUIRUPLQJDPRUHUH¿QHGDQDO\VLVWKHUHLVQRH[SOLFLWZD\RIGHWHUPLQLQJWKHGLVWULEXWLRQRIDSSOLHGFRPELQHGORDGVWRWKHFROXPQDQGPLGGOHVWULSV$FRQVHUYDWLYHDSSURDFKLQVXFKFDVHVLVWRPDNHDWOHDVWSHUFHQW of the negative reinforcing bars in the column strip continuous; this is consistent with the requirement pertaining to WZRZD\VODEVLQLQWHUPHGLDWHPRPHQWIUDPHVVHH$&, 

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κ஺ଵ 

κ஺ଶ 

κ஻ଵ 

κ஻ଶ 

κ஺ଶ 

κ஺ଷ 

κ஻ଶ 

κ஻ଷ 

൒ ‫ିܣ‬௦ଵ Τ2 

κ஺ଷ  κ஻ଷ 

൒ ‫ିܣ‬௦ଶ Τ2 

൒ ‫ିܣ‬௦ଷ Τ2  ‫ିܣ‬௦ଷ 

‫ିܣ‬௦ଶ 

‫ିܣ‬௦ଵ  ൒ 2 bars conforming to ACI 8.7.4.2

Splices permitted in this region ሺtyp.ሻ

൒ 6ԣ κ௡ଵ 

κ௡ଶ 

κ௡ଷ 

Column Strip κ஼ଵ 

κ஼ଶ 

‫ܣ‬ା௦ଵ 

൒ max ሺ‫ܣ‬ା௦ଵ Τ2͕‫ܣ‬ା௦ଶ Τ2ሻ

൒ 6ԣ κ௡ଵ 

κ஼ଷ 

κ஼ଶ 

κ஽ଵ 

κ஼ଷ 

‫ܣ‬ା௦ଶ 

κ஽ଶ 

൒ 6ԣ

κ௡ଶ 

κ஽ଶ 

κ஽ଷ  κ௡ଷ 

Middle Strip Notes:

greater of 0.30κ௡ଵ and 5݀ for two-way slabs without drop panels 1. κ஺ଵ ൒ ቐ greater of 0.33κ௡ଵ and 5݀ for two-way slabs with drop panels greater of 0.30κ௡ଵ , 0.30κ௡ଶ , and 5݀ for two-way slabs without drop panels 2. κ஺ଶ ൒ ቐ

       

greater of 0.33κ௡ଵ , 0.33κ௡ଶ , and 5݀ for two-way slabs with drop panels greater of 0.30κ௡ଶ , 0.30κ௡ଷ , and 5݀ for two-way slabs without drop panels 3. κ஺ଷ ൒ ቐ greater of 0.33κ௡ଶ , 0.33κ௡ଷ , and 5݀ for two-way slabs with drop panels 4. κ஻ଵ ൒ 0.20κ௡ଵ 5. κ஻ଶ ൒ greater of 0.20κ௡ଵ and 0.20κ௡ଶ 6. κ஻ଷ ൒ greater of 0.20κ௡ଶ and 0.20κ௡ଷ 7. κ஼ଵ ൒ 0.22κ௡ଵ 8. κ஼ଶ ൒ greater of 0.22κ௡ଵ and 0.22κ௡ଶ 9. κ஼ଷ ൒ greater of 0.22κ௡ଶ and 0.22κ௡ଷ 10. κ஽ଵ ൑ 0.15κ௡ଵ 11. κ஽ଶ ൑ 0.15κ௡ଶ 12.κ஽ଷ ൑ 0.15κ௡ଷ

Figure 5.39 Minimum bar lengths for two-way slabs without beams.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.6.7 Splices of Reinforcement

6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQWZRZD\VODEVPXVWEHLQDFFRUGDQFHZLWK$&,$&, 7KHSULmary reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to WKHFRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG /DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQW7KH LQIRUPDWLRQSURYLGHGLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUVSOLFHVRIUHLQIRUFLQJEDUVLQRQHZD\VODEVLVDOVRDSplicable to two-way slabs. 5.6.8 Structural Integrity Reinforcement

6WUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVIRUFDVWLQSODFHWZRZD\VODEVDUHJLYHQLQ$&,7KHSXUpose of this reinforcement is to improve the redundancy and ductility in the structure so in the event of damage to DPDMRUVXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFWXUH will have a higher probability of maintaining overall stability. 7ZRUHTXLUHPHQWVDUHJLYHQLQ$&,VHH)LJXUH  (1) $  OOWKHERWWRPÀH[XUDOUHLQIRUFHPHQWLQWKHFROXPQVWULSVLQERWKGLUHFWLRQVPXVWEHFRQWLQXRXVRUVSOLFHGXVLQJPHFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQODSVSOLFHVLQDFFRUGDQFH ZLWK$&,6SOLFHVPXVWEHORFDWHGLQDFFRUGDQFHZLWK$&,)LJXUH   At least two of the bottom reinforcing bars in the column strips in both directions must pass within the region bounded by the longitudinal reinforcement of the column and must be anchored at exterior supports using hooks or headed bars. 5.6.9 Shear Reinforcement Details

5HLQIRUFHPHQWGHWDLOVIRUVWLUUXSVDQGKHDGHGVKHDUVWXGVDUHJLYHQLQ$&,DQGUHVSHFWLYHO\VHH6HFWLRQ RIWKLVSXEOLFDWLRQ 7KHVHGHWDLOVDUHJLYHQLQ)LJXUHVDQGIRUVWLUUXSVDQGLQ)LJXUHVDQG IRUKHDGHGVKHDUVWXGV

5.7 Design Procedure 7KHJHQHUDOGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIWZRZD\VODEVDVVXPLQJWKH DDM is used to determine the factored bending moments at the critical sections in the column and middle strips. ,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFFDQEHIRXQGLQ this chapter.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1

EŽ

zĞƐ

EŽ

zĞƐ

1 A

Figure 5.40 Design procedure for two-way slabs.

5-54

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

EŽ

zĞƐ

1

Detail the reinforcement in accordance with Sect. 5.6

Figure 5.40 Design procedure for two-way slabs (cont.).

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.8 Examples 5.8.1

Example 5.1 – Determination of Minimum Slab Thickness: Flat Plate System, Building #1 (Framing Option A)

'HWHUPLQHWKHPLQLPXPVODEWKLFNQHVVIRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWDW\SLFDOÀRRU DQG*UDGHUHLQIRUFHPHQW ZLWKLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum slab thickness based on serviceability requirements

The longest clear span length,

ACI 8.3.1.1

measured face-to-face of columns is equal to the following: Figure 5.23

For exterior panels without edge beams: Table 5.2

For interior panels:

7U\DLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHG in ACI 8.3.1.1(a). Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements

8VH)LJXUHWRGHWHUPLQHDSUHOLPLQDU\

based on gravity load effects.

Maximum

ACI Eq. (5.3.1b)

For the edge columns along column lines A or E, tributary area,

is equal to the following:

)RUWKHHGJHFROXPQVDORQJFROXPQOLQHVRUWULEXWDU\DUHD

is equal to the following:

5-56

Sect. 5.2.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine the area ratio based on the largest tributary area:

Figure 5.3

Therefore, 3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDLQVODEWKLFNQHVVLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK $¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVVHH6HFW  Because the columns and portions of the slab form the LFRS, the total two-way shear stress is due to the combination of gravity and lateral load effects. Comments.7RLOOXVWUDWHWKHLPSDFWFROXPQVL]HKDVRQWKHUHTXLUHGVODEWKLFNQHVVGHWHUPLQHDSUHOLPLQDU\VODE WKLFNQHVVDVVXPLQJLQVTXDUHHGJHFROXPQVLQVWHDGRIWKHLQVTXDUHFROXPQVXVHGLQWKLVH[DPSOH

For exterior panels without edge beams:

Figure 5.3

Therefore, ,QWKLVFDVHSUHOLPLQDU\FDOFXODWLRQVLQGLFDWHDLQVODEWKLFNQHVVLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDU strength. 5.8.2

Example 5.2 – Determination of Minimum Slab Thickness: Flat Plate System with Edge Beams, Building #1 (Framing Option B)

'HWHUPLQHWKHPLQLPXPVODEWKLFNQHVVIRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ% DWDW\SLFDOÀRRU ZLWKLQE\LQHGJHEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum slab thickness based on serviceability requirements

The longest clear span length,

ACI 8.3.1.1

measured face-to-face of columns is equal to the following: Figure 5.23

For exterior panels with edge beams: Table 5.2

5-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$FFRUGLQJWRWKHIRRWQRWHLQ7DEOHWKLVPLQLPXPVODEWKLFNQHVVLVEDVHGRQWKHDVVXPSWLRQ HGJHEHDPV&KHFNWKLVDVVXPSWLRQIRUWKHLQE\LQHGJHEHDPVDVVXPLQJDQLQWKLFNVODE

for the

Determine the effective slab width,

Eq. (5.5), Figure 5.2

Determine the section properties of the beam: Table 5.3

Determine the moment inertia of the slab using the largest the smallest

because this results in the maximum

and hence,

Therefore, assuming the same concrete mixture is used in the slab and beams Eq. (5.2)

Thus, the assumption that

is correct.

For interior panels: Table 5.2

7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHscribed in ACI 8.3.1.1(a). Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements

Sects. 5.3.3, 5.4.4

Because of the edge beams, two-way shear is not critical at the edge and corner columns. Check two-way shear requirements at an interior column assuming the lateral load effects are resisted by the exterior frames, that is, the interior columns are subjected to gravity load effects only. The two-way shear stress due to gravity load moment transfer at an interior column is not considered at this stage.

5-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum

ACI Eq. (5.3.1b)

Section properties of the critical section: Table 5.11, Case 1

Factored shear force at the critical section: Figure 5.12

Maximum shear stress, Eq. (5.14)

Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following for a square, interior column: Eq. (5.32)

where Eq. (5.30)

and Table 5.20

3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDVODEWKLFNQHVVHTXDOWRLQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK $¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVLQFOXGLQJWKH VKHDUVWUHVVGXHWRWKHEHQGLQJPRPHQWVWUDQVIHUUHGIURPWKHVODEVHH6HFW  5.8.3

Example 5.3 – Determination of Minimum Slab Thickness: Two-way Beam-Supported Slab System, Building #1 (Framing Option C)

Determine the minimum slab thickness for the two-way beam-supported slab system in Figure 1.1 (Framing Option & DWDW\SLFDOÀRRUZLWKLQE\LQEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum slab thickness based on serviceability requirements

The longest clear span length,

ACI 8.3.1.2

measured face-to-face of beams is equal to the following:

5-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The minimum slab thickness is determined on the basis of the stiffness ratio Because the slab thickness is not and cannot be determined. In lieu of assuming a slab thickness, a minimum slab thickknown at this stage, this assumption will be checked after the slab thickness has been calculated. ness is determined assuming

Table 5.6

7U\DLQVODEWKLFNQHVVDQGFDOFXODWH

for the beams and

for the panels.

• (GJHEHDPVRQFROXPQOLQHVDQG Determine the effective slab width,

Eq. (5.5), Figure 5.2

Determine the section properties of the beam: Table 5.3

Determine the moment inertia of the slab,

Therefore, Eq. (5.2)

• Edge beams on column lines A and E %HFDXVHWKHEHDPVL]HRQFROXPQOLQHVDQGLVWKHVDPHDVWKDWRQFROXPQOLQHV$DQG(

Therefore, Eq. (5.2)

5-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• ,QWHULRUEHDPVRQFROXPQOLQHVWKURXJK Determine the effective slab width,

Eq. (5.6), Figure 5.5

Determine the section properties of the beam: Table 5.3

Determine the moment inertia of the slab,

Therefore, Eq. (5.2)

• Interior beams along column lines B through D %HFDXVHWKHEHDPVL]HRQFROXPQOLQHVWKURXJKLVWKHVDPHDVWKDWRQFROXPQOLQHV%WKURXJK'

Therefore, Eq. (5.2)

Determine • • • •

for each type of panel:

Corner panel: Edge panel on column line A or E: (GJHSDQHORQFROXPQOLQHRU Interior panel:

Because

for all panels, the initial assumption is correct.

7U\DLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHG LQ$&,

5-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements

Sects. 5.3.3, 5.4.4

Because of the column-line beams, two-way shear in the slab is not critical at any of the columns. :KHUHWKH''0LVXVHGFROXPQOLQHEHDPVPXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHIDFWRUHGVKHDUIRUFHGXH 7KLVFULWHULRQLVVDWLV¿HGIRUDOOWKHEHDPVLQWKLVH[DPSOHZKLFKPHDQVWKH to gravity loads where IDFWRUHGVKHDUIRUFHVLQWKHVODEDURXQGWKHFROXPQVDUHHTXDOWR]HUR7KHEHDPVPXVWEHGHVLJQHGIRUWKHFRPbined effects due to gravity and lateral loads. 3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDVODEWKLFNQHVVHTXDOWRLQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK 5.8.4

Example 5.4 – Determination of Minimum Slab Thickness: Flat Slab System With Edge Beams, Building #1 (Framing Option D)

'HWHUPLQHWKHPLQLPXPVODEWKLFNQHVVIRUWKHÀDWVODEV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ' DWDW\SLFDOÀRRU ZLWKVWDQGDUGGURSSDQHOVLQE\LQHGJHEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQDQG*UDGHUHLQIRUFHPHQW crete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum slab thickness based on serviceability requirements

ACI 8.3.1.1

Check the provided drop panel dimensions: • At interior columns: Figure 5.4

• At edge and corner columns:

7KXVWKHUHTXLUHPHQWVRI$&,E DUHVDWLV¿HGIRUWKHSODQGLPHQVLRQVRIWKHGURSSDQHOV2QFHDVODEWKLFNness is determined, the projection of the drop panel below the slab must be equal to at least one-fourth of the slab WKLFNQHVVLQRUGHUWRVDWLVI\WKHUHTXLUHPHQWLQ$&,D  The longest clear span length,

measured face-to-face of columns is equal to the following: Figure 5.23

For exterior panels with edge beams: Table 5.4

$FFRUGLQJWRWKHIRRWQRWHLQ7DEOHWKLVPLQLPXPVODEWKLFNQHVVLVEDVHGRQWKHDVVXPSWLRQ HGJHEHDPV&KHFNWKLVDVVXPSWLRQIRUWKHLQE\LQHGJHEHDPVDVVXPLQJDQLQWKLFNVODE

for the

Determine the effective slab width,

Eq. (5.5), Figure 5.2

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine the section properties of the beam: Table 5.3

Determine the moment inertia of the slab using the largest the smallest

because this results in the maximum

and hence,

Therefore, assuming the same concrete mixture is used in the slab and beams Eq. (5.2)

Thus, the assumption that

is correct.

For interior panels: Table 5.4

7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHscribed in ACI 8.3.1.1(b). Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements

Sects. 5.3.3, 5.4.4

Because of the edge beams, two-way shear is not critical at the edge and corner columns. Check two-way shear requirements at an interior column assuming the lateral load effects are resisted by the exterior frames, that is, the interior columns are subjected to gravity load effects only. The two-way shear stress due to gravity load moment transfer at an interior column is not considered at this stage. • Check the shear stress at the critical section located a distance equal to

from the face of the column. Figure 5.4

)RUIRUPZRUNHFRQRP\VHOHFWDLQGURSSDQHOKHLJKW

Table 5.5

5-63

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum

ACI Eq. (5.3.1b) Figure 5.13

Section properties of the critical section: Table 5.11, Case 1

Factored shear force at the critical section:

Maximum shear stress, Eq. (5.14)

Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following for a square, interior column: Eq. (5.32)

where Eq. (5.30)

and Table 5.20

• Check the shear stress at the critical section located a distance equal to

from the face of the drop panel. Figure 5.13

Section properties of the critical section: Table 5.11, Case 1

Factored shear force at the critical section: Figure 5.13

5-64

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum shear stress, Eq. (5.14)

3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDVODEWKLFNQHVVHTXDOWRLQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK $¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVLQFOXGLQJWKH VKHDUVWUHVVIURPWKHEHQGLQJPRPHQWVWUDQVIHUUHGIURPWKHVODEVHH6HFW  5.8.5

Example 5.5 – Determination of Minimum Thickness: Two-way Joist System, Building #1 (Framing Option E)

Determine the minimum overall thickness for the two-way joist system in Figure 1.1 (Framing Option E) at a typiFDOÀRRUDVVXPLQJDIWPRGXOHDLQWKLFNVODEDQGLQE\LQFROXPQV$OVRDVVXPHQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the minimum overall thickness based on serviceability requirements

ACI 8.3.1.1

7ZRZD\MRLVWV\VWHPVDUHFRQVLGHUHGDVÀDWVODEVIRUSXUSRVHVRIGHVLJQ

Sect. 5.2.5

The longest clear span length,

measured face-to-face of columns is equal to the following: Figure 5.23

For exterior panels without edge beams: Table 5.4

$LQGRPHGHSWKSOXVDLQVODESURYLGHVDQHTXLYDOHQWWKLFNQHVV

Table 5.9

The equivalent thickness is determined as follows:

Table 5.8, Figure 5.7

where

and

Eq. (5.8)

7U\DLQGRPHSOXVDLQVODEIRUDOOSDQHOV

5-65

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine a preliminary overall thickness based on two-way shear strength requirements

Check the shear stress at the critical section located a distance equal to

Maximum

Sects. 5.3.3, 5.4.4

from the face of an interior column.

ACI Eq. (5.3.1b)

Section properties of the critical section at an interior column: Table 5.11, Case 1

Factored shear force at the critical section:

Maximum shear stress, Eq. (5.14)

Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following for a square, interior column: Eq. (5.32)

where Eq. (5.30)

and Table 5.20

3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDQRYHUDOOWKLFNQHVVRILQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK $¿QDORYHUDOOWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWDOOFROXPQV%His cause the two-way joist system is also part of the LFRS and is subjected to lateral load effects, the maximum larger than that calculated above for gravity loads only. 5.8.6

Example 5.6 – Determination of Minimum Slab Thickness: Flat Plate System, Building #3

'HWHUPLQHWKHPLQLPXPVODEWKLFNQHVVIRUWKHÀDWSODWHV\VWHPLQ)LJXUHDWWKHVHFRQGÀRRUOHYHOZLWK LQE\LQLQWHULRUDQGFRUQHUFROXPQVDQGLQE\LQHGJHFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW 5-66

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the minimum slab thickness based on serviceability requirements

The longest clear span length,

ACI 8.3.1.1

is equal to the following for an exterior panel: Figure 5.23

For exterior panels without edge beams: Table 5.2

For interior panels:

7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVVOLJKWO\OHVVWKDQWKDWUHTXLUHGE\$&,DQGLV JUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHGLQ$&,D  Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements

Sects. 5.3.3, 5.4.4

&KHFNWZRZD\VKHDUUHTXLUHPHQWVDWDLQE\LQHGJHFROXPQEHQGLQJSHUSHQGLFXODUWRWKHHGJHIRUJUDYLW\ loads only (the shear walls in the building resist lateral load effects).

Maximum

ACI Eq. (5.3.1b)

7KHPD[LPXPFRPELQHGVKHDUVWUHVVRFFXUVDWHGJHFROXPQV%&%DQG&EHQGLQJSHUSHQGLFXODUWRWKH edge. Section properties of the critical section: Table 5.11, Case 3

Factored shear force at the critical section:

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Assuming the DDM can be used, the required

at this location is equal to

Sect. 5.3.4

Eq. (5.17)

ACI Eq. (8.4.4.2.2)

Maximum shear stress, Eq. (5.14)

Eq. (5.32)

where 3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDVODEWKLFNQHVVHTXDOWRLQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK $¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVVHH6HFW  5.8.7

Example 5.7 – Determination of Required Flexural Reinforcement: Flat Plate System, Building #1 (Framing Option A), SDC A

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHÀDW SODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQGLQE\LQ FROXPQVVHH)LJXUH $VVXPHWKH6LWH&ODVVLV&ZKLFKUHVXOWVLQWKHEXLOGLQJEHLQJDVVLJQHGWR6'&$>VHH DQG*UDGHUHLQIRUFHPHQW ([DPSOH3DUWD @$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

5-68

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

A

E

ᇳ

D

ᇱ

23 -6 ሺtyp.ሻ

Design strip

A

N

A

C

A

B

A

6ᇱ -7½ᇳ ½MS

11ᇱ -9ᇳ CS

6ᇱ -7½ᇳ ½MS

Figure 5.41 Design strips for the flat plate system in Example 5.7.

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Sect. 5.3.4

Check the following limitations:

Figure 5.22

• 7  KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK south and east-west directions, respectively. • Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction. • Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG« • &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset. • All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP • 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG« DVVXPLQJDQLQWKLFNVODE • )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV «7KHUHDUHQREHDPVLQWKLVÀRRUV\VWHPVRWKLV in the two perpendicular directions: limitation is not applicable. %HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

%HFDXVHWKHÀDWSODWHLVSDUWRIWKH/)56VHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVDUHGHWHUPLQHGZKLFKZLOO be combined with those due to the effects from lateral forces using the appropriate load combinations: Dead load:

Eq. (5.17)

Live load: These total factored static moments are the same for all spans of the interior design strip in the direction of analysis. Step 4 – Distribute the total factored static moment to the column strips and middle strips

Table 5.16

'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWSODWHV\VWHPZLWKRXWHGJHEHDPVDUHJLYHQLQ7DEOH$VXPPDU\RIWKH service dead and live load bending moments in the column strip and middle strip in an end span and interior span is JLYHQLQ7DEOH Table 5.25 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Flat Plate System in Example 5.7 End Span Design Strip

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Column strip

Middle strip



Step 5 – Determine the bending moments due to wind forces

ACI 6.6

'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ 7DEOH $WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFHDQGDOLQHDU¿UVWRUGHUDQDO\VLVZDV SHUIRUPHGXVLQJWKHQRUWKVRXWKZLQGORDGVLQ7DEOHDSSOLHGWRWKHFHQWURLGRIWKHEXLOGLQJIDFHDWWKHURRIDQG ÀRRUOHYHOVVHH)LJXUH ,QWKHPRGHOWKHFROXPQVDUH¿[HGDWWKHEDVH

5-70

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

AA

BA

CA

DA

EA

23ᇱ -6ᇳ ሺtyp.ሻ 27.4 kips 4 @ 11ᇱ -6ᇳ ൌ 46ᇱ -0ᇳ

R 53.4 kips 5 51.6 kips 4 49.2 kips 3

14ᇱ -0ᇳ

51.3 kips 2

Figure 5.42 Total wind loads applied to Building #1.

The following reduced moments of inertia are used to account for cracked sections:

ACI Table 6.6.3.1.1(a)

• Columns: • Slabs: $¿QLWHHOHPHQWDQDO\VLVZDVSHUIRUPHGWRGHWHUPLQHWKHH[WHQWRYHUZKLFKWKHVODEUHVLVWVWKHHIIHFWVIURPWKH wind loads assuming all the frames in the north-south direction are part of the LFRS. From the analysis, it was determined the majority of the wind load effects at an interior design strip are resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus, all the wind load effects are assigned to the column strips in the direction of analysis in this example. 7KHEHQGLQJPRPHQWVLQWKHFROXPQVWULSGXHWRZLQGORDGVDUHJLYHQLQ7DEOHIRUHQGDQGLQWHULRUVSDQVDW WKHVHFRQGÀRRUOHYHO7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWK the north direction and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). Table 5.26 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Flat Plate System in Example 5.7 End Span Design Strip

Column strip

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

—

—

Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUWD  the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered. ,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH less than those due to wind forces, and, thus, need not be considered in this example.

5-71

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 7 – Determine the combined factored bending moments due to gravity and wind forces

ACI Table 5.3.1

7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKHEHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLV$&, ,WLVHYLdent at all locations that the maximum moments are equal to those from the combination of the gravity load effects. Table 5.27 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in Example 5.7 End Span Load Combination

Location

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative









SSR

83.8



SSL

83.8







SSR





SSL









Column strip Middle strip Column strip

Middle strip Column strip

Interior Span

Middle strip







Step 8 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations where is the width of the column strip or middle strip and Eq. (5.35)

Eq. (5.36)

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected VRWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQWWKH provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area of FRUUHVSRQGLQJWRWHQVLRQFRQWUROOHGVHFWLRQVZKLFKLVGHWHUPLQHGE\(T WKHUHÀH[XUDOUHLQIRUFHPHQW IRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,

5-72

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.28 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in Example 5.7

b (in.)

As (in.2)*

Reinforcement*

































































Mu (ft-kips)

Location

Exterior Negative Column strip

Positive First Interior Negative Exterior Negative

End Span Middle strip

Positive



First Interior Negative Column strip

Positive



Negative

Interior Span Middle strip

Positive Negative



* Min. As for the column strip Min. As for the middle strip Max. spacing For For For the column strip: For the middle strip:

Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

• Edge columns Determine the required reinforcement based on

Sect. 5.3.4

This moment is greater than 86.9 ft-kips, which is the moment due to the load combination

Table 5.27

Eq.(5.9)

where Table 5.11, Case 3

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH

The moment

must be transferred over the effective slab width Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQVHH7DEOH )RUV\PPHWU\DGGEDUVDQG check the bar spacing: )RUZLWKLQWKHLQZLGWK )RUZLWKLQWKH 7KHUHIRUHSURYLGHDGGLWLRQDOEDUVZLWKLQWKHLQZLGWKWRVDWLVI\PD[LPXPVSDFLQJUHTXLUHPHQWV$WRWDO RIEDUVDUHUHTXLUHGDWWKHHGJHFROXPQVZLWKLQWKHFROXPQVWULSZLWKRIWKHEDUVFRQFHQWUDWHG ZLWKLQDZLGWKRILQ Check

within

ACI 8.6.1.2

Table 5.11, Case 3

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

and Table 5.20

5-74

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (5.37)

Provided

within

Table 5.26

Therefore,

Eq. (5.38)

Provided

within

7KHUHIRUHQRDGGLWLRQDOUHLQIRUFHPHQWLVUHTXLUHGWRVDWLVI\WKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&, 5HLQIRUFHPHQWGHWDLOVIRUWKHWRSEDUVDWWKHHGJHFROXPQVDUHJLYHQLQ)LJXUH ͳͳ̵Ǧͻԣ Ͷ̵ǦͶԣ ʹ̵ǦͲԣ

͵Ǧ͓ͷ

͹Ǧ͓ͷ

͵Ǧ͓ͷ

Figure 5.43 Reinforcement details for the top reinforcing bars at the edge columns in the flat plate system in Example 5.7.

5-75

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Interior columns Determine the required reinforcement based on the largest transfer moment at the interior columns. In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to gravity load effects and combined gravity and wind load effects can be determined from the following equations where the spans in the direction of analysis and perpendicular to the direction of analysis are equal. For gravity loads only: Eq. (5.21)

)RUJUDYLW\DQGZLQGORDGVDWWKH¿UVWLQWHULRUFROXPQV due to gravity load effects due to wind load effects

Table 5.26

Total For gravity and wind loads at the interior column: due to gravity load effects due to wind load effects Total Use

at all interior columns. Eq. (5.9)

where Table 5.11, Case 1

Determining whether The moment

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH must be transferred over the effective slab width Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

5-76

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKDXQLIRUPEDUVSDFLQJLQWKHFROXPQVWULSRIWKH EDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHPHQWLVUHTXLUHGWRVDWLVI\PRPHQWWUDQVIHUUHTXLUHPHQWVDWWKHVHORFDWLRQV6LPLODUO\RIWKHEDUVDUH ZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQVRQRDGGLWLRQDOUHLQIRUFHPHQWLVUHTXLUHGWR satisfy moment transfer requirements at this location. Check

within

ACI 8.6.1.2

Table 5.11, Case 1

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

and Table 5.20

Eq. (5.37)

$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG At the interior column, provided

within

within

Based on this load combination, additional reinforcement must be provided within PXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,

to satisfy the mini-

5-77

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$WWKH¿UVWLQWHULRUFROXPQV

Table 5.26

At the interior column, Therefore, maximum shear stress is equal to the following:

Eq. (5.38)

Eq. (5.37)

The required from this load combination is less than the required combination, so required

from the

load

To satisfy minimum reinforcement requirements within EDUVPXVWEHSODFHGZLWKLQWKH DWWKH¿UVWLQWHULRUFROXPQV in. effective slab width width of the column strip. For ZLWKWKHUHPDLQLQJEDUVWREHSODFHGZLWKLQWKH V\PPHWU\DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDW WKH¿UVWLQWHULRUFROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOVIRUWKLVFDVHDUHVKRZQLQ)LJXUH

ͳͳԢǦͻԣ ͶԢǦͶԣ

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Figure 5.44 Reinforcement details for the top reinforcing bars at the first interior columns in the flat plate system in Example 5.7.

5-78

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS Step 10 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQ)LJXUHFDQQRWEHXVHGLQWKHFROXPQVWULSEHFDXVHRIWKHHIIHFWVGXH WRZLQGORDGV7RDFFRXQWIRUWKHZLQGORDGHIIHFWVSHUFHQWRIWKHWRSEDUVLQWKHFROXPQVWULSVDUHPDGHFRQWLQXRXVRYHUWKHVSDQV,QWKLVFDVHWKHPD[LPXPDPRXQWRIWRSUHLQIRUFHPHQWRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWV EDUV VREDUVDUHPDGHFRQWLQXRXV 7KHUHPDLQLQJEDUVLQWKHFROXPQVWULSDQGWKHEDUVLQWKHPLGGOHVWULSFDQEHWHUPLQDWHGDWWKHORFDWLRQVLGHQWL¿HG LQ)LJXUHWKLV¿JXUHDOVRLQFOXGHVWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&, A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH

5-79

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

ͳͶǦ͓ͷ

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ʹǦ͓ͷ

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൒ ͸ԣ ʹͳᇱ Ǧ͸ᇱᇱ 

ʹͳᇱ Ǧ͸ᇱᇱ 

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ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ  ͷᇱ ǦͲᇱᇱ 

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‹††Ž‡–”‹’ Figure 5.45 Reinforcement details for an interior design strip in the north-south direction. for the flat plate system in Example 5.7.

5.8.8

Example 5.8 – Determination of Required Flexural Reinforcement: Flat Plate System With Edge Beams, Building #1 (Framing Option B), SDC A

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHÀDWSODWH V\VWHPLQ)LJXUH)UDPLQJ2SWLRQ% DWWKHVHFRQGÀRRUOHYHOZLWKDQLQWKLFNVODELQE\LQHGJH EHDPVDQGLQE\LQFROXPQVVHH)LJXUH $VVXPHWKH6LWH&ODVVLV&$OVRDVVXPHQRUPDOZHLJKWFRQDQG*UDGHUHLQIRUFHPHQW crete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

5-80

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

A

E

23ᇱ -6ᇳ ሺtyp.ሻ

D

Design strip

A

N

A

C

A

B

A

6ᇱ -7½ᇳ ½MS

11ᇱ -9ᇳ CS

6ᇱ -7½ᇳ ½MS

Figure 5.46 Design strips for the flat plate system in Example 5.8.

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Sect. 5.3.4

Check the following limitations:

Figure 5.22

• 7  KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK south and east-west directions, respectively. • Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction. • Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG« • &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset. • All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP • 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG« DVVXPLQJDQLQWKLFNVODE • )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV «7KHUHDUHRQO\SHULPHWHUEHDPVLQWKLVÀRRU in the two perpendicular directions: system, so this limitation is not applicable.

5-81

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips. Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

7KHÀDWSODWHV\VWHPLVQRWSDUWRIWKH/)56RQO\WKHFROXPQVDQGHGJHEHDPVIRUPWKH/)56 WKHUHIRUHLWPXVW resist the effects from gravity loads only. Maximum

ACI Eq. (5.3.1b)

Total factored static moment Eq. (5.17)

The total factored static moment is the same for all spans of the interior design strip in the direction of analysis. Step 4 – Distribute the total factored static moment to the column strips and middle strips

Table 5.17

'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWSODWHV\VWHPZLWKHGJHEHDPVDUHJLYHQLQ7DEOH7KHWDEXODWHGH[WHULRU where is determined QHJDWLYHFROXPQVWULSPRPHQWFRHI¿FLHQWVDUHDSSOLFDEOHLQFDVHVZKHUHWKHWHUP by the following equation: Eq. (5.18)

Eq. (5.5), Figure 5.2

Cross-sectional constant

is determined by the following equation: Eq. (5.19)

Calculations for

DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW

Therefore, with the slab and beam cast monolithically with the same concrete mixture (that is,

5-82

):

8.5ᇳ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

24.0ᇳ 

15.5ᇳ 

24.0 24.0ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3

‫ܥ‬஺ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

 ൅ ቈ൬1 െ 0.63

8.5ᇳ 

Case A

8.5 8.5ଷ ൈ 15.5 ൰ൈ ቉ ൌ 61,428 in.ସ  15.5 3

15.5ᇳ 

15.5ᇳ 

‫ܥ‬஻ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

15.5 15.5ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3

 ൅ ቈ൬1 െ 0.63

Case B

8.5 8.5ଷ ൈ 43.5 ൰ൈ ቉ ൌ 30,444 in.ସ  43.5 3

Figure 5.47 Calculations for cross-sectional constant C.

Because

, the exterior negative column strip bending moment is equal to the following: Table 5.17

A summary of the design bending moments in the column strip and middle strip in an end span and in an interior VSDQLVJLYHQLQ7DEOH Table 5.29 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in Example 5.8 End Span Design Strip

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Column strip Middle strip

Step 5 – Determine the bending moments due to wind forces

'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ 7DEOH Because it has been assumed all the wind load effects are resisted by the perimeter moment-resisting frames, the VODEVUHVLVWRQO\JUDYLW\ORDGHIIHFWVZKLFKDUHJLYHQLQ7DEOH

5-83

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUWD  the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered. ,WLVDOVRDVVXPHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH resisted by the perimeter moment-resisting frames, and, thus, need not be applied to the slabs. Step 7 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations where is the width of the column strip or middle strip and Eq. (5.35)

Eq. (5.36)

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected VRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQW the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area FRUUHVSRQGLQJWRWHQVLRQFRQWUROOHGVHFWLRQVZKLFKLVGHWHUPLQHGE\(T  RIÀH[XUDOUHLQIRUFHPHQW WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&, Table 5.30 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in Example 5.8

Mu (ft-kips)

Location

Column strip End Span Middle strip

Interior Span

* Min. As for the column strip Min. As for the middle strip Max. spacing For For For the column strip: For the middle strip:

5-84

Column strip Middle strip

Exterior Negative Positive First Interior Negative Exterior Negative Positive First Interior Negative Positive Negative Positive Negative





 

b (in.)

As (in.2)*

Reinforcement*









3.33



























   

   

   

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

• Edge columns The factored slab moments resisted by the edge columns need not be checked in this example because of the beams at the perimeter of the slab. These edge beams must be designed for shear forces and torsional moments transferred from the slab. • Interior columns Determine the required reinforcement based on the largest transfer moment at the interior columns. In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.21) Eq. (5.9)

where Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH must be transferred over the effective slab width

The moment

Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKDXQLIRUPVSDFLQJRIEDUVLQWKHFROXPQVWULSRI WKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQforcement is required to satisfy moment transfer requirements at these locations. Similar calculations show that QRDGGLWLRQDOUHLQIRUFHPHQWLVUHTXLUHGDWWKHLQWHULRUFROXPQZKHUHEDUVDUHSURYLGHGLQWKHFROXPQVWULS Check

within

ACI 8.6.1.2 Table 5.11, Case 1

Factored shear force at the critical section:

5-85

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore, Eq. (5.38)

where Eq. (5.30)

and Table 5.20

Eq. (5.37)

$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG At the interior column, provided

within

within

Therefore, additional reinforcement must be provided within PHQWVRI$&,

to satisfy the minimum reinforcement require-

EDUVPXVWEHSODFHGZLWKLQWKHLQHITo satisfy minimum reinforcement requirements within DWWKH¿UVWLQWHULRUFROXPQVZLWKWKH fective slab width width of the column strip. For symmetry, UHPDLQLQJEDUVWREHSODFHGZLWKLQWKH DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH¿UVWLQWHULRU FROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOIRUWKLV FDVHLVVLPLODUWRWKDWVKRZQLQ)LJXUH 6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS)RUV\PPHWU\DGGEDUWRWKH LQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKHLQWHULRUFROXPQZLWKEDUV FRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK Step 9 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only. A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows: ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW

5-86

ACI Table 25.4.2.5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH)RUVLPSOHUGHWDLOLQJWKH OHQJWKVRIWKHQHJDWLYHUHLQIRUFHPHQWDWDOOFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDUHVHWHTXDOWRSHUFHQWRIWKH VHH)LJXUH 7KHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&, clear span length, which is greater than DUHLQFOXGHG ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

ʹͳǦ͓ͷ

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ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ  ͷᇱ ǦͲᇱᇱ 

ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ 

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ͳͲǦ͓ͷ ͷǦ͓ͷ

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ͷǦ͓ͷ

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‹††Ž‡–”‹’

Figure 5.48 Reinforcement details for an interior design strip in the north-south direction for the flat plate system in Example 5.8.

5-87

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.8.9

Example 5.9 – Determination of Required Flexural Reinforcement: Two-way Beam-Supported Slab System, Building #1 (Framing Option C), SDC A

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHWZRZD\ EHDPVXSSRUWHGVODEV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ& DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODE LQE\LQEHDPVDQGLQE\LQFROXPQVVHH)LJXUH $VVXPHWKH6LWH&ODVVLV&$OVRDVVXPHQRUDQG*UDGHUHLQIRUFHPHQW malweight concrete with 1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

A

E

23ᇱ -6ᇳ ሺtyp.ሻ

D

Design strip

A

N

A

C

A

B

A

6ᇱ -7½ᇳ ½MS

11ᇱ -9ᇳ CS

6ᇱ -7½ᇳ ½MS

Figure 5.49 Design strips for the two-way beam-supported slab system in Example 5.9.

'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

5-88

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Check the following limitations:

Sect. 5.3.4

Figure 5.22

• 7  KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK south and east-west directions, respectively. • Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction. • Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG« • &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset. • All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP • 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG« DVVXPLQJDLQWKLFNVODE • )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPVLQ …Check the relative stiffnesses for interior, edge, and the two perpendicular directions: DUHGHWHUPLQHGLQ([DPSOH corner panels where Interior panel: North-south interior beam: East-west interior beam:

Edge panel (north or south face): North-south interior beam: East-west edge beam:

Edge panel (east or west face): North-south edge beam: East-west interior beam:

Corner panel: North-south edge beam: East-west edge beam:

%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips.

5-89

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

The slab in this system is not part of the LFRS (only the columns and beams form the LFRS); therefore, the slab must resist the effects from gravity loads only. Maximum

ACI Eq. (5.3.1b)

Total factored static moment Eq. (5.17)

The total factored static moment is the same for all spans of an interior design strip in the direction of analysis. Step 4 – Distribute the total factored static moment to the column strips and middle strips

7RWDOGHVLJQPRPHQWVDUHGHWHUPLQHGXVLQJ7DEOHDQGWKHLQIRUPDWLRQLQ6WHSRI6HFW$VXPPDU\RI WKHWRWDOGHVLJQVWULSPRPHQWVLVJLYHQLQ7DEOH Table 5.31 Total Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Beam-Supported Slab System in Example 5.9 End Span

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Determine the percentages for factored bending moments in the column strip:

Table 5.15

• Exterior support: Interior beams in north-south direction:

Example 5.3

Eq. (5.18)

Eq. (5.5), Figure 5.2

5-90

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Cross-sectional constant

is determined by the following equation: Eq. (5.19)

DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW 7.0ᇳ 

Calculations for

24.0ᇳ 

17.0ᇳ 

24.0 24.0ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3

‫ܥ‬஺ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

 ൅ ቈ൬1 െ 0.63

7.0ᇳ 

Case A

7.0 7.0ଷ ൈ 17.0 ൰ൈ ቉ ൌ 60,791 in.ସ  17.0 3

17.0ᇳ 

17.0ᇳ 

‫ܥ‬஻ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

 ൅ ቈ൬1 െ 0.63

Case B

17.0 17.0ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3 7.0 7.0ଷ ൈ 45.0 ൰ൈ ቉ ൌ 32,956 in.ସ  45.0 3

Figure 5.50 Calculations for cross-sectional constant C.

With the slab and beam cast monolithically with the same concrete mixture (that is,

):

Therefore,

• Positive:

• Interior support:

5-91

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine percentages for factored bending moments in the middle strip: • Exterior support: • Positive: • Interior support: Determine percentages for factored bending moments in the column-line beams: WKHEHDPVPXVWUHVLVWRIWKHWRWDOFROXPQVWULSPRPHQWV

Because • Exterior support: • Positive: • Interior support:

A summary of the design bending moments in the column strip and middle strip in an end span and interior span is JLYHQLQ7DEOH Table 5.32 Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Beam-Supported Slab System in Example 5.9 End Span Design Strip

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

Total Total Column Beam strip Slab Middle strip The following illustrates the calculation of the design bending moments at the exterior negative support:

The bending moments at the other critical sections can be obtained in a similar fashion. Step 5 – Determine the bending moments due to wind forces

ACI 6.6

'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ 7DEOH Because it has been assumed all the wind load effects are resisted by the moment-resisting frames along all the colXPQOLQHVWKHVODEVUHVLVWRQO\JUDYLW\ORDGHIIHFWVZKLFKDUHJLYHQLQ7DEOH Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUWD  the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.

5-92

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,WLVDOVRDVVXPHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH resisted by the moment-resisting frames, and, thus, need not be applied to the slabs. Step 7 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations: Eq. (5.35)

Eq. (5.36)

where

is the width of the column strip or middle strip and

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOHIRUWKHVODE7KHSURYLGHGDUHDVRIUHLQforcement are greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars DUHVHOHFWHGVRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&, It is evident the provided areas of reinforcement at all critical sections in the column strip and middle strip are less corresponding to tension-controlled sections, which is determined by WKDQWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW (T WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&, Table 5.33 Required Flexural Reinforcement at the Second-Floor Level for the Two-way Beam-Supported Slab System in Example 5.9

b (in.)

As (in.2)*

Reinforcement*













First Interior Negative







Exterior Negative











































Mu (ft-kips)

Location

Exterior Negative Column strip End Span Middle strip

Positive

Positive





First Interior Negative Column strip Interior Span Middle strip

Positive



Negative Positive Negative



* Min. As for the column strip Min. As for the middle strip Max. spacing For For For the column strip: For the middle strip:

5-93

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

The factored slab moments resisted by the columns need not be checked in this example because of the column-line beams. These beams must be designed for shear forces and torsional moments transferred from the slab. Step 9 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only. A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH)RUVLPSOHUGHWDLOLQJWKH OHQJWKVRIWKHQHJDWLYHUHLQIRUFHPHQWDWDOOFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDUHVHWHTXDOWRSHUFHQWRIWKHFOHDU 7KHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,DUHLQFOXGHG span length, which is greater than

5-94

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

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Figure 5.51 Reinforcement details for an interior design strip in the north-south direction for the two-way beam-supported slab system in Example 5.9.

5.8.10 Example 5.10 – Determination of Required Flexural Reinforcement: Flat Slab System With Edge Beams, Building #1 (Framing Option D), SDC A

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHÀDWVODE V\VWHPLQ)LJXUH)UDPLQJ2SWLRQ' DWWKHVHFRQGÀRRUOHYHOZLWKDQLQWKLFNVODELQE\LQHGJH EHDPVLQE\LQFROXPQVDQGDLQGURSSDQHOSURMHFWLRQEHORZWKHVODEVHH)LJXUH $VVXPHWKH DQG*UDGHUHLQIRUFHPHQW Site Class is C. Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

5-95

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ 8ᇱ -6ᇳ ሺtyp.ሻ

E

A

8ᇱ -6ᇳ ሺtyp.ሻ

ᇳ

D

ᇱ

23 -6 ሺtyp.ሻ

Design strip

A

N

C

A

5ᇱ -3ᇳ ሺtyp.ሻ

A

B

A

6ᇱ -7½ᇳ ½MS

11ᇱ -9ᇳ CS

6ᇱ -7½ᇳ ½MS

Figure 5.52 Design strips for the flat slab system in Example 5.10.

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Sect. 5.3.4

Check the following limitations:

Figure 5.22

• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK south and east-west directions, respectively. • Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction. • Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG« • &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset. • All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP • 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG« DVVXPLQJDQLQWKLFNVODE • )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV «7KHUHDUHRQO\SHULPHWHUEHDPVLQWKLVÀRRU in the two perpendicular directions: system, so this limitation is not applicable.

5-96

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips. Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

7KHÀDWVODEV\VWHPLVQRWSDUWRIWKH/)56RQO\WKHFROXPQVDQGHGJHEHDPVIRUPWKH/)56 WKHUHIRUHWKHÀDW slab must resist the effects from gravity loads only. Maximum

ACI Eq. (5.3.1b)

Total factored static moment Eq. (5.17)

The total factored static moment is the same for all spans of the interior design strip in the direction of analysis. Step 4 – Distribute the total factored static moment to the column strips and middle strips

Table 5.17

'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWVODEV\VWHPZLWKHGJHEHDPVDUHJLYHQLQ7DEOH7KHWDEXODWHGH[WHULRU where is determined QHJDWLYHFROXPQVWULSPRPHQWFRHI¿FLHQWVDUHDSSOLFDEOHLQFDVHVZKHUHWKHWHUP by the following equation: Eq. (5.18)

Eq. (5.5), Figure 5.2

Cross-sectional constant

is determined by the following equation: Eq. (5.19)

Calculations for

DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW

Therefore, with the slab and beam cast monolithically with the same concrete (that is,

):

5-97

8.0ᇳ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

24.0ᇳ 

16.0ᇳ 

24.0 24.0ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3

‫ܥ‬஺ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

 ൅ ቈ൬1 െ 0.63

8.0ᇳ 

Case A

8.0 8.0ଷ ൈ 16.0 ൰ൈ ቉ ൌ 61,222 in.ସ  16.0 3

16.0ᇳ 

16.0ᇳ 

‫ܥ‬஻ ൌ ቈ൬1 െ 0.63 28.0ᇳ 

16.0 16.0ଷ ൈ 28.0 ൰ൈ ቉ 28.0 3

 ൅ ቈ൬1 െ 0.63

Case B

8.0 8.0ଷ ൈ 44.0 ൰ൈ ቉ ൌ 31,116 in.ସ  44.0 3

Figure 5.53 Calculations for cross-sectional constant C.

Because

, the exterior negative column strip bending moment is equal to the following: Table 5.17

A summary of the design bending moments in the column strip and middle strip in an end span and interior span is given in Table 5.34. Table 5.34 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Slab System in Example 5.10 End Span Design Strip

1 Exterior Negative

Interior Span

2

3

4

5

Positive

First Interior Negative

Positive

Interior Negative

Column strip Middle strip Step 5 – Determine the bending moments due to wind forces

ACI 6.6

Design wind forces in the north-south direction on Building #1 are determined in Example 3.1 and are given in Table 3.10. Because it has been assumed all the wind load effects are resisted by the perimeter moment-resisting frames, the slabs resist only gravity load effects, which are given in Table 5.34.

5-98

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUWD  the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered. ,WLVDOVRDVVXPHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH resisted by the perimeter moment-resisting frames, and, thus, need not be applied to the slabs. Step 7 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations: Eq. (5.35)

Eq. (5.36)

where

is the width of the column strip or middle.

,WLVVKRZQLQ([DPSOHWKDWWKHGLPHQVLRQVRIWKHGURSSDQHOVDWLVI\WKHUHTXLUHPHQWVRI$&,7KHUHIRUHDW negative critical sections in the column strip:

Figure 5.28

where At all other critical sections, $VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH greater than or equal to then the minimum required in accordance with ACI 8.6.1.1 and the number of bars are VHOHFWHGVRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLV evident the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than corresponding to tension-controlled sections, which is determined by Eq. WKHDUHDRIÀH[XUDOUHLQIRUFHPHQW  WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,

5-99

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.35 Required Flexural Reinforcement at the Second-Floor Level for the Flat Slab System in Example 5.10

Mu (ft-kips)

Location

Exterior Negative Column strip

Positive



First Interior Negative Exterior Negative

End Span Middle strip

Positive



First Interior Negative

Interior Span

Column strip Middle strip

Positive



Negative Positive Negative



b (in.)

d (in.)

As (in.2)*

Reinforcement*

















































































* Min. As for the column strip at negative critical sections Min. As for the column strip at positive critical sections Min. As for the middle strip Max. spacing For For For the column strip at negative critical sections: For the column strip at positve critical sections: For the middle strip:

Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

• Edge columns The factored slab moments resisted by the edge columns need not be checked in this example because of the beams at the perimeter of the slab. These edges beams must be designed for shear forces and torsional moments transferred from the slab. • Interior columns Determine the required reinforcement based on the largest transfer moment at the interior columns. transferred to an interior column due to the In lieu of a more exact analysis, the factored slab moment, gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.21) Eq. (5.9)

5-100

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

where Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH

The moment

must be transferred over the effective slab width

Eq. (5.11)

Determine the required Eq. (5.36)

Eq. (5.35)

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKWKHEDUVXQLIRUPO\VSDFHGLQWKHFROXPQVWULSRI WKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVRQRDGGLWLRQDOUHLQIRUFHment is required to satisfy moment transfer requirements at these locations. Similar calculations show that no additional reinforcement is required at the interior column. Check

within

Determine the governing factored two-way shear stress,

ACI 8.6.1.2

at the interior columns. Table 5.11, Case 1

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

and Table 5.20

5-101

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Thus,

Eq. (5.37)

$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG At the interior column, provided

within

within

Additional reinforcement must be provided in 

to satisfy the minimum reinforcement requirements of ACI

EDUVPXVWEHSODFHGZLWKLQWKHLQHITo satisfy minimum reinforcement requirements within DWWKH¿UVWLQWHULRUFROXPQVZLWKWKH fective slab width width of the column strip. For symmetry, UHPDLQLQJEDUVWREHSODFHGZLWKLQWKH DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH¿UVWLQWHULRU FROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOIRUWKLV FDVHLVVLPLODUWRWKDWVKRZQLQ)LJXUH 6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS Step 9 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only. A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

5-102

ACI Table 25.4.2.5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH)RUVLPSOHUGHWDLOLQJWKH OHQJWKVRIWKHQHJDWLYHUHLQIRUFHPHQWDWDOOFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDUHVHWHTXDOWRSHUFHQWRIWKH VHH)LJXUH 7KHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&, clear span length, which is greater than DUHLQFOXGHG ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

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ʹͳᇱ Ǧ͸ᇱᇱ 

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͵ᇱ ǦͲᇱᇱ 

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‹††Ž‡–”‹’ †‰‡„‡ƒ•ƒ††”‘’’ƒ‡Ž•‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 5.54 Reinforcement details for an interior design strip in the north-south direction for the flat slab system in Example 5.10.

5-103

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.8.11 Example 5.11 – Determination of Required Flexural Reinforcement: Two-way Joist System, Building #1 (Framing Option E), SDC A

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHHDVWZHVWGLUHFWLRQIRUWKHWZRZD\ MRLVWV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ( DWWKHVHFRQGÀRRUOHYHOZLWKDIWPRGXOHDLQWKLFNVODED LQGRPHDQGLQE\LQFROXPQVVHH)LJXUH $VVXPHWKH6LWH&ODVVLV&$OVRDVVXPHQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 1A

2A

3A ᇱ

4A

ᇳ

A

50 -0 ሺtyp.ሻ

A

18ᇱ -6ᇳ

N

A

ᇱ

31 -4

ᇳ

B

7ᇱ -10ᇳ ½MS

15ᇱ -6ᇳ

ᇱ

31 -4

ᇳ

C

15ᇱ -8ᇳ CS

A

Design strip

ᇱ

31 -4

ᇳ

7ᇱ -10ᇳ ½MS

D

A

Figure 5.55 Design strips for the two-way joist system in Example 5.11.

'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Sect. 5.3.4

Check the following limitations:

Figure 5.22

•

There must be 3 or more continuous spans in each direction…There are 3 continuous spans in the north-south and east-west directions.

5-104

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

•

Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction.

•

Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG«

•

&ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset.

•

All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP

•

7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG«

•

)RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV «7KHUHDUHQREHDPVLQWKLVÀRRUV\VWHPVRWKLV in the two perpendicular directions: limitation is not applicable.

%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips. Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

Because the two-way joist system is part of the LFRS, service dead and live load bending moments are determined, which will be combined with those due to the effects from lateral forces: Dead load:

Eq. (5.17)

Live load: These total factored static moments are the same for all spans of the interior design strip in the direction of analysis. Step 4 – Distribute the total factored static moment to the column strips and middle strips

Table 5.16

$VQRWHGLQ6HFWDWZRZD\MRLVWV\VWHPLVFRQVLGHUHGWREHDÀDWVODEIRUSXUSRVHVRIGHVLJQ'HVLJQPRPHQW FRHI¿FLHQWVIRUDÀDWVODEV\VWHPZLWKRXWHGJHEHDPVDUHJLYHQLQ7DEOH$VXPPDU\RIWKHVHUYLFHGHDGDQGOLYH ORDGEHQGLQJPRPHQWVLQWKHFROXPQVWULSDQGPLGGOHVWULSLQDQHQGVSDQDQGLQWHULRUVSDQLVJLYHQLQ7DEOH

5-105

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.36 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Two-way Joist System in Example 5.11 End Span

1

Design Strip

Exterior Negative

Interior Span

2

3

4

5

Positive

First Interior Negative

Positive

Interior Negative

Column strip



Middle strip

Step 5 – Determine the bending moments due to wind forces

ACI 6.6

'HVLJQZLQGIRUFHVLQWKHHDVWZHVWGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ7DEOH  $WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFHDQGDOLQHDU¿UVWRUGHUDQDO\VLVZDV SHUIRUPHGXVLQJWKHHDVWZHVWZLQGORDGVLQ7DEOHDSSOLHGWRWKHFHQWURLGRIWKHEXLOGLQJIDFHDWWKHURRIDQG ÀRRUOHYHOVVHH)LJXUH ,QWKHPRGHOWKHFROXPQVDUH¿[HGDWWKHEDVH ͳA

ʹA

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Ͷ

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ʹ

Figure 5.56 Total wind loads applied to Building #1.

The following reduced moments of inertia are used to account for cracked sections:

ACI Table 6.6.3.1.1(a)

• Columns: • Two-way joists: $¿QLWHHOHPHQWDQDO\VLVZDVSHUIRUPHGWRGHWHUPLQHWKHH[WHQWRYHUZKLFKWKHVODEV\VWHPUHVLVWVWKHHIIHFWVIURP the wind loads. From the analysis, it was determined the majority of the wind load effects at an interior design strip are resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus, all the wind load effects are assigned to the column strips in the direction of analysis in this example.

5-106

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHEHQGLQJPRPHQWVLQWKHFROXPQVWULSGXHWRZLQGORDGVDUHJLYHQLQ7DEOHIRUWKHHQGDQGLQWHULRUVSDQV 7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQG the west direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the south elevation of the building). Table 5.37 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Two-way Joist System in Example 5.11 End Span

1

Design Strip

Exterior Negative

Interior Span

2

3

4

5

Positive

First Interior Negative

Positive

Interior Negative

ņ

Column strip

ņ

Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUWD  the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered. ,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH less than those due to wind forces, and, thus, need not be considered in this example. Step 7 – Determine the combined factored bending moments due to gravity and wind forces

ACI Table 5.3.1

7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKHEHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLV$&, ,WLVHYLdent that at all locations, the maximum moments are equal to those from the combination of the gravity load effects. Table 5.38 Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Joist System in Example 5.11 End Span Load Combination

Location

1 Exterior Negative

First Interior Negative

4 Positive







SSR





SSL









SSR





SSL









Middle strip

Middle strip Column strip

Positive

3



Column strip

Column strip

2

Interior Span

Middle strip







5 Interior Negative

5-107

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 8 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations: Eq. (5.35)

Eq. (5.36)

,QWKLVH[DPSOHWKHUHDUHULEVLQWKHFROXPQVWULSDQGULEVLQWKHPLGGOHVWULSULEVDUHLQHDFKRIWKHKDOI middle strips). • &ROXPQVWULSÀH[XUDOUHLQIRUFHPHQW is resisted by the solid area of concrete At the negative critical sections, the total negative factored moment is determined using and (the width around the column, and RIWKHVROLGKHDGSHUSHQGLFXODUWRWKHGLUHFWLRQRIDQDO\VLVLVLQZKLFKLVLQOHVVWKDQWKHZLGWKRIWKH FROXPQVVWULSWKHLQZLGWKLVXVHGWRFDOFXODWH ). At the positive critical sections, the total positive factored moment is determined for each rib using the 6 ribs, and

is shared equally between and

• 0LGGOHVWULSÀH[XUDOUHLQIRUFHPHQW At the negative critical sections, the total negative factored moment and mined using

LVUHVLVWHGE\ULEVDQG

At the positive critical sections, the total positive factored moment is determined per rib using and

is deter-

LVVKDUHGHTXDOO\EHWZHHQWKHULEV and

7KHPLQLPXPUHTXLUHGÀH[XUDOUHLQIRUFHPHQWDWQHJDWLYHFULWLFDOVHFWLRQVLQWKHFROXPQDQGPLGGOHVWULSVLV equal to the following: ACI 8.6.1.1

$WDOORWKHUFULWLFDOVHFWLRQVWKHPLQLPXPUHTXLUHGÀH[XUDOUHLQIRUFHPHQWFDQEHGHWHUPLQHGXVLQJWKHUHTXLUHments for beams:

ACI 9.6.1.2

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOHZKLFKLVDSSOLFDEOHIRUERWKWKHHQG and interior spans. It can be determined all sections are tension-controlled.

5-108

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.39 Required Flexural Reinforcement at the Second-Floor Level for the Two-way Joist System in Example 5.11

b (in.)

As (in.2)*

Reinforcement*

Exterior Negative







Positive





ULE

First Interior Negative













Positive





ULE

First Interior Negative







Mu (ft-kips)

Location

Column strip

Exterior Negative Middle strip

* Min. As for negative moment in the column strip Min. As for negative moment in the middle strip minimum number of bars Min. As for all other critical sections



;

Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements

• Edge columns Determine the required reinforcement based on

ACI 8.4.2.2

Sect. 5.3.4

7KLVPRPHQWLVJUHDWHUWKDQIWNLSVZKLFKLVWKHPRPHQWGXHWRWKHORDGFRPELQDWLRQ

Table 5.38

Eq. (5.9)

where Table 5.11, Case 3

Determining whether The moment

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH must be transferred over the effective slab width Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

5-109

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQVHH7DEOH )RUV\PPHWU\DGGEDUDQG check bar spacing: )RUZLWKLQWKHLQZLGWK )RUZLWKLQWKH Check

within

, ACI 8.6.1.2

Table 5.11, Case 3

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

Table 5.20

Eq. (5.37)

Provided within quired based on this load combination.

, so no additional reinforcement is re-

Table 5.37

5-110

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

Eq. (5.38)

Provided

within

No additional reinforcement is required to satisfy moment transfer requirements at the edge columns. • Interior columns Determine the required reinforcement based on the largest transfer moment at the interior columns. In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects and gravity and wind load effects can be determined from the following equations where the spans in the direction of analysis and perpendicular to the direction of analysis are equal. For gravity loads only: Eq. (5.21)

For gravity and wind loads: due to gravity load effects due to wind load effects

Table 5.37

Total Therefore, use

at the interior columns. Eq. (5.9)

where Table 5.11, Case 1

Determining whether The moment

can be increased to the values in A&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH must be transferred over the effective slab width Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

5-111

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKWKHEDUVXQLIRUPO\VSDFHGZLWKLQWKHFROXPQVWULS RIWKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHment is required to satisfy moment transfer requirements at this location. Check

within

ACI 8.6.1.2

Table 5.11, Case 1

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

and Table 5.20

Eq. (5.37)

Table 5.37

Therefore, the maximum shear stress is equal to the following:

Eq. (5.38)

Eq. (5.37)

5-112

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Thus, the Provided

load combination governs with respect to minimum reinforcement requirements. within

Thus, no additional reinforcement is required to satisfy moment transfer requirements at the interior columns. Step 10 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQWKHLQWKLFNVODEVKRZQLQ)LJXUHFDQQRWEHXVHGLQWKHFROXPQ VWULSEHFDXVHRIWKHHIIHFWVGXHWRZLQGORDGV7RDFFRXQWIRUWKHZLQGORDGHIIHFWVSHUFHQWRIWKHWRSEDUVLQWKH column strips are made continuous over the spans. In this case, the maximum amount of top reinforcement occurs at WKH¿UVWLQWHULRUVXSSRUWEDUV VREDUVDUHPDGHFRQWLQXRXV The remaining bars in the column strip and the top bars in the middle strip can be terminated at the locations identi¿HGLQ)LJXUH According to ACI 8.8.1.6, at least one bottom bar in each rib must be continuous and must be anchored to develop at the faces of the supports for structural integrity. The other bottom bars can be cutoff in the column strips and PLGGOHVWULSVLQDFFRUGDQFHZLWKWKHOHQJWKVVKRZQLQ)LJXUH A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5-113

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH ʹᇱ Ǧ͸ᇱᇱ 

ͳͷᇱ Ǧͻᇱᇱ 

ʹᇱ Ǧ͸ᇱᇱ 

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‘Ž—–”‹’

ʹᇱ Ǧ͸ᇱᇱ 

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ʹᇱ Ǧ͸ᇱᇱ 

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ʹͳǦ͓ͷ

ʹͳǦ͓ͷ

൒ ͸ᇱᇱ  ʹǦ͓ͺȀ”‹„ ͻᇱ ǦͲᇱᇱ  ൒ ͸ԣ

ͺᇱ ǦͲᇱᇱ  Ͷ͹ᇱ Ǧ͸ᇱᇱ 

‹††Ž‡–”‹’ Figure 5.57 Reinforcement details for an interior design strip in the east-west direction for the two-way joist system in Example 5.11.

5.8.12 Example 5.12 – Determination of Required Flexural Reinforcement: Flat Plate System, Building #1 (Framing Option A), SDC B

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHÀDW SODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQGLQE\LQ and FROXPQVVHH)LJXUH $VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW

5-114

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the widths of the column strip and middle strip

ACI 8.4.1.5, 8.4.1.6

Figure 5.8

Step 2 – Check if the Direct Design Method can be used to determine gravity load bending moments

Sect. 5.3.4

Check the following limitations:

Figure 5.22

• 7  KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK south and east-west directions, respectively. • Successive panel lengths measured center-to-center of supports in each direction must not differ by more than one-third the longer span…All the panel lengths are equal in each direction. • Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of VXSSRUWVQRWWRH[FHHG« • &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ centerlines of successive columns…None of the columns are offset. • All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP • 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG« DVVXPLQJDQLQWKLFNVODE • )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV «7KHUHDUHQREHDPVLQWKLVÀRRUV\VWHPVRWKLV in the two perpendicular directions: limitation is not applicable. %HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG middle strips. Step 3 – Determine the total factored static moment in each span

Figure 5.8 Figure 5.23

%HFDXVHWKHÀDWSODWHLVSDUWRIWKH/)56VHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVDUHGHWHUPLQHGZKLFKZLOO be combined with those due to the effects from lateral forces: Dead load:

Eq. (5.17)

Live load: These total factored static moments are the same for all spans of the interior design strip in the direction of analysis. 5-115

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Distribute the total factored static moment to the column strips and middle strips

Table 5.16

'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWSODWHV\VWHPZLWKRXWHGJHEHDPVDUHJLYHQLQ7DEOH$VXPPDU\RIWKH service dead and live load bending moments in the column strip and middle strip in an end span and interior span is JLYHQLQ7DEOH Table 5.40 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Flat Plate System in Example 5.12 End Span Design Strip

1 Exterior Negative

Interior Span

2

3

4

5

Positive

First Interior Negative

Positive

Interior Negative

Column strip

Middle strip



Step 5 – Determine the bending moments due to wind forces

ACI 6.6

'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ 7DEOH $WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFHDQGDOLQHDU¿UVWRUGHUDQDO\VLVZDV SHUIRUPHGXVLQJWKHQRUWKVRXWKZLQGORDGVLQ7DEOHDSSOLHGWRWKHFHQWURLGRIWKHEXLOGLQJIDFHDWWKHURRIDQG ÀRRUOHYHOVVHH)LJXUH ,QWKHPRGHOWKHFROXPQVDUH¿[HGDWWKHEDVH The following reduced moments of inertia are used to account for cracked sections:

ACI Table 6.6.3.1.1(a)

• Columns: • Slabs: $¿QLWHHOHPHQWDQDO\VLVZDVSHUIRUPHGWRGHWHUPLQHWKHH[WHQWRYHUZKLFKWKHVODEUHVLVWVWKHHIIHFWVIURPWKH wind loads. From the analysis, it was determined the majority of the wind load effects at an interior design strip are resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus, all the wind load effects are assigned to the column strips in the direction of analysis in this example. The bending moments in the column strip due to wind loads in an end span and interior span are given in Table 7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHQRUWKGLUHFtion and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building).

5-116

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.41 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Flat Plate System in Example 5.12 End Span Design Strip

1 Exterior Negative

Interior Span

2

3

4

5

Positive

First Interior Negative

Positive

Interior Negative

ņ

Column strip

ņ

Step 6 – Determine the bending moments due to seismic forces

$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV'IRUWKLVEXLOGLQJ)URP([DPSOH3DUWE  the building is assigned to SDC B for this case. Therefore, effects due to seismic forces must be considered. 'HVLJQVHLVPLFIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ 7DEOH 7KHWKUHHGLPHQVLRQDOPRGHOGHVFULEHGLQ6WHSZDVXVHGWRSHUIRUPWKHOLQHDU¿UVWRUGHUDQDO\VLVXVLQJWKH QRUWKVRXWKVHLVPLFORDGVLQ7DEOHZKLFKDUHDSSOLHGWRWKHFHQWHURIPDVVDWHDFKOHYHOULJLGGLDSKUDJPVDUH assigned at each level). Like in the case of wind loads, the majority of the seismic load effects at an interior design strip are resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus, all the seismic load effects are assigned to the column strips in the direction of analysis in this example. The bending moments in the column strip due to seismic loads in an end span and interior span are given in Table 7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHVHLVPLFORDGVFDQDFWLQERWKWKHQRUWK direction and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). Table 5.42 Bending Moments (ft-kips) due to Seismic Loads at the Second-Floor Level for the Flat Plate System in Example 5.12 End Span Design Strip

Column strip

Interior Span

1

2

3

4

5

Exterior Negative

Positive

First Interior Negative

Positive

Interior Negative

ņ

ņ

,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH less than those due to wind and seismic forces, and, thus, need not be considered in this example. &RPSDULQJ7DEOHVDQGLWLVHYLGHQWWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHODUJHUWKDQWKRVH GXHWRZLQGORDGHIIHFWV%HFDXVHERWKW\SHVRIHIIHFWVDUHDVVLJQHGDORDGIDFWRUHTXDOWRLQWKHGHVLJQORDG combinations, only the bending moments due to the seismic load effects are considered in the remainder of this example. Step 7 – Determine the combined factored bending moments due to gravity and seismic forces

ACI Table 5.3.1

7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHVHLVPLFORDG FRPELQDWLRQVDUHJLYHQLQ7DEOHIRU6'&%7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKH EHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLVVHH$&, 

5-117

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.43 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in Example 5.12 End Span Load Combination

Location

1

2

3

4

Exterior Negative

Positive

First Interior Negative

Positive









SSR

83.8



SSL

83.8























Column strip Middle strip Column strip

Middle strip Column strip

Interior Span

SSR



SSL

Middle strip



5 Interior Negative

Step 8 – Determine the required flexural reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ the following equations where is the width of the column strip or middle strip and Eq. (5.35)

Eq. (5.36)

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected VRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQW the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area FRUUHVSRQGLQJWRWHQVLRQFRQWUROOHGVHFWLRQVZKLFKLVGHWHUPLQHGE\(T  RIÀH[XUDOUHLQIRUFHPHQW WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,1RWHWKDWWKHERWWRPEDUV LQWKHHQGVSDQUHTXLUHGIRUWKHIDFWRUHGSRVLWLYHPRPHQWDUHDOVRDGHTXDWHWRUHVLVWWKHIWNLSSRVLWLYHPRPHQW DWWKHIDFHRIWKHH[WHULRUVXSSRUWGXHWRPRPHQWUHYHUVDOVHH7DEOH 

5-118

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.44 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in Example 5.12

b (in.)

As (in.2)*

Reinforcement*



3.11





























































Mu (ft-kips)

Location

Exterior Negative Column strip

Positive First Interior Negative Exterior Negative

End Span Middle strip

Positive



First Interior Negative Column strip

Positive



Negative

Interior Span Middle strip

Positive Negative



* Min. As for the column strip Min. As for the middle strip Max. spacing For For For the column strip: For the middle strip:

Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

• Edge columns Determine the reinforcement based on the largest transfer moment at the edge columns: Table 5.44

Eq. (5.9)

where Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH

5-119

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The moment

must be transferred over the effective slab width Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQVHH7DEOH )RUV\PPHWU\DGGEDUDQG check the bar spacing: )RUZLWKLQWKHLQZLGWK )RUZLWKLQWKH 7KHUHIRUHSURYLGHDGGLWLRQDOEDUVZLWKLQWKHLQZLGWKWRVDWLVI\PD[LPXPVSDFLQJUHTXLUHPHQWV$WRWDO RIEDUVDUHUHTXLUHGDWWKHHGJHFROXPQVZLWKLQWKHFROXPQVWULSZLWKRIWKHEDUVFRQFHQWUDWHG ZLWKLQDZLGWKRILQ Check

within

ACI 8.6.1.2

Table 5.11, Case 3

Factored shear force at the critical section:

Therefore, Eq. (5.38)

where Eq. (5.30)

Table 5.20

5-120

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (5.37)

Provided

within

Based on this load combination, no additional reinforcement must be provided to satisfy the minimum reinIRUFHPHQWUHTXLUHPHQWVRI$&,

Table 5.42

Therefore,

Eq. (5.38)

Provided

within

5HLQIRUFHPHQWGHWDLOVIRUWKHWRSEDUVDWWKHHGJHFROXPQVDUHWKHVDPHDVWKRVHVKRZQLQ)LJXUH • Interior columns Determine the required reinforcement based on the largest transfer moment at the interior columns. In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects and gravity and seismic load effects can be determined from the following equations where the spans in the direction of analysis and perpendicular to the direction of analysis are equal. For gravity loads only: Eq. (5.21)

)RUJUDYLW\DQGVHLVPLFORDGVDWWKH¿UVWLQWHULRUFROXPQV due to gravity load effects due to seismic load effects

Table 5.42

Total For gravity and seismic loads at the interior column: due to gravity load effects

5-121

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

due to seismic load effects Total Therefore, use

at all interior columns. Eq. (5.9)

where Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH must be transferred over the effective slab width

The moment

Eq. (5.10)

Determine the required Eq. (5.36)

Eq. (5.35)

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKXQLIRUPEDUVSDFLQJLQWKHFROXPQVWULSRIWKH EDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHPHQW is required to satisfy moment transfer requirements at these locations. Similarly, no additional reinforcement is required to satisfy moment transfer requirements at the interior column. Check

within

ACI 8.6.1.2

Table 5.11, Case 1

Factored shear force at the critical section:

Therefore, Eq. (5.38)

5-122

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

where Eq. (5.30)

and Table 5.20

Eq. (5.37)

$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG At the interior column, provided

within

within

Based on this load combination, additional reinforcement is required to satisfy the minimum reinforcement UHTXLUHPHQWVRI$&,

$WWKH¿UVWLQWHULRUFROXPQV

Table 5.42

At the interior column, Therefore, the maximum shear stress is equal to the following:

Eq. (5.38)

Eq. (5.37)

The required from this load combination is less than the required combination, so required

from the

load

To satisfy minimum reinforcement requirements within EDUVPXVWEHSODFHGZLWKLQWKHLQ DWWKH¿UVWLQWHULRUFROXPQVZLWK effective slab width width of the column strip. For symWKHUHPDLQLQJEDUVWREHSODFHGZLWKLQWKH PHWU\DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH ¿UVWLQWHULRUFROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOIRUWKLVFDVHLVVLPLODUWRWKDWVKRZQLQ)LJXUH

5-123

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS Step 10 – Determine the reinforcement details

ACI 8.7

7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQ)LJXUHFDQQRWEHXVHGLQWKHFROXPQVWULSEHFDXVHRIWKHHIIHFWVGXH WRVHLVPLFORDGV7RDFFRXQWIRUWKHVHLVPLFORDGHIIHFWVSHUFHQWRIWKHWRSEDUVLQWKHFROXPQVWULSVDUHPDGH FRQWLQXRXVRYHUWKHVSDQV,QWKLVFDVHWKHPD[LPXPDPRXQWRIWRSUHLQIRUFHPHQWRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWVEDUV VREDUVDUHPDGHFRQWLQXRXV 7KHUHPDLQLQJEDUVLQWKHFROXPQVWULSDQGWKHEDUVLQWKHPLGGOHVWULSFDQEHWHUPLQDWHGDWWKHORFDWLRQVLGHQWL¿HG LQ)LJXUHWKLV¿JXUHDOVRLQFOXGHVWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&, A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUHZKLFKDUHWKHVDPHDVWKRVHLQ )LJXUHIRU6'&$

5-124

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

͸ᇱ Ǧ͸ᇱᇱ  ʹᇱ ǦͲᇱᇱ  ͸ᇱ Ǧ͸ᇱᇱ 

ͳͶǦ͓ͷ

ͺǦ͓ͷ

ͳʹǦ͓ͷ

ͷǦ͓ͷ ʹᇱ Ǧʹᇱᇱ 

ʹǦ͓ͷ

ͻǦ͓ͷ

൒ ͸ԣ ʹͳᇱ Ǧ͸ᇱᇱ 

ʹͳᇱ Ǧ͸ᇱᇱ 

‘Ž—–”‹’ ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ 

ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ  ͷᇱ ǦͲᇱᇱ 

ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ 

ͻǦ͓ͷ

ͻǦ͓ͷ

ͻǦ͓ͷ ͷǦ͓ͷ

ͶǦ͓ͷ ൒ ͸ԣ ʹͳᇱ Ǧ͸ᇱᇱ 

ͷǦ͓ͷ

൒ ͸ԣ

ͶǦ͓ͷ

͵ᇱ ǦͲᇱᇱ 

͵ᇱ ǦͲᇱᇱ 

ʹͳᇱ Ǧ͸ᇱᇱ 

ͶǦ͓ͷ

͵ᇱ ǦͲᇱᇱ 

͵ᇱ ǦͲᇱᇱ 

‹††Ž‡–”‹’ Figure 5.58 Reinforcement details for an interior design strip in the north-south direction for the flat plate system in Example 5.12.

Comments. Throughout this example, increasing WRWKHYDOXHVLQ$&,7DEOHZDVQRWFRQVLGHUHG$VDQ exercise, determine if FDQEHLQFUHDVHGWRWKHPD[LPXPPRGL¿HGYDOXHRILQ7DEOHIRUWKHHGJHFROumns bending perpendicular to the edge for a typical interior design strip in the north-south direction. From Step 9, maximum

for an edge column, which corresponds to the load combination

Assuming there is no shear reinforcement in the slab, LVGHWHUPLQHGE\(T )RUDVTXDUHHGJHFROXPQ EHQGLQJSHUSHQGLFXODUWRWKHHGJHWKH¿UVWRIWKHWKUHHHTXDWLRQVJRYHUQV

Therefore,

Determine

within

EDUVDUHSURYLGHGZLWKLQ

):

5-125

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Thus,

FDQEHWDNHQDVDWWKHHGJHFROXPQVLQWKLVH[DPSOH

If DQGEDUVDUHUHTXLUHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK 3URYLGLQJWKLVDGGLWLRQDOÀH[XUDOUHLQIRUFHPHQWDWWKHHGJHFROXPQVUHVXOWV FRPSDUHGWREDUVZLWK LQWKHVKHDUVWUHVVFDXVHGE\HFFHQWULFLW\RIVKHDURQWKHVHFULWLFDOVHFWLRQVHTXDOWR]HUREHFDXVH this shear stress is usually relatively large because DWWKHHGJHFROXPQVLQÀDWSODWHV\Vand, thus, WHPVZLWKRXWHGJHEHDPVLVUHODWLYHO\ODUJHVHH([DPSOH  5.8.13 Example 5.13 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (Framing Option A), SDC B

Check the shear strength requirements at the columns in an interior design strip in the north-south direction for the ÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQGLQE\ DQG*UDGH LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. Edge Columns Step 1 – Check one-way shear strength requirements

The critical section is located a distance

ACI 8.4.3

from the face of the column.

Because the panel is rectangular, one-way shear strength requirements must be checked at critical sections A and B governs in this case. VHH)LJXUH $OVRWKHJUDYLW\ORDGFRPELQDWLRQ 

2ᇱ -0ᇱᇱ 

2ᇱ -0ᇱᇱ 

25ᇱ -0ᇱᇱ 

A

B

11ᇱ -9ᇱᇱ 

݀ ൌ 8.25ᇱᇱ 

݀ ൌ 8.25ᇱᇱ 

Figure 5.59 Critical sections for one-way shear at an edge column in Example 5.13.

Maximum

5-126

ACI Eq. (5.3.1b)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum factored shear force on critical section A:

Maximum factored shear force on critical section B:

Design shear strength: Eq. (5.29)

Eq. (5.30)

Table 5.20

)URP([DPSOHWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKHHGJHFROXPQFRQVLVWVRI EDUVEDUVLQWKHFROXPQVWULSDQGEDUVLQWKHPLGGOHVWULS LQWKHQRUWKVRXWKGLUHFWLRQVHH)LJXUH  Thus,

2QHZD\VKHDUUHTXLUHPHQWVDUHDOVRVDWLV¿HGDWFULWLFDOVHFWLRQ%FDOFXODWLRQVQRWVKRZQKHUH  Step 2 – Check two-way shear strength requirements

The critical section is located a distance

ACI 8.4.4

from the face of the column.

)DFWRUHGVKHDUIRUFHDWWKHFULWLFDOVHFWLRQVHH)LJXUH 

For the gravity load combination where the DDM is used to determine design bending moments,

Dead load:

Eq. (5.17)

5-127

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹᇱ ǦͲᇱᇱ 

ͳͳᇱ Ǧͻᇱᇱ 

ܾଶ ൌ ܿଶ ൅ ݀ ൌ ʹͶǤͲ ൅ ͺǤʹͷ ൌ ͵ʹǤʹͷᇱᇱ  

ʹᇱ ǦͲᇱᇱ 

ʹͷᇱ ǦͲᇱᇱ 

ܾଵ ൌ ܿଵ ൅ ሺ݀Ȁʹሻ ൌ ʹͶǤͲ ൅ ሺͺǤʹͷȀʹሻ  ൌ ʹͺǤͳ͵ᇱᇱ 

Figure 5.60 Critical section for two-way shear at an edge column in Example 5.13.

Live load:

Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

5-128

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.42

Total factored shear force:

For other than gravity load combinations,

is equal to that from the actual load combination: Table 5.44

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

6KHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWWKHH[WHULRUFROXPQV First interior columns Step 1 – Check one-way shear strength requirements

The critical section is located a distance

ACI 8.4.3

from the face of the column.

Because the panel is rectangular, one-way shear strength requirements must be checked at critical sections A and B governs in this case. VHH)LJXUH $OVRWKHJUDYLW\ORDGFRPELQDWLRQ Maximum

ACI Eq. (5.3.1b)

Maximum factored shear force on critical section A:

Maximum factored shear force on critical section B:

Design shear strength: Eq. (5.29)

Eq. (5.30)

Table 5.20

5-129

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



23ᇱ -6ᇱᇱ 

2ᇱ -0ᇱᇱ 

2ᇱ -0ᇱᇱ 

25ᇱ -0ᇱᇱ 

A

B

݀ ൌ 8.25ᇱᇱ 

݀ ൌ 8.25ᇱᇱ 

Figure 5.61 Critical sections for one-way shear at the first interior column in Example 5.13.

)URP([DPSOHWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKH¿UVWLQWHULRUFROXPQFRQVLVWV RIEDUVEDUVLQWKHFROXPQVWULSDQGEDUVLQWKHPLGGOHVWULS LQWKHQRUWKVRXWKGLUHFWLRQVHH )LJXUH  Thus,

$VVXPLQJWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKH¿UVWLQWHULRUFROXPQFRQVLVWVRI bars in the east-west direction:

Step 2 – Check two-way shear strength requirements

The critical section is located a distance

5-130

from the face of the column.

ACI 8.4.4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)DFWRUHGVKHDUIRUFHDWWKHFULWLFDOVHFWLRQVHH)LJXUH 



ʹ͵ᇱ Ǧ͸ᇱᇱ 

ʹᇱ ǦͲᇱᇱ 

ʹᇱ ǦͲᇱᇱ 

ʹͷᇱ ǦͲᇱᇱ 

ܾଶ ൌ ܿଶ ൅ ݀ ൌ ʹͶǤͲ ൅ ͺǤʹͷ ൌ ͵ʹǤʹͷᇱᇱ  

ܾଵ ൌ ܿଵ ൅ ݀ ൌ ʹͶǤͲ ൅ ͺǤʹͷ ൌ ͵ʹǤʹͷᇱᇱ 

Figure 5.62 Critical section for two-way shear at the first interior column in Example 5.13.

In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.21) Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

5-131

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

Table 5.42

Total factored shear force:

From gravity load effects: Eq. (5.21)

From seismic load effects:

Total Total factored shear stress: Eq. (5.14)

Eq. (5.32)

6KHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWWKH¿UVWLQWHULRUFROXPQV

5-132

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5.8.14 Example 5.14 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (Framing Option A), SDC B, Shear Cap

Check the two-way shear strength requirements at an edge column in an interior design strip in the north-south diUHFWLRQIRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDQLQWKLFNVODE DVKHDUFDSZLWKDLQSURMHFWLRQEHORZWKHVODEDQGLQE\LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVR assume normalweight concrete with Step 1 – Check two-way shear stress at the critical section around the column

Given Assume

ACI 8.4.4

Figure 5.14

at the three faces of the column.

from the face of the column (see

Š‡ƒ”…ƒ’

ʹͲᇱᇱ 

ʹͷᇱᇱ 

ʹ͸Ǥʹͷᇱᇱ 

ʹͻǤͳ͵ᇱᇱ 

The critical section is located a distance )LJXUH 

”‹–‹…ƒŽ •‡…–‹‘•

ʹͲᇱᇱ  ͵Ͳᇱᇱ  ͵ʹǤͷᇱᇱ  ͵ͺǤʹͷᇱᇱ  Figure 5.63 Critical sections for two-way shear at the edge column in Example 5.14.

Factored shear force at the critical section:

5-133

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For the gravity load combination where the DDM is used to determine design bending moments,

Figure 5.8 Figure 5.23

Dead load:

Eq. (5.17)

Live load:

Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

5-134

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH

Total factored shear force:

For other than gravity load combinations,

is equal to that from the actual load combination.

For an edge column subjected to gravity loads effects: Table 5.16

From seismic load effects,

Table 5.42

Total factored shear stress: Eq. (5.14)

Eq. (5.32) Step 2 – Check two-way shear stress at the critical section around the shear cap

ACI 8.4.4

Figure 5.63

Factored shear force at the critical section:

Table 5.11, Case 3

5-135

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

Table 5.42

Total factored shear force:

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

7ZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HGDWWKHHGJHFROXPQV 5.8.15 Example 5.15 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (Framing Option A), SDC B, Slab Opening

Check two-way shear strength requirements at an edge column in an interior design strip in the north-south direction IRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQG LQE\LQFROXPQVJLYHQWKHLQGLDPHWHUVODERSHQLQJLQ)LJXUH$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPH DQG*UDGHUHLQIRUFHPHQW normalweight concrete with Step 1 – Check if the size of the opening is permitted

ACI 8.5.4.2(b)

The 8-in.-diameter opening occurs within two intersecting column strips. Without performing an analysis of the V\VWHPZLWKWKHRSHQLQJWKHPD[LPXPRSHQLQJVL]HSHUPLWWHGLQVXFKFDVHVLVRQHHLJKWKWKHZLGWKRIWKHJRYHUQing column strip:

5-136

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 5.64 Slab opening in Example 5.15.

Figure 5.8

This opening is permitted at the indicated location because the 8-in. diameter is less than Step 2 – Check if portions of the critical section are ineffective

A portion of umn:

is considered ineffective where an opening is located closer than

ACI 22.6.4.3

from the periphery of the col-

%HFDXVHWKHRSHQLQJLVORFDWHGLQIURPWKHIDFHRIWKHRSHQLQJZKLFKLVOHVVWKDQLQDSRUWLRQRI enclosed by straight lines projecting from the centroid of the column and tangent to the boundaries of the opening must be considered ineffective. VHH)LJXUH %DVHGRQ The length of the critical section adjacent to the opening is WKHUHTXLUHPHQWVRI$&,WKHLQHIIHFWLYHOHQJWKRIWKHFULWLFDOVHFWLRQFDQEHREWDLQHGIURPVLPLODUWULDQJOHV

5-137

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܽ ༄

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༃

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ͺǤʹͷΤʹ ൌ ͶǤͳ͵ԣ ”‹–‹…ƒŽ•‡…–‹‘

‡ˆˆ‡…–‹˜‡Ž‡‰–Š‘ˆ…”‹–‹…ƒŽ•‡…–‹‘ ൌ ͸ǤͲ‹Ǥ ܽ ൌ ሺ͵ʹǤʹͷ െ ͸ǤͲሻΤʹ ൌ ͳ͵Ǥͳ͵‹Ǥ Figure 5.65 Ineffective portion of bo for the edge column in Example 5.15.

Step 3 – Check two-way shear strength requirements

ACI 8.4.4

ACI Eq. (5.3.1b)

Factored shear force at the critical section:

For the gravity load combination where the DDM is used to determine design bending moments,

Dead load:

Live load:

5-138

Eq. (5.17)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Section properties of the critical section are determined based on the straight segments of the effective portions RIWKHVHFWLRQVHH)LJXUH 

Eq. (5.15)

Segment 1:

Segment 2:

Eq. (5.16)

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

5-139

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.42

Total factored shear force:

For other than gravity load combinations,

is equal to that from the actual load combination: Table 5.44

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

6KHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGDWWKHH[WHULRUFROXPQVEHFDXVH nation

for the gravity load combi-

Comments.&RPSDULQJWKHUHVXOWVIURPWKLVH[DPSOHWRWKRVHLQ([DPSOHZKHUHWKHUHLVQRRSHQLQJDGMDFHQW WRWKHHGJHFROXPQLWLVHYLGHQWWKHLQGLDPHWHURSHQLQJKDVDVLJQL¿FDQWLPSDFWRQWZRZD\VKHDUVWUHQJWK The section properties of the critical section are determined in this example using the conservative method outlined LQ6HFWRIWKLVSXEOLFDWLRQ'HWHUPLQHWKHVHFWLRQSURSHUWLHVEDVHGRQWKHDFWXDOSURSHUWLHVRIWKHFULWLFDOVHFWLRQZLWKWKHLQLQHIIHFWLYHVHJPHQW Figure 5.65

For the two faces

For the two faces

Therefore, Total factored shear stress due to gravity loads:

5-140

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,WLVHYLGHQWVKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGXVLQJWKHDFWXDOSURSHUWLHVRIWKHFULWLFDOVHFWLRQ 1RWHWKDWVKHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGLI

If the concrete compressive strength cannot be increased, it may be possible to satisfy two-way shear strength requirements by investigating whether FDQEHLQFUHDVHGWRWKHUHE\GHFUHDVLQJ WR]HUR LQDFFRUGDQFH ZLWK$&,,WLVGHWHUPLQHGLQWKH&RPPHQWVVHFWLRQDWWKHHQGRI([DPSOHWKDW FDQEHWDNHQDV SURYLGHGWKHDPRXQWRIQHJDWLYHÀH[XUDOUHLQIRUFHPHQWDWWKHFULWLFDOVHFWLRQLVLQFUHDVHGDFFRUGLQJO\7KHUHIRUHDVthe maximum two-way shear stress for the gravity load suming ZLWKWKHRSHQLQJZKLFKPHDQVWZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HG combination ,WLVXQOLNHO\DQRSHQLQJRIWKLVVL]HZLOOKDYHDPDMRULPSDFWRQÀH[XUDORUVHUYLFHDELOLW\UHTXLUHPHQWV,QUHJDUGWR detailing requirements, a quantity of reinforcement at least equal to that interrupted by the opening must be added RQWKHVLGHVRIWKHRSHQLQJ>$&,E @ 5.8.16 Example 5.16 – Check of Shear Strength Requirements: Flat Slab System With Edge Beams, Building #1 (Framing Option D), SDC A, Circular Columns

Check two-way shear strength requirements at an interior column in an interior design strip in the north-south direcWLRQDWWKHVHFRQGÀRRUOHYHOIRUWKHÀDWVODEV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ' ZLWKDQLQWKLFNVODED GURSSDQHOZLWKDLQSURMHFWLRQEHORZWKHVODEDQGLQFLUFXODUFROXPQV$VVXPHWKH6LWH&ODVVLV&$OVR Interior design strips are not part of the LFRS. assume normalweight concrete with Step 1 – Check two-way shear stress at the critical section around the column

Maximum

ACI 8.4.4

ACI Eq. (5.3.1b) Figure 5.13

Determine the section properties of the critical section assuming a square column of equivalent area (see Figure   ACI 22.6.4.1.2

Table 5.11, Case 1

Factored shear force at the critical section:

5-141

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Figure 5.66 Critical sections for the interior column in Example 5.15.

In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.21)

where Determining whether

Figure 5.23

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Table 5.11, Case 1

5-142

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where Step 2 – Check two-way shear stress at the critical section around the drop panel

ACI 8.4.4 Figure 5.13

Section properties of the critical section: Table 5.11, Case 1

Factored shear force at the critical section:

The transfer moment stress.

is negligible at this location and is not included in the calculation of the maximum shear

Maximum shear stress, Eq. (5.14)

7ZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HGDWWKHLQWHULRUFROXPQV Comments. Instead of calculating the section properties of the critical section using an equivalent square column, calculate the section properties based on the circular section and check two-way shear requirements. Table 5.13, Case 1

Factored shear force at the critical section:

5-143

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Total factored shear stress: Eq. (5.14)

7ZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVFDQEHVDWLV¿HGE\LQFUHDVLQJWKHGURSSDQHOSURMHFWLRQEHORZWKHVODEWRLQ 5.8.17 Example 5.17 – Determination of Shear Reinforcement: Flat Plate System, Building #1 (Framing Option A), SDC B, Stirrups

&KHFNWZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDWWKHHGJHDQG¿UVWLQWHULRUFROXPQVLQDQLQWHULRUGHVLJQVWULSLQWKH QRUWKVRXWKGLUHFWLRQIRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKD LQWKLFNVODEDQGLQE\LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW3URYLGHVKHDUUHLQIRUFHPHQWLQWKHIRUPRIFORVHGVWLUUXSVLI with needed. Edge columns Step 1 – Check two-way shear strength requirements

ACI 8.4.4

Factored shear force at the critical section:

For the gravity load combination where the DDM is used to determine design bending moments, Figure 5.8 Figure 5.23

Dead load:

Live load:

5-144

Eq. (5.17)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH

Total factored shear force:

For other than gravity load combinations,

is equal to that from the actual load combination:

Total factored shear stress: Eq. (5.14)

5-145

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (5.32)

7ZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGDWWKHHGJHFROXPQVEDVHGRQERWKORDGFRPELQDWLRQV Step 2 – Determine the size and spacing of closed stirrups

ACI 8.7.6 ACI 22.6.7.1

• Step 2a – Check if stirrups can be utilized in the slab $VVXPLQJVWLUUXSV

Thus, stirrups are permitted to be used to increase the design two-way shear strength. • Step 2b – Check the maximum shear strength permitted with stirrups

ACI Table 22.6.6.3

From Step 1, maximum • Step 2c – Determine the design shear strength of the concrete with stirrups

ACI Table 22.6.6.1 Table 5.23

where

Eq. (5.30)

and Table 5.20

Note: LVSHUPLWWHGWREHWDNHQDVIRUWZRZD\VODEVZLWKVWLUUXSVWKDWVDWLVI\WKHSURYLVLRQVRI$&, D  • Step 2d – Determine the required area and spacing of the stirrups Maximum stirrup spacing

ACI 22.6.7.2 Figure 5.19

Try Total required area of stirrups, for the three-sided, rectangular critical section located a distance WKHIDFHRIWKHFROXPQVHH)LJXUH 

from

Table 5.23

8VHVWLUUXSVVSDFHGDWLQRQFHQWHU

5-146

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Figure 5.67 Closed stirrup shear reinforcement at the edge columns in Example 5.17.

• Step 2e – Determine the distance from the faces of the column where the stirrups can be terminated Stirrups can be terminated where the design strength of the concrete can resist the factored shear stress without stirrups. At the critical section located a distance

from the outermost peripheral line of stirrups: Table 5.23

Assume 6 stirrups are provided parallel to each face of the column and check if the maximum factored shear from the centerline of the outermost peripheral line of stress at the critical section located a distance of . stirrups is less than or equal to 6HFWLRQSURSHUWLHVRIWKHSRO\JRQDOFULWLFDOVHFWLRQVHH)LJXUH 

Table 5.12, Case 3

Eq. (5.15)

Segment 1:

5-147

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Segment 2:

Segment 3:

Eq. (5.16)

The transfer moment tively taken as

LVHVVHQWLDOO\]HURDWWKLVFULWLFDOVHFWLRQEXWIRUFDOFXODWLRQSXUSRVHVLVFRQVHUYD-

ACI Eq. (8.4.4.2.2)

Total factored shear stress:

Eq. (5.14)

7KHUHIRUHWKHFORVHGVWLUUXSVSURYLGHGRQHDFKIDFHRIWKHFROXPQDUHDGHTXDWH First interior columns Check two-way shear strength requirements

Factored shear force at the critical section:

5-148

ACI 8.4.4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.21) Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

Table 5.42

Total factored shear force:

From gravity load effects: Eq. (5.21)

From seismic load effects:

5-149

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Total Total factored shear stress: Eq. (5.14)

Eq. (5.32)

6KHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWWKH¿UVWLQWHULRUFROXPQV 5.8.18 Example 5.18 – Determination of Shear Reinforcement: Flat Plate System, Building #1 (Framing Option A), SDC B, Headed Shear Studs

Check two-way shear strength requirements at the edge columns in an interior design strip in the north-south direcWLRQIRUWKHÀDWSODWHV\VWHPLQ)LJXUH)UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODE DQGLQE\LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK Provide shear reinforcement in the form of headed shear studs if needed with Step 1 – Check two-way shear strength requirements

ACI 8.4.4

Table 5.11, Case 3

Factored shear force at the critical section:

For the gravity load combination where the DDM is used to determine design bending moments, : Figure 5.8 Figure 5.23

Dead load:

Live load:

5-150

Eq. (5.17)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Total factored shear stress: Eq. (5.14)

Eq. (5.32)

where

$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH

Total factored shear force:

For other than gravity load combinations,

is equal to that from the actual load combination:

Total factored shear stress: Eq. (5.14)

5-151

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (5.32)

7ZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGDWWKHHGJHFROXPQVEDVHGRQERWKORDGFRPELQDWLRQV Step 2 – Determine the size and spacing of headed shear stud reinforcement

• Step 2a – Check the maximum shear strength permitted with headed shear studs

ACI 8.7.7 ACI Table 22.6.6.3

From Step 1, maximum • Step 2b – Determine the design shear strength of the concrete with headed shear studs

ACI Table 22.6.6.1

where Eq. (5.30) Table 5.20

and and

Note: LVSHUPLWWHGWREHWDNHQDVIRUWZRZD\VODEVZLWKVPRRWKKHDGHGVKHDUVWXGUHLQIRUFHPHQWWKDW VDWLV¿HVWKHSURYLVLRQVRI$&,E WKLVSURYLVLRQLVVDWLV¿HGLQWKLVH[DPSOHVHH6WHSFEHORZ  • Step 2c – Determine the required size and spacing of the headed shear studs

ACI 22.6.8 Figure 5.20

Provide three lines of headed shear studs on each face of the column:

Assuming ½-in.-diameter studs

, the required spacing is the following: Table 5.23

In this equation, is the cross-sectional area of all the headed studs on one peripheral line parallel to the peULPHWHURIWKHFROXPQVHFWLRQVHH)LJXUH  Because

Figure 5.20

$VVXPLQJDLQVSDFLQJFKHFNWKHUHTXLUHPHQWVRI$&, ACI Eq. (22.6.8.3)

8VHòLQGLDPHWHUKHDGHGVKHDUVWXGVVSDFHGDWLQRQFHQWHU 5-152

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͺͶᇱᇱ  Ͷ̷͸ᇱᇱ ൌ ʹͶᇱᇱ 

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ᬆ ʹʹǤͲᇱᇱ  Figure 5.68 Headed shear stud reinforcement at the edge columns in Example 5.18.

• Step 2d – Determine the distance from the faces of the column where the headed shear studs can be terminated +HDGHGVKHDUVWXGVFDQEHWHUPLQDWHGZKHUHWKHGHVLJQVWUHQJWKRIWKHFRQFUHWHFDQUHVLVWWKHIDFWRUHGVKHDU stress without the headed shear studs. At the critical section located a distance

from the outermost peripheral line of headed shear studs: Table 5.23

$VVXPHKHDGHGVKHDUVWXGVDUHSURYLGHGLQHDFKOLQHSDUDOOHOWRHDFKIDFHRIWKHFROXPQDQGFKHFNLIWKHPD[from the centerline of the outermost imum factored shear stress at the critical section located a distance of . peripheral line of headed studs is less than or equal to 6HFWLRQSURSHUWLHVRIWKHSRO\JRQDOFULWLFDOVHFWLRQVHH)LJXUH  Table 5.12, Case 3

Eq. (5.15)

Segment 1:

5-153

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Segment 2:

Segment 3:

Eq. (5.16)

The transfer moment taken as

LVHVVHQWLDOO\]HURDWWKLVFULWLFDOVHFWLRQEXWIRUFDOFXODWLRQSXUSRVHVLVFRQVHUYDWLYHO\

ACI Eq. (8.4.4.2.2)

Total factored shear stress:

Eq. (5.14)

7KHUHIRUHWKHòLQGLDPHWHUKHDGHGVKHDUVWXGVVSDFHGDWLQRQFHQWHUZLWKOLQHVRIKHDGHGVWXGVRQHDFK face of the column are adequate.

5-154

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 6 BEAMS 6.1 Overview %HDPVDUHPHPEHUVVXEMHFWHGSULPDULO\WRÀH[XUHDQGVKHDUZLWKRUZLWKRXWD[LDOIRUFHRUWRUVLRQ2QHRIWKHSULPDU\IXQFWLRQVRIDEHDPLVWRWUDQVIHUURRIDQGÀRRUORDGVWRVXSSRUWLQJPHPEHUVFROXPQVZDOOVRURWKHUEHDPV WKHODWWHURIZKLFKDUHRIWHQUHIHUUHGWRDVJLUGHUV 7KHPDLQÀH[XUDOUHLQIRUFHPHQWLQDEHDPUXQVSDUDOOHOWRWKH direction of load transfer. The design and detailing of cast-in-place beams with nonprestressed reinforcement are covered in this chapter for where is the gross cross-sectional area of members subjected to a factored axial force, , less than the beam (see ACI 9.3.3.1). Provisions for beams are given in ACI Chapter 9, which are applicable to members in buildings assigned to Seismic Design Category (SDC) A and B. Nonprestressed one-way joist systems consist of a monolithic combination of regularly spaced ribs with a clear VSDFLQJEHWZHHQWKHULEVRIQRPRUHWKDQLQDQGDRQHZD\VODEPHHWLQJWKHGLPHQVLRQDOUHTXLUHPHQWVLQ$&, WKURXJK7KHULEVDQGVXSSRUWLQJEHDPVDUHGHVLJQHGLQDFFRUGDQFHZLWK$&,7KLVV\VWHPZDV once popular but is generally not used anymore. Wide-module joist systems, which are also referred to as skip joist V\VWHPVDUHIRUPHGE\LQRULQZLGHSDQIRUPVDQGKDYHHVVHQWLDOO\WDNHQWKHSODFHRIVWDQGDUGRQHZD\ joist systems. The evenly spaced joists (or, ribs) support the loads from a one-way slab cast integrally with the joists DQGEHDPVRUJLUGHUV >VHH)LJXUH@%HFDXVHWKLVV\VWHPGRHVQRWPHHWWKHFOHDUVSDFLQJUHTXLUHPHQWVEHWZHHQ ULEVLQ$&,IRUVWDQGDUGRQHZD\MRLVWV\VWHPVWKHQRPLQDOVKHDUVWUHQJWKRIWKHFRQFUHWH , is not permitWHGWREHLQFUHDVHGE\WLPHVWKHYDOXHSUHVFULEHGLQ$&,$&, WKXVWKHMRLVWVDQGEHDPVLQWKH wide-module system must be designed in accordance with the provisions in ACI Chapter 9 excluding the requirements in ACI 9.8.

Figure 6.1 A wide-module joist system.

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6-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.2 Sizing the Cross-Section 6.2.1 Determining the Beam Depth

%HDPVPXVWKDYHVXI¿FLHQWGHSWKVRDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG,QOLHXRI FDOFXODWLQJGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,DQGVXEVHTXHQWO\FKHFNLQJWKHGHÀHFWLRQVGRQRWH[FHHGWKH OLPLWVLQ$&,$&, WKHPLQLPXPRYHUDOOGHSWK , for nonprestressed beams not supporting or atWDFKHGWRSDUWLWLRQVRURWKHUW\SHVRIFRQVWUXFWLRQOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVFDQEHGHWHUPLQHGXVLQJ WKHH[SUHVVLRQVLQ$&,7DEOHZKLFKDUHDSSOLFDEOHIRUUHLQIRUFHPHQWZLWKDVSHFL¿HG\LHOGVWUHQJWK , HTXDOWRSVLDQGQRUPDOZHLJKWFRQFUHWH$&, 0RGL¿FDWLRQIDFWRUVIRUEHDPVZLWKUHLQIRUFHPHQW RWKHUWKDQSVLDQGPDGHRIOLJKWZHLJKWFRQFUHWHDUHJLYHQLQ$&,DQGUHVSHFWLYHO\ Minimum beam depths based on various support conditions are given in Table 6.1. In the expressions for minimum , is the span length of the beam in inches; for cantilevers, is the clear projection of the cantilever in inches. Also, and DUHWKHPRGL¿FDWLRQIDFWRUVIRUUHLQIRUFHPHQWJUDGHDQGOLJKWZHLJKWFRQFUHWHUHVSHFWLYHO\ Table 6.1 Minimum Depth of Nonprestressed Beams Normalweight Concrete Support Condition

ƒy = 60 ksi

ƒy other than 60 ksi(2)

Lightweight Concrete(1)

ƒy = 60 ksi(3)

ƒy other than 60 ksi(2),(3)

Simply supported One end continuous Both ends continuous Cantilever (1) (2) (3)

Applicable where equilibrium density,

is in the range of 90 to 115 lb/ft3

For the usual case of continuous construction, the depth of the beams, , must be the same for all spans and it should be determined on the basis of the span yielding the largest minimum depth; this results in economical formZRUNVHH5HIHUHQFH 7KHPLQLPXP for an end span condition governs in the case of equal end and interior times the end span, the minimum for the spans. In cases where the interior span is greater than interior span condition governs. %HDPGHSWKLVW\SLFDOO\VSHFL¿HGLQZKROHLQFKLQFUHPHQWVDOWKRXJKKDOILQFKLQFUHPHQWVDUHFRPPRQO\XVHGLQ wide-module joist construction (see below). As noted in Section 6.1, standard pan forms are used in constructing wide-module joist systems. The pan depths FRUUHVSRQGLQJWRWKHVWDQGDUGSDQZLGWKVDUHJLYHQLQ)LJXUH$LQWKLFNVODELVW\SLFDOO\VSHFL¿HGLQVXFK systems because structural requirements are minimal. Therefore, the overall depth of the system (slab thickness plus pan depth) must be greater than or equal to the applicable minimum depth given in Table 6.1 to satisfy serviceability requirements. To achieve economical formwork, the depths of the beams must be the same depth as the joists: The formwork associated with beams deeper than the joists is more complex to build and additional shoring is required to support the beam formwork, which results in unnecessary increases in construction time and cost. For economy, the joists typically span in the long direction, which means their depth usually controls the overall depth of the wide-module system. Fire-resistance requirements of the general building code must also be considered when specifying a beam depth DQGVODEWKLFNQHVV$&, ,QPRVWFDVHVWKHUHTXLUHGEHDPGHSWKEDVHGRQ¿UHUHVLVWDQFHUHTXLUHPHQWVLV smaller than that required by ACI 9.3.1 for serviceability.

6-2

ƒ†‡’–Š 

Žƒ„ –Š‹…‡••

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‹„™‹†–Š

ƒ™‹†–Š

ƒ‹†–Šሺ‹Ǥሻ ͷ͵ ͸͸

ƒ‡’–Šሺ‹Ǥሻ ͳ͸ǡʹͲǡʹͶ ͳͶǡͳ͸ǡʹͲǡʹͶ

Figure 6.2 Standard form dimensions for wide-module joist systems.

7KHGHSWKRIDQ\EHDPFDQEHGHWHUPLQHGEDVHGRQWKHGHÀHFWLRQUHTXLUHPHQWVLQ$&,)RUEHDPVVXSSRUWLQJ FRQVWUXFWLRQOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVVXFKDVJODVVSDUWLWLRQZDOOV GHÀHFWLRQFDOFXODWLRQVVKRXOG EHSHUIRUPHGLQDFFRUGDQFHZLWKWKDWVHFWLRQ0D[LPXPSHUPLVVLEOHGHÀHFWLRQVDUHJLYHQLQ$&,7DEOHIRU PHPEHUVVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVDQGIRUPHPEHUVWKDWDUHQRW0HWKRGVRQKRZWRFDOFXODWHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUEHDPVDUHJLYHQLQ 6HFWLRQRIWKLVSXEOLFDWLRQ7KLVLQIRUPDWLRQLVDOVRDSSOLFDEOHWRFDOFXODWLRQRIGHÀHFWLRQVIRURQHZD\VODEV 6.2.2 Determining the Beam Width

The width of a beam, FDQEHGHWHUPLQHGEDVHGRQÀH[XUDOVWUHQJWKUHTXLUHPHQWVWKDWLVE\VDWLVI\LQJWKHIRO$WWKLVVWDJHRIWKHGHVLJQWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW lowing strength design equation: LVW\SLFDOO\QRWNQRZQ+RZHYHUDUDQJHIRU can be determined based on required minimum and maximum DUHDVRIUHLQIRUFHPHQW7KHPLQLPXPDUHDRIÀH[XUDOUHLQIRUFHPHQWLVVSHFL¿HGLQ$&,$OVREHDPVVXEMHFWHG to relatively small or no axial loads must be designed as tension-controlled sections (ACI 9.3.3.1), which essentially VHWVDQXSSHUOLPLWRQWKHDUHDRIÀH[XUDOUHLQIRUFHPHQWSURYLGHGLQDVHFWLRQ7KHSURYLGHGDUHDRIUHLQIRUFHPHQW PXVWEHZLWKLQWKLVUDQJH)RU*UDGHUHLQIRUFHPHQWDQGDVSHFL¿HGFRQFUHWHFRPSUHVVLYHVWUHQJWK , equal to SVLPLQLPXPDQGPD[LPXP are equal to the following:

6-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Using an initial estimate of IRUWKHDUHDRIWKHÀH[XUDOUHLQIRUFHPHQWZKLFKLVDSSUR[LPDWHO\HTXDOWRWKH can be obtained by setting and solving for average of the minimum and maximum values),

For beams with

In this equation,

and

has the units of in.-kips and

:

have the units of inches.

The above equation can be rewritten in the following form where units of inches:

has the units of ft-kips and

have the

(6.1)

The largest factored bending moment along the spans should be used in Equation (6.1) to determine For where is the depth of beams with one layer of longitudinal tension reinforcement, can be taken as WKHEHDPGHWHUPLQHGIRUVHUYLFHDELOLW\VHH6HFWLRQRIWKLVSXEOLFDWLRQ DQGLQLVWKHVXPRIWKHFRQFUHWH FRYHULQDFFRUGDQFHZLWK$&,7DEOHIRUEHDPVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG . To LQ WKHGLDPHWHURIDVWLUUXSLQ DQGRQHKDOIWKHGLDPHWHURIDORQJLWXGLQDOEDU achieve economical formwork, the same VKRXOGEHXVHGIRUDOOEHDPVDORQJWKHVSDQ$OVRWKHVDPHEHDPVL]H should be used as often as possible throughout the structure. Providing a beam width greater than or equal to that determined by Equation (6.1) results in cross-sectional dimensions satsifying both strength and serviceability requirements. The following equation can be used to determine pressive strength, and reinforcement ratio

for a beam with any grade of reinforcing steel, concrete com: (6.2)

where has the units of ft-kips; and have the units of psi; and and have the units of inches. The and $&, DQGOHVV reinforcement ratio must be greater than or equal to the greater of (ACI 9.3.3.1) where the term LVGH¿QHGLQ$&,7DEOHDVIROORZV$&, than or equal to   • For • For • For

6-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(TXDWLRQV RU FDQDOVREHXVHGWRGHWHUPLQHWKHUHTXLUHGZLGWKRIWKHMRLVWVLQDZLGHPRGXOHMRLVWV\VWHP -RLVWZLGWKFDQEHWDLORUHGWRVDWLVI\YLUWXDOO\DQ\UHTXLUHPHQW,QXVXDOVLWXDWLRQVWKHWKLQQHVWSUDFWLFDOZLGWKLV W\SLFDOO\DGHTXDWHIRUVWUXFWXUDOUHTXLUHPHQWV*HQHUDOO\DLQZLGHMRLVWLVXVHGZLWKDLQSDQDQGDLQ ZLGHMRLVWLVXVHGZLWKDLQSDQUHVXOWLQJLQIWDQGIWPRGXOHVUHVSHFWLYHO\&ROXPQOLQHMRLVWVFDQEHPDGH part of the lateral force-resisting system (LFRS) and any practical width (not necessarily the same as the adjoining joists) can be provided to resist combined gravity and lateral load effects. 6.2.3 General Guidelines for Sizing Beams for Economy

7KHIROORZLQJJHQHUDOJXLGHOLQHVIRUVL]LQJEHDPVVKRXOGEHIROORZHGIRUHFRQRP\VHH5HIHUHQFH  • Use whole-inch increments for beam dimensions, except for the depth of beams in wide-module joist systems where half-inch beam depths are used for formwork economy where applicable. • 8VHEHDPZLGWKVLQPXOWLSOHVRIRULQ • 8VHFRQVWDQWEHDPVL]HIURPVSDQWRVSDQDQGYDU\WKHUHLQIRUFHPHQWDVUHTXLUHG • Use wide, shallower beams rather than narrow, deeper beams. • 8VHEHDPZLGWKVDWOHDVWLQZLGHUWKDQWKHVXSSRUWLQJFROXPQZLGWKVZKHUHYHUSRVVLEOH • Use uniform width and depth of beams throughout the building, wherever possible.

6.3 Required Strength 6.3.1 Analysis Methods Overview

7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKVHH$&,DQGUHVSHFWLYHO\  Bending Moments and Shear Forces

)LYHPHWKRGVRIDQDO\VLVDUHJLYHQLQ$&,WRGHWHUPLQHWKHIDFWRUHGEHQGLQJPRPHQWVDQGVKHDUIRUFHVDWWKH FULWLFDOVHFWLRQVLQDEHDPVHH6HFWLRQRIWKLVSXEOLFDWLRQIRUWKHORFDWLRQVRIWKHFULWLFDOVHFWLRQVIRUÀH[XUH and shear). )RUWKHFDVHRIJUDYLW\ORDGVWKHVLPSOL¿HGPHWKRGIRUDQDO\VLVRIFRQWLQXRXVEHDPVDQGRQHZD\VODEVLQ$&, LVSHUPLWWHGWREHXVHGWRGHWHUPLQHEHQGLQJPRPHQWVDQGVKHDUIRUFHVSURYLGHGWKHFRQGLWLRQVLQ$&,DUH VDWLV¿HG>$&,D @ (1) Members are prismatic   Gravity loads are uniformly distributed (3) The service live load, , is less than or equal to 3 times the service dead load,   There are at least two spans   7KHORQJHURIWZRDGMDFHQWVSDQVGRHVQRWH[FHHGWKHVKRUWHUVSDQE\SHUFHQW , and factored shear forces, LQ$&,DQGUHVSHFWLYHO\ The approximate factored bending moments, is the factored uniformly distributed gravity load on the conDUHJLYHQLQ)LJXUHRIWKLVSXEOLFDWLRQZKHUH tinuous beams. 0RPHQWUHGLVWULEXWLRQLVQRWSHUPLWWHGZKHQEHQGLQJPRPHQWVDUHFDOFXODWHGXVLQJWKHVLPSOL¿HGPHWKRG$&,   :KHUHRQHRUPRUHRIWKHFRQGLWLRQVLQ$&,LVQRWVDWLV¿HGRQHRIWKHRWKHUIRXUPHWKRGVRIDQDO\VLVJLYHQLQ and at the critical sections. $&,PXVWEHXVHGWRGHWHUPLQH

6-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Torsional Moments

Factored torsional moments, , can be obtained using the analysis methods in ACI Chapter 6. It is permitted to WDNHWKHWRUVLRQDOORDGLQJIURPDVODEDVXQLIRUPO\GLVWULEXWHGDORQJWKHVSDQRIDEHDP$&,  Once LVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVHH6HFWLRQRIWKLVSXEOLFDWLRQIRUWKHORFDWLRQRIWKHFULWLFDOVHFtion for torsion), the next step is to determine if torsional effects on a beam need to be considered or not. Torsional where is the threshold torsional moeffects on a reinforced concrete beam need not be considered if PHQW$&, )RUQRQSUHVWUHVVHGPHPEHUVZLWKVROLGFURVVVHFWLRQV LVJLYHQLQ$&,7DEOHD 

(6.3)

7KH¿UVWRIWKHVHHTXDWLRQVLVDSSOLFDEOHWRPHPEHUVQRWVXEMHFWHGWRDQD[LDOIRUFH7KHVHFRQGHTXDWLRQLVDSSOLFD(in pounds), which is taken as positive for compresble where the member is subjected to a factored axial force , and negative for tension forces. sion forces acting on the gross area of the beam, The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium density, RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[VHH$&,DQG $&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOHD @DQGYDOXHVRI based on FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOHE @1RWHWKDW is permitted to be taken as IRUOLJKWZHLJKWFRQFUHWH$&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH$&,  Table 6.2 Values of

Based on Equilibrium Density, wc

Ȝ

Equilibrium Density, wc



 Table 6.3 Values of

Based on Composition of Aggregates

Concrete

Composition of Aggregates

Ȝ

All-lightweight

)LQH$670& &RDUVH$670&



/LJKWZHLJKW¿QHEOHQG

)LQH&RPELQDWLRQRI$670&DQG& &RDUVH$670&

WR(1)

Sand-lightweight

Fine: ASTM C33 &RDUVH$670&



Sand-lightweight, coarse blend

Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670&DQG$670&

WR(2)

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

6-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term is the area enclosed by the outside perimeter of the concrete cross-section, and is the outside perimeter of the concrete cross-section. For beams cast monolithically with a slab, a portion of the slab may be able to FRQWULEXWHWRWRUVLRQDOUHVLVWDQFH7KHRYHUKDQJLQJÀDQJHZLGWK permitted to be used in the calculation of LVJLYHQLQ$&, and (6.4)

is the greater of the projection of the beam above or below the slab (see Figure 6.3). ܾ௪ ൅ ʹܾ௘ ൌ ܾ௪ ൅ ʹ݄௕ ൑ ܾ௪ ൅ ͺ݄ ݄௕ ൑ Ͷ݄

݄௕ଵ ൑ Ͷ݄

ܾ௪ 

ܾ௘ ൌ ݄௕ ൑ Ͷ݄  ൌ ܾ௪ ݄௕ ൅ ሺܾ௪ ൅ ʹܾ௘ ሻ݄

‫ܣ‬௖௣  ‫݌‬௖௣ ൌ ʹሺ݄ ൅ ݄௕ ൅ ܾ௪ ൅ ʹܾ௘ ሻ ۷‫ܚܗܑܚ܍ܜܖ‬۰‫ܕ܉܍‬

݄ ݄௕ଵ ൐ ݄௕ଶ 

݄௕ 

݄௕ଵ ൅ ݄ ൅ ݄௕ଶ 

݄

݄௕ ൑ Ͷ݄

ܾ௪ ൅ ܾ௘ ൌ ܾ௪ ൅ ݄௕ଵ ൑ ܾ௪ ൅ Ͷ݄ ݄௕ଶ 

where

ܾ௪ 

ܾ௘ ൌ ݄௕ଵ ൑ Ͷ݄  ‫ܣ‬௖௣ ൌ ܾ௪ ሺ݄௕ଵ ൅ ݄ ൅ ݄௕ଶ ሻ ൅ ݄ܾ௘   ‫݌‬௖௣ ൌ ʹሺ݄௕ଵ ൅ ݄ ൅ ݄௕ଶ ൅ ܾ௪ ൅ ܾ௘ ሻ ۳‫܍܏܌‬۰‫ܕ܉܍‬

Figure 6.3 Torsional section properties Acp and pcp for interior and edge beams cast monolithically with the slab.

Torsional section properties and for an interior beam and an upturned edge beam cast monolithically with the slab are also given in Figure 6.3. Note that the torsional section properties for the edge beam are applicable for and LQFOXGLQJDYDOXHRI]HUR corresponds to the case where the bottom of the slab is any value of FRUUHVSRQGVWRWKHFDVHZKHUHWKHWRSRIWKHVODELVÀXVKZLWKWKHWRS ÀXVKZLWKWKHERWWRPRIWKHEHDPDQG of the beam). calculated 7KHFRQWULEXWLRQRIWKHRYHUKDQJLQJÀDQJHV PXVWEHQHJOHFWHGLQFDVHVZKHUHWKHSDUDPHWHU FDOFXODWHGIRUWKHVDPHVROLGEHDPLJQRULQJWKHÀDQJHV>$&, IRUDVROLGEHDPZLWKÀDQJHVLVOHVVWKDQ for the beam with E @)RUKROORZVHFWLRQVWKHRYHUKDQJLQJÀDQJHV PXVWEHQHJOHFWHGZKHUH is the area of the concrete only and does ÀDQJHVLVOHVVWKDQWKDWIRUWKHVDPHEHDPLJQRULQJWKHÀDQJHVZKHUH not include the area of the void. For nonprestressed members with hollow cross-sections,

LVJLYHQLQ$&,7DEOHE 

(6.5)

6-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Once it has been established that torsional effects must be considered (that is, where ), the next step is to determine whether IURPDQDO\VLVFDQEHUHGXFHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&,$&,  7KHEHDPVXSSRUWLQJWKHFDQWLOHYHUVODELQ$&,)LJXUH5DLVXQDEOHWRUHGLVWULEXWHWKHHIIHFWVGXHWRWKHWRUsional moment (it is not part of an indeterminate structure where redistribution of the applied torsional moment can from analysis. This type of torsion is referred to as occur); thus, it must be designed for the torsional moment equilibrium torsion because LVUHTXLUHGWRPDLQWDLQHTXLOLEULXP$&,  7KHVSDQGUHOEHDPLQ$&,)LJXUH5ELVSDUWRIDQLQGHWHUPLQDWHVWUXFWXUHZKHUHUHGLVWULEXWLRQRIWRUVLRQDO moments can occur. This type of torsion is referred to as compatibility torsion. In this case, the beam can be deequal to the design cracking torsional moment, , which is less than that obsigned for a torsional moment is determined by the equations in ACI WDLQHGIURPDQDO\VLV$&, )RUERWKVROLGDQGKROORZVHFWLRQV 7DEOH$&, 

(6.6)

7KH¿UVWRIWKHVHHTXDWLRQVLVDSSOLFDEOHWRPHPEHUVQRWVXEMHFWHGWRDQD[LDOIRUFH7KHVHFRQGHTXDWLRQLVDSSOL(in pounds), which is taken as positive for comcable where the member is subjected to a factored axial force , and negative for tension forces. Note that for pression forces acting on the gross area of the beam, members with solid cross-sections. Adjoining members must be designed for the redistributed bending moments and shear forces due to application RIWKHFRPSDWLELOLW\WRUVLRQDOPRPHQW$&, 7KHEHDPVIUDPLQJLQWRWKHVSDQGUHOEHDPLQ$&,)LJXUH 5EPXVWEHGHVLJQHGIRUDFRQFHQWUDWHGPRPHQWDWWKHLUHQGVHTXDOWRWKHFRPSDWLELOLW\WRUVLRQDOPRPHQW ; that moment is redistributed to the positive and interior negative critical sections of the beam and are algebraically combined with the bending moments corresponding to gravity and any other applicable loads. Typically, the negative moment at the exterior support of the beam framing into the spandrel beam is reduced compared to that prior to redistribution and the positive and negative moments at the other critical sections are increased. from analysis is less than In cases where torsional effects must be considered and designed to resist the factored torsional moment from analysis.

, the section should be

6.3.2 Critical Sections for Flexure, Shear, and Torsion

,QDFFRUGDQFHZLWK$&,DQG , , and are permitted to be calculated at the faces of WKHVXSSRUWVIRUEHDPVEXLOWLQWHJUDOO\ZLWKWKHVXSSRUWV)RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVRFFXUDWWKHIDFHV of the supports where negative moments are maximum and in the span where positive moments are maximum. Beams are permitted to be designed for the shear force at a critical section located a distance VXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVHH)LJXUH 

from the face of the

(1) Support reaction, in direction of applied shear, introduces compression into the end region of the beam   Loads are applied at or near the top surface of the beam (3) No concentrated load occurs between the face of the support and the critical section The critical section for shear must be taken at the face of the support where one or more of the three conditions in $&,DUHQRWPHW)RUH[DPSOHWKHFULWLFDOVHFWLRQPXVWEHDWWKHIDFHRIWKHVXSSRUWIRUWKHIUDPLQJFRQ¿JXUDWLRQLQ)LJXUHEHFDXVHWKHVXSSRUWLQJPHPEHULVLQWHQVLRQ

6-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‘…‡–”ƒ–‡† Ž‘ƒ†ǡܲ௨ 

‫ݓ‬௨ ƒ’’Ž‹‡†ƒ–‘”‡ƒ”–‘’ •—”ˆƒ…‡‘ˆ„‡ƒ

ܸ௨ ̷݀

ܸ௨ ̷ˆƒ…‡

”‹–‹…ƒŽ•‡…–‹‘ ݀ ൐ ݀ Figure 6.4 Critical section for shear in a beam satisfying the conditions of ACI 9.4.3.2.

‡•‹‘ˆ‘”…‡ǡܶ௨ 

‫ݓ‬௨ 



ܸ௨ ̷ˆƒ…‡

”‹–‹…ƒŽ•‡…–‹‘ Figure 6.5 Critical section for shear in a beam supported by a member in tension.

Beams are permitted to be designed for the torsional moment at a critical section located a distance from the face RIWKHVXSSRUWSURYLGHGQRFRQFHQWUDWHGWRUVLRQDOPRPHQWVRFFXUZLWKLQWKDWGLVWDQFH$&, RWKHUZLVHWKH critical section must be taken at the face of the support. The edge beam in Figure 6.6 is subjected to a uniformly , from the beam framdistributed torsional moment, , from the slab and a concentrated torsional moment, ing into it. The critical section for torsion can be taken a distance from the face of the column in this case because occurs at a distance greater than from the face of the column. As such, the edge beam can be designed for ). the torsional moment at the location of the critical section (that is, at 6.3.3 Redistribution of Moments in Continuous Flexural Members

%HQGLQJPRPHQWVFDOFXODWHGLQDFFRUGDQFHZLWK$&,DWVXSSRUWVRIFRQWLQXRXVÀH[XUDOPHPEHUVDUHSHUPLWWHG WREHLQFUHDVHGRUGHFUHDVHGLQDFFRUGDQFHZLWK$&,H[FHSWZKHUHWKHPRPHQWVKDYHEHHQFRPSXWHGXVLQJWKHDSSUR[LPDWHFRHI¿FLHQWVLQ$&,,WLVFXVWRPDU\WRUHGXFHWKHQHJDWLYHPRPHQWVDWWKHVXSSRUWVZKLFK results in an increase of the positive moments in the span.

6-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

”‹–‹…ƒŽ•‡…–‹‘



݀

ܶ௨ ̷ˆƒ…‡ ܶ௨ ̷݀ ܶ௨ǡ௕௘௔௠  ‫ݐ‬௨ 

൐ ݀

Figure 6.6 Critical section for torsion in a beam where a concentrated torsional moment occurs at a distance greater than d from the face of the support.

7KHPD[LPXPSHUFHQWDJHRIUHGLVWULEXWLRQSHUPLWWHGLVWKHOHVVHURIWLPHVWKHQHWWHQVLOHVWUDLQLQWKHUHLQRUSHUFHQW forcement, Moment redistribution is dependent on adequate ductility in plastic hinge regions, which develop at points of maximum moment and which cause a shift in the elastic bending moment diagram. Thus, redistribution of negative moments is only permitted where LVJUHDWHUWKDQRUHTXDOWRDWWKHVHFWLRQLQZKLFKWKHPRPHQWLVUHGXFHG $GMXVWPHQWVRIWKHQHJDWLYHPRPHQWVDWWKHVXSSRUWVDUHPDGHIRUHDFKORDGLQJFRQ¿JXUDWLRQFRQVLGHULQJSDWWHUQ live loading. It is important to ensure that static equilibrium is maintained at all joints before and after moment redistribution. Thus, a decrease in the negative moments at the supports warrants an increase in the positive moment in the span under consideration. are equal to where Adjusted negative bending moments at the faces of the supports, is the lesser of DQGDQG is the factored negative moment at the face of the support from analysis. The adjusted positive moments are obtained based on equilibrium considering the adjusted negative bending moments at the faces of the supports.

6.4 Design Strength 6.4.1 General

)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\ VHFWLRQLQDEHDP$&,  (6.7) (6.8) (6.9) (6.10)

Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGEHDPVZLWK PXVWEHGHVLJQHGDVWHQVLRQFRQWUROOHGVHFWLRQVLQDFFRUGDQFHZLWK$&,7DEOH$&,  thus, the net tensile strain in the extreme layer of the longitudinal tension reinforcement at nominal strength, , 6-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

must be greater than or equal to and $&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPH , is equal to layer of longitudinal tension reinforcement corresponding to compression-controlled sections, $&, 7KHPRGXOXVRIHODVWLFLW\RIWKHUHLQIRUFHPHQW LVSHUPLWWHGWREHWDNHQDV $&,7DEOH  SVLUHJDUGOHVVRIWKHJUDGHRIWKHUHLQIRUFHPHQW$&, )RUVKHDUDQGWRUVLRQ Methods to determine the nominal strengths tively.

,

, and

DUHJLYHQLQ6HFWLRQVDQGUHVSHF-

6.4.2 Nominal Flexural Strength Rectangular Sections with Tension Reinforcement Only

Single Layer of Tension Reinforcement. 7KHQRPLQDOÀH[XUDOVWUHQJWK , of a rectangular section with one LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,DQGLVEDVHGRQ layer of tension reinforcement and with PRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQVHH$&,DQG)LJXUH  (6.11)

ܾ

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ

݀ െ ሺܽȀʹሻ

ܿ

ܽ ݀ ൌ ݀௧ 

݄

ܽ ൌ ߚଵ ܿ

ͲǤͺͷ݂௖ᇱ  ͲǤͲͲ͵

‫ܣ‬௦  ߝ௧ 

ܶ ൌ ‫ܣ‬௦ ݂௬ 

Figure 6.7 Strain and stress distributions at a positive moment section in a beam.

In this equation, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUHDUHDW a positive moment section and are equally applicable at negative moment sections. The depth of the equivalent stress block, , is determined from force equilibrium, that is, it is determined by set, equal to the tension force in the reinforcement, ting the resultant compressive force in the concrete, , and solving for VHH)LJXUH  (6.12)

,WLVDVVXPHGDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK , is distributed over the , where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV$&, 7KH depth XOWLPDWHVWUDLQLQWKHFRQFUHWHLVDVVXPHGWREH$&, DQGWKHWHUP is the width of the compression face of the member. Multiple Layers of Tension Reinforcement. Under certain conditions, the required tension reinforcement cannot ¿WZLWKLQDVLQJOHOD\HU)RUH[DPSOHLQEHDPVZLWKDZLGWKUHVWULFWLRQGXHWRDUFKLWHFWXUDOFRQVWUDLQWVLWPD\QRWEH SRVVLEOHWRVHOHFWDSUDFWLFDOEDUVL]HZKLOHVDWLVI\LQJWKHPLQLPXPFOHDUVSDFLQJUHTXLUHPHQWVEHWZHHQWKHEDUVLQ $&,,QVXFKFDVHVWKHORQJLWXGLQDOEDUVDUHSURYLGHGLQPRUHWKDQRQHOD\HU

6-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHQRPLQDOÀH[XUDOVWUHQJWK , for sections with multiple layers of reinforcement is determined in the same way as that for sections with a single layer of reinforcement. The main difference is that for beams with two or more layers, the reinforcement in the inner layer(s) may not yield. Therefore, it is important to check whether all the reinis then determined accordingly based on this check. forcement in the section yields or not; &RQVLGHUWKHUHFWDQJXODUEHDPLQ)LJXUHZLWKWZROD\HUVRIUHLQIRUFHPHQW$VVXPHDOOWKHEDUVDUHWKHVDPHVL]H occurs at a distance from the and are located below the neutral axis. Also assume the yield strain and : H[WUHPHFRPSUHVVLRQ¿EHU7KHIROORZLQJUHODWLRQVKLSLVHVWDEOLVKHGEHWZHHQ (6.13) ܾ

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ ݀ െ ሺܽȀʹሻ

݀௧ 

݀௧௬ 

ܿ

ܽ ݀ ݄

ߝ௧௬  ‫ܣ‬௦ 

ܽ ൌ ߚଵ ܿ

ͲǤͺͷ݂௖ᇱ  ͲǤͲͲ͵

ߝ௦  ܶ ൌ ‫ܣ‬௦ ݂௬  ߝ௧ 

Figure 6.8 A reinforced concrete beam with multiple layers of tension reinforcement.

Solving for

results in the following: (6.14)

)RU*UDGHUHLQIRUFHPHQW(TXDWLRQ UHGXFHVWRWKHIROORZLQJ (6.15)

Reinforcement located a distance equal to or greater than IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVDQG is calculated by Equation (6.11) assuming all the reinforcement is concentrated at , which is the distance from the H[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIDOOWKHUHLQIRUFHPHQWVHH)LJXUH :KHUHWKHUHLQIRUFHPHQWLQRQH or more layers does not yield, it are separated from the reinforcement in the layer(s) that yield, and the actual stress is used to calculate . in those bars Rectangular Sections with Tension and Compression Reinforcement

5HLQIRUFHPHQWLVW\SLFDOO\DGGHGWRWKHFRPSUHVVLRQ]RQHRIDEHDPVHFWLRQIRUWKHIROORZLQJUHDVRQV WR , where the dimensions of the beam are limited (due to architectural LQFUHDVHWKHQRPLQDOÀH[XUDOVWUHQJWK is greater than that correspondconstraints, for example) and the area of tensile reinforcement required to resist LQJWRDWHQVLRQFRQWUROOHGVHFWLRQDQG WRKHOSUHGXFHORQJWHUPGHÀHFWLRQVRIWKHEHDPVHH6HFWLRQRIWKLV SXEOLFDWLRQIRUDGLVFXVVLRQRQORQJWHUPGHÀHFWLRQV %HDPVZLWKERWKWHQVLRQDQGFRPSUHVVLRQUHLQIRUFHPHQWDUH commonly referred to as doubly reinforced beams.

6-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5HLQIRUFHPHQWLQWKHFRPSUHVVLRQ]RQHFRQWULEXWHVWR , though the contribution is usually relatively small. Additionally, is larger when compression reinforcement is used in a section, which results in more ductile behavior. Transverse reinforcement in the form of stirrups are typically required in beams to resist factored shear forces. Stirrups must be anchored at both the top and bottom of the section to longitudinal reinforcement to properly develop WKHPLQWHQVLRQVHH6HFWLRQRIWKLVSXEOLFDWLRQ 7KXVUHLQIRUFLQJEDUVLQWKHFRPSUHVVLRQ]RQHPXVWEH provided wherever stirrups are required. As noted above, the contribution of the compression reinforcement to LVXVXDOO\UHODWLYHO\VPDOODQGWKXVLVRIWHQQRWFRQVLGHUHGLQGHVLJQ+RZHYHULIFRPSUHVVLRQUHLQIRUFHPHQWLV . needed to make the section tension-controlled, the following information can be used to determine Nominal Flexural Strength When the Compression Reinforcement Yields. Strain and stress distributions in a doubly reinforced section are given in Figure 6.9 where the total area of compression reinforcement, , is located from the a distance IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU$VVXPLQJWKH\LHOGVWUDLQ , occurs at a distance H[WUHPHFRPSUHVVLRQ¿EHUWKHIROORZLQJUHODWLRQVKLSLVHVWDEOLVKHGEHWZHHQ and :

ͲǤͺͷ݂௖ᇱ  ݀௬ᇱ 

ܿ

ߝ௦ᇱ  ߝ௧௬ 

‫ܥ‬௦ᇱ ൌ ‫ܣ‬ᇱ௦ ݂௦ᇱ  ‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ ݀ െ ሺܽȀʹሻ

݄

݀ ൌ ݀௧ 

‫ܣ‬ᇱ௦ 

ܽ

ͲǤͲͲ͵

ܽ ൌ ߚଵ ܿ

ܾ

݀ᇱ 

(6.16)

‫ܣ‬௦  ߝ௧ 

ܶ ൌ ‫ܣ‬௦ ݂௬ 

Figure 6.9 Strain and stress distributions in a doubly reinforced concrete beam.

Solving for

results in the following: (6.17)

)RU*UDGHUHLQIRUFHPHQW(TXDWLRQ UHGXFHVWRWKHIROORZLQJ (6.18)

Compression reinforcement located a distance equal to or less than IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGV , and the depth of the equivalent stress block, , can be obtained by satisfying force equilibrium (that is, by and solving for ): setting (6.19)

6-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHQRPLQDOÀH[XUDOVWUHQJWKLQWKLVFDVHFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ (6.20)

Nominal Flexural Strength When the Compression Reinforcement Does Not Yield. When the compression rein, the depth of the stress block, , cannot be determined by Equation (6.19) because forcement does not yield is unknown. A relationship between and can be obtained from strain compatibility: the magnitude of (6.21)

LQWR(TXDWLRQ DQGVROYLQJIRU

Substituting

results in the following: (6.22)

From force equilibrium,

can be obtained from the following equation: (6.23)

where ing equations.

and

Note that

and

have the units of kips per square inch in the preced-

7KHQRPLQDOÀH[XUDOVWUHQJWKFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ (6.24)

where

and the terms

and

DUHGHWHUPLQHGE\(TXDWLRQV DQG UHVSHFWLYHO\

T-Beams and Inverted L-Beams with Tension Reinforcement

Effective Flange Width. In typical reinforced concrete construction, the beams and slabs are cast together with reinforcement extending between the members, which results in a monolithic structure. Therefore, the beams do not act alone but rather work with a portion of the slab to resist the effects from the applied loads. ,QJHQHUDOWKHHIIHFWLYHÀDQJHZLGWK , of a beam (that is, the portion of the slab that works with the beam) , slab thickness , and clear span length of the depends on the clear distance between adjoining beam webs . Dimensional limits for for nonprestressed beams supporting monolithic slabs on both sides of the beam beam web (interior beams or T-beams) and on one side of the beam web (edge beams or inverted L-beams) are JLYHQLQ$&,VHH)LJXUH  Nominal Flexural Strength – Flange in Tension. 7KHÀDQJHRID7EHDPRUDQLQYHUWHG/EHDPLVLQWHQVLRQDW locations of negative moment in a continuous system (that is, at the faces of the supports). The strain and stress distributions for a T-beam in this case are illustrated in Figure 6.11. An inverted L-beam has similar distributions. 7KHQRPLQDOÀH[XUDOVWUHQJWKLQWKLVFDVHLVGHWHUPLQHGE\(TXDWLRQ IRUDUHFWDQJXODUVHFWLRQZLWKDVLQJOH If needed, the contribution of the positive reinforcement in the web (not shown layer of reinforcement using FDQEHGHWHUPLQHGXVLQJ(TXDWLRQ RU  in Figure 6.11) can be included and

6-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ௪ଵ 

ܾ௙ଵ

݄

ܾ௙ଶ 

ܾ௙ଵ 

ܾ௪ଶ 

‫ݏ‬௪ଵ 

͸݄  ൌ ܾ௪ଵ ൅ Ž‡ƒ•–‘ˆ ‫ݏ‬௪ଵ Τʹ  ‫ ۔‬ ۖ ‫ە‬κ௡ଵ Τͳʹ ‫ۓ‬ ۖ

ܾ௙ଶ

ܾ௪ଷ 

‫ݏ‬௪ଶ 

ͳ͸݄  ൌ ܾ௪ଶ ൅ Ž‡ƒ•–‘ˆ ሺ‫ݏ‬௪ଵ Τʹሻ ൅ ሺ‫ݏ‬௪ଶ Τʹሻ ‫۔‬  ۖ ‫ ە‬κ௡ଶ ΤͶ ‫ۓ‬ ۖ

Figure 6.10 Effective flange widths for T-beams and inverted L-beams.

ܾ௙  ݄

ߝ௧ 

ܶ ൌ ‫ܣ‬௦ ݂௬ 

ܽ

ܿ

ܾ ൌ ܾ௪ 

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ܽ ൌ ߚଵ ܿ

݀ ൌ ݀௧ 

‫ܣ‬௦ 

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾ௪ ܽ

ͲǤͲͲ͵Ͳ

ܽ

ܾ ൌ ܾ௙ 

ܽ ൌ ߚଵ ܿ

Figure 6.11 Strain and stress distributions in a T-beam with the flange in tension.

ͲǤͲͲ͵Ͳ ݀ െ ሺܽȀʹሻ

݀ ൌ ݀௧ 

݄

ܿ

‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾ௙ ܽ

‫ܣ‬௦  ܾ௪ 

ߝ௧ 

ܶ ൌ ‫ܣ‬௦ ݂௬ 

Figure 6.12 Strain and stress distributions in a T-beam with the flange in compression and D”K.

Nominal Flexural Strength – Flange in Compression. At locations of positive moment in a continuous system, DSRUWLRQRIWKHÀDQJHRUWKHHQWLUHÀDQJHRID7EHDPRULQYHUWHG/EHDPLVLQFRPSUHVVLRQ7KHGHWHUPLQDWLRQRI depends on whether the depth of the equivalent stress block, LVOHVVWKDQRUJUHDWHUWKDQWKHÀDQJHWKLFNQHVV 'HSWKRI6WUHVV%ORFN/HVV7KDQRU(TXDOWRWKH)ODQJH7KLFNQHVV . The depth of the equivalent stress block, LVLQLWLDOO\GHWHUPLQHGE\(TXDWLRQ IRUVHFWLRQVZLWKRQHOD\HURIUHLQIRUFHPHQWRUE\(TXDWLRQ IRU WKHFRPSUHVVLYH]RQHLVUHFWDQJXODUZLWKDZLGWKHTXDOWR doubly reinforced sections. If it is found that for sections with one layer of reinforcement. SimiVHH)LJXUH (TXDWLRQ FDQEHXVHGWRGHWHUPLQH ODUO\(TXDWLRQ RU FDQEHXVHGLIFRPSUHVVLRQUHLQIRUFHPHQWLVLQFOXGHG

6-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HSWKRI6WUHVV%ORFN*UHDWHU7KDQWKH)ODQJH7KLFNQHVV . Where GHWHUPLQHGE\(TXDWLRQ RU  IDOOVZLWKLQWKHZHEWKHFRPSUHVVLYH]RQHLV7RU/VKDSHGDVRSSRVHGWRUHFWDQJXODUVHH)LJXUHIRUD 7EHDP 7KHQRPLQDOÀH[XUDOVWUHQJWKRIWKHVHFWLRQLQWKLVFDVHFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ (6.25)

where

and ܾ ൌ ܾ௙ 

‫ܥ‬ ݆݀

݀ ൌ ݀௧ 

ܿ

݄

ܽ ൌ ߚଵ ܿ

ͲǤͲͲ͵Ͳ

‫ܣ‬௦  ߝ௧ 

ܾ௪ 

ܶ ൌ ‫ܣ‬௦ ݂௬ 

Figure 6.13 Strain and stress distribution in a T-beam with the flange in compression and a>h.

6.4.3 Nominal Shear Strength Overview

The nominal one-way shear strength,

DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  (6.26)

where is the nominal shear strength provided by concrete and reinforcement.

is the nominal shear strength provided by shear

Nominal Shear Strength Provided by Concrete

For nonprestressed members, LVGHWHUPLQHGE\WKHDSSOLFDEOHHTXDWLRQLQ$&,7DEOH$&, 7KH HTXDWLRQVLQWKDWWDEOHDUHJLYHQLQ7DEOHDORQJZLWKWKHPD[LPXPSHUPLWWHG VSHFL¿HGLQ$&, Table 6.4 Nominal Shear Strength Provided by Concrete, Vc Area of Shear Reinforcement, Av*

*Minimum area of shear reinforcement, **

6-16

, is determined by ACI Table 9.6.3.4

Vc**

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The minimum area of shear reinforcement in a beam, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,VHH6HFWLRQRIWKLVSXEOLFDWLRQ 7KHUHTXLUHGDUHDRIVKHDUUHLQIRUFHPHQW , is often greater than or equal to at the critical section. 7KHPRGL¿FDWLRQIDFWRU WKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWR QRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ Table 6.3 based on composition of aggregates in the concrete mix. 7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength attributed to concrete in members without shear reinforcement does not increase in direct proportion with member GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (6.27)

,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK EHDPVZLWKDQRYHUDOOGHSWKOHVVWKDQDERXWLQ$VQRWHGDERYHEHFDXVH , this factor is usually not applicable when determining for beams.

This means for is often greater than or equal to

The term LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by where is the width RIWKHZHERIWKHEHDP$FFRUGLQJWR$&,5 may be taken as the sum of the areas of the longitudinal ÀH[XUDOUHLQIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUVZLWKRQHOD\HURIWHQVLRQUHLQIRUFHPHQW LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDW section. $FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken , and is to be taken as negative for tenas positive for compression forces acting on the gross area of the beam, sion forces. used to calculate DUHOLPLWHGWRSVL$&, H[FHSWDVDOORZHGLQ$&,ZKLFK Values of WREHJUHDWHUWKDQSVLSURYLGHGWKHPLQLPXPVKHDUUHLQIRUFHPHQWLQ$&,RUWKH permits values of PLQLPXPWUDQVYHUVHWRUVLRQDOUHLQIRUFHPHQWLQ$&,LIDSSOLFDEOHLVSURYLGHGLQWKHVHFWLRQ 7KLVOLPLWDWLRQ is primarily due to the fact that there is a lack of test data and practical experience with concrete having comon SUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\ IRUEHDPVVKRXOGQRWH[FHHGSVL5HIHUHQFH  7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@ (6.28)

In typical situations, beams are subjected to shear forces acting along one axis only, so the requirements in ACI DQGIRULQWHUDFWLRQRIVKHDUIRUFHVDORQJRUWKRJRQDOD[HVDUHXVXDOO\QRWDSSOLFDEOH Nominal Shear Strength Provided by Shear Reinforcement

The nominal shear strength provided by shear reinforcement, LVWREHGHWHUPLQHGE\$&,$&,  The following types of shear reinforcement are permitted in nonprestressed members: (1) 6WLUUXSVWLHVRUKRRSVSHUSHQGLFXODUWRWKHORQJLWXGLQDOD[LVRIWKHPHPEHU>$&,D @   Welded wire reinforcement with wires located perpendicular to the longitudinal axis of the member [ACI E @ (3) 6SLUDOUHLQIRUFHPHQW>$&,F @

6-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

  ,QFOLQHGVWLUUXSVPDNLQJDQDQJOHRIDWOHDVWGHJUHHVZLWKWKHORQJLWXGLQDOD[LVRIWKHPHPEHUDQGFURVVLQJ WKHSODQHRIWKHSRWHQWLDOVKHDUFUDFN$&,   /RQJLWXGLQDOUHLQIRUFHPHQWEHQWDQDQJOHRIGHJUHHVRUPRUHZLWKUHVSHFWWRWKHORQJLWXGLQDOD[LVRIWKH PHPEHU$&, 6WLUUXSVRULHQWHGSHUSHQGLFXODUWRWKHD[LVRIWKHPHPEHUDQGDQFKRUHGWRWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW are the most commonly used type of shear reinforcement for beams in buildings assigned to SDC A or B. In buildings assigned to SDC C and above, spirals or hoops in accordance with the provisions in Chapter 18 of ACI 318 PXVWEHSURYLGHGVHH&KDSWHUVDQGRIWKLVSXEOLFDWLRQ ,QFOLQHGVWLUUXSVDQGEHQWORQJLWXGLQDOEDUVDUHHVsentially not used any longer and, thus, are not covered here. 7KHQRPLQDOVKHDUVWUHQJWKSURYLGHGE\VWLUUXSVLVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (6.29)

In this equation, is the total area of shear reinforcement within the center-to-center spacing, , of the stirrups. )RUWKHWZROHJJHGVWLUUXSVGHSLFWHGLQ)LJXUHZKLFKDUHFRPPRQO\UHIHUUHGWR8VWLUUXSVEHFDXVHRIWKHLU where is the area of the stirrup bar. In general, is equal to times the number of stirrup shape), OHJVSURYLGHG$&,      ‫ܣ‬௕ 

‫ܣ‬௩ ൌ ʹ‫ܣ‬௕ 

–Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

‫ݏ‬

Figure 6.14 Two-legged U-stirrup configuration.

Locating stirrups solely around the perimeter of a wide beam is not fully effective. Thus, stirrup legs are required in the interior of a wide beam. The maximum stirrup spacing across the width of a member is given in ACI Table WKLVUHTXLUHPHQWHVVHQWLDOO\WUDQVODWHVWRDPLQLPXPQXPEHURIVWLUUXSOHJVQHHGHGDFURVVWKHZLGWKRI DEHDP,QVXFKFDVHVVKHDUUHLQIRUFHPHQWXVXDOO\FRQVLVWVRIVHWVRI8VWLUUXSV)RUWKHFRQ¿JXUDWLRQLOOXVWUDWHG LQ)LJXUHDVLQJOHRSHQVWLUUXSH[WHQGVWKHIXOOQHWZLGWKRIWKHEHDPWKHQHWZLGWKLVHTXDOWRWKHIXOOEHDP ZLGWKPLQXVWKHWRWDOVLGHFRYHU $VWLUUXSFDSFRQVLVWLQJRIDKRUL]RQWDOEDUZLWKDGHJUHHKRRNDWRQHHQGDQG DGHJUHHKRRNDWWKHRWKHUHQGLVSURYLGHGDWWKHWRSRIWKHFRQ¿JXUDWLRQZKLFKDOVRH[WHQGVWKHIXOOQHWZLGWKRI the beam. Providing a full-width stirrup helps in maintaining the required concrete cover and facilitates installation RIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDP7ZRVHWVRILGHQWLFDO8VWLUUXSVZLWKGHJUHHKRRNVDWHDFKHQG . DUHSODFHGV\PPHWULFDOO\ZLWKLQWKHLQWHULRURIWKHEHDP,QWKLVFRQ¿JXUDWLRQ In relatively narrow beams, a continuous bent bar, which forms a series of single-leg stirrups along the length of WKHPHPEHUFDQEHXVHGDVVKHDUUHLQIRUFHPHQWZKHUHUHTXLUHG$&,5VHH)LJXUH $&,F  SHUPLWVDQGEDUVWREHXVHGDVVKHDUUHLQIRUFHPHQWLQVWDQGDUGMRLVWFRQVWUXFWLRQZKLFKLVGH¿QHGLQ$&, 9.8) where the bar is anchored by a standard hook, which is not necessarily engaging any of the longitudinal rein-

6-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 6.15 Multiple-leg stirrup configuration for wide beams.



–ƒ†ƒ”†Š‘‘

‫ܣ‬௕ 

‫ݏ‬ Figure 6.16 Single-leg stirrup configuration for joists and narrow beams.

IRUFHPHQW$OWKRXJKWKLVW\SHRIVKHDUUHLQIRUFHPHQWLVSHUPLWWHGVSHFL¿FDOO\IRUVWDQGDUGMRLVWFRQVWUXFWLRQLWLV assumed that it can be used in joists that are part of a wide-module joist system or for any narrow beam where U. VWLUUXSVFDQQRWEHXVHG,QWKLVFRQ¿JXUDWLRQ 6.4.4 Nominal Torsional Strength

Where torsional effects must be considered , nominal torsional strength is determined in accordance ZLWK$&,$&, )RUQRQSUHVWUHVVHGPHPEHUV LVHTXDOWRWKHIROORZLQJ$&, 

(6.30)

The term LVWKHJURVVDUHDRIWKHEHDPFURVVVHFWLRQHQFORVHGE\WKHWRUVLRQDOVKHDUÀRZSDWKDIWHUFUDFNLQJ,Q is permitted to be approximated as where is the area enclosed by lieu of determining it by analysis, WKHFHQWHUOLQHRIWKHRXWPRVWFORVHGWUDQVYHUVHWRUVLRQDOUHLQIRUFHPHQWLQWKHVHFWLRQVHH$&,DQG$&, where and are the distances )LJXUH5 )RUWKHLQWHULRUDQGHGJHEHDPVLQ)LJXUH between the centerlines of the outmost closed stirrups in the section. Closed stirrups must be used because cracking due to torsional moments can occur on all faces of a beam. The perimeter of the centerline of the outermost closed transverse torsional reinforcement, LVDOVRJLYHQLQ)LJXUHIRUVROLGVHFWLRQV1RWHWKDWWKHGH¿QLWLRQVRI and LQ)LJXUHDUHWKHVDPHIRUKROORZVHFWLRQV The term is the area of one leg of the outermost closed stirrups, which have a center-to-center spacing of along the length of the member. Regardless of the total number of stirrup legs, only the outermost legs resist torsional effects; this is consistent with the thin-walled tube methodology that forms the basis of the torsional provisions in $&, 6-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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݀௦  ܿ ܾ௪ 

ܾ௪  ݀௦ ൌ †‹ƒ‡–‡”‘ˆ…Ž‘•‡†•–‹””—’ ܿ ൌ …Ž‡ƒ”…‘˜‡”–‘…Ž‘•‡†•–‹””—’ ۷‫ܚܗܑܚ܍ܜܖ‬۰‫ܕ܉܍‬

۳‫܍܏܌‬۰‫ܕ܉܍‬ ‫ݔ‬ଵ ൌ ܾ௪ െ ʹܿ െ ݀௦  ‫ݕ‬ଵ ൌ ݄௕ ൅ ݄ െ ʹܿ െ ݀௦  ‫ܣ‬௢௛ ൌ ‫ݔ‬ଵ ‫ݕ‬ଵ  ‫݌‬௛ ൌ ʹሺ‫ݔ‬ଵ ൅ ‫ݕ‬ଵ ሻ

Figure 6.17 Torsional section properties Aoh and ph.

The total area of longitudinal reinforcement required to resist the effects of torsion, , must be uniformly distribXWHGDURXQGWKHSHULPHWHURIWKHEHDPDQGLVFRPELQHGZLWKWKHORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHGIRUÀH[XUHVHH Section 6.5.4 of this publication). The angle is related to the orientation of the compression diagonals that are part of the space truss assumed to UHVLVWWKHHIIHFWVIURPWRUVLRQDIWHUFUDFNLQJWKHWUDQVYHUVHDQGORQJLWXGLQDOUHLQIRUFHPHQWDUHWKHRWKHUSDUWVRIWKH WUXVVVHH)LJXUH $&,D SHUPLWV to be taken as 45 degrees for nonprestressed members instead of determining it from analysis.

‘‰‹–—†‹ƒŽ„ƒ”• ”ƒ…•

Ž‘•‡†•–‹””—’•

ߠ

‫ݕ‬ଵ  ‘…”‡–‡…‘’”‡••‹‘ †‹ƒ‰‘ƒŽ•

‫ݔ‬ଵ 

 Figure 6.18 Space truss analogy for a reinforced concrete beam subjected to torsional loading.

6-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In order to help reduce unsightly cracking and to prevent crushing of the inclined concrete compression struts due to shear and torsion, the cross-sectional dimensions of solid and hollow sections are limited by ACI Equations D DQGE UHVSHFWLYHO\$&,  (6.31)

(6.32)

The terms on the left-hand side of these equations are the stresses due to shear and torsion, respectively. In solid sections, shear stresses act over the full width of the section, while those due to torsion are resisted by a WKLQZDOOHGWXEH>VHH$&,)LJXUH5E @7KXVWKHVWUHVVHVFDQQRWEHGLUHFWO\DGGHGWRJHWKHU(TXDWLRQ  represents an elliptical interaction between shear and torsional stresses. For hollow sections, the shear and torsional stresses are directly additive on one side wall of the section [see ACI )LJXUH5D @7KLVOLQHDULQWHUDFWLRQRIVWUHVVHVLVHYLGHQWLQ(TXDWLRQ )RUKROORZVHFWLRQVZKHUHWKH ZDOOWKLFNQHVVYDULHVDURXQGWKHSHULPHWHURIWKHVHFWLRQ(TXDWLRQ PXVWEHHYDOXDWHGZKHUHWKHVXPPDWLRQ RIWKHVKHDUDQGWRUVLRQDOVWUHVVHVRQWKHOHIWKDQGVLGHRIWKHHTXDWLRQLVDPD[LPXP$&, $OVRZKHUH , the torsional stress LQ(TXDWLRQ PXVW the wall thickness of a hollow section is less than where is the thickness of the wall of the hollow section at the location where stresses are be taken as EHLQJHYDOXDWHG$&,  7KHGLPHQVLRQVRIWKHFURVVVHFWLRQPXVWEHLQFUHDVHGZKHUH(TXDWLRQ RU LVQRWVDWLV¿HG,QFUHDVLQJ the cross-sectional dimensions of a section typically has a greater impact than increasing . Solid sections with an aspect ratio (where is the width of that part of the cross-section containing the closed stirrups provided to resist torsion) are permitted to be designed by a procedure other than that given in ACI 318 provided the procedure has demonstrated that the results are in substantial agreement with results from compreKHQVLYHWHVWV$&, ,QVXFKFDVHVWKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,DQGWKHGHWDLOLQJUHTXLUHPHQWVRI$&,DQGQHHGQRWEHVDWLV¿HG2QHVXFKGHVLJQSURFHGXUHLVUHIHUHQFHGLQ$&, 5

6.5 Determination of Required Reinforcement 6.5.1 Required Flexural Reinforcement Rectangular Sections with Tension Reinforcement Only

Single Layer of Tension Reinforcement. 7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW , at a critical section of a EHDPZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHPHQWLVGHWHUPLQHGEDVHGRQWKHUHTXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV and IRUWHQVLRQFRQWUROOHGVHFWLRQV8VLQJ(TXDWLRQV DQG WKHUHTXLUHG can be determined by the following equation: (6.33)

7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH

LVGH¿QHGDVIROORZV (6.34)

where

for tension-controlled sections. 6-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is important to check that calculated by Equation (6.33) is at least equal to the following minimum area of $&,  ÀH[XUDOUHLQIRUFHPHQW

(6.35)

)RUSVLFRQFUHWHDQG*UDGHUHLQIRUFHPHQW LWHGWRDPD[LPXPRISVL

governs. The value of

LQ(TXDWLRQ LVOLP-

0LQLPXPÀH[XUDOUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ QHHGQRWEHVDWLV¿HGDWDQ\VHFWLRQZKHUHDWOHDVW is provided where LVFDOFXODWHGE\(TXDWLRQ >$&,@ an area of reinforcement equal to It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear for tension-controlled sections (see strain distribution where is the depth of the neutral axis when $&,DQG)LJXUH  (6.36)

Substituting LQWR(TXDWLRQ ZLWK from Equation (6.36), the following equation can be used to , corresponding to tension-controlled sections for beams with FDOFXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW : (6.37)

)RU*UDGHUHLQIRUFHPHQW SURSHUWLHV(TXDWLRQ UHGXFHVWRWKHIROORZLQJ

and for

For these material (6.38)

If calculated by Equation (6.33) is greater than ZKLFKLVHVVHQWLDOO\WKHPD[LPXPDPRXQWRIÀH[XUDOUHLQ, and the dimensions of the beam forcement permitted at any section, the section is not tension-controlled PXVWEHPRGL¿HGDFFRUGLQJO\WRDWWDLQDWHQVLRQFRQWUROOHGVHFWLRQ is determined in the same way as for sections with Multiple Layers of Tension Reinforcement. The required one layer of tension reinforcement using and (see Figure 6.8). For beams with two layers of tension reinforceusing +RZHYHULWLVSHUPLWWHGWRXVH instead of the smaller ment, it is conservative to determine this may be advantageous in cases where the section is at or very close to the tension-controlled strain limit, The reinforcement ratio tion (

based on ; see Figure 6.8):

is equal to the following considering force equilibrium on the sec(6.39)

6-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Similarly, the reinforcement ratio to the following:

corresponding to a tension-controlled section based on

is equal (6.40)

The following relationship between

and

LVREWDLQHGIURP(TXDWLRQV DQG  (6.41)

The preceding discussion assumes the inner layer of reinforcement yields, which is advantageous for design purposIURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVZKHUH es. Reinforcement located a distance equal to or greater than LVFDOFXODWHGE\(TXDWLRQ RUE\(TXDWLRQ IRU*UDGHUHLQIRUFHPHQWVHH6HFWLRQRIWKLV publication). Unless reinforcement is provided on the sides of a beam, it is common for all layers of reinforcement to yield in normally proportioned beams. Rectangular Sections with Tension and Compression Reinforcement

For rectangular sections where it has been determined that the section is not tension-controlled (that is, ), compression reinforcement may be added to the section so that it becomes tension-controlled (see 6HFWLRQRIWKLVSXEOLFDWLRQDQG)LJXUH  which means the section is designed at the A common design method in this situation is to set , correWHQVLRQFRQWUROOHGQHWWHQVLOHVWUDLQOLPLW7KHQH[WVWHSLVWRGHWHUPLQHWKHQRPLQDOÀH[XUDOVWUHQJWK using Equation (6.11): sponding to (6.42)

where )RUDVHFWLRQZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHPHQWWKHUHLQIRUFHPHQWUDWLRWREHXVHGLQ(TXDWLRQ LV the following: (6.43)

where

LVGHWHUPLQHGE\(TXDWLRQ 

In situations where compression reinforcement is required, it is common for two layers of tension reinforcement to EHQHHGHG,QWKLVFDVHWKHUHLQIRUFHPHQWUDWLRWREHXVHGLQ(TXDWLRQ LVWKHIROORZLQJ>VHH(TXDWLRQ @ (6.44)

7KHUHTXLUHGQRPLQDOÀH[XUDOVWUHQJWKUHVLVWHGE\WKHFRPSUHVVLRQUHLQIRUFHPHQW and : UHTXLUHGQRPLQDOÀH[XUDOVWUHQJWK

, is the difference between the

(6.45)

where

6-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In general, the stress in the compression reinforcement,

, can be determined by the following equation: (6.46)

where . Compression reinforcement located a distance equal to or less than IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVZKHUH LVGHWHUPLQHGE\(TXDWLRQ RU >VHH6HFWLRQ RIWKLVSXEOLFDWLRQ@ The required area of compression reinforcement,

, is determined from strength design: (6.47)

Finally, the total required area of tension reinforcement, , is the summation of the area of reinforcement correassuming : sponding to the tension-controlled strain limit and (6.48)

where LVGHWHUPLQHGE\(TXDWLRQ RU IRUDVLQJOHOD\HURUWZROD\HUVRIWHQVLRQUHLQIRUFHPHQWUHVSHFLVGHWHUPLQHGE\(TXDWLRQ  tively, and T-Beams and Inverted L-Beams with Tension Reinforcement

Nominal Flexural Strength – Flange in Tension. 7KHÀDQJHRID7EHDPRUDQLQYHUWHG/EHDPLVLQWHQVLRQDW ORFDWLRQVRIQHJDWLYHPRPHQWLQDFRQWLQXRXVV\VWHPVHH6HFWLRQRIWKLVSXEOLFDWLRQ  2QFHWKHHIIHFWLYHÀDQJHZLGWK LVGHWHUPLQHGVHH)LJXUH WKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW is calculated using Equation (6.33) for a rectangular section with a single layer of reinforcement (see Figure 6.11). If and are determined needed, the contribution of compression reinforcement in the web can be included where E\(TXDWLRQV DQG UHVSHFWLYHO\ 7KHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVLQ$&,DUHDOVRDSSOLFDEOH>VHH(TXDWLRQ @ Nominal Flexural Strength – Flange in Compression. $WORFDWLRQVRISRVLWLYHPRPHQWDSRUWLRQRIWKHÀDQJHRU WKHHQWLUHÀDQJHRID7EHDPRULQYHUWHG/EHDPLVLQFRPSUHVVLRQ$VVXPLQJWKHVHFWLRQLVWHQVLRQFRQWUROOHGZLWK ), FDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQVHH)LJXUH  rectangular section behavior (that is, (6.49)

where

and

If GHWHUPLQHGE\(TXDWLRQ LVOHVVWKDQRUHTXDOWR , the initial assumption that the section behaves as a can be determined by Equation (6.33). rectangular section is correct and If it is found that WKHLQLWLDODVVXPSWLRQRIUHFWDQJXODUEHKDYLRULVQRWFRUUHFWDQGWKHFRPSUHVVLYH]RQHLV , and the T- or L-shaped (see Figure 6.13). In such cases, the next step is to determine the area of reinforcement, FRUUHVSRQGLQJWRWKHRYHUKDQJLQJÀDQJHV  design strength, (6.50)

6-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



(6.51)

The moment strength of the beam web, section:

, is determined by subtracting the design strength

from

at the



(6.52)

The area of flexural reinforcement, and

, required to resist

is determined by Equations (6.34) and (6.33) where

The total required area of flexural reinforcement is the sum of the areas corresponding to the contributions from the overhanging flange(s) and the web:

(6.53)

Once is determined by Equation (6.53), the assumption that the section is tension-controlled needs to be where is determined by Equation (6.37) or (6.38) with checked, that is, it must be verified that If the section is not tension-controlled, compression reinforcement may be added to the section to make it tension-controlled or the dimensions of the beam must be modified. The minimum reinforcement requirements in ACI 9.6.1.2 are also applicable [see Equation (6.35)]. 6.5.2 Required Shear Reinforcement

The required area and spacing of shear reinforcement perpendicular to the axis of a member is determined by Equation (6.8), which can be rewritten as follows:



(6.54)

The nominal shear strength provided by the concrete, , is obtained from Table 6.4 and the nominal shear strength provided by the shear reinforcement (stirrups), , is calculated by Equation (6.29).

𝑑𝑑

A summary of the shear requirements in ACI 9.6.3, 9.7.6, and 22.5 is given in Table 6.5 assuming the effects from any torsional moments, if present, need not be considered. Note that these requirements are not applicable to the beam types in ACI Table 9.6.3.1, which are cases where the minimum shear reinforcement requirements of ACI Also, “Along the length” in Table 6.5 refers to the maximum spacing of 9.6.3.4 need not be satisfied if legs of shear reinforcement along the length of the member and “Across the width” refers to the maximum spacing of legs of shear reinforcement across the width of the member (see Figure 6.19).

𝑠𝑠��� �

𝑑𝑑 ⎧lesser of � � for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑 24 ⎪

⎨ 𝑑𝑑/2 ⎪lesser of � for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑 ⎩ 12″

𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝐭𝐭𝐭𝐭𝐭𝐭 𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰

𝑠𝑠��� �

𝑑𝑑 2 ⎧lesser of � � for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑 24 ⎪

⎨ ⎪lesser of �𝑑𝑑/4 for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑 ⎩ 12″

𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝐭𝐭𝐭𝐭𝐭𝐭 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥

Figure 6.19 Maximum spacing requirements for legs of shear reinforcement.

6-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.5 Shear Reinforcement Requirements for Beams* Case

1

Required Shear Strength, Vu

Stirrup spacing, s

Required Shear Reinforcement, Av

Required

Maximum

None

ņ

ņ • Along the length:

 • Across the width:

• For ±$ORQJWKHOHQJWK

±$FURVVWKHZLGWK 3

• For ±$ORQJWKHOHQJWK

±$FURVVWKHZLGWK

*Minimum reinforcement requirements in this table are applicable in all cases except for those in ACI Table 9.6.3.1.

3ULRUWRFDOFXODWLQJWKHUHTXLUHGVKHDUUHLQIRUFHPHQWLWLVLPSRUWDQWWRFKHFNWKDW(TXDWLRQ LVVDWLV¿HG$&,  7KHVL]HRIWKHVHFWLRQPXVWEHLQFUHDVHGZKHUH )RUHFRQRP\VWLUUXSVL]HDQGVSDFLQJDORQJWKHOHQJWKDQGDFURVVWKHZLGWKRIWKHPHPEHU DUHHVWDEOLVKHGDWWKH FULWLFDOVHFWLRQ¿UVW$WVHFWLRQVDZD\IURPWKHFULWLFDOVHFWLRQWKHVSDFLQJFDQEHLQFUHDVHGEXWWKHVWLUUXSEDUVL]H and number of legs should be the same as those at the critical section. Stirrup spacing should be changed as few times as possible over the required length; it is good practice to provide no more than 3 different stirrup spacings ZKHUHUHTXLUHGDOWKRXJKGLIIHUHQWVSDFLQJVDUHSUHIHUUHGLISRVVLEOHRUSUDFWLFDO ZLWKWKH¿UVWVWLUUXSORFDWHGQR PRUHWKDQLQIURPWKHIDFHRIVXSSRUW/DUJHUVWLUUXSEDUVDWZLGHUVSDFLQJDUHXVXDOO\PRUHFRVWHIIHFWLYHWKDQ smaller bars at closer spacing; the latter require disproportionally higher costs for fabrication and placement. 7KHVHJPHQWVDORQJWKHVSDQZKHUHVKHDUUHLQIRUFHPHQWLVDQGLVQRWUHTXLUHGEDVHGRQWKHWKUHHFDVHVLQ7DEOH DUHLGHQWL¿HGLQ)LJXUHIRUDUHLQIRUFHGFRQFUHWHEHDPVXEMHFWHGWRDIDFWRUHGXQLIRUPO\GLVWULEXWHGJUDYLW\ . In this case, two stirrup spacings are provided along the length of the member assuming the three requireload, PHQWVLQ$&,SHUWDLQLQJWRWKHORFDWLRQRIWKHFULWLFDOVHFWLRQIRUVKHDUDUHVDWLV¿HG WKHVSDFLQJ corre, sponds to the factored shear force at the critical section located a distance from the face of the support corresponds to that based on minimum shear reinforcement DQG WKHVSDFLQJ

6-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ݓ‬௨ 



݀

ܸ௨ ̷ˆƒ…‡

ܸ௨ ̷݀

൑ ʹԣ

‫ ݏ‬ൌ Ž‡••‡”‘ˆ ൜ ‫ݏ‬ൌ ݀

‫ݏ‬௠௜௡ ߶‫ܣ‬௩ ݂௬௧ ݀ ൑ Ž‡••‡”‘ˆ ൜  ሺܸ௨ ̷݀ሻ െ ߶ܸ௖ ‫ݏ‬௠௔௫

‫ݏ‬௠௜௡ ‫ݏ‬௠௔௫



”‹–‹…ƒŽ•‡…–‹‘ κ௡ Ȁʹ ‫ܣ‬௩ ݂௬௧ ΤͲǤ͹ͷ—݂௖ᇱ ܾ௪ ‘–‡ǣ ‫ݏ‬௠௜௡ ൌ Ž‡••‡”‘ˆ ቐ



‫ܣ‬௩ ݂௬௧ ΤͷͲܾ௪ 

 ݀Ȁʹ ˆ‘”ሺܸ௨ ̷݀ሻ െ ߶ܸ௖ ൑ ߶Ͷ—݂௖ᇱ ܾ௪ ݀ ʹͶԣ  ݀ȀͶ Ž‡••‡”‘ˆ ቄ ˆ‘”ሺܸ௨ ̷݀ሻ െ ߶ܸ௖ ൐ ߶Ͷ—݂௖ᇱ ܾ௪ ݀ ͳʹԣ

ሺܸ௨ ̷݀ሻ െ ߶ܸ௖ 

Ž‡••‡”‘ˆ ቄ ‫ݏ‬௠௔௫ ൌ ൞ ݀

Š‡ƒ”ˆ‘”…‡”‡•‹•–‡†„›…‘…”‡–‡ሺ߶ܸ௖ ሻ ߶ܸ௖ 

ܸ௨ ̷݀

ܸ௨ ̷ˆƒ…‡

Š‡ƒ”ˆ‘”…‡”‡•‹•–‡†„›•–‹””—’•ሺ߶ܸ௦ ሻ

߶ߣ—݂௖ᇱ ܾ௪ ݀

ƒ•‡͵

ƒ•‡ʹ

Š‡ƒ””‡‹ˆ‘”…‡‡–”‡“—‹”‡†

‹‹— •Š‡ƒ” ”‡‹ˆ‘”…‡‡–

‫ݔ‬௠௦ ൌ ሾሺܸ௨ ̷ˆƒ…‡ሻ െ ߶ܸ௖ ሿΤ‫ݓ‬௨ 

ƒ•‡ͳ Š‡ƒ” ”‡‹ˆ‘”…‡‡– ‘–”‡“—‹”‡†

‫ݔ‬௡௦ ൌ ൣሺܸ௨ ̷ˆƒ…‡ሻ െ ߶ߣ—݂௖ᇱ ܾ௪ ݀൧Τ‫ݓ‬௨ 

ƒ•‡ ͳ

ࢂ࢛  ܸ௨ ൑

Š‡ƒ”‡‹ˆ‘”…‡‡–

߶ߣ—݂௖ᇱ ܾ௪ ݀

‘–”‡“—‹”‡† ͲǤ͹ͷ—݂௖ᇱ ܾ௪ ‫ݏ‬ൗ݂௬௧

ʹ

߶ߣ—݂௖ᇱ ܾ௪ ݀ ൏ ܸ௨ ൑ ߶ܸ௖ 

‫ܣ‬௩ǡ௠௜௡ ൌ ‰”‡ƒ–‡”‘ˆ ቐ

 ͷͲܾ௪ ‫ݏ‬Τ݂௬௧ 

͵

ሺܸ௨ ̷݀ሻ ൐ ߶ܸ௖ 

‫ܣ‬௩ ൌ

ሾሺܸ௨ ̷݀ሻ െ ߶ܸ௖ ሿ‫ݏ‬ ൒ ‫ܣ‬௩ǡ௠௜௡  ߶݂௬௧ ݀

Figure 6.20 Segments along the span of a reinforced concrete beam where shear reinforcement is and is not required.

6-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7RIDFLOLWDWHGHWHUPLQDWLRQRIWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHVKHDUUHLQIRUFHPHQWYDOXHVRI are given where LVWDNHQDVDQGZKLFK LQ7DEOHXVLQJWKHHTXDWLRQVLQ7DEOHIRUVWLUUXSVSDFLQJV provides a practical range for 7KHVHYDOXHVDUHEDVHGRQ*UDGHVWLUUXSVZLWKOHJVDQGDUHLQGHSHQGHQWRI PHPEHUVL]HDQGFRQFUHWHFRPSUHVVLYHVWUHQJWK)RUVWLUUXSFRQ¿JXUDWLRQVZLWKPRUHWKDQOHJVWKHWDEXODWHG YDOXHVFDQEHLQFUHDVHGDFFRUGLQJO\IRUH[DPSOHIRUDVWLUUXSFRQ¿JXUDWLRQZLWKOHJVWKHFRUUHVSRQGLQJYDOXH LVHTXDOWRWZRWLPHVWKHWDEXODWHGYDOXH $VWLUUXSVL]HDQGVSDFLQJLVVHOHFWHGIURP7DEOHWKDW of equal to or slightly greater than that required from analysis. For example, assume provides a value of )URP7DEOH8VWLUUXSVOHJV VSDFHGDW provides from analysis the required where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKH longitudinal tension reinforcement in the beam. Table 6.6 Values of Vu –ҀVc for Grade 60 Stirrups* Stirrup spacing, s

#3 U-stirrups

#4 U-stirrups

#5 U-stirrups

19.8











39.6



111.6

*Tabulated values are for 2 legs of vertical shear reinforcement. Multiply the tabulated values by total number of stirrup legs provided across the width of the member.

for stirrup arrangements with other than 2 legs where

is the

The distance from the face of the support where minimum shear reinforcement can be used is determined by setting . The following equation is apthe factored shear force, , equal to the design shear strength of the concrete, SOLFDEOHIRUWKHEHDPLQ)LJXUH (6.55)

where is the distance from the face of the support to the section where minimum shear reinforcement can be results in the following: XVHGWKDWLVWKHGLVWDQFHWRWKHRQVHWRI&DVH 6ROYLQJIRU (6.56)

Similarly, the distance, from the face of the support to the section where shear reinforcement is no longer required (that is, the distance to the onset of Case 1) can be determined from the following equation: (6.57)

In general, the following equation can be used to determine the distance, , from the face of the support to the sec, WLRQZKHUHWKHVWLUUXSFRQ¿JXUDWLRQFDQEHVSDFHGDW along the span for a given area of shear reinforcement, where is greater than that determined at the critical section:

(6.58)

6-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is important to ensure the value of XVHGLQ(TXDWLRQ LVOHVVWKDQRUHTXDOWRWKHJRYHUQLQJ and can be used is YDOXHVJLYHQLQ7DEOH)RUH[DPSOHWKHGLVWDQFHIURPWKHIDFHRIWKHVXSSRUWZKHUH determined as follows: (6.59)

6.5.3 Required Torsion Reinforcement Transverse Reinforcement

The required area and spacing of transverse torsional reinforcement perpendicular to the axis of a member are calculated by Equation (6.9) where LVGHWHUPLQHGE\WKH¿UVWRIWKHHTXDWLRQVLQ(TXDWLRQ  (6.60)

As noted previously, is the area of one leg of the outermost closed stirrups, which have a center-to-center spacing of along the length of the member. Regardless of the total number of stirrup legs across the width of the beam, only the outermost legs resist torsional effects. The maximum spacing of transverse torsional reinforcement is equal to the lesser of  

DQGLQ$&,

The minimum area of transverse reinforcement for the combined effects of shear and torsion is covered in Section RIWKLVSXEOLFDWLRQ Longitudinal Reinforcement

The longitudinal reinforcement required for torsion, , is also calculated by Equation (6.9) where PLQHGXVLQJERWKRIWKHHTXDWLRQVLQ(TXDWLRQ 

is deter-

(6.61)

In this equation,

LVFDOFXODWHGE\(TXDWLRQ 

The minimum area of longitudinal reinforcement,

LVJLYHQLQ$&,

(6.62)

It is permitted to reduce LQWKHÀH[XUDOFRPSUHVVLRQ]RQHVRIDEHDPE\DQDUHDHTXDOWR where is the factored bending moment occurring at the section simultaneously with $&, 7KHUHGXFHGDUHDRI GHWHUPLQHGE\(TXDWLRQ  longitudinal reinforcement must not be taken less than 6.5.4 Reinforcement Requirements for Combined Flexure, Shear, and Torsion

The amounts of transverse and longitudinal reinforcement required to resist the combined effects from bending moPHQWVVKHDUIRUFHVDQGWRUVLRQDOPRPHQWVDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,

6-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The total required area of transverse reinforcement per stirrup leg is equal to the required area of transverse reLQIRUFHPHQWIRUVKHDU>(TXDWLRQ @SOXVWKHUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWIRUWRUVLRQ>(TXDWLRQ  @:KHUHWRUVLRQDOUHLQIRUFHPHQWLVUHTXLUHGWKHPLQLPXPDPRXQWRIWUDQVYHUVHUHLQIRUFHPHQWSHUOHJLV GHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ$&, 

(6.63)

Longitudinal torsional reinforcement, LVDGGHGWRWKDWUHTXLUHGIRUÀH[XUH If torsional reinforcement is remust be distributed around the perimeter of the beam, so an area of steel equal to is added to each quired, IDFHRIWKHEHDP$&, 7KXVDWWKHWRSDQGERWWRPIDFHVWKHWRWDOUHTXLUHGDUHDRIUHLQIRUFHPHQWLVHTXDOWR and , respectively. Similarly, at least must be provided on the side faces or must be added to any other longitudinal reinforcement on the side faces. $SSHQGL[%LQ5HIHUHQFHFRQWDLQVWDEOHVWRIDFLOLWDWHWRUVLRQGHVLJQIRUHGJHVSDQGUHO EHDPV7RUVLRQDOVHFWLRQ SURSHUWLHVDUHJLYHQIRUFRPPRQVODEWKLFNQHVVHVDQGHGJHEHDPVL]HV$OVRLQFOXGHGDUHWKHFRUUHVSRQGLQJYDOXHV for threshold torsion, cracking torsion, and required transverse and longitudinal reinforcement. No calculations are UHTXLUHGWRREWDLQDQ\RIWKHVHLWHPVIRUDJLYHQEHDPVL]HDQGVODEWKLFNQHVV

6.6 Reinforcement Detailing 6.6.1 Concrete Cover

5HLQIRUFLQJEDUVDUHSODFHGLQEHDPVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&,$&, )RUEHDPVZLWKWUDQVYHUVHUHLQIRUFHment that enclose the longitudinal reinforcing bars, concrete cover is measured from the surface of the concrete to WKHRXWHUHGJHRIWKHWUDQVYHUVHUHLQIRUFHPHQWVHH)LJXUH 7KHPLQLPXPFRYHUWRWKHWUDQVYHUVHUHLQIRUFHPHQWLVHTXDOWRLQIRUUHLQIRUFLQJEDUVLQEHDPVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG$&, 7DEOH 

‘˜‡”

‘˜‡” Figure 6.21 Concrete cover for beams.

Cover requirements for beams subjected to fatigue in structures designed to be watertight (such as tanks) or in structures subjected to very aggressive exposures (such as concrete exposed to moisture or external sources of chlorides) DUHJLYHQLQ$&, 6.6.2 Flexural Reinforcement Spacing Minimum Spacing of Flexural Reinforcing Bars

0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&,$&,  7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJUHLQforcing bars. 6-30

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

0LQLPXPVSDFLQJUHTXLUHPHQWVDUHJLYHQLQ)LJXUHZKHUH LVWKHQRPLQDOPD[LPXPDJJUHJDWHVL]HLQWKH is the diameter of the stirrup; is the inside bend radius concrete mixture; is the clear cover to the stirrup; IRUDQGVWLUUXSVDQG IRUDQGVWLUUXSVVHH$&,7DEOH of the stirrup (which is equal to  DQG is the overall diameter of the longitudinal reinforcing bars (the overall diameter of a reinforcing bar LQFOXGHVWKHGHIRUPDWLRQVRQWKHEDUVHH7DEOH 7KHPLQLPXPFOHDUVSDFLQJEHWZHHQSDUDOOHOUHLQIRUFLQJEDUV SODFHGLQRUPRUHKRUL]RQWDOOD\HUVLVLQ$&,VHH)LJXUH ,QVXFKFDVHVORQJLWXGLQDOUHLQIRUFHment in the upper layers must be placed directly above reinforcement in the bottom layer.

൒ ͳԣ

ܿ௦ 

݀௦ 

‫ݎ‬

݀௕ 

݀௕ 

ͳԣ ‹‹—…Ž‡ƒ”•’ƒ…‡ൌ ‰”‡ƒ–‡”‘ˆ ݀௕  ሺͶΤ͵ሻ݀௔௚௚  Figure 6.22 Minimum clear spacing requirements for reinforcing bars.

Nominal Diameter

A

A Overall Diameter

Section A–A

Table 6.7 Overall Reinforcing Bar Diameter Bar Size

Approximate Diameter to Deformations (in.)















 (table continued on next page)

6-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.7 Overall Reinforcing Bar Diameter (cont.) Bar Size

Approximate Diameter to Deformations (in.)



1























 WKDWFDQ¿WLQD

The following equation can be used to determine the maximum number of reinforcing bars, VLQJOHOD\HUEDVHGRQWKHPLQLPXPVSDFLQJOLPLWVLQ$&,

(6.64)

,QWKLVHTXDWLRQ³FOHDUVSDFH´UHIHUVWRWKHJUHDWHURIWKHWKUHHPLQLPXPFOHDUVSDFHVJLYHQLQ)LJXUH7KH GHWHUPLQHGE\(TXDWLRQ LVURXQGHGGRZQWRWKHQH[WZKROHQXPEHU value of 7KHPD[LPXPQXPEHURIEDUVWKDWFDQ¿WLQDVLQJOHOD\HUEDVHGRQWKHPLQLPXPVSDFLQJUHTXLUHPHQWVLQ$&, DUHJLYHQLQ7DEOHEDVHGRQWKHIROORZLQJ • • • •

2YHUDOOEDUGLDPHWHULVXVHGIRUWKHORQJLWXGLQDOUHLQIRUFHPHQW7DEOH Clear cover to the stirrup 1RPLQDOPD[LPXPDJJUHJDWHVL]H VWLUUXSVXVHGIRUDQGORQJLWXGLQDOEDUVDQGVWLUUXSVXVHGIRUDQGODUJHUORQJLWXGLQDOEDUV

Table 6.8 Maximum Number of Reinforcing Bars Permitted in a Single Layer* Beam Width (in.)

Bar Size

12

14

16

18

20

22

24

26

28

30

36

42

48





6



8



11







16















8

9



11

13





19











6



8

9



11













3





6



8

9



11





18





3





6





8

9



11



16

19



3







6



8

8

9













3







6





8

9

11

13







3

3







6





8



11

13

*Overall bar diameter is used for the longitudinal reinforcement (Table 6.7) Clear cover to the stirrup Nominal maximum aggregate size #3 stirrups are used for #4, #5, and #6 longitudinal bars and #4 stirrups are used for #7 and larger longitudinal bars

6-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum Spacing of Flexural Reinforcing Bars for Crack Control

0D[LPXPFHQWHUWRFHQWHUVSDFLQJRIUHLQIRUFLQJEDUVLVJLYHQLQ$&,$&, 7KHLQWHQWRIWKHVHUHTXLUHPHQWVLVWRFRQWUROÀH[XUDOFUDFNLQJ,QJHQHUDODODUJHUQXPEHURI¿QHUFUDFNVDUHSUHIHUDEOHWRDIHZZLGH cracks mainly for reasons of durability and appearance. The maximum center-to-center bar spacing, LVGHWHUPLQHGE\WKHHTXDWLRQVLQ$&,7DEOH$&,  The following equation is applicable to deformed reinforcing bars:

(6.65)

In this equation, LVWKHOHDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKH The term VHFWLRQ)RUH[DPSOHIRUDEHDPZLWKDLQFRQFUHWHFRYHUWRVWLUUXSV LVWKHFDOFXODWHGVWUHVVLQWKHÀH[XUDOUHLQIRUFHPHQWFORVHWWRWKHWHQVLRQIDFHDWVHUYLFHORDGVWKDWLV is equal to the unfactored bending moment at the section divided by the product of the area of reinforcement and the internal $&, )RU*UDGHUHLQIRUFHmoment arm). In lieu of calculating , it is permitted to be taken as Assuming , the maximum center-to-center bar spacing ment, IURP(TXDWLRQ 1RWHWKDW LVLQGHSHQGHQWRIWKHVL]HRIWKHORQJLWXGLQDOUHLQIRUFLQJEDUVLQWKHVHFWLRQ The following equation can be used to determine the minimum number of reinforcing bars, VLQJOHOD\HUWRVDWLVI\WKHFUDFNFRQWUROUHTXLUHPHQWVRI$&,

, required in a

(6.66)

In this equation, LVGHWHUPLQHGE\(TXDWLRQ 7KHFDOFXODWHGYDOXHRI number.

is rounded up to the next whole

7KHPLQLPXPQXPEHURIEDUVWKDWFDQ¿WLQDVLQJOHOD\HUEDVHGRQWKHFUDFNFRQWUROUHTXLUHPHQWVRI$&,DUH given in Table 6.9 based on the following: • *UDGHUHLQIRUFHPHQWZLWK • Overall bar diameter is used for the longitudinal reinforcement 7DEOH • /HDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKHVHFWLRQ Table 6.9 Minimum Number of Reinforcing Bars Required in a Single Layer* Beam Width (in.) 12

14

16

18

20

22

24

26

28

30

36

42

48





3

3

3

3

3











6

*Grade 60 reinforcement with Overall bar diameter is used for the longitudinal reinforcement (Table 6.7) Least distance from the surface of the flexural reinforcement to the tension face of the section

Distribution of Tension Reinforcement in Flanges of T-Beams

5HTXLUHPHQWVIRUWKHFRQWURORIFUDFNLQJLQWKHÀDQJHVRI7EHDPVLQWHQVLRQWKDWLVDWQHJDWLYHEHQGLQJUHJLRQV DUH JLYHQLQ$&,VHH)LJXUH 7KHEHDPÀH[XUDOUHLQIRUFHPHQWLQWKHÀDQJHVODE  , must be uniformly

6-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

GLVWULEXWHGRYHUDZLGWKHTXDOWRWKHOHVVHURIWKHHIIHFWLYHÀDQJHZLGWK GH¿QHGLQ$&,VHH)LJXUH  where LVWKHOHQJWKRIWKHFOHDUVSDQRIWKHEHDPPHDVXUHGIDFHWRIDFHRIVXSSRUWVVHH)LJXUH  and ܾ௙ 

݄

κ௡ /10

‫ܣ‬௦ 

Additional ‫ܣ‬௦

Additional ‫ܣ‬௦

Additional ‫ܣ‬௦ ሺtotalሻ ൒ 0.0018ൣܾ௙ െ ሺκ௡ Τ10ሻ൧݄ Other reinforcement not shown for clarity

Figure 6.23 Distribution of tension reinforcement in the flange of a T-beam.

$GGLWLRQDOUHLQIRUFHPHQWVDWLVI\LQJ$&,VKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQW PXVWEHSURYLGHGLQ WKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH Crack Control Reinforcement in Deep Flexural Members

For beams with an overall depth, , greater than 36 in., longitudinal skin reinforcement must be uniformly distribfrom the tension face to help control the width of uted on both faces of the beam for a distance equal to at least WKHFUDFNVWKDWFDQIRUPRQWKHVHIDFHV$&, 7KHPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIWKHVNLQUHLQIRUFHPHQWLVHTXDOWRWKDWUHTXLUHGIRUFUDFNFRQWUROLQ$&,ZKHUH is the clear cover from the side face to the edge of the skin reinforcement.

‹”‡‹ˆ‘”…‡‡–ሺ–›’Ǥ ሻ ‫ݏ‬

      

݄ ൐ ͵͸ԣ

7KHH[WHQWRIWKHVNLQUHLQIRUFHPHQWLVLOOXVWUDWHGLQ$&,)LJXUH5DWSRVLWLYHDQGQHJDWLYHEHQGLQJUHJLRQV DORQJWKHVSDQRIWKHEHDP7KHDUHDRIVNLQUHLQIRUFHPHQWLVQRWVSHFL¿HGLQ$&,KRZHYHUDFFRUGLQJWR $&,5WREDUVDUHW\SLFDOO\SURYLGHG$QDOWHUQDWHFRQ¿JXUDWLRQLVVKRZQLQ)LJXUHZKHUHLWLV assumed skin reinforcement is provided over the entire span.

ܿ௖ 

ͶͲǡͲͲͲ ൰ െ ʹǤͷܿ௖ ‫ͳۓ‬ͷ ൬ ݂௦ ۖ ‫ ݏ‬൑ Ž‡••‡”‘ˆ

‫۔‬ ͶͲǡͲͲͲ ۖ ͳʹ ൬ ൰ ݂௦ ‫ە‬



Figure 6.24 Skin reinforcement for deep flexural members.

6-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is permitted to include the skin reinforcement when calculating the nominal moment strength of the beam, provided a strain compatibility analysis of the section is performed.

,

6.6.3 Selection of Flexural Reinforcement

7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWDQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVJLYHQLQ6HFWLRQRIWKLV publication. Selecting the number of reinforcing bars in a single layer within the limits of Table 6.8 and Table 6.9 provides autoPDWLFFRQIRUPLW\ZLWKWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\JLYHQ the assumptions noted in those tables. ,QJHQHUDORYHUDOOFRVWVDYLQJVLVXVXDOO\DFKLHYHGE\VSHFLI\LQJWKHODUJHVWSUDFWLFDOORQJLWXGLQDOEDUVL]HVWKDWVDWLVI\ERWKVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWV8VLQJWKHVDPHEDUVL]HVDQGOHQJWKVZKHUHYHUSRVVLEOHDOVRKDV a positive impact on cost savings. 6.6.4 Development of Flexural Reinforcement Overview

'HYHORSPHQWRIGHIRUPHGUHLQIRUFHPHQWLQEHDPVIRUÀH[XUHPXVWEHLQDFFRUGDQFHZLWK$&,$&,  7KHFDOFXODWHGWHQVLOHRUFRPSUHVVLYHIRUFHLQWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQWDWHDFKVHFWLRQLQDEHDPPXVW be developed on each side of that section by embedment length, hooks, headed deformed bars, mechanical devices, RUDQ\FRPELQDWLRQWKHUHRI$&,$&,  ,WLVLPSRUWDQWWRQRWHWKHIROORZLQJSURYLVLRQVJLYHQLQ$&,DQGZKHQGHWHUPLQLQJ development lengths: • +RRNHGDQGKHDGHGSRUWLRQVRIDUHLQIRUFLQJEDUPXVWQRWEHXVHGWRGHYHORSWKHUHLQIRUFLQJEDULQFRPSUHVVLRQ • Development lengths do not require a strength reduction factor, • Values of XVHGWRFDOFXODWHGHYHORSPHQWOHQJWKVDUHOLPLWHGWRSVL ,QIRUPDWLRQRQKRZWRGHWHUPLQHWKHUHTXLUHGGHYHORSPHQWOHQJWKVIRUÀH[XUDOUHLQIRUFHPHQWLQDEHDPLVJLYHQ below. Development of Deformed Bars in Tension

3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHvelopment length, LVGHWHUPLQHGXVLQJWKHHTXDWLRQVLQ$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU UHTXLUHPHQWVDUHFRYHUHG¿UVW Method 1 – ACI 25.4.2.4 The development length in tension of a deformed reinforcing bar,

LVGHWHUPLQHGE\$&,(TXDWLRQD  (6.67)

7KHWHUPVLQ(TXDWLRQ DUHDVIROORZVVHH$&,7DEOH 

6-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• 0  RGL¿FDWLRQIDFWRUIRUOLJKWZHLJKWFRQFUHWH concrete:

7KLVIDFWRUUHÀHFWVWKHORZHUWHQVLOHVWUHQJWKRIOLJKWZHLJKW

(6.68)

• 5  HLQIRUFHPHQWJUDGHIDFWRU . This factor accounts for the effect of reinforcement yield strength on the required development length:

(6.69)

• 5  HLQIRUFHPHQWFRDWLQJIDFWRU . This factor accounts for the reduced bond strength between the concrete DQGHSR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFLQJEDUV

(6.70)

• 5  HLQIRUFHPHQWVL]HIDFWRU reinforcing bars:

7KLVIDFWRUUHÀHFWVWKHPRUHIDYRUDEOHSHUIRUPDQFHRIVPDOOHUGLDPHWHU

(6.71)

• &  DVWLQJSRVLWLRQIDFWRU 7KLVIDFWRUUHÀHFWVWKHDGYHUVHHIIHFWVWKDWFDQRFFXUWRWKHWRSKRUL]RQWDO reinforcement in a member due to vertical migration of water and mortar, which collect on the underside of the bars during placement of the concrete:

(6.72)

$FFRUGLQJWRWKHIRRWQRWHLQ$&,7DEOH • Spacing or cover dimension,

QHHGQRWH[FHHG

7KLVWHUPLVGH¿QHGDVIROORZVVHH)LJXUH 

(6.73)

6-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 

݀௕ 

ܿଶ 

‫ݏ‬

‫ۓ‬ ۖ

ܿଵ

ܿ௕ ൌ Ž‡••‡”‘ˆ ܿଶ  ‫۔‬ ۖ ‫ݏە‬Τʹ

Figure 6.25 Spacing or cover dimension, cb.

• 7  UDQVYHUVHUHLQIRUFHPHQWLQGH[ 7KHWUDQVYHUVHUHLQIRUFHPHQWLQGH[UHSUHVHQWVWKHUROHRIFRQ¿QLQJ UHLQIRUFHPHQWDFURVVWKHSRWHQWLDOVSOLWWLQJSODQHVODUJHUDPRXQWVRIFRQ¿QLQJUHLQIRUFHPHQWUHGXFHVWKH potential for splitting failure, thereby reducing the overall required development length. This index is GHWHUPLQHGE\$&,(TXDWLRQE  (6.74)

where

total cross-sectional area of all transverse reinforcement within a spacing tial plane of splitting through the reinforcement being developed

that crosses the poten-

center-to-center spacing of the transverse reinforcement number of bars being developed along the plane of splitting It is permitted to conservatively use if transverse reinforcement is present or required. For reinforcing bars VSDFHGFORVHUWKDQLQRQFHQWHUWUDQVYHUVHUHLQIRUFHPHQWPXVWEHSURYLGHGDORQJWKHGHwith $&,DQG  velopment or lap splice length such that 7KHFRQ¿QLQJWHUP PXVWEHWDNHQOHVVWKDQRUHTXDOWRLQ(TXDWLRQ >$&,@,WKDV EHHQVKRZQWKDWZKHQWKLVWHUPLVOHVVWKDQVSOLWWLQJIDLOXUHVDUHOLNHO\WRRFFXU$SXOORXWIDLOXUHRIWKHUHLQIRUFHPHQWLVPRUHOLNHO\ZKHQWKLVWHUPLVJUHDWHUWKDQVRDQLQFUHDVHLQWKHDQFKRUDJHFDSDFLW\GXHWRDQ LQFUHDVHLQFRYHURUDPRXQWRIFRQ¿QLQJUHLQIRUFHPHQWLVQRWOLNHO\ The tension development length GHWHUPLQHGE\$&,(TXDWLRQD LVSHUPLWWHGWREHUHGXFHGLQFDVHV ZKHUHWKHÀH[XUDOUHLQIRUFHPHQWLVJUHDWHUWKDQWKDWUHTXLUHGIURPDQDO\VLVH[FHSWIRUWKHVL[FDVHVLQ$&, $&, 7KHUHGXFWLRQIDFWRUDSSOLHGWR LVHTXDOWRWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW GLYLGHGE\WKHSURYLGHGDUHDRIUHLQIRUFHPHQW7KHUHGXFHGGHYHORSPHQWOHQJWKPXVWQRWEHWDNHQOHVVWKDQLQ

6-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Values of • • • • • •

IRUGHIRUPHGEDUVLQEHDPVDUHJLYHQLQ7DEOHEDVHGRQWKHIROORZLQJDVVXPSWLRQV

Normalweight concrete with 8QFRDWHG*UDGHUHLQIRUFHPHQW Concrete cover controls the determination of  VWLUUXSVDUHXVHGIRUDQGORQJLWXGLQDOEDUVDQGVWLUUXSVDUHXVHGIRUDQd larger longitudinal bars

Calculated values of are rounded up to the next whole number. Values of for beams with coated bars can be obtained by multiplying the tabulated values by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI IRUEHDPVPDGHRIOLJKWZHLJKWFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFH with Equation (6.68). Table 6.10 Tension Development Length, ℓd , for Deformed Bars in Beams in Accordance with ACI 25.4.2.4*

ℓd (in.)

Bar Size

       

Top

Other

 19  33    68

  18   36  

*Normalweight concrete with Uncoated Grade 60 reinforcement Concrete cover controls the determination of #3 stirrups used for #4, #5, and #6 longitudinal bars and #4 stirrups used for #7 and larger longitudinal bars

Tabulated values for Reference 9.

EDVHGRQWKHUHTXLUHPHQWVRI$&,IRUDYDULHW\RIRWKHUFRQGLWLRQVDUHJLYHQLQ

Method 2 – ACI 25.4.2.3 7KHPHWKRGLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHVHOHFWHG 7KHWZRVSDFLQJDQGFRYHUFDVHVLQ$&,7DEOHDUHJLYHQLQ YDOXHVRIWKHFRQ¿QLQJWHUP Table 6.11.

6-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.11 Tension Development Length, ℓd , for Deformed Bars in Beams in Accordance with ACI 25.4.2.3 Spacing and Cover

Case 1

#6 and Smaller Bars

#7 and Larger Bars

Condition 1 1. Clear spacing of bars being developed or lap spliced  Clear cover 3. Stirrups or ties throughout not less than the required minimum &RQGLWLRQ 1. Clear spacing of bars being developed or lap spliced  Clear cover

&DVH

Other conditions

In order to determine XVLQJWKHHTXDWLRQVLQ&DVHWKHVSDFLQJFRYHUDQGFRQ¿QHPHQWRIWKHEDUVEHLQJ GHYHORSHGPXVWPHHWDOOWKHUHTXLUHPHQWVXQGHU&RQGLWLRQRU&RQGLWLRQVHH)LJXUH ,WLVSUHVXPHGWKDW WKHVHFRQGLWLRQVRFFXUIUHTXHQWO\LQSUDFWLFH:KHQWKHUHTXLUHPHQWVRIHLWKHU&RQGLWLRQRU&RQGLWLRQDUHPHWLW WKDWYDOXHLVVXEVWLWXWHGLQWR(TXDWLRQ ZKLFKUHVXOWVLQWKHHTXDWLRQVLQWKH is assumed ¿UVWURZRI7DEOHIRU&DVH  ൒‹‹—‫ܣ‬௩ 

݀௕ 

Ž‡ƒ”•’ƒ…‡൒ ݀௕ 

Ž‡ƒ”…‘˜‡”൒ ݀௕ 

݀௕ 

Ž‡ƒ”…‘˜‡”൒ ݀௕ 

Ž‡ƒ”•’ƒ…‡൒ ʹ݀௕  ‘†‹–‹‘ͳ

‘†‹–‹‘ʹ

Figure 6.26 Spacing, cover, and confinement conditions of ACI 25.4.2.3.

&DVHLVDSSOLFDEOHZKHUHWKHUHTXLUHPHQWVLQHLWKHU&RQGLWLRQRU&RQGLWLRQLQ&DVHDUHQRWVDWLV¿HGLQWKLV case, it is assumed Tension development lengths, , must be the greater of the values deterPLQHGE\WKHHTXDWLRQVLQ7DEOHDQGLQ Tabulated values for tions.

EDVHGRQWKHUHTXLUHPHQWVRI$&,DUHJLYHQLQ5HIHUHQFHIRUDYDULHW\RIFRQGL-

Development of Standard Hooks in Tension

+RRNVDUHSURYLGHGDWWKHHQGVRIUHLQIRUFLQJEDUVWRSURYLGHDGGLWLRQDODQFKRUDJHZKHUHGHYHORSPHQWOHQJWKFDQnot be attained with straight bars. Standard hook geometry for development of deformed bars in tension is given in $&,7DEOH$&, 

6-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The development length of a deformed reinforcing bar in tension with a standard hook, 

, is given in ACI

(6.75)

7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRNVHH)LJXUH 7KH PRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  Table 6.12 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Lightweight,

Epoxy,

&RQ¿QLQJUHLQIRUFHPHQW

Location,

Concrete strength,

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG reinforcement



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK or



Other

1.6

)RUDQGVPDOOHUKRRNHGEDUV terminating inside a column core with side cover normal to the plane of the hook or with side cover normal to the plane of the hook



Other





The term LVWKHWRWDOFURVVVHFWLRQDODUHDRIWLHVRUVWLUUXSVFRQ¿QLQJWKHKRRNHGEDUVDQG is the total crosssectional area of the hooked bars being developed at the same critical section. The term is the center-to-center are given in ACI spacing of the hooked bars, which have a nominal diameter . Detailing requirements for VHH$&,)LJXUHV5DDQG5EIRUFRQ¿QLQJUHLQIRUFHPHQWSODFHGSDUDOOHODQGSHUSHQGLFXODU to hooked bars being developed, respectively). 7KHGHWDLOLQJUHTXLUHPHQWVLQ$&,DSSO\WRKRRNHGEDUVDWGLVFRQWLQXRXVHQGVRIPHPEHUVWKDWLVDWHQGV of simply-supported members, at the free ends of cantilevers, and at exterior joints where members do not extend EH\RQGWKHMRLQW ZKHUHERWKVLGHFRYHUDQGWRSRUERWWRP FRYHUWRWKHKRRNLVOHVVWKDQLQ for hooked deformed bars in beams based on normalweight concrete with , uncoated Values of , and side cover normal to the plane of the *UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKRRNHGEDUVSDFLQJ are given in Table 6.13. Calculated values of are rounded up to the next whole number. hook

6-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.13 Tension Development Length, ℓdh , for Hooked Deformed Bars in Beams* Bar Size

ℓdh (in.)

#4

6

#5

8

#6

10

#7

13

#8

15

#9

18

#10

22

#11

25

*Normalweight concrete with Uncoated Grade 60 reinforcement Center-to-center hooked bar spacing Side cover normal to the plane of the hook

Development of Headed Deformed Bars in Tension

3URYLVLRQVIRUWKHGHYHORSPHQWRIKHDGHGGHIRUPHGEDUVLQWHQVLRQDUHJLYHQLQ$&,([DPSOHVRIKHDGHG UHLQIRUFLQJEDUVDUHJLYHQLQ)LJXUH +HDGHGGHIRUPHGEDUVDUHSHUPLWWHGWREHXVHGRQO\ZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG (a) (b) (c) (d) (e) (f)

%DUPXVWFRQIRUPWR$&, %DUVL]HPXVWEHRUVPDOOHU , must be at least where is the area of the bar Net bearing area of head, Concrete must be normalweight where is the nominal diameter of the bar Clear cover to the bar must be greater than or equal to Center-to-center spacing of the bars must be greater than or equal to

The development length of a headed deformed reinforcing bar in tension,

LVJLYHQLQ$&,

(6.76)

This development length is measured from the critical section to the bearing face of the head (see ACI Figures 5DDQG5E 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  

6-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.14 Modification Factors for Development of Headed Bars in Tension Modification Factor

Condition

Epoxy,

Parallel tie reinforcement,

Location,

Value of Factor

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG reinforcement



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK



or

Other

1.6

For headed bars (1) terminating inside a column core RU ZLWKVLGHFRYHUWR with side cover to bar bar



Other



Concrete strength,



The term DFFRXQWVIRUFRQ¿QLQJHIIHFWVSURYLGHGE\WUDQVYHUVHUHLQIRUFHPHQWRULHQWHGSDUDOOHOWRWKHGHYHORSPHQWOHQJWKRIKHDGHGEDUV&RQ¿QHPHQWUHTXLUHPHQWVIRUKHDGHGQHJDWLYHPRPHQWUHLQIRUFHPHQWLQDEHDP WHUPLQDWHGLQDMRLQWDUHJLYHQLQ$&,$WEHDPFROXPQMRLQWVWKHWRWDOFURVVVHFWLRQDODUHDRIWUDQVYHUVH must be located within of the centerline of the headed bar toward the middle of the joint reinforcement, is greater than or where LVWKHQRPLQDOGLDPHWHURIWKHKHDGHGEDU>VHH$&,)LJXUH5D @:KHUH or where the center-to center spacing of the headed bars is greater than or equal to The equal to is the total cross-sectional area of the headed bars being developed. term For other than beam-column joints,

must not be considered and

$&, 

provided

Values of for headed deformed bars in beams based on normalweight concrete with , and side cover to the bar *UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKHDGHGEDUVSDFLQJ are rounded up to the next whole number. 7DEOH&DOFXODWHGYDOXHVRI Table 6.15 Tension Development Length, ℓd t , for Headed Deformed Bars in Beams*

*Normalweight concrete with Uncoated Grade 60 reinforcement Side cover normal to the bar

6-42

Bar Size

ℓdt (in.)

#4

6

#5

6

#6

8

#7

9

#8

11

#9

14

#10

16

#11

19

, uncoated are given in

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Development of Mechanically Anchored Deformed Bars in Tension

Any mechanical attachment or device capable of developing of the deformed bars is permitted to be used providHGLWLVDSSURYHGE\WKHEXLOGLQJRI¿FLDOLQDFFRUGDQFHZLWK$&,$&, 7KHXVHRIPHFKDQLFDOGHYLFHV WKDWGRQRWPHHWWKHUHTXLUHPHQWVLQ$&,RUWKDWDUHQRWGHYHORSHGLQDFFRUGDQFHZLWK$&,PD\EH used provided test results are available that demonstrate the ability of the head and bar system to develop or anchor the required force in the bar. Development of Positive and Negative Flexural Reinforcement

Flexural reinforcement must be properly developed or anchored in a reinforced concrete beam so that the beam SHUIRUPVDVLQWHQGHGLQDFFRUGDQFHZLWKWKHVWUHQJWKGHVLJQPHWKRG&ULWLFDOVHFWLRQVIRUGHYHORSPHQWRIÀH[XUDO UHLQIRUFHPHQWRFFXUDWWKHIROORZLQJ$&,  (1) Points of maximum stress (that is, sections of maximum bending moment)   Points along the span where adjacent reinforcement is terminated because it is no longer required WRUHVLVWÀH[XUH In continuous beams subjected to uniform gravity loads, the maximum positive and negative bending moments W\SLFDOO\RFFXUQHDUPLGVSDQDQGDWWKHIDFHVRIWKHVXSSRUWVUHVSHFWLYHO\3RVLWLYHDQGQHJDWLYHÀH[XUDOUHLQIRUFLQJ EDUVPXVWEHGHYHORSHGRUDQFKRUHGRQERWKVLGHVRIWKHVHFULWLFDOVHFWLRQV$&,  7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWDWDFULWLFDOVHFWLRQFDQEHGHWHUPLQHGXVLQJWKHPHWKRGVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ ݊ ൌ ݊ଵ ൅ ݊ଶ  ݊ଵ ൒ ݊Τ3 ‫ିܣ‬ ௦ ሺ݊ barsሻ

‫ିܣ‬ ௦ଵ ሺ݊ଵ barsሻ

‫ܣ‬ା ௦ ሺ‫ ݌‬barsሻ ‫ܣ‬ା ௦ଶ ሺ‫݌‬ଶ barsሻ

‫ିܣ‬ ௦ଶ ሺ݊ଶ barsሻ

‫ ݌‬ൌ ‫݌‬ଵ ൅ ‫݌‬ଶ  ‫݌‬ଵ ൒ ‫݌‬Τ4 A

൒ 6"

B

D

C

‫ܣ‬ା ௦ଵ ሺ‫݌‬ଵ barsሻ

൒ ݀ or 12݀௕  ൒ ݀ or 12݀௕  ݊ଶ bars

‫ݔ‬஼஽  ൒ κௗ 

൒ κௗ  ‫ݔ‬஺஻ 

݊ଵ bars

൒ κௗ 

൒ ݀, 12݀௕ , or κ௡ /16

ሺ‫ܯ‬௨ା ሻ஽ 

‫݌‬ଶ bars ሺ‫ܯ‬௨ା ሻ஼ 

‫ݔ‬஻ா  E Point of inflection ሺ‫ܯ‬௨ି ሻ஻ 

ሺ‫ܯ‬௨ି ሻ஺  Figure 6.27 Development of flexural reinforcement in a beam.

6-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUWKHFRQWLQXRXVEHDPLQ)LJXUHWKHWRWDOUHTXLUHGDUHDRIQHJDWLYHUHLQIRUFHPHQWIRUWKHPD[LPXPQHJDWLYH , at critical section A is equal to , and the total number of reinforcing bars at factored bending moment, this location is equal to . Similarly, the total required area of positive reinforcement for the maximum positive fac, at critical section C is equal to , and the total number of reinforcing bars at this tored bending moment, location is equal to . Requirements for the development of these two sets of reinforcing bars are discussed next. Negative Flexural Reinforcement Assume a portion of LVFXWRIIDWVHFWLRQ%ZKHUHLWLVQRORQJHUUHTXLUHGIRUÀH[XUDOVWUHQJWK7KLVPDNHVVHFand the number of reinforcing tion B a critical section. At this location, the reinforcement has an area equal to and the correbars is equal to . The area of the remaining portion of reinforcing bars is equal to . This reinforcement must be able to resist the negative sponding number of reinforcing bars is equal to at section B. factored bending moment Because section B is a critical section, bars must be adequately developed to the right of this section. This is achieved by extending the bars a minimum distance of SDVWVHFWLRQ%DVVKRZQLQ)LJXUHZKHUH is the GHYHORSPHQWOHQJWKLQWHQVLRQRIDGHIRUPHGEDUZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  $FFRUGLQJWR$&,DWOHDVWRQHWKLUGRIWKHWRWDOQHJDWLYHUHLQIRUFHPHQWSURYLGHGDWDVXSSRUWPXVWKDYHDQ , and SDVWWKHSRLQWRILQÀHFWLRQZKHUH is the clear span embedment length equal to the greater of , length measured face-to-face of supports (which in this case are columns). Therefore, to satisfy this requirement, ZKLFKDFFRXQWVIRUSRVVLEOHVKLIWLQJRIWKHEHQGLQJPRPHQWGLDJUDPDWWKHSRLQWRILQÀHFWLRQGXHWRWKHDSSUR[L. The minimum length of bars to the right mate bending moment diagram customarily used in design, RIFULWLFDOVHFWLRQ$LVHTXDOWRWKHIROORZLQJ$&,DQG 

(6.77)

In these equations, is the distance from section A to the theoretical cutoff point at section B and GLVWDQFHIURPVHFWLRQ%WRWKHSRLQWRILQÀHFWLRQDWVHFWLRQ(

is the

Because section A is a critical section, the bars cutoff at section B must be developed a distance equal to at least EH\RQGWKDWVHFWLRQ$GGLWLRQDOO\$&,VWLSXODWHVWKHVHEDUVPXVWH[WHQGEH\RQGWKHSRLQWZKHUHWKH\DUHQR (except at supports of simply-supported spans longer required a distance equal to at least the greater of and bars to the right of critical section A is equal to the foland at free ends of cantilevers). The minimum length of lowing:

(6.78)

)OH[XUDOWHQVLRQUHLQIRUFHPHQWLVQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRIWKHIROORZLQJFRQGLWLRQVDUHVDWLV¿HG$&,  (1) At the cutoff point, WLRQRIWKLVSXEOLFDWLRQ 

6-44

where

is the design shear strength of the section at that point (see Sec-

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

  )RUDQGVPDOOHUEDUVFRQWLQXLQJUHLQIRUFHPHQWSURYLGHVDWOHDVWGRXEOHWKHDUHDUHTXLUHGIRUÀH[XUHDWWKH . cutoff point and (3) Stirrup or hoop area in excess of that required for shear and torsion is provided along each terminated bar from the cutoff point. The excess area of the stirrups or hoops must be greater over a distance equal to where the center-to-center spacing must be less than or equal to and than or equal to The development of the negative reinforcing bars to the left of section A depends on its location within the framing V\VWHP$WLQWHULRUMRLQWVOLNHWKHRQHGHSLFWHGLQ)LJXUHGHYHORSPHQWLVDFKLHYHGE\FRQWLQXLQJWKHQHJDWLYH reinforcing bars into the span to the left of the column. In an end span where the beam terminates at an edge column, a standard hook is provided at the ends of the negative reinforcing bars. Positive Flexural Reinforcement is cut off at section D. This makes section D a critical section. At this location, the reinAssume a portion of and the number of reinforcing bars is equal to . The area of the remaining forcement has an area equal to and the corresponding number of reinforcing bars is equal to portion of reinforcing bars is equal to . This reinforcement must be able to resist the negative factored bending moment at section D. The bars cut off at section D must be developed to a distance greater than or equal to beyond the critical section C. Like in the case of the negative reinforcement, these bars must extend beyond the point they are no longer $&, 7KHPLQLPXPOHQJWKRI bars to the left required by a distance equal to the greater of and of critical section C is equal to the following:

(6.79)

In this equation,

is the distance between critical section C and the theoretical cutoff point located at section D.

7KHUHTXLUHPHQWVRI$&,SHUWDLQLQJWRUHLQIRUFHPHQWWHUPLQDWHGLQDWHQVLRQ]RQHDUHDOVRDSSOLFDEOHLQWKLV case. $FFRUGLQJWR$&,DWOHDVWRQHIRXUWKRIWKHSRVLWLYHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQLQWRWKHVXS. Note that if a beam is part of the LFRS, this SRUWH[FHSWDWVLPSOHVXSSRUWVVHH$&, 7KXV reinforcement must be anchored to develop at the support. 7KHGLDPHWHURIWKHSRVLWLYHUHLQIRUFHPHQWLVOLPLWHGDWSRLQWVRILQÀHFWLRQDQGDWVLPSOHVXSSRUWV LQDFFRUGDQFH ZLWK$&,WKLVUHTXLUHPHQWKHOSVWRHQVXUHWKHEDUVDUHGHYHORSHGLQDOHQJWKVKRUWHQRXJKVXFKWKDWWKH PRPHQWFDSDFLW\LVJUHDWHUWKDQWKHDSSOLHGPRPHQWRYHUWKHHQWLUHOHQJWKRIWKHEHDPVHH$&,)LJXUH5 

(6.80)

In this equation, LVWKHQRPLQDOÀH[XUDOVWUHQJWKRIWKHEHDPVHFWLRQDVVXPLQJDOOWKHUHLQIRUFHPHQWDWWKDWVHFis the factored shear force at the section, and is the embedment length of the reinforcetion is stressed to , . Additional information on this topic can be PHQWEH\RQGWKHLQÀHFWLRQSRLQWZKLFKLVWKHJUHDWHURI and IRXQGLQ$&,5

6-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.6.5 Splices of Deformed Reinforcement Overview

6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQEHDPVPXVWEHLQDFFRUGDQFHZLWK$&,$&, 7KHSULPDU\ reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to the FRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG /DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQW Lap Splices

Lap splices are usually the most economical type of splice. There are basically two types of lap splices: contact and QRQFRQWDFW,QDFRQWDFWODSVSOLFHWKHEDUVDUHJHQHUDOO\LQFRQWDFWRYHUDVSHFL¿HGOHQJWKDQGDUHWLHGWRJHWKHU >VHH)LJXUHD @,QDQRQFRQWDFWODSVSOLFHWKHEDUVDUHQRWLQFRQWDFWDQGWKHFHQWHUWRFHQWHUVSDFLQJRIWKH EDUVEHLQJVSOLFHGPXVWQRWH[FHHGWKHOLPLWVLQGLFDWHGLQ)LJXUHE VHH$&, &RQWDFWODSVSOLFHVDUH usually preferred because the bars are tied together and are less likely to displace during construction. The minimum clear spacing between a contact lap splice and adjacent splices or bars is the same as that for individual bars (ACI VHH)LJXUH  Lap splices must be provided at locations away from maximum stress (that is, maximum bending moment) and should be staggered wherever possible. Reinforcing bars provided for negative bending moments (top bars) are spliced near the midspan of a member, and reinforcing bars provided for positive bending moments (bottom bars) are spliced over the supports. , depends on the tension development length of the bars, , the area of The required tension lap splice length, reinforcement provided over the length of the splice, and the percentage of reinforcement spliced at any one locaWLRQ%HFDXVHH[SHULPHQWDOGDWDRQODSVSOLFHVZLWKDQGEDUVDUHVSDUVHWKHXVHRIWHQVLRQODSVSOLFHVIRU WKHVHEDUVDUHSURKLELWHGH[FHSWDVSHUPLWWHGLQ$&,$&,  /DSVSOLFHVDUHFODVVL¿HGDV&ODVV$RU&ODVV%7KHOHQJWKRIDODSVSOLFHLVJLYHQDVDPXOWLSOHRI  

(ACI (6.81) (6.82)

When determining to be used in determining WKHLQPLQLPXPOHQJWKVSHFL¿HGLQ$&,E DQG WKHH[FHVVUHLQIRUFHPHQWPRGL¿FDWLRQIDFWRURI$&,DUHQRWDSSOLFDEOH$&, $OVRIRUUHLQIRUFspaced closer than 6 in. on center, transverse reinforcement must be provided along ing bars with $&,  the splice length such that 7KHGHIDXOWVSOLFHFODVVL¿FDWLRQIRUDODSVSOLFHLV&ODVV%,IERWKRIWKHIROORZLQJFRQGLWLRQVDUHVDWLV¿HGD&ODVV$ VSOLFHLVSHUPLWWHGVHH$&,7DEOH  over the entire splice length where and are the area of reinforcement provided at the splice location and the area of reinforcement required by analysis at the splice location, respectively • /HVVWKDQRUHTXDOWRSHUFHQWRIWKHUHLQIRUFHPHQWLVVSOLFHGZLWKLQWKHUHTXLUHGODSVSOLFHOHQJWK •

:KHUHEDUVRIGLIIHUHQWVL]HDUHODSVSOLFHGLQWHQVLRQ RIWKHVPDOOHUEDU$&, 

is equal to the greater of (1)

RIWKHODUJHUEDUDQG 

Mechanical Splices

Mechanical splices are, in general, a complete assembly of a coupler, a coupling sleeve, or an end-bearing sleeve, including any additional material or components, required to accomplish the splicing of the reinforcing bars. Acof the bar. This FRUGLQJWR$&,PHFKDQLFDOVSOLFHVPXVWGHYHORSLQWHQVLRQRUFRPSUHVVLRQDWOHDVW is to ensure yielding occurs in the reinforcing bar adjacent to the mechanical splice prior to the failure of the splice. 6-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,OOXVWUDWHGLQ)LJXUHDUHWKUHHRIWKHPRUHSRSXODUW\SHVRIPHFKDQLFDOVSOLFHV7KHFROGVZDJHGFRXSOLQJVOHHYH LQ)LJXUHD XVHVDK\GUDXOLFVZDJLQJSUHVVZLWKVSHFLDOGLHVWRGHIRUPWKHVOHHYHDURXQGWKHHQGVRIWKHVSOLFHG bars to produce positive mechanical interlock with the reinforcing bars. The shear screw coupling sleeve in Figure E FRQVLVWVRIDFRXSOLQJVOHHYHZLWKVKHDUKHDGVFUHZVZKLFKDUHGHVLJQHGWRVKHDURIIDWDVSHFL¿HGWRUTXH The reinforcing bars are inserted to meet at the center of the coupling sleeve and then the screws are tightened. The WLJKWHQLQJSURFHVVHPEHGVWKHSRLQWHGVFUHZVLQWRWKHEDUV7KHQRQXSVHWVWUDLJKWWKUHDGFRXSOHULQ)LJXUHF  consists of a coupler with internal straight threads at each end that joins the two reinforcing bars with matching external threads. Additional information on these and other mechanical splices can be found in Reference 8. Mechanical splices can be advantageous in a number of situations, including the following: • When long lap splices are needed • When lap splices cause reinforcement congestion • :KHUHVSDFLQJRIWKHÀH[XUDOUHLQIRUFHPHQWLVLQVXI¿FLHQWWRSHUPLWODSVSOLFHV Mechanical splices need not be staggered unless the splices are in tension tie members (that is, members having an D[LDOWHQVLOHIRUFHVXI¿FLHQWWRFUHDWHWHQVLRQRYHUWKHFURVVVHFWLRQDOHYHORIVWUHVVLQWKHUHLQIRUFHPHQWVXFKWKDW every bar should be fully effective, and limited concrete cover on all sides) where the minimum stagger between VSOLFHVLQDGMDFHQWEDUVPXVWEHDWOHDVWLQLQDFFRUGDQFHZLWK$&,$&,  Welded Splices

7KHXVHRIZHOGHGVSOLFHVDUHSHUPLWWHGWREHXVHGSURYLGHGWKH\FRQIRUPWRWKHUHTXLUHPHQWVRI$&,$&,  /LNHPHFKDQLFDOVSOLFHVDIXOOZHOGHGVSOLFHZKLFKLVJHQHUDOO\LQWHQGHGIRUDQGODUJHUEDUVPXVWEH of the bar. able to develop at least Welded splices need not be staggered unless the splices are in tension tie members. 6.6.6 Longitudinal Torsional Reinforcement

Where it has been determined the effects of torsion must be considered (see Section 6.3.1 of this publication), the GHWDLOLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HGIRUWKHORQJLWXGLQDOWRUVLRQDOUHLQIRUFHPHQW$VXPPDU\RI WKHUHTXLUHPHQWVLQ$&,DQGLVJLYHQLQ)LJXUH,QWKH¿JXUH is the center-to-center spacing of the required combined transverse reinforcement. The longitudinal torsional reinforcement is combined with that UHTXLUHGIRUÀH[XUHVHH6HFWLRQRIWKLVSXEOLFDWLRQ 

‘‰‹–—†‹ƒŽ”‡‹ˆ‘”…‡‡–†‹•–”‹„—–‡† ƒ”‘—†–Š‡’‡”‹‡–‡”‘ˆ–Š‡…Ž‘•‡†•–‹””—’• ™‹–Šƒ–Ž‡ƒ•–‘‡„ƒ”’Žƒ…‡†‹‡ƒ…Š…‘”‡” ൑ ͳʹԣ

      

ͲǤͲͶʹ‫ݏ‬ ݀௕ ൒ ‰”‡ƒ–‡”‘ˆ ൝

 ͵Τͺ‹Ǥ

ܾ௧ 

Figure 6.28 Details for longitudinal torsional reinforcement.

beyond the point is reLongitudinal torsional reinforcement must be provided for a distance equal to at least TXLUHG$&, 7KHWHUP is the width of that part of the cross-section that contains the closed stirrups resistLQJWRUVLRQVHH)LJXUH 7KLVGLVWDQFHLVODUJHUWKDQWKDWXVHGIRUVKHDUUHLQIRUFHPHQWDQGÀH[XUHUHLQIRUFHPHQW because torsional diagonal cracks develop in a helical form around a member. Also, the longitudinal reinforcement 6-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

must be fully developed for tension at both ends of the member; proper anchorage is especially important at the HQGVZKHUHWKHODUJHVWWRUVLRQDOPRPHQWVXVXDOO\RFFXU$&,  6.6.7 Transverse Reinforcement Overview

7KHGHWDLOLQJUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGIRUWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHGIRUVKHDURUIRU shear and torsion. Where shear and torsion must be considered, the most restrictive detailing requirements of ACI DSSO\$&,DQG  Shear Reinforcement

Shear reinforcement must be properly anchored and developed to be fully effective. Requirements for the developPHQWRIVWLUUXSVDUHJLYHQLQ$&,DQGDUHLOOXVWUDWHGLQ)LJXUH

݄ത

݄/2

Standard hook in accordance with ACI 25.3.2 ሺtyp. ሻ

݄/2

       

ܾ௪ 

݄ത ൒

0.014݀௕ ݂௬௧ ߣඥ݂௖ᇱ

 for #6, #7, and #8 stirrups with ݂௬௧ ൐ 40,000 psi

Figure 6.29 Anchorage details for U-stirrups.

In general, the stirrups must extend as close to the compression and tension surfaces of the member as cover and other constraints permit and must extend at least a distance IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU$&,  Each bend in the continuous portion of a single- or multiple-leg U-stirrup and each bend in a closed stirrup must HQFORVHDORQJLWXGLQDOEDU$&, 0LQLPXPLQVLGHEHQGGLDPHWHUVDQGVWDQGDUGKRRNJHRPHWU\IRUVWLUUXSV WLHVDQGKRRSVDUHJLYHQLQ$&,7DEOH 7KHEHDPGHSWKWKDWPXVWEHSURYLGHGWRVDWLVI\WKHPLQLPXPHPEHGPHQWOHQJWKIRUDQGVWLUUXSEDUV LVJLYHQLQ7DEOH>$&,E @7KHEHDPGHSWKVLQWKHWDEOHDUHEDVHGRQQRUPDOwith *UDGHVWLUUXSEDUVDQGDLQFRYHUWRWKHVWLUUXSKRRN%HDPVZLWK weight concrete with No RYHUDOOGHSWKVOHVVWKDQWKRVHLQ7DEOHDUHQRWSHUPLWWHGWRXVHDQGVWLUUXSVZLWK PLQLPXPGHSWKUHTXLUHPHQWVDUHJLYHQIRUEHDPVZLWK VWLUUXSEDUVDQGVPDOOHURIDQ\JUDGHDQG  DQGVWLUUXSVZLWK

6-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.16 Minimum Beam Depth to Anchor #6, #7, and #8 Stirrups* Stirrup Bar

Minimum Beam Depth, h (in.)

#6

23

#7

27

#8

30

*Normalweight concrete with Grade 60 stirrup bars 1.5-in. cover to the stirrup hooks

%HDPVPXVWKDYHVXI¿FLHQWZLGWK , to allow for bend radii at corners of stirrups. Minimum beam widths correVSRQGLQJWRVWLUUXSEDUVL]HDUHJLYHQLQ7DEOH Table 6.17 Minimum Beam Width for Stirrup Development Stirrup Bar

Minimum Beam Width, bw (in.)

#3

10

#4

12

#5

14

#6

18

#7

20

#8

22

Closed stirrups consisting of pairs of U-stirrups are permitted to be used except for torsion or integrity reinforcebut not PHQW$&, 7KHOHJVRIWKHVWLUUXSVPXVWEHODSVSOLFHGZLWKDVSOLFHOHQJWKHTXDOWRDWOHDVW OHVVWKDQLQZKHUHWKHWHQVLRQGHYHORSPHQWOHQJWK LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,VHH)LJXUH  ,QFDVHVZKHUHWKHUHTXLUHGODSOHQJWKFDQQRW¿WZLWKLQDPHPEHUWKDWKDVDGHSWKRIDWOHDVWLQSDLUVRI 8VWLUUXSVFDQVWLOOEHXVHGSURYLGHGWKHIRUFHLQHDFKOHJLVHTXDOWRRUOHVVWKDQOE)RU*UDGHUHLQIRUFHPHQWRQO\DVWLUUXSVDWLV¿HVWKLVUHTXLUHPHQW

Figure 6.30 A closed stirrup formed by pairs of U-stirrups.

6-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Torsion Reinforcement

5HTXLUHPHQWVIRUVWLUUXSVVXEMHFWHGWRWRUVLRQDOPRPHQWVDUHJLYHQLQ$&,$VQRWHGSUHYLRXVO\FORVHG stirrups must be used in such cases because cracking can occur on all faces of a beam. The corners of a beam are susceptible to spalling due to the inclined compressive stresses occurring due to torsion. Where spalling is not UHVWUDLQHGE\DQDGMDFHQWVODEERWKHQGVRIDVWLUUXSPXVWWHUPLQDWHZLWKGHJUHHVWDQGDUGKRRNVDURXQGORQJLWXGLQDOEDUVLQWKHVHFWLRQ>$&,D @6SDOOLQJLVHVVHQWLDOO\SUHYHQWHGZKHUHDVODELVSUHVHQWRQRQHRUERWK VLGHVRIDEHDPZHEVRGHJUHHVWDQGDUGKRRNVLQDFFRUGDQFHZLWK$&,DUHSHUPLWWHGLQVXFKFDVHV>$&, E @'HWDLOVIRUWUDQVYHUVHWRUVLRQUHLQIRUFHPHQWZKHUHVSDOOLQJLVXQUHVWUDLQHGDQGUHVWUDLQHGDUHJLYHQLQ Figure 6.31. ”‘••–‹‡

ͳ͵ͷǦ†‡‰Š‘‘ሺ–›’Ǥሻ

”‘••–‹‡

ͻͲǦ†‡‰Š‘‘

Ǧ•–‹””—’

ሺƒሻ

Ǧ•–‹””—’

ሺ„ሻ

Figure 6.31 Details for transverse torsion reinforcement. (a) Spalling is unrestrained. (b) Spalling is restrained.

7ZRSLHFHFORVHGVWLUUXSVDUHSHUPLWWHGWREHXVHG$&, 7KHFORVHGVWLUUXSLQ)LJXUHD FRQVLVWVRI D8VWLUUXSZLWKGHJUHHKRRNVDWERWKHQGVDQGDFURVVWLHZLWKGHJUHHKRRNVDWERWKHQGV7KHGHJUHH hooks are required on both ends of both pieces of reinforcement because spalling is unrestrained. Similarly, Figure E FRQVLVWVRID8VWLUUXSZLWKGHJUHHKRRNVDWERWKHQGVDQGDFURVVWLHZLWKDGHJUHHKRRNRQRQHHQG DQGDGHJUHHKRRNRQWKHRWKHUHQG7KHGHJUHHKRRNRIWKHFURVVWLHLVORFDWHGRQWKHVLGHDGMDFHQWWRWKHVODE which provides restraint against spalling. Using two-piece closed stirrups are preferred because closed, one-piece VWLUUXSVPDNHLWGLI¿FXOWWRSODFHWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDPZKHUHWKHUHLQIRUFHPHQWLVEXLOWLQ place (as opposed to preassembled cages). Like longitudinal reinforcement required for torsion, transverse torsional reinforcement must extend a distance of at EH\RQGWKHSRLQWLWLVUHTXLUHGE\DQDO\VLV$&,  least 6.6.8 Structural Integrity Reinforcement

6WUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVIRUFDVWLQSODFHEHDPVDUHJLYHQLQ$&,7KHSXUSRVHRIWKLV reinforcement is to improve the redundancy and ductility in the structure so that in the event of damage to a major VXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFWXUHZLOOKDYH a higher probability of maintaining overall stability. A summary of the structural integrity reinforcement requirements is given in Table 6.18.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.18 Structural Integrity Reinforcement Requirements for Beams Requirement

ACI Section No.

Perimeter Beams

1. At least one-quarter of the maximum positive moment reinforcement, but not OHVVWKDQEDUVPXVWEHFRQWLQXRXV  At least one-sixth of the negative moment reinforcement at a support, but not OHVVWKDQEDUVPXVWEHFRQWLQXRXVDQG 3. Longitudinal structural integrity reinforcement must be enclosed by closed VWLUUXSVLQDFFRUGDQFHZLWK$&,RUE\KRRSVDORQJWKHFOHDUVSDQRI the beam.



Other Than Perimeter Beams

1. At least one-quarter of the maximum positive moment reinforcement, but not OHVVWKDQEDUVPXVWEHFRQWLQXRXVRU  Longitudinal reinforcement must be enclosed by closed stirrups in accordance ZLWK$&,RUE\KRRSVDORQJWKHFOHDUVSDQRIWKHEHDP



Longitudinal structural integrity reinforcement must pass through the region bounded by the longitudinal reinforcement in the column.



Longitudinal structural integrity reinforcement at noncontinuous supports must be anchored to develop at the face of the support.



Splices for longitudinal structural integrity reinforcement must occur at or near a support for positive moment reinforcement and at or near midspan for negative moment reinforcement.



Splices for longitudinal structural integrity reinforcement must be mechanical or welded splices LQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQODSVSOLFHVLQDFFRUGDQFHZLWK$&,



6.6.9 Flexural Reinforcement Requirements for SDC B

For beams in buildings assigned to SDC B that are part of an ordinary moment frame resisting seismic load effects, WKHGHWDLOLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG7ZRFRQWLQXRXVÀH[XUDOEDUVPXVWEHSURYLGHGDWERWK WKHWRSDQGERWWRPIDFHVRIWKHEHDP7KHFRQWLQXRXVERWWRPEDUVPXVWKDYHDQDUHDHTXDOWRDWOHDVWSHUFHQWWKH maximum area of the bottom bars along the span and must be anchored to develop in tension at the face of the support. These provisions are intended to improve continuity and improve lateral force resistance and structural integrity. 6.6.10 Recommended Flexural Reinforcement Details

5HFRPPHQGHGÀH[XUDOUHLQIRUFHPHQWGHWDLOVIRUEHDPVEDVHGRQWKHUHTXLUHPHQWVFRYHUHGLQWKHSUHYLRXVVHFWLRQVRIWKLVFKDSWHULQFOXGLQJWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&,DUHJLYHQLQ)LJXUH7KHEDU OHQJWKVJLYHQLQWKH¿JXUHDUHEDVHGRQDEHDPVXEMHFWHGWRXQLIRUPO\GLVWULEXWHGJUDYLW\ORDGVZKHUHWKHVKDSHRI WKHPRPHQWGLDJUDPLVNQRZQWKHVHOHQJWKVFDQEHXVHGIRUEHDPVGHVLJQHGXVLQJWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,VHH6HFWLRQRIWKLVSXEOLFDWLRQ )RUEHDPVVXEMHFWHGWRWKHHIIHFWVIURPRWKHUW\SHVRIORDGV required bar lengths must be determined by calculations. Also, for beams that are part of an ordinary moment frame LQEXLOGLQJVDVVLJQHGWR6'&%WKHUHTXLUHPHQWVRI$&,PXVWDOVREHVDWLV¿HG

6.7 Deflections 6.7.1 Overview

7ZRPHWKRGVFDQEHXVHGIRUFRQWUROOLQJEHDPGHÀHFWLRQV SURYLGLQJDEHDPGHSWKJUHDWHUWKDQRUHTXDOWRWKH PLQLPXPEHDPGHSWKSUHVFULEHGLQ$&,VHH6HFWLRQRIWKLVSXEOLFDWLRQ DQG SURYLGLQJDEHDPGHSWK

6-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κ௡ଵ Τ3



൒ greater of ൝

κ௡ଶ Τ3



൒ greater of ൝

κ௡ଶ Τ3

κ௡ଷ Τ3

൒ κ௡ଵ Τ4

Note 1 ሺtyp.ሻ

൒ ‫ܣ‬ା௦ଵ /4

Note 2

Note 2 ሺsim.ሻ

‫ܣ‬ା௦ଵ 

‫ܣ‬ା௦ଶ 

൑ 0.125κ௡ଵ 

൒ 6ԣ

൑ 0.125κ௡ଶ 

൑ 0.125κ௡ଷ 

κ௡ଶ 

κ௡ଵ 

‫ܣ‬ା௦ଷ 

κ௡ଷ 

Beams Other Than Perimeter Beams

κ௡ଵ Τ3



൒ greater of ൝

κ௡ଶ Τ3 ൒ greater of ൝

κ௡ଶ Τ3



κ௡ଷ Τ3

൒ κ௡ଵ Τ4

‫ିܣ‬௦ଵ 

Note 1 ሺtyp.ሻ

൒ ‫ܣ‬ା௦ଵ /4 ‫ܣ‬ା௦ଵ 

൒ 6ԣ

‫ିܣ‬௦ଶ 

Note 3

൑ 0.125κ௡ଵ 

‫ିܣ‬௦ଷ 

Note 3 ሺsim.ሻ

Note 2

Note 2 ሺsim.ሻ ‫ܣ‬ା௦ଶ  Note 4 ሺtyp.ሻ ൑ 0.125κ௡ଶ 

κ௡ଵ 

κ௡ଶ 

‫ܣ‬ା௦ଷ  ൑ 0.125κ௡ଷ  κ௡ଷ 

Perimeter Beams

Notes 1. Reinforcement to be anchored to develop ݂௬ at the face of the support. Standard hooks are shown in this figure. ା 2. At least the larger of ሺ‫ܣ‬ା ௦ଵ /4 ሻ or ሺ‫ܣ‬௦ଶ /4ሻ, but not less than 2 bars, must be continuous or spliced with Class B tension lap splices or mechanical or welded splices. ି 3. At least the larger of ሺ‫ିܣ‬ ௦ଵ /6ሻ or ሺ‫ܣ‬௦ଶ /6ሻ, but not less than 2 bars, must be continuous or spliced with Class B tension lap splices or mechanical or welded splices.

4. Closed stirrups in accordance with ACI Section 25.7.1.6 or hoops must be provided along the clear span. 5. Where the requirements in Note 2 are not satisfied for beams other than perimeter beams, closed stirrups in accordance with ACI Section 25.7.1.6 or hoops along the clear span must be provided.

Figure 6.32 Recommended flexural reinforcement details for beams.

6-52



Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

EDVHGRQFDOFXODWHGGHÀHFWLRQVDQGGHÀHFWLRQOLPLWDWLRQV$&, ,PPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQV FDQEHFDOFXODWHGIRUDQ\UHLQIRUFHGFRQFUHWHEHDPLQDFFRUGDQFHZLWK$&,UHJDUGOHVVRIZKHWKHUWKHPLQLPXP GHSWKUHTXLUHPHQWVRI$&,DUHVDWLV¿HGRUQRW+RZHYHUWKHSURYLVLRQVRI$&,PXVWEHXVHGZKHUHPHPEHUVDUHVXSSRUWLQJHOHPHQWVOLNHO\WREHGDPDJHGE\UHODWLYHO\ODUJHGHÀHFWLRQV &DOFXODWHGGHÀHFWLRQVPXVWEHOHVVWKDQRUHTXDOWRWKHOLPLWLQJGHÀHFWLRQVLQ$&,LQRUGHUWRVDWLVI\VHUYLFHability requirements. 3URFHGXUHVIRUGHWHUPLQLQJGHÀHFWLRQVRIEHDPVDUHJLYHQEHORZWKHVHSURFHGXUHVFDQDOVREHXVHGWRGHWHUPLQH GHÀHFWLRQVRIRQHZD\VODEV $OWKRXJKQXPHURXVPHWKRGVDUHDYDLODEOHWRGHWHUPLQHGHÀHFWLRQVLWLVLPSRUWDQWWR QRWHWKDWWKHVHPHWKRGVFDQRQO\HVWLPDWHGHÀHFWLRQVZLWKLQDQDFFXUDF\UDQJHRIWRSHUFHQW7KLVLVSULPDULO\ due to the variability in the constituent materials of concrete and to tolerances in construction. This range of accuUDF\VKRXOGEHFRQVLGHUHGFDUHIXOO\ZKHQFDOFXODWLQJGHÀHFWLRQVIRUGHÀHFWLRQVHQVLWLYHPHPEHUV 6.7.2 Immediate Deflections Uncracked Sections

For beams loaded such that maximum bending moment at service loads, is less than or equal to the VHFWLRQLVDVVXPHGWREHXQFUDFNHGDQGWKHLPPHGLDWHGHÀHFWLRQRIWKHEHDPFDQEHFDOFXODWHGXVLQJDQ\HODVWLF method of analysis. In such cases, the gross moment of inertia, , of the section (neglecting the longitudinal reinIRUFHPHQW FDQEHXVHGLQWKHGHÀHFWLRQFDOFXODWLRQV The cracking moment,

LVGHWHUPLQHGE\$&,(TXDWLRQ  (6.83)

where is the distance from the centroidal axis of the gross section, neglecting longitudinal reinforcement, to the tension face of the member and the modulus of rupture, LVGHWHUPLQHGE\$&,(TXDWLRQ  (6.84)

7KHPRGL¿FDWLRQIDFWRU LVGHWHUPLQHGE\7DEOHRU7DEOHLQWKLVSXEOLFDWLRQ 7KHFUDFNLQJPRPHQWLVPXOWLSOLHGE\WRDFFRXQWIRU UHVWUDLQWWKDWFDQUHGXFHWKHHIIHFWLYHFUDFNLQJPRPHQW DQG UHGXFHGWHQVLOHVWUHQJWKRIFRQFUHWHGXULQJFRQVWUXFWLRQWKDWFDQOHDGWRFUDFNLQJZKLFKFRXOGVXEVHTXHQWO\ KDYHDQLPSDFWRQVHUYLFHGHÀHFWLRQV Cracked Sections

2QFHDVHFWLRQFUDFNVWKHJURVVPRPHQWRILQHUWLDFDQQRORQJHUEHXVHGWRGHWHUPLQHLPPHGLDWHGHÀHFWLRQV , must be used. In general, is determined by transforming the longituInstead, the cracked moment of inertia, dinal reinforcing steel into an equivalent area of concrete (see Figure 6.33); the transformation is attained by mulwhere is the modulus of elasticity of tiplying the area of reinforcement, , by the modular ratio WKHUHLQIRUFLQJVWHHO$&, DQG LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWH$&, 7KHFRQFUHWH IURPWKHH[WUHPHFRPSUHVVLRQ¿EHULVFUDFNHGDQGLV below the neutral axis, which is located a distance equal to ignored in the analysis. Taking moments of areas about the neutral axis results in the following equation: (6.85)

6-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݇݀

ܾ

݄

݀

N. A.

‫ܣ‬௦ 

݊‫ܣ‬௦ 

ܾ

݊ ൌ ‫ܧ‬௦ /‫ܧ‬௖ 

Figure 6.33 Cracked transformed section of a rectangular beam with tension reinforcement.

ܾ

݄

݀

݇݀

ܾ

݇݀ ൌ

݀ᇱ 

݊‫ܣ‬௦ 

ܾሺ݇݀ሻଷ ൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ  3



ᇱ

N.A.

݀

‫ܣ‬ᇱ௦ 

‫ܫ‬௖௥ ൌ

ܾ

݇݀

ܾ

݄



N.A.

‫ܣ‬௦ 

ඥ2ܽଵ ݀ ൅ 1 െ 1  ܽଵ

‫ܣ‬௦ 

ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦  ݊‫ܣ‬௦ 

ට2ܽଵ ݀ ቀ1 ൅ ܽଶ ݀ ቁ ൅ ሺ1 ൅ ܽଶ ሻଶ െ ሺ1 ൅ ܽଶ ሻ ݀ ݇݀ ൌ  ܽଵ  ܾሺ݇݀ሻଷ ‫ܫ‬௖௥ ൌ ൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ ൅ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ሺ݇݀ െ ݀ ᇱ ሻଶ  3

Notes:

݊ ൌ ‫ܧ‬௦ /‫ܧ‬௖  Notes: ‫ܫ‬௚ ൌ ܾ݄ଷ /12 ܽଵ ൌ ܾ/݊‫ܣ‬௦  Notes: ܽଶ ൌ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ /݊‫ܣ‬௦  Figure 6.34 Cracked section properties for rectangular reinforced concrete beams.

(TXDWLRQ FDQEHVROYHGIRU

: (6.86)

where

6-54

݄

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ௙ 

ܾ௙ 

‫ݕ‬௧ 

݄௕ 

݀

݇݀

݇݀ ൌ



N.A.

݊‫ܣ‬௦ 

‫ܣ‬௦ 

݀ᇱ

݄

൫ܾ௙ െ ܾ௪ ൯݄ଷ ܾ௪ ሺ݇݀ሻଷ ݄ ଶ ൅ ൅ ൫ܾ௙ െ ܾ௪ ൯݄ ൬݇݀ െ ൰ ൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ  12 3 2

ܾ௙ 

݇݀ ൌ

݇݀

ܾ௙ 

݀

‫ܣ‬ᇱ௦  ‫ݕ‬௧ 

‫ܫ‬௖௥ ൌ



ܾ௪ 

݄௕ 

ඥܽଷ ሺ2݀ ൅ ݄ܽସ ሻ ൅ ሺ1 ൅ ܽସ ሻଶ െ ሺ1 ൅ ܽସ ሻ  ܽଷ

N.A.

‫ܣ‬௦ 

ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦  ݊‫ܣ‬௦ 

ඥܽଷ ሺ2݀ ൅ ݄ܽସ ൅ 2ܽଶ ݀ ᇱ ሻ ൅ ሺ1 ൅ ܽଶ ൅ ܽସ ሻଶ െ ሺ1 ൅ ܽ2 ൅ ܽ4 ሻ  ܽଷ

 ‫ܫ‬௖௥ ൌ

൫ܾ௙ െ ܾ௪ ൯݄ଷ ܾ௪ ሺ݇݀ሻଷ ݄ ଶ ൅ ൅ ൫ܾ௙ െ ܾ௪ ൯݄ ൬݇݀ െ ൰  12 3 2



ܾ௪ 

൅݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ ൅ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ሺ݇݀ െ ݀ ᇱ ሻଶ  

Notes: Notes:

݊ ൌ ‫ܧ‬௦ Τ‫ܧ‬௖ ‫ݕ‬௧ ൌ ൣሺܾ௙ െ ܾ௪ ሻሺ݄௕ ൅ 0.5݄ሻ݄ ൅ 0.5ܾ௪ ሺ݄௕ ൅ ݄ሻଶ ൧/ሾ൫ܾ௙ െ ܾ௪ ൯݄ ൅ ܾ௪ ሺ݄௕ ൅ ݄ሻሿ ଷ ଶ ଷ ଶ ‫ܫ‬Notes: ௚ ൌ ሾሺܾ௙ െ ܾ௪ ሻ݄ Τ12ሿ ൅ ൫ܾ௙ െ ܾ௪ ൯݄ሺ݄௕ ൅ 0.5݄ െ ‫ݕ‬௧ ሻ ൅ ሾܾ௪ ሺ݄௕ ൅ ݄ሻ Τ12ሿ ൅ ܾ௪ ሺ݄௕ ൅ ݄ሻሾ‫ݕ‬௧ െ 0.5ሺ݄௕ ൅ ݄ሻሿ 

ܽଶ ൌ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ /݊‫ܣ‬௦  ܽଷ ൌ ܾ௪ /݊‫ܣ‬௦  ܽସ ൌ ݄ሺܾ௙ െ ܾ௪ ሻ/݊‫ܣ‬௦  Figure 6.35 Cracked section properties for flanged reinforced concrete beams.

Thus,

is equal to the following: (6.87)

where

is determined by Equation (6.86).

Similar equations can be derived for rectangular sections with tension and compression reinforcement and for ÀDQJHGVHFWLRQV&UDFNHGVHFWLRQSURSHUWLHVIRUUHFWDQJXODUDQGÀDQJHGUHLQIRUFHGFRQFUHWHEHDPVDUHJLYHQLQ)LJXUHDQG)LJXUHUHVSHFWLYHO\ Effective Moment of Inertia

It is evident from the preceding discussion that two different moments of inertia would be required for calculating ). To eliminate the need for two moments of inertia, the effective moment LPPHGLDWHGHÀHFWLRQVWKDWLV and of inertia, FDQEHXVHGLQOLHXRIDPRUHFRPSUHKHQVLYHDQDO\VLVVHH$&,7DEOH 

(6.88)

6-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In this equation, lated.

LVWKHPD[LPXPPRPHQWLQWKHEHDPGXHWRVHUYLFHORDGVDWWKHVWDJHGHÀHFWLRQVDUHFDOFX-

)RUFRQWLQXRXVEHDPVLWLVSHUPLWWHGWRFDOFXODWHLPPHGLDWHGHÀHFWLRQVXVLQJ as the average of the values calcuODWHGE\(TXDWLRQ DWWKHFULWLFDOSRVLWLYHDQGQHJDWLYHPRPHQWVHFWLRQV$&, ,WLVDOVRSHUPLWWHGWR FDOFXODWHGHÀHFWLRQVXVLQJ at midspan for prismatic beams with simple and continuous spans, and at the support IRUFDQWLOHYHUV$&,  Approximate Immediate Deflections

7KHIROORZLQJHTXDWLRQZKLFKLVLQ$&,&RPPHQWDU\6HFWLRQFDQEHXVHGWRGHWHUPLQHLPPHGLDWH GHÀHFWLRQ , at the tips of cantilevers and at the midspan of simply supported and continuous members: (6.89)

In this equation, is the support bending moment at service loads for cantilevers and the midspan bending moPHQWDWVHUYLFHORDGVIRUVLPSO\VXSSRUWHGDQGFRQWLQXRXVEHDPV9DOXHVRIWKHGHÀHFWLRQFRHI¿FLHQW , are given in Table 6.19 for beams subjected to uniformly distributed service loads, , and different span conditions. Table 6.19 Deflection Coefficient, K Span Condition

K

Cantilever



Simple



Continuous 6.7.3 Time-Dependent Deflections

,QRQHZD\ÀH[XUDOPHPEHUVFUHHSDQGVKULQNDJHGXHWRVXVWDLQHGORDGVFDXVHWLPHGHSHQGHQWGHÀHFWLRQVWKDWFDQ EHWRWLPHVJUHDWHUWKDQWKHLPPHGLDWHGHÀHFWLRQV1RWDFFRXQWLQJIRUWLPHGHSHQGHQWGHÀHFWLRQVFDQUHVXOWLQD VLJQL¿FDQWXQGHUHVWLPDWLRQRIWRWDOGHÀHFWLRQ 7LPHGHSHQGHQWGHÀHFWLRQVDUHGHWHUPLQHGE\PXOWLSO\LQJWKHLPPHGLDWHGHÀHFWLRQ ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ the factor

, due to sustained loads by (6.90)

In this equation, LVWKHWLPHGHSHQGHQWIDFWRUIRUVXVWDLQHGORDGVJLYHQLQ$&,7DEOHVHH7DEOH  is the reinforcement ratio for any compression reinforcement in the section. The term and DFFRXQWVIRUWKHHIIHFWRIFRPSUHVVLRQUHLQIRUFHPHQWLQUHGXFLQJWLPHGHSHQGHQWGHÀHFWLRQV,WLVSHUPLWWHGWRFDOculate DWPLGVSDQIRUVLPSOHDQGFRQWLQXRXVVSDQVDQGDWWKHVXSSRUWIRUFDQWLOHYHUV$&,  Table 6.20 Time-Dependent Factor for Sustained Loads, ȟ

6-56

Sustained Load Duration (Months)

ȟ

3



6







RUPRUH



Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Only the dead load and a portion of the sustained live load (if applicable) need to be included in the determination ZKHQFDOFXODWLQJWKHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJH : of (6.91)

6.7.4 Maximum Permissible Calculated Deflections

'HÀHFWLRQVFDOFXODWHGXVLQJWKHPHWKRGVLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQPXVWEHOHVVWKDQRUHTXDOWR WKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQVJLYHQLQ$&,7DEOHVHH7DEOH  Table 6.21 Maximum Permissible Calculated Deflections Member

Flat roofs

Condition

Not supporting or attached to nonstructural HOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV

Floors

Roof or ÀRRUV

Deflection to be Considered

,PPHGLDWHGHÀHFWLRQGXHWRWKH maximum of

Deflection Limitation (1)

,PPHGLDWHGHÀHFWLRQGXHWR Supporting or attached to nonstructural elements OLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV Supporting or attached to nonstructural elements QRWOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV

7KHSDUWRIWKHWRWDOGHÀHFWLRQ occurring after attachment of nonstructural elements(2)

(3)

(4)

(1) This limit is not intended to safeguard against ponding. Ponding must be checked by deflection calculations, including (1) additional deflections due to ponded water and (2) consideration of time-dependent effects of sustained load, camber, construction tolerances, and reliability of provisions for drainage. (2) The time-dependent deflection is to be determined in accordance with ACI 24.2.4; this deflection is permitted to be reduced by the amount of deflection calculated to occur before attachment of nonstructural elements. This amount must be calculated based on accepted engineering data relating time-dependent characteristics of members similar to those being considered. (3) This limit is permitted to be exceeded if measures are taken to prevent damage to supported or attached elements. (4) This limit must not exceed the tolerance provided for the nonstructural elements.

7KHOLPLWDWLRQVJLYHQLQ7DEOHUHODWHRQO\WRVXSSRUWHGRUDWWDFKHGQRQVWUXFWXUDOHOHPHQWV:KHUHVWUXFWXUDO PHPEHUVDUHOLNHO\WREHDIIHFWHGE\GHÀHFWLRQRUGHIRUPDWLRQRIWKHPHPEHUVWRZKLFKWKH\DUHDWWDFKHGLQVXFKD ZD\DVWRDGYHUVHO\DIIHFWWKHVWUHQJWKRIWKHVWUXFWXUHWKHVHGHÀHFWLRQVDQGWKHUHVXOWLQJIRUFHVPXVWEHFRQVLGHUHG H[SOLFLWO\LQWKHDQDO\VLVDQGGHVLJQRIWKHVWUXFWXUDOPHPEHUVDVUHTXLUHGE\$&,

6.8 Design Procedure The design procedure in Figure 6.36 can be used in the design and detailing of rectangular beam sections with one OD\HURIORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQW,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVWDEOHQXPEHUVDQG¿JXUH QXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFFDQEHIRXQGLQWKLVFKDSWHU

6-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1

EŽ

zĞƐ

A

Figure 6.36 Design procedure for beams.

6-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

EŽ

zĞƐ

1

B

Figure 6.36 Design procedure for beams. (cont.)

6-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

B

EŽ

zĞƐ

Torsion effects can be neglected. EŽ

zĞƐ

C

EŽ

zĞƐ

EŽ

zĞƐ

Modify section dimensions.

1 Figure 6.36 Design procedure for beams. (cont.)

6-60

D

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

C

zĞƐ

EŽ

No shear reinforcement is required at the section.

EŽ

zĞƐ

EŽ

zĞƐ

zĞƐ

EŽ

Modify section dimensions.

1 Detail the flexural and shear reinforcement in accordance with Sect. 6.6

Figure 6.36 Design procedure for beams. (cont.)

6-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

D

Detail the longitudinal and transverse reinforcement in accordance with Sect. 6.6

Figure 6.36 Design procedure for beams. (cont.)

6-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.9 Examples 6.9.1

Example 6.1 – Determination of Beam Size: Building #1 (Framing Option C), Beam is Not Part of the LFRS, SDC A

'HWHUPLQHWKHVL]HRIDW\SLFDOEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DWDW\SLFDOÀRRUOHYHO DVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHDLQVODELQE\LQFROXPQV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the beam depth based on serviceability requirements

ACI 9.3.1.1

Assume the beam is not supporting or attached to partitions or other construction likely to be damaged by large GHÀHFWLRQV7KXVGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,QHHGQRWEHFDOFXODWHG Because the length is the same for all spans, minimum beam depth is based on the support condition of one end continuous: Table 6.1

For formwork economy, determine minimum depth for the beams in the east-west direction because the span lengths are longer than those in the north-south direction:

8VHDLQGHHSEHDPZKLFKLVDGHTXDWHIRUWKHEHDPVLQERWKGLUHFWLRQVVHH([DPSOHIRUWKHGHWHUPLQDWLRQRIWKHPLQLPXPVODEWKLFNQHVVZKLFKGHSHQGVRQWKHEHDPVL]HV  Step 2 – Determine the beam width based on strength requirements

Assuming , the required *UDGHUHLQIRUFHPHQW

can be determined by the following equation with

Sect. 6.2.2

and

Eq. (6.1)

The largest

along the spans is used to determine

.

The factored bending moments in the beam due to the weight of the slab and the superimposed dead and live loads DUHGHWHUPLQHGLQ6WHSRI([DPSOHIRUWKLVEHDPVXSSRUWHGWZRZD\VODEXVLQJWKH'LUHFW'HVLJQ0HWKRGVHH )URP7DEOHWKH 7DEOH 7KHVHIDFWRUHGEHQGLQJPRPHQWVDUHEDVHGRQ$&,(TE  LQWKHEHDPRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQDQGLVHTXDOWRIWNLSV largest &RQVHUYDWLYHO\HVWLPDWHWKHIDFWRUHGGHDGORDGRIWKHEHDPZHEDVNLSIW 7KHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVEHDPDQGWKHPD[LPXPIDFWRUHGEHQGLQJPRPHQWGXHWRWKHZHLJKWRIWKHEHDPVWHPDOVRRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQ Figure 4.3

Total

6-63

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Approximate Therefore,

8QOHVVWKHZLGWKLVUHVWULFWHGIRUDUFKLWHFWXUDOUHDVRQVDLQZLGHEHDPLVLPSUDFWLFDOWRXVH$Q\IXQFWLRQDO EHDPZLGWKZLGHUWKDQLQVDWLV¿HVVWUHQJWKUHTXLUHPHQWV8VHDLQZLGHEHDPZKLFKLVLQZLGHUWKDQ WKHLQFROXPQLWIUDPHVLQWRVHH6HFWRIWKLVSXEOLFDWLRQRQJXLGHOLQHVIRUVL]LQJEHDPVIRUHFRQRP\  Check the assumption regarding the weight of the beam web:

8VHDLQE\LQEHDP 6.9.2

Example 6.2 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C), Beam is Not Part of the LFRS, SDC A, Single Layer of Tension Reinforcement

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHLQE\LQEHDPDORQJFROXPQOLQHLQ%XLOGLQJ )UDPLQJ2SWLRQ&DWDW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPH DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the factored bending moments along the span

The factored bending moments in the beam due to the weight of the slab, the superimposed dead load, and the live ORDGRQWKHVODEDUHGHWHUPLQHGLQ6WHSRI([DPSOHIRUWKLVEHDPVXSSRUWHGWZRZD\VODEVHH7DEOH  7KHVHIDFWRUHGEHQGLQJPRPHQWVDUHEDVHGRQ$&,(TE  *LYHQDLQWKLFNVODEWKHIDFWRUHGGHDGORDGRIWKHEHDPZHELVHTXDOWRNLSIWVHH([DPSOH  7KHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVEHDP7KHIDFWRUHGEHQGLQJ PRPHQWVGXHWRWKHGHDGORDGRIWKHEHDPZHEDUHGHWHUPLQHGXVLQJ)LJXUHDQGDUHJLYHQLQ7DEOHZKHUH for all spans. Table 6.22 Factored Bending Moments due to Dead Load of Beam Web (ft-kips) End Span Exterior Negative

Positive

Interior Span First Interior Negative

Positive

Interior Negative

7RWDOGHVLJQEHQGLQJPRPHQWVDUHJLYHQLQ7DEOHVHH7DEOH  Table 6.23 Design Bending Moments (ft-kips) for the Beam in Example 6.2 End Span Exterior Negative

6-64

Positive

Interior Span First Interior Negative

Positive

Interior Negative

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the effective flange width, bf , of the beam

ACI 6.3.2.1

7KHHIIHFWLYHÀDQJHZLGWK , of the beam is equal to the following:

Figure 6.10

Step 3 – Determine the required flexural reinforcement at the critical sections

$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQWHQVLRQ UHTXLUHG single layer of tension reinforcement:

is determined by the following assuming a

Eq. (6.34)

Eq. (6.33)

where

Figure 6.11

$WSRVLWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQFRPSUHVVLRQ GHWHUPLQHWKHGHSWKRIWKHHTXLYDOHQWVWUHVVEORFN , LQ7DEOHZKLFKUHVXOWVLQWKHODUJHVW ): assuming rectangular section behavior using the largest positive

Eq. (6.49)

where Because with

WKHVHFWLRQEHKDYHVDVDUHFWDQJXODUVHFWLRQDQG(TV DQG FDQEHXVHGWRGHWHUPLQH DWSRVLWLYHPRPHQWVHFWLRQVVHH)LJXUH 

$VXPPDU\RIWKHUHTXLUHGUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWPLQLPXPUHLQIRUFHPHQWLVUH. quired at all sections. Also, all sections are tension-controlled because

6-65

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.24 Required Flexural Reinforcement for the Beam in Example 6.2 Location

Mu (ft-kips)

Rn (psi)

As (in2)*

















161



Exterior negative End span



Positive First interior negative

Interior span *Min.



Positive Negative [Eq. (6.35)]

Max.

[Eq. (6.38)]

Step 4 – Select the flexural reinforcement

Sect. 6.6.3

6HOHFWWKHÀH[XUDOUHLQIRUFHPHQWLQDVLQJOHOD\HUEDVHGRQWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ $&,DQGUHVSHFWLYHO\ • Negative reinforcement

Because

and

distribute the required

within

Figure 6.23

)RUWKHLQZLGHEHDP Minimum number of reinforcing bars 7U\EDUV

Table 6.9

).

0D[LPXPQXPEHURIEDUVLQDLQZLGHVHFWLRQ

Table 6.8

The reinforcement provided in the slab is at least equal to the shrinkage and temperature reinforcement preVFULEHGLQ$&,VHH7DEOHLQ([DPSOH WKLVUHLQIRUFHPHQWVDWLV¿HVWKHUHTXLUHPHQWLQ$&, IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH • Positive reinforcement 6WUHQJWKDQGVSDFLQJUHTXLUHPHQWVDWWKHSRVLWLYHFULWLFDOVHFWLRQVDUHVDWLV¿HGXVLQJEDUVZKHUHWKHZLGWK RIWKHEHDPLVHTXDOWRLQ 8VHEDUVDWDOOFULWLFDOVHFWLRQV 6.9.3

Example 6.3 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C), Beam is Not Part of the LFRS, SDC A

'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUWKHEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DW DW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG

6-66

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the factored shear forces along the span

,WLVGHWHUPLQHGLQ6WHSRI([DPSOHWKDW for the interior beams in the north-south direction. 7KHUHIRUHWKHVHEHDPVPXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHVKHDUIRUFHVLQWKLVVODEV\VWHPVHH6HFW of this publication). The tributary area that must be used to determine IRUWKHEHDPLQWKLVH[DPSOHLVJLYHQLQ)LJXUHZKLFKLV WKHVDPHIRUDOOEHDPVDORQJDQ\LQWHULRUFROXPQOLQHLQWKHQRUWKVRXWKGLUHFWLRQVHH6HFWRIWKLVSXEOLFDWLRQ   ͵A

ͶA

ͷA

ʹͷᇱ ǦͲԣ

ʹͷᇱ ǦͲԣ

A

ʹ͵ᇱ Ǧ͸ԣ

”‹„—–ƒ”›ƒ”‡ƒ

Ͷͷ†‡‰ A

ͳͳᇱ Ǧͻԣ

ͳͳᇱ Ǧͻԣ

Figure 6.37 Tributary area for shear forces on the beam in Example 6.3.

Determine the factored dead and live loads on the beam:

For simplicity, the uniformly distributed dead load of the beam web is transformed into a triangular distributed load. The shear force at the face of the support due to the uniformly distributed dead load is equal to 7KHWULDQJXODUGLVWULEXWHGORDGWKDWSURGXFHVDVKHDUIRUFHRINLSVDWWKHIDFHRIWKHVXSSRUWLVHTXDO to

Table 3.3

7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUWLQDQHQGVSDQDQGLVREWDLQHGIURPHTXLOLEULXPFRQVLGHULQJWKHNLSVIWWULDQJXODUORDGDQGWKHIWNLSDQGIWNLSQHJDWLYHPRPHQWVDWWKHH[WHULRU VXSSRUWDQGWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUWUHVSHFWLYHO\VHH)LJXUHDQG7DEOH ,WLVDVVXPHGIRUVLP6-67

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

݀ ൌ ʹͳǤͷԣ

͸ǤͶ‹’•Ȁˆ–

ͻǦ͓͵•–‹””—’•ሺ͵Ž‡‰•ሻ̷ͳͲԣ‘Ǥ…Ǥ

ʹԣ

ሺ‡ƒǤ‡†ሻ

”‹–‹…ƒŽ•‡…–‹‘

ʹᇱ ǦͲԣ

݀ ൌ ʹͳǤͷԣ

κ௡ Τʹ ൌ ͳͲǤ͹ͷԢ

۳‫ܖ܉ܘ܁܌ܖ‬

ܸ௨ ̷ˆƒ…‡ ൌ ͵ͻǤͺ‹’•

ܸ௨ ̷݀ ൌ ͵ͺǤͻ‹’•

ܸ௨ ൌ ߶ߣ—݂௖ᇱ ܾ௪ ݀ ൌ ʹͺǤ͸‹’•

‫ݔ‬௡௦ ൌ ͸ǤͳԢ

Figure 6.38 Partial shear diagram for the beam in Example 6.3.

SOLFLW\WKDWWKHWULDQJXODUORDGLVHTXDOWR]HURDWWKHIDFHVRIWKHVXSSRUWVLQVWHDGRIDWWKHFHQWHUVRIWKHFROXPQV this assumption has a negligible impact on the results.

6-68

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH from the face of the support:

Step 2 – Determine the required shear reinforcement

Assuming at least minimum shear reinforcement is provided, : with the axial force

can be determined from the following equation Table 6.4

where

for normalweight concrete.

Because

Table 6.2

, provide minimum shear reinforcement:

Table 6.5

For

: along the length of the beam.

Maximum spacing Assuming

,

Figure 6.19

is equal to the following:

Across the width of the beam, maximum spacing

Figure 6.19

$VVXPLQJ8VWLUUXSVRXWHUOHJV ZLWKLQFRYHU

Therefore, 3 stirrup legs must be provided to satisfy maximum spacing requirements across the width of the beam. 8VHVWLUUXSVZLWKOHJVVSDFHGDWLQRQFHQWHU Stirrups are no longer required at the section where The length from the face of the sup, can be obtained from the following equation (see Figure 6.38): port where stirrups are no longer required,

8VHVWLUUXSVOHJV VSDFHGDWLQRQFHQWHUDWERWKHQGVRIWKHEHDPZLWKWKH¿UVWVWLUUXSORFDWHGLQIURP the face of the column. 6.9.4

Example 6.4 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C), Beam is Not Part of the LFRS, SDC A

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUWKHEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DW DW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with

6-69

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the flexural reinforcement details

,WLVGHWHUPLQHGLQ([DPSOHWKDWEDUVDUHUHTXLUHGDWWKHQHJDWLYHDQGSRVLWLYHFULWLFDOVHFWLRQVLQWKHEHDP 7KLVQXPEHURIUHLQIRUFLQJEDUVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQG UHVSHFWLYHO\VHH6WHSLQ([DPSOH  ), the negative reinforce%H\RQGWKHSRLQWRILQÀHFWLRQLQWKHPRPHQWGLDJUDPWKDWLVDWWKHVHFWLRQZKHUH PHQWLVQRORQJHUUHTXLUHGIRUVWUHQJWK7KHPD[LPXPOHQJWKWRWKHSRLQWRILQÀHFWLRQRFFXUVLQWKHHQGVSDQVHH Figure 6.39):

A

͸ǤͶ‹’•Ȁˆ–

݀ ൌ ʹͳǤͷԣ

ͷǦ͓͸

‫ۓ‬ ۖ

݀ ൌ ʹͳǤͷᇳ 

‰”‡ƒ–‡”‘ˆ ͳʹ݀௕ ൌ ͹Ǥͷᇳ  ‫۔‬ ۖ ‫ە‬κ௡ Τͳ͸ ൌ ͳ͸Ǥͳԣ

ͷǦ͓͸

ͶǤ͵ͷ ൅ ሺʹͳǤͷΤͳʹሻ ൌ ͸ǤͳԢ ͹ᇱ ǦͲԣ κ௡ Τʹ ൌ ͳͲǤ͹ͷԢ

ʹᇱ ǦͲԣ

۳‫ܖ܉ܘ܁܌ܖ‬

‘‹–‘ˆ‹ˆŽ‡…–‹‘

‫ܯ‬௨  ൌ ͳ͸ͷǤͲˆ–Ǧ‹’•

‫ݔ‬௣௜ ൌ ͶǤ͵ͷԢ

Figure 6.39 Location of the point of inflection for the beam in Example 6.4

6-70

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHQHJDWLYHUHLQIRUFHPHQWPXVWKDYHDQHPEHGPHQWOHQJWKEH\RQGWKHSRLQWRILQÀHFWLRQHTXDOWRWKHJUHDWHURI the following:

ACI 9.7.3.8.4

7KHUHIRUHWKHWRSEDUVDUHSHUPLWWHGWREHFXWRIIDGLVWDQFHHTXDOWR from the face of WKHVXSSRUW+RZHYHUIURP([DPSOHVWLUUXSVOHJV VSDFHGDWLQRQFHQWHUPXVWEHSURYLGHGDWERWK HQGVRIWKHEHDPZLWKWKH¿UVWVWLUUXSORFDWHGLQIURPWKHIDFHRIWKHFROXPQ7KLVPHDQVWKHODVWVWLUUXSLVORFDWHG IURPWKHIDFHRIWKHVXSSRUW7KHUHIRUHSURYLGHDIWLQHPEHGPHQWOHQJWKIRUWKH WRSEDUV7KLVUHLQIRUFHPHQWLVQRWWHUPLQDWHGLQDWHQVLRQ]RQHVHH)LJXUH DQGFDQEHFRQVHUYDWLYHO\XVHGDW all support locations. )RUVLPSOHUGHWDLOLQJWKHERWWRPEDUVDUHPDGHFRQWLQXRXVLQVWHDGRIFXWWLQJRIIDSRUWLRQRIWKHPWKHODWWHURI ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH supports. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK

6-71

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the shear reinforcement details

,WLVGHWHUPLQHGLQ([DPSOHWKDWVWLUUXSVZLWKOHJVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDU strength requirements. This arrangement consists of (1) a U-stirrup extending across the effective width of the beam ZLWKGHJUHHKRRNVDWHDFKHQGDQG DFURVVWLHZLWKGHJUHHDQGGHgree hooks at the ends tied to the top and bottom longitudinal reinforcement. 7KHPLQLPXPEHDPZLGWKUHTXLUHGIRUGHYHORSPHQWRIWKHVWLUUXSVLVLQLQ

Table 6.17

5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWV LQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOEHDPVDORQJWKHLQWHULRUFROXPQOLQHVLQ the north-south direction that are not part of the LFRS.         

2ᇱ -0ᇱᇱ 

7ᇱ -0ᇱᇱ 

A

5-#6 

A

5-#6 2ᇱᇱ 

9-#3 stirrups @ 10ᇱᇱ  ሺtyp. ሻ

2ᇱ -0ᇱᇱ 

7ᇱ -0ᇱᇱ 

2ᇱ -0ᇱᇱ 

7ᇱ -0ᇱᇱ 

5-#6 

5-#6

21ᇱ -6ᇱᇱ 

7ᇱᇱ 

5-#6 

1Ԣ-5ᇱᇱ 

Other beams and reinforcement not shown for clarity

#3 stirrups @ 10ԣ  5-#6  2ᇱ -4ᇱᇱ 

‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬

Figure 6.40 Reinforcement details for the beam in Examples 6.1 through 6.4.

6.9.5

Example 6.5 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is Not Part of the LFRS, SDC A

'HWHUPLQHWKHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUWKHEHDPLQWKHHQGVSDQDORQJFROXPQOLQHLQ %XLOGLQJ)UDPLQJ2SWLRQ&DWDW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH  $VVXPHWKHEHDPLVQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV DQG*UDGH DQGSHUFHQWRIWKHOLYHORDGLVVXVWDLQHG$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK

6-72

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the service loads and bending moments

The Direct Design Method can be used to determine the bending moments in the beam due to the weight of the slab, WKHVXSHULPSRVHGGHDGORDGDQGWKHOLYHORDGVHH6WHSVDQGRI([DPSOH 

7KHEHQGLQJPRPHQWFRHI¿FLHQWVDWWKHFULWLFDOVHFWLRQVLQWKHEHDPLQWKHHQGVSDQDUHJLYHQLQ7DEOH Exterior negative:

Positive:

First interior negative:

7KHEHQGLQJPRPHQWVGXHWRWKHZHLJKWRIWKHEHDPVWHPFDQEHGHWHUPLQHGXVLQJWKHVLPSOL¿HGPHWKRGLQ$&, VHH6WHSLQ([DPSOH  Exterior negative:

Positive:

6-73

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

First interior negative:

7KHVXVWDLQHGSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVDUHWKHIROORZLQJZKHUHSHUFHQWRIWKHOLYHORDGLVVXVtained:

At the exterior support: $WWKH¿UVWLQWHULRUVXSSRUW Step 2 – Determine the material properties of the concrete and reinforcing steel Eq.(6.84)

where

for normalweight concrete.

Table 6.2 ACI Eq. (19.2.2.1b) ACI 20.2.2.2

Modular ratio Step 3 – Determine the gross and cracked moments of inertia

(IIHFWLYHÀDQJHZLGWKRIWKHEHDP

Step 2, Example 6.2

ܾ௙ ൌ ͻʹǤͷᇱᇱ 

‫ݕ‬௧ ൌ ͳͷǤͶᇱᇱ 

݄௕ ൌ ͳ͹ᇱᇱ 

݄ ൌ ͹ᇱᇱ 

• Positive moment section 7KHJURVVVHFWLRQRIWKHEHDPDWWKHSRVLWLYHPRPHQWVHFWLRQLVVKRZQLQ)LJXUH

ܾ௪ ൌ ʹͺᇱᇱ 

Figure 6.41 Gross section of the beam at a positive moment section.

The gross section properties of the beam are the following:

Figure 6.35

6-74

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)URP([DPSOHERWWRPEDUVDUHUHTXLUHG

.

$VVXPLQJDUHFWDQJXODUFRPSUHVVLRQDUHDWKHFUDFNHGVHFWLRQSURSHUWLHVDUHWKHIROORZLQJVHH)LJXUH 

݀ ൌ ʹͳǤͷ 

ᇱᇱ

ܾ௙ ൌ ͻʹǤͷᇱᇱ 

݇݀ ൌ ʹǤ͹ᇱᇱ 

Figure 6.34

݊‫ܣ‬௦ 

Figure 6.42 Cracked transformed section of the beam at a positive moment section.

Therefore, the assumption of a rectangular section is correct. Figure 6.34

• Negative moment section 7KHQHJDWLYHPRPHQWVHFWLRQRIWKHEHDPLVWKHE\LQEHDPZHE)URP([DPSOHWRSEDUV . DUHUHTXLUHGDWWKHH[WHULRUDQG¿UVWLQWHULRUVHFWLRQV Figure 6.34

6-75

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the effective moment of inertia

The effective moment of inertia,

, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.

• Positive moment section

Therefore,

Therefore,

Therefore, Eq. (6.88)

• Negative moment section

At the exterior support:

Therefore,

Therefore,

Therefore, $WWKH¿UVWLQWHULRUVXSSRUW

Therefore,

6-76

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

Therefore,

• Average effective moment of inertia Dead load: ACI 24.2.3.6

Sustained load:

Dead plus live load:

Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads

Assume the loads due to the dead load of the slab, the superimposed dead load, and the live load are uniformly distributed over the span instead of the actual triangular load distribution. The following equation can be used to GHWHUPLQHLPPHGLDWHGHÀHFWLRQV Eq. (6.89)

For continuous members: Table 6.19

where

and

is the service load moment at midspan. Also assume

• Dead load

• Sustained load

6-77

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Dead plus live load

• Live load

)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV WKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ Table 6.21 Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection

$VVXPLQJDPRQWKGXUDWLRQRIORDGLQJZLWK

: Eq. (6.90), Table 6.20

7KHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJHLVHTXDOWRWKHIROORZLQJ Eq. (6.91) Table 6.21

7KHLQE\LQEHDPLVDGHTXDWHIRUGHÀHFWLRQ 6.9.6

Example 6.6 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C), Beam is Part of the LFRS, SDC A, Multiple Layers of Tension Reinforcement

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ )UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP DUHOLPLWHGWRLQE\LQVHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW and the DQG*UDGHUHLQIRUFHFROXPQVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK ment. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored bending moments, Mu along the span

,WFDQEHGHWHUPLQHGXVLQJWKHUHTXLUHPHQWVLQ$&,WKDWDLQWKLFNVODELVDGHTXDWHIRUVHUYLFHDELOLW\ EDVHGRQWKHLQE\LQEHDPV$OVRWKH'LUHFW'HVLJQ0HWKRGFDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWV LQWKHFROXPQDQGPLGGOHVWULSVRIWKHVODEVHH6HFWRIWKLVSXEOLFDWLRQ 

6-78

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$FFRUGLQJWRWKH'LUHFW'HVLJQ0HWKRGWKHEHDPVLQWKLVH[DPSOHPXVWUHVLVWSHUFHQWRIWKHFROXPQVWULSEHQGing moments due to the weight of the slab, the superimposed dead load, and the live load.

7KHEHQGLQJPRPHQWVGXHWRZHLJKWRIWKHEHDPVWHPFDQEHGHWHUPLQHGXVLQJWKHVLPSOL¿HGPHWKRGLQ$&, A summary of the service bending moments due to the dead and live loads at the critical sections of the beam in the HQGVSDQLVJLYHQLQ7DEOH Table 6.25 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Beam in the End Span in Example 6.6 Exterior Negative

Positive

First Interior Negative

7KHEHQGLQJPRPHQWVLQWKHEHDPGXHWRZLQGORDGVDUHJLYHQLQ7DEOH7KH³SOXVPLQXV´VLJQSUHFHGLQJWKH WDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHQRUWKGLUHFWLRQDQGWKHVRXWKGLUHFWLRQWKDWLVVLGHVZD\ to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). Table 6.26 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Beam in the End Span in Example 6.6 Exterior Negative

Positive

First Interior Negative

ņ 7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH Table 6.27 Design Bending Moments (ft-kips) at the Second-Floor Level for the Beam in the End Span in Example 6.6 Load Combination

Exterior Negative

Positive

First Interior Negative

 SSR



SSL



SSR SSL



 

6-79

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the effective flange width, bf , of the beam

ACI 6.3.2.1

7KHHIIHFWLYHÀDQJHZLGWK , is equal to the following:

Figure 6.10

Step 3 – Determine the required flexural reinforcement at the critical sections

$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQWHQVLRQ UHTXLUHG is determined by the following where for beams with one layer of tension reinforcement: Eq. (6.34)

Eq. (6.33)

where

Figure 6.11

$WSRVLWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQFRPSUHVVLRQ GHWHUPLQHWKHGHSWKRIWKHHTXLYDOHQWVWUHVVEORFN , LQ7DEOHZKLFKUHVXOWVLQWKHODUJHVW ): assuming rectangular section behavior using the largest positive

Eq. (6.49)

where and tension reinforcement are provided because the beam is relatively narrow). Because

ZKHUHLWLVDVVXPHGOD\HUVRI

the section behaves as a rectangular section.

Determine if the reinforcing steel in the inner layer located IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVDVVXPLQJ DLQFOHDUFRYHUWRVWLUUXSVORQJLWXGLQDOEDUVDQGDLQFOHDUVSDFLQJEHWZHHQWKHOD\HUVRIORQJLWXGLnal reinforcement:

Therefore, the reinforcing steel in the inner layer yields and

6-80

FDQEHGHWHUPLQHGXVLQJ(TV DQG ZLWK

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VXPPDU\RIWKHUHTXLUHGUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWDOOVHFWLRQVDUHWHQVLRQFRQWUROOHG because Table 6.28 Required Flexural Reinforcement for the Beam in Example 6.6 Location

Mu (ft-kips)

Rn (psi)

As (in2)*





89







Exterior negative 

Positive First interior negative *Min.

[Eq. (6.35)]

Max.

[Eq. (6.38)]

Step 4 – Select the flexural reinforcement

Sect. 6.6.3

6HOHFWWKHÀH[XUDOUHLQIRUFHPHQWEDVHGRQWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQG respectively. • Negative reinforcement

Because

, distribute the required negative

within

.

Figure 6.23

Exterior negative: :LWKLQFRYHUVWLUUXSV*UDGHUHLQIRUFLQJEDUVDQG and the maximum center-to-center bar spacing is equal to the following:

Eq. (6.65)

8VHEDUV

Table 6.7

The reinforcement provided in the slab is at least equal to the shrinkage and temperature reinforcement in ACI VHH7DEOHLQ([DPSOH WKLVUHLQIRUFHPHQWVDWLV¿HVWKHUHTXLUHPHQWLQ$&,IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH First interior negative: 8VHEDUV

Table 6.7

• Positive reinforcement 0LQLPXPQXPEHURIUHLQIRUFLQJEDUVUHTXLUHGZLWKLQDLQZLGWK

Table 6.9

0D[LPXPQXPEHURIEDUVLQDLQZLGHVHFWLRQ

Table 6.8

6-81

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8VHEDUVLQOD\HUVZLWKEDUVSHUOD\HU 7KHEDUVDUHDOVRDGHTXDWHIRUWKHIWNLSSRVLWLYHPRPHQWDWWKHH[WHULRUVXSSRUWIRU665VHH7DEOH   Comments.,QVWHDGRIXVLQJEDUVLQOD\HUVIRUWKHSRVLWLYHUHLQIRUFHPHQWEDUVLQDVLQJOHOD\HUFDQEH ZKLFKVDWLV¿HVPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV7KHSULPDU\ used purpose of this example is to illustrate the design procedure with multiple layers of tension reinforcement. 6.9.7

Example 6.7 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C), Beam is Part of the LFRS, SDC A

'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ )UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP DUHOLPLWHGWRLQE\LQVHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW. Also asDQG*UDGHUHLQIRUFHPHQW sume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFWSee Example 6.6. Step 1 – Determine the factored shear forces along the span

It can be determined that for the interior beams in the north-south direction. Therefore, these beams PXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHVKHDUIRUFHVLQWKLVVODEV\VWHPVHH6HFWRIWKLVSXEOLFDWLRQ  )RUWKHEHDPLQWKLVH[DPSOHWKHWULEXWDU\DUHDLVJLYHQLQ)LJXUHWKLVWULEXWDU\DUHDLVWKHVDPHIRUDOOEHDPV along any interior column line in the north-south direction. In addition to the shear forces due to the gravity loads, the beam must resist the shear forces due to the wind loads. Determine the factored dead and live loads on the beam:

For simplicity, the uniformly distributed dead load of the beam web is transformed into a triangular distributed load. The shear force at the face of the support due to the uniformly distributed dead load is equal to 7KHWULDQJXODUGLVWULEXWHGORDGWKDWSURGXFHVDVKHDUIRUFHRINLSVDWWKHIDFHRIWKHVXSSRUWLVHTXDO to

7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWLQDQHQGVSDQIRUWKHORDGFRPELQDWLRQ VHH)LJXUHDQG7DEOH ,WLVDVVXPHGIRUVLPSOLFLW\WKDWWKHWULDQJXODUORDGLVHTXDOWR ]HURDWWKHIDFHVRIWKHVXSSRUWVLQVWHDGRIDWWKHFHQWHUVRIWKHFROXPQVWKLVDVVXPSWLRQKDVQHJOLJLEOHLPSDFWRQ the results. Table 3.3

6-82

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

݀ ൌ ʹͳǤͷԣ

ͻǤͻ‹’•Ȁˆ–

ͳͷǦ͓ͶǦ•–‹””—’•̷ͻᇳ ‘Ǥ…Ǥሺ‡ƒǤ‡†ሻ

ʹԣ

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݀ ൌ ʹͳǤͷԣ κ௡ Τʹ ൌ ͳͲǤ͹ͷԢ

۳‫ܖ܉ܘ܁܌ܖ‬

ܸ௨ ̷ˆƒ…‡ ൌ ͸ͳǤͶ‹’•

ܸ௨ ̷݀ ൌ ͷͻǤͻ‹’•

ܸ௨ ൌ ߶ߣ—݂௖ᇱ ܾ௪ ݀ ൌ ͳʹǤʹ‹’•

‫ݔ‬௡௦ ൌ ͳͲǤ͵Ԣ

Figure 6.43 Partial shear diagram for the beam in Example 6.7.

7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH IURPWKHIDFHRIWKHVXSSRUWVHH)LJXUH 

6-83

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the required shear reinforcement

Assuming at least minimum shear reinforcement is provided, with the axial force

can be determined from the following equation Table 6.4

where

for normalweight concrete.

Table 6.2

For Maximum spacing 8VWLUUXSVVSDFHGDW

along the length of the beam. provides

Figure 6.19 Table 6.6

Table 6.5

Across the width of the beam, maximum spacing

Figure 6.19

)RU8VWLUUXSVZLWKLQFRYHU

8VH8VWLUUXSVVSDFHGDWLQRQFHQWHU Stirrups are no longer required at the section where The length from the face of the supFDQEHREWDLQHGIURPWKHIROORZLQJHTXDWLRQVHH)LJXUH  port where stirrups are no longer required,

)RUVLPSOHUGHWDLOLQJXVH8VWLUUXSVVSDFHGDWLQRQFHQWHUDWERWKHQGVRIWKHEHDPZLWKWKH¿UVWVWLUUXS ORFDWHGLQIURPWKHIDFHRIWKHFROXPQ Comments. This shear reinforcement arrangement can be used for all beams in all spans along the column lines in the north-south direction for the frames that are part of the LFRS. ,QVWHDGRIXVLQJ7DEOHWRGHWHUPLQHWKHVL]HDQGVSDFLQJRIWKHVKHDUUHLQIRUFHPHQWXVH&DVHLQ7DEOH$VVXPLQJ8VWLUUXSVWKHUHTXLUHGVSDFLQJLVHTXDOWRWKHIROORZLQJ

$LQVWLUUXSVSDFLQJLVXVHGDWHDFKHQGLQOLHXRIVSHFLI\LQJDXQLIRUPVWLUUXSVSDFLQJDFURVVWKHHQWLUHOHQJWKRI WKHEHDPEHFDXVHWKHODWWHUUHVXOWVLQDVSDFLQJWKDWLVQRWFRQYHQLHQWWRPHDVXUHLQWKH¿HOG

6-84

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.9.8

Example 6.8 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C), Beam is Part of the LFRS, SDC A

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ )UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP DUHOLPLWHGWRLQE\LQVHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW. Also asDQG*UDGHUHLQIRUFHPHQW sume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the flexural reinforcement details

It is determined in Example 6.6 that the following reinforcement is required at the critical sections: • ([WHULRUQHJDWLYHEDUVZLWKLQLQ • 3RVLWLYHEDUVOD\HUV • )LUVWLQWHULRUQHJDWLYHEDUVZLWKLQLQ 7KHSURYLGHGQXPEHURIUHLQIRUFLQJEDUVDWDOOFULWLFDOVHFWLRQVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\VHH6WHSLQ([DPSOH  The top reinforcing bars are made continuous over the entire length of the end span because stirrups are required over the entire length and the beam is subjected to wind loads. $&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGQHDUPLGVSDQ$FFRUGLQJWR$&,ZKHUHEDUVRIGLIIHUHQWVL]HDUH of lap spliced in tension, the required lap splice length is based on the greater of the tension development length of the smaller bar. the larger bar and the tension lap splice length 7KHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVLVGHWHUPLQHGDVIROORZV ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUPRUHWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHQHJDWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

6-85

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

6LPLODUO\IRUWKHEDUV

ACI Table 25.5.2.1

7KHUHIRUHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVJRYHUQ 8VHDIWLQODSVSOLFHOHQJWKIRUWKHWRSUHLQIRUFLQJEDUVLQWKHHQGVSDQ )RUVLPSOHUGHWDLOLQJWKHERWWRPEDUVDUHPDGHFRQWLQXRXVLQVWHDGRIFXWWLQJRIIDSRUWLRQRIWKHPWKHODWHURI ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH supports. The required lap splice length is determined as follows: ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK

6-86

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the shear reinforcement details

,WLVGHWHUPLQHGLQ([DPSOHWKDW8VWLUUXSVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDUVWUHQJWK with requirements. The U-stirrups extends over the effective width of the beam GHJUHHKRRNVDWHDFKHQGDQGDUHWLHGWRWKHWRSDQGERWWRPORQJLWXGLQDOUHLQIRUFHPHQW Minimum

IRUVWLUUXSGHYHORSPHQW

which is equal to the provided

Table 6.17

5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWV LQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOEHDPVDORQJWKHLQWHULRUFROXPQOLQHVLQ the north-south direction that are part of the LFRS.         

2ᇱ -0ᇱᇱ 

2ᇱ -0ᇱᇱ 

4-#5 

3ᇱ -2ᇱᇱ 

A

4-#7

4-#8 

A

30-#4 U-stirrups @ 9ᇱᇱ 

2ᇱᇱ 

21ᇱ -6ᇱᇱ  Other beams and reinforcement not shown for clarity

7ᇱᇱ 

4-#8 

1Ԣ-5ᇱᇱ 

2ᇱ -2ᇱᇱ 

#4 U-stirrups @ 9ԣ  4-#7  1ᇱ -0ᇱᇱ 

‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬

Figure 6.44 Reinforcement details for the beam in Examples 6.6 through 6.8.

6.9.9

Example 6.9 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is Part of the LFRS, SDC A, Includes Compression Reinforcement

'HWHUPLQHWKHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUWKHEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDPDUH OLPLWHGWRLQE\LQVHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW, the beam is QRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVDQGSHUFHQWRIWKH DQG*UDGHUHLQIRUFHPHQW live load is sustained. Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK

6-87

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the service loads and bending moments

7KHVHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVDUHJLYHQLQ7DEOHIRUWKHEHDPLQWKHHQGVSDQVHH6WHSRI ([DPSOH DQGDUHUHSHDWHGLQ7DEOHIRUFRQYHQLHQFH Table 6.29 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Beam in the End Span in Example 6.9 Exterior Negative

Positive

First Interior Negative

7KHVXVWDLQHGSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVDUHWKHIROORZLQJZKHUHSHUFHQWRIWKHOLYHORDGLVVXVtained:

At the exterior support: $WWKH¿UVWLQWHULRUVXSSRUW Step 2 – Determine the material properties of the concrete and reinforcing steel Eq. (6.84)

where

for normalweight concrete.

Table 6.2 ACI Eq. (19.2.2.1b) ACI 20.2.2.2

Modular ratio Step 3 – Determine the gross and cracked moments of inertia

(IIHFWLYHÀDQJHZLGWKRIWKHEHDP

Step 2, Example 6.6

‫ݕ‬௧ ൌ ͳ͹Ǥʹᇱᇱ 

ܾ௙ ൌ ͹͸Ǥͷᇱᇱ 

݄௕ ൌ ͳ͹ᇱᇱ 

݄ ൌ ͹ᇱᇱ 

• Positive moment section 7KHJURVVVHFWLRQRIWKHEHDPDWWKHSRVLWLYHPRPHQWVHFWLRQLVVKRZQLQ)LJXUH

ܾ௪ ൌ ͳʹᇱᇱ 

Figure 6.45 Gross section of the beam at a positive moment section.

6-88

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The gross section properties of the beam are the following:

Figure 6.35

)URP([DPSOHERWWRPEDUVDUHUHTXLUHG

.

$VVXPLQJDUHFWDQJXODUFRPSUHVVLRQDUHDWKHFUDFNHGVHFWLRQSURSHUWLHVDUHWKHIROORZLQJVHH)LJXUH 

݊‫ܣ‬௦ 

݀ ൌ ʹͲǤ͸ᇱᇱ 

ܾ௙ ൌ ͹͸Ǥͷᇱᇱ 

݇݀ ൌ ͵ǤͲᇱᇱ 

Figure 6.34

Figure 6.46 Cracked transformed section of the beam at a positive moment section.

Therefore, the assumption of a rectangular section is correct. Figure 6.34

• Negative moment section 7KHQHJDWLYHPRPHQWVHFWLRQRIWKHEHDPLVWKHE\LQEHDPZHE)URP([DPSOHWRS DQGEDUVDUHUHTXLUHGDWWKH¿UVWLQWHULRUVXSSRUW bars are required at the exterior support

6-89

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 6.34

At the exterior support:

$WWKH¿UVWH[WHULRUVXSSRUW

Step 4 – Determine the effective moment of inertia

The effective moment of inertia,

, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.

• Positive moment section

Therefore, Eq. (6.88)

Therefore,

6-90

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

• Negative moment section

At the exterior support:

Therefore,

Therefore,

Therefore,

$WWKH¿UVWLQWHULRUVXSSRUW

Therefore,

Therefore,

6-91

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

• Average effective moment of inertia Dead load: ACI 24.2.3.6

Sustained load:

Dead plus live load:

Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads

Assume the loads due to the dead load of the slab, the superimposed dead load, and the live load are uniformly distributed over the span instead of the actual triangular load distribution. The following equation can be used to GHWHUPLQHLPPHGLDWHGHÀHFWLRQV Eq. (6.89)

For continuous members: Table 6.19

where

and

• Dead load

• Sustained load

• Dead plus live load

• Live load

6-92

is the service load moment at midspan. Also assume

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV WKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ Table 6.21 Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection

$VVXPLQJDPRQWKGXUDWLRQRIORDGLQJZLWK

: Eq. (6.90), Table 6.20

7KHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJHLVHTXDOWRWKHIROORZLQJ Eq. (6.91) Table 6.21

7KHLQE\LQEHDPLVQRWDGHTXDWHIRUWLPHGHSHQGHQWGHÀHFWLRQV Step 7 – Recalculate the deflections using compression reinforcement

5HFDOFXODWHWKHGHÀHFWLRQVLQFOXGLQJWKHFRQWULEXWLRQRIFRPSUHVVLRQUHLQIRUFHPHQWLQWKHVHFWLRQ)RUSXUSRVHVRI DQDO\VLVDVVXPHWKHWRSEDUVH[WHQGDFURVVWKHHQWLUHOHQJWKRIWKHEHDP • Step 7a – Determine the cracked moments of inertia Positive moment section Assuming a rectangular compression area, the cracked section properties are the following: Figure 6.34

Therefore, the assumption of a rectangular section is correct.

Figure 6.34

6-93

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Negative moment section $WWKHH[WHULRUDQG¿UVWLQWHULRUVXSSRUWV Figure 6.34

Figure 6.34

• Step 7b – Determine the effective moment of inertia Positive moment section Eq. (6.88)

Negative moment sections At the exterior support:

6-94

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$WWKH¿UVWLQWHULRUVXSSRUW

Average effective moment of inertia Dead load: ACI 24.2.3.6 Sustained load:

Dead plus live load:

• 6WHSF±'HWHUPLQHWKHLPPHGLDWHGHÀHFWLRQVDQGFKHFNWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQ for live loads Dead load:

Sustained load:

6-95

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Dead plus live load:

Live load:

)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVWKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ Table 6.21

• 6WHSG±'HWHUPLQHWKHWLPHGHSHQGHQWGHÀHFWLRQVDQGFKHFNWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQ $VVXPLQJDPRQWKGXUDWLRQRIORDGLQJ ACI 24.2.4.1.2 Eq. (6.90), Table 6.20

7KHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJHLVHTXDOWRWKHIROORZLQJ Eq. (6.91) Table 6.21

7KHLQE\LQEHDPLVQRWDGHTXDWHIRUWLPHGHSHQGHQWGHÀHFWLRQV Comments.7KHGHÀHFWLRQFDOFXODWLRQVDUHGHWHUPLQHGDVVXPLQJDXQLIRUPO\GLVWULEXWHGORDGHTXDOWRWKHPD[Lmum value of the actual triangular load, which is conservative. As such, it is safe to conclude this beam is adequate IRUWLPHGHSHQGHQWGHÀHFWLRQVZKHQWKHFRQWULEXWLRQRIWKHFRPSUHVVLRQUHLQIRUFHPHQWLVLQFOXGHG 6.9.10 Example 6.10 – Determination of Joist Size: Joist in Building #2, Joist is Not Part of the LFRS, SDC C

'HWHUPLQHWKHVL]HRIDW\SLFDOMRLVWLQ%XLOGLQJDWDW\SLFDOÀRRUOHYHOZKHUHWKHMRLVWLVQRWSDUWRIWKH/)56 DQG*UDGH VHH)LJXUH $VVXPHDLQWKLFNVODE$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the joist depth based on serviceability requirements

ACI 9.3.1.1

Assume the joist (beam) is not supporting or attached to partitions or other construction likely to be damaged by ODUJHGHÀHFWLRQV7KXVGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,QHHGQRWEHFDOFXODWHG The minimum joist depth is based on the end spans (longest spans) with a support condition of one end continuous: Table 6.1

8VHDLQGHHSSDQIRUPZKLFKUHVXOWVLQDQRYHUDOOGHSWKHTXDOWR

6-96

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the joist width

:KHUHLQZLGHSDQIRUPVDUHXVHGDLQZLGHMRLVWULELVW\SLFDOO\VSHFL¿HGUHVXOWLQJLQDIWFHQWHUWR FHQWHUVSDFLQJRIWKHULEVVHH)LJXUH 7KHLQZLGWKRFFXUVDWWKHERWWRPRIWKHULEDQGWKHZLGWKW\SLFDOO\ LQFUHDVHVDWDWRVORSHWRWKHXQGHUVLGHRIWKHVODE 7KHDGHTXDF\RIWKHLQZLGHMRLVWULELVFKHFNHGIRUPRPHQWVWUHQJWKLQ([DPSOH 8VHDLQE\LQMRLVW 6.9.11 Example 6.11 – Determination of Flexural Reinforcement: Joist in Building #2, Joist Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUDW\SLFDOMRLVWLQ%XLOGLQJDWDW\SLFDOÀRRUOHYHODVVXPLQJWKH MRLVWLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHLQE\LQHGJHFROXPQVLQE\LQFRUQHUFROXPQVLQZLGHLQWHULRUEHDPVLQZLGHHGJHEHDPVQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the factored bending moments along the span

&KHFNLIWKHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVV\VWHP • Members are prismatic • Loads are uniformly distributed • • 7KHUHDUHVSDQVZKLFKLVJUHDWHUWKDQVSDQV • 7KHUHIRUHWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVDQGVKHDU forces. Table 3.3

7KHIDFWRUHGEHQGLQJPRPHQWVDUHJLYHQLQ7DEOH

Figure 4.3

6-97

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.30 Factored Bending Moments for a Typical Joist in Example 6.11 (ft-kips) End Span Exterior Negative

Positive

Interior Span First Interior Negative

Positive

Step 2 – Determine the effective flange width of the joist

Interior Negative

ACI 6.3.2.1

7KHHIIHFWLYHÀDQJHZLGWK , is equal to the following:

Figure 6.10

Step 3 – Determine the required flexural reinforcement at the critical sections

$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQWHQVLRQ UHTXLUHG single layer of tension reinforcement:

is determined by the following assuming a

Eq. (6.34)

Eq. (6.33)

where

Figure 6.11

$WSRVLWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQFRPSUHVVLRQ GHWHUPLQHWKHGHSWKRIWKHHTXLYDOHQWVWUHVVEORFN , LQ7DEOHZKLFKUHVXOWVLQWKHODUJHVW ): assuming rectangular section behavior using the largest positive

Eq. (6.49)

where Because with

WKHVHFWLRQEHKDYHVDVDUHFWDQJXODUVHFWLRQDQG(TV DQG FDQEHXVHGWRGHWHUPLQH VHH)LJXUH 

A summary of the required reinforcement is given in Table 6.31. All sections are tension-controlled because

6-98

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.31 Required Flexural Reinforcement for the Joist in Example 6.11 Location

Mu (ft-kips)

Exterior negative Positive

End span



First interior negative Interior span

Positive



Negative

*Min.

Rn (psi)

As (in2)*











1.89









[Eq. (6.35)]

Max.

[Eq. (6.38)]

Step 4 – Select the flexural reinforcement

6HOHFWWKHÀH[XUDOUHLQIRUFHPHQWLQDVLQJOHOD\HUEDVHGRQWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ $&,DQGUHVSHFWLYHO\ • Negative reinforcement

Because

, distribute the required

within

Figure 6.23

Exterior negative: :LWKLQFRYHU*UDGHUHLQIRUFLQJEDUVDQG spacing is equal to the following:

the maximum center-to-center bar

Eq. (6.65)

5HTXLUHGUHLQIRUFHPHQWZLWKLQWKHLQZLGWK 8VH#LQRQFHQWHU 7KLVUHLQIRUFHPHQWDOVRVDWLV¿HVWKHUHTXLUHPHQWLQ$&,IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKH RXWHUSRUWLRQVRIWKHÀDQJHZKHUH First interior negative: 5HTXLUHGUHLQIRUFHPHQWZLWKLQWKHLQZLGWK 8VH#LQRQFHQWHU 7KLVUHLQIRUFHPHQWDOVRVDWLV¿HVWKHUHTXLUHPHQWLQ$&,IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKH RXWHUSRUWLRQVRIWKHÀDQJHZKHUH • Positive reinforcement 8VHEDUV 6-99

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPLQJDLQPD[LPXPDJJUHJDWHVL]H

Figure 6.22

Check minimum clear space: Table 6.7

Comments. The reactions at the ends of the joists in the end spans cause torsional moments on the edge beams. No DGMXVWPHQWRIPRPHQWVLQDFFRUGDQFHZLWK$&,KDVEHHQFRQVLGHUHGLQWKLVH[DPSOH6HH([DPSOHIRU the design of the joists considering redistribution of torsional moments. 6.9.12 Example 6.12 – Determination of Shear Reinforcement: Joist in Building #2, Joist is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUDW\SLFDOMRLVWLQ%XLOGLQJDWDW\SLFDOÀRRUOHYHODVVXPLQJWKH DQG*UDGH MRLVWLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the factored shear forces along the span

7KHVLPSOL¿HGPHWKRGRIDQDO\VLVFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFHVIRUWKLVV\VWHPVHH6WHSLQ([DPSOH  

7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUW

7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH IURPWKHIDFHRIWKHWUDQVYHUVHEHDPVHH)LJXUH 

Step 2 – Determine the required shear reinforcement

Assuming at least minimum shear reinforcement is provided, with the axial force

can be determined from the following equation Table 6.4

where

6-100

for normalweight concrete.

Table 6.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For Maximum spacing

along the length of the joist.

Because of the narrow width of the joist, provide single-leg stirrups. $VLQJOHOHJVWLUUXSVSDFHGDW

provides

Figure 6.19 Figure 6.16 Table 6.6

Table 6.5

8VHVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHU Stirrups are no longer required at the section where The length from the face of the sup, can be obtained from the following equation: port where stirrups are no longer required, Eq. (6.58)

8VHVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHUDWERWKHQGVRIWKHMRLVWZLWKWKH¿UVWVWLUUXSORFDWHGLQ from the face of the transverse beam. , which is the minimum width of the sloping Comments. The shear calculations were performed using ZHE$QDYHUDJHRIWKHWRSDQGERWWRPULEZLGWKVFDQEHXVHGLQWKHFDOFXODWLRQVEXWEHFDXVHWKHVORSHLVWR little is gained by doing so. The reactions at the ends of the joists in the end spans cause torsional moments on the edge beams. No adjustment RIVKHDUIRUFHVLQDFFRUGDQFHZLWK$&,KDYHEHHQFRQVLGHUHGLQWKLVH[DPSOH6HH([DPSOHIRUWKH design of the joists considering redistribution of torsional moments. 6.9.13 Example 6.13 – Determination of Reinforcement Details: Joist in Building #2, Joist is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUDW\SLFDOMRLVWLQ%XLOGLQJDWDW\SLFDOÀRRUOHYHODVVXPLQJWKH DQG*UDGH MRLVWLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the flexural reinforcement details

It is determined in Example 6.11 that the following reinforcement is required at the critical sections: • ([WHULRUQHJDWLYH#LQRQFHQWHU • 3RVLWLYHEDUV • )LUVWLQWHULRUQHJDWLYH#LQRQFHQWHU 7KHSURYLGHGQXPEHURIUHLQIRUFLQJEDUVDWDOOFULWLFDOVHFWLRQVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\VHH6WHSLQ([DPSOH 

6-101

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVFDQEHGHWHUPLQHGE\)LJXUHIRUEHDPVRWKHUWKDQSHULPHWHUEHDPVEHFDXVH the joist is subjected to uniformly distributed gravity loads. )RUVLPSOHUGHWDLOLQJWKHERWWRPEDUVDUHPDGHFRQWLQXRXVLQVWHDGRIFXWWLQJRIIDSRUWLRQRIWKHPWKHODWWHURI ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH supports. The required lap splice length is determined as follows: ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK )OH[XUDOUHLQIRUFHPHQWGHWDLOVDUHJLYHQLQ)LJXUH Step 2 – Determine the shear reinforcement details

,WLVGHWHUPLQHGLQ([DPSOHWKDWVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDU VWUHQJWKUHTXLUHPHQWVZKLFKLVEHQWLQWRWKHSUR¿OHVKRZQLQ)LJXUH,WLVDWWDFKHGWRRQHRIWKHERWWRPEDUVRI the joist and has a standard hook at the top. 6KHDUUHLQIRUFHPHQWGHWDLOVIRUWKHMRLVWLQWKHHQGVSDQDUHJLYHQLQ)LJXUH 7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQ be used for all joists.

6-102

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



10Ԣ-0ԣ

13Ԣ-4ԣ A

#5 @ 8ԣ

2Ԣ-4½ԣ

#4 @ 10ԣ

Standard hook ሺtyp.ሻ 2ԣ

2-#8

5Ԣ-2ԣ

A

19-#4 stirrups @ 12ԣ ሺtyp.ሻ

2Ԣ-6ԣ

2Ԣ-8ԣ

39Ԣ-10ԣ

4½ԣ

Other reinforcement not shown for clarity

#5 @ 8ԣ

2Ԣ-0ԣ

#4 single-leg stirrup

2-#8 7ԣ Section A-A Figure 6.47 Flexural reinforcement details for the joist in Example 6.13.

 ʹͺǤͷԣ

–ƒ†ƒ”†Š‘‘ሺ–›’Ǥሻ

ʹԣሺ–›’Ǥሻ ʹͲǦ͓Ͷ•‹‰Ž‡ǦŽ‡‰•–‹””—’•̷ͳʹԣ‘Ǥ…Ǥ ሺ–›’Ǥሻ ʹԢǦ͸ԣ

͵ͻԢǦͳͲԣ

ʹƍǦͺԣ

–Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 6.48 Shear reinforcement details for the joist in Example 6.13.

6.9.14 Example 6.14 – Determination of Deflections: Joist in Building #2, Typical Floor, Joist is Not Part of the LFRS, SDC C

'HWHUPLQHWKHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUDW\SLFDOMRLVWLQDQHQGVSDQLQ%XLOGLQJDWDW\SLFDO ÀRRUOHYHODVVXPLQJWKHMRLVWLVQRWSDUWRIWKH/)56VHH)LJXUH $VVXPHWKHMRLVWLVQRWVXSSRUWLQJRUDWWDFKHG WRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVDQGSHUFHQWRIWKHOLYHORDGLVVXVWDLQHG$OVR DQG*UDGHUHLQIRUFHPHQW assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK

6-103

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the service loads and bending moments

7KHVLPSOL¿HGPHWKRGRIDQDO\VLVFDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVIRUWKLVV\VWHPVHH6WHSLQ ([DPSOH  $VXPPDU\RIWKHVHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVLVJLYHQLQ7DEOH Table 6.32 Service Dead and Live Load Bending Moments (ft-kips) for the Joist in an End Span in Example 6.14 Exterior Negative

Positive

First Interior Negative

The sustained positive and negative bending moments are the following:

At the exterior support: $WWKH¿UVWLQWHULRUVXSSRUW Step 2 – Determine the material properties of the concrete and reinforcing steel Eq. (6.84)

where

for normalweight concrete.

Table 6.2 ACI Eq. (19.2.2.1b) ACI 20.2.2.2

Modular ratio Step 3 – Determine the gross and cracked moments of inertia

(IIHFWLYHÀDQJHZLGWKRIWKHMRLVW • Positive moment section 7KHJURVVVHFWLRQRIWKHMRLVWDWWKHSRVLWLYHPRPHQWVHFWLRQLVVKRZQLQ)LJXUH

6-104

Step 2, Example 6.11

ܾ௙ ൌ ͸Ͳԣ

‫ݕ‬௧ ൌ ʹͲǤͺԣ

݄௕ ൌ ʹͶԣ

݄ ൌ ͶΦԣ

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ௪ ൌ ͹ԣ Figure 6.49 Gross section of the joist at a positive moment section.

The gross section properties of the joist are the following (not including the sloped portions of the web):

Figure 6.35

)URP([DPSOHERWWRPEDUVDUHUHTXLUHG $VVXPLQJDUHFWDQJXODUFRPSUHVVLRQDUHDWKHFUDFNHGVHFWLRQSURSHUWLHVDUHWKHIROORZLQJVHH)LJXUH  Figure 6.34

6-105

݀ ൌ ʹ͸ǤͲԣ

ܾ௙ ൌ ͸Ͳԣ

݇݀ ൌ ͵Ǥͳᇱᇱ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݊‫ܣ‬௦ 

Figure 6.50 Cracked transformed section of the joist at a positive moment section.

Therefore, the assumption of a rectangular section is correct. Figure 6.34

• Negative moment section 7KHQHJDWLYHPRPHQWVHFWLRQRIWKHEHDPLVWKHLQE\LQMRLVWZHE)URP([DPSOHWRSEDUVDW DQGWRSEDUVDWLQ LQRQFHQWHUDUHUHTXLUHGDWWKHH[WHULRUVXSSRUW RQFHQWHUDUHUHTXLUHGDWWKH¿UVWLQWHULRUVXSSRUW Figure 6.34

At the exterior support:

$WWKH¿UVWLQWHULRUVXSSRUW

6-106

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the effective moment of inertia

The effective moment of inertia,

, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.

• Positive moment section

Therefore, Eq. (6.88)

Therefore,

Therefore, Eq. (6.88)

• Negative moment section

At the exterior support:

Therefore,

6-107

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

Therefore,

AtWKH¿UVWLQWHULRUVXSSRUW

Therefore,

Therefore,

Therefore,

• Average effective moment of inertia Dead load: ACI 24.2.3.6

Sustained load:

Dead plus live load:

6-108

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads

7KHIROORZLQJHTXDWLRQFDQEHXVHGWRGHWHUPLQHLPPHGLDWHGHÀHFWLRQV Eq. (6.89)

For continuous members: Table 6.19

where

and

Therefore,

• Dead load

• Sustained load

• Dead plus live load

• Live load

)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV WKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ Table 6.21 Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection

$VVXPLQJDPRQWKGXUDWLRQRIORDGLQJZLWK

: Eq. (6.90), Table 6.20

7KHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJHLVHTXDOWRWKHIROORZLQJ Eq. (6.91) Table 6.21

7KHMRLVWLVQRWDGHTXDWHIRUWLPHGHSHQGHQWGHÀHFWLRQV Comments.,WLVDVVXPHGLQWKLVH[DPSOHWKDWWKHFRQWULEXWLRQRIWKHFRPSUHVVLRQUHLQIRUFHPHQWLV]HURWKDWLV ,WFDQEHVKRZQWKHMRLVWLVDGHTXDWHIRUWLPHGHSHQGHQWGHÀHFWLRQVZKHQWKHFRPSUHVVLRQUHLQIRUFHPHQWLV included in the analysis.

6-109

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.9.15 Example 6.15 – Determination of Beam Size: Edge Beam in Building #2, Beam is Not Part of the LFRS, SDC C

'HWHUPLQHWKHVL]HRIDQHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHOZKHUHWKHEHDPLVQRWSDUWRIWKH/)56 MRLVWVLQE\LQFRUQHUFROXPQVDQGLQE\LQHGJHFROVHH)LJXUH $VVXPH DQG*UDGHUHLQIRUFHPHQW umns. Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the beam depth based on serviceability requirements

ACI 9.3.1.1

Assume the beam is not supporting or attached to partitions or other construction likely to be damaged by large GHÀHFWLRQV7KXVGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,QHHGQRWEHFDOFXODWHG The minimum beam depth is based on the end spans with a support condition of one end continuous: Table 6.1

%HFDXVHWKHEHDPVDUHSDUWRIDZLGHPRGXOHMRLVWV\VWHPZKHUHWKHRYHUDOOGHSWKRIWKHMRLVWVDUHLQWKH GHSWKRIWKHEHDPVLVWDNHQDVLQIRUIRUPZRUNHFRQRP\ Step 2 – Determine the beam width

Assuming an initial estimate of determined by the following equation:

with

DQG*UDGHUHLQIRUFHPHQWWKHUHTXLUHG

is

Eq. (6.1)

The largest

along the spans is used to determine

&RQVHUYDWLYHO\HVWLPDWHWKHIDFWRUHGGHDGORDGRIWKHEHDPZHEDVNLSIW

Table 3.3

7KHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVEHDPVHH6WHSLQ([DPSOH 6.16). Exterior spans: Interior spans: Maximum

RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQLQDQHQGVSDQ Figure 4.3

6-110

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Approximate Therefore,

)RUIRUPZRUNHFRQRP\XVHDLQZLGHEHDPZKLFKLVLQZLGHUWKDQWKHLQFRUQHUFROXPQLWIUDPHV LQWRVHH6HFWRIWKLVSXEOLFDWLRQRQJXLGHOLQHVIRUVL]LQJEHDPVIRUHFRQRP\  8VHDLQE\LQEHDP 6.9.16 Example 6.16 – Determination of Flexural Reinforcement: Edge Beam in Building #2, Beam is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUDQHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHOZKHUHWKH DQG*UDGH EHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the factored bending moments along the span

&KHFNLIWKHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVV\VWHP • • • • •

Members are prismatic Loads are uniformly distributed 7KHUHDUHVSDQVZKLFKLVJUHDWHUWKDQVSDQV All spans have the same length

7KHUHIRUHWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVDQGVKHDU forces. Exterior spans:

6-111

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Interior spans:

The factored bending moments are given in Table 6.33.

Figure 4.3

Table 6.33 Factored Bending Moments for an Edge Beam in Example 6.16 (ft-kips) End Span Exterior Negative

Positive

Interior Span First Interior Negative

Positive

Step 2 – Determine the effective flange width of the beam

Interior Negative

ACI 6.3.2.1

)RUDQHGJHEHDPWKHHIIHFWLYHÀDQJHZLGWK , is equal to the following:

Figure 6.10

Step 3 – Determine the required flexural reinforcement at the critical sections

$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQWHQVLRQ UHTXLUHG single layer of tension reinforcement:

is determined by the following assuming a

Eq. (6.34)

Eq. (6.33)

where

Figure 6.11

$WSRVLWLYHPRPHQWFULWLFDOVHFWLRQVÀDQJHLQFRPSUHVVLRQ GHWHUPLQHWKHGHSWKRIWKHHTXLYDOHQWVWUHVVEORFN , in Table 6.33 (which results in the largest ): assuming rectangular section behavior using the largest positive

Eq. (6.33)

where Because with

6-112

WKHVHFWLRQEHKDYHVDVDUHFWDQJXODUVHFWLRQDQG(TV DQG FDQEHXVHGWRGHWHUPLQH VHH)LJXUH 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VXPPDU\RIWKHUHTXLUHGUHLQIRUFHPHQWLVJLYHQLQ7DEOH$OOVHFWLRQVDUHWHQVLRQFRQWUROOHGEHFDXVH Table 6.34 Required Flexural Reinforcement for the Edge Beam in Example 6.16 Location

Mu (ft-kips)

Exterior negative End span

Positive



First interior negative Positive Interior span *Min.

Max.

363.3

Negative

Rn (psi)

As (in2)*







3.68













[Eq. (6.35)]

[Eq. (6.38)]

,WLVGHWHUPLQHGLQ([DPSOHWKDWWRUVLRQDOHIIHFWVPXVWEHFRQVLGHUHGIRUWKLVHGJHEHDP7KXVWKHÀH[XUDO UHLQIRUFHPHQWLQ7DEOHPXVWEHFRPELQHGZLWKWKHORQJLWXGLQDOWRUVLRQDOUHLQIRUFHPHQWZKLFKLVGHWHUPLQHGLQ ([DPSOH7KHVL]HDQGVSDFLQJRIWKHWRWDOUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWDUHJLYHQLQ([DPSOH 6.9.17 Example 6.17 – Determination of Shear Reinforcement: Edge Beam in Building #2, Beam is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUDQHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHOZKHUHWKH DQG*UDGH EHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the factored shear forces along the span

7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWLQDQHQGVSDQIRUWKHORDGFRPELQDWLRQ VHH)LJXUH 

7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH IURPWKHIDFHRIWKHVXSSRUWVHH)LJXUH 

6-113

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the required transverse reinforcement for shear

Assuming at least minimum shear reinforcement is provided, force

is determined from the following with the axial Table 6.4

where

for normalweight concrete.

Table 6.2

For Maximum spacing

along the length of the beam.

Figure 6.19

It is determined in Example 6.18 that torsional effects must be considered for this edge beam. Thus, the shear reinforcement calculated in this example must be combined with the transverse torsional reinforcement, which is deWHUPLQHGLQ([DPSOH7KHVL]HDQGVSDFLQJRIWKHWRWDOUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWDUHJLYHQLQ([DPSOH 6.19. 6.9.18 Example 6.18 – Determination of Torsion Reinforcement: Edge Beam in Building #2, Second-Floor Level, Beam is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGWRUVLRQUHLQIRUFHPHQWIRUDQHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHOZKHUHWKH DQG*UDGH EHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the factored torsional moments along the span

The wide-module joist system framing into the edge beam produces a uniform torsional load along the length of the beam. The torsional load is caused by the shear forces and bending moments transferred from the joists to the beam VHH)LJXUH 7KHGHDGORDGRIWKHMRLVWVDQGWKHVXSHULPSRVHGGHDGORDGDQGOLYHORDGRQWKHMRLVWVPXVWEH considered:

The factored shear and bending moments at the end of the joists per foot width of joists are determined using Figure 

The total uniform torsional load on the edge beam is equal to the sum of the torsional loads due to

6-114

and

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Figure 6.51 Reactions on the edge beam in Example 6.18.

$VVXPLQJWKHHQGVRIWKHHGJHEHDPDUH¿[HGDJDLQVWURWDWLRQE\WKHFROXPQVWKHIDFWRUHGWRUVLRQDOPRPHQWDWWKH face of each end of the beam due to is equal the following:

where the longest clear span is used to calculate ACI 9.4.4.3 Step 2 – Determine if torsional effects must be considered

Torsion can be neglected where the factored torsional moment from analysis is less than the threshold torsion: Eq. (6.3)

'HWHUPLQHWKHRYHUKDQJLQJÀDQJHZLGWK , permitted to be used in the calculation of

and

: Eq. (6.4)

6-115

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7RUVLRQDOVHFWLRQSURSHUWLHVLQFOXGLQJWKHRYHUKDQJLQJÀDQJH Figure 6.3

7RUVLRQDOVHFWLRQSURSHUWLHVZLWKRXWWKHRYHUKDQJLQJÀDQJH

Because ÀDQJHXVH

IRUWKHEHDPZLWKWKHRYHUKDQJLQJÀDQJHLVOHVVWKDQWKDWIRUWKHEHDPZLWKRXWWKHRYHUKDQJLQJ >$&,E @

Therefore, torsional effects must be considered. Because the edge beam is part of indeterminate system in which redistribution of internal forces can occur following torsional cracking, the maximum factored torsional moment at the critical section need not exceed the cracking torsional moment: ACI 22.7.3.2

This redistribution is equivalent to reducing the factored distributed torsional loading on the edge beam to the following:

Step 3 – Adjust the bending moments and shear forces in the joists in the end spans

7KHEHQGLQJPRPHQWVLQWKHMRLVWVLQWKHHQGVSDQVPXVWEHDGMXVWHGLQDFFRUGDQFHZLWK$&, 7KHEHQGLQJPRPHQWVDWWKHFULWLFDOVHFWLRQVLQDQHQGVSDQDUHJLYHQLQ7DEOHLQ([DPSOHDQGDUHUHSURGXFHGLQ)LJXUHSHUIRRWZLGWKRIMRLVWWKDWLVWKHWDEXODWHGYDOXHVRIWKHEHQGLQJPRPHQWVDUHGLYLGHGE\WKH IWVSDFLQJEHWZHHQMRLVWV  3ULRUWRDGMXVWPHQWWKHWRUVLRQDOPRPHQWDWWKHFHQWURLGRIWKHHGJHEHDPLVHTXDOWRIWNLSVIWZKLFKLV WKHEHQGLQJPRPHQWLQWKHMRLVWVDWWKLVORFDWLRQVHH6WHSDQG)LJXUH 'XHWRUHGLVWULEXWLRQRIWKHIDFWRUHGWRUVLRQDOPRPHQWVWKHWRUVLRQDOPRPHQWLVUHGXFHGWRIWNLSVIWDWWKHFHQWURLGRIWKHHGJHEHDPVHH 6WHS 7KHUHIRUHWRDFFRXQWIRUUHGLVWULEXWLRQRIWKHWRUVLRQDOPRPHQWVDWRUVLRQDOPRPHQWHTXDOWR PXVWEHDSSOLHGDWWKHFHQWURLGRIWKHHGJHEHDPLQWKHRSSRVLWHGLUHFWLRQWRWKHIWNLSIWEHQGLQJPRPHQWVHH)LJXUH 2QHKDOIRIWKHIWNLSVIWWRUVLRQDOPRPHQWLVFDUULHGRYHUWRWKHFHQWURLGRIWKH ¿UVWLQWHULRUEHDP The design bending moments in the joists per foot width of joist after adjustment are obtained at the critical sections by algebraically combining the bending moments prior to adjustment with those due to redistribution (see Figure   6-116

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Figure 6.52 Bending moments in a joist in an end span (ft-kips/ft).

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWEDVHGRQWKHGHVLJQEHQGLQJPRPHQWVLQ)LJXUHDUHJLYHQLQ7DEOH WKHQXPEHUVLQSDUHQWKHVHVXQGHUWKH FROXPQDUHWKHUHTXLUHGDUHDVRIÀH[XUDOUHLQIRUFHPHQWSULRUWR DGMXVWPHQWRIWKHWRUVLRQDOPRPHQWVVHH([DPSOH $VH[SHFWHGWKHÀH[XUDOUHLQIRUFHPHQWGHFUHDVHVDWWKH H[WHULRUQHJDWLYHFULWLFDOVHFWLRQDQGLQFUHDVHVDWWKHRWKHUWZRFULWLFDOVHFWLRQV7KHQXPEHUDQGVL]HRIWKHUHLQIRUFLQJEDUVDUHDOVRJLYHQLQ7DEOHDORQJZLWKWKHUHLQIRUFLQJEDUVVHOHFWHGLQ([DPSOHZKLFKDUHJLYHQLQ SDUHQWKHVHV$WWKHH[WHULRUQHJDWLYHFULWLFDOVHFWLRQEDUVDUHXVHGWRVDWLVI\VSDFLQJUHTXLUHPHQWV

6-117

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 6.35 Required Flexural Reinforcement for the Joist in the End Span in Example 6.18 Rn (psi)

As (in2)*

Reinforcement

Exterior negative





#Ǝ#Ǝ

Positive







First interior negative





#Ǝ#Ǝ

Location

*Min.

Mu (ft-kips)

[Eq. (6.35)]

Max.

Eq. [(6.38)]

7KHPD[LPXPGHVLJQVKHDUIRUFHDIWHUUHGLVWULEXWLRQRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUW

Assuming at least minimum shear reinforcement is provided, with the axial force

can be determined from the following equation Table 6.4

where

for normalweight concrete.

Table 6.2

For Maximum spacing

along the length of the joist.

Because of the narrow width of the joist, provide single-leg stirrups. $VLQJOHOHJVWLUUXSVSDFHGDW

provides

Figure 6.19 Figure 6.16 Table 6.6

Table 6.5

8VHVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHU determined prior to redistribution.

6-118

which is the same shear reinforcement

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Check the adequacy of the cross-sectional dimensions of the beam

$VVXPLQJLQFOHDUFRYHUWRFORVHGVWLUUXSV Figure 6.17

)URP6WHS )URP6WHSLQ([DPSOH

at the critical section. at the critical section.

)URP6WHSLQ([DPSOH

Eq. (6.31)

Therefore, the cross-sectional dimensions are adequate. Step 5 – Determine the required transverse reinforcement for torsion Eq. (6.60)

Step 6 – Determine the required longitudinal reinforcement for torsion Eq. (6.61)

Eq. (6.62)

Use The transverse and longitudinal for torsion are combined with the transverse reinforcement required for shear and WKHORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHGIRUÀH[XUHVHH([DPSOH 

6-119

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6.9.19 Example 6.19 – Design for Combined Flexure, Shear, and Torsion: Edge Beam in Building #2, Beam is Not Part of the LFRS, SDC C

'HVLJQWKHHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHOIRUWKHFRPELQHGHIIHFWVGXHWRÀH[XUHVKHDUDQG WRUVLRQDVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the total transverse reinforcement

7KHWRWDOUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWSHUVWLUUXSOHJLVHTXDOWRWKDWUHTXLUHGIRUVKHDU([DPSOH  plus that required for torsion (Example 6.18):

)RUFORVHGVWLUUXSVWKHUHTXLUHGVSDFLQJDWWKHFULWLFDOVHFWLRQLVHTXDOWRWKHIROORZLQJ

,WLVGHWHUPLQHGLQ([DPSOHWKDW Therefore, the maximum spacing of the transverse reinforcement is equal to the following:

Determine the location where the maximum stirrup spacing can be used. 7U\DVHFWLRQORFDWHGIWIURPWKHIDFHRIWKHVXSSRUW

6-120

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHIRUHWKHFORVHGVWLUUXSVFDQEHVSDFHGDWLQRQFHQWHUDWDGLVWDQFHHTXDOWRDSSUR[LPDWHO\IWIURP the face of the column. The distance from the face of the support where transverse reinforcement is no longer required for shear is deterPLQHGIURPWKHIROORZLQJHTXDWLRQVHH)LJXUH  Eq. (6.57)

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ͳͲ͸ǤͶ ͳʹ͵Ǥ͵

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‫ܖܗܑܛܚܗ܂‬ሺ܎‫ܜ‬ǦǦ‫ܛܘܑܓ‬ሻ ܶ௨ ൌ ߶ܶ௧௛ ൌ ʹͶǤ͹

ͻͺǤͺ

Ž‘•‡†•–‹””—’•”‡“—‹”‡† Figure 6.53 Shear and torsion diagrams in an end span for the edge beam in Example 6.19.

6-121

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Similarly, the distance from the face of the support where transverse reinforcement is no longer required for torsion LVGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVHH)LJXUH 

The transverse reinforcement for torsion must extend a distance

past the location it is no longer required: ACI 9.7.6.3.2

Therefore, torsion requirements govern over the entire length of the beam.

Because

closed stirrups are required

Step 2 – Determine the total longitudinal reinforcement

7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWDWWKHFULWLFDOVHFWLRQVLVJLYHQLQ7DEOH From Step 6 in Example 6.18, This reinforcement must be distributed around the perimeter of the EHDPZLWKDPD[LPXPVSDFLQJRILQDQGPXVWEHFRPELQHGZLWKWKDWUHTXLUHGIRUÀH[XUH7KXVDVVLJQ to each face. 8VHEDUVRQHDFKVLGHIDFHRIWKHEHDP $VVXPLQJWRSDQGERWWRPORQJLWXGLQDOUHLQIRUFHPHQWZLWKFORVHGVWLUUXSVDQGLQFOHDUFRYHU

Figure 6.28

&KHFNWKHUHTXLUHPHQWRI$&, Figure 6.28

where the largest spacing of the transverse reinforcement,

is used to check this requirement.

The remaining longitudinal reinforcement for torsion is distributed equally between the top and bottom of the section: • Exterior negative:

8VHEDUV • Positive:

8VHEDUV • First interior negative:

8VHEDUV 6-122

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The provided number of bars at all locations also satisfy maximum and minimum spacing requirements (see Table 6.8 and Table 6.9, respectively). 6.9.20 Example 6.20 – Determination of Reinforcement Details: Edge Beam in Building #2, Beam is Not Part of the LFRS, SDC C

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUWKHHGJHEHDPLQ%XLOGLQJDWWKHVHFRQGÀRRUOHYHODVVXPand LQJWKHEHDPLVQRWSDUWRIWKH/)56VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the flexural reinforcement details

It is determined in Example 6.19 that the following reinforcement is required at the critical sections in the end span: • ([WHULRUQHJDWLYHEDUV • 3RVLWLYHEDUV • )LUVWLQWHULRUQHJDWLYHEDUV $OVREDUVDUHUHTXLUHGRQHDFKVLGHIDFH 7KHSURYLGHGQXPEHURIUHLQIRUFLQJEDUVDWDOOFULWLFDOVHFWLRQVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\VHH6WHSLQ([DPSOH  It is determined in Example 6.19 that the transverse torsion reinforcement must be provided over the entire span of the beam. Consequently, the longitudinal torsional reinforcement must also be provided over the entire span (ACI  7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHDUHQRWDSSOLFDEOHLQWKLVH[DPSOHEHFDXVHRIWRUVLRQ $OVRWKHWRSEDUVUHTXLUHGDWWKH¿UVWLQWHULRUVXSSRUWDUHPDGHFRQWLQXRXVRYHUWKHHQWLUHVSDQ $&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKHVXSSRUWVIRUWKHERWWRPEDUV7KHUHTXLUHGODSVSOLFHOHQJWKLV determined as follows:

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

6-123

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK Step 2 – Determine the transverse reinforcement details

,WLVGHWHUPLQHGLQ([DPSOHWKDWFORVHGVWLUUXSVPXVWEHSURYLGHGRYHUWKHHQWLUHVSDQRIWKHEHDP $VVXPLQJRQO\RXWHUVWLUUXSOHJVDUHSURYLGHGFKHFNWKHVSDFLQJDFURVVWKHZLGWKRIWKHEHDP

Figure 6.19

7KHUHIRUHSURYLGHDFURVVWLHWRVDWLVI\WKHUHTXLUHPHQWVIRUVSDFLQJRIWKHVWLUUXSVDFURVVWKHZLGWKRIWKHEHDP Based on the calculations in Example 6.19, provide the following closed stirrup arrangements over the length of the beam: • FORVHGVWLUUXSVZLWKFURVVWLHVSDFHGDWLQRQFHQWHUZLWKWKH¿UVWFORVHGVWLUUXSORFDWHGLQ from the face of the column • FORVHGVWLUUXSVZLWKFURVVWLHVSDFHGDWLQRQFHQWHUZLWKLQWKHFHQWHUSRUWLRQRIWKHEHDP 5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPLQWKHHQGVSDQDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,IRUSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOHGJHEHDPV

6-124

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

2ᇱ -8ᇱᇱ 

2ᇱ -4ᇱᇱ 

6-#9 

5-#9 16-#4 stirrups @ 5ԣ

A

4ᇱ -0ᇱᇱ 

2-#5 15-#4 stirrups @ 11ԣ ᇱ

16-#4 stirrups @ 5ԣ

2ᇱᇱ 

ᇱᇱ

27 -6 

4.5ᇱᇱ 

Other reinforcement not shown for clarity

6-#9  2-#5 

2Ԣ-0ᇱᇱ 

2ᇱᇱ 

A

#4 stirrups 5-#9  ᇱ

ᇱᇱ

2 -6  ‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬

Figure 6.54 Reinforcement details for the edge beam in Examples 6.15 through 6.20.

6-125

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6-126

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 7 COLUMNS 7.1 Overview 7KHSULPDU\IXQFWLRQRIDFROXPQLVWRVXSSRUWD[LDOIRUFHVIURPÀRRUDQGURRIJUDYLW\ORDGVDQGWRWUDQVIHUWKRVH forces to a structural element below (such as a transfer beam or a foundation system). Columns not part of the lateral force-resisting system (LFRS) of the building resist primarily axial compression forces due to gravity loads; bending moments and shear forces from gravity loads must also be resisted, but these reactions are usually relatively small, especially for interior columns. Columns in moment frames designated to part of the LFRS must be designed IRUWKHFRPELQHGHIIHFWVRIÀH[XUHDQGD[LDOIRUFHVDQGWKHDFFRPSDQ\LQJVKHDUIRUFHV6OHQGHUQHVVHIIHFWVPD\ need to be considered in the design of a column regardless of whether the column is part of the LFRS or not. The design and detailing of cast-in-place concrete columns with nonprestressed reinforcement (longitudinal and WUDQVYHUVH DUHFRYHUHGLQWKLVFKDSWHU3URYLVLRQVIRUFROXPQVDUHJLYHQLQ$&,&KDSWHUZKLFKDUHDSSOLFDEOHWR members in buildings assigned to Seismic Design Category (SDC) A and B.

7.2 Dimensional Limits 1RPLQLPXPVL]HVDUHH[SOLFLWO\UHTXLUHGLQ$&,IRUFROXPQVLQEXLOGLQJVDVVLJQHGWR6'&$WKURXJK&+RZever, there are some practical constructability issues that need to be considered, which have a direct impact on the FURVVVHFWLRQDOGLPHQVLRQV,WLVLPSRUWDQWWRHQVXUHWKDWWKHORQJLWXGLQDOEDUVFDQ¿WZLWKLQWKHFURVVVHFWLRQFRQVLGHULQJWKHPLQLPXPFRYHUUHTXLUHPHQWVRI$&,DQGWKHPLQLPXPVSDFLQJUHTXLUHPHQWVRI$&,7KH FURVVVHFWLRQDOGLPHQVLRQVDQGWKHORQJLWXGLQDOEDUVL]HVPXVWEHVHOHFWHGWRPLQLPL]HUHLQIRUFHPHQWFRQJHVWLRQ especially at beam-column or slab-column joints. It is also important to verify that the requirements for transverse UHLQIRUFHPHQWWLHVRUVSLUDOV DUHVDWLV¿HG7KHUHIRUHLWLVUHFRPPHQGHGWRXVHDWOHDVWDLQFROXPQGLPHQVLRQ IRUUHFWDQJXODUFROXPQVDQGDWOHDVWDLQGLDPHWHUIRUFLUFXODUFROXPQV For columns that have cross-sectional dimensions larger than required to resist the factored loads (like those in the XSSHUÀRRUVRIDEXLOGLQJ LWLVSHUPLWWHGWRFDOFXODWHWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUDWLREDVHGRQWKH UHTXLUHGFURVVVHFWLRQDODUHDUDWKHUWKDQWKHSURYLGHGFURVVVHFWLRQDODUHD$&, 7KHSURYLGHGDUHDRIORQJLWXGLQDOUHLQIRUFHPHQWLQVXFKFDVHVPXVWQRWEHOHVVWKDQSHUFHQWRIWKHDFWXDOFURVVVHFWLRQDODUHDRIWKHFROumn. This requirement is not applicable to columns that are part of a special moment frame or that must be designed LQDFFRUGDQFHZLWK$&,FROXPQVQRWSDUWRIWKHVHLVPLFIRUFHUHVLVWLQJV\VWHP  Where columns are built integrally with concrete walls, the cross-sectional area to be used in the design of the colwhere and are the cross-sectional dimensions (in inches) umn must not exceed an area equal to RIWKHFROXPQFRUHPHDVXUHGWRWKHRXWVLGHHGJHVRIWKHWUDQVYHUVHUHLQIRUFHPHQW$&,VHH)LJXUH  ,QFDVHVZKHUH$&,RUKDYHEHHQDSSOLHGWKHDFWXDOFURVVVHFWLRQDOGLPHQVLRQVRIWKHFROXPQ must be used in the structural analysis of the building and in the design of any members that interact with the colXPQ$&, 

7.3 Required Strength 7.3.1 Analysis Methods Overview

7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKVHH$&,DQGUHVSHFWLYHO\  Axial forces, bending moments, shear forces, and torsional moments on a reinforced concrete column, including the HIIHFWVRIVOHQGHUQHVVZKHUHDSSOLFDEOH$&, FDQEHGHWHUPLQHGE\WKHIROORZLQJDQDO\VLVPHWKRGVLQ$&, 

7-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

൑ ܾଵ ൅ ͵ԣ

ܾଶ 

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Figure 7.1 Reinforced concrete column built integrally with a concrete wall.

Linear Elastic First-Order Analysis

,QDOLQHDUHODVWLF¿UVWRUGHUDQDO\VLVUHDFWLRQVLQWKHVWUXFWXUDOPHPEHUVDUHGHWHUPLQHGEDVHGRQWKHXQGHIRUPHG JHRPHWU\RIWKHVWUXFWXUHVOHQGHUQHVVVHFRQGRUGHU HIIHFWVZKLFKFDQKDYHDVLJQL¿FDQWLQÀXHQFHRQWKHGHsign strength of columns, are not explicitly accounted for in the analysis. Where applicable, slenderness effects in FROXPQVPD\EHGHWHUPLQHGE\WKHPDJQL¿HGPRPHQWPHWKRGLQ$&,XVLQJWKHUHVXOWVIURPWKH¿UVWRUGHU DQDO\VLVVHH6HFWLRQRIWKLVSXEOLFDWLRQ  Linear Elastic Second-Order Analysis

In a linear elastic second-order analysis, it is assumed the structure remains elastic and second-order effects are LQKHUHQWO\DFFRXQWHGIRULQWKHDQDO\VLVWKDWLVHTXDWLRQVRIHTXLOLEULXPDUHVDWLV¿HGXVLQJWKHGHIRUPHGVKDSHRI the structure). The effects of cracking along the length of the members and creep must be considered in the analysis VHH$&, :KHUHDSSOLFDEOHVOHQGHUQHVVHIIHFWVRQFROXPQVPD\EHGHWHUPLQHGE\WKHPRPHQWPDJQL¿FDWLRQPHWKRGIRUQRQVZD\IUDPHVLQ$&,XVLQJWKHUHVXOWVIURPWKHVHFRQGRUGHUDQDO\VLV$&,  The method for nonsway frames is applicable because a second-order analysis inherently accounts for the relative displacements of the ends of the members. Inelastic Analysis

In an inelastic analysis, nonlinear stress-strain response of the materials in the structure and compatibility of deforPDWLRQVPXVWEHFRQVLGHUHG$&, (TXLOLEULXPPXVWEHVDWLV¿HGLQWKHXQGHIRUPHGFRQ¿JXUDWLRQZKHUH DQLQHODVWLF¿UVWRUGHUDQDO\VLVLVXVHGRULQWKHGHIRUPHGFRQ¿JXUDWLRQZKHUHDQLQHODVWLFVHFRQGRUGHUDQDO\VLV is used. It must be demonstrated that the results from an inelastic analysis are in substantial agreement with results IURPWHVWV$&, /LNHLQWKHFDVHRIDOLQHDUHODVWLFVHFRQGRUGHUDQDO\VLVVOHQGHUQHVVHIIHFWVRQFROXPQV DUHSHUPLWWHGWREHGHWHUPLQHGE\WKHPRPHQWPDJQL¿FDWLRQPHWKRGIRUQRQVZD\IUDPHVLQ$&,XVLQJWKH results from the inelastic analysis (ACI 6.8.1.3). Finite Element Analysis

$¿QLWHHOHPHQWDQDO\VLVFRQIRUPLQJWRWKHSURYLVLRQVRI$&,FDQEHXVHGWRDQDO\]HHVVHQWLDOO\DQ\UHLQIRUFHG concrete structure provided an appropriate model is constructed (ACI 6.9).

7-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRPHQWPDJQL¿FDWLRQSURFHGXUHWHQGVWRJLYHPRUHFRQVHUYDWLYHUHVXOWVWKDQWKRVHREWDLQHGIURPDPRUHUHat the ends of the ¿QHGDQDO\VLV5HJDUGOHVVRIWKHDQDO\VLVPHWKRGWKHPD[LPXPIDFWRUHGEHQGLQJPRPHQWV of any structural PHPEHUVRIDIUDPHPXVWEHOLPLWHGWRHQVXUHWKHVWUXFWXUHLVVWDEOH$FFRUGLQJWR$&, PHPEHUREWDLQHGIURPDQDQDO\VLVLQFOXGLQJVHFRQGRUGHUHIIHFWVGXHWRVOHQGHUQHVVPXVWQRWH[FHHGSHUFHQWRI WKHPRPHQWREWDLQHGIURPD¿UVWRUGHUDQDO\VLV Section Properties

:KHQSHUIRUPLQJD¿UVWRUGHURUDQHODVWLFVHFRQGRUGHUODWHUDOORDGDQDO\VLVWKHVHFWLRQSURSHUWLHVLQ$&, may be used in lieu of a more sophisticated analysis to account for member cracking and other effects. The reduced moments of inertia in ACI Table 6.6.3.1.1(a) and the alternative moments of inertia in ACI Table 6.6.3.1.1(b) for YDULRXVVWUXFWXUDOPHPEHUVDUHJLYHQLQ7DEOHDQG7DEOHUHVSHFWLYHO\7KHUHGXFHGPRPHQWVRILQHUWLDLQ 7DEOHDUHEDVHGRQDQDQDO\VLVZKHUHVWUHQJWKOHYHOIDFWRUHG ORDGVDUHXVHG)RUDQDQDO\VLVEDVHGRQVHUYLFH OHYHOORDGVLWLVSHUPLWWHGWRXVHUHGXFHGPRPHQWVRILQHUWLDHTXDOWRWLPHVWKHYDOXHVLQ7DEOHSURYLGHGWKH moments of inertia do not exceed the gross moments of inertia $&,  Table 7.1 Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using StrengthLevel Loads Member

Moment of Inertia

Columns :DOOVņXQFUDFNHG :DOOVņFUDFNHG Beams )ODWSODWHVDQGÀDWVODEV

Table 7.2 Alternative Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using Strength-Level or Service-Level Loads Member

Moment of Inertia*

Columns and walls

%HDPVÀDWSODWHVDQGÀDWVODEV *For a service-level load analysis, use service-level axial force and moment effects in the equation for columns and walls.

7KHPRUHUH¿QHGHTXDWLRQVIRUUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHLQFOXGHWKH IDFWRUHGD[LDOIRUFH RQDFROXPQIRUWKHORDGFRPELQDWLRQXQGHUFRQVLGHUDWLRQ QRPLQDOD[LDOVWUHQJWKDW and factored moment, These ]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ DQG DUHDRIORQJLWXGLQDOUHLQIRUFHPHQW equations are applicable for strength-level and service-level loading, even though they are presented in terms of factored load effects. For service-level analysis, the factored load effects are replaced by the corresponding servicelevel load effects. Regardless of the equations used to determine moment of inertias, the gross area of the section, the analysis. Also, the cross-sectional area for calculation of shear deformations is

is to be used in

7-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QOLHXRIWKHUHTXLUHPHQWVRI$&,LWLVSHUPLWWHGWRDVVXPHLQD¿UVWRUGHUIDFWRUHGODWHUDOORDGDQDO\VLVWKDW DOOWKHPHPEHUVLQDVWUXFWXUHKDYHDUHGXFHGPRPHQWRILQHUWLDHTXDOWRSHUFHQWRIWKHJURVVPRPHQWRILQHUWLD $&, $PRUHGHWDLOHGDQDO\VLVFRQVLGHULQJWKHHIIHFWLYHVWLIIQHVVRIDOOPHPEHUVLVDOVRSHUPLWWHG 7.3.2 Factored Axial Force and Moment

Axial forces and bending moments at the ends of a column are obtained from an analysis of the structure. When designing a reinforced concrete column, it is not always obvious which load combination for combined factored and bending moment governs. Therefore, each applicable factored load combination must be axial force FRQVLGHUHGLQGHVLJQ$&,  7.3.3 Slenderness Effects Overview

&ROXPQVFDQJHQHUDOO\EHFODVVL¿HGDV³VKRUWFROXPQV´DQG³VOHQGHUFROXPQV´7KHWHUP³VKRUWFROXPQ´LVRIWHQ used to indicate a column with a strength equal to that computed for its cross-section. In such cases, strength can be represented by an interaction diagram, which is constructed based on the geometric and material properties of the VHFWLRQVHH)LJXUHDQG6HFWLRQRIWKLVSXEOLFDWLRQ ,IDFROXPQGRHVQRWGHÀHFWODWHUDOO\DQ\FRPELQDtion of axial force and bending moment falling outside of the interaction curve is assumed to cause failure. This is commonly referred to as a material failure. For purposes of design, a column has adequate strength when all the IDFWRUHGFRPELQDWLRQVRID[LDOIRUFHDQGEHQGLQJPRPHQWREWDLQHGIURPDQHODVWLF¿UVWRUGHUDQDO\VLVRIWKHIUDPH fall within or on the design strength interaction curve. ܲ݁



ܲȟ

Axial Force, P

Short column Slender column, material failure

Slender column, instability failure

Bending Moment, M

Figure 7.2 Impact of slenderness effects on design strength of columns.

$³VOHQGHUFROXPQ´LVGH¿QHGDVDFROXPQZKRVHVWUHQJWKLVUHGXFHGE\VHFRQGRUGHUGHIRUPDWLRQVGXHWRKRUL]RQWDOGLVSODFHPHQWV6HFRQGDU\HIIHFWVFDXVHGE\KRUL]RQWDOGHÀHFWLRQDUHFRPPRQO\UHIHUUHGWRDV3GHOWDHIIHFWV&RQVLGHUWKHLGHDOL]HGFROXPQLQ)LJXUHZKLFKLVSDUWRIDIUDPH7KHEHQGLQJPRPHQWVDWWKHHQGVRI where is the axial force on the column and is the corresponding eccentricity. the column are equal to %HFDXVHRIWKHDSSOLHGORDGLQJWKHFROXPQGLVSODFHVKRUL]RQWDOO\E\DQDPRXQWHTXDOWR 7KLVGHÀHFWLRQFDXVHV

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

an additional (or secondary) moment, which is equal to Thus, the total moment in the column is equal to the SOXVWKHPRPHQWFDXVHGE\WKHKRUL]RQWDOGHÀHFWLRQRIWKHPHPEHU moment caused by the applied loading 7KLVWRWDOPRPHQWDQGLWVHIIHFWRQFROXPQVWUHQJWKLVGHSLFWHGLQ)LJXUHGHQRWHGDV³6OHQGHUFROXPQ PDWHULDOIDLOXUH´LQWKH¿JXUH  ܲ

ܲ ‫ ܯ‬ൌ ܲ݁

‫ ܯ‬ൌ ܲ݁

ȟ

‫ ܯ‬ൌ ܲሺ݁ ൅ ȟሻ ܲ

‫ ܯ‬ൌ ܲ݁ ܲ

Figure 7.3 P-delta effects in a column.

,QWKHFDVHRIYHU\VOHQGHUFROXPQVLWLVSRVVLEOHIRUWKHGHÀHFWLRQ WRLQFUHDVHLQGH¿QLWHO\ZLWKDQLQFUHDVHLQ the axial force 7KLVW\SHRIIDLOXUHLVNQRZQDV³LQVWDELOLW\IDLOXUH´ZKLFKJHQHUDOO\RFFXUVDWDQD[LDOIRUFHOHVV WKDQWKDWFRUUHVSRQGLQJWRPDWHULDOIDLOXUHRIWKHVHFWLRQVHH)LJXUH  Slenderness effects can be neglected in columns with slenderness ratios less than or equal to the limiting values in $&,,QIRUPDWLRQRQKRZWRGHWHUPLQHVOHQGHUQHVVUDWLRVDQGKRZWRGHVLJQIRUWKHHIIHFWVRIVOHQGHUQHVVDUH covered below. Columns in Nonsway and Sway Frames

6ZD\LQEXLOGLQJVGXHWRODWHUDORURWKHUIRUFHVFDQKDYHDVXEVWDQWLDOLQÀXHQFHRQVHFRQGRUGHUHIIHFWVLQFROXPQV It is common for secondary effects to increase with increasing sway. Distinguishing between columns in nonsway frames and those in sway frames can usually be done by comparing the total lateral stiffness of the columns in a VWRU\WRWKDWRIWKHEUDFLQJHOHPHQWV0RPHQWIUDPHEXLOGLQJVDUHW\SLFDOO\ODWHUDOO\ÀH[LEOHDQGWKHFROXPQVDUH more susceptible to secondary effects than those in a frame with structural walls. Methods to determine whether a column is in a nonsway frame (braced against sidesway) or in a sway frame (not EUDFHGDJDLQVWVLGHVZD\ DUHJLYHQLQ$&,DQG7KHFRQGLWLRQVIRUFROXPQVLQDQRQVZD\IUDPHDUH JLYHQLQ7DEOH

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.3 Conditions for Columns in a Nonsway Frame (Braced Against Sidesway) ACI Section No.

Condition



Bracing elements (for example, moment frames or walls) resisting lateral movement of a story have a total stiffness times the gross lateral stiffness of the columns in that story in the direction of analysis. Increase in column end moments due to second-order effects end moments.

SHUFHQWRIWKH¿UVWRUGHU

:KHUHIDFWRUHGORDGVDUHXVHGWRFDOFXODWHWKHODWHUDOGHÀHFWLRQRIDIUDPHVWDELOLW\LQGH[ IRUDVWRU\VDWLV¿HVWKHIROORZLQJHTXDWLRQ

where 

total factored vertical load in the story ¿UVWRUGHUUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQGERWWRPRIWKHVWRU\ IDFWRUHGKRUL]RQWDOVWRU\VKHDULQDVWRU\ length of the columns measured center-to-center of the joints in the frame

:KHUHVHUYLFHORDGVDUHXVHGWRFDOFXODWHWKHODWHUDOGHÀHFWLRQRIDIUDPHVWDELOLW\LQGH[ IRUDVWRU\VDWLV¿HVWKHIROORZLQJHTXDWLRQ

where

total service dead and live axial loads in the story ¿UVWRUGHUVHUYLFHOHYHOUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQG bottom of the story service-level story shear

in the equation is the moment in a story due to the axial loads being The numerator GXHWRWKHKRUL]RQWDOVWRU\VKHDU VHH)LJXUH 7KHVXPRIWKHIDFWRUHGD[LDOORDGV displaced an amount must correspond to the lateral loading case for which this sum is the greatest. The denominator in the story, in the equation for is the overall moment in a story due to ȭܲ௨  ȟ௢ 

κ௖ 

ܸ௨௦ 

Figure 7.4 Definition of stability index,

:KHUHWKHFRQGLWLRQVLQ7DEOHDUHQRWPHWWKHFROXPQVDUHFRQVLGHUHGWREHLQDVZD\IUDPHWKDWLVLQDIUDPH not braced against sidesway). In general, a frame may contain both nonsway and sway columns and stories.

7-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Consideration of Slenderness Effects

Whether slenderness effects need to be considered or not for nonsway and sway frames depends on the slenderness 7KHWHUPVLQWKHVOHQGHUQHVVUDWLRDUHGH¿QHGLQ7DEOHVHH$&,  ratio of the column, Table 7.4 Slenderness Ratio, kℓu / r

Nonsway Frames

Sway Frames Unsupported length, LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRU other members capable of providing lateral support in the direction of analysis. Radius of gyration,

is permitted to be calculated by the following:

for rectangular columns where is the dimension of the column in the direction of analysis for circular columns where is the diameter of the column is determined based on the stiffness ratio, at the ends of the column. This ratio is The effective length factor, GHWHUPLQHGE\GLYLGLQJWKHVXPRIWKHVWLIIQHVVHVRIWKHFROXPQVDWDMRLQWE\WKHVXPRIWKHVWLIIQHVVHVRIWKHÀH[ural members framing into that joint in the direction of analysis: (7.1)

In this equation, LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWHZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, LVWKHPRPHQWRILQHUWLDRIWKHFROXPQRUÀH[XUDOPHPEHU is the length of the columns measured center-tocenter of the joints in the frame; and LVWKHOHQJWKRIWKHÀH[XUDOPHPEHUVPHDVXUHGFHQWHUWRFHQWHURIWKHMRLQWV The stiffness ratio LVGHWHUPLQHGDWERWKWKHWRSDQGERWWRPRIDFROXPQ7KHFKDUWLQ$&,)LJXUH5D IRU QRQVZD\IUDPHVDQGLQ$&,)LJXUH5E IRUVZD\IUDPHVFDQEHXVHGWRJUDSKLFDOO\REWDLQ by drawing a at the top of the column to the stiffness ratio at the bottom of the column. The line from the stiffness ratio The charts in ACI Figure value of is obtained from the chart at the location where the line crosses the 5DUHEDVHGRQWKHHTXDWLRQVLQ7DEOH Table 7.5 Equations for Stiffness Ratio, Ȍ

Nonsway frames

Sway frames The unsupported length of a column, LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRURWKHUHOHPHQWVFDSDEOH RISURYLGLQJODWHUDOVXSSRUWLQWKHGLUHFWLRQRIDQDO\VLV)RUH[DPSOHWKHEHDPLQ)LJXUHSURYLGHVODWHUDOVXSSRUW which to the top of the column in the direction parallel to the x-axis, and the unsupported length is equal to

7-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ሺκ௨ ሻଶ 

‫ݕ‬

ሺκ௨ ሻଵ 

‫ݔ‬

Figure 7.5 Unsupported column length with beams and slabs.

is the distance from the bottom of the beam to the top of the slab at the bottom of the column. This beam does not is the distance from provide lateral support to the column parallel to the y-direction, so the unsupported length the bottom of the slab at the top of the column to the top of the slab at the bottom of the column. 7KHXQVXSSRUWHGFROXPQOHQJWKIRUFROXPQVZLWKFDSLWDOVRUKDXQFKHVLVLOOXVWUDWHGLQ)LJXUHIRUWKHFDVHRID is measured from the top of the slab at the bottom of the column to the lower excolumn capital. In this case, tremity of the capital. 6OHQGHUQHVVOLPLWVDUHJLYHQLQ$&,D IRUFROXPQVQRWEUDFHGDJDLQVWVLGHVZD\DQGLQ$&,E IRU FROXPQVEUDFHGDJDLQVWVLGHVZD\6OHQGHUQHVVHIIHFWVDUHSHUPLWWHGWREHQHJOHFWHGZKHQWKHHTXDWLRQVLQ7DEOH DUHVDWLV¿HG Table 7.6 Limits for Slenderness Effects Column Bracing Condition

Slenderness Ratios Where Slenderness is Permitted to Be Neglected

Not braced against sidesway (sway)

Braced against sidesway (nonsway)

*Ratio

7-8

is negative if a column is bent in single curvature and is positive if a column is bent in double curvature.

κ௨ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 7.6 Unsupported length of a column with a capital.

Because columns braced against sidesway are more stable when bent in double curvature, the limiting slenderness ratio is larger than that for columns bent in single curvature. 7KHOLPLWLQJYDOXHVRIWKHVOHQGHUQHVVUDWLRVLQ7DEOHDUHEDVHGRQDSHUFHQWLQFUHDVHLQPRPHQWVGXHWRVOHQderness effects, which is considered to be acceptable. Minimum column dimensions so that slenderness effects need not be considered as a function of the unsupported DUHJLYHQLQ7DEOH7KHPLQLPXPGLPHQVLRQVIRUFROXPQVEUDFHGDJDLQVWVLGHVZD\DUHEDVHG column length, for columns bent in double curvature where For columns not braced against sidesway, on Tabulated minimum column dimensions are rounded up to the minimum column dimensions are based on the next whole even numbers. Table 7.7 Minimum Column Dimensions to Neglect Slenderness Unsupported Column Length, ℓu (ft)

 8 9 

Minimum Column Dimension (in.) Column Bracing Condition Rectangular (c1 )

Circular

Braced against sidesway

8



Not braced against sidesway





Braced against sidesway

8



Not braced against sidesway





Braced against sidesway





Not braced against sidesway





Braced against sidesway





Not braced against sidesway





7-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Minimum column dimensions to neglect slenderness for columns braced against sidesway are small; the dimensions DUHDWPRVWHTXDOWRWKHPLQLPXPSUDFWLFDOGLPHQVLRQVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ7KHUHIRUHVOHQGHUness effects typically do not need to be considered for columns braced against sidesway in buildings with typical VWRU\KHLJKWV,QWDOOHUEXLOGLQJVVKHDUZDOOVRUDFRPELQDWLRQRIVKHDUZDOOVDQGPRPHQWIUDPHVDUHRIWHQXWLOL]HG which essentially brace the columns against sidesway. Columns not braced against sidesway most often occur in low-rise buildings without walls where the lateral displacements are usually relatively small. Thus, second-order effects are typically negligible even if column dimenVLRQVDUHVPDOOHUWKDQWKHPLQLPXPGLPHQVLRQVLQ7DEOH Slenderness effects usually need to be considered where the cross-sectional dimensions of a column are limited due WRDUFKLWHFWXUDORURWKHUFRQVWUDLQWVDQGRUWKHXQVXSSRUWHGOHQJWKRIWKHFROXPQLVUHODWLYHO\ORQJ Moment Magnification Method

Overview 7KHPRPHQWPDJQL¿FDWLRQPHWKRGLQ$&,LVDQDSSUR[LPDWHPHWKRGWKDWDFFRXQWVIRUVOHQGHUQHVVHIIHFWVE\ PDJQLI\LQJWKHEHQGLQJPRPHQWVDWWKHHQGVRIDFROXPQREWDLQHGIURPD¿UVWRUGHUDQDO\VLV,QJHQHUDODVOHQGHU FROXPQPXVWEHGHVLJQHGWRUHVLVWWKHFRPELQHGHIIHFWVIURPIDFWRUHGD[LDOFRPSUHVVLRQIRUFHVDQGPDJQL¿HGEHQGLQJPRPHQWVZKLFKDUHREWDLQHGE\PXOWLSO\LQJWKH¿UVWRUGHUIDFWRUHGEHQGLQJPRPHQWVE\DPRPHQWPDJQL¿HU that is a function of the factored axial force and the critical buckling load of the column. The column is adequate ZKHUHDOOWKHIDFWRUHGORDGFRPELQDWLRQVIRUFRPELQHGD[LDOIRUFHDQGPDJQL¿HGEHQGLQJPRPHQWIDOOZLWKLQRURQ the design strength interaction diagram, which is obtained from a strain compatibility analysis of the section. Nonsway Frames 3URYLVLRQVIRUVOHQGHUFROXPQVLQQRQVZD\IUDPHVDUHJLYHQLQ$&,&ROXPQVPXVWEHGHVLJQHGIRUFRPELDQGIDFWRUHGPDJQL¿HGPRPHQWV where is determined by ACI Equation nations of factored axial forces,   (7.2)

In this equation, LVWKHODUJHURI WKHWZRIDFWRUHGFROXPQHQGPRPHQWVIURPD¿UVWRUGHUDQDO\VLVDQG  where is the dimension of the column in the direction of analysis minimum moment $&,  7KHPDJQL¿FDWLRQIDFWRU

LVGHWHUPLQHGE\$&,(TXDWLRQ  (7.3)

The moment factor, relates the actual moment diagram to an equivalent uniform moment diagram. It is asVXPHGLQWKHPRPHQWPDJQL¿HUPHWKRGWKDWWKHPD[LPXPPRPHQWLVDWRUQHDUWKHPLGKHLJKWRIWKHFROXPQ,IWKH (that maximum moment occurs at one end of the column, design is based on an equivalent uniform moment is assumed to act at both ends of the column). This leads to the same maximum moment at or near midis, KHLJKWRIWKHFROXPQZKHQPDJQL¿HG depends on whether transverse loads are applied between the supports of the column or not (see The value of 7DEOH ,Q$&,(TXDWLRQD IRUWKHFDVHRIFROXPQVZLWKRXWWUDQVYHUVHORDGVDSSOLHGEHWZHHQVXSSRUWV is the ratio of the smaller to the larger factored end moment; this ratio is negative if the column is bent in VLQJOHFXUYDWXUHDQGSRVLWLYHLIWKHFROXPQLVEHQWLQGRXEOHFXUYDWXUHVHH)LJXUH )RUFROXPQVZLWKWUDQVYHUVH loads applied between supports, it is possible for the maximum moment to occur at a section away from the end of LQWKLVFDVH>$&,(TXDWLRQE @ the member; thus,

7-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.8 Values of Cm

Cm

Column Transverse Load Condition

Transverse loads not applied between supports 

Transverse loads applied between supports       

ܲ

ܲ ‫ܯ‬ଵ 

‫ܯ‬ଵ 

െͳ ൑ ‫ܯ‬ଵ Τ‫ܯ‬ଶ ൑ Ͳ

Ͳ ൑ ‫ܯ‬ଵ Τ‫ܯ‬ଶ ൑ ͳ

ͳǤͲ ൒ ‫ܥ‬௠ ൒ ͲǤ͸

ͲǤ͸ ൒ ‫ܥ‬௠ ൒ ͲǤʹ

‫ܯ‬ଶ 

‫ܯ‬ଶ 

ܲ

Figure 7.7 Values of

ܲ for columns without transverse loads applied between supports.

, the factored column end moments and from analysis are to be used in ACI In cases where (TXDWLRQD WRGHWHUPLQH $&, $OWHUQDWLYHO\ LVSHUPLWWHGWREHWDNHQHTXDOWR The critical buckling load,

LVGHWHUPLQHGE\$&,(TXDWLRQ  (7.4)

7KHHIIHFWLYHÀH[XUDOVWLIIQHVVRIDFROXPQ LVSHUPLWWHGWREHFDOFXODWHGE\$&,(TXDWLRQD  E RUF >$&,VHH7DEOH@7KHQXPHUDWRUVLQWKHVHHTXDWLRQVUHSUHVHQWWKHVKRUW term column stiffness. Creep effects due to sustained loads are accounted for by dividing the short-term stiffness by where is the ratio of the maximum factored sustained axial load to the maximum factored axial load associated with the same load combination. This factor is typically equal to the factored axial dead load divided by the total factored axial load on a column.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.9 Effective Flexural Stiffness, (EI)eff

(EI)eff

ACI Equation No.

D

E

F *

(see Table 7.2)

$&,(TXDWLRQD LVDVLPSOL¿HGYHUVLRQRI$&,(TXDWLRQE DQGGRHVQRWGLUHFWO\DFFRXQWIRU , which is the moment of longitudinal reinforcement in the column like the latter equation does through the term inertia of the longitudinal reinforcement about the centroidal axis of the cross-section in the direction of analysis. 7KXVWKLVHTXDWLRQLVQRWDVDFFXUDWHDV$&,(TXDWLRQE HVSHFLDOO\LQFROXPQVZLWKJUHDWHUDPRXQWVRI longitudinal reinforcement. and large axial load ratios $&,(TXDWLRQE ZDVRULJLQDOO\GHULYHGIRUVPDOOHFFHQWULFLW\UDWLRV and represents the lower limit of the practical range of stiffness values, especially for columns with larger DPRXQWVRIORQJLWXGLQDOUHLQIRUFHPHQW$&,(TXDWLRQF XVXDOO\\LHOGVPRUHDFFXUDWHYDOXHVRIWKHHIIHFWLYHÀH[XUDOVWLIIQHVVWKDQWKRVHGHWHUPLQHGE\WKHRWKHUWZRHTXDWLRQVZKHUH in this equation is the moment of inertia for columns and walls in ACI Table 6.6.3.1.1(b). 7KHIDFWRULQWKHGHQRPLQDWRURI(TXDWLRQ LVDVWLIIQHVVUHGXFWLRQIDFWRUVLPLODUWRWKHRQHGHULYHGIRUWKH UHGXFHGPRPHQWRILQHUWLDZKLFKLVHTXDOWR$GGLWLRQDOLQIRUPDWLRQRQWKHGHYHORSPHQWRIWKLVVWLIIQHVVIDFWRULVJLYHQLQ$&,5 Sway Frames 3URYLVLRQVIRUVOHQGHUFROXPQVLQVZD\IUDPHVDUHJLYHQLQ$&,&ROXPQVPXVWEHGHVLJQHGIRUFRPELQDand factored total moments and at the ends of the column, which are the tions of factored axial forces summation of the following: and due to loads causing no appreciable sidesway (like gravity 1. 8  QPDJQL¿HGIDFWRUHGPRPHQWV loads), and and due to loads causing appreciable sidesway (like wind and  0DJQL¿HGIDFWRUHGPRPHQWV HDUWKTXDNHORDGV >VHH$&,(TXDWLRQVD DQGE @ Therefore, (7.5) (7.6)

The moments DUHGHWHUPLQHGIURPD¿UVWRUGHUHODVWLFDQDO\VLVRIWKHVWUXFWXUHXVLQJUHGXFHGVHFWLRQSURSHUWLHVRIWKHPHPEHUVLQDFFRUGDQFHZLWK$&,7DEOHD >VHH7DEOH@

7-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRPHQWPDJQL¿FDWLRQIDFWRU for sway frames, is determined by either a linear elastic second-order elastic DQDO\VLVVHH6HFWLRQRIWKLVSXEOLFDWLRQ RUE\$&,(TXDWLRQD RUE >$&,@ (7.7)

(7.8)

(TXDWLRQ LVWKHVROXWLRQRIDQLQ¿QLWHVHULHVUHSUHVHQWLQJWKHLWHUDWLYH3GHOWDDQDO\VLVDQGFORVHO\SUHGLFWVWKH (TXDWLRQ RUDOLQHDUHODVWLFVHFRQGRUGHU magnitude of second-order moments in a sway frame where HODVWLFDQDO\VLVLQDFFRUGDQFHZLWK$&,PXVWEHXVHGLQFDVHVZKHUH FDOFXODWHGE\(TXDWLRQ H[FHHGV 7KHVWDELOLW\LQGH[ LVGHWHUPLQHGE\$&,(TXDWLRQ >VHH7DEOH@DQGLVEDVHGRQGHÀHFWLRQV REWDLQHGIURPD¿UVWRUGHUHODVWLFDQDO\VLVXVLQJUHGXFHGVHFWLRQSURSHUWLHVRIWKHPHPEHUV (TXDWLRQ ZDVSDUWRIWKHPRPHQWPDJQL¿HUPHWKRGWKDWDSSHDUHGLQSUHYLRXVHGLWLRQVRI$&,7KHIDFare summed over the entire story, and the critical buckling loads, are summed over the tored axial loads, used in the sway-resisting columns in that story. Like in the case of nonsway frames, the effective stiffness which is the ratio of the maximum calculation of LVGHWHUPLQHGE\$&,)RUVZD\IUDPHVWKHWHUP factored sustained story shear to the maximum factored story shear, is used in the denominator of the equations of instead of Note that LVQRUPDOO\HTXDOWR]HUREHFDXVHZLQGDQGVHLVPLFORDGVDUHQRWVXVWDLQHG loads. An example of a sustained lateral load is lateral earth pressure. 7KHPDJQL¿HGPRPHQWVDWWKHHQGVRIDFROXPQPXVWEHFRQVLGHUHGLQWKHGHVLJQRIWKHEHDPVIUDPLQJLQWRWKH FROXPQV$&, %HDPVWLIIQHVVSOD\VDNH\UROHLQWKHVWDELOLW\RIFROXPQVDQGWKLVSURYLVLRQHQVXUHVWKH EHDPVZLOOKDYHDGHTXDWHVWUHQJWKWRUHVLVWWKHPDJQL¿HGFROXPQPRPHQWV It is possible for the maximum moment in a slender column in a sway frame to occur at a section away from its HQGV,WLVSHUPLWWHGWRPDJQLI\VXFKPRPHQWVXVLQJWKHSURYLVLRQVIRUQRQVZD\IUDPHVLQ$&,ZKHUH and GHWHUPLQHGE\(TXDWLRQV DQG >$&,@ calculated using the total moments

is

7.3.4 Required Shear Strength for Columns in Buildings Assigned to Seismic Design Category B

For certain columns in buildings assigned to SDC B that are part of an ordinary moment frame resisting seismic ORDGHIIHFWVWKHVKHDUVWUHQJWKUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGLQDGGLWLRQWRDOORWKHUDSSOLFDEOH LVOHVVWKDQRUHTXDOWR¿YHWLPHVWKHSODQ requirements. In cases where the unsupported length of a column, in the direction of analysis, the design shear strength of the column must be taken as dimension of the column, the lesser of the following: associated with the development of nominal moment strengths, of the column at each • The shear force, UHVWUDLQHGHQGRIWKHXQVXSSRUWHGOHQJWKGXHWRUHYHUVHFXUYDWXUHEHQGLQJVHH)LJXUH ,QWKH¿JXUH DUHWKHQRPLQDOÀH[XUDOVWUHQJWKVDWWKHWRSDQGERWWRPRIWKHFROXPQUHVSHFWLYHO\7KHVHÀH[XUDO and consistent with the direction of analysis resulting strengths must be calculated for the factored axial force, LQWKHKLJKHVWÀH[XUDOVWUHQJWK6LGHVZD\WRWKHULJKWDQGVLGHVZD\WRWKHOHIWPXVWERWKEHFRQVLGHUHG • 7KHPD[LPXPVKHDUIRUFHREWDLQHGIURPWKHGHVLJQORDGFRPELQDWLRQVRI$&,&KDSWHULQFOXGLQJWKHHDUWKwith substituted for in the load combinations. The term is the overstrength factor quake effect, IRURUGLQDU\PRPHQWIUDPHVRIUHLQIRUFHGFRQFUHWHZKLFKLVHTXDOWRVHH$6&(6(,7DEOH  The provisions for the design shear strength are intended to provide additional capacity to resist shear in relatively squat columns, which are vulnerable to shear failure under earthquake loading.

7-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ܯ‬௡௧  ܿଵ 



ܸ௨ 

ܸ௨ ൌ

‫ܯ‬௡௧ ൅ ‫ܯ‬௡௕  κ௨

κ௨ ൑ ͷܿଵ 

ܸ௨  ‫ܯ‬௡௕  Figure 7.8 Design shear strength of reinforced concrete columns in buildings assigned to SDC B.

7.4 Design Strength 7.4.1 General

)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\ VHFWLRQLQDFROXPQ$&,  (7.9) (7.10) (7.11) (7.12)

Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ IDFWRUVSHUWLQHQWWRWKHGHVLJQRIUHLQIRUFHGFRQFUHWHFROXPQVLVJLYHQLQ7DEOH7KHVWUDLQXVHGWRGH¿QHD LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHUHLQIRUFHPHQW divided by the compression-controlled section, ZKLFKFDQEHWDNHQDVSVLIRUDQ\JUDGHRIUHLQIRUFHmodulus of elasticity of the reinforcing steel, PHQW$&,  Table 7.10 Strength Reduction Factors, Ҁ Strength Reduction Factor, Ҁ Net Tensile Strain, İt

Classification

Compression-controlled

Type of Transverse Reinforcement Spirals Conforming to ACI 25.7.3

Other









Transition* Tension-controlled *Sections classified as transition are permitted to use

7-14

corresponding to compression-controlled sections.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUFROXPQVVXEMHFWHGSULPDULO\WRXQLD[LDOFRPSUHVVLRQIRUFHV(TXDWLRQ LVDSSOLFDEOH)RUFROXPQVVXEMHFWHG WRFRPELQHGPRPHQWDQGD[LDOIRUFHV(TXDWLRQV DQG PXVWERWKEHVDWLV¿HGIRUDOODSSOLFDEOHIDFWRUHG ORDGFRPELQDWLRQV7KHVKHDUDQGWRUVLRQDOVWUHQJWKHTXDWLRQV>(TXDWLRQV DQG UHVSHFWLYHO\@PXVWEH VDWLV¿HGIRUDQ\FROXPQ combined Methods to determine the nominal strengths DQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\

and

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7.4.2 Nominal Axial Strength Nominal Axial Compressive Strength

The nominal axial compressive strength, of a nonslender, reinforced concrete column is determined in accorGDQFHZLWK$&,)RUQRQSUHVWUHVVHGFROXPQVZLWKWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRI WLHVFRQIRUPLQJWR$&,DQG VSLUDOVFRQIRUPLQJWR$&,VHH)LJXUH  must be less than or equal to which is determined using the equations in ACI Table the maximum nominal axial compressive strength, DQGWKHQRPLQDOD[LDOVWUHQJWKDW]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ 

(a) tied column

(b) spiral column

Figure 7.9 Reinforced concrete columns. (a) With tied transverse reinforcement. (b) With spiral transverse reinforcement.

• For columns with ties: (7.13)

• For columns with spirals: (7.14)

These equations account for any accidental eccentricities that may exist in a column that were not accounted for in DQDO\VLV,QWKHFDVHRIWLHGFROXPQVWKHHFFHQWULFLW\LVDSSUR[LPDWHO\SHUFHQWRIWKHFROXPQGHSWKZKLOHIRU VSLUDOFROXPQVWKHHFFHQWULFLW\LVDSSUR[LPDWHO\SHUFHQW WKHYDOXHRIWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHORQJLWXGLQDOUHLQIRUFHPHQW is limited When calculating WRSVLHYHQWKRXJK for the longitudinal reinforcement used in the column may be larger than that (ACI  

7-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Nominal Axial Tensile Strength

The nominal axial tensile strength of a nonprestressed, reinforced concrete column, must be less than or equal ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ  to the maximum nominal axial tensile strength, (7.15)

It is evident that the entire tensile force must be resisted by the longitudinal reinforcement in a column. 7.4.3 Nominal Strength of Columns Subjected to Moment and Axial Forces Overview

The nominal strength of a reinforced concrete column subjected to both moment and axial force is determined using HTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ$&,ZKLFKDUHVXPPDUL]HGLQ7DEOH Table 7.11 Design Assumptions for Concrete and Nonprestressed Reinforcement Material

Assumptions

1. 0D[LPXPVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHULVDVVXPHGHTXDOWR  7HQVLOHVWUHQJWKRIFRQFUHWHLVQHJOHFWHGLQÀH[XUDODQGD[LDOVWUHQJWKFDOFXODWLRQV 3. The relationship between concrete compressive stress and strain is to be represented by DUHFWDQJXODUWUDSH]RLGDOSDUDEROLFRURWKHUVKDSHWKDWUHVXOWVLQSUHGLFWLRQRIVWUHQJWK in substantial agreement with results of comprehensive tests.  7  KHHTXLYDOHQWUHFWDQJXODUFRQFUHWHVWUHVVGLVWULEXWLRQLQDFFRUGDQFHZLWK$&, WKURXJKVDWLV¿HVWKHUHTXLUHPHQWRI$&,IRUWKHUHODWLRQVKLSEHWZHHQ concrete compressive stress and strain. Concrete

is to be assumed uniformly distributed over an equivalent  A concrete stress of FRPSUHVVLRQ]RQHERXQGHGE\WKHHGJHVRIWKHFURVVVHFWLRQDQGDOLQHSDUDOOHOWRWKH neutral axis located a distance IURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQZKHUH 7KHGLVWDQFHIURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQWRWKHQHXWUDOD[LV is to be measured perpendicular to the neutral axis. Values of

DUHDVIROORZV$&,7DEOH 

• • •

Nonprestressed reinforcement

1. Deformed reinforcement used to resist tensile or compressive forces must conform to $&,  The stress-strain relationship and the modulus of elasticity of the deformed reinforcement DUHWREHLGHDOL]HGLQDFFRUGDQFHZLWK$&,DQG

Rectangular Sections

7KHJHQHUDOSULQFLSOHVDQGDVVXPSWLRQVRIWKHVWUHQJWKGHVLJQPHWKRGLQ7DEOHDUHDSSOLHGWRWKHUHFWDQJXODU UHLQIRUFHGFRQFUHWHFROXPQLQ)LJXUH)RUSXUSRVHVRIGLVFXVVLRQLWLVDVVXPHGWKHORQJLWXGLQDOUHLQIRUFHPHQW LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ

7-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͲǤͲͲ͵ ߝ௦ଵ 

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݀ଶ 

ܿ

‫ܣ‬௦ଵ 

‫ܣ‬௦ଶ 

‫ܣ‬௦ଷ 

ߝ௦ଶ 

ߝ௦ଷ ൌ ߝ௧ 

ܽ ൌ ߚଵ ܿ

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‫ܨ‬௦ଷ 

ܾ Figure 7.10 Rectangular reinforced concrete column subjected to moment and axial compression.

The nominal axial strength, DQGWKHQRPLQDOÀH[XUDOVWUHQJWK RIWKHFROXPQLQ)LJXUHFDQEHREXVLQJWKHHTXDWLRQVLQ)LJXUHZKHUH is the net tensile strain in the extreme layer tained for a given strain, of tension reinforcement at nominal strength, excluding strains due to creep, shrinkage, and temperature. Circular Sections

7KHJHQHUDOSULQFLSOHVDQGDVVXPSWLRQVRIWKHVWUHQJWKGHVLJQPHWKRGLQ7DEOHDUHDSSOLHGWRWKHFLUFXODU UHLQIRUFHGFRQFUHWHFROXPQLQ)LJXUH)RUSXUSRVHVRIGLVFXVVLRQLWLVDVVXPHGWKHORQJLWXGLQDOUHLQIRUFHPHQW LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ)RUFLUFXODUVHFWLRQVWKHVKDSHRIWKH FRPSUHVVLRQ]RQHLVUHODWHGWRDVHJPHQWRIDFLUFOHVHH)LJXUH  DQGWKHQRPLQDOÀH[XUDOVWUHQJWK The nominal axial strength, XVLQJWKHHTXDWLRQVLQ)LJXUH tained for a given strain,

RIWKHFROXPQLQ)LJXUHFDQEHRE-

Interaction Diagrams

An interaction diagram for a reinforced concrete column is a collection of and values determined for a series of strain distributions using the methods above for rectangular and circular sections. Nominal strengths for a given strain distribution represent a single point on the interaction diagram. Such diagrams are useful in establishing the adequacy of a column subjected to a combination of axial forces and bending moments. Nominal and design strength interaction diagrams for a tied, rectangular reinforced concrete column with a symmetULFDOGLVWULEXWLRQRIORQJLWXGLQDOUHLQIRUFHPHQWDUHJLYHQLQ)LJXUH7KHSRUWLRQVRIWKHGLDJUDPVFRUUHVSRQGLQJ WRFRPSUHVVLYHFRQWUROOHGVHFWLRQVWHQVLRQFRQWUROOHGVHFWLRQVDQGVHFWLRQVLQWKHWUDQVLWLRQ]RQHDUHLGHQWL¿HGLQ WKH¿JXUH The design strength for combined moment and axial forces must be greater than or equal to the corresponding UHTXLUHGVWUHQJWKVWRVDWLVI\VWUHQJWKUHTXLUHPHQWV>VHH(TXDWLRQV DQG @'HVLJQVWUHQJWKYDOXHVDUHREWDLQHGE\PXOWLSO\LQJWKHQRPLQDOVWUHQJWKYDOXHVZLWKWKHVWUHQJWKUHGXFWLRQIDFWRUVLQ$&,VHH6HFWLRQ of this publication). Factored axial force-bending moment combinations falling on or within the boundaries of the design strength interaction diagram can be safely carried by the column. This column is adequate for the combination denoted by Point 1 LQ)LJXUHEXWLWLVQRWDGHTXDWHIRUWKHFRPELQDWLRQGHQRWHGE\3RLQW Interaction diagrams about the major axis and the minor axis of a rectangular column can be different; this must be taken into consideration when designing the column. For circular reinforced concrete columns with 6 longitudinal EDUVRUOHVVWKHGHVLJQÀH[XUDOVWUHQJWKGHSHQGVRQWKHEDUDUUDQJHPHQWDQGWKHFROXPQPXVWEHFKHFNHGXVLQJWKH 7-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿൌ

ͲǤͲͲ͵݀‫ݐ‬  ߝ‫ ݐ‬൅ ͲǤͲͲ͵

 ͲǤ͸ͷ ൑ ߚͳ ൌ ͲǤͺͷ െ

ͲǤͲͷሺ݂ܿԢ െ ͶǡͲͲͲሻ ൑ ͲǤͺͷ ͳǡͲͲͲ

ܽ ൌ ߚͳ ܿ ൑ ݄

‫ ܥ‬ൌ ͲǤͺͷ݂ܿԢ ܾܽ

݅ ൌ ͳ



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Figure 7.11 Combined flexural and axial strength of a rectangular reinforced concrete column.

DUUDQJHPHQWWKDWJLYHVPLQLPXPGHVLJQÀH[XUDOVWUHQJWK8WLOL]LQJRUPRUHORQJLWXGLQDOEDUVHVVHQWLDOO\HOLPLQDWHVWKHQHHGWRGHWHUPLQHGLIIHUHQWGHVLJQÀH[XUDOVWUHQJWKVEDVHGRQEDUDUUDQJHPHQW 5HIHUHQFHFRQWDLQVGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHIROORZLQJ • • • • •

7LHGUHFWDQJXODUFROXPQVUDQJLQJLQVL]HIURPWRLQLQFOXVLYH 7LHGDQGVSLUDOFLUFXODUFROXPQVUDQJLQJLQGLDPHWHUIURPWRLQLQFOXVLYH *UDGHDQG*UDGHORQJLWXGLQDOUHLQIRUFHPHQW &RQFUHWHFRPSUHVVLYHVWUHQJWKVIURPSVLWRSVLLQFOXVLYHDQG /RQJLWXGLQDOUHLQIRUFHPHQWUDWLRVIURPSHUFHQWWROHVVWKDQSHUFHQW

7-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



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EŽ

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݂‫ ݅ݏ‬ൌ ‫ ݅ݏߝ ݏܧ‬

•݀݅ ൐ ܽǫ

EŽ

‫ ݅ݏܨ‬ൌ ሺ݂‫ ݅ݏ‬െ ͲǤͺͷ݂ܿԢ ሻ‫ ݅ݏܣ‬

zĞƐ

‫ ݅ݏܨ‬ൌ ݂‫ ݅ݏܣ ݅ݏ‬

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zĞƐ

݊

ܲ݊ ൌ ‫ ܥ‬൅ ෍ ‫ ݅ݏܨ‬ 

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‫ ݊ܯ‬ൌ ͲǤͷ‫ܥ‬ሺ݄ െ ܽሻ ൅ ෍ ‫ ݅ݏܨ‬ሺͲǤͷ݄ െ ݀݅ ሻ ݅ൌͳ

Figure 7.11 (cont.) Combined flexural and axial strength of a rectangular reinforced concrete column.

7-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿ

‫ܣ‬௦ଷ 

ߝ௦ଶ 

݀ହ 

݄

݀ସ 

݀ଷ 

‫ܣ‬௦ଶ 

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ͲǤͲͲ͵ ߝ௦ଵ 

ܽ ൌ ߚଵ ܿ

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ͲǤͺͷ݂௖ᇱ 

‫ܨ‬௦ଷ 

ߝ௦ଷ  ߝ௦ସ 

‫ܣ‬௦ସ  ‫ܣ‬௦ହ 

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ߝ௦ହ ൌ ߝ௧ 

Figure 7.12 Circular reinforced concrete column subjected to moment and axial compression.

‫ݕ‬ത

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ܽ

‫ ܣ‬ൌ ݄ଶ ሺߠ െ •‹ ߠሻΤͺ

ߠȀʹ ߠȀʹ

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ߠ ൌ ʹ …‘•ିଵ ൬ͳ െ

ʹܽ ൰ ݄

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Figure 7.13 Section properties of the compression zone for a circular reinforced concrete column.

Biaxial Loading

Overview Biaxial bending of columns occurs where loading causes bending simultaneously about both principal axes. This is often encountered in corner columns. The strength of an axially loaded column subjected to bending moments DERXWERWKSULQFLSDOD[HVFDQEHUHSUHVHQWHGE\DELD[LDOVWUHQJWKLQWHUDFWLRQVXUIDFHVHH)LJXUH 7KLVVXUIDFH is formed by a series of uniaxial interaction diagrams drawn radially from the vertical axis. Intermediate interaction diagrams between the angle HTXDOWR]HURGHJUHHVXQLD[LDOEHQGLQJDERXWWKHxD[LV DQGGHJUHHVXQLD[LDO bending about the yD[LV DUHREWDLQHGE\YDU\LQJWKHDQJOHRIWKHQHXWUDOD[LVIRUDVVXPHGVWUDLQFRQ¿JXUDWLRQV A biaxial strength interaction surface is constructed by performing a series of involved and time-consuming strain compatibility analyses. For example, the strains and forces for a rectangular column subjected to an axial compresand biaxial bending DUHJLYHQLQ)LJXUH'HVLJQLQJDFROXPQIRU sion force WKLVORDGLQJFRQGLWLRQFDQEHDYHU\OHQJWK\SURFHVVZLWKRXWWKHXVHRIDFRPSXWHUSURJUDP+RZHYHUFRQVHUYDWLYH results for columns subjected to biaxial bending can be obtained using some approximate methods.

7-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿൌ

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 ͲǤ͸ͷ ൑ ߚͳ ൌ ͲǤͺͷ െ

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ܽ ൌ ߚͳ ܿ ൑ ݄

ߠ ൌ ʹ …‘• െͳ ൬ͳ െ

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ʹ݄•‹͵ ሺߠΤʹሻ  ͵ሺߠ െ •‹ ߠሻ

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݅ ൌ ͳ



ߝ‫ ݅ݏ‬ൌ

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Figure 7.14 Combined flexural and axial strength of a circular reinforced concrete column.

7-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



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‫ ݊ܯ‬ൌ ‫ݕܥ‬ത ൅ ෍ ‫ ݅ݏܨ‬ሺͲǤͷ݄ െ ݀݅ ሻ ݅ൌͳ

Figure 7.14 (cont.) Combined flexural and axial strength of a circular reinforced concrete column.

7-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܲ௢ 

‫ܔ܉ܑܠۯ‬۴‫܍܋ܚܗ‬

ܲ௡ǡ௠௔௫ ൌ ͲǤͺͲܲ௢ 

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‘’”‡••‹‘Ǧ …‘–”‘ŽŽ‡†

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۰‫ܜܖ܍ܕܗۻ܏ܖܑ܌ܖ܍‬ Figure 7.15 Nominal and design strength interaction diagrams.

ܲ ܲ௢ 

Žƒ‡‘ˆ…‘•–ƒ–ܲ௡ 

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Figure 7.16 Biaxial strength interaction surface.

$SSUR[LPDWH0HWKRGIRU&ROXPQVZLWK(TXDO%HQGLQJ0RPHQW&DSDFLWLHV$ERXW%RWK3ULQFLSDO$[HV For circular columns or for square columns with the same longitudinal reinforcement on each face, the uniaxial is the same as that about the y-axis, $KRUL]RQWDOVOLFHWKURXJKWKH bending capacity about the x-axis, LVVKRZQLQ)LJXUH biaxial interaction surface of a square column at an axial compression force where the dash-dot line represents the nominal moment strengths and the solid line represents the design moment

7-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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‫ܯ‬௨௫ ൌ ܲ௨ ݁௬  ‫ܯ‬௨௬ ൌ ܲ௨ ݁௫  Figure 7.17 Rectangular column subjected to biaxial bending.

‫ݕ‬ ‫ܯ‬௡௢௬ 

‫ܯ‬௡ 

߶‫ܯ‬௡௢௬  ߶‫ܯ‬௡ 

Approx. ߶‫ܯ‬௡ ൅ 1 ൫‫ܯ‬௨௫ , ‫ܯ‬௨௬ ൯

߶‫ܯ‬௡௢௫  ‫ݔ‬ ‫ܯ‬௡௢௫  ߶‫ܯ‬௢௡௫ ൌ ߶‫ܯ‬௡௢௬ 

Figure 7.18 Biaxial capacity approximation for circular columns and square columns with the same longitudinal reinforcement on each face.

strengths. The moments and computer software or design aid.

can be determined using the methods described above or by any other

This line 7KHVWUDLJKWGDVKHGOLQHLQ)LJXUHFRQQHFWVWKHXQLD[LDOGHVLJQPRPHQWVWUHQJWKV always lies inside the actual biaxial capacity curve, so it can be used as a conservative (but not overly conservative) representation of the actual capacity diagram. Therefore, a circular column or a square column with the same longiis adequate for any combinatudinal reinforcement on each face subjected to a factored axial compression force and that lies within the approximate biaxial capacity surface (an example of an adequate bending tion of PRPHQWFRPELQDWLRQLVUHSUHVHQWHGE\SRLQWLQ)LJXUH 

7-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The approximate capacity surface can be represented by the following equation: (7.16)

(TXDWLRQ FDQEHUHZULWWHQDVIROORZV

Because

(7.17)

Therefore, for a given

a column is adequate where

Reciprocal Load Method The Reciprocal Load Method provides a simple and conservative estimate of the strength of a column under biaxial corresponding to eccentricities about both prinORDGLQJFRQGLWLRQV5HIHUHQFH 7KHQRPLQDOD[LDOVWUHQJWK cipal axes of a column is calculated by the following equation: (7.18)

where

nominal axial strength when the column is subjected to the uniaxial moment nominal axial strength when the column is subjected to the uniaxial moment  QRPLQDOD[LDOVWUHQJWKDW]HURHFFHQWULFLW\

Determination of the items needed to calculate is relatively straightforward. The nominal axial strengths can be obtained from uniaxial nominal strength strain compatibility analyses, and is determined by ACI and (TXDWLRQ  is obtained by multiplying GHWHUPLQHGE\(TXDWLRQ ZLWKWKHDSSURSULThe design axial strength ate strength reduction factor based on the strain in the reinforcing bars farthest from the compression face of the is greater than or equal to the factored axial column. The column is adequate where the design axial strength force 7KH5HFLSURFDO/RDG0HWKRGSURGXFHVUHDVRQDEO\DFFXUDWHUHVXOWVZKHQÀH[XUHGRHVQRWJRYHUQWKHGHVLJQRID and are greater than the axial strength corresponding to balanced failure. In cases column, that is, when ZKHUHÀH[XUHJRYHUQVDQRWKHUPHWKRGOLNHWKH/RDG&RQWRXU0HWKRGGLVFXVVHGEHORZVKRXOGEHXWLOL]HG Load Contour Method )RUFROXPQVZLWKXQHTXDOEHQGLQJPRPHQWFDSDFLWLHVDERXWWKHSULQFLSDOD[HVDQGZKHUHÀH[XUHJRYHUQVWKHGHVLJQ (that is, where axial forces are relatively small), the Load Contour Method can be used to determine biaxial strength 5HIHUHQFH 7KHORDGFRQWRXUWKDWLVWKHKRUL]RQWDOVOLFHWKURXJKWKHELD[LDOLQWHUDFWLRQVXUIDFH FRUUHVSRQGLQJWR can be approximated by the following nondimensional interaction equation: a nominal axial compression force, (7.19)

In this equation, the column, and ly. The exponents

and are the nominal biaxial moments that occur simultaneously on and are the nominal uniaxial moment strengths about the x-axis and y-axis, respectiveand are a function of the amount, distribution, and location of the longitudinal reinforcement

7-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

in the column; the dimensions of the column; and the strength and elastic properties of the concrete and reinforcing Interaction curves for variVWHHO,WLVGHPRQVWUDWHGLQ5HIHUHQFHWKDWLWLVUHDVRQDEO\DFFXUDWHWRDVVXPH DUHJLYHQLQ)LJXUH,QFOXGHGLVWKHVWUDLJKWOLQHFRUUHVSRQGLQJWR which results in the ous following interaction equation: (7.20)

This equation always yields conservative results because it underestimates the capacity of the column. The coland the applied bending moments, and umn is adequate for the applied axial force, that lie within the approximate biaxial capacity surface, that is, where the following equation is VDWLV¿HG (7.21)

The PCA Load Contour Method is an extension of the Load Contour Method and information on this method can be found in Reference 18. Slenderness effects must also be considered in the design of columns subjected to biaxial loading, where applicable. A review of several methods to evaluate slenderness effects in columns subjected to axial compression forces and biaxial bending is given in Reference 19.  ϭ͘Ϭ

ࡹ࢔࢟ Ȁࡹ࢔࢕࢟ 

Ϭ͘ϴ

Ϭ͘ϲ

Ϭ͘ϰ

Ϭ͘Ϯ

Ϭ͘Ϭ Ϭ͘Ϭ

Ϭ͘Ϯ

Ϭ͘ϰ

Ϭ͘ϲ

Ϭ͘ϴ

ϭ͘Ϭ

ࡹ࢔࢞ Τࡹ࢔࢕࢞  Figure 7.19 Interaction curves for the Load Contour Method.

7-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.4.4 Nominal Shear Strength Overview

The nominal one-way shear strength,  

DWDVHFWLRQLQDFROXPQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&, (7.22)

where is the nominal shear strength provided by concrete and reinforcement.

is the nominal shear strength provided by shear

Nominal Shear Strength Provided by Concrete

For nonprestressed members, LVGHWHUPLQHGE\WKHDSSOLFDEOHHTXDWLRQLQ$&,7DEOH$&, 7KH HTXDWLRQVLQWKDWWDEOHDUHJLYHQLQ7DEOHDORQJZLWKWKHPD[LPXPSHUPLWWHGQRPLQDOVKHDUVWUHQJWK VSHFL¿HGLQ$&, Table 7.12 Nominal Shear Strength Provided by Concrete, Vc Area of Shear Reinforcement, Av*

*Minimum area of shear reinforcement, ** (ACI 22.5.5.1.2)

Vc**

, is determined by ACI 10.6.2.2

The minimum area of shear reinforcement in a column, (TXDWLRQ EHORZ@

LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,>VHH

7KHPRGL¿FDWLRQIDFWRU UHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ7DEOH EDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[VHH$&,  Table 7.13 Values of

Based on Equilibrium Density, wc

Equilibrium Density, wc

Ȝ 



7-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.14 Values of

Based on Composition of Aggregates

Concrete

All-lightweight

/LJKWZHLJKW¿QHEOHQG

Sand-lightweight

Sand-lightweight, coarse blend

Composition of Aggregates

)LQH$670& &RDUVH$670& )LQH&RPELQDWLRQRI$670&DQG& &RDUVH$670& Fine: ASTM C33 &RDUVH$670& Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670& and ASTM C33

Ȝ  WR(1)  WR

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength attributed to concrete in members without shear reinforcement does not increase in direct proportion with member GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (7.23)

,WLVHYLGHQWIURP(TXDWLRQ WKDW

LVOHVVWKDQIRUPHPEHUVZLWK

The term LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by where is the width of a rectangular column perpendicular to the direction of analysis or the diameter of a circular column [ACI E @7KHHIIHFWLYHGHSWK can be obtained from a strain compatibility analysis for each applicable load combination; in the case of circular columns, it is permitted to take DVSHUFHQWRIWKHGLDPHWHURIWKHFROXPQ >$&,D @$FFRUGLQJWR$&,5 may be taken as the sum of the areas of the longitudinal reinIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU $FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken , and is to be taken as negative for tenas positive for compression forces acting on the gross area of the column, sion forces. used to calculate DUHOLPLWHGWRSVL$&,WKHH[FHSWLRQLQ$&,LVQRWDSValues of is primarily due to the fact that there is a lack of test data and practical plicable to columns). This limitation on H[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL In rare occasions where axial tension acts on a column,

FDQEHWDNHQDV]HUR

7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@ (7.24)

7-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Nominal Shear Strength Provided by Shear Reinforcement

7KHQRPLQDOVKHDUVWUHQJWKRIWKHWUDQVYHUVHUHLQIRUFHPHQWLVGHWHUPLQHGE\$&,(TXDWLRQ  (7.25)

where s is the center-to-center spacing of ties or the spiral pitch. For columns with rectilinear ties and crossties, is equal to the area of all the ties and crossties within the spacing LQWKHGLUHFWLRQRIDQDO\VLV$&, ,Q is equal to two times the area of the tie or spiral within the spacing (ACI the case of circular ties or spirals,   A minimum area of shear reinforcement,

must be provided at sections where

$&, 

(7.26)

)RUFROXPQVVXEMHFWHGWRUHODWLYHO\VPDOOIDFWRUHGVKHDUIRUFHVWKHFRQFUHWHDORQHPD\EHVXI¿FLHQWWRVDWLVI\VKHDU is typically calculated using strength requirements. When more than just the concrete shear strength is needed, WKHUHTXLUHPHQWVLQ$&,DQGIRUWUDQVYHUVHEDUVL]HDQGVSDFLQJ,IVKHDUVWUHQJWKUHTXLUHPHQWV DUHQRWVDWLV¿HGEDVHGRQWKRVHUHTXLUHPHQWVWKHDPRXQWDQGRUVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWPXVWEH adjusted appropriately, or the compressive strength of the concrete can be increased, although the latter is generally QRWWKHPRVWHIIHFWLYHZD\WRLQFUHDVHVKHDUVWUHQJWK,QFDVHVZKHUHWKHVKHDUGHPDQGLVYHU\ODUJHWKHVL]HRIWKH column may need to be increased. Biaxial Shear Strength

Reinforced concrete columns may be subjected to the effects from biaxial shear forces (this commonly occurs in corner columns). For symmetrically reinforced circular columns, the nominal one-way shear strength is the same along any axis of the member, and shear strength can be evaluated based on the requirements outlined above using the resultant of the shear forces acting on the section. For rectangular sections, it is not practical to determine the nominal shear strength based on the direction of the UHVXOWDQWVKHDUIRUFH$VDQDOWHUQDWLYHWKHELD[LDOVKHDUIRUFHUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG and are calculated in the orthogonal x and y directions of Required shear stresses and are the factored shear forces in the x and y directions, respectively, is the crossthe section where sectional dimension of the column perpendicular to the direction of analysis, and is the distance from the extreme FRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWLQWKHGLUHFWLRQRIDQDO\VLVVHH)LJXUH   and in the x and y directions are determined by dividing the nominal shear strength, Nominal shear stresses FDOFXODWHGLQDFFRUGDQFHZLWK(TXDWLRQ E\WKHDSSOLFDEOH and in each direction. $FFRUGLQJWR$&,LQWHUDFWLRQRIRUWKRJRQDOVKHDUIRUFHVQHHGQRWEHFRQVLGHUHGZKHUHRQHRUERWK RI WKHIROORZLQJHTXDWLRQVLVVDWLV¿HG (7.27)

7-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଶ 

݀௬ 

ܿଵ 

ܸ௨ǡ௫  ‫ݒ‬௨ǡ௫ ൌ

ܸ௨ǡ௫  ܿଶ ݀௫

݀௫ 

ܸ௨ǡ௬ 

‫ݒ‬௨ǡ௬ ൌ

ܸ௨ǡ௬  ܿଵ ݀௬

Figure 7.20 Determination of biaxial shear stresses on a rectangular reinforced concrete column.

(7.28)

In cases where and WLRQ PXVWEHVDWLV¿HG

, then interaction effects must be considered and ACI Equa-

(7.29)

:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHFURVVVHFWLRQDOGLPHQVLRQVRIWKHFROXPQWKHDPRXQWRIVKHDUUHLQIRUFHment, or both must be increased accordingly. The compressive strength of the concrete can also be increased but doing so does has a smaller impact than increasing the other items. 7.4.5 Nominal Torsional Strength

Columns in building structures are rarely subjected to torsional moments. In cases where such effects must be conVLGHUHGWKHSURYLVLRQVLQ$&,&KDSWHUPXVWEHVDWLV¿HG$&, 7KHVHUHTXLUHPHQWVDUHFRYHUHGLQ6HFWLRQV DQGRIWKLVSXEOLFDWLRQIRUEHDPV

7.5 Reinforcement Limits 7.5.1 Longitudinal Reinforcement

For columns supporting only gravity loads or that are part of an ordinary or intermediate moment frame, the miniDUHJLYHQLQ$&,ZKLFKPXVWEHVDWLV¿HG mum and maximum areas of longitudinal reinforcement, regardless of the type of transverse reinforcement used in the column: • • where

7-30

is the gross cross-sectional area of the column.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The 1 percent lower limit is meant to provide resistance to any bending moments not accounted for in the analysis, such as those caused by construction tolerances or misalignments. It is also meant to help reduce creep and shrinkage LQWKHFRQFUHWHXQGHUVXVWDLQHGFRPSUHVVLRQVWUHVVHV7KHXSSHUOLPLWVKHOSPLQLPL]HFRQJHVWLRQDQGWKHGHYHORSLVJHQHUDOO\PRUHHI¿FLHQW PHQWRIKLJKHUVKHDUVWUHVVHVDOWKRXJKDWRSHUFHQWUHLQIRUFHPHQWUDWLR DQGFRVWHIIHFWLYHVHH5HIHUHQFH $WORFDWLRQVZKHUHODSVSOLFHVDUHXVHGWKHORQJLWXGLQDOUHLQIRUFHPHQWUDWLRPD\ EHHTXDOWRWZRWLPHVWKDWDWVHFWLRQVDZD\IURPWKHODSVSOLFHDQGWKDWUDWLRPXVWEHOHVVWKDQRUHTXDOWR $VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQDFROXPQLVSHUPLWWHGWREHGHVLJQHGRIVXI¿FLHQWVL]HWRFDUU\WKH required factored loads and additional concrete is permitted to be added to the section without having to increase WKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWWRVDWLVI\WKHSHUFHQWOLPLWLQ$&,$&, ,WLVDVVXPHG WKHDGGLWLRQDOFRQFUHWHGRHVQRWFDUU\DQ\ORDGKRZHYHUWKHDFWXDOFROXPQVL]HPXVWEHLQFOXGHGLQWKHVWUXFWXUDO analysis to correctly account for its actual stiffness. The columns in the upper stories of a building where the factored loads are usually relatively small can typically be designed using a reinforcement ratio less than 1 percent based on this provision. 7.5.2 Shear Reinforcement

0LQLPXPVKHDUUHLQIRUFHPHQWUHTXLUHPHQWVIRUFROXPQVDUHJLYHQLQ$&,DQG(TXDWLRQ )RUFROXPQV is equal to the area of all the ties and crossties within the spacing in the with rectilinear ties and crossties, is equal to two times the area of the GLUHFWLRQRIDQDO\VLV$&, ,QWKHFDVHRIFLUFXODUWLHVRUVSLUDOV tie or spiral within the spacing $&, 

7.6 Sizing the Cross-Section 7.6.1 Axial Compression

For columns subjected primarily to axial compression forces, the cross-sectional dimension of a column can be equal to the design axial load strength, obtained by setting the total factored axial compression force, JLYHQLQ$&,7DEOHIRUFROXPQVZLWKWLHVFRQIRUPLQJWR$&,DQGZLWKVSLUDOVFRQIRUPLQJWR$&, >VHH(TXDWLRQV DQG LQWKLVSXEOLFDWLRQ@ • For tied columns: (7.30)

• For spiral columns: (7.31)

where $SUHOLPLQDU\FROXPQVL]HFDQEHREWDLQHGIURPWKHVHHTXDWLRQVIRUDJLYHQ assuming a value of QRWHGSUHYLRXVO\VKRXOGEHLQWKHUDQJHRIWRSHUFHQWWRDFKLHYHRYHUDOOHFRQRP\

which as

7KHFKDUWVLQ$SSHQGL[%RI5HIHUHQFHFDQEHXVHGWRTXLFNO\GHWHUPLQHDSUHOLPLQDU\VL]HRIDQRQVOHQGHUWLHG with the following: square column for a given factored axial compression force • *UDGHRU*UDGHORQJLWXGLQDOUHLQIRUFHPHQW • &RQFUHWHFRPSUHVVLYHVWUHQJWKVIURPSVLWRSVLDQG • /RQJLWXGLQDOUHLQIRUFHPHQWUDWLRVIURPWRSHUFHQW

7-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The required gross column area, LVDOVRSURYLGHGLQWKHFKDUWVZKLFKFDQEHXVHGWRREWDLQSUHOLPLQDU\VL]HVIRU UHFWDQJXODURUFLUFXODUWLHGFROXPQV$PRGL¿FDWLRQIDFWRULVJLYHQLQWKHDSSHQGL[WRDFTXLUHSUHOLPLQDU\VL]HVIRU spiral columns. 7KHXVHRIKLJKVWUHQJWKORQJLWXGLQDOUHLQIRUFHPHQWFDQKDYHDQLPSDFWRQWKHUHTXLUHGVL]HRIDFROXPQDOWKRXJK WKHLPSDFWLVJHQHUDOO\QRWVLJQL¿FDQWEHFDXVHWKHFRQFUHWHLVUHVLVWLQJPRVWRIWKHFRPSUHVVLRQIRUFH+RZHYHU when high-strength longitudinal reinforcement is combined with high-strength concrete, less congestion in the structure and more usable space in the building may be possible. 7.6.2 Combined Moment and Axial Force

Columns that are part of the LFRS in a moment frame resisting the effects from lateral forces are typically subjected to combined moment and axial forces. Because of the inherent complexity of this situation, a preliminary column VL]HLVRIWHQGHWHUPLQHGEDVHGRQD[LDOFRPSUHVVLRQIRUFHVIURPJUDYLW\ORDGVRQO\DQGWKHQDGMXVWHGODWHULIUHquired, to account for the effects due to the bending moments. $VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ$SSHQGL[&LQ5HIHUHQFHFDQEHXVHGWRIDFLOLWDWHVHOHFWLRQRID SUHOLPLQDU\FROXPQVL]HIRUFROXPQVVXEMHFWHGWRFRPELQHGPRPHQWDQGD[LDOIRUFHV 7.6.3 Slenderness Effects

Minimum column dimensions so that slenderness effects need not be considered as a function of the unsupported DUHJLYHQLQ7DEOHRIWKLVSXEOLFDWLRQ column length,

7.7 Determination of Required Reinforcement 7.7.1 Required Longitudinal Reinforcement Axial Compression

For nonslender reinforced concrete columns subjected to primarily uniaxial compression forces, the required area FDQEHREWDLQHGE\XVLQJWKHDSSURSULDWHHTXDWLRQVLQ$&,7DEOHIRU of longitudinal reinforcement, and are known, the following equation can be used to determine the tied and spiral columns. Where longitudinal reinforcement ratio

(7.32)

In this equation, the strength reduction, LVHTXDOWRIRUWLHGFROXPQVDQGIRUVSLUDOFROXPQVDQGWKH term LVHTXDOWRIRUWLHGFROXPQVDQGIRUVSLUDOFROXPQV$VQRWHGSUHYLRXVO\RYHUDOOHFRQRP\LVW\SLFDOO\DFKLHYHGE\XVLQJDORQJLWXGLQDOUHLQIRUFHPHQWUDWLREHWZHHQDSSUR[LPDWHO\DQG The charts in Appendix B of Reference 16 can be used to determine

for nonslender, tied and spiral columns.

Combined Moment and Axial Force

The required longitudinal reinforcement can be acquired from an interaction diagram for columns subjected to combined moment and axial force. The design strength interaction diagrams in Appendix C of Reference 16 can be used to quickly obtain the longituGLQDOUHLQIRUFHPHQWIRUDZLGHUDQJHRIUHFWDQJXODUDQGFLUFXODUFROXPQVL]HVORQJLWXGLQDOEDUFRQ¿JXUDWLRQVDQG material properties. 7-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.7.2 Required Shear Reinforcement

For a given the spacing of the transverse reinforcement in a column, HTXDWLRQZKLFKLVEDVHGRQ(TXDWLRQ 

can be determined by the following

(7.33)

0D[LPXPVSDFLQJRIVKHDUUHLQIRUFHPHQWLQFROXPQVLVJLYHQLQ$&,DQGLVVXPPDUL]HGLQ7DEOH Table 7.15 Maximum Spacing of Shear Reinforcement in Columns

Vs (lb)

For a given

Maximum Spacing, s (in.)

can be determined from the following equation: (7.34)

For columns with rectilinear ties and crossties, is equal to the area of all the ties and crossties within the spacing LQWKHGLUHFWLRQRIDQDO\VLV$&, ,QWKHFDVHRIFLUFXODUWLHVRUVSLUDOV is equal to two times the area of the tie or spiral within the spacing $&, 

7.8 Reinforcement Detailing 7.8.1 Concrete Cover

5HLQIRUFLQJEDUVDUHSODFHGLQFROXPQVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&,$&, )RUFROXPQVZLWKWLHVDQGVSLUDOV concrete cover is measured from the surface of the concrete to the outer edge of the transverse reinforcement (see )LJXUHIRUDUHFWDQJXODUFROXPQZLWKWLHVDQGFURVVWLHV 7KHPLQLPXPFRYHUWRWKHWUDQVYHUVHUHLQIRUFHPHQW LVHTXDOWRLQIRUUHLQIRUFLQJEDUVLQFROXPQVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG$&,7DEOH   

‘˜‡”

Figure 7.21 Concrete cover for columns.

7-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.8.2 Minimum Number of Longitudinal Bars

7KHPLQLPXPQXPEHURIORQJLWXGLQDOEDUVWKDWPXVWEHSURYLGHGLQDFROXPQLVJLYHQLQ$&,DQGLVEDVHG RQWKHW\SHRIWUDQVYHUVHUHLQIRUFHPHQWVHH7DEOH  Table 7.16 Minimum Number of Longitudinal Bars in a Reinforced Concrete Column Type of Transverse Reinforcement

Minimum Number of Longitudinal Bars

Triangular ties

3

Rectangular or circular ties



Spirals

6

7.8.3 Spacing of Longitudinal Bars 0LQLPXPFOHDUVSDFLQJEHWZHHQWKHORQJLWXGLQDOEDUVLQDFROXPQPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&, $&, 7KHSXUSRVHRIWKHVHUHTXLUHPHQWVLVWRHQVXUHFRQFUHWHFDQHDVLO\ÀRZEHWZHHQWKHORQJLWXGLQDO bars. The minimum clear spacing, between longitudinal bars at any section in a column, including at locations ZKHUHODSVSOLFHVDUHXVHGLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQVHH$&,DQG)LJXUH 

(7.35)

In this equation, is the diameter of the longitudinal bars in the column and coarse aggregate in the concrete mix.

LVWKHQRPLQDOPD[LPXPVL]HRI



‫ݏ‬௠௜௡ 

GE 

‫ۓ‬ ۖ

ͳǤͷ‹Ǥ

‫ݏ‬௠௜௡ ൌ ‰”‡ƒ–‡”‘ˆ ͳǤͷ݀௕   ‫۔‬ ۖ ‫ە‬ሺͶΤ͵ሻ݀௔௚௚ Figure 7.22 Minimum clear spacing requirements for longitudinal bars in columns.

Minimum face dimensions of rectangular, tied columns with bearing splices, normal lap splices, and tangential lap VSOLFHVEDVHGRQWKHPLQLPXPFOHDUVSDFLQJUHTXLUHPHQWVRI$&,DUHJLYHQLQ7DEOHVHH)LJXUHIRU WKHVSOLFHW\SHV 7KHIDFHGLPHQVLRQVDUHURXQGHGXSWRWKHQHDUHVWLQFKDQGDUHEDVHGRQD LQFOHDUFRYHU WRWLHVIRUWKURXJKORQJLWXGLQDOEDUVDQGWLHVIRUDQGORQJLWXGLQDOEDUVDQG DPD[LPXPDJJUHJDWHVL]HRILQ

7-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ƒ…‡†‹‡•‹‘ǡܾ

ƒ…‡†‹‡•‹‘ǡܾ

ܿ

ƒ…‡†‹‡•‹‘ǡܾ

ܿ

ܿ

݀௧௜௘  ‫ݏ‬௠௜௡ 

݀௧௜௘ 

݀௕ 

݀௧௜௘  ‫ݏ‬௠௜௡ 

݀௕ 

‫ݏ‬௠௜௡ 

(a)

(b)

݀௕  (c)

Figure 7.23 Column splices for columns with longitudinal bars arranged in a rectangular pattern. (a) Bearing. (b) Normal lap. (c) Tangential lap.

Table 7.17 Minimum Face Dimensions (Inches) of Rectangular, Tied Columns Number of Bars Per Face Splice Type

Bearing

Normal Lap

Tangential Lap

Bar Size 2

3

4

5

6

7

8

9

10





9

11

13



18











9





16

18



















19









8



13



18











8

11





19







31



9





18







31





9

13

16







31



38







19







36









18

















8







16

18









8

11

13















9

11

13

16

18











9







19













13



18



























33

36



11



18







33

36





13











38







16







38







61



8



13

16

19











8

11

















8





18







31





9



16

19







33





9

13









33











19















11

16





31

36











18





36

















38











7-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHPLQLPXPIDFHGLPHQVLRQVIRUDQ\UHFWDQJXODUWLHGFROXPQ 7KHLWHPVLQ7DEOHIRUQRUPDOODSVSOLFHVDUHGH¿QHGLQ)LJXUH Table 7.18 Equations to Determine Minimum Face Dimensions (Inches) of Rectangular, Tied Columns Minimum Face Dimension, b*

Splice Type

Bearing

where

Normal Lap

Tangential Lap

‫ݏ‬ҧ

ܽത

ܿ ݀௧௜௘  ݀௕ Ȁʹ ܽത

ߠ

ܾത ‫ݏ‬௠௜௡ 

݀௕ ሺ–›’Ǥ ሻ

ܽത ൌ ݀௕ Ȁξʹ

ߠ ൌ ƒ”…•‹ሾܾതȀሺ‫ݏ‬௠௜௡ ൅ ݀௕ ሻሿ

ܾത ൌ ሺͳ െ ͳȀξʹሻ݀௕ 

‫ݏ‬ҧ ൌ ሺ‫ݏ‬௠௜௡ ൅ ݀௕ ሻ …‘• ߠ ൅ ܽത

‫ݏ‬௠௜௡

ͳǤͷ‹Ǥ  ൌ ‰”‡ƒ–‡”‘ˆ ͳǤͷ݀௕   ‫۔‬  ۖ ‫ە‬ሺͶΤ͵ሻ ݀௔௚௚ ‫ۓ‬ ۖ

Figure 7.24 Items for normal lap splices.

7-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݀௕ ሺ–›’Ǥሻ

݀௕ ሺ–›’Ǥሻ ܿ

ܿ ݀௧௜௘ ‘”݀௦ 

݀௧௜௘ ‘”݀௦  ‫ݏ‬ҧ ‫ݏ‬௠௜௡  ‫ݏ‬ҧ ‫ݏ‬௠௜௡  ߠ

ߠ ߠ

ߠ

݄Ȁʹ

݄Ȁʹ ͳ ͵͸Ͳ ߠൌ ൬ ൰ ʹ ݊

݀௕ ሺ–›’Ǥ ሻ



ߠൌ

ܿ

ͳ ͵͸Ͳ ൬ ൰ ʹ ݊

ܿ

 ‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕ 

‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕ 

(a)

(b)

݀௕ ሺ–›’Ǥሻ ܿ ”݀௦ 

‫ݏ‬ҧ ‫ݏ‬௠௜௡ 

ߠ

ߠ

݄Ȁʹ

ߠ ൌ ʹƒ”…•‹ ൤

݀௕ ൨ ܿ ݊ െ ʹሺܿ ൅ ݀௧௜௘ ሻ െ ݀௕

 ‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕  (c)

Figure 7.25 Column splices for columns with longitudinal bars arranged in a circular pattern. (a) Bearing. (b) Normal lap. (c) Tangential lap.

7-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The maximum number of longitudinal bars in a column where the bars are arranged in a circle with bearing splices, QRUPDOODSVSOLFHVDQGWDQJHQWLDOODSVSOLFHVLVJLYHQLQ7DEOHVHH)LJXUHIRUWKHVSOLFHW\SHVZKHUH is the spiral diameter). In addition to satisfying the minimum clear spacing requirements is the tie diameter and RI$&,WKHUHTXLUHPHQWVIRUPLQLPXPDQGPD[LPXPDUHDRIORQJLWXGLQDOUHLQIRUFHPHQWLQ$&, DUHDOVRVDWLV¿HGDWVHFWLRQVDZD\IURPODSVSOLFHORFDWLRQVZKHUHDOOWKHEDUVDUHVSOLFHG+RZHYHULQPDQ\FDVHV HLWKHUWKHSURYLGHGUHLQIRUFHPHQWUDWLRLVPXFKODUJHUWKDQWKHUHFRPPHQGHGPD[LPXPYDOXHRISHUFHQWRUWKHUH is an inordinately large number of smaller longitudinal bars. The number of bars have been rounded down to the QHDUHVWZKROHQXPEHUDQGDUHEDVHGRQDLQFOHDUFRYHUWRWLHVRUVSLUDOV Table 7.19 Maximum Number of Longitudinal Bars in Columns with Longitudinal Bars Arranged in a Circle Bar Size Splice Type

Bearing

Normal Lap

h (in.)

#5

#6

#7

#8

#9

#10

#11

#14

#18





9

9

8



6





ņ



13



11

11

9

8





ņ

16

16





13

11



9





18

19

18



16





11

8









19

18

16







6











18

16



11















18

16

13

9



31











18





















16

11







33

31









13





38



33







19









38

36











36







38









16

38









36



















38







18













36





19













38









61











36





















31





8



6

6



ņ

ņ

ņ

ņ



11



9

8







ņ

ņ

16



13



11

9



6



ņ

18



16



13

11

9

8

6

ņ





19



16

13

11















18

16

13



9













18



13























8

(table continued on next page)

7-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.19 Maximum Number of Longitudinal Bars in Columns with Longitudinal Bars Arranged in a Circle (cont.) Bar Size Splice Type

Normal Lap (cont.)

Tangential Lap

h (in.)

#5

#6

#7

#8

#9

#10

#11

#14

#18













19



13

9





33









19







38



33

31







16

11





38

36

33







18



36





38

36

31





19

13

38







38





















36

31





16











38

33





















31



18















33



19













39









8



6

6







ņ

ņ





9

8



6

6





ņ

16



11



9

8



6



ņ

18



13



11



8



6











13

11



9







19

18

16



13

11



8









18





13

11

9

6









18

16



13















18

16



11

8











19







9



31









18

16

13





33











18





36













19

16

11

38

38



31

















36

33

31







18

13





38



33







19





























39

36



















38









16

*Indicates the tabulated value is not applicable to columns where a minimum of 6 longitudinal bars are required. **Indicates the longitudinal reinforcement ratio is less than 1 percent. A dash (ņ) indicates the maximum number of longitudinal bars is less than 4.

7-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHPD[LPXPQXPEHURIORQJLWXGLQDOEDUVIRUDQ\FROXPQ where the longitudinal reinforcement is arranged in a circle. Table 7.20 Equations to Determine the Maximum Number of Longitudinal Bars in a Column with Longitudinal Bars Arranged in a Circle Maximum Number of Longitudinal Bars, n*

Splice Type

Bearing

Normal Lap

Lap Splice Length

Tangential Lap

Offset Bent Longitudinal Bar (typ.)

max. 6 1

Figure 7.26 Offset bent longitudinal reinforcement in a column.

7-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.8.4 Offset Bent Longitudinal Reinforcement

Offset bent longitudinal reinforcement is typically provided at locations where the longitudinal bars are spliced in a FROXPQVHH)LJXUH $FFRUGLQJWR$&,WKHVORSHRIWKHLQFOLQHGSRUWLRQRIDQRIIVHWEHQWEDUUHODWLYH to the longitudinal axis of the column must not exceed 1 to 6, and the portions of the bar above and below the offset must be parallel to the axis of the column. In cases where the face of a column is offset by 3 in. or more, offset bent longitudinal reinforcement is not permitted $&, ,QVWHDGWKHRIIVHWORQJLWXGLQDOUHLQIRUFLQJEDUVRQHDFKIDFHPXVWEHODSVSOLFHGZLWKVHSDUDWHGRZHO EDUV6KRZQLQ)LJXUHLVWKHFDVHZKHUHVHSDUDWHGRZHOEDUVPXVWEHSURYLGHGDWDORFDWLRQZKHUHDFKDQJHLQ column cross-section occurs.

൒ ͵ԣ

‘™‡Ž„ƒ”•

Figure 7.27 Separate dowel bars at offset column faces.

7.8.5 Splices of Longitudinal Reinforcement Overview

6SOLFHUHTXLUHPHQWVIRUORQJLWXGLQDOUHLQIRUFHPHQWLQFROXPQVDUHJLYHQLQ$&,IRUODSVSOLFHVPHFKDQLFDO splices, butt-welded splices, and end-bearing splices. Splice lengths must be provided that satisfy the factored load combinations on a column. Where the stress in all the longitudinal bars due to factored loads is compressive, all the splice types listed above may be used. All but end-bearing splices are permitted when the longitudinal bar stress is WHQVLOH5HTXLUHPHQWVIRUVSOLFHVDUHJLYHQLQ$&,$&,  Lap Splices

Lap splices, which are the most popular and usually the most economical type of splices used in columns, are permitted to occur immediately above the top of the slab for columns that are not part of special moment frames (see )LJXUH 7KLVLVWKHSUHIHUUHGORFDWLRQIRUHDVHRIFRQVWUXFWLRQ7KHORQJLWXGLQDOEDUVIURPWKHFROXPQEHORZ extend above the slab a distance greater than or equal to the required lap splice length, and the longitudinal bars in WKHFROXPQDERYHDUHWLHGWRWKHVHEDUVDIWHUWKHÀRRUEHORZKDVEHHQFRQVWUXFWHG

7-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

/DSVSOLFHVDUHQRWSHUPLWWHGIRUEDUVODUJHUWKDQEDUVH[FHSWLQWKHFDVHRIFRPSUHVVLRQODSVSOLFHVZKHUH RUEDUVDUHSHUPLWWHGWREHVSOLFHGWRRUVPDOOHUEDUV$&,DQG  The type of lap splice that must be used (compressive or tensile) depends on the stress in the longitudinal bars due WRWKHIDFWRUHGORDGFRPELQDWLRQVVHH)LJXUH 5HTXLUHGODSVSOLFHOHQJWKVIRUWKHWKUHH]RQHVLQGLFDWHGLQ )LJXUHDUHJLYHQLQ7DEOH

‫ܔ܉ܑܠۯ‬۴‫܍܋ܚܗ‬

݂௦ ൌ Ͳ ݂௦ ൌ ͲǤͷ݂௬ 

‫ͳ܍ܖܗ܈‬ ͳ ‫ʹ܍ܖܗ܈‬ ʹ

‫͵܍ܖܗ܈‬ ͵

۰‫ܜܖ܍ܕܗۻ܏ܖܑ܌ܖ܍‬ Figure 7.28 Splice requirements for reinforced concrete columns. 

݄ଶ 

݄ଵ 

‹”‡…–‹‘‘ˆƒƒŽ›•‹• ͵‫ܣ‬௧௜௘ ൒ ͲǤͲͲͳͷ݄ଶ ‫ݏ‬

‫ܣ‬௧௜௘ ൌ ƒ”‡ƒ‘ˆ–‹‡„ƒ”

‹”‡…–‹‘‘ˆƒƒŽ›•‹• ʹ‫ܣ‬௧௜௘ ൒ ͲǤͲͲͳͷ݄ଵ ‫ݏ‬ Figure 7.29 Application of ACI 10.7.5.2.1(a).

7-42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.21 Required Lap Splices Lengths Zone

1



3

Stress in Longitudinal Bars

Minimum Lap Splice Length

All longitudinal bars are in compression

Stress in the longitudinal bars on the tension face is tensile and

Stress in the longitudinal bars on the tension face is tensile and

Class A tension lap splice

Notes

ACI Section No.

±

 

±

 

±

 

Class B tension lap splice length

Class B tension lap splice length

Notes: 1. For tied columns where ties throughout the lap splice length have an effective area in both directions, is permitted to be multiplied by 0.83. Tie legs perpendicular to are considered in calculating the effective tie area [ACI 10.7.5.2.1(a); see Figure 7.29]. For spiral columns where spirals satisfying ACI 25.7.3 are provided throughout the lap splice length, is permitted to be multiplied by 0.75 [ACI 10.7.5.2.1(b)]. must be increased by one-third [ACI 25.5.5.1]. 2. For 3. Compression lap splices must not be used for bars larger than #11 except as permitted in ACI 25.5.5.3 [ACI 25.5.5.2]. 4. Compression lap splices of #14 and #18 bars to #11 or smaller bars are permitted in accordance with ACI 25.5.5.4 (see Note 5) [ACI 25.5.5.3]. 5. Where bars of different size are lap spliced in compression, is the longer of (a) of the larger bar calculated by ACI 25.4.9.1 of the smaller bar calculated by ACI 25.5.5.1 [ACI 25.5.5.4]. (see below) and (b)

where

(see Table 7.22)

6. 7.

Reduction of development length in accordance with ACI 25.4.10.1 is not permitted in calculating lap splice lengths (ACI 25.5.1.4). 8. Requirements for Class A and Class B tension lap splices are given in ACI Table 10.7.5.2.2 (see Table 7.23). is determined in accordance with ACI 25.4.2.1(a). 9. Tension development length is the greater of (a) of the larger bar and (b) of the smaller bar 10. Where bars of different size are lap spliced together, (ACI 25.5.2.2).

7-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.22 Modification Factors for Deformed Bars in Compression Modification Factor

Lightweight,

&RQ¿QLQJUHLQIRUFHPHQW

Condition

Value of Factor

Lightweight concrete Normalweight concrete

 

Longitudinal reinforcement enclosed in the following: 1. A spiral  $  FLUFXODUFRQWLQXRXVO\ZRXQGWLHZLWK and a pitch   WLHVLQDFFRUGDQFHZLWK$&,VSDFHG on center  +  RRSVLQDFFRUGDQFHZLWK$&,VSDFHG on center





Other

Table 7.23 Requirements for Class A and Class B Tension Lap Splices Stress in the longitudinal bars on the tension face

*Tension development length

Splice Details

Splice Type

There are percent of the longitudinal bars spliced at any section and the lap splices on adjacent bars are staggered by at least VHH)LJXUH

Class A

Other

Class B

All cases

Class B

Tension Lap Splice Length, ℓst*

is determined in accordance with ACI 25.4.2.1(a).

3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHOopment length, LVGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU UHTXLUHPHQWVDUHFRYHUHG¿UVW Method 1 – ACI 25.4.2.4 The development length in tension of a deformed reinforcing bar,

LVGHWHUPLQHGE\$&,(TXDWLRQD  (7.36)

7-44

Stagger distance ൒ κௗ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Class A lap splice length

Clear space to determine κௗ ݊௕,௦௣௟௜௖௘ௗ ൑ 0.5݊௕,௧௢௧௔௟ 

Figure 7.30 Class A Tension Lap Splice

7KHIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHWKHVHIDFWRUVDUHJLYHQLQ7DEOHDORQJZLWK must be taken less than or equal GH¿QLWLRQVIRUWKHRWKHUWHUPVLQ(TXDWLRQ 7KHFRQ¿QLQJWHUP WRLQ(TXDWLRQ >$&,@ Table 7.24 Factors for Development of Deformed Bars in Tension Factor

Lightweight, Reinforcement grade,

Epoxy,

6L]H

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



*UDGHRU*UDGH



*UDGH



*UDGH

1.3

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG or clear reinforcement with clear cover spacing



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG reinforcement for all other conditions



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQforcement



DQGODUJHUEDUV



DQGVPDOOHUEDUV

 (table continued on next page)

7-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.24 Factors for Development of Deformed Bars in Tension (cont.) Factor

Casting position,

Condition

Value of Factor

0RUHWKDQLQRIIUHVKFRQFUHWHSODFHG EHORZKRUL]RQWDOUHLQIRUFHPHQW

Spacing or cover dimension,

Transverse reinforcement index,

1.3

Other



All

Lesser of: 1. The distance from the center of a longitudinal bar to the nearest concrete surface.  2  QHKDOIWKHFHQWHUWRFHQWHUVSDFLQJRI the longitudinal bars being developed (see )LJXUH 

All

where: total cross-sectional area of all 1. transverse reinforcement within a spacing that crosses the potential plane of splitting through the longitudinal reinforcement being developed center-to-center spacing of the trans  verse reinforcement number of bars being developed 3. along the plane of splitting

*The product need not exceed 1.7. **It is permitted to conservatively use if transverse reinforcement is present or required. For reinforcing bars with center, transverse reinforcement must be provided along the development length such that (ACI 25.4.2.2).

spaced closer than 6.0 in. on

Method 2 – ACI 25.4.2.3 7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHTwo sets of spacing and cover cases are given in ACI Table VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP VHH7DEOH 7KHPRGL¿FDWLRQVIDFWRUVLQWKHVHHTXDWLRQVDUHREWDLQHGIURP$&,7DEOHVHH 7DEOH 

7-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 

݀௕ 

ܿଶ 

‫ݏ‬

ܿଵ ‫ ۓ‬ ۖ ܿ௕ ൌ Ž‡••‡”‘ˆ ܿଶ  ‫ ۔‬ ۖ ‫ݏە‬Ȁʹ Figure 7.31 Spacing or cover dimension, cb.

Table 7.25 Tension Development Length, ℓd, for Deformed Bars in Accordance with ACI 25.4.2.3 Spacing and Cover

#6 and Smaller Bars

#7 and Larger Bars

Condition 1 1. Clear spacing of bars being developed or lap spliced  &OHDUFRYHU 3. Stirrups or ties throughout not less than the required minimum Case 1

&RQGLWLRQ 1. Clear spacing of bars being developed or lap spliced  &OHDUFRYHU

&DVH

Other conditions

7-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

End-Bearing Splices

End-bearing splices, which are also referred to as compression-only mechanical splices, are permitted to be used to splice the longitudinal bars in a column where the force in all the bars is compressive for all load combinations (ACI  &RPSUHVVLRQIRUFHVDUHWUDQVPLWWHGIURPRQHORQJLWXGLQDOEDUWRDQRWKHUE\EHDULQJRIVTXDUHFXWHQGV KHOGLQFRQFHQWULFFRQWDFWE\DVXLWDEOHSURSULHWDU\GHYLFH7ROHUDQFHVIRUWKHEDUHQGVDUHJLYHQLQ$&, End-bearing splices are permitted to be used only in columns where closed ties, closed stirrups, spirals, or hoops are SURYLGHG$&, 7KLVHQVXUHVPLQLPXPVKHDUUHVLVWDQFHLVSURYLGHGLQWKHPHPEHU $FFRUGLQJWR$&,HQGEHDULQJVSOLFHVPXVWKDYHDWHQVLOHVWUHQJWKRIDWOHDVWSHUFHQWRI of the continuing longitudinal reinforcement area on each face of a column. This is typically achieved by either staggering the location of the end-bearing splices or adding additional longitudinal reinforcement through the splice locations. Additional information and various types of end-bearing splices can be found in Reference 9. Mechanical and Welded Splices

0HFKDQLFDORUZHOGHGVSOLFHVDUHSHUPLWWHGLQFROXPQVDQGPXVWPHHWWKHUHTXLUHPHQWVRI$&,7KHVHW\SHV of splices may be used in both tension and compression. A variety of proprietary mechanical devices are available that can be used to splice longitudinal reinforcing bars in DFROXPQVHH5HIHUHQFH 7KHVHW\SHVRIVSOLFHVDUHFRPPRQO\VSHFL¿HGDWORFDWLRQVZKHUHLQRUGLQDWHO\ORQJODS splices would be required or where lap splices would cause congestion. Mechanical splices must be able to develop LQWHQVLRQRUFRPSUHVVLRQDWOHDVWSHUFHQWRIWKH\LHOGVWUHQJWKRIWKHORQJLWXGLQDOEDUV of the longitudinal bars. These types of splices are primarily Welded splices must also be able to develop LQWHQGHGIRUEDUVDQGODUJHU:HOGLQJRIUHLQIRUFLQJEDUVPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&, 0HFKDQLFDORUZHOGHGVSOLFHVLQFROXPQVQHHGQRWEHVWDJJHUHG$&,  7.8.6 Transverse Reinforcement Overview

The main purposes of transverse reinforcement in columns are to provide lateral support of the longitudinal reinIRUFHPHQWDQGWRSURYLGHVKHDUUHVLVWDQFHZKHUHUHTXLUHG7KHPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,IRU WLHVDQGLQ$&,IRUVSLUDOVDUHPHDQWWRDFKLHYHWKHVHSXUSRVHV$OVRLQFOXGHGLQWKHVHVHFWLRQVDUHPLQLPXP VSDFLQJUHTXLUHPHQWVZKLFKKHOSWRHQVXUHFRQFUHWHFDQHDVLO\ÀRZEHWZHHQWKHWUDQVYHUVHUHLQIRUFLQJEDUV 7KHW\SHVRIWUDQVYHUVHUHLQIRUFHPHQWW\SLFDOO\XVHGLQFROXPQVDUH WLHVUHFWLOLQHDURUFLUFXODU  VSLUDOVRU  KRRSV5HTXLUHPHQWVIRUVL]HDQGVSDFLQJRIWLHVDQGVSLUDOVDUHFRYHUHGEHORZ Tie Reinforcement

5HTXLUHPHQWVIRUFROXPQVZLWKWLHUHLQIRUFHPHQWDUHJLYHQLQ$&,DQG7LHVFRQVLVWRIDFORVHGORRS RIGHIRUPHGUHLQIRUFLQJEDUV>VHH)LJXUHD IRUUHFWLOLQHDUWLHV@0LQLPXPLQVLGHEHQGGLDPHWHUVDQGVWDQGDUG KRRNJHRPHWU\IRUWLHVDUHJLYHQLQ$&,7DEOH,QIRUPDWLRQIRUGHJUHHKRRNVLVLQFOXGHGLQWKDWWDEOHEXW VXFKKRRNVDUHQRWUHFRPPHQGHGPDLQO\GXHWRWKHGLI¿FXOWLHVDVVRFLDWHGZLWKSODFLQJWKHPLQWKH¿HOG 7KHUHTXLUHPHQWVIRUPLQLPXPDQGPD[LPXPWLHVSDFLQJ$&, DQGPLQLPXPWLHEDUVL]HEDVHGRQWKH VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ$&, DUHJLYHQLQ)LJXUH7DEXODWHGYDOXHVRIWKHPD[LPXPFHQWHUWRFHQWHUWLHVSDFLQJDUHDOVRSURYLGHGLQ)LJXUHIRUYDULRXVFROXPQORQJLWXGLQDOEDUVL]HVWKH smallest cross-sectional dimension of the column must be compared to the appropriate tabulated value and the lesser of the two is to be used for the tie spacing. 7KHSURYLVLRQVLQ$&,SHUWDLQWRUHFWLOLQHDUWLHFRQ¿JXUDWLRQVDQGWKHPD[LPXPFOHDUVSDFLQJSHUPLWWHG between laterally supported longitudinal bars in a column. Corner bars and every other longitudinal bar in the sec-

7-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 

ܿଶ 



ܿଵ ൏ ܿଶ 

݀௕ሺ௖௢௟௨௠௡ሻ 

݀௕ሺ௧௜௘ሻ 

ͳ͸݀௕ሺ௖௢௟௨௠௡ሻ  ‫ ݏ‬൑ Ž‡ƒ•–‘ˆ Ͷͺ݀௕ሺ௧௜௘ሻ   ‫۔‬  ۖ ‫ܿ ە‬ଵ   ൒ ሺͶΤ͵ሻ݀௔௚௚ ൅ ݀௕ሺ௧௜௘ሻ   ‫ۓ‬ ۖ

൑ ‫ݏ‬Ȁʹ

‹‡ƒ” ͓͵

‘Ž—‘‰Ǥƒ” ƒšǤ࢙ሺ‹Ǥሻȗ ͓ͷ ͳͲ ͓͸ ͳʹ ͓͹ ͳͶ ͓ͺ ͳ͸ ͓ͻ ͳͺ ͓ͳͲ ͳͺ ͓Ͷ ͓ͳͳ ʹʹ ͓ͳͶ ʹͶ ͓ͳͺ ʹͶ ȗƒšǤ‫ ݏ‬൑ ܿଵ —•–ƒŽ•‘„‡…‘•‹†‡”‡† Figure 7.32 Tie requirements for reinforced concrete columns.

7-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

WLRQPXVWKDYHODWHUDOVXSSRUWSURYLGHGE\WKHFRUQHURIDWLHWKDWKDVDKRRNRIQRWPRUHWKDQGHJUHHVZKHUH the clear spacing between the longitudinal bars on each side is less than or equal to 6 in. This is illustrated in FigXUHD IRUWKHFDVHRIDVTXDUHWLHGFROXPQZLWKORQJLWXGLQDOEDUV%HFDXVHWKHFOHDUVSDFLQJEHWZHHQWKH EDUVLVOHVVWKDQLQFURVVWLHVZLWKGHJUHHDQGGHJUHHKRRNVRQHDFKHQGDUHSURYLGHGDURXQGHYHU\RWKHU ORQJLWXGLQDOEDU7KHFOHDUVSDFHEHWZHHQORQJLWXGLQDOEDUVLVJUHDWHUWKDQLQIRUWKHFROXPQLQ)LJXUHE ,Q WKLVFDVHFURVVWLHVDUHUHTXLUHGDURXQGHYHU\ORQJLWXGLQDOEDULQDFFRUGDQFHZLWK$&,E >VHH$&,)LJXUH 5D@ 

൑ ͸ԣሺ–›’Ǥሻ

൐ ͸ԣሺ–›’Ǥሻ ”‘••–‹‡ሺ–›’Ǥሻ

ሺƒሻ

”‘••–‹‡ሺ–›’Ǥሻ

ሺ„ሻ

Figure 7.33 Lateral support requirements for longitudinal bars in tied columns.

$GGLWLRQDOUHTXLUHPHQWVIRUODWHUDOVXSSRUWRIORQJLWXGLQDOEDUVZLWKWLHVDUHJLYHQLQ$&,7KHVHUHTXLUHPHQWVDUHJLYHQLQ)LJXUH Circular ties may be used in cases where the longitudinal bars are located around the perimeter of a circle, like in FLUFXODURUVTXDUHFROXPQV$&, $QFKRUDJHUHTXLUHPHQWVIRUFLUFXODUWLHVDUHJLYHQLQ$&,VHH $&,)LJXUH5  Spiral Reinforcement

5HTXLUHPHQWVIRUFROXPQVZLWKVSLUDOUHLQIRUFHPHQWDUHJLYHQLQ$&,DQG6WDQGDUGVSLUDOVL]HV DUHWRDQGWKHFOHDUVSDFLQJEHWZHHQFRQVHFXWLYHWXUQVRQDVSLUDOPXVWQRWH[FHHGLQRUEHOHVVWKDQWKH used in the concrete mix. JUHDWHURILQRU WLPHVWKHGLDPHWHURIWKHODUJHVWDJJUHJDWHVL]H The minimum volumetric spiral reinforcement ratio,

LVJLYHQE\$&,(TXDWLRQ  (7.37)

In this equation, is the area of the column core enclosed by the spiral, which is equal to where WKHGLDPHWHURIWKHFROXPQFRUHPHDVXUHGWRWKHRXWVLGHHGJHVRIWKHVSLUDOUHLQIRUFHPHQWVHH)LJXUH 

7-50

is

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.2ሻ Offset bent bars ሺACI 10.7.4ሻ

Additional ties within 6ԣ of bend ሺACI 10.7.6.4ሻ

Lap splice

Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ

൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.1ሻ

Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ

൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.1ሻ

Beams on all four  sides of column

൑ 3ᇳ ሺACI 10.7.6.2.2ሻ Additional ties within 6ԣ of bend ሺACI 10.7.6.4ሻ Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ

Figure 7.34 Tie and splice details for reinforced concrete columns.

7-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‘…”‡–‡…‘”‡

’‹”ƒŽ„ƒ”

’‹”ƒŽ’‹–…Šǡ‫ݏ‬

‫ܦ‬௖௛ 

‘˜‡”ሺ–›’Ǥ ሻ

݄

Figure 7.35 Spiral reinforcement.

The volumetric spiral reinforcement ratio,

is equal to the following: (7.38)

where is the area of the spiral reinforcing bar and monly referred to as the spiral pitch). An equation for

is the center-to-center spacing of the spirals (which is com-

can be obtained by substituting Equation (7.38) into Equation (7.37): (7.39)

The limitations in ACI 25.7.3.1 must also be considered when determining Recommended standard spirals for circular columns with various reinforcement yield strengths and concrete compressive strengths are given in Table 7.26. Table 7.26 Recommended Standard Spirals for Circular Columns

fyt (psi)

Iүc (psi)

Column Size (in.)

Spiral Size and Pitch

12 – 24

#3 @Ǝ

26 – 48

#3 @Ǝ

12 – 16

#4 @Ǝ

18 – 48

#4 @Ǝ

10,000

20 – 48

#5 @Ǝ

14,000

36 – 48

#5 @Ǝ

4,000

60,000

7,000

(table continued on next page)

7-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.26 Recommended Standard Spirals for Circular Columns (cont.)

fyt (psi)

Iүc (psi)

Column Size (in.)

Spiral Size and Pitch

±

#Ǝ

±

#Ǝ



#Ǝ

±

#Ǝ

±

#Ǝ

±

#Ǝ

±

#Ǝ



±

#Ǝ



±

#Ǝ



±

#Ǝ



±

#Ǝ



±

#Ǝ











6SLUDOVPXVWEHDQFKRUHGDWHDFKHQGE\SURYLGLQJRQHDQGRQHKDOIH[WUDWXUQVRIWKHVSLUDOEDU$&,VHH $&,)LJXUH5  5HTXLUHPHQWVIRUVSLUDOVSOLFHVDUHJLYHQLQ$&,0HFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&, Lap splice lengths must DUHSHUPLWWHGDVDUHODSVSOLFHVLQDFFRUGDQFHZLWK$&,IRU EHWKHJUHDWHURILQRUWKHODSOHQJWKGHWHUPLQHGE\$&,7DEOHZKLFKLVHTXDOWRDPXOWLSOHRIWKHGLDPHWHURIWKHVSLUDOEDU/DSVSOLFHOHQJWKVDUHJLYHQLQ$&,7DEOHIRUXQFRDWHGEDUVZLWKRXWKRRNVDQGIRU coated bars with and without hooks and are based on HTXDOWRSVLUHJDUGOHVVRIWKHDFWXDO\LHOGVWUHQJWK RIWKHEDU>VHH$&,E @ • 8QFRDWHGRU]LQFFRDWHGJDOYDQL]HGGHIRUPHGEDUVZLWKRXWKRRNV/DSOHQJWK • (SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGGHIRUPHGEDUVZLWKRXWKRRNV/DSOHQJWK • (SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGGHIRUPHGEDUVZLWKDVWDQGDUGKRRNFRQIRUPLQJWR$&, Lap length Requirements for lateral support of longitudinal bars using spirals at the top and bottom of a story are given in ACI DQGDUHVXPPDUL]HGLQ)LJXUH

7.9 Connections to Foundations 7.9.1 Overview

9HUWLFDODQGKRUL]RQWDOIRUFHVPXVWEHWUDQVIHUUHGDWWKHLQWHUIDFHEHWZHHQDFROXPQDQGDIRXQGDWLRQ$&,  Vertical compression forces are transferred by bearing on the concrete or by a combination of bearing and interface reinforcement. Tension forces must be transferred entirely by reinforcement, which may consist of extended longituGLQDOEDUVGRZHOVDQFKRUEROWVRUPHFKDQLFDOFRQQHFWRUV+RUL]RQWDOIRUFHVDUHWUDQVIHUUHGXVLQJWKHVKHDUIULFWLRQ SURYLVLRQVRI$&,RURWKHUDSSURSULDWHPHWKRGV

7-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄



”‘˜‹†‡–‹‡•–‘—†‡”•‹†‡‘ˆ•Žƒ„ ™Š‡”‡„‡ƒ•†‘‘–ˆ”ƒ‡‹–‘ ƒŽŽ•‹†‡•‘ˆ–Š‡…‘Ž—

’‹”ƒŽ–‡”‹ƒ–‡†ƒ––‘’‘ˆ•Žƒ„

š–‡†•’‹”ƒŽ–‘–Š‡Ž‘™‡•–Š‘”‹œ‘–ƒŽ ”‡‹ˆ‘”…‡‡–‹–Š‡‡„‡”•—’’‘”–‡†ƒ„‘˜‡

’‹”ƒŽ–‡”‹ƒ–‡†ƒ––‘’‘ˆ•Žƒ„

ʹ݄ ‘Ž—…ƒ’‹–ƒŽ š–‡†•’‹”ƒŽ–‘–Š‡Ž‡˜‡Žƒ–™Š‹…Š–Š‡†‹ƒ‡–‡”‘” ™‹†–Š‘ˆ–Š‡…ƒ’‹–ƒŽ‹•–™‹…‡–Šƒ–‘ˆ–Š‡…‘Ž—

Figure 7.36 Lateral support of longitudinal bars using spirals.

7.9.2 Vertical Transfer Compression

:KHUHYHUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGWRDIRXQGDWLRQEHDULQJVWUHQJWKUHTXLUHPHQWVRI$&,PXVW EHVDWLV¿HGIRUERWKWKHFROXPQDQGWKHIRXQGDWLRQ$&, )RUEHDULQJRQDFROXPQWKHIDFWRUHGEHDULQJ must be less than or equal to the design bearing strength, where the nominal bearing strength, force, LVJLYHQLQ$&,7DEOH (7.40)

In this equation,

7-54

LVHTXDOWRIRUEHDULQJ$&,7DEOH DQG

is the gross area of the column.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUEHDULQJRQDIRXQGDWLRQZLGHURQDOOVLGHVWKDQWKHORDGHGDUHDWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG (7.41)

The term is the area of the lower base of the largest frustrum of a pyramid, cone, or tapered wedge contained and having side slopes of 1 vertical to wholly within the foundation and having for its upper base the loaded area KRUL]RQWDO$UHDV and DUHJLYHQLQ)LJXUHIRUWKHFDVHRIDFROXPQVXSSRUWHGE\DVSUHDGIRRWLQJ

ʹ

ͳ

‫ܮ‬

ܿଶ  ܾ

ܿଵ 

‫ܤ‬

Ͷͷ‫ ל‬

‫ܣ‬ଵ ൌ ܿଵ ܿଶ 

ʹ

݄

ͳ

‫ܣ‬ଶ ൌ ܾ‫ܮ‬

‫ܮ‬

 Figure 7.37 Determination of areas for bearing strength.

Where the excess compression stress from the column must be transferred by reinforcement to the founis determined by the following equation: dation. The required area of interface reinforcement, (7.42)

In this equation, is the gross area of the column. The minimum amount of interface reinforcement, $&,  be provided even where

must

Dowel bars emanating from the foundation are the type of interface reinforcement commonly used because of ease of construction. The dowel bars are set in the foundation prior to casting the foundation concrete and are subsequently spliced to the column longitudinal bars. Dowels should be positioned so as not to interfere with the longitudinal bars or the tie hooks in the column. ,OOXVWUDWHGLQ)LJXUHDUHGRZHOEDUVDFURVVWKHLQWHUIDFHEHWZHHQDUHLQIRUFHGFRQFUHWHFROXPQDQGVSUHDGIRRWing where all the longitudinal bars in the column are in compression for all factored load combinations (Zone 1 in deter)LJXUH 7KHGRZHOEDUVPXVWH[WHQGLQWRWKHIRRWLQJDWOHDVWDFRPSUHVVLRQGHYHORSPHQWOHQJWK PLQHGLQDFFRUGDQFHZLWK$&,VHH6HFWLRQRIWKLVSXEOLFDWLRQ ,WLVQRWUHTXLUHGWRSURYLGHGRZHO bars for all the longitudinal bars in such cases.

7-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



൒ κௗ௖ 

‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ 

–ƒ†ƒ”†Š‘‘ ሺ–›’Ǥሻ

‘‘–‹‰„ƒ”•ǡሺ݀௕ ሻ௙ ሺ–›’Ǥሻ

͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ

Š

‘’”‡••‹‘Žƒ’ •’Ž‹…‡

‘Ž—Ž‘‰‹–—†‹ƒŽ„ƒ”•

Figure 7.38 Dowel bars at the interface of a reinforced concrete column and footing where all the longitudinal bars in the column are in compression.

Standard hooks are typically provided at the ends of the dowel bars, which are tied to the reinforcing bars in the footing; this detail is far more economical than terminating the dowel bars above the footing reinforcing bars. The hooked portion of the dowels cannot be considered effective for developing the dowels bars in compression (ACI  7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKH footing: (7.43)

In this equation, LVWKHUDGLXVRIWKHGRZHOEDUEHQGVHH$&,7DEOHZKLFKFRQWDLQVPLQLPXPLQVLGHEHQG GLDPHWHUVIRUEDUVZLWKVWDQGDUGGHJUHHKRRNV :KHUH(TXDWLRQ LVQRWVDWLV¿HGHLWKHUDJUHDWHUQXPEHURI smaller dowel bars can be used (if possible) or the depth of the footing must be increased. The information above pertaining to the development of dowel bars in spread footings is also applicable to the development of dowel bars in pile caps. For columns supported on drilled piers or caissons, the dowel bars in the drilled pier or caisson usually do not have hooks at the ends. The dowel bars must also be fully developed in the column; this is typically achieved by lap splicing the dowel bars WRWKHORQJLWXGLQDOEDUVLQWKHFROXPQVHH)LJXUH :KHUHWKHGRZHOEDUVDUHWKHVDPHVL]HDVWKHFROXPQEDUV

7-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

WKHPLQLPXPFRPSUHVVLRQODSVSOLFHOHQJWKLVXVHGVHH7DEOH :KHUHWKHGRZHOEDUVDUHGLIIHUHQWLQVL]HWKDQ WKHFROXPQEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWKPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,/DSVSOLFHVRI DQGORQJLWXGLQDOEDUVLQFRPSUHVVLRQIRUDOOIDFWRUHGORDGFRPELQDWLRQVZLWKDQGVPDOOHUGRZHOEDUVDUH SHUPLWWHGSURYLGHGWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG$&,  Tension

Tension forces transferred from a column to a foundation must be resisted entirely by reinforcement across the inWHUIDFH$&,E DQG DQGGRZHOEDUVPXVWEHSURYLGHGIRUDOOWKHORQJLWXGLQDOEDUVLQWKHFROXPQ ,QVXFKFDVHVIDFWRUHGORDGFRPELQDWLRQVIDOOZLWKLQ=RQHVRURIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVHH )LJXUH  7HQVLOHDQFKRUDJHRIWKHGRZHOEDUVLQWRDIRRWLQJRUSLOHFDSLVW\SLFDOO\DFFRPSOLVKHGE\SURYLGLQJGHJUHH determined in acstandard hooks at the ends of the dowel bars with the development length of the hooked bar, FRUGDQFHZLWK$&,VHH)LJXUH 



͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ

‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ 

൒ κௗ௛ 

Š

Žƒ••–‡•‹‘ Žƒ’•’Ž‹…‡

‘Ž—Ž‘‰‹–—†‹ƒŽ„ƒ”•

–ƒ†ƒ”†Š‘‘ ሺ–›’Ǥሻ

‘‘–‹‰„ƒ”•ǡሺ݀௕ ሻ௙ ሺ–›’Ǥሻ

Figure 7.39 Dowel bars at the interface of a reinforced concrete column and footing where the longitudinal bars in the column are in compression and tension.

7-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(7.44)

7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRNVHH)LJXUHRIWKLV SXEOLFDWLRQ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHVHH7DEOH  Table 7.27 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Lightweight, Epoxy,

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK &RQ¿QLQJUHLQIRUFHPHQW

Location,

or

Other )RUDQGVPDOOHUKRRNHGEDUV 1. terminating inside a column core with side cover normal to the plane of the hook or  ZLWKVLGHFRYHUQRUPDOWRWKHSODQHRIWKHKRRN Other

Concrete strength,

 1.6







7KHFRQ¿QLQJUHLQIRUFHPHQWIDFWRU LVW\SLFDOO\HTXDOWRIRUKRRNHGGRZHOEDUVLQIRRWLQJVEHFDXVHFRQ¿Qing reinforcement is usually not provided in such cases. 7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKHIRRWLQJFRQVLGering the hooked portion of the dowel bars can be developed in tension): (7.45)

:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHGHSWKRIWKHIRRWLQJXVXDOO\PXVWEHLQFUHDVHG For development of the dowel bars into the column, a tension lap splice or a mechanical connection in accordance ZLWK$&,RUUHVSHFWLYHO\PXVWEHSURYLGHGEHWZHHQWKHGRZHOEDUVDQGWKHORQJLWXGLQDOEDUV,QWKH FDVHRIODSVSOLFHVD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGVHH7DEOHDQG)LJXUH 

7-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.9.3 Horizontal Transfer

7KHVKHDUIULFWLRQPHWKRGRI$&,LVSHUPLWWHGWREHXVHGWRGHWHUPLQHWKHQRPLQDOVKHDUVWUHQJWK at the FRQWDFWVXUIDFHEHWZHHQWKHFROXPQDQGWKHIRXQGDWLRQ$&, 7KHUHTXLUHGDUHDRIUHLQIRUFHPHQW across the interface between the column and the foundation is determined by the following equation, which is apSOLFDEOHWRVKHDUIULFWLRQUHLQIRUFHPHQWSHUSHQGLFXODUWRWKHLQWHUIDFH$&,  (7.46)

In this equation, is the factored shear force due to the lateral force effects at the interface, the strength reduction factor LVHTXDOWRDQG LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP$&,7DEOHVHH7DEOH   Table 7.28 Coefficient of Friction, μ

μ

Contact Surface Condition

Concrete placed monolithically Concrete placed against hardened concrete that is clean, free of laitance, and intentionally roughened to a full amplitude of approximately ¼ in. Concrete placed against hardened concrete that is clean, free of laitance, and not intentionally roughened 8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOHVHH7DEOH 7KHWHUP is the area of For example, where a column is supported by a footing, is equal to the gross area of the concrete resisting column. Where the concrete strengths of the column and the foundation are different, the smaller of the two must be XVHGLQWKHVHHTXDWLRQV$&,  Table 7.29 Maximum Vn Across the Assumed Shear Plane Contact Surface Condition

Maximum Vn

= Vu / Ҁ*

Normalweight concrete placed monolithically or placed against hardened concrete intentionally roughened to a full amplitude of approximately ¼ in.

Other cases

*

area of concrete resisting

The required amount of shear friction reinforcement, LVSHUPLWWHGWREHGHWHUPLQHGE\(TXDWLRQ EDVHG where is a permanent net compression force transmitted across the ason a net force equal to VXPHGVKHDUSODQH$&, $FFRUGLQJWR$&,WKHDUHDRIUHLQIRUFHPHQWUHTXLUHGWRUHVLVWDQHW factored tension force across the assumed shear plane is to be added to the area of reinforcement required for shear friction that crosses the assumed shear plane. For columns transmitting a bending moment across the shear plane, WKHÀH[XUDOFRPSUHVVLRQDQGWHQVLRQIRUFHVDUHLQHTXLOLEULXPDQGGRQRWFKDQJHWKHUHVXOWDQWFRPSUHVVLRQIRUFH acting across the shear plane or the shear-friction resistance. Thus, it is not necessary to provide additional UHLQIRUFHPHQWWRUHVLVWWKHÀH[XUDOWHQVLRQVWUHVVHVXQOHVVWKHUHTXLUHGÀH[XUDOWHQVLRQUHLQIRUFHPHQWH[FHHGVWKH DPRXQWRIVKHDUWUDQVIHUUHLQIRUFHPHQWSURYLGHGLQWKHÀH[XUDOWHQVLRQ]RQH 7-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

EŽ

zĞƐ

EŽ

zĞƐ

A

Figure 7.40 Design procedure for columns.

7-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

Provide dowel bars to satisfy the requirements in ACI 16.3 for connections to foundations ሺSect. 7.9ሻ Figure 7.40 Design procedure for columns (cont.).

Full tension anchorage of the shear-friction reinforcement must be provided into the foundation and into the column. The lengths of these bars are determined in the same way as those for vertical transfer where tension forces are present. The area of the dowel bars is usually determined initially based on the requirements for vertical transfer. That area is FRPSDUHGZLWKWKHDUHDUHTXLUHGIRUKRUL]RQWDOWUDQVIHUDQGWKHODUJHURIWKHWZRLVSURYLGHGDWWKHLQWHUIDFH

7.10 Design Procedure 7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHFROXPQVVXEMHFWHGWRD[LDOFRPSUHVVLRQIRUFHVDQGXQLD[LDOEHQGLQJ,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVWDEOHQXPEHUV DQG¿JXUHQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQLQWKLVFKDSWHUFDQEHIRXQG 7-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.11 Examples 7.11.1

Example 7.1 – Determination of Preliminary Column Size: Building #1 (Framing Option B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJDUHFWDQJXODUWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored axial forces

Table 3.3

Roof:

Floors:

This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible. $VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOH Table 7.30 Summary of Axial Forces for Column C3 Load Case

Axial Force (kips)

Dead



Roof live

11.8

Live



Load Combination $&,(TD



$&,(TE

661.6

$&,(TF



Step 2 – Determine the gross area of the column

Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement a tied column is the following using the largest LQ7DEOH

the required gross area of

Eq. (7.30)

With a 1 percent reinforcement ratio, an 18-in. square column is adequate

7-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG column.

is equal to

ZKLFKFRUUHVSRQGVWRDLQVTXDUH

An 18-in. square column is selected. Comments.,WLVVKRZQLQ([DPSOHWKDWDQLQWKLFNVODELVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK UHTXLUHPHQWVEDVHGRQLQVTXDUHFROXPQV7KHUHTXLUHGVODEWKLFNQHVVEDVHGRQLQFROXPQVLVHTXDOWR ,WFDQEHVKRZQWKDWWZRZD\VKHDUUHTXLUHPHQWVDUHQRWVDWLV¿HGZLWK LQFROXPQVDQGDQLQWKLFNVODE7KHLQFROXPQVFDQEHXVHGKRZHYHULIVKHDUUHLQIRUFHPHQWLQWKHVODE is provided. 7.11.2

Example 7.2 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ %DVVXPLQJDQLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the required area of longitudinal reinforcement

Based on an 18-in. square tied column, the required reinforcement ratio,

is equal to the following: Eq. (7.32)

6HOHFWEDUV Step 2 – Check the minimum number of longitudinal bars and the minimum face dimension of the column

For rectangular, tied columns: Minimum number of longitudinal bars with rectangular ties Minimum face dimension

IRUEDUVSHUIDFHDQGQRUPDOODSVSOLFHV

Table 7.16 Table 7.17

8VHEDUV 7.11.3

Example 7.3 – Determination of Transverse Reinforcement: Building #1 (Framing Option B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% DVVXPLQJDQLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the required tie bar size

ACI 25.7.2.2

%HFDXVHWKHVKHDUIRUFHGXHWRWKHIDFWRUHGJUDYLW\ORDGVLVQHJOLJLEOHWKHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ $&, )RUORQJLWXGLQDOEDUVXVHWLHV 7-63

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the required tie bar spacing

ACI 25.7.2.1

Because the shear force due to the factored gravity loads is negligible, the required tie bar spacing is determined usLQJ$&,

8VHWLHVVSDFHGDWLQRQFHQWHUWKLVPDWFKHVWKHYDOXHLQ)LJXUH 7.11.4

Example 7.4 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing Option B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% spread footing (see Figure 1.1). Also assuming an 18-in. square tied column supported by a and DVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Check the bearing loads on the column and footing

ACI 16.3.3.4

• Bearing strength of the column: Eq. (7.40)

• Bearing strength of the footing: Eq. (7.41)

Footing thickness +RUL]RQWDOSURMHFWLRQIRUDVORSH

Figure 7.37

Projected length Therefore,

Step 2 – Determine the required interface reinforcement

ACI 16.3.4.1

Because the design bearing strengths are adequate for the column and footing, provide minimum area of reinforcement across the interface:

3URYLGHGRZHOEDUV 7-64

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Check the development of the dowel bars in the footing

Determine the compression development length,

RIWKHGRZHOEDUV

Table 7.21, Note 5

Table 7.22 Table 7.13

Therefore,

$VVXPLQJWZROD\HUVRIEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJWKLFNQHVV for the development of the dowel bars:

Eq. (7.43)

Because the minimum LVOHVVWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQWKHKRRNHGGRZHOEDUV can be fully developed in the footing for compression. Step 4 – Determine the development length of the dowel bars in the column

The dowel bars are lap spliced to the longitudinal reinforcement in the column. Because the dowel bars are smaller must be greater than or equal to the in diameter than the longitudinal bars, the compression lap splice length, larger of the following: 1. Development length in compression,

RIWKHORQJLWXGLQDOEDUV

7-65

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

 Compression lap splice length,

RIWKHGRZHOEDUV

Table 7.21

3URYLGHDODSVSOLFHOHQJWKRIIWLQ 1HJOLJLEOHKRUL]RQDOIRUFHVDUHWUDQVIHUUHGIURPWKHFROXPQWRWKHIRRWLQJVRKRUL]RQWDOIRUFHWUDQVIHULVQRWLQYHVWLJDWHG 5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH ͳԢǦ͸ԣ ͳԢǦ͸ԣ



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Figure 7.41 Reinforcement details for column C3 in Examples 7.1 through 7.4.

7.11.5

Example 7.5 – Determination of Preliminary Column Size: Building #1 (Framing Option B), Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJWKHFROXPQLVFLUFXODUZLWKWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIWLHVVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODE DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW

7-66

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the factored axial forces

Table 3.3

Roof:

Floors:

This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible. $VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHLQ([DPSOH Step 2 – Determine the gross area of the column

Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement a tied column is the following using the largest LQ7DEOH

the required gross area of

Eq. (7.30)

:LWKDSHUFHQWUHLQIRUFHPHQWUDWLRDLQFLUFXODUFROXPQLVDGHTXDWH )RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG LQFLUFXODUFROXPQ

is equal to

which corresponds approximately to a

$LQFLUFXODUFROXPQLVVHOHFWHG 7.11.6

Example 7.6 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B), Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ %DVVXPLQJDLQFLUFXODUWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the required area of longitudinal reinforcement

%DVHGRQDLQFLUFXODUWLHGFROXPQWKHUHTXLUHGUHLQIRUFHPHQWUDWLR

is equal to the following: Eq. (7.32)

6HOHFWEDUV

7-67

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Check the minimum and maximum number of longitudinal bars

For circular, tied columns: Minimum number of longitudinal bars with circular ties

Table 7.16

0D[LPXPQXPEHURIEDUV

Table 7.19

IRUDLQFLUFXODUFROXPQDQGQRUPDOODSVSOLFHV

8VHEDUV Comments.,WFDQEHVKRZQWKDWWKHVL]HDQGVSDFLQJRIWKHFLUFXODUWLHVDQGWKHQXPEHUDQGVL]HRIWKHGRZHOEDUV IRUWKHLQFLUFXODUWLHGFROXPQDUHWKHVDPHDVWKRVHGHWHUPLQHGLQ([DPSOHVDQGIRUWKHLQVTXDUH WLHGFROXPQVHH)LJXUH  ͳԢǦͺԣ



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Figure 7.42 Reinforcement details for column C3 in Examples 7.5 and 7.6.

7.11.7

Example 7.7 – Determination of Preliminary Column Size: Building #1 (Framing Option B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJDFLUFXODU and VSLUDOFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW

7-68

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored axial forces

Table 3.3

Roof:

Floors:

This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible. $VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHLQ([DPSOH Step 2 – Determine the gross area of the column

Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement a spiral column is the following using the largest LQ7DEOH

the required gross area of

Eq. (7.31)

With a 1 percent reinforcement ratio, a 19-in. circular column is adequate )RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG 18-in. circular column.

is equal to

which corresponds approximately to an

An 18-in. circular column is selected. 7.11.8

Example 7.8 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ %DVVXPLQJDQLQFLUFXODUVSLUDOFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQDQG*UDGHUHLQIRUFHPHQW crete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the required area of longitudinal reinforcement

Based on an 18-in. circular spiral column, the required reinforcement ratio,

is equal to the following: Eq. (7.32)

6HOHFWEDUV

7-69

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Check the minimum and maximum number of longitudinal bars

For circular, spiral columns: Minimum number of longitudinal bars with spirals

Table 7.16

0D[LPXPQXPEHURIEDUV

Table 7.19

for an 18-in. circular column and normal lap splices

8VHEDUV 7.11.9

Example 7.9 – Determination of Transverse Reinforcement: Building #1 (Framing Option B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% DVVXPLQJDQLQFLUFXODUVSLUDOFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the required spiral bar size

ACI 25.7.3.2

%HFDXVHWKHVKHDUIRUFHGXHWRWKHIDFWRUHGJUDYLW\ORDGVLVQHJOLJLEOHWKHUHTXLUHGVSLUDOEDUVL]HLVGHWHUPLQHGXVLQJ$&, 6SLUDOEDUVPXVWKDYHDGLDPHWHURIDWOHDVWLQVRXVHDVSLUDO Step 2 – Determine the required spiral bar spacing

ACI 25.7.3.1, 25.7.3.3

Because the shear force due to the factored gravity loads is negligible, the required spiral bar spacing is determined XVLQJ$&,

Eq. (7.39)

8VHDVSLUDODWDLQSLWFKWKLVPDWFKHVWKHYDOXHLQ7DEOH 7.11.10 Example 7.10 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing Option B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces

'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVspread footing (see Figure 1.1). Also suming an 18-in. circular spiral column supported by a and DVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG

7-70

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Check the bearing loads on the column and footing

ACI 16.3.3.4

• Bearing strength of the column: Eq. (7.40)

Therefore, transfer the excess compression force by interface reinforcement: Eq. (7.42)

• Bearing strength of the footing: Eq. (7.41)

Footing thickness +RUL]RQWDOSURMHFWLRQIRUDVORSH Projected diameter of cone Therefore,

Step 2 – Determine the required interface reinforcement

ACI 16.3.4.1

It is determined in Step 1 that the required area of interface reinforcement is 3URYLGHGRZHOEDUV Step 3 – Check the development of the dowel bars in the footing

Determine the compression development length,

RIWKHGRZHOEDUV

Table 7.21, Note 5

(the dowel bars are not enclosed by any transverse reinforcement in the footing)

Table 7.22 Table 7.13

7-71

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPLQJWZROD\HUVRIEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJWKLFNQHVV IRUWKHGHYHORSPHQWRIWKHGRZHOEDUV

Eq. (7.43)

Because the minimum LVOHVVWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQWKHKRRNHGGRZHOEDUV can be fully developed in the footing for compression. Step 4 – Determine the development length of the dowel bars in the column

7KHGRZHOEDUVDUHODSVSOLFHGWRWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQ%HFDXVHWKHGRZHOEDUVDUH must be greater than or equal to the WKHVDPHVL]HDVWKHORQJLWXGLQDOEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWK following:

Table 7.21

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Figure 7.43 Reinforcement details for column C3 in Examples 7.7 through 7.10.

7-72

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ͳԢǦ͸ԣ

͹Ǧ͓͸†‘™‡Ž„ƒ”•

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For spiral columns,

LVSHUPLWWHGWREHPXOWLSOLHGE\ ACI 10.7.5.2.1(b)

Provide a lap splice length of 1 ft-6 in. 1HJOLJLEOHKRUL]RQDOIRUFHVDUHWUDQVIHUUHGIURPWKHFROXPQWRWKHIRRWLQJVRKRUL]RQWDOIRUFHWUDQVIHULVQRWLQYHVWLgated. 5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH 7.11.11 Example 7.11 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1 (Framing Option B), Rectangular, Tied Column, Grade 60 Longitudinal Reinforcement

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ʹǤͶͶԣ

'HWHUPLQHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHLQVTXDUHWLHGFROXPQLQ([DPSOHV WKURXJK7KHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGDFOHDUFRYHURILQLVSURYLGHGWRWLHV DQG*UDGHUHLQIRUFHPHQW VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK

ͳͺԣ Figure 7.44 Reinforced concrete column in Example 7.11.

Nominal and design axial forces and bending moments are determined at the following points: A. B. C. D. E.

Pure axial compression Balanced point Stress in reinforcement farthest from compression face Stress in reinforcement farthest from compression face Pure bending

Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for all the other points. In the calculations, it is assumed compression is positive. A. Pure axial compression

The nominal axial compressive strength for a tied column is determined by the following: Eq. (7.13)

7-73

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The section is compression-controlled, so

Table 7.10

B. Balanced point Step 1 – Determine the neutral axis depth

Table 7.11

%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQ (Note: ACI in the reinforcement farthest from the compression face is IRU*UDGHGHIRUPHGUHLQIRUFHPHQWEXWWKDWYDOXHLVQRWXVHGLQWKLVH[DPSOH 7KH SHUPLWV neutral axis depth, is determined by the following equation: Figure 7.10

Step 2 – Determine the depth of the equivalent stress block Table 7.11 Step 3 – Determine the resultant compression force

Step 4 – Determine the strain in the reinforcement

Layer 1:

/D\HU Step 5 – Determine the stress in the reinforcement

Layer 1: /D\HU Step 6 – Determine the force in the reinforcement

Layer 1 /D\HU

7-74

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 7 – Determine the nominal and design axial strengths

Step 8 – Determine the nominal and design flexural strengths

C. Stress in reinforcement farthest from the compression face = 0

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH Table 7.31 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)

















D. Stress in reinforcement farthest from the compression face = 0.5fy

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH

7-75

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.32 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0.5fy

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)











E. Pure bending

There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH

1,400 1,200

Axial Force ሺkipsሻሻ

1,000

݂௦ ൌ 0

800

݂௦ ൌ 0.5݂௬ 600 400 200 0 0

100

200

300

400

Bending Moment ሺft-kipsሻ

Figure 7.45 Nominal and design strength interaction diagrams for the column in Example 7.11.

7-76

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.33 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)

 

7KHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH 7.11.12 Example 7.12 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1 (Framing Option B), Rectangular, Tied Column, Grade 100 Longitudinal Reinforcement

'HWHUPLQHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHLQVTXDUHWLHGFROXPQLQ([DPSOHV WKURXJK7KHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGDFOHDUFRYHURILQLVSURYLGHGWRWLHV DQG*UDGHUHLQIRUFHPHQW VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK Nominal and design axial forces and bending moments are determined at the following points: A. B. C. D. E.

Pure axial compression Balanced point Stress in reinforcement farthest from compression face Stress in reinforcement farthest from compression face Pure bending

Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for all the other points. In the calculations, it is assumed compression is positive. A. Pure axial compression

The nominal axial compressive strength for a tied column is determined by the following:

Eq. (7.13)

Note that is used in the equation for LQDFFRUGDQFHZLWKWKHUHTXLUHPHQWLQ$&,ZKLFK limits WRNVLIRUPHPEHUVVXEMHFWHGWRSXUHD[LDOFRPSUHVVLRQ The section is compression-controlled, so

Table 7.10

7-77

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

B. Balanced point Step 1 – Determine the neutral axis depth

Table 7.11

%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQLQ The neutral the reinforcement farthest from the compression face is axis depth, is determined by the following equation: Figure 7.10

Step 2 – Determine the depth of the equivalent stress block Table 7.11 Step 3 – Determine the resultant compression force

Step 4 – Determine the strain in the reinforcement

Layer 1:

/D\HU Step 5 – Determine the stress in the reinforcement

Layer 1: /D\HU Step 6 – Determine the force in the reinforcement

Layer 1 /D\HU Step 7 – Determine the nominal and design axial strengths

7-78

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 8 – Determine the nominal and design flexural strengths

C. Stress in reinforcement farthest from the compression face = 0

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH Table 7.34 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)

















D. Stress in reinforcement farthest from the compression face = 0.5fy

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH

7-79

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.35 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0.5fy

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)











E. Pure bending

There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH  1,600

Point A 1,400

Axial Force ሺkipsሻሻ

1,200

݂௦ ൌ 0

1,000 800 600

݂௦ ൌ 0.5݂௬ 

400 200 0 0

100

200

300

400

Bending Moment ሺft-kipsሻ

Figure 7.46 Nominal and design strength interaction diagrams for the column in Example 7.12.

7-80

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.36 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)











7KHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH3RLQW$RQWKHQRPLwhich corresponds nal strength interaction diagram represents the maximum nominal axial force WRWKHNVLOLPLWIRUSXUHD[LDOFRPSUHVVLRQLQ$&,$KRUL]RQWDOOLQHFRQQHFWVWKDWSRLQWWRWKHSRLQW and where 7.11.13 Example 7.13 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1 (Framing Option B), Circular, Tied Column, Grade 60 Longitudinal Reinforcement

ʹǤͶͶԣ

'HWHUPLQHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHLQFLUFXODUWLHGFROXPQLQ([DPSOHV DQG7KHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGDFOHDUFRYHURILQLVSURYLGHGWRWLHVVHH DQG*UDGHUHLQIRUFHPHQW )LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK

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ʹͲԣ Figure 7.47 Reinforced concrete column in Example 7.13.

Nominal and design axial forces and bending moments are determined at the following points: A. B. C. D. E.

Pure axial compression Balanced point Stress in reinforcement farthest from compression face Stress in reinforcement farthest from compression face Pure bending

7-81

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for all the other points. In the calculations, it is assumed compression is positive. A. Pure axial compression

The nominal axial compressive strength for a tied column is determined by the following:

Eq. (7.13)

where The section is compression-controlled, so

Table 7.10

B. Balanced point Step 1 – Determine the neutral axis depth

Table 7.11

%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQ (Note: ACI in the reinforcement farthest from the compression face is IRU*UDGHGHIRUPHGUHLQIRUFHPHQWEXWWKDWYDOXHLVQRWXVHGLQWKLVH[DPSOH 7KH SHUPLWV neutral axis depth, is determined by the following equation: Figure 7.12

Step 2 – Determine the depth of the equivalent stress block Table 7.11 Step 3 – Determine the section properties of the compression zone Figure 7.13

Step 4 – Determine the resultant compression force

Step 5 – Determine the strain in the reinforcement

7-82

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Layer 1:

/D\HU

Layer 3: Step 6 – Determine the stress in the reinforcement

Layer 1: /D\HU Layer 3: Step 7 – Determine the force in the reinforcement

Layer 1 /D\HU Layer 3 Step 8 – Determine the nominal and design axial strengths

Step 9 – Determine the nominal and design flexural strengths

C. Stress in reinforcement farthest from the compression face = 0

7-83

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 7.13

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH Table 7.37 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)

























D. Stress in reinforcement farthest from the compression face = 0.5fy

Figure 7.13

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH

7-84

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.38 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest from Compression Face = 0.5fy

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)



















E. Pure bending

There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that

Figure 7.13

7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH Table 7.39 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending

dsi (in.)

İsi

fsi (ksi)

Fsi (kips)







36.6

 

7-85

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH  1,400 1,200

݂௦ ൌ 0

Axial Force ሺkipsሻሻ

1,000 800

݂௦ ൌ 0.5݂௬  600 400 200 0 0

50

100

150

200

250

300

Bending Moment ሺft-kipsሻ

Figure 7.48 Nominal and design strength interaction diagrams for the column in Example 7.13.

7.11.14 Example 7.14 – Determination of Preliminary Column Size: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Column Subjected to Uniaxial Bending and Axial Forces

'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ&IRUZLQGIRUFHVLQ WKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDUHFWDQJXODUWLHGFROXPQVHH)LJXUH $OVRDVVXPHDLQWKLFNVODE DQG*UDGHUHLQIRUFHPHQW E\LQEHDPVQRUPDOZHLJKWFRQFUHWHZLWK 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored axial forces and bending moments

Roof:

Floors:

7-86

Table 3.3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

This column is part of the LFRS and the bending moments due to gravity load effects are small compared to the bending moments due to wind load effects in the north-south direction, so the former are not considered. $VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ&LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW665 DQGVLGHVZD\WRWKHOHIW66/  Table 7.40 Summary of Axial Forces and Bending Moments for Column C1 Bending Moment (ft-kips) Load Case

Axial Force (kips) Top

Bottom

Dead







Roof live







Live







Wind



Load Combination $&,(TD







$&,(TE



















63.9



63.9

$&,(TF

SSR



SSL



SSR



SSL



SSR



SSL



$&,(TG

$&,(TI

Step 2 – Determine the gross area of the column

$SUHOLPLQDU\FROXPQVL]HLVLQLWLDOO\GHWHUPLQHGEDVHGRQIDFWRUHGD[LDOIRUFHRQO\ $VVXPLQJDSHUFHQWUHLQIRUFHPHQWUDWLRIRUWKHORQJLWXGLQDOUHLQIRUFHPHQW a tied column is the following using the largest LQ7DEOH

the required gross area of

Eq. (7.30)

$LQVTXDUHFROXPQKDVDJURVVDUHD +RZHYHUDLQVTXDUHFROXPQLVVHOHFWHGIRUWKHIROORZLQJUHDVRQV WKHFROXPQLVVXEMHFWHGWREHQGLQJPRPHQWVWKDWKDYHQRWEHHQDFFRXQWHGIRULQWKHLQLWLDOGHVLJQ LWKDVQRWEHHQGHWHUPLQHGZKHWKHUWKHFROXPQLV part of a nonsway or sway frame and whether slenderness effects need to be considered or not and (3) preliminary DQDO\VLVLQGLFDWHVWKDWDWOHDVWLQVTXDUHFROXPQVDUHQHHGHGWRSURYLGHVXI¿FLHQWRYHUDOOODWHUDOVWLIIQHVVIRUWKH EXLOGLQJ7KHFROXPQVL]HFDQEHDGMXVWHGDFFRUGLQJO\RQFHPRUHFDOFXODWLRQVDUHSHUIRUPHG

7-87

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.11.15 Example 7.15 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A

'HWHUPLQHZKHWKHUWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ&LVDQRQVZD\RUVZD\IUDPHDVVXPLQJDOOWKH FROXPQVDUHLQVTXDUHVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHIRUDVXPPDU\RIWKHORDGVDWWKHYDULRXVOHYHOV Step 1 – Determine the total loads in the first story

• Roof:

• 7\SLFDOÀRRU

The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI (TG SURGXFHVWKHODUJHVWORDG

Step 2 – Determine the stability index for the first story

An elastic analysis of the structure was performed with the building subjected to the north-south wind forces in 7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRU the columns, beams, and slabs. The following results were obtained from the analysis:

The stability index,

is determined from the following equation: Table 7.3

Because

7-88

WKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPHVHH$&, 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.11.16 Example 7.16 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame

&KHFNLIVOHQGHUQHVVHIIHFWVPXVWEHFRQVLGHUHGIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ &DVVXPLQJWKHFROXPQLVLQVTXDUHVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG ,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVQRQVZD\LQWKHQRUWKVRXWKGLUHFWLRQ7KHUHIRUH VOHQGHUQHVVHIIHFWVFDQEHQHJOHFWHGZKHQWKHIROORZLQJHTXDWLRQLVVDWLV¿HG

Table 7.6

For all load combinations including wind,

Table 7.40

The column is bent in double curvature for all load combinations; therefore, slenderness can be neglected where the IROORZLQJLVVDWLV¿HG

Assuming the effective length factor, 7DEOH 

LVHTXDOWRWKHZKLFKLVWKHPD[LPXPYDOXHIRUQRQVZD\IUDPHVVHH

Therefore, slenderness effects need not be considered. 7.11.17 Example 7.17 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Uniaxial Bending and Axial Forces

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ &DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the required area of longitudinal reinforcement

Eq. (7.32)

%DVHGRQDLQVTXDUHWLHGFROXPQWKHUHTXLUHGUHLQIRUFHPHQWUDWLR is equal to the following for the largest IDFWRUHGD[LDOIRUFHLQ7DEOHZKLFKRFFXUVZLWKRXWDQ\DSSUHFLDEOHEHQGLQJPRPHQWV

Therefore, use the minimum required

7-89

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6HOHFWEDUV Step 2 – Check if the selected longitudinal reinforcement is adequate for all load combinations

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUVLVJLYHQLQ)LJXUH$OVR VKRZQLQWKH¿JXUHDUHORDGFRPELQDWLRQSRLQWVIRUWKHIDFWRUHGD[LDOIRUFHVDQGEHQGLQJPRPHQWVLQ7DEOH  900 800

݂௦ ൌ 0

Axial Force ሺkipsሻሻ

700 600

݂௦ ൌ 0.5݂௬  500

 300      200 400

 

100 0 0

50

100

150

200

250

300

350

Bending Moment ሺft-kipsሻ

Figure 7.49 Design strength interaction diagram for column C1.

,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHFROXPQLVDGHTXDWHIRUDOOIDFWRUHGORDGFRPELQDWLRQV Step 3 – Check the minimum number of longitudinal bars and the minimum face dimension of the column

For rectangular, tied columns: Minimum number of longitudinal bars with rectangular ties Minimum face dimension

IRUEDUVSHUIDFHDQGQRUPDOODSVSOLFHV

Table 7.16 Table 7.17

8VHEDUV Comments.,WLVHYLGHQWIURP)LJXUHWKDWWKHLQFROXPQZLWKEDUVSRVVHVVHVVLJQL¿FDQWO\PRUH VWUHQJWKWKDQUHTXLUHGEDVHGRQWKHIDFWRUHGORDGFRPELQDWLRQV+RZHYHUDVQRWHGLQ([DPSOHLQFROXPQV are needed primarily for lateral stiffness of the moment frames. Furthermore, the column is also part of the LFRS in the east-west direction and must also be designed for combined gravity and wind load effects for wind in that direcWLRQ'HFUHDVLQJWKHDPRXQWRIORQJLWXGLQDOUHLQIRUFHPHQWEHORZSHUFHQWLQDFFRUGDQFHZLWK$&,LVQRW done in this example.

7-90

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.11.18 Example 7.18 – Determination of Transverse Reinforcement: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Uniaxial Bending and Axial Forces

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ &DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the required tie bar size

ACI 25.7.2.2

7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&, )RUORQJLWXGLQDOEDUVXVHWLHV Step 2 – Determine the required tie bar spacing

ACI 25.7.2.1

7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR determine if they are adequate for shear strength requirements.

7U\WLHVVSDFHGDWLQRQFHQWHU Step 3 – Check the adequacy of the tie bar size and spacing for shear

ACI 22.5

Maximum factored shear force, LVREWDLQHGIURP$&,(TVG DQGI EDVHGRQWKHIDFWRUHGEHQGLQJ PRPHQWGXHWRWKHZLQGORDGHIIHFWVDQGLVHTXDOWRNLSVVHH7DEOHDQG)LJXUH  Check if the design shear strength of the concrete, LVSURYLGHGRUQRWVHH7DEOH 

alone is adequate, which is determined based on whether

Eq. (7.26)

The minimum area of shear reinforcement is greater than SURYLGHGE\WKHWLHVVSDFHG 18 in. on center. Because LVUHODWLYHO\VPDOOLWLVQRWQHFHVVDU\WRLQFUHDVHWKHWLHEDUVL]HDQGRUGHFUHDVHWKH is determined based on spacing, so Table 7.12

7-91

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܲ௨  ‫ܯ‬௨ǡ௧௢௣ ൌ ʹͲǤͶˆ–Ǧ‹’• κ௨ ൌ ͳͶǤͲ െ ሾʹͶǤͲȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͵ǤͲˆ–

ܸ௨ 

ܸ௨ ൌ

ʹͲǤͶ ൅ ͸͵Ǥͻ ൌ ͸Ǥͷ‹’• ͳ͵ǤͲ

ܸ௨  ‫ܯ‬௨ǡ௕௢௧௧௢௠ ൌ ͸͵Ǥͻˆ–Ǧ‹’• ܲ௨  Figure 7.50 Factored shear force for column C1.

Eq. (7.23)

Table 7.13

WKHUHDUHEDUVORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWK IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU  Use minimum

FRUUHVSRQGLQJWR$&,(TI EHFDXVHWKLVUHVXOWVLQPLQLPXP ACI 22.5.5.1.2

Also,

so minimum shear reinforcement is not required.

7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU

7-92

ACI 10.6.2.1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7.11.19 Example 7.19 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Uniaxial Bending and Axial Forces

'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ&DVspread footing (see Figure 1.1). Also assume VXPLQJDLQVTXDUHWLHGFROXPQVXSSRUWHGE\D DQG*UDGH DLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Check the bearing stresses on the column and footing

ACI 16.3.3.4

Bearing strength of the column. is determined for pure compression using the factored axial force determined by ACI Factored bearing stress, (TVE DQGIRUFRPSUHVVLRQSOXVEHQGLQJXVLQJWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\ $&,(TVG DQGI >VHH7DEOH@ • $&,(TE 

• $&,(TG 

• $&,(TI 

Design bearing strength:

Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQVLRQIRUFHV are adequately transferred from the column into the footing.

7-93

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Bearing strength of the footing. There is no need to check the bearing strength of the footing because interface reinforcement must be provided due WRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(TI  Step 2 – Determine the required interface reinforcement

ACI 16.3.4.1

7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHVWKDWERWK the compression and tension forces are adequately transferred through the interface. Check minimum area of reinforcement across the interface:

Step 3 – Check the development of the dowel bars in the footing

,WLVHYLGHQWIURP)LJXUHWKDWWKHVWUHVVLQWKHORQJLWXGLQDOUHLQIRUFHPHQWIDUWKHVWIURPWKHH[WUHPHFRPSUHVVLRQ IDFHLVWHQVLOHIRURQHRIWKHORDGFRPELQDWLRQVWKDWLVWKHORDGFRPELQDWLRQIDOOVZLWKLQ=RQHRIWKHLQWHUDFWLRQ GLDJUDPVHH)LJXUH 7KHUHIRUHWKHGRZHOEDUVPXVWEHGHYHORSHGIRUWHQVLRQLQWRWKHIRRWLQJDQGLQWRWKH column). $VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSPHQW RIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN length,

Eq. (7.44)

Table 7.27

$VVXPLQJWZROD\HUVRIÀH[XUDOEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJ thickness for the development of the dowel bars: Eq. (7.45)

7-94

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The minimum LVJUHDWHUWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQ7KXVLQFUHDVHWKHIRRWLQJ WKLFNQHVVWRLQWRDFFRPPRGDWHGHYHORSPHQWRIWKHGRZHOEDUV Step 4 – Determine the development length of the dowel bars in the column

The dowel bars must be lap spliced to the longitudinal reinforcement in the column using a tension lap splice. Because all the dowel bars are spliced at the same location, a Class B tension lap splice is required (ACI Table   Development length in tension,

RIWKHEDUV Eq. (7.36)

Table 7.24

Set

ACI 25.4.2.4

Therefore,

3URYLGHDODSVSOLFHOHQJWKRIIWLQ Step 5 – Check horizontal force transfer

,WLVGHWHUPLQHGLQ([DPSOHWKDW LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGIRRWLQJ7KH required area of shear-friction reinforcement is determined as follows assuming the surface between the column and footing has not been intentionally roughened: Eq. (7.46), Table 7.28

7KHGRZHOEDUVSURYLGHV

across the interface, which is greater than the required amount of

7-95

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Check the upper shear limit: Table 7.29

5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH ͳԢǦͺԣ ͳԢǦͺԣ



ͶǦ͓ͻ

ͶԢǦʹԣ

ͶǦ͓ͻ†‘™‡Ž„ƒ”•

͓ͻˆ‘‘–‹‰„ƒ”•ሺ–›’Ǥሻ ͺԢǦ͸ԣ

͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ

͵ԢǦͲԣ

͓͵–‹‡•̷ͳԢǦ͸ԣ‘Ǥ…Ǥ

Figure 7.51 Reinforcement details for column C1 in Examples 7.14 through 7.19.

7.11.20 Example 7.20 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Biaxial Bending and Axial Forces

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ(LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ &DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW

7-96

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the required area of longitudinal reinforcement

This corner column is subjected to axial forces and biaxial bending, so there is no direct way of determining the required area of longitudinal reinforcement. Therefore, initially try the minimum required

6HOHFWEDUV Step 2 – Check if the selected longitudinal reinforcement is adequate for all load combinations

An elastic analysis of the structure was performed with the building subjected to the north-south and east-west wind IRUFHVLQ7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHFULWLFDOZLQGORDGLQJIRUDFRUQHUFROXPQRFFXUVZKHQSHUFHQWRIWKHZLQGIRUFHVLQERWKGLUHFWLRQVDUHDSSOLHGVLPXOWDQHRXVO\WRWKHVWUXFWXUH/RDG&DVH LQ$6&(6(,)LJ 7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRUWKHFROXPQVEHDPVDQGVODEV in the analysis. $VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ(LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHIRUVLGHsway to the right (SSR) and sidesway to the left (SSL) in both principal directions. Table 7.41 Summary of Axial Forces and Bending Moments for Column E1 Bending Moment (ft-kips) Axial Force (kips)

Load Case

North-South Top

East-West

Bottom

Top





ņ

ņ

Dead



Roof live



Live

38.3

6.8

11.9

$&,(TD





38.8

$&,(TE













ņ

Bottom

ņ

Wind Load Combination

$&,(TF

$&,(TG $&,(TI

SSR



SSL



SSR



SSL



SSR



SSL



 

 







 



9.1

,WLVGHWHUPLQHGLQ([DPSOHWKDWWKH¿UVWVWRU\RIWKLVEXLOGLQJLVQRQVZD\,WFDQDOVREHGHWHUPLQHGWKDWVOHQderness effects need not be considered.

7-97

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

:LWKORQJLWXGLQDOEDUVWKLVFROXPQKDVWKHVDPHEHQGLQJPRPHQWFDSDFLW\DERXWERWKSULQFLSDOD[HV7KXV WKHDSSUR[LPDWHPHWKRGGHSLFWHGLQ)LJXUHFDQEHXVHGWRGHWHUPLQHWKHDGHTXDF\RIWKHFROXPQIRUHDFKORDG FRPELQDWLRQLQ7DEOH To illustrate the design procedure, the column will be checked for the factored axial force and bending moments REWDLQHGIURP$&,(TG IRUVLGHVZD\WRWKHULJKW WKHXQLD[LDOGHVLJQPRPHQWVWUHQJWKVDUHWKHIROORZLQJVHH)LJXUH 

At an axial force

 900 800

Axial Force ሺkipsሻ

700 600 500 400 300 ܲ௨ ൌ 204.3 kips

200 100 0 0

50

100

150

200

Bending Moment ሺft-kipsሻ

250 ߶‫ܯ‬௡௢

300 ேௌ

ൌ ߶‫ܯ‬௡௢

350 ாௐ

ൌ 274.0 ft-kips

Figure 7.52 Uniaxial design strength interaction diagram for column E1.

The approximate biaxial strength diagram corresponding to LVJLYHQLQ)LJXUH,WLVHYLGHQW IURPWKH¿JXUHWKDWWKHFROXPQLVDGHTXDWHIRUWKHDSSOLHGIDFWRUHGELD[LDOPRPHQWV It can be determined in a similar fashion that the column is adequate for all the other load combinations. 8VHORQJLWXGLQDOEDUV Comments. )RUFRPSDULVRQSXUSRVHV5HIHUHQFHZDVXVHGWRFKHFNWKHDGHTXDF\RIWKLVFROXPQ$VH[SHFWHG the column was found to be adequate.

7-98

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ሺ߶‫ܯ‬௡௢ ሻாௐ ൌ ʹ͹ͶǤͲˆ–Ǧ‹’•

ܲ௨ ൌ ʹͲͶǤ͵‹’•

ሺ‫ܯ‬௨ ሻேௌ ൌ ʹͳǤ͹ˆ–Ǧ‹’• ሺ‫ܯ‬௨ ሻாௐ ൌ ͶͺǤ͹ˆ–Ǧ‹’•

н

н ሺ‫ܯ‬௨ ሻேௌ ൌ ͶͶǤʹˆ–Ǧ‹’• ሺ‫ܯ‬௨ ሻாௐ ൌ ͵ͶǤͺˆ–Ǧ‹’• ሺ߶‫ܯ‬௡௢ ሻேௌ ൌ ʹ͹ͶǤͲˆ–Ǧ‹’•

Figure 7.53 Approximate biaxial strength diagram for column E1.

7.11.21 Example 7.21 – Determination of Transverse Reinforcement: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Biaxial Bending and Axial Forces

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ(LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ& DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the required tie bar size

ACI 25.7.2.2

7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&, )RUORQJLWXGLQDOEDUVXVHWLHV Step 2 – Determine the required tie bar spacing

ACI 25.7.2.1

7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR determine if they are adequate for shear strength requirements.

7U\WLHVVSDFHGDWLQRQFHQWHU

7-99

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Check the adequacy of the tie bar size and spacing for shear

ACI 22.5

Maximum factored shear force, is obtained from ACI Eq. (5.3.1d) based on the factored bending moments at the top and bottom of the column (see Table 7.41): • North-south wind and SSL:

• East-west wind and SSR:

alone is adequate, which is determined based on whether



Check if the design shear strength of the concrete, is provided or not (see Table 7.12).

Eq. (7.26)

The minimum area of shear reinforcement is greater than Av = 0.15 in.2 provided by the #3 ties spaced 18 in. on cenis relatively small in both directions, it is not necessary to increase the tie bar size and/or decrease ter. Because is determined based on the spacing, so



Table 7.12



Eq. (7.23)

where

Table 7.13

(there are 2-#9 bars located more than two-thirds of the overall member depth from the extreme compression fiber). which corresponds to ACI Eq. (5.3.1d) and sidesway to the left.

• North-south wind: Use axial force

7-100

ACI 22.5.5.1.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• East-west wind: Use axial force

ZKLFKFRUUHVSRQGVWR$&,(TG DQGVLGHVZD\WRWKHULJKW ACI 22.5.5.1.2

Check if interaction of shear forces acting along both the north-south and east-west directions must be considered VHH$&, 

%HFDXVHWKHUDWLRVDUHOHVVWKDQLQWHUDFWLRQRIVKHDUIRUFHVDUHSHUPLWWHGWREHQHJOHFWHG 7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU 7.11.22 Example 7.22 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Uniaxial Bending and Axial Forces, Slenderness Effects

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ &DVVXPLQJDLQVTXDUHWLHGFROXPQPXVWEHXVHGIRUWKLVFROXPQRQO\GXHWRDUFKLWHFWXUDOUHDVRQVVHH)LJXUH and  $OVRDVVXPHDLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored axial forces

Table 3.3

Roof:

7-101

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Floors:

This column is part of the LFRS for wind in the north-south direction and the bending moments due to gravity load effects are negligible. $VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ&LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW665 DQGVLGHVZD\WRWKHOHIW66/  Table 7.42 Summary of Axial Forces and Bending Moments for Column C1 Bending Moment (ft-kips) Load Case

Axial Force (kips) Top

Bottom

Dead







Roof live







Live







Wind



Load Combination $&,(TD







$&,(TE













SSR





SSL



SSR



SSL



SSR



SSL



$&,(TF

$&,(TG

$&,(TI

  6.3  6.3

Step 2 – Check if slenderness effects must be considered

,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVQRQVZD\ZKLFKLVVWLOOYDOLGHYHQWKRXJKWKHVL]H of this one column has been reduced). Therefore, slenderness effects can be neglected when the following equation LVVDWLV¿HG

Table 7.6

For all load combinations including wind,

7-102

Table 7.42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The column is bent in double curvature for all load combinations; therefore, slenderness can be neglected where the IROORZLQJLVVDWLV¿HG

Assuming the effective length factor, 7DEOH 

LVHTXDOWRWKHZKLFKLVWKHPD[LPXPYDOXHIRUQRQVZD\IUDPHVVHH

Therefore, slenderness effects must be considered. Step 3 – Determine the magnified moments

0DJQL¿HGPRPHQWV tion:

ACI 6.6.4.5

for each load combination are determined for this nonsway frame by the following equaEq. (7.2)

$VXPPDU\RIWKHPDJQL¿FDWLRQIDFWRUV LVJLYHQLQ7DEOHIRUHDFKORDGFRPELQDWLRQZLWKQRQ]HURYDOXHV ,WLVDVVXPHGWKDWWKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUV DQGWLHV7KH of results for sidesway to the right and sidesway to the left are equal because the gravity bending moments are equal to ]HUR Table 7.43 Magnified Moments for Column C1

(EI)eff

M2,min

Mc

(ft-kips)

(ft-kips)



















16.8

16.8

ȕdns

(kip-in2)

Pc (kips)

Cm

į

ACI Eq. F









ACI Eq. G







ACI Eq. I







Load Combination

,WLVHYLGHQWIURP7DEOHWKDW in all load combinations, which means the structural system is stable and is greater than in all load QHHGQRWEHUHYLVHG$&, $OVRWKHPLQLPXPPRPHQW FRPELQDWLRQVVHH7DEOH  6DPSOHFDOFXODWLRQVIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(TG DUHDVIROORZV ACI Eq. (19.2.2.1a)

7-103

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(IIHFWLYHÀH[XUDOVWLIIQHVV used in this example:

FDQEHFDOFXODWHGE\DQ\RIWKHHTXDWLRQVLQ7DEOH$&,(TE LV

The moment of inertia of the reinforcement about the centroidal axis of the column, is determined by multiplying the area of the reinforcing bars by the square of the distance from the centroid of the reinforcing bars to the centroid of the column (the moment of inertia of a reinforcing bar about its centroidal axis is negligible):

Eq. (7.4)

For the column bent in double curvature: Table 7.8  400 350





Axial Force ሺkipsሻሻ

300



݂௦ ൌ 0

250



200

݂௦ ൌ 0.5݂௬

150 100 50 0 0

25

50

75

Bending Moment ሺft-kipsሻ

Figure 7.54 Design strength interaction diagram for column C1.

7-104

100

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (7.3)

Step 4 – Check the adequacy of the assumed longitudinal reinforcement

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUFROXPQ&LVJLYHQLQ)LJXUH$OVRVKRZQLQWKH¿JXUHDUHWKH load combination points for the factored axial forces LQ7DEOHDQGWKHFRUUHVSRQGLQJIDFWRUHGEHQGLQJPRLQ7DEOHZKHUHDSSOLFDEOH%HFDXVHDOOORDGFRPELQDWLRQVIDOOZLWKLQWKHGHVLJQVWUHQJWKLQWHUDFWLRQ ments diagram, the column is adequate. 8VHORQJLWXGLQDOEDUV 7.11.23 Example 7.23 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option B), Rectangular, Tied Column is Part of the LFRS, SDC A

'HWHUPLQHZKHWKHUWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%LVDQRQVZD\RUVZD\IUDPHDVVXPLQJDOOWKH FROXPQVDUHLQVTXDUHVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the total loads in the first story

Roof:

7\SLFDOÀRRU

The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI (TG SURGXFHVWKHODUJHVWORDG

7-105

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the stability index for the first story

An elastic analysis of the structure was performed with the building subjected to the north-south wind forces in 7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRU the columns, beams, and slabs. The following results were obtained from the analysis:

The stability index,

is determined from the following equation: Table 7.3

Because

WKHIUDPHLQWKH¿UVWVWRU\LVDVZD\IUDPHVHH$&, 

7.11.24 Example 7.24 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame

&KHFNLIVOHQGHUQHVVHIIHFWVPXVWEHFRQVLGHUHGIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% DVVXPLQJWKHFROXPQLVLQVTXDUHVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH ,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVDVZD\IUDPH7KHUHIRUHVOHQGHUQHVVHIIHFWVFDQ EHQHJOHFWHGZKHQWKHIROORZLQJHTXDWLRQLVVDWLV¿HG Table 7.6

The effective length factor, LVGHWHUPLQHGIURP$&,)LJ5IRUVZD\IUDPHV,QWKHDQDO\VLVWKHFROXPQVDUH At the top of the column, is determined using the DVVXPHGWREH¿[HGDWWKHEDVHVRVWLIIQHVVUDWLR UDWLRRIWKHVWLIIQHVVRIWKHFROXPQVWRWKHVWLIIQHVVRIWKHEHDPV7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHDUH used in calculating the stiffnesses.

ACI Eq. (19.2.2.1a)

&ROXPQEHORZWKHVHFRQGÀRRUOHYHO

&ROXPQDERYHWKHVHFRQGÀRRUOHYHO

7-106

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore, Eq. (7.1)

)URPWKHDOLJQPHQWFKDUWLQ$&,)LJ5IRUVZD\IUDPHVZLWK HTXDOWR

and

is approximately

Therefore, slenderness effects must be considered. 7.11.25 Example 7.25 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to Uniaxial Bending and Axial Forces, Slenderness Effects

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ %DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the factored axial forces and bending moments

Roof:

Floors:

This column is part of the LFRS. $VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ'LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW665 DQGVLGHVZD\WRWKHOHIW66/ 

7-107

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.44 Summary of Axial Forces and Bending Moments for Column D1 Bending Moment (ft-kips) Load Case

Axial Force (kips) Top

Bottom

Dead





Roof live



Live





$&,(TD



3.9

$&,(TE





316.9



ņ

ņ

Wind Load Combination

$&,(TF

SSR



SSL



SSR



SSL



SSR



SSL



$&,(TG

$&,(TI













Step 2 – Determine the total moments including slenderness effects

The total moments,

ACI 6.6.4.6

at the end of the column are determined by the following equation: Eq. (7.6)

These moments are used because they are greater than the

PRPHQWVGHWHUPLQHGE\(T 

In this example, the nonsway moments are due to gravity load effects and the sway moments ZLQGORDGHIIHFWV7KHVHPRPHQWVDUHJLYHQLQ7DEOHIRUDOOORDGFRPELQDWLRQV

are due to

Table 7.45 Summary of Nonsway and Sway Moments (ft-kips) for Column D1

M1

Load Combination

M2

M1ns

M2ns

M1s

M2s

$&,(TD

3.9

3.9

ņ

ņ

$&,(TE





ņ

ņ





ņ

ņ



116.1

$&,(TF



SSR SSL







(table continued on next page)

7-108

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.45 Summary of Nonsway and Sway Moments (ft-kips) for Column D1 (cont.)

M1

Load Combination

M2

M1ns

SSL







M2s

69.1



69.1





SSR $&,(TI SSL

M1s



SSR $&,(TG

M2ns







7KHIROORZLQJHTXDWLRQLVXVHGWRGHWHUPLQHWKHPRPHQWPDJQL¿FDWLRQIDFWRU

for sway frames: Eq. (7.7)

The stability index,

LVGHWHUPLQHGE\$&,(T IRUHDFKORDGFRPELQDWLRQWKDWLQFOXGHV

$VXPPDU\RIWKHVOHQGHUQHVVFDOFXODWLRQVIRUFROXPQ'LVJLYHQLQ7DEOH Table 7.46 Summary of Slenderness Calculations for Column D1

Ȉ3u

ǻo

Vus

(kips)

(in.)

(kips)

$&,(TD

ņ

ņ

$&,(TE

ņ

Q

įs

ņ

ņ

ņ

ņ

ņ

ņ

ņ

ņ

ņ

ņ

ņ

ņ

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1.11

SSL









1.11

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Load Combination

$&,(TF

$&,(TG

$&,(TI

M2 (ft-kips)







6DPSOHFDOFXODWLRQVIRUWKHORDGFRPELQDWLRQLQ$&,(TG IRUVLGHVZD\WRWKHULJKWDUHDVIROORZV Example 7.23, Step 1 Example 7.23, Step 2 Example 7.23, Step 2

7-109

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,WFDQEHGHWHUPLQHGEDVHGRQWKHUHTXLUHPHQWVLQ$&,WKDWPD[LPXPPRPHQWVRFFXUDWWKHHQGVRIWKH columns. Step 3 – Determine the required area of longitudinal reinforcement

A 1 percent reinforcement ratio is initially selected for this column. Therefore,  1,400 1,200

݂௦ ൌ 0 Axial Force ሺkipsሻሻ

1,000 800

݂௦ ൌ 0.5݂௬

600 400

 

200

 



0 0

100

200

300

400

500

600

700

Bending Moment ሺft-kipsሻ

Figure 7.55 Design strength interaction diagram for column D1.

7U\EDUV 7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUFROXPQ'UHLQIRUFHGZLWKEDUVLVJLYHQLQ)LJXUH$OVR VKRZQLQWKH¿JXUHDUHWKHORDGFRPELQDWLRQSRLQWVIRUIDFWRUHGD[LDOIRUFHV LQ7DEOHDQGWKHFRUUHVSRQGLQ7DEOH7KHFROXPQLVDGHTXDWHEHFDXVHDOOWKHORDGFRPELQDWLRQSRLQWVIDOO ing total bending moments within the design strength interaction diagram. 8VHEDUV 7.11.26 Example 7.26 – Determination of Transverse Reinforcement: Building #1 (Framing Option B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to Uniaxial Bending and Axial Forces, Slenderness Effects

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ %DVVXPLQJDLQVTXDUHWLHGFROXPQVHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV DQG*UDGHUHLQIRUFHPHQW normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK

7-110

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the required tie bar size

ACI 25.7.2.2

7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&, )RUORQJLWXGLQDOEDUVXVHWLHV Step 2 – Determine the required tie bar spacing

ACI 25.7.2.1

7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR determine if they are adequate for shear strength requirements.

7U\WLHVVSDFHGDWLQRQFHQWHU Step 3 – Check the adequacy of the tie bar size and spacing for shear

ACI 22.5

Maximum factored shear force, LVREWDLQHGIURP$&,(TG IRUVLGHVZD\WRWKHULJKWDQGPXVWLQFOXGHWKH HIIHFWVIURPVOHQGHUQHVVVHH7DEOHVDQG 

Check if the design shear strength of the concrete, LVSURYLGHGRUQRWVHH7DEOH 

alone is adequate, which is determined based on whether

Eq. (7.26)

The minimum area of shear reinforcement is greater than LQRQFHQWHU'HWHUPLQH based on

SURYLGHGE\WKHWLHVVSDFHG

Table 7.12

Eq. (7.23)

where

7-111

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 7.13

WKHUHDUHEDUVORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWK IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU  Use

ZKLFKFRUUHVSRQGVWR$&,(TG  ACI 22.5.5.1.2

Also,

VRPLQLPXPVKHDUUHLQIRUFHPHQWQHHGQRWEHSURYLGHG$&, 

7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU 7.11.27 Example 7.27 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing Option B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to Uniaxial Bending and Axial Forces, Slenderness Effects

'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% DVVXPLQJDLQVTXDUHWLHGFROXPQVXSSRUWHGE\DPDWIRXQGDWLRQZLWKDWKLFNQHVVRIIWLQVHH)LJXUH  $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Check the bearing stresses on the column and mat foundation

ACI 16.3.3.4

Bearing strength of the column. Factored bearing stress, is determined for compression plus bending using the factored axial forces and bending PRPHQWVGHWHUPLQHGE\$&,(TVE DQGG >VHH7DEOHVDQG@ • $&,(TE 

7-112

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• $&,(TG 

Design bearing strength:

Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQVLRQIRUFHV are adequately transferred from the column into the footing. Bearing strength of the mat foundation. There is no need to check the bearing strength of the mat foundation because interface reinforcement must be proYLGHGGXHWRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(TG  Step 2 – Determine the required interface reinforcement

ACI 16.3.4.1

7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHVWKDW both the compression and tension forces are adequately transferred through the interface. Check minimum area of reinforcement across the interface:

Step 3 – Check the development of the dowel bars in the footing

,WLVHYLGHQWIURP)LJXUHWKDWWKHVWUHVVLQWKHORQJLWXGLQDOUHLQIRUFHPHQWIDUWKHVWIURPWKHH[WUHPHFRPSUHVVLRQIDFHLVWHQVLOHIRUDQXPEHURIORDGFRPELQDWLRQVWKDWLVWKHORDGFRPELQDWLRQVIDOOZLWKLQ=RQHVDQGRIWKH LQWHUDFWLRQGLDJUDPVHH)LJXUH 7KHUHIRUHWKHGRZHOEDUVPXVWEHGHYHORSHGIRUWHQVLRQLQWRWKHIRRWLQJDQG into the column). $VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSPHQW RIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN length,

Eq. (7.44)

Table 7.27

7-113

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPLQJWZROD\HUVRIEDUVLQWKHPDWIRXQGDWLRQZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPPDW thickness for the development of the dowel bars: Eq. (7.45)

The minimum LVOHVVWKDQWKHSURYLGHGPDWWKLFNQHVVRILQ7KHUHIRUHWKHGRZHOEDUVFDQEH adequately developed in the mat. Step 4 – Determine the development length of the dowel bars in the column

The dowel bars must be lap spliced to the longitudinal reinforcement in the column using a tension lap splice. Because all the dowel bars are spliced at the same location, a Class B tension lap splice is required (ACI Table   Development length in tension,

RIWKHEDUV Eq. (7.36)

Table 7.24

Set

7-114

ACI 25.4.2.4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

3URYLGHDODSVSOLFHOHQJWKRIIWLQ Step 5 – Check horizontal force transfer

,WLVGHWHUPLQHGLQ([DPSOHWKDW LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGPDWIRXQdation. The required area of shear-friction reinforcement is determined as follows assuming the surface between the column and the mat has not been intentionally roughened: Eq. (7.46), Table 7.28

7KHGRZHOEDUVSURYLGHV

across the interface, which is greater than the required amount of

Check the upper shear limit:

Table 7.29

5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH

7-115

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹԢǦͲԣ ʹԢǦͲԣ



ͶǦ͓ͳͳ

ͷԢǦͻԣ

ͶǦ͓ͳͳ†‘™‡Ž„ƒ”•

͓ͳͲƒ–„ƒ”•ሺ–›’Ǥሻ

͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ

ͶԢǦͲԣ

͓Ͷ–‹‡•̷ͳԢǦͳͲԣ‘Ǥ…Ǥ

Figure 7.56 Reinforcement details for column D1 in Examples 7.23 through 7.27.

7-116

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 8 WALLS 8.1 Overview $ZDOOLVGH¿QHGDVDYHUWLFDOVWUXFWXUDOHOHPHQWZLWKDKRUL]RQWDOOHQJWKWRWKLFNQHVVUDWLRJUHDWHUWKDQZKLFK LVGHVLJQHGWRUHVLVWD[LDOIRUFHVODWHUDOIRUFHVRUERWK$&, 6WUXFWXUDOZDOOVDUHGHVLJQHGWRUHVLVWWKHHIIHFWV IURPYHUWLFDOIRUFHVDQGODWHUDOIRUFHVLQWKHSODQHRIWKHZDOOWKDWLVLQWKHGLUHFWLRQSDUDOOHOWRWKHKRUL]RQWDO length of the wall) and are commonly referred to as shear walls. $FFRUGLQJWR5HIHUHQFHDORDGEHDULQJRUEHDULQJ FRQFUHWHZDOOLVDZDOOVXSSRUWLQJPRUHWKDQSRXQGVSHU linear foot of vertical load in addition to its own weight. A nonload-bearing (or, nonbearing) wall is any wall that is not a load-bearing wall; such walls support essentially their own weight and possibly out-of-plane loads like those from wind. $[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVDUHUHVLVWHGE\RQHOD\HURUWZROD\HUVRIYHUWLFDODQGKRUL]RQWDO reinforcement in a wall. The design and detailing of cast-in-place concrete walls with nonprestressed reinforcement are covered in this chapter. Provisions for walls are given in ACI Chapter 11 and are applicable to walls in buildings assigned to Seismic 'HVLJQ&DWHJRU\6'& $%DQG&&DQWLOHYHUUHWDLQLQJZDOOVDUHQRWFRYHUHGKHUHVHH5HIHUHQFHIRUFRPSUHKHQVLYHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWKHVHW\SHVRIZDOOV6SHFL¿FGHVLJQUHFRPPHQGDWLRQVIRUFDVWLQ place walls constructed with insulating concrete forms are given in the references in ACI R11.1.6.

8.2 Design Limits 8.2.1 Minimum Wall Thickness

Minimum thicknesses of bearing walls, nonbearing walls, and exterior basement and foundation walls are given in ACI 11.3.1.1 (see Figure 8.1 for bearing and nonbearing walls). These minimum thicknesses need not be applied to EHDULQJZDOOVDQGH[WHULRUEDVHPHQWDQGIRXQGDWLRQZDOOVGHVLJQHGE\WKHSURYLVLRQVLQ$&,RUDQDO\]HGE\ the alternative analysis method in ACI 11.8. 8.2.2 Intersecting Elements

For cast-in-place concrete walls where the factored axial force, exceeds the portion of the wall within of the wall WKHWKLFNQHVVRIWKHÀRRUV\VWHPPXVWKDYHDVSHFL¿HGFRQFUHWHFRPSUHVVLYHVWUHQJWKRIDWOHDVW $&, 7KHIDFWRUUHÀHFWVWKHUHGXFHGFRQ¿QHPHQWLQÀRRUZDOOMRLQWVFRPSDUHGZLWKÀRRUFROXPQ joints for members subjected to gravity loads.

8.3 Required Strength 8.3.1 Analysis Methods

7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKVHH$&,DQGUHVSHFWLYHO\  $[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVFDQEHGHWHUPLQHGE\DQ\RIWKHDQDO\VLVPHWKRGVLQ$&, :KHQSHUIRUPLQJD¿UVWRUGHURUDQHODVWLFVHFRQGRUGHUODWHUDOORDGDQDO\VLVWKHVHFWLRQSURSHUWLHVLQ$&, may be used in lieu of a more sophisticated analysis to account for member cracking and other effects. The reduced moments of inertia in ACI Table 6.6.3.1.1(a) and the alternative moments of inertia in ACI Table 6.6.3.1.1(b) for YDULRXVVWUXFWXUDOPHPEHUVDUHJLYHQLQ7DEOHDQG7DEOHUHVSHFWLYHO\7KHUHGXFHGPRPHQWVRILQHUWLDLQ Table 8.1 are based on an analysis where strength-level (factored) loads are used. For an analysis based on serviceOHYHOORDGVLWLVSHUPLWWHGWRXVHUHGXFHGPRPHQWVRILQHUWLDHTXDOWRWLPHVWKHYDOXHVLQ7DEOHSURYLGHGWKH moments of inertia do not exceed the gross moments of inertia $&, 

8-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



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‫ ۓ‬lesser of ൝ ‫۔‬ ‫ە‬4ᇳ

κ௪ Τ25 κ௨ Τ25 

۰‫ כܔܔ܉ܟ ܏ܖܑܚ܉܍‬ *Applicable to walls

݄ ൒ greater of

‫ ۓ‬lesser of ൝ ‫۔‬ ‫ە‬4ᇳ

κ௪ Τ30 κ௨ Τ30 

‫ܔܔ܉ܟ ܏ܖܑܚ܉܍܊ܖܗۼ‬

designed by ACI 11.5.3 only

Figure 8.1 Minimum thicknesses for bearing and nonbearing walls.

Table 8.1 Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using Strength-Level Loads Member

Moment of Inertia

Columns Walls — uncracked Walls — cracked Beams )ODWSODWHVDQGÀDWVODEV Table 8.2 Alternative Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using Strength-Level or Service-Level Loads Member

Moment of Inertia*

Columns and walls

%HDPVÀDWSODWHVDQGÀDWVODEV *For a service-level load analysis, use service-level axial force and moment effects in the equation for columns and walls.

8-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRUHUH¿QHGHTXDWLRQVIRUUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHLQFOXGHWKH IDFWRUHGD[LDOIRUFH RQDZDOOIRUWKHORDGFRPELQDWLRQXQGHUFRQVLGHUDWLRQ QRPLQDOD[LDOVWUHQJWKDW and factored moment, ]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ DQG DUHDRIORQJLWXGLQDOYHUWLFDO UHLQIRUFHPHQW These equations are applicable for strength-level and service-level loading, even though they are presented in terms of factored load effects. For service-level analysis, the factored load effects are replaced by the corresponding service-level load effects. Regardless of the equations used to determine moments of inertia, the gross area of the section, the analysis. Also, the cross-sectional area for calculation of shear deformations in a wall is

is to be used in

,QOLHXRIWKHUHTXLUHPHQWVRI$&,LWLVSHUPLWWHGWRDVVXPHLQD¿UVWRUGHUIDFWRUHGODWHUDOORDGDQDO\VLVWKDW DOOWKHPHPEHUVLQDVWUXFWXUHKDYHDUHGXFHGPRPHQWRILQHUWLDHTXDOWRSHUFHQWRIWKHJURVVPRPHQWRILQHUWLD $&, $PRUHGHWDLOHGDQDO\VLVFRQVLGHULQJWKHHIIHFWLYHVWLIIQHVVRIDOOPHPEHUVLVDOVRSHUPLWWHG 8.3.2 Factored Axial Force, Moment, and Shear

:DOOVPXVWEHGHVLJQHGIRUIDFWRUHGD[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVGXHWRLQSODQHDQGRURXW of-plane forces, including the effects of slenderness where appropriate, for each applicable load combination (ACI DQG ,QSODQHDQGRXWRISODQHIRUFHVDQGWKHLUHIIHFWVRQDEHDULQJZDOODUHLOOXVWUDWHGLQ )LJXUHDQG)LJXUHUHVSHFWLYHO\ Axial force, ܲ௨

In-plane forces

In-plane shear force, ܸ௨ In-plane bending moment, ‫ܯ‬௨

Figure 8.2 In-plane forces on a bearing wall.

8.3.3 Slenderness Effects Overview

Like columns, walls must be designed for the effects of slenderness where applicable. Determining whether slenderness effects need to be considered or not depends on whether the frame is nonsway or sway. According to ACI DIUDPHLVFRQVLGHUHGWREHQRQVZD\ZKHUHWKHVWDELOLW\LQGH[ LVOHVVWKDQZKHUH is determined E\$&,(TXDWLRQ  (8.1)

8-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Axial force, ܲ௨



Axial force, ܲ௨

Eccentricity, ݁

Out-of-plane forces

Out-of-plane shear force, ܸ௨

Out-of-plane bending moment, ‫ܯ‬௨

Out-of-plane bending moment, ‫ܯ‬௨

Figure 8.3 Out-of-plane forces on a bearing wall.

where Ȉ3u = total factored vertical load in the story ǻo ¿UVWRUGHUUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQGERWWRPRIWKHVWRU\ Vus IDFWRUHGKRUL]RQWDOVWRU\VKHDULQDVWRU\ ℓc = length of the walls measured center-to-center of the joints in the frame More often than not, the frame in the direction of analysis can be considered nonsway where structural walls are used to resist the lateral forces in that direction. Whether slenderness effects need to be considered or not for nonsway and sway frames depends on the slenderness LQWKHGLUHFWLRQRIDQDO\VLV7KHWHUPVLQWKHVOHQGHUQHVVUDWLRDUHGH¿QHGLQ7DEOHVHH ratio of the wall, $&,  Table 8.3 Slenderness Ratio, kℓu / r

Nonsway Frames Sway Frames Unsupported length, LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRURWKHU members capable of providing lateral support in the direction of analysis. Radius of gyration,

is permitted to be calculated by the following:

1.



8-4

for rectangular walls where is the dimension of the wall in the direction of analysis (that is, either for slenderness about the minor axis or for slenderness about the major axis)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6OHQGHUQHVVHIIHFWVDUHSHUPLWWHGWREHQHJOHFWHGZKHQWKHHTXDWLRQVLQ7DEOHDUHVDWLV¿HG$&,  Table 8.4 Limits for Slenderness Effects Wall Bracing Condition

Slenderness Ratios Where Slenderness is Permitted to Be Neglected

Not braced against sidesway (sway)

Braced against sidesway (nonsway)

*Ratio

is negative if a wall is bent in single curvature and is positive if a wall is bent in double curvature.

,WLVPRUHOLNHO\WKHRXWRISODQHEHQGLQJPRPHQWVLQDZDOOZRXOGQHHGWREHPDJQL¿HGWKDQWKHLQSODQHEHQGLQJ moments: In the direction parallel to the length of the wall, slenderness effects can usually be neglected because the radius of gyration of the wall, in that direction is relatively large, which results in a slenderness ratio less than WKHOLPLWVLQ7DEOH,QWKHGLUHFWLRQSDUDOOHOWRWKHWKLFNQHVVRIWKHZDOOWKHVOHQGHUQHVVUDWLRLVPRUHOLNHO\WREH greater than the prescribed slenderness limits, which means factored bending moments about the minor axis of the ZDOOZRXOGQHHGWREHPDJQL¿HG 7KHIROORZLQJPHWKRGVDUHSHUPLWWHGWREHXVHGWRDFFRXQWIRUVOHQGHUQHVVHIIHFWV$&,  ƒ ƒ ƒ ƒ

0RPHQWPDJQL¿FDWLRQPHWKRG$&, /LQHDUHODVWLFVHFRQGRUGHUDQDO\VLV$&, Inelastic analysis (ACI 6.8) Alternative method for out-of-plane slender wall analysis (ACI 11.8)

7KHPRPHQWPDJQL¿FDWLRQPHWKRGDQGWKHDOWHUQDWLYHPHWKRGDUHFRYHUHGEHORZ,QIRUPDWLRQRQWKHOLQHDUHODVWLF VHFRQGRUGHUDQDO\VLVDQGWKHLQHODVWLFDQDO\VLVLVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ Moment Magnification Method

'HWDLOVRIWKHPRPHQWPDJQL¿FDWLRQPHWKRGDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ:KHQGHWHUPLQLQJWKH E\$&,(TXDWLRQ WKHHIIHFWLYHVWLIIQHVV determined by ACI Equation critical buckling load, F LVXVHGIRUZDOOV (8.2) LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWHGHWHUPLQHGLQDFFRUGDQFHZLWK$&, is the In this equation, is the ratio of PRPHQWRILQHUWLDRIWKHZDOOGHWHUPLQHGE\WKHHTXDWLRQIRUFROXPQVDQGZDOOVLQ7DEOHDQG the maximum factored sustained axial load to the maximum factored axial load associated with the same load combination, which is typically equal to the factored axial dead load divided by the total factored axial load on a wall. As noted previously, frames are typically nonsway when structural walls are part of the frame; in such cases, the GHWHUPLQHGE\$&,(TXDWLRQ ZKHUHDSSOLFDEOH ZDOOLVWREHGHVLJQHGXVLQJWKHPDJQL¿HGPRPHQWV

8-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Alternative Method for Out-of-Plane Slender Wall Analysis

Limitations The design procedure given in ACI 11.8 can be used for the out-of-plane design of slender, reinforced concrete ZDOOVSURYLGHGWKHIROORZLQJOLPLWDWLRQVDUHVDWLV¿HG$&,  1. The cross-section of the wall is constant over the entire height.  The wall is tension-controlled for out-of-plane moment effects. is greater than or equal to the cracking moment, 3. 7KHGHVLJQÀH[XUDOVWUHQJWKRIWKHZDOO and VHH$&, 

where

 The factored axial force, at the mid-height section of the wall is less than or equal to  7KHFDOFXODWHGRXWRISODQHGHÀHFWLRQGXHWRVHUYLFHORDGV including P-delta effects, is less than or equal where is the length of the wall measured center-to-center of the joints. to :KHQRQHRUPRUHRIWKHVH¿YHOLPLWDWLRQVDUHQRWVDWLV¿HGWKLVPHWKRGFDQQRWEHXVHG)RUH[DPSOHZDOOVZLWK relatively large window and door openings are not considered to have a constant cross-section over their full height, VRWKHDOWHUQDWLYHPHWKRGLVQRWDSSOLFDEOHEHFDXVHWKH¿UVWOLPLWDWLRQLVQRWVDWLV¿HG Modeling 7KHIROORZLQJPRGHOLQJUHTXLUHPHQWVPXVWEHVDWLV¿HGZKHQXVLQJWKHDOWHUQDWLYHPHWKRG$&,  1. 7  KHZDOOPXVWEHDQDO\]HGDVDVLPSO\VXSSRUWHGD[LDOO\ORDGHGPHPEHUVXEMHFWHGWRDXQLIRUPO\GLVWULEXWHG ODWHUDOORDGZLWKWKHPD[LPXPPRPHQWDQGGHÀHFWLRQRFFXUULQJDWWKHPLGKHLJKWRIWKHZDOOVHH)LJXUH  applied to the wall above any section are assumed to be distributed over a  Concentrated gravity loads, ZLGWKHTXDOWRWKHEHDULQJZLGWKSOXVDZLGWKRQHDFKVLGHWKDWLQFUHDVHVDWDVORSHRIYHUWLFDOWRKRUL]RQWDO EXWQRWH[WHQGLQJEH\RQGWKHVSDFLQJRIWKHFRQFHQWUDWHGORDGVRUWKHHGJHVRIWKHZDOOSDQHOVHH)LJXUH 

ܲ௨௚ 

ȟ௨ 

κ௖ Ȁʹ

‫ݓ‬௨ 

Figure 8.4 Wall analysis in accordance with ACI 11.8.

8-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κ௖ Ȁʹ

κ௖ Ȁʹ

‫ܧ‬

ܵ

ܵ

‫ܤ‬ ሺ–›’Ǥ ሻ

ʹ ʹ ͳ ͳ ‫ݓ‬ଵ 

‫ݓ‬ଶ 

‫ݓ‬ଶ 

‫ݓ‬ଵ ൌ ‫ ܤ‬൅ ሺκ௖ ΤͶሻ ൅ ‫ ܧ‬൑ ܵ ‫ݓ‬ଶ ൌ ‫ ܤ‬൅ ሺκ௖ Τʹሻ ൑ ܵ Figure 8.5 Distribution of concentrated loads in the alternative design method for walls.

Factored Moment 7KHIROORZLQJVWUHQJWKHTXDWLRQPXVWEHVDWLV¿HGDWWKHPLGKHLJKWRIDZDOO (8.3)

Two methods to determine which includes the effects due to slenderness, are given in ACI 11.8.3: (1) by iteraWLYHFDOFXODWLRQDQG E\GLUHFWFDOFXODWLRQ Iterative Calculation Method In the iterative calculation method, the factored moment,

FRQVLVWVRIWZRSDUWV>$&,(TXDWLRQD @ (8.4)

The moment LVWKHPD[LPXPIDFWRUHGPRPHQWDWWKHPLGKHLJKWRIWKHZDOOGXHWRD ODWHUDOORDGVDQGRUE  applied at an eccentricity from the centroid of the wall. Note that does not include vertical factored loads, 3GHOWDHIIHFWV7KHGHÀHFWLRQ LVWKHWRWDOGHÀHFWLRQDWWKHPLGKHLJKWRIWKHZDOOGXHWRWKHIDFWRUHGORDGV7KLV GHÀHFWLRQLVGHWHUPLQHGE\$&,(TXDWLRQE  (8.5)

In this equation, is the length of the wall, measured center-to-center of joints, and is the moment of inertia of the cracked wall section transformed to concrete, which is determined by ACI Equation (11.8.3.1c): (8.6)

In this equation, is the modulus of elasticity of the reinforcing steel, is the modulus of elasticity of the conis the area of vertical reinforcement in the wall, LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWR crete, LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LVDQG is the length of the centroid of must be taken greater than or equal to 6 in this equation. the wall. The ratio In the strength design method, the neutral axis depth, is related to the depth of the equivalent rectangular stress In general, is the force in the tension reinforcement divided by the equivalent comblock, by

8-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

pressive stress dinal reinforcement,

times the width of the section. In the alternative design method, an effective area of longituis used in the determination of and , and is calculated by the following equation: (8.7)

This effective area of longitudinal reinforcement is used in the determination of LQ(TXDWLRQ @

>VHHWKH¿UVWWHUPLQSDUHQWKHVHV

Therefore, the depth of the equivalent stress block for bending about the minor axis of the wall can be determined by the following equation: (8.8)

$OVRWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LVLVWKHIROORZLQJ (8.9)

,WLVHYLGHQWIURP(TXDWLRQ WKDWWKHGHÀHFWLRQ is a function of +RZHYHU is a function of ZKLFKLVDSSDUHQWIURP(TXDWLRQ 7KLVFOHDUO\LOOXVWUDWHVWKHLWHUDWLYHQDWXUHLQKHUHQWWRWKLVPHWKRG7KXV and then performing several calculation iterations until convergence can be determined by assuming a value of occurs. Direct Calculation Method In the direct calculation method,

is determined by ACI Equation (11.8.3.1d): (8.10)

Regardless of which method is used to determine

is determined by the following equation: (8.11)

2XWRI3ODQH'HÀHFWLRQ ,QDGGLWLRQWRVDWLVI\LQJVWUHQJWKUHTXLUHPHQWVWKHIROORZLQJGHÀHFWLRQUHTXLUHPHQWPXVWEHVDWLV¿HG ZKLFKLVWKH¿IWKOLPLWDWLRQLQ$&, 7KHPD[LPXPGHÀHFWLRQGXHWRVHUYLFHORDGV which includes second-order effects, depends on the magnitude >$&,(TXDWLRQ @ of the service load moment, (8.12)

The terms respectively.

and

DUHWKH¿UVWRUGHUVHUYLFHOHYHOEHQGLQJPRPHQWDQGD[LDOORDGDWWKHPLGKHLJKWRIWKHZDOO

exceeds two-thirds of the cracking moment, 2XWRISODQHGHÀHFWLRQVLQFUHDVHUDSLGO\ZKHQ GHWHUPLQHGE\RQHRIWKHIROORZLQJWZRHTXDWLRQVLQ$&,7DEOH

8-8

Thus,

is

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(8.13)

(8.14)

In these equations, and WKHQRPLQDOÀH[XUDOVWUHQJWK E @

DUHWKHPLGKHLJKWGHÀHFWLRQVFRUUHVSRQGLQJWRWKHFUDFNLQJPRPHQW and UHVSHFWLYHO\DQGDUHGHWHUPLQHGDVIROORZV>VHH$&,(TXDWLRQVD DQG

(8.15)

(8.16)

Because is a function of >VHH(TXDWLRQ RU @DQG is a function of >VHH(TXDWLRQ @ A value of is obtained by iteration, that is, an initial value of is asthere is no closed-form solution for sumed and calculations are performed until convergence occurs. 6HUYLFHOHYHOORDGFRPELQDWLRQVDUHQRWJLYHQLQ$&,&KDSWHUEXWDUHGLVFXVVHGLQ$SSHQGL[&&RI5HIHUHQFH 3. For structures subjected to wind, the following service-level load combination is recommended for calculating VHUYLFHOHYHOODWHUDOGHÀHFWLRQVRIZDOOVDQGIUDPHV (8.17)

where are the effects due to wind forces obtained from serviceability wind speeds, which are given in Figures &&&&&&DQG&&IRU\HDU\HDU\HDUDQG\HDUPHDQUHFXUUHQFHLQWHUYDOVUHVSHFWLYHO\8VLQJZLQGVSHHGVEDVHGRQPRUHWKDQD\HDUPHDQUHFXUUHQFHLQWHUYDOLQFKHFNLQJODWHUDOGHÀHFWLRQVLV excessively conservative. For slender walls subjected to strength-level earthquake effects, HYDOXDWHVHUYLFHOHYHOODWHUDOGHÀHFWLRQV

the following load combination can be used to (8.18)

8.4 Design Strength 8.4.1 General

)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\ VHFWLRQLQDZDOO$&,  (8.19) (8.20) (8.21)

8-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ IDFWRUVSHUWLQHQWWRWKHGHVLJQRIUHLQIRUFHGFRQFUHWHZDOOVLVJLYHQLQ7DEOHIRUWUDQVYHUVHUHLQIRUFHPHQWRWKHU LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWK WKDQVSLUDOV7KHVWUDLQXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQ divided by the modulus of elasticity of the reinforcing steel, which can be taken as of the reinforcement, SVLIRUDQ\JUDGHRIUHLQIRUFHPHQW$&,  Table 8.5 Strength Reduction Factors, Ҁ Net Tensile Strain, İt

Classification

Strength Reduction Factor, Ҁ*

Compression-controlled



Transition** 

Tension-controlled * Applicable to transverse reinforcement other than spirals. **Sections classified as transition are permitted to use corresponding to compression-controlled sections.

For bearing walls subjected primarily to uniaxial compression forces, Equation (8.19) is applicable. For bearing ZDOOVVXEMHFWHGWRFRPELQHGD[LDOIRUFHVDQGEHQGLQJPRPHQWLQSODQHRURXWRISODQH (TXDWLRQV DQG  PXVWERWKEHVDWLV¿HGIRUDOODSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQV$&, $OWHUQDWLYHO\EHDULQJZDOOV VXEMHFWHGWRD[LDOIRUFHVDQGRXWRISODQHEHQGLQJPRPHQWVDUHSHUPLWWHGWREHGHVLJQHGXVLQJWKHVLPSOL¿HGGHVLJQ PHWKRGLQ$&,SURYLGHGWKHOLPLWDWLRQVRIWKDWPHWKRGDUHVDWLV¿HGVHH6HFWLRQRIWKLVSXEOLFDWLRQ  )RUQRQEHDULQJZDOOV(TXDWLRQ PXVWEHVDWLV¿HGZKHUH VWUHQJWKUHTXLUHPHQWVRI$&,$&, 

LVGHWHUPLQHGLQDFFRUGDQFHZLWKWKHÀH[XUDO

(TXDWLRQ PXVWEHVDWLV¿HGIRUERWKLQSODQHDQGRXWRISODQHVKHDUIRUFHVRQDZDOO Methods to determine the nominal strengths DQGUHVSHFWLYHO\

combined

and

DUHJLYHQLQ6HFWLRQV

8.4.2 Nominal Axial Strength Nominal Axial Compressive Strength

The nominal axial compressive strength, of a nonslender, reinforced concrete wall is determined in accordance must be less than or equal to the maximum nominal ZLWK$&,7KHQRPLQDOD[LDOFRPSUHVVLYHVWUHQJWK ZKLFKLVGHWHUPLQHGXVLQJ$&,7DEOHDQGWKHQRPLQDOD[LDOVWUHQJWKDW axial compressive strength, ]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ  (8.22)

When calculating WKHYDOXHRIWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHORQJLWXGLQDOUHLQIRUFHPHQW is limited to SVLHYHQWKRXJK for the longitudinal (vertical) reinforcement used in the wall may be larger than that (ACI   Nominal Axial Tensile Strength

The nominal axial tensile strength of a nonprestressed, reinforced concrete wall, must be less than or equal to ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ  the maximum nominal axial tensile strength, (8.23)

It is evident that the entire tensile force must be resisted by the longitudinal reinforcement in a wall.

8-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8.4.3 Nominal Strength of Walls Subjected to Moment and Axial Forces Overview

The nominal strength of a reinforced concrete wall subjected to both moment and axial force is determined using HTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ$&,ZKLFKDUHVXPPDUL]HGLQ7DEOH Table 8.6 Design Assumptions for Concrete and Nonprestressed Reinforcement Material

Assumptions

1. 0D[LPXPVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHULVDVVXPHGHTXDOWR  7HQVLOHVWUHQJWKRIFRQFUHWHLVQHJOHFWHGLQÀH[XUDODQGD[LDOVWUHQJWKFDOFXODWLRQV 3. The relationship between concrete compressive stress and strain is to be represented by a UHFWDQJXODUWUDSH]RLGDOSDUDEROLFRURWKHUVKDSHWKDWUHVXOWVLQSUHGLFWLRQRIVWUHQJWKLQ substantial agreement with results of comprehensive tests.  7  KHHTXLYDOHQWUHFWDQJXODUFRQFUHWHVWUHVVGLVWULEXWLRQLQDFFRUGDQFHZLWK$&, WKURXJKVDWLV¿HVWKHUHTXLUHPHQWRI$&,IRUWKHUHODWLRQVKLSEHWZHHQ concrete compressive stress and strain. Concrete

is to be assumed uniformly distributed over an equivalent  A concrete stress of FRPSUHVVLRQ]RQHERXQGHGE\WKHHGJHVRIWKHFURVVVHFWLRQDQGDOLQHSDUDOOHOWRWKH neutral axis located a distance IURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQZKHUH . 7KHGLVWDQFHIURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQWRWKHQHXWUDOD[LV is to be measured perpendicular to the neutral axis. Values of

are as follows:

• • •

Nonprestressed reinforcement

1. Deformed reinforcement used to resist tensile or compressive forces must conform to $&,  The stress-strain relationship and the modulus of elasticity of the deformed reinforcePHQWDUHWREHLGHDOL]HGLQDFFRUGDQFHZLWK$&,DQG

Rectangular Sections

The general principles and assumptions of the strength design method in Table 8.6 are applied to the rectangular reinforced concrete wall given in Figure 8.6. For purposes of discussion, it is assumed the longitudinal reinforcement LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ 7KHHTXDWLRQVLQ)LJXUHRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUFROXPQVHFWLRQVFDQEHXVHGWRGHWHUPLQHWKH DQGWKHQRPLQDOÀH[XUDOVWUHQJWK of the rectangular wall in Figure 8.6 for a given nominal axial strength, which is the net tensile strain in the extreme layer of tension reinforcement at nominal strength, excluding strain, strains due to creep, shrinkage, and temperature. I-, T-, and L-Shaped Sections

)RUZDOOVZLWKÀDQJHVZKHUHWKHÀDQJHVDUHLQFRPSUHVVLRQWKHQRPLQDOD[LDODQGÀH[XUDOVWUHQJWKVGHSHQGRQ whether the depth of the equivalent stress block, LVOHVVWKDQRUJUHDWHUWKDQWKHÀDQJHWKLFNQHVV

8-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͲǤͺͷ݂௖ᇱ 

݀ଵ 

݄

‫ܨ‬௦ଵ  ‫ܥ‬

ߝ௦ଶ 

‫ܨ‬௦ଶ 

ߝ௦ଷ 

‫ܨ‬௦ଷ 

݀଺ 

݀ହ 

݀ସ 

݀ଷ 

ܿ

݀ଶ 

ߝ௦ଵ 

ܽ ൌ ߚଵ ܿ

ͲǤͲͲ͵



κ௪ 

݀଻ 

ߝ௦ସ 

ߝ௦ହ 

ߝ௦଺ 

ߝ௦଻ ൌ ߝ௧ 

‫ܨ‬௦ସ 

‫ܨ‬௦ହ 

‫ܨ‬௦଺ 

‫ܨ‬௦଻ 

Figure 8.6 Reinforced concrete wall subjected to moment and axial compression.

Where LVOHVVWKDQRUHTXDOWRWKHÀDQJHWKLFNQHVV IRUDUHFWDQJXODUZDOOVHFWLRQ

and

can be determined using the equations in Figure

are determined based on a T- or L-shaped compression Where LVJUHDWHUWKDQWKHÀDQJHWKLFNQHVV and ]RQH&RQVLGHUWKH,VKDSHGZDOOVHFWLRQLQ)LJXUHZKHUHWKHHTXLYDOHQWVWUHVVEORFNIDOOVLQWKHZHE7KHUHVXOcan be determined by the following equation: tant compression force,

‫ݐ‬௙ 

(8.24)

κ௪ 

ܽ

ܾ௙ 

݄

Figure 8.7 An I-shaped wall where the equivalent stress block falls in the web.

The strains, stresses, and forces in the longitudinal reinforcement and are determined using the expressions in )LJXUH7KHQRPLQDOÀH[XUDOVWUHQJWKLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ (8.25)

8-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ZKHUHWKHGLVWDQFHIURPWKHFHQWURLGRIWKHZDOOWRWKHFHQWURLGRIWKHFRPSUHVVLRQ]RQH ing:

is equal to the follow-

(8.26)

6LPLODUH[SUHVVLRQVFDQEHGHULYHGIRU/VKDSHGFRPSUHVVLRQ]RQHV ,WLVDVVXPHGLQWKHDERYHGLVFXVVLRQWKDWWKHHQWLUHZLGWKRIWKHÀDQJHLVHIIHFWLYHLQUHVLVWLQJWKHFRPELQHGD[LDO IRUFHDQGEHQGLQJPRPHQW$OWKRXJKQRWVSHFL¿FDOO\UHTXLUHGLQ$&,&KDSWHULWPD\EHZDUUDQWHGWRXVHD ÀDQJHZLGWKOHVVWKDQWKHDFWXDOÀDQJHZLGWKLQWKHDQDO\VLV$PHWKRGWRGHWHUPLQHWKHSRUWLRQRIWKHÀDQJHFRQVLGHUHGWREHHIIHFWLYHIRUZDOOVVXEMHFWHGWRHDUWKTXDNHHIIHFWVLVJLYHQLQ$&,WKDWHIIHFWLYHÀDQJHZLGWK FDQEHXVHGIRUDQ\ZDOOZLWKÀDQJHVLQOLHXRISHUIRUPLQJDPRUHUH¿QHGDQDO\VLV$QRWKHURSWLRQLVWRFRPSOHWHO\ and thickness GLVUHJDUGWKHÀDQJHVDQGWRGHVLJQWKHZDOODVDUHFWDQJXODUVHFWLRQRIOHQJWK Simplified Design Method

7KHVLPSOL¿HGGHVLJQPHWKRGLQ$&,FDQEHXVHGWRGHWHUPLQHWKHQRPLQDOD[LDOFRPSUHVVLYHVWUHQJWK RIDZDOOZKHUHWKHIROORZLQJWZROLPLWDWLRQVDUHVDWLV¿HG 1. The wall has a solid, rectangular cross-section.  The resultant of all applicable factored loads falls within the middle third of the wall thickness. acting at an eccentricity, The second limitation is illustrated in Figure 8.8 for the case of a factored axial force, less than or equal to In general, the total eccentricity caused by all applicable factored load effects, including those from lateral loads, must be less than or equal to ܲ௨  ݁ ൑ ݄Ȁ͸

݄

݄Ȁ͸

Figure 8.8 Eccentricity limitation in the simplified design method.

$&,(TXDWLRQ FDQEHXVHGWRGHWHUPLQH

ZKHQERWKOLPLWDWLRQVDUHVDWLV¿HG (8.27)

In this equation, 7DEOH 

is the gross area of the wall and

LVWKHHIIHFWLYHOHQJWKIDFWRUJLYHQLQ$&,7DEOHVHH

8-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.7 Effective Length Factor for Walls Designed by the Simplified Design Method

k

Boundary Conditions

Walls braced top and bottom against lateral translation

Restrained against rotation at one or both ends (top, bottom, or both)



Unrestrained against rotation at both ends

 

Walls not braced against lateral translation

A value of HTXDOWRLPSOLHVWKHHQGRIWKHZDOOLVDWWDFKHGWRDPHPEHUZLWKDÀH[XUDOVWLIIQHVVDWOHDVWHTXDO to that of the wall in the direction of analysis. A value of HTXDOWRZRXOGEHDSSOLFDEOHIRUH[DPSOHWRIUHHVWDQGLQJFDQWLOHYHU ZDOOVRUWRZDOOVFRQQHFWHGWRGLDSKUDJPVWKDWXQGHUJRVLJQL¿FDQWGHÀHFWLRQVZKHQVXEMHFWHG to lateral loads. (TXDWLRQ WDNHVLQWRFRQVLGHUDWLRQERWKHFFHQWULFLW\DQGVOHQGHUQHVVHIIHFWV,QRUGHUWRVDWLVI\VWUHQJWKUHmust be greater than or equal to the factored axial force, where quirements, the design strength, for compression-controlled sections. ,WLVLQKHUHQWO\DVVXPHGLQWKHVLPSOL¿HGGHVLJQPHWKRGWKDWWKHZDOOVDUHGHVLJQHGFRQVLGHULQJ as a concentric axial compression force. As such, it is best suited for relatively short walls subjected to vertical loads. Because the the application of this method becomes very limited when lateral loads need to eccentricity must not exceed be considered. 8.4.4 Nominal Shear Strength In-Plane Shear

For walls subjected to in-plane shear forces, the nominal shear strength, ZLWK$&,DQG$&, 

, at a section is determined in accordance (8.28)

The term is the nominal shear strength of the concrete and WUDQVYHUVHKRUL]RQWDO UHLQIRUFHPHQWLQWKHZDOO

is the nominal shear strength of the

7KHFRHI¿FLHQW GH¿QHVWKHUHODWLYHFRQWULEXWLRQRIWKHFRQFUHWHVWUHQJWKWRWKHQRPLQDOVKHDUVWUHQJWKDQGLV determined by the following:

(8.29)

For walls where a net tension force is determined for the entire wall section,  >$&,@

is calculated by ACI Equation

(8.30)

The axial tension force,

8-14

is negative and has the units of pounds.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRGL¿FDWLRQIDFWRU UHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOweight concrete of the same compressive strength is given in Table 8.8 based on equilibrium density and in Table EDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[WXUHVHH$&,  Table 8.8 Values of Ȝ Based on Equilibrium Density, wc

Ȝ

Equilibrium Density, wc





Table 8.9 Values of Ȝ Based on Composition of Aggregates Concrete

All-lightweight

/LJKWZHLJKW¿QHEOHQG

Sand-lightweight

Sand-lightweight, coarse blend

Composition of Aggregates

)LQH$670& &RDUVH$670& )LQH&RPELQDWLRQRI$670& and C33 &RDUVH$670& Fine: ASTM C33 &RDUVH$670& Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670& and ASTM C33

Ȝ  WR(1)  WR

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

The term LVWKHUDWLRRIWKHGLVWULEXWHGWUDQVYHUVHKRUL]RQWDO UHLQIRUFHPHQWWRWKHJURVVDUHDRIWKHFRQFUHWHDUHD perpendicular to that reinforcement. is the gross area of the cross-section of the wall resisting the shear force. For a rectangular wall with The area Illustrated in Figure 8.9 is the area WREHXVHGLQ(TXDWLRQ ZKHUHWKHHQGVRIWKH no openings, ZDOODUHHQODUJHGRUIUDPHLQWRFROXPQVWKLVLVRIWHQUHIHUUHGWRDVD³EDUEHOO´ZDOOEHFDXVHRILWVVKDSH  For walls with one or more openings, the area of the opening(s) must not be included in

(see Figure 8.9).

For walls with WKHVWUXWDQGWLHPHWKRGLQ$&,&KDSWHUPD\EHXVHGWRGHVLJQIRULQSODQHVKHDU IRUFHVLQVWHDGRIWKHPHWKRGGHVFULEHGDERYH$&,  Out-of-Plane Shear

For walls subjected to out-of-plane shear forces, the nominal shear strength, is determined by the one-way shear VWUHQJWKUHTXLUHPHQWVLQ$&,$&, %HFDXVHVKHDUUHLQIRUFHPHQWLVJHQHUDOO\QRWSURYLGHGWRVDWLVI\ where the nominal shear strength of the concrete, is deterone-way out-of-plane shear requirements, PLQHGE\WKHIROORZLQJHTXDWLRQIURP$&,7DEOHZKHUH (8.31)

8-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄

κ௪ 

‫ܣ‬௖௩ ൌ ݄κ௪ 

κ௪ 

݄

κ௢௣௘௡ 

‫ܣ‬௖௩ ൌ ݄ሺκ௪ െ κ௢௣௘௡ ሻ Figure 8.9 Determination of Acv

7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength attributed to concrete in members without shear reinforcement does not increase in direct proportion with member GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@ (8.32)

,WLVHYLGHQWIURP(TXDWLRQ WKDW

LVOHVVWKDQIRUPHPEHUVZLWK

The term LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by where is commonly WDNHQDVRQHIRRWVHH)LJXUH $FFRUGLQJWR$&,5 may be taken as the sum of the areas of the lonJLWXGLQDOÀH[XUDOUHLQIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPH FRPSUHVVLRQ¿EHU)RUW\SLFDOZDOOVHFWLRQV is the area of longitudinal tension reinforcement within the one-foot GHVLJQVWULSVHH)LJXUHIRUWKHFDVHRIDZDOOZLWKWZROD\HUVRIORQJLWXGLQDOUHLQIRUFHPHQW  $FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken , and is to be taken as negative for tension as positive for compression forces acting on the gross area of the wall, $&,  forces. Also, Values of used to calculate DUHOLPLWHGWRSVLWKHH[FHSWLRQLQ$&,LVQRWDSSOLFDEOHWRZDOOV VHH$&, 7KLVOLPLWDWLRQRQ is primarily due to the fact that there is a lack of test data and practical H[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL 7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@ (8.33)

:KHUHRXWRISODQHVKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGWKHWKLFNQHVVRIWKHZDOOPXVWW\SLFDOO\EHLQcreased. Increasing the compressive strength of the concrete is usually not as effective as increasing

8-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄

ͳʹԣ

κ௪ 



ܸ௨ 

‫ܣ‬௦  ݀ ߩ௪ ൌ

‫ܣ‬௦  ͳʹ݀

Figure 8.10 Out-of-plane shear force.

8.5 Reinforcement Limits %RWKORQJLWXGLQDOYHUWLFDO DQGWUDQVYHUVHKRUL]RQWDO UHLQIRUFHPHQWPXVWEHSURYLGHGLQDUHLQIRUFHGFRQFUHWH wall. Minimum longitudinal and transverse reinforcement requirements are given in ACI Table 11.6.1 and are sumand The longitudinal PDUL]HGLQ7DEOHIRUGHIRUPHGUHLQIRUFLQJEDUVZLWK corresponds to the vertical bars in the wall and is equal to the area of the vertical reinforcreinforcement ratio, ing bars divided by the gross area of the wall perpendicular to those bars, that is, the thickness of the wall times FRUUHVSRQGVWRWKHKRUL]RQWDOEDUVLQWKH the length of the wall. Similarly, the transverse reinforcement ratio, ZDOODQGLVHTXDOWRWKHDUHDRIWKHKRUL]RQWDOUHLQIRUFLQJEDUVGLYLGHGE\WKHJURVVDUHDRIWKHZDOOSHUSHQGLFXODUWR those bars, that is, the thickness of the wall times the height of the wall. For regularly spaced reinforcing bars, the reinforcement ratios can be determined on the basis of the applicable bar spacing instead of on the overall length or height of the wall. Table 8.10 Minimum Reinforcement for Walls in Accordance with ACI Table 11.6.1 Bar Size

Minimum longitudinal, ȡℓ

Minimum transverse, ȡt









WKHUHLQIRUFHPHQWUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG7KHVHUHTXLUHPHQWV Where along with the minimum reinforcement requirements of ACI 11.6.1, are given in Figure 8.11.

8.6 Determining the Wall Thickness Because structural walls in buildings are typically located around elevator and stair openings, the length of a wall LVRIWHQGLFWDWHGE\DUFKLWHFWXUDOUHTXLUHPHQWV7KHWKLFNQHVVRIDZDOOLQFRQMXQFWLRQZLWKWKHVSHFL¿HGOHQJWK DQGPDWHULDOSURSHUWLHVQHHGVWREHGHWHUPLQHGVRWKDWVWUHQJWKD[LDOIRUFHÀH[XUHDQGVKHDU DQGVHUYLFHDELOLW\ UHTXLUHPHQWVDUHVDWLV¿HG

8-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ۓ‬ ۖ

18ᇳ  ‫۔‬ ۖ ‫ە‬κ௪ Τ3‫כ‬

‫ ݏ‬൑ lesser of

           

3݄

ߩκ  ߩ௧ 

3݄ ‫ۓ‬ ۖ ‫ ݏ‬൑ lesser of 18ᇳ  ‫۔‬ ۖ ‫ە‬κ௪ Τ5‫כ‬ * Applicable where shear reinforcement is required for in-plane shear strength All other reinforcement not shown for clarity

κ௪  Ž‡˜ƒ–‹‘ Longitudinal reinforcement, ࣋र Longitudinal reinforcement, ܸ௨ ൑ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩ ൑ #5 bars

൐ #5 bars

0.0012

0.0015

ܸ௨ ൐ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩ 0.0025 ൅ 0.5ሺ2.5 െ ݄௪ Τκ௪ ሻሺߩ௧ െ 0.0025ሻ ൒ 0.0025‫ככ‬

Transverse reinforcement, ࢚࣋ Transverse reinforcement, ܸ௨ ൑ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩ ൑ #5 bars

൐ #5 bars

0.0020

0.0025

ܸ௨ ൐ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩ 0.0025

**ߩκ need not exceed ߩ௧ required by ACI 11.5.4.3

Figure 8.11 Minimum reinforcement requirements for cast-in-place reinforced concrete walls.

)RUZDOOVZKHUHVKHDUVWUHQJWKUHTXLUHPHQWVJRYHUQ(TXDWLRQ FDQEHXVHGWRGHWHUPLQHDPLQLPXP on a maximum in-plane shear force,

based

(8.34)

where

8-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUZDOOVVDWLVI\LQJWKHOLPLWDWLRQVRIWKHVLPSOL¿HGGHVLJQPHWKRGLQ$&,WKHIROORZLQJHTXDWLRQFDQEH used to determine minimum >VHH(TXDWLRQ @ (8.35)

where

for compression-controlled sections.

5HODWLYHO\WDOODQGRUVOHQGHUZDOOVFDQEHJRYHUQHGE\DFRPELQDWLRQRID[LDOIRUFHDQGLQSODQHEHQGLQJPRPHQWV $ZDOOWKLFNQHVVDQGORQJLWXGLQDOZDOOUHLQIRUFHPHQWVL]HDQGVSDFLQJ PXVWEHVHOHFWHGVRWKDWDQLQWHUDFWLRQGLDgram can be constructed. Because there is no closed-form solution for multiple factored load cases, iterations must and PXVWEHVDWLV¿HGIRUDOO EHSHUIRUPHGXQWLODOOGHVLJQVWUHQJWKFULWHULDDUHVDWLV¿HGWKDWLV factored load combinations.

8.7 Determination of Required Reinforcement 8.7.1 Longitudinal Reinforcement

$VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQDZDOOPXVWEH VHOHFWHGVRWKDWDOOGHVLJQVWUHQJWKFULWHULDDUHVDWLV¿HGIRUFRPELQHGÀH[XUHDQGD[LDOIRUFH1RFORVHGIRUPVROXfor a given set of factored load combinations; it is usually determined by tion exists to determine the required SHUIRUPLQJDQXPEHURILWHUDWLRQVZKHUHDWOHDVWWKHPLQLPXPUHLQIRUFHPHQWSUHVFULEHGLQ$&,DQG DUHXVHGLQWKH¿UVWLWHUDWLRQ Minimum longitudinal reinforcement per foot length of wall as a function of wall thickness for cases where is given in Table 8.11 IRUEDUVDQGVPDOOHUVHH7DEOH DQG)LJXUH $OVRLQFOXGHGLQWKHWDEOHLVVXJJHVWHGUHLQIRUFLQJEDUVL]HDQGVSDFLQJ7ZROD\HUVRIUHLQIRUFHPHQWPXVWEHSURYLGHGZKHUHWKHWKLFNQHVVRIWKHZDOOLVJUHDWHUWKDQLQH[FHSWIRUVLQJOHVWRU\EDVHPHQW ZDOOVDQGFDQWLOHYHUUHWDLQLQJZDOOV$&, (DFKOD\HURIUHLQIRUFHPHQWVKRXOGEHSODFHGQHDUHDFKIDFH — Table 8.11 Minimum Required Longitudinal Reinforcement for Walls where Vu”‫׋‬ťc Ȝ¥ f'c Acv Wall Thickness, h (in.)

As (in.2/ft)

Suggested Reinforcement

6



#Ǝ

8



#Ǝ





#Ǝ





#Ǝ





#Ǝ

16*



#Ǝ

18*



#Ǝ

*Two layers of reinforcement are required.

It is common for the walls in a low-rise building to have minimum longitudinal reinforcement over the entire height of the wall. WKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG0LQLPXP Where assumlongitudinal reinforcement per foot length of wall as a function of wall thickness where LVJLYHQLQ7DEOH see Figure 8.11). ing

8-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

—

Table 8.12 Minimum Required Longitudinal Reinforcement for Walls where Vu!‫׋‬ťc Ȝ¥ f'c Acv Wall Thickness, h (in.)

As (in.2/ft)

Suggested Reinforcement

6



#Ǝ

8



#Ǝ





#Ǝ





#Ǝ





#Ǝ

16*



#Ǝ

18*



#Ǝ

*Two layers of reinforcement are required.

8.7.2 Transverse Reinforcement

The required amount of transverse reinforcement, depends on the maximum factored in-plane shear force, ZKLFKDFWVRQWKHZDOO(TXDWLRQ FDQEHVROYHGIRU (8.36)

Where minimum determined by ACI 11.6.1 must be provided. Minimum transverse reinis given in Table 8.13 forcement per foot length of wall as a function of wall thickness where IRUEDUVDQGVPDOOHUVHH7DEOHDQG)LJXUH  — Table 8.13 Minimum Required Transverse Reinforcement for Walls where Vu”‫׋‬ťc Ȝ¥ f'c Acv Wall Thickness, h (in.)

As (in.2/ft)

Suggested Reinforcement

6



#Ǝ

8



#Ǝ





#Ǝ





#Ǝ





#Ǝ

16*



#Ǝ

18*



#Ǝ

*Two layers of reinforcement are required.

Where WKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG0LQLPXP is given transverse reinforcement per foot length of wall as a function of wall thickness where see Figure 8.11). LQ7DEOHWKDWLV

8-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

—

Table 8.14 Minimum Required Transverse Reinforcement for Walls where Vu!‫׋‬ťc Ȝ¥ f'c Acv Wall Thickness, h (in.)

As (in.2/ft)

Suggested Reinforcement

6



#Ǝ

8



#Ǝ





#Ǝ





#Ǝ





#Ǝ

16*



#Ǝ

18*



#Ǝ

*Two layers of reinforcement are required.

8.8 Reinforcement Detailing 8.8.1 Concrete Cover

5HLQIRUFLQJEDUVDUHSODFHGLQZDOOVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&,$&, &RQFUHWHFRYHUIRUZDOOVLVPHDsured from the surface of the concrete to the outer edge of the layer of reinforcement closest to the wall surface (see )LJXUHIRUDZDOOZKHUHWKHWUDQVYHUVHUHLQIRUFHPHQWLVFORVHVWWRWKHFRQFUHWHVXUIDFH 7KHPLQLPXPFRYHULV HTXDOWRLQIRUDQGVPDOOHUUHLQIRUFLQJEDUVLQZDOOVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG $&,7DEOH )RUDQGEDUVWKHPLQLPXPFRYHULVHTXDOLQ 

݄ …‘˜‡”

”ƒ•˜‡”•‡ ”‡‹ˆ‘”…‡‡–

Figure 8.12 Concrete cover for walls.

8.8.2 Splices of Reinforcement Overview

Splice lengths for longitudinal and transverse reinforcement in walls must be in accordance with the provisions in $&,$&, /DSVSOLFHVDQGPHFKDQLFDOVSOLFHVDUHFRPPRQO\XVHGLQZDOOV)RUWKHORQJLWXGLQDOEDUV compression or tension splice lengths must be provided that satisfy the factored load combinations resisted by the wall. Tension lap and mechanical splices are used for the transverse reinforcement.

8-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Lap Splices

Lap splices are the most popular and usually the most economical type of splices used in walls. The longitudinal reinforcement in a wall is permitted to be lap spliced immediately above the top of the slab (see Figure 8.13). This is the preferred location for ease of construction: The longitudinal bars from the wall below extend above the slab a distance that is greater than or equal to the required lap splice length, and the longitudinal bars in the wall above are WLHGWRWKHVHEDUVDIWHUWKHÀRRUEHORZKDVEHHQFRQVWUXFWHG

ƒ’•’Ž‹…‡Ž‡‰–Š



Ȁ Ž‘‘” Žƒ„”‡‹ˆ‘”…‡‡–‘– •Š‘™ˆ‘”…Žƒ”‹–›

ƒŽŽŽ‡˜ƒ–‹‘

 Figure 8.13 Location of lap splices for longitudinal bars in a wall.

/DSVSOLFHVDUHQRWSHUPLWWHGIRUEDUVODUJHUWKDQEDUVH[FHSWLQWKHFDVHRIFRPSUHVVLRQODSVSOLFHVZKHUH RUEDUVDUHSHUPLWWHGWREHVSOLFHGWRRUVPDOOHUEDUV$&,DQG  The type of lap splice that must be used (compressive or tensile) depends on the stress in the longitudinal bars due WRWKHIDFWRUHGORDGFRPELQDWLRQVVHH)LJXUHRIWKLVSXEOLFDWLRQZKLFKLVDOVRDSSOLFDEOHWRZDOOV 5HTXLUHG ODSVSOLFHOHQJWKVIRUWKHWKUHH]RQHVLQGLFDWHGLQ)LJXUHDUHJLYHQLQ7DEOH

8-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.15 Required Lap Splices Lengths Zone

Stress in Longitudinal Bars

1

All longitudinal bars are in compression



3

Stress in the longitudinal bars on the tension face is tensile and

Stress in the longitudinal bars on the tension face is tensile and

Notes

ACI Section No.

±



Class B tension lap splice length

±



Class B tension lap splice length

±



Minimum Lap Splice Length

Class A tension lap splice

Notes: 1. For must be increased by (4/3) [ACI 25.5.5.1]. 2. Compression lap splices must not be used for bars larger than #11 except as permitted in ACI 25.5.5.3 [ACI 25.5.5.2]. 3. Compression lap splices of #14 and #18 bars to #11 or smaller bars are permitted in accordance with ACI 25.5.5.4 (see Note 4) [ACI 25.5.5.3]. 4. Where bars of different size are lap spliced in compression, is the longer of (a) of the larger bar and (b) of the smaller bar where is calculated by ACI 25.4.9.1 [ACI 25.5.5.4]:

where

5. 6. 7. 8. 9.

(see Table 8.16)

Reduction of development length in accordance with ACI 25.4.10.1 is not permitted in calculating lap splice lengths (ACI 25.5.1.4). Requirements for Class A and Class B tension lap splices are given in ACI Table 10.7.5.2.2 (see Table 8.17). Tension development length is determined in accordance with ACI 25.4.2.1(a). Where bars of different size are lap spliced together, is the greater of (a) of the larger bar and (b) of the smaller bar (ACI 25.5.2.2).

8-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.16 Modification Factors for Deformed Bars in Compression Modification Factor

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



Lightweight, Longitudinal reinforcement enclosed in the following: 1. A spiral  A circular continuously wound tie with and a pitch 3. WLHVLQDFFRUGDQFHZLWK$&,VSDFHG on center on center  +RRSVLQDFFRUGDQFHZLWK$&,VSDFHG

&RQ¿QLQJ reinforcement,





Other

Table 8.17 Requirements for Class A and Class B Tension Lap Splices Stress in the longitudinal bars on the tension face

Splice Details

Splice Type

There are percent of the longitudinal bars spliced at any section and the lap splices on adjacent bars are staggered by at least

Class A

Other

Class B

All cases

Class B

*Tension development length

Tension Lap Splice Length, ℓst*

is determined in accordance with ACI 25.4.2.1(a).

3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHOopment length, LVGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU UHTXLUHPHQWVDUHFRYHUHG¿UVW Method 1 – ACI 25.4.2.4 The development length in tension of a deformed reinforcing bar,

LVGHWHUPLQHGE\$&,(TXDWLRQD  (8.37)

8-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHWKHVHIDFWRUVFDQEHIRXQGLQ7DEOHDORQJZLWK GH¿QLWLRQVIRUWKHRWKHUWHUPVLQ(TXDWLRQ :KHQFDOFXODWLQJ E\(TXDWLRQ WKHFRQ¿QLQJWHUP PXVWEHWDNHQOHVVWKDQRUHTXDOWR>$&,@ Table 8.18 Factors for Development of Deformed Bars in Tension Factor

Lightweight, Reinforcement grade,

Epoxy,

6L]H Casting position,

Spacing or cover dimension,

Transverse reinforcement index,

Condition

Value of Factor

Lightweight concrete Normalweight concrete *UDGHRU*UDGH

  

*UDGH



*UDGH

1.3

(SR[\FRDWHGRU]LQFDQGHSR[\GXDO coated reinforcement with clear cover or clear spacing



(SR[\FRDWHGRU]LQFDQGHSR[\GXDO coated reinforcement for all other conditions



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG  reinforcement



DQGODUJHUEDUV



DQGVPDOOHUEDUV



0RUHWKDQLQRIIUHVKFRQFUHWH SODFHGEHORZKRUL]RQWDOUHLQIRUFHPHQW

1.3

Other



All

Lesser of: 1. The distance from the center of a longitudinal bar to the nearest concrete surface.  One-half the center-to-center spacing of the lonJLWXGLQDOEDUVEHLQJGHYHORSHGVHH)LJXUH for the case of the longitudinal bars in a wall being developed).

All

where: total cross-sectional area of all transverse 1. reinforcement within a spacing that crosses the potential plane of splitting through the longitudinal reinforcement being developed center-to-center spacing of the transverse  reinforcement number of bars being developed along the 3. plane of splitting

*The product need not exceed 1.7. **It is permitted to conservatively use if transverse (confining) reinforcement is present or required. For reinforcing bars with than 6.0 in. on center, transverse reinforcement must be provided along the development length such that (ACI 25.4.2.2).

spaced closer

8-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄



‫ݏ‬

ܿͳ ൌ …‘˜‡” ൅ ܾ݀ǡ‫ ݏ݊ܽݎݐ‬൅ ൫ܾ݀ǡ݈‫ ݃݊݋‬Τʹ൯

ܿͳ  ܾܿ ൌ Ž‡••‡”‘ˆ ൝   ‫ ݏ‬Τʹ

”ƒ•˜‡”•‡ ”‡‹ˆ‘”…‡‡–

Figure 8.14 Spacing or cover dimension, cb.

Method 2 – ACI 25.4.2.3 7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHTwo sets of spacing and cover cases are given in ACI Table VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP VHH7DEOH 7KHPRGL¿FDWLRQVIDFWRUVLQWKHVHHTXDWLRQVDUHREWDLQHGIURP$&,7DEOHVHH Table 8.18). Table 8.19 Tension Development Length, ℓd, for Deformed Bars in Accordance with ACI 25.4.2.3 Spacing and Cover

Case 1

#6 and Smaller Bars

#7 and Larger Bars

Condition 1 1. Clear spacing of bars being developed or lap spliced  Clear cover 3. Stirrups or ties throughout not less than minimum required &RQGLWLRQ 1. Clear spacing of bars being developed or lap spliced  Clear cover

&DVH

Other conditions

Mechanical and Welded Splices

0HFKDQLFDORUZHOGHGVSOLFHVDUHSHUPLWWHGLQZDOOVDQGPXVWPHHWWKHUHTXLUHPHQWVRI$&,7KHVHW\SHVRI splices may be used in both tension and compression. A variety of proprietary mechanical devices are available that can be used to splice longitudinal reinforcing bars in DZDOOVHH5HIHUHQFH 7KHVHW\SHVRIVSOLFHVDUHFRPPRQO\VSHFL¿HGDWORFDWLRQVZKHUHLQRUGLQDWHO\ORQJODS splices would be required or where lap splices would cause congestion. Mechanical splices must be able to develop LQWHQVLRQRUFRPSUHVVLRQDWOHDVWSHUFHQWRIWKH\LHOGVWUHQJWKRIWKHORQJLWXGLQDOEDUV

8-26

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Welded splices must also be able to develop of the longitudinal bars. These types of splices are primarily LQWHQGHGIRUEDUVDQGODUJHU:HOGLQJRIUHLQIRUFLQJEDUVPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&, 8.8.3 Spacing of Longitudinal Reinforcement

0D[LPXPVSDFLQJRIORQJLWXGLQDOUHLQIRUFHPHQWLQZDOOVLVJLYHQLQ$&,VHH)LJXUH 

(8.38)

The

limit in Equation (8.38) is applicable where shear reinforcement is required for in-plane strength.

8.8.4 Spacing of Transverse Reinforcement

0D[LPXPVSDFLQJRIWUDQVYHUVHUHLQIRUFHPHQWLQZDOOVLVJLYHQLQ$&,VHH)LJXUH 

(8.39)

The

limit in Equation (8.39) is applicable where shear reinforcement is required for in-plane strength.

8.8.5 Lateral Support of Longitudinal Reinforcement

Longitudinal reinforcement in a wall must be laterally supported with transverse ties where the following two condiWLRQVDUHVDWLV¿HG$&,  1. The longitudinal reinforcement is required to resist compression (that is, the longitudinal reinforcement is required as compression reinforcement). is greater than where is the gross cross-sectional area  The area of longitudinal reinforcement, of the wall. ݄ ͓͵…”‘••–‹‡•™‹–Š͓ͳͲƒ†•ƒŽŽ‡”Ž‘‰Ǥ„ƒ”•   ൝ ͓Ͷ…”‘••–‹‡•™‹–Š͓ͳͳƒ†Žƒ”‰‡”Ž‘‰Ǥ„ƒ”•

Figure 8.15 Transverse ties in a wall in accordance with ACI 11.7.4.1.

8-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHVL]HDQGH[WHQWRIWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWLVQRWJLYHQLQ$&,6LPLODUWRFROXPQVLWLV UHFRPPHQGHGWRVSHFLI\WUDQVYHUVHWLHVIRUDQGVPDOOHUORQJLWXGLQDOEDUVLQWKHZDOODQGWUDQVYHUVHWLHVIRU larger longitudinal bars. The transverse ties must engage all longitudinal bars subjected to compression considering all DSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQV,QRUGHUWRGHYHORSWUDQVYHUVHWLHVLQWHQVLRQDPLQLPXPLQZDOOWKLFNQHVVLVUHTXLUHG6LPLODUO\DPLQLPXPLQWKLFNZDOOLVUHTXLUHGLQRUGHUWRGHYHORSWUDQVYHUVHWLHVLQWHQVLRQ $QH[DPSOHRIDZDOOZKHUHFURVVWLHVDUHSURYLGHGDVWKHUHTXLUHGWUDQVYHUVHWLHVLVJLYHQLQ)LJXUH7KLVLVWKH SUHIHUUHGW\SHRIWUDQVYHUVHUHLQIRUFHPHQWIRUZDOOVZLWKOD\HUVRIORQJLWXGLQDOUHLQIRUFHPHQWFRPSDUHGWRUHFWDQgular ties that enclose four of the longitudinal bars in a wall.  ൒ κௗ  ‘–‡ͳ

൒ κௗ 

‘–‡ʹ

‡•‹‘Žƒ’•’Ž‹…‡

‘–‡͵

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Žƒ„ƒ†‘–Š‡”™ƒŽŽ”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

‘–‡• ͳǤ ʹǦ͓ͷൈͶԢǦͲԣ†‹ƒ‰‘ƒŽ„ƒ”•ƒ–‡ƒ…Š…‘”‡”Ǥ ʹǤ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†Ž‘‰‹–—†‹ƒŽ™ƒŽŽ ”‡‹ˆ‘”…‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ–‡ƒ…Š•‹†‡‘ˆ–Š‡‘’‡‹‰Ǥ ͵Ǥ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†–”ƒ•˜‡”•‡™ƒŽŽ

”‡‹ˆ‘”…‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ––Š‡–‘’ƒ†„‘––‘‘ˆ–Š‡‘’‡‹‰Ǥ Figure 8.16 Recommended reinforcement details around a door opening in a reinforced concrete wall with two layers of reinforcement.

8-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8.8.6 Reinforcement Around Openings

)RUZDOOVZLWKZLQGRZGRRUDQGVLPLODUO\VL]HGRSHQLQJVWKHIROORZLQJDGGLWLRQDOUHLQIRUFHPHQWPXVWEHSURYLGHG DURXQGWKHSHULPHWHURIWKHRSHQLQJV$&,  • $PLQLPXPRIEDULQZDOOVZLWKRQHOD\HURIUHLQIRUFHPHQW • $PLQLPXPRIEDUVLQZDOOVZLWKWZROD\HUVRIUHLQIRUFHPHQW The additional reinforcement must be anchored to develop in tension at the corners of the openings. These bars, which are commonly referred to as trim bars, essentially replace the reinforcement interrupted by the opening. More than the minimum amounts of trim bars may need to be provided based on an analysis of the wall with the opening(s). A recommended detail for the reinforcement around a typical door opening is given in Figure 8.16 for a ZDOOZLWKOD\HUVRIUHLQIRUFHPHQW$VLPLODUGHWDLOIRUZLQGRZDQGRWKHUZDOORSHQLQJVLVJLYHQLQ)LJXUH  ൒ κௗ 

‘–‡ͳ

൒ κௗ 

‘–‡ʹ ‘–‡͵

‘–‡͵

–Š‡”™ƒŽŽ”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

‘–‡• ͳǤ ʹǦ͓ͷൈͶԢǦͲԣ†‹ƒ‰‘ƒŽ„ƒ”•ƒ–‡ƒ…Š…‘”‡”Ǥ ʹǤ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†Ž‘‰‹–—†‹ƒŽ™ƒŽŽ ”‡‹ˆ‘”…‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ–‡ƒ…Š•‹†‡‘ˆ–Š‡‘’‡‹‰Ǥ ͵Ǥ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†–”ƒ•˜‡”•‡™ƒŽŽ

”‡‹ˆ‘”…‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ––Š‡–‘’ƒ†„‘––‘‘ˆ–Š‡‘’‡‹‰Ǥ Figure 8.17 Recommended reinforcement details around an opening in a reinforced concrete wall with two layers of reinforcement.

of the longitudinal and transverse The preferred reference point for measuring the tension development length, WULPEDUVLVWKHFRUQHURIWKHRSHQLQJEHFDXVHLWLVD¿[HGSRLQWVHH)LJXUHRU)LJXUH 0HDVXULQJ from a longitudinal or transverse trim bar can be done, but the development length may end up being too short if the perpendicular trim bars adjacent to the opening are not at their intended locations.

8-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For openings occurring close to the top of a wall or where openings are stacked on top of each other, the typical straight diagonal bar detail may not be possible. In order to properly develop the bars, either a standard hook needs to be provided at one end of the bar or the bar can be bent, as shown in Figure 8.18 for the case of stacked openings. 

κௗ௛ 

–Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 8.18 Diagonal bar development at stacked wall openings.

8.9 Connections to Foundations 8.9.1 Overview

9HUWLFDODQGKRUL]RQWDOIRUFHVPXVWEHWUDQVIHUUHGDWWKHLQWHUIDFHEHWZHHQDZDOODQGDIRXQGDWLRQLQDFFRUGDQFH ZLWK$&,$&, 9HUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGE\EHDULQJRQWKHFRQFUHWHRUE\DFRPELnation of bearing and interface reinforcement. Tension forces must be transferred entirely by reinforcement, which PD\FRQVLVWRIH[WHQGHGORQJLWXGLQDOEDUVGRZHOVDQFKRUEROWVRUPHFKDQLFDOFRQQHFWRUV+RUL]RQWDOIRUFHVDUH WUDQVIHUUHGXVLQJWKHVKHDUIULFWLRQSURYLVLRQVRI$&,RURWKHUDSSURSULDWHPHWKRGV 8.9.2 Vertical Transfer Compression

:KHUHYHUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGWRDIRXQGDWLRQEHDULQJVWUHQJWKUHTXLUHPHQWVRI$&,PXVW EHVDWLV¿HGIRUERWKWKHZDOODQGWKHIRXQGDWLRQ$&, )RUEHDULQJRQDZDOOWKHIDFWRUHGEHDULQJIRUFH must be less than or equal to the design bearing strength, where the nominal bearing strength, is given LQ$&,7DEOH (8.40)

In this equation,

LVHTXDOWRIRUEHDULQJ$&,7DEOH DQG

is the gross area of the wall.

)RUEHDULQJRQDIRXQGDWLRQZLGHURQDOOVLGHVWKDQWKHORDGHGDUHDWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG (8.41)

8-30

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term is the area of the lower base of the largest frustrum of a pyramid, cone, or tapered wedge contained and having side slopes of 1 vertical to wholly within the foundation and having for its upper base the loaded area are depicted in Figure 8.19 for the case of a wall supported by a reinforced concrete KRUL]RQWDO$UHDV and foundation.

ʹ

݄

ͳ

‫ܮ‬

κ௪ 

‫ܤ‬

ܾ

‫ܤ‬

Ͷͷ‫ ל‬

‫ܣ‬ଵ ൌ ݄κ௪ 

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‫ܣ‬ଶ ൌ ܾ‫ܮ‬ ‫ܮ‬ Figure 8.19 Determination of areas for bearing strength.

the excess compression stress from the wall must be transferred by reinforcement to the foundaWhere is determined by the following equation: tion. The required area of interface reinforcement, (8.42)

In this equation, LVWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUDWLRVDWLVI\LQJ$&,$&,VHH must be provided even where 7DEOH 7KHPLQLPXPDPRXQWRILQWHUIDFHUHLQIRUFHPHQW Dowel bars emanating from the foundation are the type of interface reinforcement commonly used due to ease of FRQVWUXFWLRQ7KHGRZHOEDUVZKLFKDUHW\SLFDOO\RIWKHVDPHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQ the wall, are set in the foundation prior to placing the foundation concrete and are subsequently spliced to the longitudinal bars. ,OOXVWUDWHGLQ)LJXUHDUHGRZHOEDUVDFURVVWKHLQWHUIDFHEHWZHHQDUHLQIRUFHGFRQFUHWHZDOODQGIRXQGDWLRQ where all the longitudinal bars in the wall are in compression for all factored load combinations. The dowel bars determined in accordance with ACI must extend into the foundation at least a compression development length, VHH6HFWLRQRIWKLVSXEOLFDWLRQ  6WDQGDUGKRRNVDUHW\SLFDOO\SURYLGHGDWWKHHQGVRIWKHGRZHOEDUVZKLFKDUHWLHGWRWKHÀH[XUDOUHLQIRUFLQJEDUV LQWKHIRXQGDWLRQWKLVGHWDLOLVIDUPRUHHFRQRPLFDOWKDQWHUPLQDWLQJWKHGRZHOEDUVDERYHWKHÀH[XUDOUHLQIRUFLQJ bars. The hooked portion of the dowels cannot be considered effective for developing the dowels bars in compresVLRQ$&, 7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHOoped into the foundation: (8.43)

8-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄

ƒŽŽŽ‘‰‹–—†‹ƒŽ„ƒ”•

൒ κௗ௖ 

–ƒ†ƒ”†Š‘‘ ሺ–›’Ǥሻ

‘—†ƒ–‹‘„ƒ”•ǡሺ݀௕ ሻ௙ ሺ–›’Ǥሻ

͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ



‘’”‡••‹‘Žƒ’ •’Ž‹…‡

‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ 

Figure 8.20 Dowel bars at the interface of a reinforced concrete wall and foundation where all the longitudinal bars in the wall are in compression.

In this equation, LVWKHUDGLXVRIWKHGRZHOEDUEHQGVHH$&,7DEOHZKLFKFRQWDLQVPLQLPXPLQVLGHEHQG GLDPHWHUVIRUEDUVZLWKVWDQGDUGGHJUHHKRRNV :KHUH(TXDWLRQ LVQRWVDWLV¿HGRQHW\SLFDOZD\WRHQVXUH adequate development is to increase the thickness of the foundation, The dowel bars must also be fully developed in the wall; this is typically achieved by lap splicing the dowel bars to WKHORQJLWXGLQDOEDUVLQWKHZDOOVHH)LJXUH :KHUHWKHGRZHOEDUVDUHWKHVDPHVL]HDVWKHZDOOORQJLWXGLQDO EDUVWKHPLQLPXPFRPSUHVVLRQODSVSOLFHOHQJWKLVXVHGVHH7DEOH :KHUHWKHGRZHOEDUVDUHODUJHULQGLDPHWHUWKDQWKHZDOOORQJLWXGLQDOEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWKPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&, /DSVSOLFHVRIDQGORQJLWXGLQDOEDUVLQFRPSUHVVLRQIRUDOOIDFWRUHGORDGFRPELQDWLRQVZLWKDQGVPDOOHU GRZHOEDUVDUHSHUPLWWHGSURYLGHGWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG$&,  Tension

Tension forces transferred from a wall to a foundation must be resisted entirely by reinforcement across the interIDFH>$&,E DQG@ 7HQVLOHDQFKRUDJHRIWKHGRZHOEDUVLQWRDIRXQGDWLRQLVW\SLFDOO\DFFRPSOLVKHGE\SURYLGLQJGHJUHHVWDQGDUG determined in accordance hooks at the ends of the dowel bars with the development length of the hooked bar, ZLWK$&,VHH)LJXUH 

(8.44)

8-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄

ƒŽŽŽ‘‰‹–—†‹ƒŽ„ƒ”•

൒ κௗ௛ 



Žƒ••–‡•‹‘ Žƒ’•’Ž‹…‡

‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ 

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–ƒ†ƒ”†Š‘‘ ሺ–›’Ǥሻ

‘—†ƒ–‹‘„ƒ”•ǡሺ݀௕ ሻ௙ ሺ–›’Ǥሻ

Figure 8.21 Dowel bars at the interface of a reinforced concrete wall and foundation where the longitudinal bars in the wall are in compression and tension.

7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRNVHH)LJXUHRIWKLV SXEOLFDWLRQ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHVHH7DEOH  Table 8.20 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Lightweight,

Epoxy,

&RQ¿QLQJ reinforcement,

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK Other

or

 1.6 (table continued on next page)

8-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.20 Modification Factors for Development of Hooked Bars in Tension (cont.) Modification Factor

Location,

Condition

Value of Factor

)RUDQGVPDOOHUKRRNHGEDUV 1. terminating inside a column core with side cover normal to the plane of the hook or  with side cover normal to the plane of the hook



Other



Concrete strength,



7KHFRQ¿QLQJUHLQIRUFHPHQWIDFWRU is typically equal to 1.6 for hooked dowel bars in foundations because con¿QLQJUHLQIRUFHPHQWLVXVXDOO\QRWSURYLGHGLQVXFKFDVHV 7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKHIRXQGDWLRQ (considering that the hooked portion of the dowel bars can be developed in tension): (8.45)

:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHGHSWKRIWKHIRXQGDWLRQXVXDOO\PXVWEHLQFUHDVHG For development of the dowel bars into the wall, a tension lap splice or a mechanical connection in accordance with $&,RUUHVSHFWLYHO\PXVWEHSURYLGHGEHWZHHQWKHGRZHOEDUVDQGWKHORQJLWXGLQDOEDUV,QWKHFDVH RIODSVSOLFHVD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGVHH7DEOHDQG)LJXUH  8.9.3 Horizontal Transfer

7KHVKHDUIULFWLRQPHWKRGRI$&,LVSHUPLWWHGWREHXVHGWRGHWHUPLQHWKHQRPLQDOVKHDUVWUHQJWK at the across FRQWDFWVXUIDFHEHWZHHQWKHZDOODQGWKHIRXQGDWLRQ$&, 7KHUHTXLUHGDUHDRIUHLQIRUFHPHQW the interface is determined by the following equation, which is applicable to shear-friction reinforcement perpenGLFXODUWRWKHLQWHUIDFH$&,  (8.46)

In this equation, is the factored shear force due to the lateral force effects at the interface (in-plane or out-ofplane), the strength reduction factor LVHTXDOWRDQG LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP $&,7DEOHVHH7DEOH  Table 8.21 Coefficient of Friction, μ Contact Surface Condition

Concrete placed monolithically Concrete placed against hardened concrete that is clean, free of laitance, and intentionally roughened to a full amplitude of approximately ¼ in. Concrete placed against hardened concrete that is clean, free of laitance, and not intentionally roughened

8-34

μ

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOHVHH7DEOH 7KHWHUP is the area of which is equal to the gross cross-sectional area of the wall. Where the concrete strengths of concrete resisting WKHZDOODQGWKHIRXQGDWLRQDUHGLIIHUHQWWKHVPDOOHURIWKHWZRPXVWEHXVHGLQWKHVHHTXDWLRQV$&,  Table 8.22 Maximum Vn Across the Assumed Shear Plane Contact Surface Condition

Maximum Vn

= Vu / Ҁ*

Normalweight concrete placed monolithically or placed against hardened concrete intentionally roughened to a full amplitude of approximately ¼ in.

Other cases

*

area of concrete resisting

Full tension anchorage of the shear-friction reinforcement must be provided into the foundation and into the wall. The lengths of these bars are determined in the same way as those for vertical transfer where tension forces are present. The area of the dowel bars is usually determined initially based on the requirements for vertical transfer. That area is FRPSDUHGZLWKWKHDUHDUHTXLUHGIRUKRUL]RQWDOWUDQVIHUDQGWKHODUJHURIWKHWZRLVSURYLGHGDWWKHLQWHUIDFH

8.10 Design Procedure 7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHZDOOVVXEMHFWHG WRD[LDOFRPSUHVVLRQIRUFHVDQGXQLD[LDOEHQGLQJLQSODQHRURXWRISODQH ,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQ QXPEHUVWDEOHQXPEHUVDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQLQWKLVFKDSWHUFDQEHIRXQG

8.11 Examples 8.11.1 Example 8.1 – Design of Reinforced Concrete Wall: Building #2, Interior Wall is Not Part of the SFRS, Simplified Design Method

Design a solid, interior reinforced concrete wall that is not part of the seismic force-resisting system (SFRS) in the ¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJD[LDOIRUFHVZKLFKDUHDVVXPHGWRDFWDWDLQHFFHQWULFLW\ and Assume normalZLWKUHVSHFWWRWKHFHQWURLGRIWKHZDOOVHH)LJXUH  DQG*UDGHUHLQIRUFHPHQW weight concrete with Step 1 – Determine the factored axial force

Table 3.3

The following load combination governs:

Step 2 – Determine a trial wall thickness

$VVXPHWKH6LPSOL¿HG'HVLJQ0HWKRGFDQEHXVHG7KLVDVVXPSWLRQLVYHUL¿HGLQ6WHSEHORZ

8-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Provide dowel bars to satisfy the requirements in ACI 16.3 for connections to foundations ሺSect. 8.9ሻ

Figure 8.22 Design procedure for walls.

A minimum wall thickness can be determined by the following equation: Eq. (8.35)

8-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܲ஽ ǡ ܲ௅ 

κ௖ ൌ  ͳͺǤͲᇱ 

ͳᇳ 

Figure 8.23 Interior bearing wall of Example 8.1.

Assuming the wall is braced at the top and bottom against lateral translation and unrestrained against rotation at both ends: Table 8.7

Therefore,

For a bearing wall, the minimum thickness is equal to the following: Figure 8.1

Try Step 3 – Determine if the Simplified Design Method can be used

ACI 11.5.3

&KHFNWKHOLPLWDWLRQVRIWKH6LPSOL¿HG'HVLJQ0HWKRG 1. The wall is solid and rectangular.  Eccentricity

Figure 8.8

7KHUHIRUHWKH6LPSOL¿HG'HVLJQ0HWKRGFDQEHXVHG

8-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the design strength of the wall

Eq. (8.27)

Step 5 – Determine the minimum reinforcement in the wall

$VVXPHEDUVDUHXVHGIRUWKHORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQW Minimum longitudinal reinforcement

Table 8.10

3URYLGHDVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU

Minimum transverse reinforcement 3URYLGHDVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU Maximum spacing

Figure 8.11

8.11.2 Example 8.2 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS, Moment Magnification Method, Out-of-Plane Forces

'HVLJQDVROLGH[WHULRUUHLQIRUFHGFRQFUHWHZDOOWKDWLVQRWSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJIRUFHVVHH)LJXUH 

(it can be determined that the effects due to out-of-plane seismic effects in accordance with $6&(6(,DUHOHVVWKDQWKDWGXHWRZLQGORDGHIIHFWV DQG*UDGHUHLQIRUFHPHQWDQGWKHIUDPHLQWKH¿UVWVWRU\LV Assume normalweight concrete with nonsway. Also, the gravity loads are assumed to act concentrically on the wall. Step 1 – Determine a trial wall thickness

There is no closed-form solution to determine

for a wall subjected to these types of loads.

A trial wall thickness of 8 in. is selected. The adequacy of this wall thickness will be checked in the following steps. Step 2 – Determine the factored load combinations

Table 3.3

Weight of the wall at mid-height Total dead load The gravity loads do not cause any bending moments in the wall because they act through the centroid of the wall.

8-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

κ௨ ൌ ͳͺǤͲ െ ሾʹͺǤͷȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͸Ǥͺᇱ 

ܲ஽ ǡ ܲ௅ 

‫ݓ‬ௐ 

Figure 8.24 Exterior bearing wall of Example 8.2.

Assuming the wall is pinned at the top and bottom, the bending moment at the mid-height of the wall due to the uniformly distributed wind load is equal to the following:

$VXPPDU\RIWKHIDFWRUHGORDGFRPELQDWLRQVLVJLYHQLQ7DEOH Table 8.23 Summary of Axial Forces and Bending Moments for the Wall in Example 8.2 Load Case

Axial Force (kips/ft)

Bending Moment (in.-kips/ft)

Dead





Live





Wind



Load Combination $&,(TD





$&,(TE





$&,(TG





$&,(TI





8-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine if slenderness effects need to be considered

%HFDXVHWKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPHVOHQGHUQHVVHIIHFWVQHHGQRWEHFRQVLGHUHGZKHUH$&,(TV E DQGF DUHVDWLV¿HGVHH7DEOHRIWKLVSXEOLFDWLRQ 

Therefore, slenderness effects must be considered in the design of this wall. Step 4 – Determine the magnified moments in the wall using the moment magnification method

ACI 6.6.4

7KHPRPHQWPDJQL¿FDWLRQPHWKRGLVFRYHUHGLQGHWDLOLQ6HFWRIWKLVSXEOLFDWLRQ )RUQRQVZD\IUDPHVIDFWRUHGPDJQL¿HGPRPHQWV

DUHGHWHUPLQHGE\(T 

where LVWKHIDFWRUHGPLGKHLJKWEHQGLQJPRPHQWLQWKHZDOODQGWKHPDJQL¿FDWLRQIDFWRU (T 

Because there are transverse loads between the supports, The critical buckling load,

The effective stiffness,

is determined by

ACI 6.6.4.5.3(b)

LVGHWHUPLQHGE\(T 

is determined by the following equation: Eq. (8.2)

Table 8.2

It is evident that the area of longitudinal reinforcement in the wall, is needed to calculate purposes, the minimum amount of longitudinal reinforcement is assumed:

8-40

For calculation

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7U\DVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU checked in the following steps.

The adequacy of this reinforcement will be

&DOFXODWLRQVDUHSURYLGHGIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(TG LQ7DEOH For a 1-foot design strip: ACI Eq. (22.4.2.2)

ACI Eq. (19.2.2.1a)

Eq. (8.2)

Therefore,

Determine the minimum factored bending moment,

6LPLODUFDOFXODWLRQVFDQEHSHUIRUPHGIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(TI  $VXPPDU\RIWKHUHVXOWVIURPWKHPRPHQWPDJQL¿FDWLRQPHWKRGIRUWKHORDGFRPELQDWLRQVWKDWLQFOXGHZLQGORDG HIIHFWVDUHJLYHQLQ7DEOH

8-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.24 Summary of Results from the Moment Magnification Method for the Wall in Example 8.2 Load Combination

ȕdns

I

(EI)eff x 10 3

Pc

(in.4)

(kip-in.2)

(kips)





























Pu

Mu

(kips)

(in.-kips)

 

į

Step 5 – Check the adequacy of the section for combined flexure and axial forces

Mc (in.-kips)

ACI 22.4

The lower portion of the design strength interaction diagram for a 1-foot-wide section of the wall reinforced with EDUVVSDFHGDWLQRQFHQWHULVVKRZQLQ)LJXUH%HFDXVHDOOORDGFRPELQDWLRQVIDOOZLWKLQWKHGHVLJQ strength interaction diagram, the wall is adequate. 

80 70

Axial Force ሺkipsሻሻ

60 50 40 30 20 10 0

  0

25

50

75

100

125

150

175

200

Bending Moment ሺin.-kipsሻ Figure 8.25 Design strength interaction diagram for the wall in Example 8.2.

Step 6 – Check the out-of-plane shear strength requirements

The required shear strength,

is determined based on the design wind force:

The design shear strength,

is equal to the design shear strength of the concrete,

ACI 22.5

Eq. (8.31)

Eq. (8.32)

for normalweight concrete

8-42

Table 8.8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

The axial force smaller value of

FRUUHVSRQGLQJWR$&,(TI LVXVHGLQGHWHUPLQLQJ because it results in a than that determined using FRUUHVSRQGLQJWR$&,(TG 

The wall is adequate for out-of-plane shear. 8.11.3 Example 8.3 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS, Alternative Method for Out-of-Plane Forces

'HVLJQDVROLGH[WHULRUUHLQIRUFHGFRQFUHWHZDOOWKDWLVQRWSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJIRUFHVVHH)LJXUH 

(it can be determined that the effects due to out-of-plane seismic effects in accordance with $6&(6(,DUHOHVVWKDQWKDWGXHWRZLQGORDGHIIHFWV ܲ஽ ǡ ܲ௅ 

κ௨ ൌ ͳͺǤͲ െ ሾʹͺǤͷȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͸Ǥͺᇱ 

݁ ൌ ʹǤͲԣ

‫ݓ‬ௐ 

Figure 8.26 Exterior bearing wall in Example 8.3.

8-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Assume normalweight concrete with DQG*UDGHUHLQIRUFHPHQW$OVRDVVXPHDQHFFHQWULFLW\RI in. between the dead and live load axial forces and the centroid of the wall. Step 1 – Determine a trial wall thickness

There is no closed-form solution to determine

for a wall subjected to these types of loads.

A trial wall thickness of 8 in. is selected. The adequacy of this wall thickness will be checked in the following steps. Step 2 – Determine the factored load combinations at the mid-height of the wall

Weight of the wall from top to mid-height

Service dead load moment mid-height)

(only the axial dead load causes a bending moment at

Service live load moment Wind load moment $VXPPDU\RIWKHIDFWRUHGORDGFRPELQDWLRQVLVJLYHQLQ7DEOH Table 8.25 Summary of Axial Forces and Bending Moments for the Wall in Example 8.3 Load Case

Axial Force (kips/ft)

Bending Moment (in.-kips/ft)

Dead





Live





Wind



Load Combination $&,(TD

3.9



$&,(TE





$&,(TG

3.8



$&,(TI





8-44

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the factored load combinations at the mid-height of the wall, including slenderness effects

8VLQJWKHGLUHFWFDOFXODWLRQPHWKRGRIWKHDOWHUQDWLYHPHWKRG>$&,E @WKHWRWDOIDFWRUHGPRPHQW which includes slenderness effects, is determined by the following equation: Eq. (8.10)

where

DUHWKHIDFWRUHGPRPHQWVLQWKHZDOODWPLGKHLJKWVHH7DEOH 

A summary of the factored axial forces and bending moments, which include slenderness effects, is given in Table  Table 8.26 Summary of Factored Axial Forces and Bending Moments, Including Slenderness Effects, for the Wall in Example 8.3 Load Combination

Pu

Mua

Ase,w

c

Icr

Mu

(kips)

(in.-kips)

(in.2)

(in.)

(in.4)

(in.-kips)

3.9



















31.6



3.8























&DOFXODWLRQVDUHSURYLGHGIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(TG 

Maximum longitudinal bar spacing

Figure 8.11

$VVXPHWKHZDOOLVUHLQIRUFHGZLWKRQHOD\HURIEDUVVSDFHGDWLQRQFHQWHU Eq. (8.7)

Eq. (8.9) ACI 20.2.2.2

Check if the section is tension-controlled:

ACI Table 21.2.2

8-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore, the section is tension-controlled. It can be determined that this wall section is tension-controlled for all ORDGFRPELQDWLRQV7KLVVDWLV¿HVWKHVHFRQGOLPLWDWLRQLQ$&, ACI Eq. (19.2.2.1a) ACI 11.8.3.1 Eq. (8.6)

Therefore,

Similar calculations can be performed for the other load combinations. Step 4 – Determine the cracking moment ACI Eq. (19.2.3.1)

ACI Eq. (24.2.3.5)

Step 5 – Determine the design flexural strength of the wall and check the adequacy of the wall

7KHGHVLJQÀH[XUDOVWUHQJWKRIWKHZDOOLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ Eq. (8.11)

,WLVGHWHUPLQHGLQ6WHSWKDWWKHZDOOVHFWLRQLVWHQVLRQFRQWUROOHGIRUDOOORDGFRPELQDWLRQVVR 7DEOH 

(see

so the wall is adequate 7KHGHVLJQÀH[XUDOVWUHQJWKVDUHJLYHQLQ7DEOH)RUDOOORDGFRPELQDWLRQV IRUDOOORDGFRPELQDWLRQVZKLFKVDWLV¿HVWKHWKLUGOLPLWDWLRQLQ$&, IRUÀH[XUH$OVR 11.8.1.1.

8-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.27 Summary of Design Flexural Strengths for the Wall in Example 8.3 Load Combination

Ase,w

c

a

ҀMn

Mu

(in.2)

(in.)

(in.2)

(in.-kips)

(in.-kips)







68.9



















66.9









63.1



Step 6 – Check the maximum axial force at the mid-height of the wall section

According to the fourth limitation in ACI 11.8.1.1,

at the mid-height of the wall must be less than or equal to

,WLVHYLGHQWWKDWWKLVOLPLWDWLRQLVVDWLV¿HGIRUDOOORDGFRPELQDWLRQVVHH7DEOH  Step 7 – Determine the maximum service-level at the mid-height of the wall section

Because there is no closed-form solution for for the initial iteration:

The maximum service-level bending moment,

assume

Also assume the following value of

at the mid-height section of the wall is equal to the following: Eq. (8.17)

where the strength-level bending moment due to wind loads is divided by 1.6 to convert it to a service-level bending moment. 7KHGHÀHFWLRQ

is determined by the following equation: Eq. (8.15)

Therefore,

Determine

from the following equation: Eq. (8.12)

Because it has been assumed that

determine

from the following equation: Eq. (8.13)

8-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

This value of

is the same as the value initially assumed for

so no additional iterations are required.

Check the initial assumption:

&KHFNWKDWWKH¿IWKOLPLWDWLRQRI$&,LVVDWLV¿HG

Step 8 – Check the out-of-plane shear strength requirements

ACI 22.5

The required shear strength,

LVGHWHUPLQHGEDVHGRQWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(TG 

The design shear strength,

is equal to the design shear strength of the concrete, Eq. (8.31)

Eq. (8.32)

for normalweight concrete

Table 8.8

Therefore,

The wall is adequate for out-of-plane shear. 8.11.4 Example 8.4 – Determination of Trial Wall Thickness of Reinforced Concrete Wall: Building #2, SDC C, Interior Wall is Part of the SFRS

Determine a trial wall thickness for the interior reinforced concrete wall along column line F that is part of the 6)56LQWKH¿UVWVWRU\RI%XLOGLQJFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQVHH)LJXUH $VVXPH and Grade 60 reinforcement. normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Solution 7KLVLQWHULRUZDOOLVVXEMHFWHGWRD[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHV,WLVHYLGHQWIURP7DEOHVDQG RIWKLVSXEOLFDWLRQWKDWZLQGORDGHIIHFWVDUHOHVVWKDQWKRVHIURPVHLVPLFHIIHFWV

8-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

There is no closed-form solution to determine h for a wall subjected to these combined load effects. However, a trial wall thickness can be obtained based on shear strength requirements: Eq. (8.34)

)URP7DEOHWKHEDVHVKHDUGXHWRVHLVPLFHIIHFWVLVHTXDOWRNLSV7KHZDOOVDORQJFROXPQOLQHV%DQG )HDFKUHVLVWSHUFHQWRIWKLVEDVHVKHDUVRWKHIDFWRUHGVKHDUIRUFH LQHDFKZDOOLQWKH¿UVWVWRU\LVHTXDOWR NLSV Eq. (8.29)

for normalweight concrete $VVXPLQJ

Table 8.8

VHH)LJXUH 

$WULDOZDOOWKLFNQHVVRILQLVVHOHFWHGFRQVLGHULQJWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVWKDWWKLVZDOOLVVXEMHFWHGWRVHH([DPSOH  8.11.5 Example 8.5 – Design of Reinforced Concrete Wall for Combined Flexure and Axal Forces: Building #2, SDC C, Interior Wall is Part of the SFRS

'HVLJQWKHLQWHULRUUHLQIRUFHGFRQFUHWHZDOODORQJFROXPQOLQH)WKDWLVSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJIRUFRPELQHGÀH[XUHDQGD[LDOIRUFHVFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQVHH)LJXUH  and Grade $VVXPHDLQWKLFNZDOOVHH([DPSOH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK 60 reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored load combinations

Table 3.3

$WKUHHGLPHQVLRQDOHODVWLFDQDO\VLVRIWKHVWUXFWXUHZDVSHUIRUPHGXVLQJ5HIZLWKWKHZDOOVVXEMHFWHGWRWKH HDVWZHVWVHLVPLFIRUFHVLQ7DEOHDVVXPLQJWKHZDOOVDUH¿[HGDWWKHEDVHZLQGORDGHIIHFWVDUHOHVVWKDQWKRVH IURPVHLVPLFHIIHFWVDQGDUHQRWFRQVLGHUHGLQWKLVH[DPSOHVHH7DEOH 5LJLGGLDSKUDJPVZHUHDVVLJQHGDW ZDVDVVLJQHGWRWKHZDOOVLQDFFRUGDQFHZLWK$&, each level of the building and a moment of inertia equal to 7DEOHD DVVXPLQJWKHZDOOVDUHFUDFNHG7KHDQDO\VLVVKRZHGWKDWWKHODWHUDOORDGHIIHFWVDUHHVVHQWLDOO\ UHVLVWHGE\WKHIWZDOOZHEVLQWKHGLUHFWLRQRIDQDO\VLVVRQRHIIHFWLYHÀDQJHZLGWKVDUHFRQVLGHUHGLQGHVLJQ that is, all the lateral load effects are assigned to the webs. $VXPPDU\RIWKHIDFWRUHGORDGFRPELQDWLRQVLVJLYHQLQ7DEOHIRUWKHZDOODORQJFROXPQOLQH) Table 8.28 Summary of Axial Forces and Bending Moments for the Wall in Example 8.4 Load Case

Axial Force (kips)

Bending Moment (ft-kips)

Dead



0

Roof live



ņ

Live



0

Seismic

0 (table continued on next page)

8-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 8.28 Summary of Axial Forces and Bending Moments for the Wall in Example 8.4 (cont.) Load Case

Axial Force (kips)

Bending Moment (ft-kips)

$&,(TD





$&,(TE





$&,(TF





$&,(TH



$&,(TJ



Load Combination

Further explanation is needed on how the load factors in the load combinations that include (TH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR $6&(6(, FDQEHWDNHQDVIRUVWUXFWXUHVDVVLJQHGWR6'&&WKHUHIRUH Therefore, the load combination becomes the following:

are obtained. In ACI According to

,Q$&,(TJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR and the load combination becomes the following:

Step 2 – Determine if the first story is nonsway or sway

The stability index,

LVXVHGWRGHWHUPLQHZKHWKHUWKH¿UVWVWRU\LVQRQVZD\RUVZD\

7KHIROORZLQJUHVXOWVDUHREWDLQHGIURPWKHDQDO\VLVLQWKH¿UVWVWRU\ Total

Total

The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI (TH SURGXFHVWKHODUJHVWORDG

8-50

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

7KHUHIRUHWKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPH Step 3 – Determine if slenderness effects must be considered

%HFDXVHWKH¿UVWVWRU\IUDPHLVDQRQVZD\IUDPHVOHQGHUQHVVHIIHFWVQHHGQRWEHFRQVLGHUHGZKHUHWKHIROORZLQJ HTXDWLRQLVVDWLV¿HG Table 8.4

7KHPRPHQWDWWKHWRSRIWKHZDOOLQWKH¿UVWVWRU\LVHTXDOWRIWNLSVDQGWKHZDOOLVEHQWLQVLQJOHFXUYDture. Therefore,

Therefore,

Thus, slenderness effects need not be considered in the design of this wall in the direction of analysis. Step 4 – Determine if the wall is adequate for combined flexure and axial forces

An estimate of the longitudinal reinforcement in the wall is needed in order to determine if the section is adequate RUQRWIRUWKHIDFWRUHGORDGFRPELQDWLRQVLQ7DEOH The minimum amount of longitudinal reinforcement depends on the magnitude of (see Figure 8.11).

with respect to the limit

From the analysis, Eq. (8.29)

Preliminary analysis indicates that minimum transverse reinforcement can be used (that is, 8.11 and Example 8.6).

see Figure

Therefore, Minimum

Figure 8.11

8-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Check the adequacy of the wall using minimum longitudinal reinforcement:

Maximum longitudinal bar spacing

Figure 8.11

7U\OD\HUVRIEDUVVSDFHGDWLQRQFHQWHU 7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVZDOOUHLQIRUFHGZLWKOD\HUVRIEDUVVSDFHGDWLQRQFHQWHU LVJLYHQLQ)LJXUH,WZDVFRQVWUXFWHGEDVHGRQHTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ $&,VHH6HFWLRQDQG)LJXUHRIWKLVSXEOLFDWLRQRQKRZWRFRQVWUXFWLQWHUDFWLRQGLDJUDPVIRUUHFWDQJXODUVHFWLRQV 7KHORDGFRPELQDWLRQVIURP7DEOHDUHDOVRJLYHQLQWKH¿JXUH%HFDXVHDOOORDGFRPELQDWLRQ points fall within the design strength interaction diagram, the wall is adequate. 

15,000

݂௦ ൌ 0

Axial Force ሺkipsሻሻ

12,000

݂௦ ൌ 0.5݂௬ 

 9,000  



6,000



3,000

0 0

15,000

30,000

45,000

60,000

75,000

90,000

105,000

Bending Moment ሺft-kipsሻ Figure 8.27 Design strength interaction diagram for the wall in Example 8.5

8.11.6 Example 8.6 – Design of Reinforced Concrete Wall for Shear Forces: Building #2, SDC C, Interior Wall is Part of the SFRS

'HVLJQWKHLQWHULRUUHLQIRUFHGFRQFUHWHZDOODORQJFROXPQOLQH)WKDWLVSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJIRUVKHDUIRUFHVFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQVHH)LJXUH $VVXPHQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH

8-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the transverse reinforcement

An estimate of the transverse reinforcement in the wall is needed in order to determine if the section is adequate or not for the factored shear force The minimum amount of transverse reinforcement depends on the magnitude of (see Figure 8.11).

with respect to the limit

Eq. (8.29)

Therefore, minimum

Figure 8.11

Maximum transverse bar spacing

Figure 8.11

7U\OD\HUVRIEDUVVSDFHGDWLQRQFHQWHU Step 2 – Check the adequacy of the section

Eq. (8.28)

7KHUHIRUHWKHVHFWLRQLVDGHTXDWHIRUVKHDUXVLQJWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIOD\HUVRIEDUVVSDFHG at 18 in. on center. 8.11.7 Example 8.7 – Determination of Dowel Reinforcement at the Foundation of a Reinforced Concrete Wall: Building #2, SDC C, Interior Wall is Part of the SFRS

Determine the required dowel reinforcement for the interior reinforced concrete wall along column line F that is SDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQDVVXPLQJ WKHZDOOLVVXSSRUWHGE\DIWWKLFNPDWIRXQGDWLRQVHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK and IRUWKHZDOODQGIRXQGDWLRQUHVSHFWLYHO\DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Check the bearing stresses on the wall and mat foundation

ACI 16.3.3.4

Bearing strength of the wall. Factored bearing stress, is determined for compression and for compression and bending using the factored axial IRUFHVDQGEHQGLQJPRPHQWVGHWHUPLQHGE\$&,(TVE DQG$&,(TH UHVSHFWLYHO\VHH7DEOH 

8-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• $&,(TE 

• $&,(TH 

Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the VL]HDQGVSDFLQJRIWKHORQJLWXGLQDOEDUVLQWKHZDOO7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQsion forces are adequately transferred from the wall into the mat foundation. Bearing strength of the mat foundation. There is no need to check the bearing strength of the mat foundation because dowel bars will be provided that match WKHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDOEDUVLQWKHZDOO Step 2 – Determine the required interface reinforcement

ACI 16.3.4.1

)URP([DPSOHWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHZDOOLVOD\HUVRIEDUVVSDFHGDWLQRQFHQWHUDWWKLV VSDFLQJURZVRIUHLQIRUFHPHQWDUHSURYLGHGLQWKHZDOO Check minimum area of reinforcement across the interface: ACI 16.3.4.2

7KHORQJLWXGLQDOUHLQIRUFHPHQWUDWLRRILVXVHGWRGHWHUPLQHWKHPLQLPXPDUHDRILQWHUIDFHUHLQIRUFHPHQWLQ DFFRUGDQFHZLWK$&,EHFDXVHORQJLWXGLQDOEDUV*UDGH DUHXVHGLQWKHZDOOVHH$&,7DEOH  Step 3 – Check horizontal force transfer

)URP([DPSOHWKHIDFWRUHGVKHDUIRUFHDWWKHEDVHRIWKHZDOOLVHTXDOWRNLSV7KLVIRUFHLVWUDQVIHUUHG KRUL]RQWDOO\EHWZHHQWKHZDOODQGWKHIRXQGDWLRQ7KHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVGHWHUPLQHGDV follows assuming the surface between the wall and the foundation has not been intentionally roughened: Eq. (8.46)

7KLVDUHDRIUHLQIRUFHPHQWLVODUJHUWKDQWKHDUHDRIUHLQIRUFHPHQWSURYLGHGE\WKHGRZHOEDUVWKDWPDWFKWKHVL]H and spacing of the longitudinal reinforcement in the wall, which is equal to 7KHUHIRUHWU\GRZHOEDUVVSDFHGDWLQRQFHQWHU

8-54

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Check the upper shear limit: Table 8.22

8VHGRZHOEDUVVSDFHGDWLQRQFHQWHU Step 4 – Check the development of the dowel bars in the mat foundation

,WLVHYLGHQWIURP)LJXUHWKDWWKHVWUHVVLQWKHORQJLWXGLQDOUHLQIRUFHPHQWIDUWKHVWIURPWKHH[WUHPHFRPSUHVVLRQ face is tensile for the load combinations that include seismic effects. Therefore, the dowel bars must be developed for tension into the mat (and into the wall). $VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSPHQW RIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN length,

Eq. (8.44)

Table 8.20

$VVXPLQJWZROD\HUVRIEDUVLQWKHPDWIRXQGDWLRQZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPPDW WKLFNQHVVIRUWKHGHYHORSPHQWRIWKHGRZHOEDUV Eq. (8.45)

The minimum

LVOHVVWKDQWKHSURYLGHGPDWWKLFNQHVVRILQ

Step 5 – Determine the development length of the dowel bars in the wall

7KHGRZHOEDUVPXVWEHODSVSOLFHGWRWKHORQJLWXGLQDOEDUVLQWKHZDOOXVLQJDWHQVLRQODSVSOLFH:KHUHEDUV RIGLIIHUHQWVL]HDUHODSVSOLFHGWRJHWKHUWKHUHTXLUHGWHQVLRQODSVSOLFHOHQJWKLVWKHJUHDWHURID  of the larger RIWKHVPDOOHUEDUVHH1RWHLQ7DEOH %HFDXVHDOOWKHGRZHOEDUVDUHVSOLFHGDWWKHVDPHORFDbar and (b) WLRQD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHG$&,7DEOH 

8-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Development length in tension,

RIWKHGRZHOEDUV

Eq. (8.37)

Table 8.18

Figure 8.14

Set

ACI 25.4.2.4

Therefore,

Tension lap splice length,

RIWKHORQJLWXGLQDOEDUV

Eq. (8.37)

Table 8.18

Figure 8.14

Set

8-56

ACI 25.4.2.4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

3URYLGHDODSVSOLFHOHQJWKRIIWLQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVZDOODUHJLYHQLQ)LJXUH ͳᇱ ǦͲԣ

͓ͷ̷ͳᇱ Ǧ͸ԣ

͵ԢǦͲԣ

͵ԣ…Ž‡ƒ”…‘˜‡”

ʹԢǦʹԣ

͓͹†‘™‡Ž„ƒ”•

͓ͳͳƒ–„ƒ”• Figure 8.28 Reinforcement details for the wall in Examples 8.4 through 8.7.

8-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 9 DIAPHRAGMS 9.1 Overview 5HLQIRUFHGFRQFUHWHGLDSKUDJPVDUHURRIDQGÀRRUV\VWHPVZLWKLQDVWUXFWXUHWKDWUHVLVWDQGWUDQVIHUJUDYLW\DQG ODWHUDOIRUFHV7\SLFDOGLDSKUDJPDFWLRQVLQDUHLQIRUFHGFRQFUHWHEXLOGLQJDUHGHSLFWHGLQ$&,)LJXUH5 Diaphragms must support and transfer gravity loads to columns, walls, and other supporting elements in a building. The weight of the structure, superimposed dead loads, and live loads are common out-of-plane gravity loads applied to the surface of a diaphragm. Roof diaphragms must also be able to support effects due to rain, snow, and wind, to name a few. Lateral forces from wind, earthquakes, and soil pressure are common types of forces transferred from a diaphragm to the lateral force-resisting system (LFRS) in a building. The load path from the application of the force on the diaphragm to the vertical elements of the LFRS depends on several factors, including the type of applied force and the ULJLGLW\RUÀH[LELOLW\ RIWKHGLDSKUDJP7KHLQSODQHIRUFHVJHQHUDWHLQSODQHVKHDUIRUFHVEHQGLQJPRPHQWDQG axial forces in a diaphragm. The connections between the diaphragm and the vertical elements of the LFRS are very important; these connections must be properly designed and detailed otherwise there is no mechanism for proper load transfer to the LFRS. Collectors, which are part of a diaphragm and are sometimes referred to as drag struts, are required where the LFRS does not extend the full depth of a diaphragm or where there is a discontinuity in the LFRS. These elements are parallel to the applied force and their main functions are to collect and transfer diaphragm shear forces to the vertical elements of the LFRS, distribute forces within the diaphragm, or both. In general, diaphragms must be designed and detailed for the combined effects due to in-plane and out-of-plane load effects. Reinforcement is provided to resist the effects from in-plane and out-of-plane axial forces, bending moments, and shear forces. The design and detailing of cast-in-place diaphragms with nonprestressed reinforcement are covered in this chapter. 3URYLVLRQVIRUGLDSKUDJPVDUHJLYHQLQ$&,&KDSWHUZKLFKDUHDSSOLFDEOHWRGLDSKUDJPVLQEXLOGLQJVDVVLJQHGWR Seismic Design Category (SDC) A, B, and C.

9.2 Minimum Diaphragm Thickness 'LDSKUDJPVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRWKDWDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG$&, 7KHIROORZLQJORDGHIIHFWVPXVWEHLQYHVWLJDWHGIRUVWUHQJWK RXWRISODQHPRPHQWVDQG VKHDUVGXHWRJUDYLW\IRUFHVRUFRPELQDWLRQVRIJUDYLW\DQGODWHUDOIRUFHVDQG LQSODQHPRPHQWVLQSODQHVKHDU forces, and axial forces due to wind, seismic, and other applicable lateral forces. For buildings assigned to SDC A, B, or C, the minimum diaphragm thickness must be the largest of the following $&,  (1) 7  KLFNQHVVEDVHGRQWKHVHUYLFHDELOLW\UHTXLUHPHQWVRI$&,IRURQHZD\VODEVRU$&,IRU two-way slabs.   Thickness based on out-of-plane strength requirements. (3) Thickness based on in-plane strength requirements. 6HUYLFHDELOLW\UHTXLUHPHQWVIRURQHZD\VODEVDQGWZRZD\VODEVDUHFRYHUHGLQ6HFWLRQVDQGRIWKLVSXEOLFDtion, respectively. Methods are given in those sections that can be used to determine minimum slab thicknesses that VDWLVI\GHÀHFWLRQUHTXLUHPHQWV 2XWRISODQHÀH[XUDOUHTXLUHPHQWVUDUHO\KDYHDQLPSDFWRQWKHUHTXLUHGVODEWKLFNQHVVRIRQHZD\DQGWZRZD\ VODEV\VWHPV5HDVRQDEOHDPRXQWVRIÀH[XUDOUHLQIRUFHPHQWFDQXVXDOO\EHSURYLGHGDWWKHFULWLFDOVHFWLRQVXVLQJD 9-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

slab thickness determined based on serviceability requirements. Two-way shear requirements may have an impact RQWKHWKLFNQHVVRIÀDWSODWHV\VWHPV)LJXUHLQWKLVSXEOLFDWLRQFDQEHXVHGWRGHWHUPLQHDSUHOLPLQDU\VODE WKLFNQHVVIRUÀDWSODWHVFRQVLGHULQJWZRZD\VKHDUVWUHVVHV$PRUHUH¿QHGDQDO\VLVQHHGVWREHSHUIRUPHGWRHQVXUH WKDWWZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HG,QOLHXRILQFUHDVLQJWKHVODEWKLFNQHVVRUFROXPQVL]HWZRZD\VKHDU UHTXLUHPHQWVFDQEHVDWLV¿HGE\SURYLGLQJVKHDUUHLQIRUFHPHQWVHH6HFWLRQRIWKLVSXEOLFDWLRQ  A slab thickness based on serviceability requirements is usually adequate to satisfy in-plane strength requirements. In-plane moments are resisted by chord reinforcement placed perpendicular to the direction of the lateral force. In PRVWFDVHVWKLVUHLQIRUFHPHQWLVGHWHUPLQHGXVLQJDVODEWKLFNQHVVVXI¿FLHQWIRUVHUYLFHDELOLW\RURWKHUVWUHQJWK UHTXLUHPHQWVWKHUHLVXVXDOO\QRQHHGWRLQFUHDVHWKHWKLFNQHVVRIDVODEEDVHGRQLQSODQHÀH[XUDOUHTXLUHPHQWV It is common for the design shear strength of the concrete alone to satisfy in-plane shear strength requirements of a diaphragm. Fire-resistance requirements of the governing building code must also be considered when selecting a minimum VODEWKLFNQHVV)RUORFDOMXULVGLFWLRQVWKDWKDYHDGRSWHGWKH,%&PLQLPXPVODEWKLFNQHVVIRUYDULRXV¿UHUHVLVWDQFH UDWLQJVEDVHGRQFRQFUHWHW\SHLVJLYHQLQ,%&7DEOH,WLVFRPPRQIRU¿UHUHVLVWDQFHUHTXLUHPHQWVWREH VDWLV¿HGXVLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHDELOLW\UHTXLUHPHQWVVWUHQJWKUHTXLUHPHQWVRUERWK

9.3 Required Strength 9.3.1 General

7KHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWREHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWKRIGLDSKUDJPVFROOHFWRUVDQGWKHLUFRQQHFWLRQVVHH$&, $FFRUGLQJWR$&,PHPEHUVPXVWEHGHVLJQHGIRUWKH simultaneous effects from out-of-plane and in-plane loading. For example, collectors are commonly designed for ÀH[XUHDQGVKHDUIRUFHVGXHWRJUDYLW\ORDGVLQFRPELQDWLRQZLWKD[LDOFRPSUHVVLRQDQGWHQVLRQIRUFHVGXHWRLQ plane forces in the diaphragm. 9.3.2 Diaphragm Design Forces Overview

,QJHQHUDOGLDSKUDJPVPXVWEHGHVLJQHGWRUHVLVWWKHHIIHFWVIURPWKHIROORZLQJIRUFHVVHH$&,DQG$&, )LJXUH5  (a) (b) (c) (d) (e)

Diaphragm in-plane forces due to lateral loads acting on the building Diaphragm transfer forces Connection forces between the diaphragm and vertical framing or nonstructural elements Forces resulting from bracing vertical or sloped building elements Diaphragm out-of-plane forces due to gravity and other loads applied to the surface of the diaphragm

In-Plane Forces due to Lateral Loads

0HWKRGVWRGHWHUPLQHLQSODQHZLQGIRUFHVDQGVHLVPLFIRUFHVRQGLDSKUDJPVDUHJLYHQLQ6HFWLRQVDQGRI at level in a building is equal to the total this publication. In the case of wind loads, the resultant wind force, at that level (which is sum of the windward pressure, and the leeward pressure, ) times the wind pressure, times the tributary story height, (see Figure 9.1). This force is applied at the centroid of width of the building, the face of the building in the direction of analysis. is applied at the center of mass (CM) of the diaphragm at level because it is an The seismic design force, LQHUWLDOIRUFHVHH)LJXUH )RUULJLGGLDSKUDJPVVHH6HFWLRQRIWKLVSXEOLFDWLRQIRUFODVVL¿FDWLRQRIGLDphragms), the diaphragm translates and rotates about the center of rigidity (CR).

9-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ܮ‬

‫ܤ‬

‫݌‬௫  ‫ܤ‬Τʹ ܹ௫ ൌ ‫݌‬௫ ‫݄ܤ‬௫  Figure 9.1 Resultant wind force acting on a diaphragm.

݁௫ 

‫ܮ‬

 

݁௬ 

‫ܨ‬௣௫ 

‫ܤ‬

Figure 9.2 Seismic force acting on a rigid diaphragm.

For buildings with one or more basement levels, lateral forces are generated by soil pressure bearing against the basement walls (see Figure 9.3). The magnitude of the soil pressure is usually obtained from a geotechnical investigation. In cases where the results of such an investigation are not available, the lateral soil loads in IBC Table FDQEHXVHGVLPLODUGHVLJQODWHUDOORDGVDUHSURYLGHGLQ$6&(6(,7DEOH ,WLVHYLGHQWIURP)LJXUH 9.3 that a portion of the soil pressure is transferred to the foundation system and a portion is transferred to the reinforced concrete slab, which acts as a diaphragm. The diaphragm must be designed for the in-plane effects due to the soil pressure (plus any other applicable in-plane load effects) and must transfer the forces to the basement walls parallel to the direction of the soil pressure.

9-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‡‹ˆ‘”…‡†…‘…”‡–‡•Žƒ„

‡‹ˆ‘”…‡†…‘…”‡–‡ „ƒ•‡‡–™ƒŽŽ

‹•–”‹„—–‹‘‘ˆ †‡•‹‰Žƒ–‡”ƒŽ •‘‹Ž’”‡••—”‡ǡ‫ܪ‬

‡‹ˆ‘”…‡†…‘…”‡–‡™ƒŽŽ ˆ‘—†ƒ–‹‘

Figure 9.3 Distribution of at-rest soil pressure on a basement wall.

$FFRUGLQJWR,%&DQG$6&(6(,EDVHPHQWZDOOVDQGUHWDLQLQJZDOOVVXSSRUWLQJPRUHWKDQIW RIEDFN¿OOKHLJKWPXVWEHVXEMHFWHGWRG\QDPLFVHLVPLFODWHUDOHDUWKSUHVVXUHVGXHWRGHVLJQHDUWKTXDNHJURXQG PRWLRQVIRUVWUXFWXUHVDVVLJQHGWR6'&'(RU)+RZHYHUQRSURYLVLRQVRUJXLGHOLQHVDUHSURYLGHGRQKRZWR GHWHUPLQHWKHVHG\QDPLFSUHVVXUHV$PHWKRGWRGHWHUPLQHVXFKIRUFHVLVJLYHQLQ5HIHUHQFH In-Plane Forces due to Transfer Forces

$W\SLFDOFDVHZKHUHDGLDSKUDJPLVVXEMHFWHGWRWUDQVIHUIRUFHVLVLOOXVWUDWHGLQ)LJXUH,QDGGLWLRQWRWKHHIIHFWV IURPZLQGVHLVPLFDQGRWKHUDSSOLFDEOHLQSODQHORDGVZKLFKDUHFROOHFWLYHO\ODEHOOHGDV³'LDSKUDJPIRUFH´LQWKH ¿JXUH WKHSRGLXPVODEPXVWEHGHVLJQHGIRUWKHHIIHFWVGXHWRWKHIRUFHVWUDQVIHUUHGIURPWKHVKHDUZDOOVDERYH The podium slab, in turn, transfers all the in-plane forces to the basement walls parallel to the direction of analysis.

‹ƒ’Š”ƒ‰ˆ‘”…‡

‡‹ˆ‘”…‡†…‘…”‡–‡ •Š‡ƒ”™ƒŽŽሺ–›’Ǥሻ

”ƒ•ˆ‡”ˆ‘”…‡

‡‹ˆ‘”…‡†…‘…”‡–‡ „ƒ•‡‡–™ƒŽŽሺ–›’Ǥሻ

Figure 9.4 Transfer forces to a podium slab.

Connection Forces Between the Diaphragm and Vertical Framing or Nonstructural Elements

Nonstructural components, such as cladding, is commonly attached to reinforced concrete slabs over the height of a building. These components transfer wind pressures, inertial forces from earthquake shaking, or both, to the diaphragm through the connections between the two.

9-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

”‡…ƒ•–…‘…”‡–‡™ƒŽŽ’ƒ‡Ž

‘‡…–‘”ƒ••‡„Ž›

‡‹ˆ‘”…‡†…‘…”‡–‡•Žƒ„

Ƭ™‹†’”‡••—”‡

Žƒ„”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

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Figure 9.5 Wind forces from a nonstructural element transferred to a diaphragm.

,OOXVWUDWHGLQ)LJXUHLVDJHQHULFWLHEDFNFRQQHFWLRQEHWZHHQDSUHFDVWFRQFUHWHZDOOSDQHODQGDUHLQIRUFHGFRQcrete slab, which is representative of the many types of connections between cladding and diaphragms. Component and cladding (C&C) wind pressure normal to the surface of the panel is transferred from the precast concrete wall panel to the diaphragm via the connector assembly. The diaphragm must be designed for this force in combination with all other applicable forces. A similar force transfer to a diaphragm occurs for nonstructural wall panels subjected to out-of-plane seismic forces. Forces Resulting from Bracing Vertical or Sloped Building Elements

Diaphragms must be designed for forces resulting from bracing vertical structural elements in a building. For example, in the case of a building frame system, it is assumed that all the lateral forces are resisted by the shear walls. :KHQDQDO\]HGVHSDUDWHO\VKHDUZDOOVDQGIUDPHVKDYHGLIIHUHQWGLVSODFHPHQWSUR¿OHVRYHUWKHKHLJKWRIDEXLOGLQJ when subjected to lateral forces (see Figure 9.6). In an actual building, the diaphragms connect the columns and shear walls together, which means these elements must displace the same amount at each level. Thus, the resulting transfer forces caused by displacement compatibility between the columns that are not part of the LFRS and the shear walls must be resisted by the diaphragms. Diaphragms must also be designed to resist forces from any sloped structural members in a building. An inclined FROXPQVXFKDVWKHRQHLOOXVWUDWHGLQ$&,)LJXUH5WUDQVIHUVWKHKRUL]RQWDOFRPSRQHQWVRIWKHVXSSRUWHG gravity and lateral forces, where applicable, into the reinforced concrete slab (diaphragm) in cases where there are no other local structural elements to counteract these forces. Out-of-Plane Forces

One of the main functions of a diaphragm is to support and transfer gravity loads to the supporting structural members in a building. The weight of the structure, superimposed dead loads, and live loads are typical out-of-plane JUDYLW\ORDGVDSSOLHGWRWKHVXUIDFHRIDGLDSKUDJP'HDGDQGOLYHORDGVDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$6&( 6(,&KDSWHUVDQGUHVSHFWLYHO\ 6QRZDQGUDLQORDGVFDQRFFXURQURRIGLDSKUDJPVWRQDPHDIHZ$6&(6(,&KDSWHUVDQG :LQGXSOLIWORDGV are also applicable, as are vertical accelerations due to seismic effects.

9-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‹ƒ’Š”ƒ‰ሺ–›’Ǥሻ

•‘Žƒ–‡†ˆ”ƒ‡

•‘Žƒ–‡†•Š‡ƒ”™ƒŽŽ

”ƒ‡ƒ†•Š‡ƒ”™ƒŽŽ …‘‡…–‡†„›†‹ƒ’Š”ƒ‰•

Figure 9.6 Diaphragm force transfer due to displacement compatibility.

Collector Design Forces

Where vertical elements of the LFRS do not extend the full depth of a diaphragm, collectors are needed to transfer IRUFHVEHWZHHQWKHGLDSKUDJPDQGWKHYHUWLFDOHOHPHQWVRIWKH/)56VHH$6&(6(,)LJXUH 'HSHQGLQJRQ several factors, a collector can be part of the slab or can be a beam. ,QJHQHUDOFROOHFWRUVPXVWEHGHVLJQHGIRUWKHFRPELQHGHIIHFWVGXHWRJUDYLW\ORDGVÀH[XUHDQGVKHDU DQGODWHUDO loads (axial tension and compression forces caused by in-plane shear transfer in the diaphragm due to wind forces, seismic forces, or both). The in-plane diaphragm forces are determined using the methods presented in Section 9.3.3 RIWKLVSXEOLFDWLRQ2QFHWKHIRUFHVDUHGHWHUPLQHGWKHGLDSKUDJPLVPRGHOHGDQGDQDO\]HGXVLQJWKHPHWKRGVLQ $&,DQGWKHD[LDOIRUFHVLQDFROOHFWRUFDQEHGHWHUPLQHGEDVHGRQWKHVHOHFWHGPHWKRGRIDQDO\VLV6HFWLRQ RIWKLVSXEOLFDWLRQFRYHUVWKHDQDO\VLVSURFHGXUHVLQ$&,7KHVHSURFHGXUHVFDQEHXVHGWRGHWHUPLQH collector axial forces in diaphragms subjected to wind and seismic forces in buildings assigned to SDC A, B, or C. 9.3.3 Diaphragm Modeling and Analysis Overview

*HQHUDOPRGHOLQJDQGDQDO\VLVUHTXLUHPHQWVIRUGLDSKUDJPVDUHJLYHQLQ$&,7KHSURYLVLRQVLQ$&, WKURXJKDUHWREHXVHGZKHUHWKHUHTXLUHPHQWVRIWKHJHQHUDOEXLOGLQJFRGHVXFKDVWKH,%&RU $6&(6(,DUHQRWDSSOLFDEOH Some of the general analysis requirements in ACI Chapter 6 are applicable to diaphragms. For example, the provisions for elastic analysis in ACI 6.6.1 through 6.6.3 can be applied because diaphragms are designed to remain essentially elastic when subjected to in-plane wind and seismic forces. In-plane stiffness modeling and analysis methods for diaphragms are covered below. In-Plane Stiffness Modeling

It is permitted to use any set of reasonable and consistent assumptions for in-plane diaphragm stiffness (ACI  'LVWULEXWLRQRIIRUFHVLQWKHGLDSKUDJPDQGGLVSODFHPHQWVDQGIRUFHVLQWKHYHUWLFDOHOHPHQWVRIWKH/)56 are contingent on the in-plane stiffness of the diaphragm.

9-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$GLDSKUDJPLVFRQVLGHUHGWREHULJLGZKHUHWKHLQSODQHGHÀHFWLRQGXHWRODWHUDOIRUFHVLVUHODWLYHO\VPDOOFRPSDUHG to that of the vertical elements of the LFRS. For purposes of analysis, rigid diaphragms displace and rotate as a rigid body when subjected to lateral forces, and the vertical elements of the LFRS move together accordingly (displacement compatibility). Also, lateral forces are distributed to the vertical elements of the LFRS in proportion to their relative rigidities (stiffnesses) and their location with respect to the center of rigidity (CR), which is the stiffness centroid within a diaphragm. ,QFRQWUDVWWKHLQSODQHGHÀHFWLRQRIDÀH[LEOHGLDSKUDJPLVUHODWLYHO\ODUJHFRPSDUHGWRWKDWRIWKHYHUWLFDOHOHments of the LFRS, and the distribution of lateral forces to the vertical elements of the LFRS is independent of their relative rigidities. In such cases, lateral forces are distributed based on the tributary mass of the diaphragm to the vertical elements of the LFRS. For diaphragms of uniform material and weight, lateral forces can be distributed by tributary areas. Flexible diaphragms do not undergo rigid body rotation like rigid diaphragms. $GLDSKUDJPLVFODVVL¿HGDVVHPLULJLGZKHUHWKHLQSODQHGHÀHFWLRQRIWKHGLDSKUDJPDQGWKHGHÀHFWLRQRIWKHYHUWLcal elements of the LFRS are of the same order of magnitude. $FDVWLQSODFHUHLQIRUFHGFRQFUHWHGLDSKUDJPLVSHUPLWWHGWREHFODVVL¿HGDVULJLGZKHUHWKHIROORZLQJFULWHULDDUH VDWLV¿HG • :KHQVXEMHFWHGWRODWHUDOZLQGIRUFHV$6&(6(, 6SDQWRGHSWKUDWLR • :KHQVXEMHFWHGWRODWHUDOVHLVPLFIRUFHV$6&(6(,  and (a) Span-to depth ratio (b) 6WUXFWXUHKDVQRQHRIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH

݄

When determining the span-to-depth ratio, the span is equal to the distance between lines of lateral resistance (such DVZDOOVDQGIUDPHV LQWKHGLUHFWLRQRIDQDO\VLVVHH)LJXUH 7KHRYHUDOOGHSWKRIWKHGLDSKUDJPLQWKHGLUHFWLRQ of analysis is used to determine the span-to-depth ratio. κଵ 

κଶ 

κଵ ’ƒ ൌ  ‡’–Š ݄

’ƒ κଶ ൌ  ‡’–Š ݄

Figure 9.7 Definition of span-to-depth ratio for a diaphragm.

7KHUHLQIRUFHGFRQFUHWHGLDSKUDJPLOOXVWUDWHGLQ$&,)LJXUH5DPD\QRWEHFRQVLGHUHGULJLGLQWKHGLUHFwhere is the distance between the tion of the lateral force because of its relatively large span-to-depth ratio, walls at each end, which have been designated to be part of the LFRS (the interior moment frames are not part of the LFRS), and is the depth of the diaphragm in the direction of analysis. Similarly, ramps in parking structures PD\QRWEHFODVVL¿HGDVULJLGEHFDXVHRIUHODWLYHO\ORQJVSDQWRGHSWKUDWLRVZKLFKDUHFRPPRQLQVXFKVWUXFWXUHV

9-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$FFRUGLQJWR$6&(6(,DGLDSKUDJPLVSHUPLWWHGWREHLGHDOL]HGDVÀH[LEOHZKHUH LVWKHPD[LPXPLQSODQHGHÀHFWLRQRIWKHGLDSKUDJPDQG is >$6&(6(,(TXDWLRQ @7KHWHUP the average drift of adjoining vertical elements of the LFRS over the story below the diaphragm under consideration ZKHQVXEMHFWHGWRWKHLQSODQHIRUFH$ULJLGGLDSKUDJPLVGH¿QHGLQ,%&EDVHGRQGHÀHFWLRQVDQGGULIWV as well: A diaphragm is rigid for the purpose of distribution of story shear and torsional moment when the lateral is less than or equal to two times the average story drift, deformation of the diaphragm, )RUGLDSKUDJPVWKDWFDQQRWEHLGHDOL]HGDVHLWKHUULJLGRUÀH[LEOHWKHLQSODQHVWLIIQHVVRIWKHGLDSKUDJPPXVWEH explicitly accounted for in the analysis. A three-dimensional analysis considering the stiffnesses of the diaphragms and the elements of the LFRS usually provides the most accurate distribution of forces in such cases. 7RDFFRXQWIRUFUDFNLQJLQUHLQIRUFHGFRQFUHWHGLDSKUDJPVDVWLIIQHVVPRGL¿HUVKRXOGEHDSSOLHGWRWKHJURVV in-plane stiffness of the diaphragm. In the case of wind forces, the structure is typically assumed to respond in the HODVWLFUDQJHVRXVLQJSHUFHQWRIWKHJURVVLQSODQHPRPHQWRILQHUWLDRIWKHGLDSKUDJPZRXOGEHDSSURSULDWH VHH$&, )RUVHLVPLFIRUFHVLWKDVEHHQVKRZQWKDWVWLIIQHVVPRGL¿HUVW\SLFDOO\IDOOLQWKHUDQJHRI WR Analysis Methods

Once the diaphragm forces have been determined and the type of diaphragm has been established based on in-plane VWLIIQHVVULJLGVHPLULJLGRUÀH[LEOH WKHGLDSKUDJPDQGDQ\FROOHFWRUVPXVWEHDQDO\]HGIRUWKHHIIHFWVGXHWRLQ plane forces in combination with out-of-plane forces. $FFRUGLQJWR$&,LQSODQHGHVLJQPRPHQWVVKHDUIRUFHVDQGD[LDOIRUFHVLQGLDSKUDJPVPXVWEHGHWHUPLQHGXVLQJDQ\PHWKRGWKDWVDWLV¿HVHTXLOLEULXPDQGWKHGHVLJQERXQGDU\FRQGLWLRQV0HWKRGVRIDQDO\VLVEDVHG on the following models are permitted to be used: • $ULJLGGLDSKUDJPPRGHOZKHUHGLDSKUDJPVFDQEHLGHDOL]HGDVULJLG • $ÀH[LEOHGLDSKUDJPPRGHOZKHUHGLDSKUDJPVFDQEHLGHDOL]HGDVÀH[LEOH • A bounding analysis in which the design values are the envelope of values obtained by assuming upper bound and lower bound in-plane stiffnesses for the diaphragm in two or more separate analyses. • $¿QLWHHOHPHQWPRGHOFRQVLGHULQJGLDSKUDJPÀH[LELOLW\ • $VWUXWDQGWLHPRGHOLQDFFRUGDQFHZLWK$&, The applicability of these models and other pertinent information are given in Table 9.1. Table 9.1 Diaphragm Models Model

Applicability

Notes

Rigid diaphragm

• For lateral wind forces, reinforced concrete slabs ZLWKDVSDQWRGHSWKUDWLRRIRUOHVV • For lateral seismic forces, when the following two FRQGLWLRQVDUHVDWLV¿HG UHLQIRUFHGFRQFUHWH VODEVZLWKDVSDQWRGHSWKUDWLRRIRUOHVVDQG  VWUXFWXUHSRVVHVVHVQRQHRIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH

Beam models are commonly used to determine diaphragm internal forces in buildings without major irregularities DQGRUWUDQVIHUIRUFHV

Flexible diaphragm

$GLDSKUDJPLVSHUPLWWHGWREHLGHDOL]HGDVÀH[LEOH ZKHUHWKHUDWLRRIWKHPD[LPXPLQSODQHGHÀHFWLRQ and the average story drift, of the diaphragm, LVJUHDWHUWKDQ

Beam models are commonly used to determine diaphragm internal forces in buildings without major irregularities DQGRUWUDQVIHUIRUFHV (table continued on next page)

9-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.1 Diaphragm Models (cont.) Model

Bounding analysis

Applicability

Notes

Suitable for semirigid diaphragms

Diaphragm internal forces are taken as the envelope of values from the following two analyses: (1) rigid diaphragm on ÀH[LEOHVXSSRUWVDQG ÀH[LEOHGLDphragm on rigid supports

Finite element

Suitable for any diaphragm

This model is especially useful for irregularly shaped diaphragms and diaSKUDJPVZLWKODUJHWUDQVIHUIRUFHVDQG or openings

Strut-and-tie

Suitable for any diaphragm

Model should include consideration of force reversals that may occur under design load combinations

%HDPPRGHOVDUHZLGHO\XVHGWRGHWHUPLQHLQWHUQDOIRUFHVLQULJLGDQGÀH[LEOHGLDSKUDJPVLQEXLOGLQJVZLWKRXW PDMRULUUHJXODULWLHVDQGRUWUDQVIHUIRUFHV'LDSKUDJPVDUHWUHDWHGDVEHDPVWKDWVSDQEHWZHHQWKHYHUWLFDOHOHPHQWV RIWKH/)56ZKLFKDUHLGHDOL]HGDVHLWKHUULJLGRUÀH[LEOHVSULQJ VXSSRUWV%HDPPRGHOVIRUULJLGGLDSKUDJPVDUH covered in detail below. Equivalent Beam Model with Rigid Supports

This model is often used for low-rise buildings with regular geometries where two or more lines of lateral forceresisting systems are provided in a given direction that have approximately the same relative rigidity. Diaphragm Forces Illustrated in Figure 9.8 is a reinforced concrete diaphragm and a LFRS consisting of structural (shear) walls (it is assumed that the perimeter frames are not part of the LFRS). In the direction of analysis, the diaphragm is modeled The walls, which have the same in-plane stiffness, as a beam whose depth is equal to the full diaphragm depth, are assumed to be rigid supports in the direction of analysis. This method can also be used where walls are at the perimeter of the building or where more than two walls are in the direction of analysis (which means the equivalent beam is continuous). Because the walls do not extend the full depth of the diaphragm, collectors are needed to collect the shear from the diaphragm and to transfer it to the walls. in Figure 9.8 is uniformly distributed over the width of the diaphragm. The reacThe equivalent in-plane load, are the diaphragm forces transferred to the vertical elements of the LFRS, and WLRQVLQZDOOVDQG and are obtained from statics. It is permitted to assume that the diaphragm resists in-plane moment and axial force in DFFRUGDQFHZLWK$&,DQGZKLFKLQFOXGHVWKHDVVXPSWLRQWKDWVWUDLQVYDU\OLQHDUO\RYHUWKHGHSWKRIWKH diaphragm. Shear and moment diagrams for the diaphragm are also given in Figure 9.8. ,QWKLVPRGHOURRIDQGÀRRUV\VWHPVDFWDVWKHZHERIWKHHTXLYDOHQWEHDPZKLFKUHVLVWVWKHGHVLJQVKHDUIRUFHV must be uniform over the depth )RUZDOO )RUGLDSKUDJPVDQDO\]HGE\WKLVPHWKRGWKHVKHDUÀRZ )RUZDOOWKHPD[LPXPXQLIRUPVKHDUÀRZLVHTXDOWR which must be less than RUHTXDOWRWKHGHVLJQVKHDUVWUHQJWKRIWKHGLDSKUDJP6KHDUÀRZLVWUDQVIHUUHGIURPWKHGLDSKUDJPWRWKHYHUWLFDO elements of the LFRS by shear friction. 7KHGLDSKUDJPERXQGDULHVSHUSHQGLFXODUWRWKHODWHUDOIRUFHDFWDVWKHÀDQJHVRIWKHHTXLYDOHQWEHDPFRPPRQO\ referred to as chords) and resist the tension and compression forces induced in the diaphragm due to bending. These

9-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

          

‫ܤ‬ ࢀ࢛ 

ᬇ

Š‘”†ˆ‘”…‡ ሺ–›’Ǥሻ –”—…–—”ƒŽ™ƒŽŽሺ–›’Ǥሻ ܴଵ 

ܴଶ 

‹ˆ‘”•Š‡ƒ”ˆŽ‘™ǡ ሺ‫ݒ‬௨ ሻଶ ൌ ܴଶ Τ‫ܮ‬

‫ܮ‬

ᬅ

ᬆ



‘ŽŽ‡…–‘”ሺ–›’Ǥሻ ࡯࢛ 

ܴଵ 

ᬈ

‫ݓ‬௨ 

ܴଶ 

“—‹˜ƒŽ‡–„‡ƒ ‫ݓ‬௨  ܴଶ 

ܸ௨ 

ܸ௨ǡ௠௔௫ 

‫ܯ‬௨ 

‫ܯ‬௨ǡ௠௔௫  Figure 9.8 Equivalent beam model with rigid supports.

9-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

forces can be determined by dividing the maximum bending moment in the diaphragm by the distance between the forces (that is, the moment arm): (9.1) In this equation, LVWKHSHUSHQGLFXODUGLVWDQFHEHWZHHQWKHFKRUGIRUFHVZKLFKLVFRPPRQO\WDNHQDVSHUFHQW ). Reinforcement must be provided of the total diaphragm depth in the direction of analysis (in this case, in the chords to resist the tension forces generated by this bending moment. Because wind and seismic forces can act in any direction, chord reinforcement is required at both boundaries of the diaphragm perpendicular to the direction of analysis. Collector Forces A reinforced concrete beam or a portion of a reinforced concrete slab in line with the LFRS can be used as a collector. In cases where the beam or slab has the same width as the vertical element of the LFRS it frames in to (for example, the width of the collector is equal to the thickness of the structural wall), all the collected forces are WUDQVIHUUHGGLUHFWO\LQWRWKHPHPEHURIWKH/)56DWLWVHQGV7KLVLVLOOXVWUDWHGLQ)LJXUHIRUZDOOZKLFKLV part of the LFRS of the building in Figure 9.8. The uniform factored shear force in the diaphragm at this location, is equal to the reaction in the wall, divided by the depth of the diaphragm in the direction of analysis, The net factored Similarly, the uniform factored shear force in the wall along its length is equal to shear force at any point is equal to the difference between the unit factored shear forces in the wall and diaphragm at that point. ܴଶ  ‫ܮ‬

ܴଶ  ‫ܮ‬



κଷ 

Collector ሺtyp.ሻ

ܴଶ  κଶ

൫‫ݒ‬௨,ௐ ൯ଶ ൌ ܴଶ /κଶ 

൬

ܴଶ ൰κ  ‫ ܮ‬ଷ

‫ܮ‬

κଶ 

ᬆ ሺ‫ݒ‬௨ ሻଶ ൌ ܴଶ /‫ܮ‬

ܴଶ ܴଶ െ  κଶ ‫ܮ‬

κଵ 

ܴଶ ൬ ൰ κଵ  ‫ܮ‬  ܴଶ  ‫ܮ‬ Unit Shear Forces

Net Shear Forces

Collector Axial Forces

Figure 9.9 Unit shear forces, net shear forces, and collector forces in a diaphragm.

The axial forces in the collectors are determined by summing the areas in the net shear force diagram. At the edges RIWKHGLDSKUDJPWKHD[LDOIRUFHLQWKHFROOHFWRUVLVHTXDOWR]HUR1RWH,QFDVHVZKHUHZLQGDQGRUVHLVPLFIRUFHV IURPSHULPHWHUHOHPHQWVVXFKDVFODGGLQJPXVWEHFROOHFWHGWKHIRUFHVDWWKHHQGVRIDFROOHFWRUPD\QRWEH]HUR  The axial force increases linearly as shear is transferred from the diaphragm to the collectors. At the end of the wall from the edge of the diaphragm, the axial tension force in the collector is equal to located a distance at a distance from the edge of the At the other end of the wall, the axial compression force is equal to diaphragm. Because wind and seismic forces can act in any direction, the axial forces in the collectors change from compression to tension and from tension to compression, so a collector must be designed for the most critical effects. Similar force diagrams can be obtained for wall 1.

9-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Where a collector is assumed to be wider than the member of the LFRS element it frames in to, a part of the collector force is transferred directly into the member of the LFRS at its ends and a part is transferred through shearIULFWLRQDORQJWKHOHQJWKRIWKHYHUWLFDOHOHPHQWRIWKH/)56VHH$&,)LJXUH5IRUWKHFDVHRIDZDOOORFDWHG at the edge of a diaphragm). Wider collectors are usually required in buildings where it is not practical or possible (from a design or constructability perspective) to provide collector elements concentric with the vertical elements of the LFRS. An example of where wider collectors may be necessary is in a building with relatively thin walls where only a narrow strip of slab is available to resist relatively large collector forces. adjacent to the shear wall or frame is used to resist the collector forces. Requirements An effective slab width, DUHQRWFRYHUHGLQWKHFRGHVHFWLRQVRI$&,RULQ$6&(6(,5HIHUHQFHUHFRPPHQGVXVLQJ for equal to the width of the vertical element of the LFRS plus a width on either side of the vertical element equal to RQHKDOIWKHFRQWDFWOHQJWKEHWZHHQWKHGLDSKUDJPDQGWKHYHUWLFDOHOHPHQWVHH)LJXUHIRUWKHHIIHFWLYHFROOHF $&,5UHFRPPHQGVDQHIIHFWLYHVODEZLGWKHTXDOWREDVLWRUZLGWKIRUZDOOLQ)LJXUHZKHUH FDOO\WKHVDPHDVWKDWLQ5HIHUHQFH ܾ௘௙௙  ݁

ܾ௘௙௙ ൌ ‫ ݐ‬൅ ሺκଶ Τʹሻ

κଷ 



ܶ௩ 

‫ݐ‬Τʹ

ܶௗ 

‫ݐ‬

ᬆ

ܴଶ Ȁκଶ 

‫ܮ‬

‘ŽŽ‡…–‘”

κଶ 

ሺܶ௩ ൅ ‫ܥ‬௩ ሻΤκଶ 

κଵ 

ܴଶ Ȁ‫ܮ‬ ‫ܥ‬௩ ‫ܥ‬ௗ 

Figure 9.10 Effective slab width for collectors wider than the vertical elements of the LFRS.

$OVRLOOXVWUDWHGLQ)LJXUHLVWKHIRUFHWUDQVIHUWRZDOO7KHWHQVLRQDQGFRPSUHVVLRQIRUFHVWUDQVIHUUHGGLUHFWO\ respectively. The tension force, is resisted by reinforcement in LQWRWKHHQGVRIZDOODUHGHVLJQDWHG and is resisted by the concrete. The remainthe wall parallel to the direction of analysis and the compression force, and respectively, are transferred through shear along which is the ing tension and compression forces this reinforceOHQJWKRIZDOO6KHDUIULFWLRQUHLQIRUFHPHQWLVUHTXLUHGWRUHVLVWWKHWRWDOVKHDUIRUFH PHQWLVSODFHGSHUSHQGLFXODUWRWKHZDOOVHH$&,)LJXUH5  act at an eccentricity with respect to the centerline of wall ,WLVHYLGHQWIURP)LJXUHWKDWWKHIRUFHV and ,ILWLVDVVXPHGWKHUHLQIRUFHPHQWLQWKHFROOHFWRULVXQLIRUPO\GLVWULEXWHGLQWKHGLUHFWLRQRIDQDO\VLVZLWKLQ and the in-plane bending OLNHWKDWVKRZQLQ$&,)LJXUH5 WKHQWKHHFFHQWULFLW\ can be taken as moment resulting from this eccentricity and the portion of the collector force not transferred directly into the end of WKHZDOOPXVWEHFRQVLGHUHGLQGHVLJQVHH6HFWLRQRIWKLVSXEOLFDWLRQ  In general, the section of the collector concentric with the vertical element of the LFRS should be designed for a reasonable portion of the total collector force considering overall design and construction limitations. Reinforcing bar congestion at the ends of a shear wall, for example, is a practical limitation to consider when selecting the portion of the total tension force transferred directly into the end of a wall. In some cases, the total collector force will need to be resisted entirely by the shear in the slab adjacent to the vertical element of the LFRS.

9-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Corrected Equivalent Beam Method with Spring Supports

This model is best suited for buildings with rigid diaphragms and lateral force-resisting systems with different stiffnesses. Effects of torsion are automatically accounted for in this method. Diaphragm Forces and Center of Rigidity 7KH¿UVWVWHSLQGHWHUPLQLQJWKHLQWHUQDOLQSODQHIRUFHVLQDGLDSKUDJPEDVHGRQWKHFRUUHFWHGHTXLYDOHQWEHDP model is to obtain the diaphragm forces transferred to the vertical elements of the LFRS. In multistory buildings, a three-dimensional model of the building is usually constructed with the appropriate lateral forces applied at each level over the height of the building. The forces in the vertical elements of the LFRS are obtained from an analysis of the structure using this model. In low-rise buildings with rigid diaphragms, the stiffnesses of the elements of the /)56FDQEHREWDLQHGXVLQJDSSUR[LPDWHDQDO\VHVZKLFKGRQRWUHVXOWLQDVLJQL¿FDQWORVVRIDFFXUDF\0HWKRGVWR GHWHUPLQHDSSUR[LPDWHLQSODQHVWLIIQHVVHVIRUZDOOVDQGIUDPHVDUHJLYHQLQ5HIHUHQFH2WKHUZLVHVWLIIQHVVHV FDQEHREWDLQHGIURPDPRUHUH¿QHGDQDO\VLV :KHQXWLOL]LQJDQDSSUR[LPDWHPHWKRGRIDQDO\VLVWRGHWHUPLQHWKHIRUFHVLQWKHYHUWLFDOHOHPHQWVRIWKH/)56 WKHORFDWLRQRIWKHFHQWHURIULJLGLW\&5 QHHGVWREHGHWHUPLQHG7KH&5LVWKHSRLQWRQDÀRRUURRIOHYHOZKHUH the equivalent story stiffness is assumed to be located. For buildings with rigid diaphragms, application of a lateral force through the CR produces only rigid body displacement of the story. Displacement and rotation occur where the lateral force is applied at any other point. The following equations can be used to locate the CR in the x-direction and y-direction from a selected origin (see Figure 9.11 where the origin is selected at the centroid of the column located at the bottom left of the framing system): (9.2)

(9.3)

where

in-plane lateral stiffness of lateral-force-resisting element i in the y-direction in-plane lateral stiffness of lateral-force-resisting element i in the x-direction distance in the x-direction from the origin to the centroid of lateral-force-resisting element i distance in the y-direction from the origin to the centroid of lateral-force-resisting element i

All the elements of the LFRS parallel to the y- and xGLUHFWLRQVRIDQDO\VHVDUHLQFOXGHGLQ(TXDWLRQV DQG  respectively; it is commonly assumed that out-of-plane resistance of a lateral-force-resisting element is negligible compared to its in-plane resistance. Thus, when determining the location of the CR in the x-direction by Equation  RQO\WKHVWLIIQHVVHVRIZDOOVDQGDUHFRQVLGHUHG1RWHWKDWDSRUWLRQRIZDOOWKDWLVDQHIIHFWLYHÀDQJH width) may be included when determining the stiffness of wall 1. Similarly, when determining the location of the CR in the yGLUHFWLRQE\(TXDWLRQ RQO\WKHVWLIIQHVVRIZDOOLQFOXGLQJDQHIIHFWLYHÀDQJHZLGWKRIZDOOLI GHVLUHG DQGZDOODUHFRQVLGHUHG which is Once the location of the CR has been determined, the following equation can be used to determine resisted by element i considering both direct and torsional shear forces: the portion of the total story shear, (9.4)

where

perpendicular distance from the centroid of element i to the CR parallel to the x-axis

9-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ᬈ

‫ݕ‬ସ 



‫ݕ‬௖௥ 

ᬆ

‫ݕ‬ଷ 

ᬅ ᬇ ‫ݔ‬ଵ  ‫ݔ‬௖௥  ‫ݔ‬ଶ 

Figure 9.11 Location of center of rigidity (CR).

perpendicular distance from the centroid of element i to the CR parallel to the y-axis perpendicular distance between

and the CR

Similarly, the following equation can be used to determine which is the portion of the total story shear, resisted by element i considering both direct and torsional shear forces: (9.5)

where

perpendicular distance between

and the CR.

7KH¿UVWDQGVHFRQGWHUPVLQWKHVHHTXDWLRQVUHSUHVHQWWKHGLUHFWVKHDUIRUFHVDQGWKHWRUVLRQDOVKHDUIRUFHVWUDQVferred to the vertical elements of the LFRS, respectively. Once the diaphragm forces transferred to the vertical elements of the LFRS have been obtained using either an DSSUR[LPDWHRUPRUHUH¿QHGDQDO\VLVDVRXWOLQHGDERYHWKHVHFRQGVWHSLVWRGHWHUPLQHDQHTXLYDOHQWLQSODQH distributed load on the diaphragm that is in equilibrium with these forces (reactions). The distributed load is usuat each end of DOO\WUDSH]RLGDOZKLFKDFFRXQWVIRUDQ\WRUVLRQDOPRPHQWVVHH)LJXUH 7KHORDGV and the equivalent beam can be obtained by using the equations for force equilibrium and moment equilibrium and then and solving these two equations for the two unknowns )RUWKHULJLGGLDSKUDJPLQ)LJXUHWKHIRUFHDQGPRPHQWHTXDWLRQVRIHTXLOLEULXPLQWKHGLUHFWLRQRIDQDO\VLV are the following where moments are summed about the left edge of the diaphragm: (9.6)

(9.7)

9-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ᬉ

‫ܮ‬

ᬈ

ᬅ

ᬇ

ᬆ

ᬊ

ᬋ ܾଶ 

ܾଵ 

ܾଷ 

ܸ௬  ܴଵ 

ܴଶ 

ܴଷ 

‫ݓ‬ଵ  ‫ݓ‬ଶ  ܸ௨ǡ௠௔௫ 

Š‡ƒ”

‘‡–

‫ܯ‬௨ǡ௠௔௫  Figure 9.12 Equivalent distributed load, shear diagram, and moment diagram for a rigid diaphragm.

9-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHVXOWDQWIRUFHRIWKHWUDSH]RLGDOGLVWULEXWHGORDGLVHTXDOWRWKHDSSOLHGODWHUDOIRUFHDWWKDWOHYHOREWDLQHGIURP DQDO\VLVZKLFKLVUHÀHFWHGLQ(TXDWLRQ 7KHPRPHQWFDXVHGE\WKHIRUFHVLQWKHHOHPHQWVRIWKH/)56LQWKH GLUHFWLRQSHUSHQGLFXODUWRWKHLQSODQHIRUFHLVRIWHQLJQRUHGLQRYHUDOOKRUL]RQWDOIRUFHGLVWULEXWLRQLWFDQEHLQFRUSRUDWHGLQWRWKHWUDSH]RLGDOORDGLIGHVLUHG and KDYHEHHQGHWHUPLQHGWKH¿QDOVWHSLVWRFRQVWUXFWVKHDUDQGPRPHQWGLDJUDPVIRUWKHGLDSKUDJP Once VHH)LJXUH 7KHVKHDUGLDJUDPLVXVHGLQ FKHFNLQJWKHGHVLJQVKHDUVWUHQJWKRIWKHGLDSKUDJP GHVLJQing the connections of the diaphragm to the vertical elements of the LFRS, and (3) determining the axial compression and tension forces in the collectors, if any. As noted previously, the moment diagram is used in determining the tension and compression forces in the chords, the former of which is needed to calculate the required area of chord reinforcement near the edges of the diaphragm. Diaphragms with Openings

$FFRUGLQJWR$&,WKHHIIHFWVRIVODERSHQLQJVDQGVODEYRLGVPXVWEHFRQVLGHUHGLQWKHDQDO\VLVDQGGHVLJQ of diaphragms. The corrected equivalent beam method can also be used for diaphragms with relatively large openings. For purposes of analysis, the diaphragm segments (commonly referred to as subdiaphragms) above and below the opening can be LGHDOL]HGDVEHDPV¿[HGDWHDFKHQG,WLVDVVXPHGWKDWWKHFROOHFWRUHOHPHQWRQRQHVLGHRIWKHRSHQLQJFROOHFWVWKH uniform diaphragm shear on that side and transfers it to the subdiaphragms above and below the opening in proportion to their relative stiffness or mass. The collector on the other side of the opening then collects the shear from the subdiaphragms and transfers it to the portion of the diaphragm on that side of the opening. Thus, the loading on a subdiaphragm is based on the total applied force at that level and the relative stiffness or mass of the subdiaphragm. Secondary chord forces occur in each subdiaphragm due to local bending caused by this loading.

ᬆ

ᬇ

κଶ 

ᬅ

‫ܣ‬ଵ 

κ௢௣௘௡ 

‫ܮ‬

κଵ 

7KHGLDSKUDJPGHSLFWHGLQ)LJXUHLVWKHVDPHDVWKHRQHLQ)LJXUHEXWZLWKDUHODWLYHO\ODUJHRSHQLQJLQLW and ZKLFKFDQEHREWDLQHGIURPWKHRYHUDOOWUDSH]RLThe forces at each edge of the opening are designated dal force distribution.

‫ܣ‬ଶ 

ܾଶ 

ܾଵ 

ܴଵ 

ܾଷ 

ܴଶ 

ܴଷ  ܾ௢௣௘௡ 

‫ݓ‬ଵ 

‫ݓ‬௅ 

‫ݓ‬ோ 

Figure 9.13 Diaphragm with a relatively large opening.

9-16

‫ݓ‬ଶ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

When a diaphragm is subjected to wind forces, the forces on the subdiaphragms above and below the opening can be approximately determined using the in-plane stiffness ratios of these subdiaphragms. The mass of the subdiaphragms is used to determine the forces when a diaphragm is subjected to seismic forces. The stiffness and mass raand DUHJLYHQLQ7DEOH7KHPDVVUDWLRVLQ7DEOHIRUXVH tios for the top and bottom subdiaphragms, and where it is assumed that the diaphragm with seismic forces are based on the areas of the subdiaphragms, has the same thickness and material properties everywhere. ܴ௧ǡ௅ 

ܾ௢௣௘௡ 

ܴ௧ǡோ 

ି ‫ܯ‬௨ǡ௧ǡ௅ 

ି ‫ܯ‬௨ǡ௧ǡோ 

‫ݓ‬௧ǡ௅ ൌ ݂௧௢௣ ‫ݓ‬௅  ‫ݓ‬௧ǡோ ൌ ݂௧௢௣ ‫ݓ‬ோ 

ା ‫ܯ‬௨ǡ௧ 

‘‡–

ି ‫ܯ‬௨ǡ௧ǡ௅  ି ‫ܯ‬௨ǡ௧ǡோ 

ሺƒሻ ‘’•—„†‹ƒ’Š”ƒ‰

ܴ௕ǡ௅ 

ܾ௢௣௘௡ 

ܴ௕ǡோ 

ି ‫ܯ‬௨ǡ௕ǡ௅ 

ି ‫ܯ‬௨ǡ௕ǡோ 

‫ݓ‬௕ǡ௅ ൌ ݂௕௢௧ ‫ݓ‬௅ 

‫ݓ‬௕ǡோ ൌ ݂௕௢௧ ‫ݓ‬ோ 

ା ‫ܯ‬௨ǡ௕ 

‘‡–

ି ‫ܯ‬௨ǡ௕ǡ௅ 

ሺ„ሻ‘––‘•—„†‹ƒ’Š”ƒ‰

ି ‫ܯ‬௨ǡ௕ǡோ 

Figure 9.14 Free-body diagrams and moment diagrams for the subdiaphragms in Figure 9.13.

9-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.2 Stiffness and Mass Ratios for Diaphragms with Openings In-Plane Force

ftop

fbot

Wind Seismic )UHHERG\GLDJUDPVRIWKHWRSDQGERWWRPVXEGLDSKUDJPVDUHJLYHQLQ)LJXUHDVQRWHGSUHYLRXVO\WKHHQGVRI WKHVHHOHPHQWVDUHDVVXPHGWREH¿[HG7KHPRPHQWGLDJUDPVLQWKH¿JXUHFDQEHREWDLQHGIURPVWDWLFVZKHUHWKH and GHWHUPLQHGIURP7DEOHDUHXVHGWRREWDLQWKHIRUFHVDWHDFKHQGRIWKHVXEGLDSKUDJP appropriate and compression chord forces are determined using Equation The secondary tension is the maximum positive moment in each subdiaphragm. (9.1) where The larger of the following two tension chord forces is used to determine the required area of chord reinforcement is the bending moment in the diaphragm at the center of the opening (see at the edges of the diaphragm where )LJXUH 

(9.8)



ᬆ

κଶ 

ᬅ

ܾଵ 

ܾଶ 

ࢀ࢛ǡ૚  ࡯࢛ǡ૚  ᬇ

κ௢௣௘௡ 

‫ܮ‬

κଵ 

ࢀ࢛ǡ࢕࢖ࢋ࢔ 

ࢀ࢛ǡ૛  ࡯࢛ǡ૛  ࡯࢛ǡ࢕࢖ࢋ࢔  ܾଷ 

‘‡–

‫ܯ‬௨ǡ௠௔௫  ‫ܯ‬௨ 

Figure 9.15 Determination of chord forces in a diaphragm with an opening.

9-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Secondary chord forces also develop at the corners of the opening due to the negative moments at these locations. along the bottom edge of the opening (that is, at the )RUWKHGLDSKUDJPLQ)LJXUHWKHWHQVLRQFKRUGIRUFH and Reinforcing bars top of the bottom subdiaphragm) is equal to the larger of must be provided along the entire top and bottom faces of the opening to resist the larger of the tension forces due to lateral forces acting in either direction.

9.4 Design Strength 9.4.1 General

)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHDSSOLFDEOHGHVLJQVWUHQJWKV of diaphragms and connections must be greater than or equal to the applicable required strengths, $&, $V noted previously, interaction between load effects must be considered in design. Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ factors pertinent to the design of diaphragms and collectors subject to moment, axial force, or combined moment DQGD[LDOIRUFHLVJLYHQLQ7DEOHIRUPHPEHUVZLWKRXWVSLUDOWUDQVYHUVHUHLQIRUFHPHQW7KHVWUDLQXVHGWRGH¿QH LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHUHLQIRUFHPHQW divided by the a compression-controlled section, ZKLFKFDQEHWDNHQDVSVLIRUDQ\JUDGHRIUHLQIRUFHmodulus of elasticity of the reinforcing steel, PHQW$&,  Table 9.3 Strength Reduction Factors, ‫׋‬ Net Tensile Strain, İt

Classification

Strength Reduction Factor, Ҁ*

Compression-controlled



Transition** 

Tension-controlled *Applicable to transverse reinforcement other than spirals. **Sections classified as transition are permitted to use corresponding to compression-controlled sections.

For diaphragms and collectors subjected to shear forces,

$&,7DEOH 

The design strength of a diaphragm depends on the type of model used to determine the internal force distribution (see Section 9.3.3 of this publication for permitted analysis models). Design strength requirements based on the PRGHOW\SHDUHJLYHQLQ$&,VHH7DEOH  Table 9.4 Design Strength Requirements for Diaphragms Model

Design Strength Requirements

Beam

$&,WKURXJK

Strut-and-tie

$&,

Finite element

$&,&KDSWHU

Alternative

$&,G

9-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

9.4.2 Nominal Moment and Axial Force Strength

A diaphragm modeled as a beam is permitted to be designed for in-plane moment and axial force using the assumpWLRQVLQ$&,ÀH[XUDOVWUHQJWK DQGD[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWK ZKLFKDUH WKHVDPHDVVXPSWLRQVXVHGLQWKHGHVLJQRIUHLQIRUFHGFRQFUHWHEHDPVFROXPQVDQGZDOOV$&, 7KLV includes the assumption that strains vary linearly over the depth of the diaphragm when it is subjected to in-plane forces. $FFRUGLQJWR$&,QRQSUHVWUHVVHGUHLQIRUFHPHQWDQGPHFKDQLFDOFRQQHFWRUVXVHGWRUHVLVWWHQVLRQFKRUG IRUFHVLQDGLDSKUDJPDUHSHUPLWWHGWREHORFDWHGZLWKLQ]RQHVH[WHQGLQJIURPWKHWHQVLRQHGJHRIWKHGLDSKUDJPD GLVWDQFHHTXDOWRSHUFHQWRIWKHGHSWKRIWKHGLDSKUDJPLQWKHGLUHFWLRQRIDQDO\VLVVHH$&,)LJXUH5  ,WLVDVVXPHGWKDWE\SODFLQJWKHFKRUGUHLQIRUFHPHQWZLWKLQWKHVH]RQHVDQHVVHQWLDOO\XQLIRUPVKHDUÀRZRYHUWKH depth of the diaphragm occurs, just like the case where the chord reinforcing bars are concentrated along the edges of the diaphragm. Where the depth of the diaphragm changes along its span, it is permitted to develop chord reinIRUFHPHQWLQWRDGMDFHQWVHFWLRQVHYHQZKHUHWKHUHLQIRUFHPHQWIDOOVRXWVLGHWKHSHUFHQWGHSWKOLPLWRIWKHDGMDcent section. 9.4.3 Nominal Shear Strength Nominal Shear Strength of Diaphragms

Nominal shear strength,  

for cast-in-place reinforced concrete diaphragms is determined by ACI Equation (9.9)

where

gross area of the diaphragm (diaphragm thickness times width in the direction of analysis) PRGL¿FDWLRQIDFWRUDFFRXQWLQJIRUWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWH ratio of distributed slab transverse reinforcement to gross concrete area

7KHPRGL¿FDWLRQIDFWRU WKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWR QRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ 7DEOHEDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[VHH$&,  Table 9.5 Values of Ȝ Based on Equilibrium Density, wc

Ȝ

Equilibrium Density, wc





Table 9.6 Values of Ȝ Based on Composition of Aggregates Concrete

Composition of Aggregates

Ȝ

All-lightweight

)LQH$670& &RDUVH$670&



/LJKWZHLJKW¿QHEOHQG

)LQH&RPELQDWLRQRI$670&DQG& &RDUVH$670&

WR(1) (table continued on next page)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.6 Values of Ȝ Based on Composition of Aggregates (cont.) Concrete

Composition of Aggregates

Ȝ

Sand-lightweight

Fine: ASTM C33 &RDUVH$670&



Sand-lightweight, coarse blend

Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670& and ASTM C33

WR

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

The transverse reinforcement ratio, is determined using the slab reinforcement parallel to the in-plane shear by Equation (9.9), it is conservative to check the design shear strength requirements asforce. When calculating LQWKLVHTXDWLRQLVOLPLWHGWRSVL$&,  suming LVHTXDOWR]HUR$OVR The factored shear force,  

must not exceed the maximum design shear strength determined by ACI Equation (9.10)

Shear Transfer

6KHDUWUDQVIHUEHWZHHQGLDSKUDJPVFROOHFWRUVDQGYHUWLFDOHOHPHQWVRIWKH/)56PXVWEHVDWLV¿HGXVLQJWKHVKHDU IULFWLRQSURYLVLRQVRI$&,RUE\PHFKDQLFDOFRQQHFWRUVRUGRZHOV$&, ,QFDVWLQSODFHFRQVWUXFWLRQ VKHDUWUDQVIHUUHTXLUHPHQWVPXVWW\SLFDOO\EHVDWLV¿HGDWFRQVWUXFWLRQMRLQWVEHWZHHQWKHHOHPHQWV Where shear-friction provisions are used, the nominal shear strength across the assumed shear plane is determined E\$&,(TXDWLRQ  (9.11)

In this equation, LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP$&,7DEOHVHH7DEOH DQG is the area of shear-friction reinforcement crossing the shear plane with the reinforcing bars oriented perpendicular to that plane. Table 9.7 Coefficient of Friction, μ Contact Surface Condition

ȝ

Concrete placed monolithically Concrete placed against hardened concrete that is clean, free of laitance, and intentionally roughened to a full amplitude of approximately ¼ in. Concrete placed against hardened concrete that is clean, free of laitance, and not intentionally roughened 8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOHVHH7DEOH 7KHWHUP is the area of which is equal to the gross cross-sectional area of the diaphragm in the direction of analysis. concrete resisting

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.8 Maximum Vn Across the Assumed Shear Plane Contact Surface Condition

Maximum Vn

= Vu / Ҁ*

Normalweight concrete placed monolithically or placed against hardened concrete intentionally roughened to a full amplitude of approximately ¼ in.

Other cases

*

area of concrete resisting

9.4.4 Collectors

Collectors are to be designed as tension members, compression members, or both in accordance with the design proYLVLRQVIRUD[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWKLQ$&,$&, ,QFDVHVZKHUHFROOHFWRUVDUHVXEMHFWHGWRUHODWLYHO\VLJQL¿FDQWHIIHFWVIURPJUDYLW\IRUFHVLQDGGLWLRQWRWKRVHIURPWHQVLRQDQGFRPSUHVsion axial forces due to in-plane lateral forces, design strength interaction diagrams are often constructed to check the adequacy of a collector for all applicable load combinations, just like for columns. Unlike typical columns, both the axial compression and tension parts of the interaction diagram are generally needed for the design of collectors. ,QDGGLWLRQWRWKHVWUHQJWKUHTXLUHPHQWVQRWHGDERYHWKHVKHDUVWUHQJWKUHTXLUHPHQWVLQ$&,IRURQHZD\VODEV DQGLQ$&,IRUWZRZD\VODEVPXVWEHVDWLV¿HGZKHUHDSRUWLRQRIDVODELVXVHGDVDFROOHFWRU:KHUHEHDPV DUHXWLOL]HGDVFROOHFWRUVWKHVWUHQJWKUHTXLUHPHQWVIRUVKHDURUIRUFRPELQHGVKHDUDQGWRUVLRQLQ$&,DQG PXVWDOVREHVDWLV¿HG

9.5 Reinforcement Limits The area of reinforcement in diaphragms must be at least equal to the area corresponding to shrinkage and tem$&, WKLVLVDOVRWKHPLQLPXPDUHD SHUDWXUHUHLQIRUFHPHQWJLYHQLQ$&,ZKLFKLVHTXDOWR RIÀH[XUDOUHLQIRUFHPHQWIRURQHZD\VODEVVHH$&,DQG )RUWZRZD\VODEVWKHPLQLPXPÀH[XUDO H[FHSWLQWKHHIIHFWLYHVODEZLGWKVSHFL¿HGLQ$&,ZKHUHWKHPLQLreinforcement is also equal to DQGWKHDUHDGHWHUPLQHGE\$&,(TXDWLRQ LQ mum area of reinforcement is equal to the larger of FDVHVZKHUHWKHIDFWRUHGWZRZD\VKHDUVWUHVVH[FHHGVWKHYDOXHJLYHQLQ$&, Reinforcement required to resist in-plane forces in diaphragms must be in addition to the reinforcement required to UHVLVWRWKHUORDGHIIHFWVVXFKDVWKRVHIURPJUDYLW\$&, 6KULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQWKRZever, is permitted to also resist diaphragm in-plane forces.

9.6 Determination of Required Reinforcement 9.6.1 Chord Reinforcement

The maximum tension chord force, which is determined by Equation (9.1), must be resisted by reinforcement must be less perpendicular to the direction of the applied in-plane force. At any location within the diaphragm, than or equal to the design tension strength of the chord reinforcement: (9.12)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In this equation, for reinforcing bars in tension and (TXDWLRQ FDQEHXVHGWRGHWHUPLQH For a given

is the required area of chord reinforcement.

(9.13)

For diaphragms with openings, the maximum tension chord force is determined by Equation (9.8) and that force DWWKHHGJHVRIWKHGLDSKUDJPVHH)LJXUH 7KHUHTXLUHGDUHD is used in Equation (9.13) to determine of chord reinforcement along the edges of the opening perpendicular to the direction of analysis can also be deterand VHH)LJXUHDQG6HFWLRQ mined by Equation (9.13) using the subdiaphragm tension chord forces 9.3.3 of this publication). Because wind and seismic forces can act in any direction, chord reinforcement is required along all edges of diaphragms and openings. For diaphragms in buildings assigned to Seismic Design Category (SDC) A, B, or C, diaphragm chords subjected to FRPSUHVVLRQIRUFHVQHHGQRWEHGHVLJQHGDQGGHWDLOHGDVFROXPQV$&,5 +RZHYHULQFDVHVZKHUHGLDphragm boundaries (edges of a diaphragm or edges of an opening in a diaphragm) are subjected to relatively large compression forces compared to the axial compression strength of the boundary element, transverse reinforcement, VXFKDVKRRSVVKRXOGEHXVHGWRFRQ¿QHWKHFRQFUHWH6XFKWUDQVYHUVHUHLQIRUFHPHQWLVXVXDOO\IHDVLEOHZKHUHWKHUHDUH beams along the diaphragm boundaries; without beams, the hoops must be developed in the slab, which is not always possible, especially for relatively thin slabs. In such cases, a thicker slab may be required if beams are not an option. 9.6.2 Diaphragm Shear Reinforcement

7KHIROORZLQJUHTXLUHPHQWVPXVWEHVDWLV¿HGIRUGHVLJQVKHDUVWUHQJWKRIGLDSKUDJPVVHH$&,DQG6HFWLRQ RIWKLVSXEOLFDWLRQ 

(9.14)

Required shear strength, due to in-plane forces is determined using one of the models in Section 9.3.3 of this publication. For diaphragms modeled as beams with chord reinforcement concentrated near it edges, the factored is uniformly distributed over the depth of the diaphragm (see Figure 9.8). VKHDUÀRZ The shear reinforcement ratio, is equal to the area of the uniformly distributed slab reinforcement oriented parallel to the shear force divided by the gross area of the slab perpendicular to that reinforcement. Equation (9.9) can be solved for the required (9.15)

,WLVHYLGHQWIURP(TXDWLRQ WKDWVKHDUUHLQIRUFHPHQWLVQRWUHTXLUHGZKHUH Providing a slab thickness adequate for serviceability or, where applicable, for two-way shear is usually adequate for in-plane shear strength as well. :KHUHLQSODQHVKHDUUHLQIRUFHPHQWLVUHTXLUHGLWLVXVXDOO\FRPELQHGZLWKWKHÀH[XUDOUHLQIRUFHPHQWLQWKHVODELQ WKHGLUHFWLRQRIDQDO\VLVZKHUHWZROD\HUVRIÀH[XUDOUHLQIRUFHPHQWDUHSURYLGHGLQDVODEWKHFRPELQDWLRQXVXDOO\ occurs with the bottom reinforcement). Thus, the total area of reinforcement in such cases is equal to the area of VKHDUUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ SOXVWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW&RPELQLQJWKHUHLQIRUFHPHQWLQWKLVIDVKLRQJHQHUDOO\UHVXOWVLQVLPSOHUGHWDLOLQJDQGPRUHHI¿FLHQWSODFHPHQWRIWKHUHLQIRUFLQJEDUV LQWKH¿HOG 9-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

9.6.3 Shear Transfer Reinforcement

Shear transfer reinforcement must be provided between diaphragms, collectors, and vertical elements of the LFRS. In cast-in-place construction, shear transfer reinforcement is provided mainly at construction joints between the elewhich depends on the overall layout of the elements, ments. The required area of shear-friction reinforcement, the magnitude of the factored shear force at the location of interest, the width of the collector with respect to the width of the vertical element of the LFRS it frames into, and the construction sequence, can be determined by Equation (9.11): (9.16)

In this equation, is the factored shear force at the interface of interest, is the length of the interface of interest, has the and LVGHWHUPLQHGE\7DEOHFRQVLGHULQJWKHFRQWDFWVXUIDFHFRQGLWLRQDWWKHLQWHUIDFH1RWHWKDW units of square inches per unit length in Equation (9.16). Shear transfer reinforcement is typically required at the interface between the diaphragm and the vertical elements of the LFRS and at the interface between the diaphragm and the collectors (if any). The equations in Table 9.9, which are based on Equation (9.16), can be used to determine the required area of shear-friction reinforcement at these locations. Table 9.9 Required Shear Transfer Reinforcement in Diaphragms Case

Shear Transfer Interface

1

Between the diaphragm and the vertical elements of the LFRS



Between the diaphragm and the collectors

Required Avf

In Case 1, is equal to the reaction in the vertical element of the LFRS where there are no collectors or where a collector is required and the collector has the same width as the vertical element of the LFRS it frames in (see Section 9.3.3 of this publication). to. Where a collector is wider than the vertical element, is the length of the vertical element of the LFRS in the direction of analysis (for example, the length Also, of the structural wall or the total length of the moment-resisting frames minus the length of any openings in the diaor dependphragm adjacent to these elements). For buildings assigned to SDC A, B, or C, is equal to LQJRQWKHFRQWDFWVXUIDFHFRQGLWLRQSURYLGHGDWWKHLQWHUIDFHVHH7DEOH  for concrete ,Q&DVH is the overall depth of the diaphragm in the direction of analysis and is equal to placed monolithically because the collector is either part of the slab or, where beams are used as collectors, the beams and slab are placed at the same time (that is, there is no cold joint between these two elements). 7KHFRQVWUXFWLRQPHWKRGKDVDQLPSDFWRQWKHGHWHUPLQDWLRQRIVKHDUWUDQVIHUUHLQIRUFHPHQW7KH¿UVWW\SHRI FRQVWUXFWLRQPHWKRGLVLOOXVWUDWHGLQ)LJXUHZKHUHVWUXFWXUDOZDOOVDUHXWLOL]HGDVWKHYHUWLFDOHOHPHQWVRIWKH /)56,QWKLVPHWKRGWKHZDOOEHORZLVFRQVWUXFWHG¿UVWWRWKHXQGHUVLGHRIWKHVODEWKHVODEDQGWKHQWKHZDOO above are subsequently constructed in that order. Cold joints typically occur at the top and bottom surfaces of the The shear-friction reinforcement in Figure 9.16 consists of dowel slab (diaphragm) along the length of the wall, is calculated by the bars connecting the slab to the wall below. In this case, the required area of the dowel bars, equation in Table 9.9 for Case 1: 7KHVHFRQGW\SHRIFRQVWUXFWLRQPHWKRGLVVKRZQLQ)LJXUH,QWKLVFDVHWKHZDOOLVFRQVWUXFWHGDKHDGRIWKH slab (for example, where slip forms are used to construct the wall). A cold joint occurs at the face of the wall and shear transfer reinforcement must be provided at that joint. Proprietary form saver systems are commonly used in WKLVW\SHRIFRQVWUXFWLRQPDLQO\IRUHFRQRPLFUHDVRQV7KHVKHDUIULFWLRQUHLQIRUFHPHQWLQ)LJXUHFRQVLVWVRI

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‘Ž†Œ‘‹– ‘™‡Ž„ƒ”•ǣ ‫ܣ‬௩௙ ൒

ܸ௨ଵ Τκ௪  ߶ߤ݂௬

൒ κ ሺ–›’Ǥ ሻ

—’’‘”–„ƒ”

‘Ž†Œ‘‹–

–”—…–—”ƒŽ™ƒŽŽ Figure 9.16 Shear transfer between a diaphragm and a structural wall utilizing dowel bars – Construction Method 1.

‘”•ƒ˜‡”ƒ••‡„Ž›Ȃ†‡ˆ‘”‡† „ƒ”™‹–ŠƒͻͲǦ†‡‰Š‘‘

‘™‡Ž„ƒ”•ǣ ‫ܣ‬௩௙ ൒

ܸ௨ଵ Τκ௪  ߶ߤ݂௬

‘Ž†Œ‘‹–

Figure 9.17 Shear transfer between a diaphragm and a structural wall utilizing dowel bars – Construction Method 2.

dowel bars, each of which requires its own form saver. The required area of the dowel bars, equation in Table 9.9 for Case 1:

is calculated by the

0HWKRGVXWLOL]LQJVKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQWRUWKHERWWRPUHLQIRUFHPHQWLQDVODEDVVKHDUIULFWLRQ UHLQIRUFHPHQWDUHJLYHQLQ5HIHUHQFH 7KHDERYHGLVFXVVLRQLVHTXDOO\YDOLGIRUPRPHQWUHVLVWLQJIUDPHVWKDWDUHXWLOL]HGDVSDUWRIWKH/)56LQVWHDGRI or in combination with, structural walls. 9.6.4 Reinforcement Due to Eccentricity of Collector Forces

For collectors wider than the width of the vertical elements of the LFRS that they frame in to, an in-plane bending and which act moment is generated in a diaphragm adjacent to the vertical elements due in part to the forces DWDQHFFHQWULFLW\IURPWKHFHQWHUOLQHRIWKHYHUWLFDOHOHPHQWVVHH)LJXUH $PHWKRGWRGHWHUPLQHWKHUHTXLUHG UHLQIRUFHPHQWWRUHVLVWWKLVEHQGLQJPRPHQWLVJLYHQEHORZZKLFKLVEDVHGRQWKHPHWKRGJLYHQLQ5HIHUHQFH Illustrated in Figure 9.18 is a portion of a diaphragm with a structural wall that is part of the LFRS located at its edge and a collector wider than the thickness of the wall. The internal forces acting on the free-body diagram of the

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

GLDSKUDJPDGMDFHQWWRWKHZDOODUHLQGLFDWHGLQWKH¿JXUHEDVHGRQVRPHVLPSOLI\LQJFRQVHUYDWLYHDVVXPSWLRQV7KH due to these internal forces can be approximated by the following equation: eccentric bending moment, (9.17)

‫ݐ‬Τʹ

݁ ܶௗ  ܶ௩  ܸ௦ 

κ௪ 

‫ݐ‬ ܸ௨ 

‫ܯ‬௘  ܸ௦  ‫ܥ‬ௗ  ‫ܥ‬௩  ܾ௘௙௙ 

Figure 9.18 Internal forces in a diaphragm where the collector is wider than the vertical element of the LFRS.

In this equation, is the shear strength of the diaphragm based on only the reinforcing bars in the slab (that is, ). The nominal strength of the concrete, is not included because tension forces are present, which makes The area of tension reinforcement,

required to resist

can be determined by the following equation: (9.18)

where for reinforcing bars in tension. This reinforcement is placed perpendicular to the face of the wall at both ends and must be developed into the slab and into the wall (see Figure 9.19).

κ௪ 

‫ܣ‬௦ሺ‡……ሻ ሺ–›’Ǥሻ

–Š‡””‡‹ˆ‘”…‡‡–‘– •Š‘™ˆ‘”…Žƒ”‹–›

Figure 9.19 Required tension reinforcement where the collector is wider than the vertical element of the LFRS.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

9.6.5 Collector Reinforcement Overview

In general, collectors must be designed as tension members, compression members, or both in accordance with WKHSURYLVLRQVRI$&,IRUPHPEHUVVXEMHFWHGWRD[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWK$&,   7KHW\SHVDQGDPRXQWRIUHLQIRUFHPHQWDQGWKHGHWDLOLQJUHTXLUHPHQWVWKDWPXVWEHVDWLV¿HGGHSHQGPRVWO\RQ  WKHW\SHRIFROOHFWRUWKDWLVDFROOHFWRUWKDWLVDSRUWLRQRIWKHVODERULVDEHDP DQG WKHZLGWKRIWKHFROOHFWRU with respect to the width of the vertical element of the LFRS that it frames in to. Slabs

:KHUHVODEVDUHXWLOL]HGDVFROOHFWRUVWKHIROORZLQJHTXDWLRQFDQEHXVHGWRGHWHUPLQHWKHUHTXLUHGDUHDRIORQJLWXin a collector to resist the factored axal tension force, dinal reinforcement, (9.19)

where IRUUHLQIRUFLQJEDUVLQWHQVLRQ7KLVUHLQIRUFHPHQWLVSURYLGHGLQDGGLWLRQWRWKHÀH[XUDOUHLQIRUFHment in the slab required for gravity loads. A typical detail where the collector is the same width as the vertical elePHQWRIWKH/)56LVJLYHQLQ)LJXUH

‘ŽŽ‡…–‘””‡‹ˆ‘”…‡‡–ǡ‫ܣ‬௦ሺ…‘ŽŽ‡…–‘”ሻ 

Figure 9.20 Longitudinal reinforcement in a collector that has the same width as the vertical element of the LFRS.

For the case of axial compression forces, WKHVHFWLRQDW]HURHFFHQWULFLW\

must be less than or equal to the design axial compression strength of (9.20)

In this equation, IRUFRPSUHVVLRQFRQWUROOHGVHFWLRQV$&,7DEOH  is equal to the thickness of is the area of longitudinal reinforcement in The design strength the slab times the width of the collector, and requirements for axial compression rarely govern.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For slabs wider than the vertical element of the LFRS, the total area of longitudinal tension reinforcement, required to resist can also be determined by Equation (9.19). The axial tension force and the corin the width of the vertical element of the LFRS is selected responding required area of tension reinforcement, FRQVLGHULQJGHVLJQDQGFRQVWUXFWLRQOLPLWDWLRQV2QFHWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVDUHFKRVHQEDVHGRQ the area of reinforcement, required in the effective slab width outside the width of the vertical elements minus the area of longitudinal reinforcement provided in the width of the vertical elements of is equal to are usually uniformly distributed over the effective width of the LFRS. The reinforcing bars corresponding to WKHFROOHFWRUVHH)LJXUH  

ܾ௘௙௙ 

‫ܣ‬௦ሺௗሻ 

‫ܣ‬௦ሺ௩ሻ   

‫ܣ‬௦ሺ…‘ŽŽ‡…–‘”ሻ ൌ ‫ܣ‬௦ሺௗሻ ൅ ‫ܣ‬௦ሺ௩ሻ 

Figure 9.21 Longitudinal reinforcement in a collector wider than the vertical element of the LFRS.

)RUFROOHFWRUVZLGHUWKDQWKHYHUWLFDOHOHPHQWRIWKH/)56WKHGHVLJQD[LDOVWUHQJWKGH¿QHGLQ(TXDWLRQ PXVW and using the appropriate and for each segment. be checked for Beams

%HDPVGHVLJQDWHGDVFROOHFWRUHOHPHQWVPXVWEHGHVLJQHGIRUWKHFRPELQHGHIIHFWVIURPÀH[XUHVKHDUWRUVLRQD[LDO compression forces, and axial tension forces due to gravity and lateral loads. Applicable design and detailing proviVLRQVLQ$&,&KDSWHUPXVWEHVDWLV¿HGDORQJZLWKWKHDSSOLFDEOHGHWDLOLQJUHTXLUHPHQWV /RQJLWXGLQDOUHLQIRUFHPHQWPXVWEHGHWHUPLQHGIRUWKHFRPELQHGHIIHFWVGXHWRÀH[XUHWRUVLRQDQGD[LDOFRPSUHVsion and tension forces, and transverse reinforcement must be determined for combined effects due to shear and torsion, where applicable. Generally, a design strength interaction diagram is constructed, which includes both the axial compression and tension portions, to determine whether the collector is adequate for all the combined factored ÀH[XUHDQGD[LDOORDGHIIHFWV7KHLQIRUPDWLRQSURYLGHGDERYHIRUVODEVZLGHUWKDQWKHYHUWLFDOHOHPHQWVRIWKH LFRS is also applicable to beams.

9.7 Reinforcement Detailing Requirements for concrete cover, development lengths, splices, bar spacing, and reinforcement detailing for diaSKUDJPVDQGFROOHFWRUVDUHJLYHQLQ$&,$VXPPDU\RIWKHVHUHTXLUHPHQWVLVJLYHQLQ7DEOH

9-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.10 Reinforcement Detailing Requirements for Diaphragms and Collectors Requirement

ACI Section No.

Concrete cover



Development lengths



Splices



Spacing Reinforcement detailing

Minimum



Maximum



One-way slabs



Two-way slabs



At critical sections in diaphragms and collectors, tension and compression forces in the reinforcement must be DGHTXDWHO\GHYHORSHGRQHDFKVLGHRIWKRVHVHFWLRQV$&, 7HQVLRQUHLQIRUFHPHQWPXVWH[WHQGEH\RQGWKH determined in point at which it is no longer required a distance equal to at least the tension development length, DFFRUGDQFHZLWK$&,H[FHSWDWGLDSKUDJPHGJHVDQGDWH[SDQVLRQMRLQWV$&,  Longitudinal reinforcement in collectors must extend into the vertical elements of the LFRS a length equal to at OHDVWWKHJUHDWHURIWKHWZROHQJWKVJLYHQLQ$&,7KLVUHTXLUHPHQWLVLOOXVWUDWHGLQ$&,)LJXUH5 $GGLWLRQDOLQIRUPDWLRQRQGHWDLOLQJUHTXLUHPHQWVIRUGLDSKUDJPVDQGFROOHFWRUVLVJLYHQLQ5HIHUHQFH

9.8 Design Procedure 7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHGLDSKUDJPV ,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFLQWKLV FKDSWHUFDQEHIRXQG&RPSUHKHQVLYHÀRZFKDUWVDVVRFLDWHGZLWKHDFKRIWKHVWHSVLQ)LJXUHDUHJLYHQLQ5HIHUHQFH

9.9 Examples 9.9.1

Example 9.1 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A, Collectors Not Required

'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQG IRUFHVLQWKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDOOWKHIUDPHVDORQJFROXPQOLQHVDQGDUHSDUWRIWKH/)56VHH )LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVDQGE\LQFROXPQV 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the diaphragm design in-plane forces

)URP7DEOHLQ([DPSOHWKHZLQGIRUFHLQWKHQRUWKVRXWKGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR kips. This force is applied at the centroid of the building face at this level. Step 2 – Determine the classification of the diaphragm

Sect. 9.3.3

For forces in the north-south direction, the span-to-depth ratio is equal to %HFDXVHWKHVSDQWRGHSWKUDWLRLVOHVVWKDQWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG

ASCE/SEI 26.2

9-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine the diaphragm thickness ሺSect. 9.2ሻ

Determine the diaphragm design forces ሺSect. 9.3.2ሻ

Determine the classification of the diaphragm, select the diaphragm model, and determine the diaphragm internal forces ሺSect. 9.3.3ሻ

Determine the combined load effects ሺACI Table 5.3.1ሻ

Determine the chord reinforcement ሾEq. ሺ9.13ሻሿ

Determine the diaphragm shear reinforcement ሾEq. ሺ9.15ሻሿ

Determine the shear transfer reinforcement ሾEq. ሺ9.16ሻሿ

Determine the reinforcement due to eccentricity of collector forces ሺwhere applicableሻ ሾEq. ሺ9.18ሻሿ

Determine the collector reinforcement ሺSect. 9.6.5ሻ

Figure 9.22 Design procedure for diaphragms.

9-30

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Select the diaphragm model

Table 9.1

Because this building has no irregularities and is not subjected to any transfer forces, and because the frames along FROXPQOLQHVDQGKDYHWKHVDPHODWHUDOVWLIIQHVVWKDWLVWKHVL]HVRIWKHEHDPVDQGFROXPQVDUHWKHVDPHLQERWK frames), an equivalent beam model with rigid supports is selected to determine the internal forces for this diaphragm. 1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

E

A

‫ܥ‬௨ ൌ 10.8 kips

ሺtyp.ሻ

ሺ‫ݒ‬௨ ሻଵ ൌ

25.7 ൌ 0.27 kips/ft 94.0

ሺ‫ݒ‬௨ ሻ଻ ൌ

25.7 ൌ 0.27 kips/ft 94.0

N

A

23ᇱ -6ᇳ

D

A

C

B

961.9 ൌ 10.8 kips 0.95 ൈ 94.0

A

ܶ௨ ൌ A

‫ݓ‬ௌ ൌ 0.342 kipsΤft

150.0Ԣ ሺܴଵ ሻ௬ ൌ 25.7 kips

ሺܴ଻ ሻ௬ ൌ 25.7 kips

25.7

‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ

961.9

25.7

‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬-‫ܛܘܑܓ‬ሻ

Figure 9.23 Equivalent beam model for the diaphragm in Example 9.1 for wind in the south direction.

9-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the diaphragm internal forces

Sect. 9.3.3

7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ)LJXUHIRUZLQGLQWKH VRXWKGLUHFWLRQ%HFDXVHWKHPRPHQWUHVLVWLQJIUDPHVRQFROXPQOLQHVDQGDUHLGHQWLFDOWKH&5LVORFDWHG from either end of the diaphragm, which coincides with the location where the wind force is applied at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an equivalent uniformly distributed wind load equal to • 5HDFWLRQVLQHDFKRIWKHPRPHQWUHVLVWLQJIUDPHVDORQJFROXPQOLQHVDQG 7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH • Chord forces at the edges of the diaphragm perpendicular to the direction of analysis Eq. (9.1)

• Maximum unit shear forces in the diaphragm

Collectors are not required because the moment-resisting frames extend the entire depth of the diaphragm in the direction of analysis. 9.9.2

Example 9.2 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A, Collectors Not Required

'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDOOWKHIUDPHVDORQJFROXPQOLQHVDQGDUHSDUWRIWKH DQG*UDGH /)56VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the chord reinforcement

From Example 9.1, Eq. (9.13)

$WFROXPQOLQHV$DQG(SURYLGHEDUORFDWHGMXVWRXWVLGHRIWKHFURVVVHFWLRQRIWKHEHDPVLQWKHPRPHQW IUDPHVVHH)LJXUH  Step 2 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHVDQGDQGDUHHTXDOWRNLSVIWVHH([DPSOH 9.1).

9-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Žƒ„–‘’”‡‹ˆ‘”…‡‡–

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡–

ͺΦԣ

ͳǦ͓Ͷ…Š‘”†„ƒ”

ʹԢǦͲԣ

Žƒ„„‘––‘”‡‹ˆ‘”…‡‡–

‡ƒ–”ƒ•˜‡”•‡”‡‹ˆ‘”…‡‡–

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡– ʹԢǦͶԣ

Figure 9.24 Location of chord reinforcement in the diaphragm in Example 9.2 for wind forces in the north-south direction.

Check shear strength requirements assuming

Eq. (9.14)

Therefore, no shear reinforcement is required to satisfy shear strength requirements. Step 3 – Determine the shear transfer reinforcement

Only shear transfer reinforcement between the diaphragm and the moment-resisting frames is required: Eq. (9.16)

A value of for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPHVHH7DEOH  %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the momentUHVLVWLQJIUDPHVDORQJFROXPQOLQHVDQG

9-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

9.9.3

Example 9.3 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS

'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQG forces in the north-south direction assuming only the frames between column lines B and D along column lines 1 DQGDUHSDUWRIWKH/)56VHH)LJXUH ,WLVDVVXPHGWKDWWKHZLGWKRIWKHFROOHFWRUVEHDPV DUHHTXDOWRWKH ZLGWKVRIWKHEHDPVLQWKHPRPHQWUHVLVWLQJIUDPHV$OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVDQG E\LQFROXPQV 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the diaphragm design in-plane forces

)URP7DEOHLQ([DPSOHWKHZLQGIRUFHLQWKHQRUWKVRXWKGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR kips. This force is applied at the centroid of the building face at this level. Step 2 – Determine the classification of the diaphragm

Sect. 9.3.3

For forces in the north-south direction, the span-to-depth ratio is equal to %HFDXVHWKHVSDQWRGHSWKUDWLRLVOHVVWKDQWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG Step 3 – Select the diaphragm model

ASCE/SEI 26.2 Table 9.1

Because this building has no irregularities and is not subjected to any transfer forces, and because the frames EHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQGKDYHWKHVDPHODWHUDOVWLIIQHVVWKDWLVWKHVL]HVRIWKH beams and columns are the same in both frames), an equivalent beam model with rigid supports is selected to determine the internal forces for this diaphragm. Step 4 – Determine the diaphragm internal forces

Sect. 9.3.3

7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ)LJXUHIRUZLQGLQWKHVRXWK GLUHFWLRQ%HFDXVHWKHPRPHQWUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQGDUH from either end of the diaphragm, which coincides with the location identical, the CR is located where the wind force is applied at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an equivalent uniformly distributed wind load equal to • 5HDFWLRQVLQHDFKRIWKHPRPHQWUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQG 7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH • Chord forces at the edges of the diaphragm perpendicular to the direction of analysis Eq. (9.1)

• Maximum unit shear forces in the moment-resisting frames between column lines B and D

• Maximum unit shear force in the diaphragm

9-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

4A

5A ᇱ

6A

7A

ᇳ

A

25 -0 ሺtyp.ሻ

E

‫ܥ‬௨ ൌ 10.8 kips

Collector

A

Collector

23ᇱ -6ᇳ ሺtyp.ሻ

D

25.7 ൌ 0.27 kips/ft 94.0

ሺ‫ݒ‬௨,ி ሻଵ ൌ

25.7 ൌ 0.55 kips/ft 47.0

ሺ‫ݒ‬௨ ሻ଻ ൌ

25.7 ൌ 0.27 kips/ft 94.0

N

A

ሺ‫ݒ‬௨ ሻଵ ൌ

C

25.7 ൌ 0.55 kips/ft 47.0

A

ሺ‫ݒ‬௨,ி ሻ଻ ൌ

B

Collector

Collector 961.9 ൌ 10.8 kips 0.95 ൈ 94.0

A

ܶ௨ ൌ A

‫ݓ‬ௌ ൌ 0.342 kipsΤft

150.0Ԣ ሺܴ଻ ሻ௬ ൌ 25.7 kips

ሺܴଵ ሻ௬ ൌ 25.7 kips

25.7

‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ

961.9

25.7

‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬-‫ܛܘܑܓ‬ሻ

Figure 9.25 Equivalent beam model for the diaphragm in Example 9.3 for wind in the south direction.

• Collector forces Collectors are required because the moment-resisting frames do not extend the entire depth of the diaphragm in the direction of analysis.

9-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHXQLWVKHDUIRUFHVQHWVKHDUIRUFHVDQGFROOHFWRUIRUFHVDORQJFROXPQOLQHDUHJLYHQLQ)LJXUHZKHUH the collectors are the beams between column lines A and B and between column lines D and E. Note that in FigXUHWKHIRUFHVDUHVKRZQZLWKPRUHVLJQL¿FDQW¿JXUHVDIWHUWKHGHFLPDOSODFHWKDQDUHXVHGLQWKHDERYH calculations. The main reason for this is to demonstrate that the maximum axial force in the collector is equal to the same value regardless of which end of the net shear force diagram is used to calculate it. Using a smaller QXPEHURIVLJQL¿FDQW¿JXUHVUHVXOWVLQWZRVOLJKWO\GLIIHUHQWYDOXHVRIWKHPD[LPXPFROOHFWRUIRUFHZKLFKLV due to due roundoff only. 1A

2A

3A

4A

5A

A

25ᇱ -0ᇳ 0.273 kips/ftሺtyp.ሻ

6A

7A

0.273 kips/ft

E

‫ܥ‬௨ ൌ 10.8 kips

Collector

Collector

A

0.546 kips/ft

0.273 kips/ft

23ᇱ -6ᇳ ሺtyp.ሻ

D

ሺ‫ݒ‬௨ ሻଵ ൌ

25.65 ൌ 0.273 kips/ft 94.0

ሺ‫ݒ‬௨,ி ሻଵ ൌ

25.65 ൌ 0.546 kips/ft 47.0

ሺ‫ݒ‬௨ ሻ଻ ൌ

25.8 ൌ 0.27 kips/ft 96.0

A

N

6.4 kips

C

25.8 ൌ 0.53 kips/ft 49.0

A

ሺ‫ݒ‬௨,ி ሻ଻ ൌ

B

6.4 kips Collector

A

Collector

A

0.273 kips/ft ‫ ܚ܉܍ܐ܁ ܜܑܖ܃‬۴‫ܛ܍܋ܚܗ‬

‫ ܚ܉܍ܐ܁ ܜ܍ۼ‬۴‫ܛ܍܋ܚܗ‬

۱‫ ܚܗܜ܋܍ܔܔܗ‬۴‫ܛ܍܋ܚܗ‬

0 34 k Τf forces in the diaphragm in Example 9.3. Figure 9.26 Unit shear forces, net shear forces, and collector

9.9.4

Example 9.4 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS

'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2Stion B for wind forces in the north-south direction assuming only the frames between column lines B and D along FROXPQOLQHVDQGDUHSDUWRIWKH/)56VHH)LJXUH ,WLVDVVXPHGWKDWWKHZLGWKRIWKHFROOHFWRUVEHDPV DUH HTXDOWRWKHZLGWKVRIWKHEHDPVLQWKHPRPHQWUHVLVWLQJIUDPHV$OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKW DQG*UDGHUHLQIRUFHPHQW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the chord reinforcement

From Example 9.3, Eq. (9.13) -

9-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$WFROXPQOLQHV$DQG(SURYLGHEDUORFDWHGMXVWRXWVLGHRIWKHFURVVVHFWLRQRIWKHEHDPVLQWKHPRPHQW IUDPHVVHH)LJXUH  Step 2 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHVDQGDQGDUHHTXDOWRNLSVIWVHH([DPSOH  Check shear strength requirements assuming

Eq. (9.14)

Therefore, no shear reinforcement is required to satisfy shear strength requirements. Step 3 – Determine the shear transfer reinforcement

Shear transfer reinforcement between the diaphragm and the moment-resisting frames between column lines B and D: Eq. (9.16)

A value of for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPHVHH7DEOH  %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the momentUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQG Shear transfer reinforcement between the diaphragm and the collectors: Eq. (9.16)

A value of for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPHVHH7DEOH  %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the collectors. Step 4 – Determine the collector reinforcement

The collector beams must be designed for the combined effects due to gravity loads and the axial force due to wind ORDGVZKLFKLVREWDLQHGIURP)LJXUH A summary of the axial forces, bending moments, and shear forces for a typical collector beam is given in Table 9.11.

9-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.11 Summary of Axial Forces, Bending Moments, and Shear Forces for a Typical Collector Beam in Example 9.4 Axial Force (kips)

Load Case

Dead Live

Bending Moment (ft-kips) Exterior Negative

Positive

First Interior Negative

Shear (kips)



63.8









ņ

Wind

ņ

ņ

ņ

Load Combination $&,(TD



89.3

36.3

$&,(TE







$&,(TG





$&,(TI





The collector beams are also subjected to a uniform torsional load from the slab. Because the torsional moments can be redistributed, the beam can be designed for the compatibility torsional moment (see Sect. 6.3.1 of this publication). The longitudinal reinforcement in the beam is determined based on the factored axial forces, bending moments, DQGWRUVLRQDOPRPHQWVFDOFXODWLRQVQRWJLYHQKHUHVHH&KDSWHURIWKLVSXEOLFDWLRQDQG)LJXUH 7KHGHVLJQ VWUHQJWKLQWHUDFWLRQGLDJUDPLQFOXGLQJWKHWHQVLRQSRUWLRQ LVJLYHQLQ)LJXUH,QFOXGHGLQWKH¿JXUHDUHWKH factored load combinations in Table 9.11. It is evident that the longitudinal reinforcement is adequate because all the factored load combination points fall within the diagram.

ͺΦԣ

Žƒ„–‘’”‡‹ˆ‘”…‡‡–

ʹԢǦͲԣ

ͶǦ͓ͺ

Žƒ„„‘––‘”‡‹ˆ‘”…‡‡– ʹǦ͓ͷሺ–›’Ǥሻ ͓ͷ̷ͺԣ‘…‡–‡”

ͶǦ͓ͺ ʹԢǦͶԣ

Figure 9.27 Required reinforcement for the collector beam in Example 9.4.

9-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1,500 1,250

Axial Force ሺkipsሻሻ

1,000 750 500 250 0 0

   100

200

300

400

500

600

700

-250 -500 Bending Moment ሺft-kipsሻ

Figure 9.28 Design strength interaction diagram for the collector beam in Example 9.4.

9.9.5

Example 9.5 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option C), SDC A, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS

'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ&IRUZLQG forces in the north-south direction assuming the lateral forces are resisted by the following moment-resisting frames (see Figure 1.1): ƒ )UDPHVRQFROXPQOLQHVDQGEHWZHHQFROXPQOLQHV$DQG( ƒ Frames on column line 3 between column lines B and D ƒ )UDPHVRQFROXPQOLQHV$DQG(EHWZHHQFROXPQOLQHVDQG It is assumed that the width of the collectors (beams) are equal to the widths of the beams in the moment-resisting IUDPHV$OVRDVVXPHDLQWKLFNVODEE\LQEHDPVDQGE\LQFROXPQV 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the diaphragm design in-plane forces

)URP7DEOHLQ([DPSOHWKHZLQGIRUFHLQWKHQRUWKVRXWKGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR kips. This force is applied at the centroid of the building face at this level. Step 2 – Determine the classification of the diaphragm

For forces in the north-south direction, the span-to-depth ratios are equal to %HFDXVHWKHVSDQWRGHSWKUDWLRVDUHOHVVWKDQWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG

Sect. 9.3.3

and ASCE/SEI 26.2

9-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Select the diaphragm model

Table 9.1

This building has no irregularities and is not subjected to any transfer forces. Because the frame on column line 3 KDVDGLIIHUHQWODWHUDOVWLIIQHVVWKDQWKHIUDPHVRQFROXPQOLQHVDQGWKHFRUUHFWHGHTXLYDOHQWEHDPPRGHOZLWK spring supports is selected to determine the internal forces in this diaphragm. Step 4 – Determine the diaphragm internal forces

Sect. 9.3.3

7KHFRUUHFWHGHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVGHSLFWHGLQ)LJXUHIRUZLQG in the south direction. • Frame stiffnesses ,WFDQEHGHWHUPLQHGWKDWWKHVWLIIQHVVHVRIWKHIUDPHVDORQJFROXPQOLQHVDQGDUHHTXDOWRWKHIROORZLQJ 5HI 

Assume the frames along column lines A and E are designated as the LFRS for wind forces in the east-west direction. The stiffnesses of these frames are equal to the following:

• Location of the CR The location of the CR in the x-direction is determined by the following equation where the origin is taken at column line 1: Eq. (9.2)

The eccentricity between the location of the load application and the CR is equal to From symmetry,

from column line A or E.

• Reactions in the frames The reactions in the frames in the north-south direction of analysis are obtained by the following equation: Eq. (9.4)

The reactions in the frames perpendicular to the direction of analysis are obtained by the following equation:

5HDFWLRQVLQWKHIUDPHVGXHWRWKHNLSZLQGIRUFHLQWKHVRXWKGLUHFWLRQDUHJLYHQLQ7DEOH

9-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

5A

4A

6A

7A

A

25ᇱ -0ᇳ ሺtyp.ሻ

E

A

‫ܥ‬௨ ൌ 7.9 kips Collector

ሺtyp.ሻ

ሺ‫ݒ‬௨,ி ሻଷ ൌ 11.1Τ47.0 ൌ 0.24 kips/ft

CR

N

A

23ᇱ -6ᇳ

D

C

ሺ‫ݒ‬௨ ሻଵ ൌ 18.9Τ94.0 ൌ 0.20 kips/ft

ሺ‫ݒ‬௨ ሻ଻ ൌ 21.3Τ94.0 ൌ 0.23 kips/ft

ሺ‫ݒ‬௨ ሻଷ,ா ൌ 12.0Τ94.0 ൌ 0.13 kips/ft

A

ሺ‫ݒ‬௨ ሻଷ,ௐ ൌ 0.9Τ94.0 ൌ 0.01 kips/ft

B

Collector

701.1 ൌ 7.9 kips 0.95 ൈ 94.0

A

ܶ௨ ൌ A

‫ݔ‬௖௥ ൌ 69.5Ԣ

݁௫ ൌ 5.5Ԣ

75.0Ԣ ܸ௬ ൌ 51.3 kips

‫ݓ‬ଵ,ௌ ൌ 0.368 kipsΤft

100.0Ԣ

50.0Ԣ ሺܴଵ ሻ௬ ൌ 18.9 kips

‫ݓ‬ଶ,ௌ ൌ 0.316 kipsΤft

ሺܴଷ ሻ௬ ൌ 11.1 kips

ሺܴ଻ ሻ௬ ൌ 21.3 kips

18.9 12.0 0.9

‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ

34.9Ԣ 21.3 701.1 492.3

‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬--‫ܛܘܑܓ‬ሻ

Figure 9.29 Equivalent beam model for the diaphragm in Example 9.5 for wind in the south direction

9-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 9.12 Reactions in the Frames of Example 9.5 for the Wind Force in the South Direction

Frame

xi

yi

(ki )y

(ki )x

xi

yi

xi2(ki )y

yi2(ki )x

(ft)

(ft)

Ec

Ec

(ft)

(ft)

Ec

Ec

(ki )yVy

x (k ) V e

*

Ȉxi(ki )y

i i y y x ‫ۓ‬ ۗ 2 ‫ ە‬Ȉxi(ki )y + Ȉyi (ki )xۙ

(kips)*

(kips)

Ri (kips)

1



—



—



—



—



18.9

3



—



—



—



—

11.3

11.1





—



—



—



—



1.3



A

—



—



—



—

1,688

—

1.1

1.1

E

—



—



—



—

1,688

—

Ȉ

1.396







*For frames along column lines A and E, replace **





with



in this equation.

• Equivalent in-plane distributed loads 7KHHTXLYDOHQWLQSODQHORDGIRUZLQGLQWKHVRXWKGLUHFWLRQLVWUDSH]RLGDOZKLFKDFFRXQWVIRUWKHHFFHQWULFLW\ between the location of the load application and the CR. The equations for force and moment equilibrium are and where moments are summed about column line 1: solved simultaneously for the two unknowns Eq. (9.6)

Eq. (9.7)

Therefore,

and

7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH • Chord forces at the edges of the diaphragm perpendicular to the direction of analysis: Eq. (9.1)

• Maximum unit shear forces in the moment-resisting frames occurs along column line 3:

• 0D[LPXPXQLWVKHDUIRUFHLQWKHGLDSKUDJPRFFXUVDORQJFROXPQOLQH

• Collector forces Collectors are required on column line 3 because the moment-resisting frames do not extend the entire depth of the diaphragm in the direction of analysis. 7KHXQLWVKHDUIRUFHVQHWVKHDUIRUFHVDQGFROOHFWRUIRUFHVDORQJFROXPQOLQHDUHJLYHQLQ)LJXUHZKHUH the collectors are the beams between column lines A and B and between column lines D and E.

9-42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

Collector

E

ሺ‫ݒ‬௨,ி ሻଷ ൌ 0.24 kips/ft

A

3A 0.12 kips/ft

0.12 kips/ft

0.24 kips/ft

0.12 kips/ft

2.8 kips

A

C

ሺ‫ݒ‬௨ ሻଷ,ா ൌ 0.13 kips/ft

A

N

ሺ‫ݒ‬௨ ሻଷ,ௐ ൌ 0.01 kips/ft

23ᇱ -6ᇳ ሺtyp.ሻ

D

B

A

Collector

2.8 kips

A

0.12 kips/ft ‫ ܚ܉܍ܐ܁ ܜܑܖ܃‬۴‫ܛ܍܋ܚܗ‬

‫ ܚ܉܍ܐ܁ ܜ܍ۼ‬۴‫ܛ܍܋ܚܗ‬

۱‫ ܚܗܜ܋܍ܔܔܗ‬۴‫ܛ܍܋ܚܗ‬

Figure 9.30 Unit shear forces, net shear forces, and collector forces in the diaphragm in Example 9.5.

9.9.6

Example 9.6 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option C), SDC A, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS

'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2Stion C for wind forces in the north-south direction assuming the lateral forces are resisted by the following momentresisting frames (see Figure 1.1): • )UDPHVRQFROXPQOLQHVDQGEHWZHHQFROXPQOLQHV$DQG( • Frames on column line 3 between column lines B and D • )UDPHVRQFROXPQOLQHV$DQG(EHWZHHQFROXPQOLQHVDQG It is assumed that the width of the collectors (beams) are equal to the widths of the beams in the moment-resisting IUDPHV$OVRDVVXPHDLQWKLFNVODEE\LQEHDPVE\LQFROXPQVQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the chord reinforcement

)URP([DPSOH Eq. (9.13)

9-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳǦ͓Ͷ…Š‘”†„ƒ”

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡–

͹ԣ

Žƒ„–‘’”‡‹ˆ‘”…‡‡–

ʹԢǦͲԣ

Žƒ„„‘––‘”‡‹ˆ‘”…‡‡–

‡ƒ–”ƒ•˜‡”•‡”‡‹ˆ‘”…‡‡–

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡–

ʹԢǦͶԣ

Figure 9.31 Location of chord reinforcement in the diaphragm in Example 9.6 for wind forces in the north-south direction.

$WFROXPQOLQHV$DQG(SURYLGHEDUORFDWHGMXVWRXWVLGHRIWKHFURVVVHFWLRQRIWKHEHDPVLQWKHPRPHQW frames (see Figure 9.31). Step 2 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHRFFXUVDORQJFROXPQOLQHDQGLVHTXDOWRNLSVIWVHH([DPSOH  Check shear strength requirements assuming

Eq. (9.14)

Therefore, no shear reinforcement is required to satisfy shear strength requirements. Step 3 – Determine the shear transfer reinforcement

Shear transfer reinforcement between the diaphragm and the moment-resisting frames on column line 3 between column lines B and D: Eq. (9.16)

A value of for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPHVHH7DEOH  %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the momentresisting frames on column line 3 between column lines B and D. Shear transfer reinforcement between the diaphragm and the collectors: Eq. (9.16)

9-44

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A value of for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPHVHH7DEOH  %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the collectors. Step 4 – Determine the collector reinforcement

The collector beams must be designed for the combined effects due to gravity loads and the axial force due to wind ORDGVZKLFKLVREWDLQHGIURP)LJXUH A summary of the axial forces, bending moments, and shear forces for a typical collector beam is given in Table 9.13. Table 9.13 Summary of Axial Forces, Bending Moments, and Shear Forces for a Typical Collector Beam in Example 9.6 Load Case

Dead Live

Axial Force (kips)

Bending Moment (ft-kips) Exterior Negative

Positive

First Interior Shear (kips) Negative













ņ

Wind

ņ

ņ

ņ

Load Combination $&,(TD







$&,(TE







$&,(TG





$&,(TI



19.1

ͶǦ͓͹

͹ԣ

Žƒ„–‘’”‡‹ˆ‘”…‡‡–

ʹԢǦͲԣ

Žƒ„„‘––‘”‡‹ˆ‘”…‡‡–

͓Ͷ̷ͺԣ‘…‡–‡”

ͶǦ͓͹

ʹԢǦͶԣ

Figure 9.32 Required reinforcement for the collector beam in Example 9.6.

9-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1,500 1,250

Axial Force ሺkipsሻሻ

1,000 750 500 250 0 0

   100

200

300

400

500

600

700

-250 -500 Bending Moment ሺft-kipsሻ

Figure 9.33 Design strength interaction diagram for the collector beam in Example 9.6.

The longitudinal reinforcement in the beam is determined based on the factored axial forces and bending moments FDOFXODWLRQVQRWJLYHQKHUHVHH&KDSWHURIWKLVSXEOLFDWLRQDQG)LJXUH 7KHGHVLJQVWUHQJWKLQWHUDFWLRQ GLDJUDPLQFOXGLQJWKHWHQVLRQSRUWLRQ LVVKRZQLQ)LJXUH,QFOXGHGLQWKH¿JXUHDUHWKHIDFWRUHGORDGFRPbinations in Table 9.13. It is evident that the longitudinal reinforcement is adequate because all the factored load combination points fall within the diagram. 9.9.7

Example 9.7 – Determination of Diaphragm In-Plane Forces: Building #2, SDC C, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the SFRS

'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJIRUVHLVPLFIRUFHVLQWKHHDVW ZHVWGLUHFWLRQDVVXPLQJRQO\WKHZDOOVRQFROXPQOLQHV%DQG)DUHSDUWRIWKH6)56VHH)LJXUH $OVRDVVXPH DLQWKLFNVODEDQGLQWKLFNZDOOV,WLVDVVXPHGWKDWWKHSRUWLRQVRIWKHVODELQOLQHZLWKWKHZDOOVDUHWKH collectors and the widths of the collectors are equal to the thicknesses of the walls. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the diaphragm design in-plane forces

)URP7DEOHLQ([DPSOHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHLQWKHHDVWZHVWGLUHFWLRQDWWKHVHFRQGÀRRU OHYHOLVHTXDOWRNLSV7KLVIRUFHLVDSSOLHGDWWKH&0ZKLFKLVORFDWHGDWWKHFHQWURLGRIWKHÀRRUSODWHDWWKLV level due to symmetry. Step 2 – Determine the classification of the diaphragm

Sect. 9.3.3

For forces in the east-west direction, the span-to-depth ratio is equal to $OVRWKHVWUXFWXUHGRHVQRWKDYHDQ\RIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH 7KHUHIRUHWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG

9-46

ASCE/SEI 12.3.1.2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Select the diaphragm model

Table 9.1

Because this building has no irregularities and is not subjected to any transfer forces, and because the walls on column lines B and F have the same lateral stiffness (that is, the thicknesses and the lengths of both walls are the same), an equivalent beam model with rigid supports is selected to determine the internal forces in this diaphragm. Step 4 – Determine the diaphragm internal forces

Sect. 9.3.3

7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ)LJXUHIRUVHLVPLFIRUFHVLQ the west direction. Because the walls on column lines B and F have the same thickness and length (that is, the same from either end of the diaphragm, which coincides with the in-plane stiffness), the CR is located location of the CM at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an equivalent uniformly distributed seismic load equal to

GA

AA

‫ݓ‬ௐ ൌ 1.92 kipsΤft E

30.0Ԣ

120.0Ԣ

30.0Ԣ ሺܴ஻ ሻ௫ ൌ 172.8 kips

ሺܴி ሻ௫ ൌ 172.8 kips

115.2

57.6

‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ

57.6

115.2 2,592.0

CL opening 37.5Ԣ ‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬--‫ܛܘܑܓ‬ሻ

54.0

864.0

864.0

Figure 9.34 Equivalent beam model for the diaphragm in Example 9.7 for seismic forces in the west direction.

9-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Reactions in each of the walls on column lines B and F

7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH • Chord forces The maximum tension chord force based on the maximum bending moment in the diaphragm is equal to the following: Eq. (9.1)

7KHSULPDU\WHQVLRQFKRUGIRUFHDWWKHFHQWHURIWKHRSHQLQJZKLFKORFDWHGIWIURPFROXPQOLQH$RU*LV FDOFXODWHGXVLQJWKHPRPHQWLQWKHGLDSKUDJPDWWKLVORFDWLRQVHH)LJXUH  Eq. (9.8)

LVGHWHUPLQHGIURPWKHPRPHQWGLDJUDPLQ)LJXUH

where

1A

2A

3A

4A

35ᇱ -6ᇳ

42ᇱ -6ᇳ

A

42ᇱ -6ᇳ

‫ܥ‬௨,௢௣௘௡ ൌ 0.5 kips

‫ܥ‬௨,ଶ ൌ 0.5 kips

ܶ௨,ଶ ൌ 0.5 kips

‫ܥ‬௨,ଵ ൌ 0.2 kips

ܶ௨,ଵ ൌ 0.2 kips

A

F

ܶ௨,௢௣௘௡ ൌ 0.5 kips

A

G

‫ܥ‬௨ ൌ 22.6 kips

N

A

D

ܶ௨ ൌ 22.6 kips

A

E

C

Figure 9.35 Chord forces in the diaphragm of Example 9.7 for the seismic force in the west direction.

9-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The secondary tension chord force at the center of the opening,

is determined by the following equation: Eq. (9.8)

In this equation,

is the positive bending moment in the subdiaphragm to the west of the opening.

,WLVDVVXPHGWKDWWKHVXEGLDSKUDJPVDUH¿[HGDWERWKHQGVDQGDUHVXEMHFWHGWRDSRUWLRQRIWKHWRWDOXQLIRUP based on the mass of the segment. Because each segment has the same area, each segment diaphragm load, From statics, is equal to the following: UHVLVWVSHUFHQWRI

Therefore, the secondary tension chord force at the center of the opening is equal to the following:

The total tension chord force at the center of the opening is equal to NLSWHQVLRQFKRUGIRUFHGHWHUPLQHGIRUWKHRYHUDOOGLDSKUDJP

which is less than the

Secondary tension chord forces develop at the corners of the openings due to the negative bending moments in the subdiaphragm to the east of the opening. The tension chord force occurring along column line 3 is equal to the following:

• Unit shear forces $ORQJFROXPQOLQHV%DQG)WKHVKHDUIRUFHLQWKHVODELVHTXDOWRNLSVVHH)LJXUH ZKLFKLVGLVWULEXWHGRYHUDOHQJWKRIIW7KHUHIRUHWKHXQLWVKHDUIRUFHLVHTXDOWRWKHIROORZLQJ

-XVWWRWKHQRUWKRIFROXPQOLQH%DQGWRWKHVRXWKRIFROXPQOLQH)WKHVKHDUIRUFHLVHTXDOWRNLSVZKLFK due to the openings. Thus, the unit shear forces at these is distributed over a length of locations are equal to the following:

• Collector forces Collectors are required because the walls along column lines B and F do not extend the entire depth of the diaphragm in the direction of analysis. In this example, the portions of the slab in line with the walls are the collectors and the widths of the collectors are equal to the thicknesses of the walls. The unit shear forces, net shear forces, and collector forces along column line F are given in Figure 9.36. In buildings assigned to SDC C, collectors and their connections must be designed for the maximum of the three IRUFHVJLYHQLQ$6&(6(,

9-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳA

ʹA

͵A

ͶA

͵ͷᇱ Ǧ͸ᇳ

Ͷʹᇱ Ǧ͸ᇳ

A

Ͷʹᇱ Ǧ͸ᇳ

͵Ͳᇱ ǦͲᇳ ሺ–›’Ǥሻ



ͷ͹Ǥ͸ ൌ ͲǤͶ͹ͺ ‹’•Τˆ– ͳʹͲǤͷ

A

 

ͳͳͷǤʹ ൌ ͳǤ͵ͷͷ ‹’•Τˆ– ͺͷǤͲ

ͳ͹ʹǤͺ ൌ ͶǤͺ͸ͺ ‹’•Τˆ– ͵ͷǤͷ

ͶǤͺ͸ͺ‹’•Ȁˆ–

‫ܚ܉܍ܐ܁ܜܑܖ܃‬۴‫ܛ܍܋ܚܗ‬ ͲǤͶ͹ͺ‹’•Ȁˆ– ͳǤͺ͵͵‹’•Ȁˆ–

ͳǤͺ͵͵‹’•Ȁˆ–

ͶǤ͵ͻͲ‹’•Ȁˆ–

‫ܚ܉܍ܐ܁ܜ܍ۼ‬۴‫ܛ܍܋ܚܗ‬ ͳǤͺ͵͵‹’•Ȁˆ–

ͳǤͺ͵͵‹’•Ȁˆ–

͹͹Ǥͻ‹’•

۱‫ܚܗܜ܋܍ܔܔܗ‬۴‫ܛ܍܋ܚܗ‬

͹͹Ǥͻ‹’•

Figure 9.36 Unit shear forces, net shear forces, and collector forces in the diaphragm in Example 9.7.

1. )  RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKWKHVHLVPLF IRUFHVGHWHUPLQHGE\WKH(TXLYDOHQW/DWHUDO)RUFH3URFHGXUHRI$6&(6(,RUWKHPRGDOUHVSRQVHVSHFWUXPDQDO\VLVSURFHGXUHRI$6&(6(, For the ordinary building frame system with ordinary reinforced concrete shear walls in the east-west direcLVHTXDOWRVHH$6&(6(,7DEOH 7KHUHIRUHWKHUHTXLUHGLQSODQH tion, the overstrength factor, VHH7DEOHLQ([DPSOH  GLDSKUDJPIRUFHDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR

9-50

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

 )  RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKWKHVHLVPLF IRUFHVGHWHUPLQHGE\$6&(6(,(T  DWWKHVHFRQGÀRRUOHYHOLVHTXDOWRWKHIROORZLQJ Using the information in Table 3.31, the diaphragm force,

The required in-plane force including overstrength is equal to 3. )  RUFHVFDOFXODWHGXVLQJWKHORDGFRPELQDWLRQVRI$6&(6(,ZLWKWKHVHLVPLFIRUFHVGHWHUPLQHGE\ $6&(6(,(T  DWWKHVHFRQGÀRRUOHYHOEDVHGRQ$6&(6(,(T LVHTXDOWRNLSV The diaphragm force, (see Table 3.31). Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the HIIHFWVGXHWRWKHNLSLQSODQHIRUFHVWLSXODWHGLQWKHVHFRQGDQGWKLUGUHTXLUHPHQWV7KXVWKHUHTXLUHG D[LDOIRUFHLQWKHFROOHFWRULVHTXDOWRNLSVVHH)LJXUH  9.9.8

Example 9.8 – Determination of Diaphragm Reinforcement: Building #2, SDC C, Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the SFRS

'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJIRUVHLVPLF forces in the east-west direction assuming only the walls on column lines B and F are part of the SFRS (see Figure DQG*UDGH  $OVRDVVXPHDLQWKLFNVODELQWKLFNZDOOVQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. It is assumed that the portions of the slab in line with the walls are the collectors and the widths of the collectors are equal to the thicknesses of the walls. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the chord reinforcement

)URP([DPSOH

DWFROXPQOLQHVDQG Eq. (9.13)

3URYLGHEDU slab.

ORFDWHGQHDUWKHHGJHVRIWKHGLDSKUDJPDWPLGGHSWKRIWKHLQWKLFN

At the edges of the openings:

The required amount of secondary chord reinforcement along the slab edges adjacent to the openings at column OLQHVDQGLVQRPLQDO7KHFROOHFWRUUHLQIRUFHPHQWIRUVHLVPLFIRUFHWUDQVIHULQWKHQRUWKVRXWKGLUHFWLRQZLOOEH provided along these edges of the opening, which is greater than Step 2 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHV%DQG)DQGDUHHTXDOWRNLSVIWVHH)LJXUH 

9-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Check shear strength requirements assuming

Eq. (9.14)

Therefore, no shear reinforcement is required to satisfy shear strength requirements. Step 3 – Determine the shear transfer reinforcement

• Shear transfer reinforcement between the diaphragm and walls on column lines B and F assuming the slab and walls are not placed at the same time and the interface is not intentionally roughened (see Figure 9.36 and 7DEOH  Eq. (9.16)

3URYLGHGRZHOEDUVVSDFHGDWLQRQFHQWHU

over the lengths of the walls.

• Shear transfer reinforcement between the diaphragm and the collectors (see Figure 9.36):

A value of for monolithic concrete is used because the collectors are part of the slab and have the VDPHZLGWKVDVWKHWKLFNQHVVRIWKHZDOOVVHH7DEOH  %HFDXVHWKHDUHDRIVKHDUIULFWLRQLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDOUHLQIRUFHPHQWLQWKH slab oriented in the north-south direction can be used as the shear transfer reinforcement between the diaphragm and the collectors. Step 4 – Determine the collector reinforcement

7KHPD[LPXPD[LDOIRUFHLQWKHFROOHFWRUVDORQJFROXPQOLQHV%DQG)LVHTXDOWRNLSVVHH)LJXUH 7KH required area of longitudinal collector reinforcement is equal to the following: Eq. (9.19)

3URYLGHEDUV along column lines B and F. These reinforcing bars are in addition to any RWKHUUHLQIRUFHPHQWLQWKHVODEDQGPXVWEHSODFHGZLWKLQWKHLQWKLFNQHVVRIWKHZDOOV&ROOHFWRUUHLQIRUFHPHQWPXVWH[WHQGDORQJWKHOHQJWKRIWKHZDOOVLQDFFRUGDQFHZLWK$&,([WHQGLQJDOOWKHFROOHFWRUEDUVWKH IXOOOHQJWKRIWKHZDOOVHQVXUHVXQLIRUPVKHDUÀRZDFURVVWKHZDOOVHJPHQWEHORZWKHVODEZKLFKPXVWUHVLVWWKH shear force from the wall above plus the tension force transferred by the collector. &KHFNWKHD[LDOFRPSUHVVLRQIRUFHLQWKHFROOHFWRUVDVVXPLQJWKHVHFWLRQVDUHUHLQIRUFHGZLWKORQJLWXGLQDOEDUV Eq. (9.20)

5HLQIRUFHPHQWGHWDLOVIRUWKHVHFRQGÀRRUGLDSKUDJPDUHJLYHQLQ)LJXUHIRUVHLVPLFIRUFHVLQWKHHDVWZHVW GLUHFWLRQ)RUVLPSOHUGHWDLOLQJWKHFROOHFWRUEDUVRQFROXPQOLQHV%DQG)DUHH[WHQGHGWKHIXOOGHSWKRIWKH diaphragm. Similar details can be obtained for seismic forces in the north-south direction.

9-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1A

2A

3A

35ᇱ -6ᇳ

4A

42ᇱ -6ᇳ

A

42ᇱ -6ᇳ G

ᬆ

A

ᬇ

A

F

30ᇱ -0ᇳ ሺtyp.ሻ

E

ᬅ

ᬅ

A

N

A

D

A

C

B

ᬇ

A

ᬆ

A

1.ᬅ ᪫ Chord reinforcement: 1-#6 bar 2.ᬆ ᪫ Shear transfer reinforcement: #4 @ 12ԣ dowel bars 3.ᬇ ᪫ Collector reinforcement: 3-#7 bars 4.Provide standard 90-deg hooks at the ends of all bars 5.Provide Class B lap splices or Type 2 mechanical connectors where required 6.Other reinforcement not shown for clarity

Figure 9.37 Reinforcement details for the diaphragm at the second-floor level in Examples 9.7 and 9.8.

Comments. It is important to ensure that the collector reinforcement does not cause any congestion issues within the wall. If congestion issues or any other problems occur, the collectors can be made wider than the thickness of WKHZDOOVVHH([DPSOHVDQGEHORZ 

9-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

9.9.9

Example 9.9 – Determination of Diaphragm In-Plane Forces: Building #2, SDC C, Collectors Required, Collector Width Wider than the Width of the Vertical Elements of the SFRS

'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJIRUVHLVPLFIRUFHVLQWKHHDVW ZHVWGLUHFWLRQDVVXPLQJRQO\WKHZDOOVRQFROXPQOLQHV%DQG)DUHSDUWRIWKH6)56VHH)LJXUH $OVRDVVXPH DLQWKLFNVODEDQGLQWKLFNZDOOV,WLVDVVXPHGWKDWWKHSRUWLRQVRIWKHVODELQOLQHZLWKWKHZDOOVDUHWKH collectors and the widths of the collectors are wider than the thicknesses of the walls. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the diaphragm design in-plane forces

)URP7DEOHLQ([DPSOHWKHVHLVPLFIRUFHLQWKHHDVWZHVWGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR NLSV7KLVIRUFHLVDSSOLHGDWWKH&0ZKLFKLVORFDWHGDWWKHFHQWURLGRIWKHÀRRUSODWHDWWKLVOHYHOGXHWR symmetry. Step 2 – Determine the classification of the diaphragm

Sect. 9.3.3

For forces in the east-west direction, the span-to-depth ratio is equal to $OVRWKHVWUXFWXUHGRHVQRWKDYHDQ\RIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH 7KHUHIRUHWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG Step 3 – Select the diaphragm model

ASCE/SEI 12.3.1.2 Table 9.1

Because this building has no irregularities and is not subjected to any transfer forces, and because the walls on column lines B and F have the same lateral stiffness (that is, the thicknesses and the lengths of both walls are the same), an equivalent beam model with rigid supports is selected to determine the internal forces in this diaphragm. Step 4 – Determine the diaphragm internal forces

Sect. 9.3.3

7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVGHSLFWHGLQ)LJXUHIRUVHLVPLFIRUFHV in the west direction. Because the walls on column lines B and F have the same thickness and length (that is, infrom either end of the diaphragm, which coincides with the loplane stiffness), the CR is located cation of the CM at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an equivalent uniformly distributed wind load equal to • Reactions in each of the walls on column lines B and F

7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH • Chord forces The maximum tension chord force based on the maximum bending moment in the diaphragm is equal to the following: Eq. (9.1)

9-54

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHSULPDU\WHQVLRQFKRUGIRUFHDWWKHFHQWHURIWKHRSHQLQJZKLFKORFDWHGIWIURPFROXPQOLQH$RU*LV HTXDOWRWKHIROORZLQJVHH)LJXUH  Eq. (9.8)

where

LVGHWHUPLQHGIURPWKHPRPHQWGLDJUDPLQ)LJXUH

The secondary tension chord force at the center of the opening,

is determined by the following equation: Eq. (9.8)

In this equation,

is the positive bending moment in the subdiaphragm to the west of the opening.

,WLVDVVXPHGWKDWWKHVXEGLDSKUDJPVDUH¿[HGDWERWKHQGVDQGDUHVXEMHFWHGWRDSRUWLRQRIWKHWRWDOXQLIRUP based on the mass of the segment. Because each segment has the same area, each segment diaphragm load, From statics, is equal to the following: UHVLVWVSHUFHQWRI

Therefore, the secondary tension chord force at the center of the opening is equal to the following:

The total tension chord force at the center of the opening is equal to NLSWHQVLRQFKRUGIRUFHGHWHUPLQHGIRUWKHRYHUDOOGLDSKUDJP

which is less than the

Secondary tension chord forces develop at the corners of the openings due to the negative bending moments in the subdiaphragm to the east of the opening. The tension chord force occurring along column line 3 is equal to the following:

• Unit shear forces $ORQJFROXPQOLQHV%DQG)WKHVKHDUIRUFHLQWKHVODELVHTXDOWRNLSVVHH)LJXUH ZKLFKLVGLVWULEXWHGRYHUDOHQJWKRIIW7KHUHIRUHWKHXQLWVKHDUIRUFHLVHTXDOWRWKHIROORZLQJ

-XVWWRWKHQRUWKRIFROXPQOLQH%DQGWRWKHVRXWKRIFROXPQOLQH)WKHVKHDUIRUFHLVHTXDOWRNLSVZKLFK because of the openings. Thus, the unit shear forces at these is distributed over a length of locations are equal to the following:

9-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Collector forces Collectors are required because the walls along column lines B and F do not extend the entire depth of the diaphragm in the direction of analysis. In this example, the portions of the slab in line with the walls are the collectors and the widths of the collectors are taken greater than the thicknesses of the walls. The effective slab is equal to the following: width, Sect. 9.3.3

In buildings assigned to SDC C, collectors and their connections must be designed for the maximum of the three IRUFHVJLYHQLQ$6&(6(, 1. )  RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKWKHVHLVPLF IRUFHVGHWHUPLQHGE\WKH(TXLYDOHQW/DWHUDO)RUFH3URFHGXUHRI$6&(6(,RUWKHPRGDOUHVSRQVHVSHFWUXPDQDO\VLVSURFHGXUHRI$6&(6(, For the ordinary building frame system with ordinary reinforced concrete shear walls in the east-west direcLVHTXDOWRVHH$6&(6(,7DEOH 7KHUHIRUHWKHUHTXLUHGLQSODQH tion, the overstrength factor, VHH7DEOHLQ([DPSOH  GLDSKUDJPIRUFHDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR  )  RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKWKHVHLVPLF IRUFHVGHWHUPLQHGE\$6&(6(,(T  DWWKHVHFRQGÀRRUOHYHOLVHTXDOWRWKHIROORZLQJ Using the information in Table 3.31, the diaphragm force,

The required in-plane force including overstrength is equal to 3. )  RUFHVFDOFXODWHGXVLQJWKHORDGFRPELQDWLRQVRI$6&(6(,ZLWKWKHVHLVPLFIRUFHVGHWHUPLQHGE\ $6&(6(,(T  DWWKHVHFRQGÀRRUOHYHOEDVHGRQ$6&(6(,(T LVHTXDOWRNLSV The diaphragm force, (see Table 3.31). Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the HIIHFWVGXHWRWKHNLSLQSODQHIRUFHVWLSXODWHGLQWKHVHFRQGDQGWKLUGUHTXLUHPHQWV7KXVWKHWRWDOD[LDO IRUFHLQWKHFROOHFWRULVHTXDOWRNLSVVHH)LJXUH  )RUSXUSRVHVRIDQDO\VLVLWLVDVVXPHGWKDWSHUFHQWRIWKHWRWDOD[LDOIRUFHLVWUDQVIHUUHGGLUHFWO\LQWRWKH DQGSHUFHQWLVWUDQVIHUUHGE\VKHDUIULFWLRQDGMDFHQWWRWKHZDOO wall

9.9.10 Example 9.10 – Determination of Diaphragm Reinforcement: Building #2, SDC C, Collectors Required, Collector Width Wider than the Width of the Vertical Elements of the SFRS

'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJIRUVHLVPLF forces in the east-west direction assuming only the walls on column lines B and F are part of the SFRS (see Figure DQG*UDGH  $OVRDVVXPHDLQWKLFNVODELQWKLFNZDOOVQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. It is assumed that the portions of the slab in line with the walls are the collectors and the widths of the collectors are wider than the thicknesses of the walls. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH

9-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the chord reinforcement

From Example 9.9,

DWFROXPQOLQHVDQG Eq. (9.13)

3URYLGHEDU slab.

ORFDWHGQHDUWKHHGJHVRIWKHGLDSKUDJPDWPLGGHSWKRIWKHLQWKLFN

At the edges of the openings:

The required amount of secondary chord reinforcement along the slab edges adjacent to the openings at column OLQHVDQGLVQRPLQDO7KHFROOHFWRUUHLQIRUFHPHQWIRUVHLVPLFIRUFHWUDQVIHULQWKHQRUWKVRXWKGLUHFWLRQZLOOEH provided along these edges of the opening, which is greater than Step 2 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHV%DQG)DQGDUHHTXDOWRNLSVIWVHH)LJXUH 9.36). Check shear strength requirements assuming

Eq. (9.14)

Therefore, no shear reinforcement is required to satisfy shear strength requirements. Step 3 – Determine the shear transfer reinforcement

Shear transfer reinforcement between the diaphragm and walls on column lines B and F assuming the slab and walls are not placed at the same time and the interface is not intentionally roughened. 7KHWRWDOVKHDUIRUFHWUDQVIHUUHGEHWZHHQWKHGLDSKUDJPVDQGWKHZDOOVFRQVLGHULQJSHUFHQWRIWKHFROOHFWRUD[LDO IRUFHLVWUDQVIHUUHGE\VKHDULVHTXDOWRWKHIROORZLQJVHH)LJXUH7DEOHDQG6WHSLQ([DPSOH 

The required area of shear-friction reinforcement is equal to the following: Eq. (9.16)

3URYLGHGRZHOEDUVVSDFHGDWLQRQFHQWHU

over the lengths of the walls.

9-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Shear transfer reinforcement between the diaphragm and the collectors (see Figure 9.36):

A value of

IRUPRQROLWKLFFRQFUHWHLVXVHGEHFDXVHWKHFROOHFWRUVDUHSDUWRIWKHVODEVHH7DEOH 

%HFDXVHWKHDUHDRIVKHDUIULFWLRQLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDOUHLQIRUFHPHQWLQWKH slab oriented in the north-south direction can be used as the shear transfer reinforcement between the diaphragm and the collectors. Step 4 – Determine the collector reinforcement

The total area of collector reinforcement is equal to the following: Eq. (9.19)

$VVXPLQJSHUFHQWRIWKHPD[LPXPD[LDOIRUFHLQWKHFROOHFWRUVDORQJFROXPQOLQHV%DQG)LVWUDQVIHUUHGGLUHFWO\ is equal to the following: into the ends of the walls, the required area of longitudinal collector reinforcement, Eq. (9.19)

3URYLGHEDU along column lines B and F. These reinforcing bars are in addition to any RWKHUUHLQIRUFHPHQWLQWKHVODEDQGPXVWEHSODFHGZLWKLQWKHLQWKLFNQHVVRIWKHZDOOV The remaining required collector reinforcement is uniformly distributed within

3URYLGHEDUV

uniformly spaced within

Check the axial compression force in the collectors. )RUWKHLQZLGHVHFWLRQUHLQIRUFHGZLWKEDU

Eq. (9.20)

For the

9-58

VHFWLRQUHLQIRUFHGZLWKEDUV

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the reinforcement due to eccentricity of collector forces

The in-plane bending moment, due to the eccentricity, between the portion of the collector force not transferred directly in to the ends of the walls and the centerline of the walls can be determined by the following equation: Eq. (9.17) 1A

2A

3A

35ᇱ -6ᇳ

4A

42ᇱ -6ᇳ

A

42ᇱ -6ᇳ G

ᬈ

ᬆ

A

ᬇ

A

F

ሺtyp.ሻ

30ᇱ -0ᇳ

E

ᬅ

ᬅ

A

N

A

D

A

C

B

ᬇ ᬆ

A

ᬈ

A

1.ᬅ ᪫ Chord reinforcement: 1-#6 bar 2.ᬆ ᪫ Shear transfer reinforcement: #5 @ 12ԣ dowel bars 3.ᬇ ᪫ Collector reinforcement: 5-#5 bars 4.ᬈ ᪫ Reinforcement due to eccentricity of collector forces: 2-#6 bars 5.Provide standard 90-deg hooks at the ends of all bars 6.Provide Class B lap splices or Type 2 mechanical connectors where required 7.Other reinforcement not shown for clarity

Figure 9.38 Reinforcement details for the diaphragm at the second-floor level in Examples 9.9 and 9.10.

9-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In this equation, the reinforcement,

HTXDOWR]HUR

The required reinforcement,

Conservatively taking the shear strength of the diaphragm due to is equal to the following:

is equal to the following: Eq. (9.18)

3URYLGHEDUV place perpendicular to the face of the wall at both ends; these bars must be developed for tension into the slab and into the wall. 5HLQIRUFHPHQWGHWDLOVIRUWKHVHFRQGÀRRUGLDSKUDJPDUHJLYHQLQ)LJXUHIRUVHLVPLFIRUFHVLQWKHHDVWZHVW direction. For simpler detailing, all the collector bars on column lines B and F are extended the full depth of the diaphragm. Similar details can be obtained for seismic forces in the north-south direction.

9-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 10 FOUNDATIONS 10.1 Overview The main function of a foundation is to transmit the loads from the structure above to the soil below. In building structures, the loads are transmitted directly or indirectly by columns and walls. Foundations must be located on a soil or rock stratum with adequate strength to support the loads; in other words, the loads must be spread out over a VXI¿FLHQWDUHDVRWKHUHVXOWLQJSUHVVXUHLVQRWJUHDWHUWKDQWKHDOORZDEOHEHDULQJFDSDFLW\RIWKHVRLORUURFN,QDGGLtion to strength, the total settlement and the differential settlement between adjoining foundations must be limited to tolerable amounts in order to prevent possible damage to the structure. Overturning, sliding, and rotation must also be limited to tolerable values. Numerous types of shallow and deep foundations are available to support the loads from the superstructure of a building. The design and detailing of the following foundation systems are covered in this chapter: • • • •

Isolated spread footings Walls footings Combined spread footings Drilled piers (caissons)

Provisions for foundations are given in ACI Chapter 13, which are applicable to foundations supporting buildings assigned to Seismic Design Category (SDC) A and B. 'HWDLOHGLQIRUPDWLRQRQWKHGHVLJQRISLOHFDSVLVJLYHQLQ5HIHUHQFH'HVLJQDQGGHWDLOLQJPHWKRGVIRUFDQWLOHYHUHGUHWDLQLQJZDOOVDUHJLYHQLQ5HIHUHQFH

10.2 Design Criteria Foundations must be proportioned such that the requirements in the governing building code for bearing effects, VWDELOLW\DJDLQVWRYHUWXUQLQJDQGVOLGLQJDWWKHVRLOIRXQGDWLRQLQWHUIDFHDUHVDWLV¿HG$&, 7\SLFDOO\WKH base area of a spread footing or the number and arrangement of deep foundation members are determined using allowable strengths of the soil or rock beneath the foundation (which have usually been established by a geotechniFDOLQYHVWLJDWLRQ DQGDOORZDEOHVWUHVVORDGFRPELQDWLRQV+RZHYHUQRPLQDOJHRWHFKQLFDOVWUHQJWKVZLWKUHVLVWDQFH factors and strength design load combinations may also be used. When determining the base dimensions of footings, WKHPLQLPXPPRPHQWUHTXLUHPHQWIRUVOHQGHUQHVVFRQVLGHUDWLRQVJLYHQLQ$&,QHHGQRWEHFRQVLGHUHGIRU transfer of forces and moments to a footing; only the calculated end moment at the base of a column needs to be transferred. )DFWRUHGUHDFWLRQVDUHXVHGWRGHVLJQIRXQGDWLRQPHPEHUVH[FHSWDVSHUPLWWHGE\$&,IRUGHHSIRXQGDWLRQ PHPEHUV$&, )RUH[DPSOHWKHWKLFNQHVVRIDIRRWLQJDQGWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWDUH GHWHUPLQHGXVLQJIDFWRUHGEHQGLQJPRPHQWV6LPLODUO\VKHDUVWUHQJWKUHTXLUHPHQWVPXVWEHVDWLV¿HGXVLQJIDFWRUHG shear forces. As note above, it is permitted to design deep foundation members using allowable stress load combinaWLRQVDQGDOORZDEOHVWUHQJWKV$&,  )RXQGDWLRQGHVLJQLVSHUPLWWHGWREHEDVHGRQDQ\SURFHGXUHWKDWVDWLV¿HVHTXLOLEULXPDQGJHRPHWULFFRPSDWLELOLW\ $&, 7KHVWUXWDQGWLHPHWKRGLQ$&,&KDSWHUPD\EHXVHGWRGHVLJQIRXQGDWLRQV$&, 

10.3 Footings 10.3.1 Overview

In general, footings are shallow foundations that transfer the load from the superstructure to a soil stratum beneath WKHVWUXFWXUHUHODWLYHO\FORVHWRWKHJURXQGVXUIDFH$VRLOVWUDWXPPXVWEHLGHQWL¿HGZLWKDGHTXDWHEHDULQJFDSDFLW\ 10-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

to support the loads. To avoid possible frost heave, shallow foundations must be placed below the frost line at the site (where applicable). A geotechnical report provides guidance on the soil bearing capacity and the appropriate depth for particular site conditions. 6SUHDGIRRWLQJVW\SLFDOO\VXSSRUWRQHRUPRUHYHUWLFDOHOHPHQWV7KHLVRODWHGVSUHDGIRRWLQJLQ)LJXUHVXSSRUWV a single column. Its function is to spread the column load to the soil so that the maximum pressure beneath the footing is less than or equal to the permissible bearing capacity of the soil. Flexural reinforcement is provided in two orthogonal directions at the bottom of the footing.    ‘Ž—  ’”‡ƒ†ˆ‘‘–‹‰            Figure 10.1 An isolated spread footing.

:DOOIRRWLQJVDUHXVXDOO\FRQWLQXRXVXQGHUWKHOHQJWKRIDZDOODQGDUHVLPLODUWRVSUHDGIRRWLQJVVHH)LJXUH  Flexural reinforcement is placed at the bottom of the footing perpendicular to the face of the wall and shrinkage and temperature reinforcement is provided parallel to the length of the wall.   ƒŽŽ       ƒŽŽˆ‘‘–‹‰          Figure 10.2 A wall footing.

10-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A combined footing is a spread footing that supports multiple columns or walls on the same footing (see Figure  7KLVW\SHRIIRRWLQJLVFRPPRQO\XVHGZKHUHWKHVSDFHEHWZHHQDGMRLQLQJIRRWLQJVLVVPDOORUZKHUHDEXLOGing is close to a property line. If the load at the interior column is greater than the load at the edge column for the DUUDQJHPHQWVKRZQLQWKH¿JXUHWKHIRRWLQJFDQEHVL]HGVRWKDWWKHSUHVVXUHDWWKHEDVHRIWKHIRRWLQJLVXQLIRUP :KHUHDQH[WHULRUFROXPQKDVDJUHDWHUORDGWKDQDQDGMDFHQWLQWHULRUFROXPQDWUDSH]RLGDOIRRWLQJFDQEHXVHGWR DFKLHYHDXQLIRUPGLVWULEXWLRQRISUHVVXUHEHQHDWKWKHIRRWLQJVHH)LJXUH  ‘Ž—ሺ–›’Ǥሻ ‘„‹‡†ˆ‘‘–‹‰

Figure 10.3 A rectangular combined footing.

‘Ž—ሺ–›’Ǥሻ

‘„‹‡†ˆ‘‘–‹‰

Figure 10.4 A trapezoidal spread footing.

A strap footing (or, cantilever footing) consists of an eccentrically loaded footing connected to an adjacent footing E\DVWUDSEHDPDVVKRZQLQ)LJXUH7KHSXUSRVHRIWKHVWUDSEHDPLVWRWUDQVPLWWKHPRPHQWIURPWKHHFFHQtrically loaded footing to the adjacent footing so that a uniform soil pressure is achieved beneath both footings. The strap beam must be rigid to avoid rotation of the exterior footing; providing a strap beam with a moment of inertia of at least two times that of the footing will usually accomplish this. The design and detailing of reinforced concrete footings generally involves determining the following: 1.  3. 

Base area of the footing Thickness of the footing 5HTXLUHGDPRXQWRIÀH[XUDODQGLQWHUIDFHUHLQIRUFHPHQW Reinforcement details

Methods on how to determine these items are given in the following sections. 10-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‘Ž—ሺ–›’Ǥሻ ‘‘–‹‰ሺ–›’Ǥሻ

–”ƒ’„‡ƒ

Ž‡˜ƒ–‹‘

Ž‡˜ƒ–‹‘

Žƒ Figure 10.5 A strap footing. Žƒ ܲ

‫ݍ‬ൌ

ܲ  ‫ܣ‬௙

Figure 10.6 Soil pressure distribution beneath a concentrically loaded footing.

10.3.2 Determining the Base Area of a Footing Overview

The base area of a footing subjected to sustained gravity loads is usually determined using allowable stress load combinations and an allowable soil bearing capacity determined from a geotechnical investigation (the presumpWLYHORDGEHDULQJYDOXHVLQ,%&7DEOHPD\DOVREHXVHGZKHUHSHUPLWWHG 7KHDOORZDEOHVRLOVWUHVVSURYLGHG in the geotechnical report is either the stress associated with soil bearing failure (including a factor of safety) or an equivalent stress meant to control settlement of the footing based on the anticipated sustained loads. It is common for footings to be designed in the elastic range for the case of sustained gravity loads without any uplift. %HDULQJIDLOXUHLVWKHSULPDU\GHVLJQFRQVLGHUDWLRQZKHQIRRWLQJVDUHVXEMHFWHGWRZLQGDQGRUVHLVPLFHIIHFWV)RU WKHVHW\SHVRIWUDQVLHQWORDGVLWLVFRPPRQSUDFWLFHWRXVHDOORZDEOHVWUHVVORDGFRPELQDWLRQVWRVL]HWKHIRRWLQJ with an allowable bearing capacity equal to the static bearing capacity multiplied by a factor to account for the transient nature of the loads. Footings are typically designed in the elastic range when subjected to the effects from wind loads, and uplift is commonly permitted provided the maximum bearing stress is less than or equal to the allowable value. 10-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Isolated Spread Footing

&RQVLGHUWKHIRRWLQJLQ)LJXUHZKLFKLVVXEMHFWHGWRDVHUYLFHD[LDOORDG acting through the centroid of the For purposes of design, the footing is assumed to be rigid and the resulting soil pressure, footing base area, at the base of the footing is assumed to be uniform. The required area of the footing is equal to the total service load which is equal to the allowable bearing capacity of the soil, divided by the net permissible soil pressure, minus the weight of the surcharge above the base of the footing (the surcharge consists of the weight of the soil and concrete above the base plus any additional service surcharge applied at the surface): (10.1)

In this equation,

and

are the base dimensions of the footing.

‫ܤ‬

‫ܮ‬

‫ܣ‬௙ ൌ ‫ܮܤ‬

ܲ ‫ܯ‬

ሺƒሻ݁ ൌ ‫ܯ‬Ȁܲ ൏ ‫ܮ‬Ȁ͸ ‫ݍ‬ൌ

ܲ ͸‫ܯ‬ േ  ‫ܣ‬௙ ‫ܮܤ‬ଶ

ሺ„ሻ݁ ൌ ‫ܯ‬Ȁܲ ൌ ‫ܮ‬Ȁ͸ ‫ݍ‬௠௔௫ ൌ

ʹܲ  ‫ܣ‬௙

‫ݍ‬௠௔௫ ൌ

ʹܲ  ͵ሺͲǤͷ‫ ܮ‬െ ݁ሻ‫ܤ‬

͵ሺͲǤͷ‫ ܮ‬െ ݁ሻ ሺ…ሻ݁ ൌ ‫ܯ‬Ȁܲ ൐ ‫ܮ‬Ȁ͸

Figure 10.7 Soil pressure distribution beneath a footing subjected to axial force and moment. (a) e < L / 6. (b) e = L / 6. (c) e > L / 6.

10-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For footings subjected to an axial force and a moment, or to an axial force at an eccentricity, from the centroid of the footing, the total stress at the base of the footing is the sum of the stresses due to the axial force, (stress D[LDOIRUFHIRRWLQJDUHD DQGWKHEHQGLQJPRPHQW (stress PRPHQWVHFWLRQPRGXOXVRIIRRWLQJ 7KHWRWDO SUHVVXUHLVDVVXPHGWRYDU\OLQHDUO\DVVKRZQLQ)LJXUHD IRUWKHFDVHZKHUHWKHD[LDOIRUFHIDOOVZLWKLQWKH The minimum and maximum pressures at the edges of the kern of the footing area, that is, where footing in this case can be determined from the following equation: (10.2)

Where WKHSUHVVXUHDWRQHHGJHRIWKHIRRWLQJLVHTXDOWR]HURDQGWKHPD[LPXPSUHVVXUHDWWKHRWKHU >VHH)LJXUHE @ edge is equal to

The base dimensions of the footing,

can be obtained by the following equation where (10.3)

,WLVHYLGHQWWKDWEHDULQJFDSDFLW\UHTXLUHPHQWVDUHVDWLV¿HGIRUDQ\FRPELQDWLRQRI equation.

and

WKDWVDWLV¿HVWKLV

WKHFRPELQHGPLQLPXPVWUHVVGHWHUPLQHGE\(TXDWLRQ LVOHVVWKDQ]HURZKLFKPHDQVWKH Where VWUHVVLVWHQVLOH%HFDXVHWHQVLRQFDQQRWEHWUDQVPLWWHGEHWZHHQWKHIRRWLQJDQGWKHVRLO(TXDWLRQ LVQRORQJHU YDOLGDQGWKHSUHVVXUHGLVWULEXWLRQVKRZQLQ)LJXUHF PXVWEHXVHG,QVXFKFDVHV and can be obtained by the following equation: (10.4)

,QWKHFDVHRIVTXDUHIRRWLQJV(TXDWLRQ FDQEHVROYHGIRU

directly: (10.5)

Combined Footings

It is desirable to design combined footings so that the pressure is uniform over the entire area of the footing; among RWKHUWKLQJVWKLVKHOSVSUHYHQWWKHIRRWLQJIURPURWDWLQJ)RUWKHFRPELQHGUHFWDQJXODUIRRWLQJLQ)LJXUHWKH and at the locations and measured from the edge of the column on columns are supporting axial forces the left. must act through the In order for the pressure beneath the footing to be uniform, the resultant force centroid of the footing. The distance from the edge of the footing to is obtained by summing moments about this point and solving for (10.6)

10-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܲଵ 

‫ݔ‬ଵ 

ܲ ൌ ܲଵ ൅ ܲଶ 

‫ݔ‬ൌ

ܲଶ 

ܲଵ ‫ݔ‬ଵ ൅ ܲଶ ‫ݔ‬ଶ  ܲ ‫ݔ‬ଶ 

‫ݍ‬ൌ

ܲ  ‫ܣ‬௙

‫ ܮ‬ൌ ʹ‫ݔ‬

Figure 10.8 Uniform soil pressure distribution for a combined rectangular footing.

A uniform pressure is obtained by setting the footing dimension width of the footing, FDQEHFDOFXODWHGE\(TXDWLRQ 

equal to

Once

has been determined, the

(10.7)

7KHGLPHQVLRQVRIDFRPELQHGWUDSH]RLGDOIRRWLQJFDQEHGHWHUPLQHGXVLQJVLPLODUPHWKRGV&RQVLGHUWKHWUDSH]RLGDOIRRWLQJLQ)LJXUH$VVXPLQJWKHOHQJWKRIWKHIRRWLQJ LVJLYHQEDVHGRQWKHVL]HDQGVSDFLQJRIWKH and can be determined so that a uniform pressure occurs beneath the footing. columns, the dimensions ‫ܮ‬ ‫ܮ‬Ԣ ‫ݔ‬ҧ 

‫ܤ‬ଶ 

‫ܤ‬ଵ 

ܿଵ Ȁʹ

ܿଵ 

Figure 10.9 Dimensions of a combined trapezoidal footing.

The term is the distance from the center of the heavier-loaded column on the left to the location where the ,WLVHYLGHQWWKDWDWUDSH]RLcentroid of the footing coincides with the resultant of the column loads, in such cases, the result is a triangular footing with the column on dal footing is not possible if the right not fully supported on the foundation. A combined rectangular footing occurs where The following equation can 7KHUHIRUHDFRPELQHGWUDSH]RLGDOIRRWLQJLVSRVVLEOHZKHUH be used to determine the location of the centroid of the footing with respect to the left edge where

10-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(10.8)

The distance from the center of the left column to the resultant axial force, about this point and solving for

is obtained by summing moments

(10.9)

where

is the distance between

and

(that is, the distance between the centerlines of the columns).

The pressure at the base of the footing must be less than or equal to the permissible soil pressure: (10.10)

(TXDWLRQV DQG FDQEHVROYHGIRUWKHWZRXQNQRZQV

and

10.3.3 Determining the Thickness of a Footing Overview

Once the required area of a footing has been established on the basis of the service loads and the permissible bearing capacity of the soil, the thickness, FDQEHGHWHUPLQHGFRQVLGHULQJERWKÀH[XUHDQGVKHDU Isolated spread footings and wall footings must be designed for bending moments caused by the pressure developed DWWKHEDVHRIWKHIRRWLQJGXHWRIDFWRUHGORDGV5HTXLUHPHQWVIRURQHZD\DQGWZRZD\VKHDUPXVWDOVREHVDWLV¿HG (two-way shear is not applicable to wall footings). Methods on how to determine the thickness of a footing based on these three criteria are given below. $PLQLPXPGHSWKRILQLVUHTXLUHGDERYHWKHERWWRPUHLQIRUFHPHQWRIIRRWLQJV$&, $FFRUGLQJWR$&, 7DEOHWKHPLQLPXPFRQFUHWHFRYHUWRWKHUHLQIRUFHPHQWLVHTXDOWRLQIRUFRQFUHWHFDVWDJDLQVWDQG SHUPDQHQWO\H[SRVHGWRHDUWK7KXVWKHPLQLPXPWKLFNQHVVLVHTXDOWRDSSUR[LPDWHO\LQ It is also important to consider the minimum footing thickness based on the development of the dowel bars emanating from the footing, which are spliced to the longitudinal reinforcement in the columns or walls. Information on the UHTXLUHGGHYHORSPHQWOHQJWKVLVFRYHUHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ ܲ௨ 

‘…”‡–‡…‘Ž—ǡ ’‡†‡•–ƒŽǡ‘”™ƒŽŽ

݀

݄

ܿ

”‹–‹…ƒŽ•‡…–‹‘

‫ݍ‬௨ ൌ

ܲ௨  ‫ܣ‬௙

Figure 10.10 Factored pressure at the base of a concentrically loaded, isolated spread footing.

10-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Minimum Thickness Based on Flexural Requirements

A footing must be designed for bending moments induced by the pressure developed at the base of the footing from WKHIDFWRUHGORDGV$FRQFHQWULFDOO\ORDGHGLVRODWHGVSUHDGIRRWLQJLVLOOXVWUDWHGLQ)LJXUH7KHIDFWRUHGSUHVis equal to the factored axial force, divided by the area of the footing, sure, 7KHFULWLFDOVHFWLRQIRUÀH[XUHIRUDQLVRODWHGIRRWLQJVXSSRUWLQJDFRQFUHWHFROXPQSHGHVWDORUZDOOLVORFDWHGDW at this critiWKHIDFHRIWKHVXSSRUWHGPHPEHU$&,7DEOH 7KHPD[LPXPIDFWRUHGEHQGLQJPRPHQW cal section can be calculated by the following equation: (10.11)

where

is the distance from the critical section to the edge of the footing (that is, the length of the cantilever).

In cases where the factored pressure at the base of the footing is nonuniform, can be determined from statics, MXVWOLNHLQWKHFDVHRIIDFWRUHGXQLIRUPSUHVVXUHRULWFDQEHFRQVHUYDWLYHO\GHWHUPLQHGE\(TXDWLRQ DVVXPing the maximum pressure at the edge of the footing is uniformly distributed across the entire width of the footing. &ULWLFDOVHFWLRQORFDWLRQVIRUWZRRWKHUFDVHVDUHJLYHQLQ$&,7DEOH IRRWLQJVVXSSRUWLQJPDVRQU\ZDOOV DQG IRRWLQJVVXSSRUWLQJFROXPQVZLWKDVWHHOEDVHSODWH,QWKH¿UVWFDVHWKHFULWLFDOVHFWLRQLVORFDWHGKDOIZD\ between the center and the face of the wall, and for the second case, it is located halfway between the face of the column and the edge of the base plate. DWWKHFULWLFDOVHFWLRQKDVEHHQGHWHUPLQHGWKHUHTXLUHGDUHDRIÀH[XUDO Once the maximum bending moment, LVGHWHUPLQHGLQWKHVDPHZD\DVDQ\RWKHUÀH[XUDOPHPEHU7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRI reinforcement, is calculated by the following equation: resistance, (10.12)

where 7KHUHTXLUHGWKLFNQHVVRIWKHIRRWLQJEDVHGRQÀH[XUDOUHTXLUHPHQWVFDQEHREWDLQHGE\DVVXPLQJWKHPLQLPXP DUHDRIUHLQIRUFLQJVWHHOSUHVFULEHGLQ$&,DQGLVXVHGLQWKHIRRWLQJZKLFKLVHTXDOWR WLPHVWKHJURVVDUHDRIWKHIRRWLQJ%HFDXVHWKHGHVLJQVWUHQJWKHTXDWLRQIRUÀH[XUHLVEDVHGRQWKHHIIHFWLYHGHSWK DQGQRWWKHRYHUDOOWKLFNQHVVRIWKHPHPEHUPXVWEHPRGL¿HGWRDFFRXQWIRUWKLV$VVXPLQJ then Assuming *UDGHUHLQIRUFHPHQWDWHQVLRQFRQWUROOHGVHFWLRQ VWULS(TXDWLRQ UHGXFHVWRWKHIROORZLQJ

and a one-foot-wide design

(10.13)

where

has the units of ksi.

Thus, for a factored bending moment at the critical section of the footing, the minimum effective depth of the footing, WKDWVDWLV¿HVÀH[XUDOUHTXLUHPHQWVFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ (10.14)

In this equation,

has the units of inches,

has the units of ksi, and

has the units of ft-kips.

10-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

6XEVWLWXWLQJ(TXDWLRQ LQWR(TXDWLRQ UHVXOWVLQWKHIROORZLQJHTXDWLRQIRUFRQFHQWULFDOO\ORDGHGIRRWLQJV (10.15)

where

is in inches,

For

is in feet,

is in kips,

is in square feet, and

is in ksi.

(TXDWLRQ UHGXFHVWRWKHIROORZLQJ (10.16)

Minimum Thickness Based on Shear Requirements

Overview 7KHRQHZD\DQGWZRZD\VKHDUUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\PXVWEHVDWLV¿HGIRU two-way footings with the critical sections for factored shear forces measured from the locations of the critical LQ$&,7DEOH$&, 1RWHWKDWWKHVL]HHIIHFWPRGL¿FDWLRQIDFWRU GH¿QHGLQ sections for $&,IRURQHZD\VWUHQJWKDQGLQ$&,IRUWZRZD\VWUHQJWKQHHGQRWEHFRQVLGHUHGZKHQGHWHUPLQLQJWKH design shear strength of one-way shallow footings, two-way isolated footings, or two-way combined footings (ACI   One-way Shear at the critical section, determined using the factored pressure at the For one-way shear, the factored shear force, must be less than or equal to the design shear strength, determined in accordance with base of the footing $&, for a concentrically loaded, isolated spread footing supporting a The tributary area used in the calculation of FROXPQLVJLYHQLQ)LJXUH7KHFULWLFDOVHFWLRQLVORFDWHGDGLVWDQFH from the face of the column, and the IROORZLQJHTXDWLRQPXVWEHVDWLV¿HGDWWKLVORFDWLRQ (10.17) ‫ܮ‬

”‹–‹…ƒŽ•‡…–‹‘

‫ܤ‬

ܿଵ  ܿଶ 

ܿ

݀

”‹„—–ƒ”›ƒ”‡ƒ

Figure 10.11 Critical section for one-way shear in an isolated spread footing.

10-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In this equation, the design shear strength corresponds to members with less than the required minimum shear UHLQIRUFHPHQWVHH$&,7DEOH 7KHPRGL¿FDWLRQIDFWRU WKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHV RIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOH EDVHGRQHTXLOLEULXPGHQVLW\DQGLQ7DEOHEDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[VHH$&,   Table 10.1 Values of Ȝ Based on Equilibrium Density, wc

Ȝ

Equilibrium Density, wc





Table 10.2 Values of Ȝ Based on Composition of Aggregates Concrete

All-lightweight

/LJKWZHLJKW¿QHEOHQG

Sand-lightweight

Sand-lightweight, coarse blend

Composition of Aggregates

)LQH$670& &RDUVH$670& )LQH&RPELQDWLRQRI$670& and C33 &RDUVH$670& Fine: ASTM C33 &RDUVH$670& Fine: ASTM C33 &RDUVH&RPELQDWLRQRI$670& and ASTM C33

Ȝ  WR(1)  WR

(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate. (2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.

The term LVWKHUDWLRRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHJURVVDUHDRIWKHFRQFUHWHDUHD$VVXPLQJPLQLPXP (TXDWLRQ FDQEHVROYHGIRUWKHPLQLPXP reÀH[XUDOUHLQIRUFHPHQW quired to satisfy one-way shear requirements: (10.18)

Two-way Shear For two-way shear, the factored shear stress, at the critical section, determined using the factored pressure at must be less than or equal to the design shear strength, determined in accordance the base of the footing, ZLWK$&,)RUDFRQFHQWULFDOO\ORDGHGIRRWLQJWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGDWWKHFULWLFDOVHFWLRQ IURPWKHIDFHRIWKHFROXPQVHH)LJXUH  which is located (10.19)

where

is the perimeter of the critical section and

LVWKHOHDVWRIWKHYDOXHVGH¿QHGLQ$&,7DEOH

10-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ܮ‬

‫ܤ‬

ܿଵ 

”‹–‹…ƒŽ•‡…–‹‘ ܿଶ 

݀Ȁʹሺ–›’Ǥሻ

”‹„—–ƒ”›ƒ”‡ƒ Figure 10.12 Critical section for two-way shear in an isolated spread footing.

ƒŽŽ

‘Ž—

Š”‹ƒ‰‡ƒ†–‡’‡”ƒ–—”‡ ”‡‹ˆ‘”…‡‡–ሺ–›’Ǥሻ

Ž‡š—”ƒŽ”‡‹ˆ‘”…‡‡–ሺ–›’Ǥሻ

Figure 10.13 Distribution of flexural reinforcement in one-way footings and square two-way footings.

For footings supporting square columns, typically governs. Therefore, the following equation can be used to determine the minimum required to satisfy two-way shear requirements for a square column supported by an isolated footing: (10.20)

10-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

where

The largest FDOFXODWHGE\(TXDWLRQV  DQG LVXVHGLQGHWHUPLQLQJWKHRYHUDOOWKLFNQHVVRI a footing. Providing an effective depth equal to at least the largest of these three calculated values automatically VDWLV¿HVDOOUHTXLUHPHQWVIRUÀH[XUHRQHZD\VKHDUDQGWZRZD\VKHDU7KHRYHUDOOGHSWKRIWKHIRRWLQJ can be REWDLQHGE\DGGLQJLQLQFRYHUSOXVWKHGLDPHWHURIDÀH[XUDOUHLQIRUFLQJEDU WRWKHODUJHVWFDOFXODWHGHIIHFtive depth, 10.3.4 Determining the Flexural Reinforcement

By determining the effective depth XVLQJWKHPHWKRGVRXWOLQHGLQ6HFWLRQRIWKLVSXEOLFDWLRQPLQLPXPRU FORVHWRPLQLPXP ÀH[XUDOUHLQIRUFHPHQWZLOOEHSURYLGHGLQWKHIRRWLQJ,WLVHYLGHQWWKDWPLQLPXPUHLQIRUFHPHQW is applicable if GHWHUPLQHGE\(TXDWLRQ RU LVODUJHUWKDQWKRVHGHWHUPLQHGE\(TXDWLRQV  DQG $OVRLI GHWHUPLQHGE\(TXDWLRQ RU LVWKHODUJHVWWKHQPLQLPXPUHLQIRUFHPHQW is also applicable because GHWHUPLQHGE\(TXDWLRQ RU ZKLFKLVVPDOOHULVEDVHGRQPLQLPXP reinforcement. In general, minimum reinforcement must be calculated using a footing thickness based on the largest HIIHFWLYHGHSWKGHWHUPLQHGE\(TXDWLRQV  DQG  10.3.5 Detailing the Flexural Reinforcement

5HTXLUHPHQWVIRUWKHGLVWULEXWLRQRIÀH[XUDOUHLQIRUFHPHQWIRURQHZD\VKDOORZIRRWLQJVDUHJLYHQLQ$&, DQGIRUWZRZD\LVRODWHGIRRWLQJVLQ$&,DQG)RUZDOOIRRWLQJVDQGVTXDUHWZRZD\IRRWLQJV UHLQIRUFHPHQWLVWREHXQLIRUPO\GLVWULEXWHGDFURVVWKHHQWLUHZLGWKRIWKHIRRWLQJVHH)LJXUH 7KHVDPHUHLQforcement is used in both orthogonal directions in square two-way footings because the factored bending moments at the critical sections are the same. 5HTXLUHPHQWVIRUGLVWULEXWLRQRIÀH[XUDOUHLQIRUFHPHQWLQUHFWDQJXODUWZRZD\IRRWLQJVDUHJLYHQLQ$&, Reinforcement in the long direction is uniformly distributed across the entire width of the footing. In the short direcmust be uniformly distributed over a band width centered on the tion, a portion of the total reinforcement, ‫ܣ‬௦௅  ‫ܣ‬௦ଵ ̷‫ݏ‬ଵ 

‫ܣ‬௦஻ ̷‫ݏ‬ଷ 

‫ܤ‬

‫ܣ‬௦ଶ ̷‫ݏ‬ଶ 

‫ܣ‬௦ଶ ̷‫ݏ‬ଶ 

‫ܮ‬ ߚൌ  ‫ܤ‬

‫ܤ‬Ȁʹ

‫ܤ‬Ȁʹ ‫ܮ‬

‫ܣ‬௦ଵ ൌ

ʹ ‫ ܣ‬ ߚ ൅ ͳ ௦௅

‫ܣ‬௦ଶ ൌ

‫ܣ‬௦௅ െ ‫ܣ‬௦ଵ  ʹ

Figure 10.14 Distribution of flexural reinforcement in rectangular footings.

10-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

column or pedestal equal to the length of the short side of the footing where and is the ratio of WKHORQJVLGHWRWKHVKRUWVLGHRIWKHIRRWLQJVHH)LJXUH 7KHUHPDLQGHURIWKHUHLQIRUFHPHQWRXWVLGHRIWKH center band must be uniformly distributed. )RUHDVLHUSODFHPHQWLQWKH¿HOGLWLVFRPPRQSUDFWLFHWRLQFUHDVHWKHUHTXLUHGDPRXQWRIUHLQIRUFHPHQWLQWKHVKRUW and to space it uniformly across the long dimension of the footing instead of distributing direction by WKHUHLQIRUFLQJEDUVDVVKRZQLQ)LJXUH 10.3.6 Development of Flexural Reinforcement

'HYHORSPHQWRIUHLQIRUFHPHQWLQIRRWLQJVPXVWEHLQDFFRUGDQFHZLWK$&,&KDSWHU$&, 7KHFULWLFDOVHFWLRQVIRUGHYHORSPHQWRIUHLQIRUFHPHQWDUHWKHVDPHDVWKRVHJLYHQLQ$&,IRUWKHIDFWRUHGEHQGLQJ PRPHQWDQGDWDOORWKHUYHUWLFDOSODQHVZKHUHFKDQJHVRIVHFWLRQRUUHLQIRUFHPHQWRFFXU$&, ,QRWKHU EH\RQGWKHFULWLFDOVHFWLRQIRUÀH[words, reinforcing bars must extend at least a tension development length, ure. must be less than or equal to the available development

For a concrete column supported by an isolated footing, length:

(10.21)

where and are the lengths of the footing and the column in the direction of analysis, respectively, and the cover LVQROHVVWKDQLQVHH)LJXUH  ܿଵ 

”‹–‹…ƒŽ•‡…–‹‘

κௗ 

…‘˜‡”

‫ܮ‬ Figure 10.15 Development length of flexural reinforcement in footings.

(TXDWLRQ FDQDOVREHXVHGIRUZDOOIRRWLQJVZKHUHWKHWKLFNQHVVRIWKHZDOO

is substituted for

5HTXLUHPHQWVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQ LVGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RULQFRQMXQFWLRQZLWKWKH development length, PRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKH ODWWHUUHTXLUHPHQWVDUHFRYHUHG¿UVW Method 1 – ACI 25.4.2.4 The development length in tension of a deformed reinforcing bar,

LVGHWHUPLQHGE\$&,(TXDWLRQD  (10.22)

10-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHWHUPVLQ(TXDWLRQ DUHDVIROORZV$&,7DEOH  • 0  RGL¿FDWLRQIDFWRUIRUOLJKWZHLJKWFRQFUHWH concrete:

7KLVIDFWRUUHÀHFWVWKHORZHUWHQVLOHVWUHQJWKRIOLJKWZHLJKW

(10.23)

• 5  HLQIRUFHPHQWJUDGHIDFWRU quired development length:

. This factor accounts for the effect of reinforcement yield strength on the re-

(10.24)

• 5  HLQIRUFHPHQWFRDWLQJIDFWRU . This factor accounts for the reduced bond strength between the concrete and HSR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFLQJEDUV

(10.25)

• 5  HLQIRUFHPHQWVL]HIDFWRU bars:

7KLVIDFWRUUHÀHFWVWKHPRUHIDYRUDEOHSHUIRUPDQFHRIVPDOOHUGLDPHWHUUHLQIRUFLQJ

(10.26)

• &  DVWLQJSRVLWLRQIDFWRU 7KLVIDFWRUUHÀHFWVWKHDGYHUVHHIIHFWVWKDWFDQRFFXUWRWKHWRSKRUL]RQWDOUHLQIRUFHment in a member due to vertical migration of water and mortar, which collect on the underside of the bars during placement of the concrete: (10.27)

$FFRUGLQJWRWKHIRRWQRWHLQ$&,7DEOH • Spacing or cover dimension,

QHHGQRWH[FHHG

7KLVWHUPLVGH¿QHGDVIROORZV (10.28)

10-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• 7  UDQVYHUVHUHLQIRUFHPHQWLQGH[ 7KHWUDQVYHUVHUHLQIRUFHPHQWLQGH[UHSUHVHQWVWKHUROHRIFRQ¿QLQJUHLQIRUFHPHQWDFURVVWKHSRWHQWLDOVSOLWWLQJSODQHV7KLVLQGH[LVGHWHUPLQHGE\$&,(TXDWLRQE  (10.29)

where

total cross-sectional area of all transverse reinforcement within a spacing potential plane of splitting through the reinforcement being developed

crossing the

center-to-center spacing of the transverse reinforcement number of bars being developed across the plane of splitting It is permitted to conservatively use if transverse reinforcement is present or required. For footings, LVW\SLFDOO\]HUREHFDXVHWUDQVYHUVHUHLQIRUFHPHQWLVQRWXVHGLQVXFKPHPEHUV 7KHFRQ¿QLQJWHUP PXVWEHWDNHQOHVVWKDQRUHTXDOWRLQ(TXDWLRQ >$&,@ :KHQWKLVWHUPLVOHVVWKDQVSOLWWLQJIDLOXUHVDUHOLNHO\WRRFFXU$SXOORXWIDLOXUHRIWKHUHLQIRUFHPHQWLV PRUHOLNHO\ZKHQWKLVWHUPLVJUHDWHUWKDQVRDQLQFUHDVHLQWKHDQFKRUDJHFDSDFLW\GXHWRDQLQFUHDVHLQ FRYHURUDPRXQWRIFRQ¿QLQJUHLQIRUFHPHQWLVQRWOLNHO\ 7KHWHQVLRQGHYHORSPHQWOHQJWKLVSHUPLWWHGWREHUHGXFHGLQFDVHVZKHUHWKHÀH[XUDOUHLQIRUFHPHQWLVJUHDWHUWKDQ WKDWUHTXLUHGIURPDQDO\VLVH[FHSWIRUWKHVL[FDVHVLQ$&,$&, 7KHUHGXFWLRQIDFWRUDSSOLHG to LVHTXDOWRWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWGLYLGHGE\WKHSURYLGHGDUHDRIUHLQIRUFHPHQW Method 2 – ACI 25.4.2.3 7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP VHH7DEOH 

LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHTwo sets of spacing and cover cases are given in ACI Table

Table 10.3 Tension Development Length, ℓd, in Accordance with ACI 25.4.2.3 Spacing and Cover

Case 1

#6 and Smaller Bars

#7 and Larger Bars

Condition 1 1. Clear spacing of bars being developed or lap spliced  Clear cover 3. Stirrups or ties throughout not less than ACI 318 minimum &RQGLWLRQ 1. Clear spacing of bars being developed or lap spliced  Clear cover

&DVH

Other conditions

,Q&DVHWKHVSDFLQJFRYHUDQGFRQ¿QHPHQWRIWKHEDUVEHLQJGHYHORSHGPHHWWKHUHTXLUHPHQWVXQGHU&RQGLWLRQRU&RQGLWLRQ:KHQWKHUHTXLUHPHQWVRIHLWKHU&RQGLWLRQRU&RQGLWLRQDUHPHWLWLVDVVXPHG WKDWYDOXHLVVXEVWLWXWHGLQWR(TXDWLRQ ZKLFKUHVXOWVLQWKHHTXDWLRQVLQWKH¿UVWURZRI 7DEOH 10-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

&DVHLVDSSOLFDEOHZKHUHWKHUHTXLUHPHQWVLQHLWKHU&RQGLWLRQRU&RQGLWLRQLQ&DVHDUHQRWVDWLV¿HGLQWKLV Tension development lengths, , must be the greater of the values detercase, it is assumed PLQHGE\WKHHTXDWLRQVLQ7DEOHDQGLQ and ,QW\SLFDOIRRWLQJVWKHFOHDUVSDFLQJEHWZHHQWKHÀH[XUDOUHLQIRUFLQJEDUVLVXVXDOO\JUHDWHUWKDQRUHTXDOWR Therefore, can be calculated by the equations for Case 1 the clear cover is typically greater than or equal to LQ7DEOH7KHGHYHORSPHQWOHQJWKVFDOFXODWHGE\WKHVHHTXDWLRQVDUHXVXDOO\ORQJHUWKDQWKRVHFDOFXODWHGE\ (TXDWLRQ  10.3.7 Force Transfer at the Base of Supported Members Overview

9HUWLFDODQGKRUL]RQWDOIRUFHVPXVWEHWUDQVIHUUHGDWWKHLQWHUIDFHEHWZHHQWKHVXSSRUWHGPHPEHUDQGDIRRWLQJ$&,  5HTXLUHPHQWVIRUIRUFHWUDQVIHUIURPDFROXPQZDOORUSHGHVWDOWRDIRRWLQJDUHJLYHQLQ$&, Vertical compressive forces are transferred by bearing on the concrete or by a combination of bearing and interface reinforcement. Tensile forces must be transferred entirely by reinforcement, which may consist of extended longitudinal bars, dowels, anchor bolts, or mechanical connectors. /DWHUDOIRUFHVDUHWUDQVIHUUHGXVLQJWKHVKHDUIULFWLRQSURYLVLRQVRI$&,RUDQ\RWKHUDSSURSULDWHPHWKRGV Vertical Transfer – Compression

:KHUHYHUWLFDOFRPSUHVVLYHIRUFHVDUHWUDQVIHUUHGWRDIRRWLQJEHDULQJVWUHQJWKUHTXLUHPHQWVRI$&,PXVWEH VDWLV¿HGIRUERWKWKHVXSSRUWHGPHPEHUDQGWKHIRRWLQJ$&, )RUEHDULQJRQWKHVXSSRUWHGPHPEHUWKH $&,7DEOH  factored bearing force must be less than or equal to the design bearing strength, (10.30)

ʹ

ͳ

‫ܮ‬

ܿଶ  ܾ

ܿଵ 

‫ܤ‬

Ͷͷ‫ ל‬

‫ܣ‬ଵ ൌ ܿଵ ܿଶ 

ʹ

݄

ͳ

‫ܣ‬ଶ ൌ ܾ‫ܮ‬

‫ܮ‬

 Figure 10.16 Determination of areas A1 and A2.

10-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In this equation, the strength reduction factor IRUEHDULQJLVHTXDOWR$&,7DEOH DQG the column, wall, or pedestal supported by the footing.

is the area of

)RUEHDULQJRQDIRRWLQJWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG (10.31)

The term LVGH¿QHGDVWKHDUHDRIWKHORZHUEDVHRIWKHODUJHVWIUXVWUXPRIDS\UDPLGFRQHRUWDSHUHGZHGJH and having side slopes of 1 contained wholly within the footing and having for its upper base the loaded area YHUWLFDOWRKRUL]RQWDOVHH)LJXUH  the excess compressive stress from the supported member must be transferred by reinIn cases where forcement to the footing. The required area of interface reinforcement can be calculated by the following equation: (10.32)

A minimum area of reinforcement, must be provided across the interface even where RequireDUHGHWHUPLQHGRQWKHEDVLVRIWKHW\SHRIPHPEHUVXSSRUWHGE\WKHIRRWLQJ$&,  ments for Cast-in-place columns or pedestal supported on footings. The minimum area of interface reinforcement in this FDVHLVHTXDOWRSHUFHQWRIWKHJURVVDUHDRIWKHVXSSRUWHGPHPEHU$&, 7KHLQWHUIDFHUHLQIRUFHPHQW can consist of (a) extended longitudinal reinforcing bars from the column or wall into the footing or (b) dowels emanating from the footing. For ease of construction, dowel bars are the type of interface reinforcement commonly used. Cast-in-place walls supported on footings. The minimum area of interface reinforcement is equal to the minimum ORQJLWXGLQDOZDOOUHLQIRUFHPHQWJLYHQLQ$&,7DEOH$&, )RU

(10.33)

,OOXVWUDWHGLQ)LJXUHRIWKLVSXEOLFDWLRQDUHGRZHOEDUVDFURVVWKHLQWHUIDFHEHWZHHQDUHLQIRUFHGFRQFUHWHFROumn and footing for the case where all of the longitudinal bars in the column are in compression. The dowel bars are set in the footing prior to casting the footing concrete and are subsequently spliced to the column or wall longitudinal bars. Dowels should be positioned so as not to interfere with the longitudinal bars or the tie hooks from the column. Where all the longitudinal bars of the supported member are in compression for all the factored load combinations, determined in accorthe dowel bars must extend into the footing at least a compression development length, GDQFHZLWK$&,

(10.34)

The terms

10-18

VHH7DEOH 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 10.4 Modification Factors for Deformed Bars in Compression Modification Factor

Lightweight,

&RQ¿QLQJUHLQIRUFHPHQW

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



Longitudinal reinforcement enclosed in the following: 1. A spiral  A circular continuously wound tie with and a pitch 3. WLHVLQDFFRUGDQFHZLWK$&, on center spaced  +RRSVLQDFFRUGDQFHZLWK$&, on center spaced





Other

Standard hooks are typically provided at the ends of the dowel bars, which are tied to the reinforcing bars in the footing; this detail is far more economical than terminating the dowel bars above the footing reinforcing bars. The hooked portion of the dowels cannot be considered effective for developing the dowels bars in compression (ACI  7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKH footing: (10.35)

In this equation, is the radius of the dowel bar bend; minimum inside bend diameters for bars with standard and are the diameters of the dowel bars and footing GHJUHHKRRNVDUHJLYHQLQ$&,7DEOH$OVR EDUVUHVSHFWLYHO\,QFDVHVZKHUH(TXDWLRQ LVQRWVDWLV¿HGHLWKHUDJUHDWHUQXPEHURIVPDOOHUGRZHOEDUVFDQ be used or the depth of the footing must be increased. )RUZDOOIRRWLQJVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHG into the footing: (10.36)

The dowel bars must also be fully developed in the supported member and are typically lap spliced to the longituGLQDOEDUV:KHUHWKHGRZHOEDUVDUHWKHVDPHVL]HDVWKHORQJLWXGLQDOEDUVWKHPLQLPXPFRPSUHVVLRQODSVSOLFH OHQJWKLVHTXDOWRWKHIROORZLQJ>$&,D @

(10.37)

The lap splice length must be increased by one-third where

$&, 

:KHUHWKHGRZHOEDUVDUHGLIIHUHQWLQVL]HWKDQWKHORQJLWXGLQDOEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWK JUHDWHUWKDQRUHTXDOWRWKHORQJHURIWKHIROORZLQJ$&, 

must be

10-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1. The development length in compression, RIWKHODUJHUEDUGHWHUPLQHGLQDFFRUGDQFHZLWK$&, RIWKHVPDOOHUEDUGHWHUPLQHGLQDFFRUGDQFHZLWK$&,  The compression lap splice length, 5HTXLUHPHQWVWKDWSHUPLWDQGFROXPQEDUVLQFRPSUHVVLRQWREHODSVSOLFHGWRDQGVPDOOHUGRZHOEDUV DUHJLYHQLQ$&, Vertical Transfer – Tension

Tensile forces transferred from a supported member to a footing must be resisted entirely by reinforcement across the LQWHUIDFH>$&,E DQG@DQGGRZHOEDUVPXVWEHSURYLGHGIRUDOOWKHEDUVLQWKHVXSSRUWHGPHPEHU 7HQVLOHDQFKRUDJHRIWKHGRZHOEDUVLQWRWKHIRRWLQJLVW\SLFDOO\DFFRPSOLVKHGE\SURYLGLQJGHJUHHVWDQGDUG determined in accordance hooks at the ends of the dowel bars with the development length of the hooked bar, ZLWK$&,VHH)LJXUHRIWKLVSXEOLFDWLRQIRUWKHFDVHRIDFROXPQVXSSRUWHGE\DIRRWLQJ 

(10.38)

7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRNVHH)LJXUHRIWKLV SXEOLFDWLRQ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHVHH7DEOH  Table 10.5 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



Lightweight,

Epoxy, &RQ¿QLQJ reinforcement,

Location,

Concrete strength,

)RUDQGVPDOOHUEDUVZLWK

or



Other

1.6

)RUDQGVPDOOHUKRRNHGEDUV 1. terminating inside a column core with side cover normal to the plane of the hook or  with side cover normal to the plane of the hook



Other





7KHFRQ¿QLQJUHLQIRUFHPHQWIDFWRU LVW\SLFDOO\HTXDOWRIRUKRRNHGGRZHOEDUVLQIRRWLQJVEHFDXVHFRQ¿Qing reinforcement is usually not provided in such members.

10-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKHIRRWLQJ (considering that the hooked portion of the dowel bars can be developed in tension): (10.39)

Where Equation LVQRWVDWLV¿HGWKHGHSWKRIWKHIRRWLQJXVXDOO\PXVWEHLQFUHDVHG For development of the dowel bars into the supported member, a tension lap splice or a mechanical connection in DFFRUGDQFHZLWK$&,RUUHVSHFWLYHO\PXVWEHSURYLGHGEHWZHHQWKHGRZHOEDUVDQGWKHORQJLWXGLQDO bars. In the case of lap splices, a Class B tension lap splice is required. Horizontal Transfer

7KHVKHDUIULFWLRQPHWKRGRI$&,LVSHUPLWWHGWREHXVHGWRGHWHUPLQHWKHQRPLQDOVKHDUVWUHQJWK at the FRQWDFWVXUIDFHEHWZHHQWKHVXSSRUWHGPHPEHUDQGWKHIRXQGDWLRQ$&, 7KHUHTXLUHGDUHDRIUHLQIRUFHacross the interface between the supported member and the footing is determined by the following equament, WLRQZKLFKLVDSSOLFDEOHWRVKHDUIULFWLRQUHLQIRUFHPHQWSHUSHQGLFXODUWRWKHLQWHUIDFH$&,  (10.40)

In this equation, is the factored shear force due to the lateral force effects at the interface, the strength reduction factor LVHTXDOWRDQG LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP$&,7DEOHVHH7DEOH  Table 10.6 Coefficient of Friction, μ

ȝ

Contact Surface Condition

Concrete placed monolithically Concrete placed against hardened concrete that is clean, free of laitance, and intentionally roughened to a full amplitude of approximately ¼ in. Concrete placed against hardened concrete that is clean, free of laitance, and not intentionally roughened 8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOHVHH7DEOH 7KHWHUP is the area of For example, where a column is supported by a footing, is equal to the gross area of the concrete resisting column. Where the concrete strengths of the supported member and the footing are different, the smaller of the two PXVWEHXVHGLQWKHVHHTXDWLRQV$&,  Table 10.7 Maximum Vn Across the Assumed Shear Plane Contact Surface Condition

Maximum Vn

= Vu / Ҁ*

Normalweight concrete placed monolithically or placed against hardened concrete intentionally roughened to a full amplitude of approximately ¼ in.

Other cases

*

area of concrete resisting

10-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The required amount of shear friction reinforcement, LVSHUPLWWHGWREHGHWHUPLQHGE\(TXDWLRQ EDVHG where is a permanent net compression force transmitted across the on a net force equal to DVVXPHGVKHDUSODQH$&, $FFRUGLQJWR$&,WKHDUHDRIUHLQIRUFHPHQWUHTXLUHGWRUHVLVWDQHW factored tension force across the assumed shear plane is to be added to the area of reinforcement required for shear friction that crosses the assumed shear plane. For supported members transmitting a bending moment across the VKHDUSODQHWKHÀH[XUDOFRPSUHVVLRQDQGWHQVLRQIRUFHVDUHLQHTXLOLEULXPDQGGRQRWFKDQJHWKHUHVXOWDQWFRPSUHVacting across the shear plane or the shear-friction resistance. Thus, it is not necessary to provide sion force, DGGLWLRQDOUHLQIRUFHPHQWWRUHVLVWWKHÀH[XUDOWHQVLRQVWUHVVHVXQOHVVWKHUHTXLUHGÀH[XUDOWHQVLRQUHLQIRUFHPHQW H[FHHGVWKHDPRXQWRIVKHDUWUDQVIHUUHLQIRUFHPHQWSURYLGHGLQWKHÀH[XUDOWHQVLRQ]RQH

Determine the base area of the footing ሺSect. 10.3.2ሻ

A

Figure 10.17 Design procedure for footings.

10-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

Determine the thickness of the footing ሺSect. 10.3.3ሻ

Detail and develop the flexural reinforcement ሺSects. 10.3.5 and 10.3.6ሻ

Determine the force transfer reinforcement ሺSect. 10.3.7ሻ

Figure 10.17 Design procedure for footings (cont.).

10-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Full tension anchorage of the shear-friction reinforcement must be provided into the footing and into the supported member. The lengths of these bars are determined in the same way as those for vertical transfer where tension forces are present. The area of the dowel bars is usually determined initially based on the requirements for vertical transfer. That area is FRPSDUHGZLWKWKHDUHDUHTXLUHGIRUKRUL]RQWDOWUDQVIHUDQGWKHODUJHURIWKHWZRLVSURYLGHGDWWKHLQWHUIDFH 10.3.8 Design Procedure

7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHIRRWLQJV,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQLQWKLVFKDSWHUFDQEH found.

10.4 Drilled Piers 10.4.1 Overview

A drilled pier, which is also referred to as a pier or caisson, is a type of deep foundations that transfers the loads from the superstructure to a soil or rock stratum usually well below the ground surface. The bottom of the pier may EHEHOOHGWRSURYLGHDODUJHUHQGEHDULQJDUHDVHH)LJXUH  ‘Ž—ǡ™ƒŽŽ‘” ‰”ƒ†‡„‡ƒ

‹‡”

‡ŽŽ

Figure 10.18 Typical drilled pier with a bell.

The loads from the supported members are transferred to the pier by bearing. Skin resistance (or, friction), point bearing, or a combination of the two are ways in which the load is transferred to the soil surrounding and below the SLHURUEHOO'ULOOHGSLHUVDUHFDWHJRUL]HGEDVHGRQWKHPDQQHULQZKLFKWKHORDGVDUHWUDQVIHUUHGWRWKHVRLORUURFN 7KHGLVFXVVLRQEHORZIRFXVHVRQGULOOHGSLHUVVXEMHFWHGSULPDULO\WRD[LDOIRUFHVWKDWWUDQVIHUWKHIRUFHVWRWKHVRLO rock by end bearing.

10-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

10.4.2 Design Methods Overview

7KHGHVLJQRIGULOOHGSLHUVLVSHUPLWWHGWREHLQDFFRUGDQFHZLWKWKHDOORZDEOHD[LDOVWUHQJWKPHWKRGLQ$&, IRUPHPEHUVVXEMHFWHGSULPDULO\WRD[LDOIRUFHVRUWKHVWUHQJWKGHVLJQPHWKRGLQ$&,$&, )RU GULOOHGSLHUVZLWKRXWEHOOVWKHGLDPHWHURIWKHSLHULVGHWHUPLQHGEDVHGRQ $&,RURU WKHDOORZDEOHEHDULQJFDSDFLW\RIWKHVRLOURFN)RUGULOOHGSLHUVZLWKEHOOVWKHGLDPHWHURIWKHSLHULVGHWHUPLQHGEDVHGRQ $&,RUZKLOHWKHGLDPHWHURIWKHEHOOLVEDVHGRQWKHDOORZDEOHEHDULQJFDSDFLW\RIWKHVRLOURFN Allowable Axial Strength

'ULOOHGSLHUVPD\EHGHVLJQHGXVLQJWKHORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,DQGWKH DOORZDEOHFRPSUHVVLYHVWUHQJWKVJLYHQLQ$&,7DEOHVHH7DEOH SURYLGHGWKHIROORZLQJFRQGLWLRQVDUH VDWLV¿HG • The drilled pier is laterally supported its entire length. • The applied forces cause bending moments in the drilled pier less than the moment due to an accidental where is the diameter HFFHQWULFLW\RISHUFHQWRIWKHPHPEHUGLDPHWHUWKDWLV of the pier). Drilled piers not meeting one or both of these conditions must be designed using the strength design method in ACI $&,  Table 10.8 Maximum Allowable Compressive Strength for Drilled Piers Type of Drilled Pier

Maximum Allowable Compressive Strength, Pa

Uncased cast-in-place concrete drilled pier Cast-in-place concrete drilled pier in rock or within a pipe, tube, or other permanent metal casing that does not VDWLVI\WKHFRQGLWLRQVLQ$&, 0HWDOFDVHGFDVWLQSODFHFRQFUHWHGULOOHGSLHUFRQ¿QHG LQDFFRUGDQFHZLWK$&, ,Q7DEOH is the gross cross-sectional area of the concrete in the pier. Where temporary or permanent casing is the area of the longitudinal is used, the inside face of the casing is considered to be the concrete surface. Also, reinforcement in the drilled pier; it must not include the area of any casing, pipes, or tubes present. ,WLVHYLGHQWIURP7DEOHWKDWWKHDOORZDEOHFRPSUHVVLYHVWUHQJWKGHSHQGVRQZKHWKHUWKHGULOOHGSLHUKDVPHWDO FDVLQJRUQRWDQGLIPHWDOFDVLQJLVSURYLGHGZKHWKHUWKHSLHUFDQEHFRQVLGHUHGWREHFRQ¿QHGRUQRWE\WKHFDVLQJ LQDFFRUGDQFHZLWK$&, Strength Design

7KHVWUHQJWKGHVLJQSURYLVLRQVLQ$&,FDQEHXVHGWRGHVLJQDQ\GULOOHGSLHU$&, )RUGULOOHGSLHUV subjected to axial compression loads only, the pier is designed using the nominal strengths for columns, which are JLYHQLQ$&,DQGWKHVWUHQJWKUHGXFWLRQIDFWRUVLQ$&,7DEOHVHH7DEOH )RUGULOOHGSLHUVVXEMHFWHGWRFRPELQHGD[LDOIRUFHDQGPRPHQWWHQVLRQDQGVKHDUWKHVWUHQJWKUHGXFWLRQIDFWRUVLQ$&,7DEOH must be used instead. The design strengths must be less than or equal to the required strengths determined by the DSSURSULDWHIDFWRUHGORDGFRPELQDWLRQVLQ$&,

10-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 10.9 Compressive Strength Reduction Factors Ҁ for Drilled Piers Type of Drilled Pier

Compressive Strength Reduction Factor, Ҁ

Uncased cast-in-place concrete drilled pier



Cast-in-place concrete drilled pier in rock or within a pipe, tube, or other permanent metal casing that does not VDWLVI\WKHFRQGLWLRQVLQ$&,



0HWDOFDVHGFDVWLQSODFHFRQFUHWHGULOOHGSLHUFRQ¿QHG LQDFFRUGDQFHZLWK$&,



$FFRUGLQJWR$&,WKHSURYLVLRQVRI$&,DQGSHUWDLQLQJWRWLHUHLQIRUFHPHQWDQGVSLUDOUHinforcement for lateral support of longitudinal reinforcement in compression members, respectively, need not apply to the lateral reinforcement in drilled piers. Longitudinal reinforcement must be provided in a drilled pier subjected to uplift or where the factored maximum is greater than where is the cracking moment determined by ACI Equation bending moment,  >$&,@ (10.41)

The term is the distance from the centroidal axis of the gross section, neglecting longitudinal reinforcement, to the tension face of the member and the modulus of rupture, LVGHWHUPLQHGE\$&,(TXDWLRQ  (10.42)

7KHUHTXLUHPHQWIRUORQJLWXGLQDOUHLQIRUFHPHQWQHHGQRWEHVDWLV¿HGLIWKHGULOOHGSLHULVHQFORVHGE\DVWUXFWXUDO steel pipe or tube. Where soil cannot provide lateral resistance or where a pier extends above the soil surface or through subsurface OD\HUVRIDLURUZDWHUWKHSLHUPXVWEHGHVLJQHGDVDFROXPQXVLQJWKHDSSOLFDEOHSURYLVLRQVLQ$&,&KDSWHU$&,  7KLVLQFOXGHVFKHFNLQJZKHWKHUWKHHIIHFWVRIVOHQGHUQHVVPXVWEHFRQVLGHUHGRUQRW 10.4.3 Determining the Pier Size Allowable Axial Strength

For drilled piers designed using the allowable axial strength method, the diameter of the pier, can be deterfrom the applicable load combinations for allowable stress mined by setting the maximum governing axial load, GHVLJQLQ$6&(6(,HTXDOWRWKHDSSURSULDWHDOORZDEOHFRPSUHVVLYHVWUHQJWK LQ7DEOHDQGVROYLQJIRU which is equal to 7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQH LQWKH¿UVWWZRURZVRIWKHWDEOH

10-26

where it is assumed

for the drilled piers

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 10.10 Required Pier Diameter Based on Allowable Axial Strength Required Pier Diameter, dpier

Type of Drilled Pier

Uncased cast-in-place concrete drilled pier* Cast-in-place concrete drilled pier in rock or within a pipe, tube, or other permanent metal casing that does not satisfy the conditions in ACI  0HWDOFDVHGFDVWLQSODFHFRQFUHWHGULOOHGSLHUFRQ¿QHGLQDFFRUGDQFHZLWK $&, *It is assumed

for this type of drilled pier.

Strength Design

For drilled piers subjected to axial compression loads only, can be determined by setting the maximum govIURPWKHDSSOLFDEOHORDGFRPELQDWLRQVLQ$&,HTXDOWRWKHGHVLJQD[LDOORDGVWUHQJWK erning axial load, JLYHQLQ$&,7DEOHXVLQJWKHDSSURSULDWHVWUHQJWKUHGXFWLRQIDFWRULQ7DEOH The following equation can be used to determine reinforcement in the pier:

where

is the assumed percentage of longitudinal

(10.43)

Initially,

FDQEHWDNHQDVDQGFDQEHVXEVHTXHQWO\DGMXVWHGLIUHTXLUHG

:KHUHVLJQL¿FDQWEHQGLQJPRPHQWVDQGVKHDUIRUFHVDUHWUDQVIHUUHGLQFRPELQDWLRQZLWKD[LDOORDGVWKHSLHULV typically designed considering the lateral restraint provided by the soil (which is usually modeled as spring supSRUWV $SSUR[LPDWHPHWKRGVRIDQDO\VLVDUHJLYHQLQ5HIHUHQFHLQFOXGHGLVDSURFHGXUHWRFDOFXODWHWKHODWHUDO GHÀHFWLRQDWWKHWRSRIWKHSLHU 10.4.4 Determining the Bell Diameter

For the case of end-bearing drilled piers, the diameter of the bell, and the allowable bearing capacity of the soil or rock, load,

can be determined using the service axial

(10.44)

The height of the bell depends on the diameter of the pier and the slope of the bell. The thickness of the lower porWLRQRIWKHEHOOWKDWLVQRWVORSHGLVW\SLFDOO\IRRW7KHDQJOHRIWKHVORSHGSRUWLRQLVXVXDOO\GHJUHHVRUPRUHVR the effects of vertical shear do not need to be considered in the design. ,QOLHXRIXVLQJVHUYLFHOHYHOORDGVIDFWRUHGD[LDOIRUFHVDQGQRPLQDOVRLOURFNEHDULQJVWUHQJWKVDQGVWUHQJWKUHGXFtion factors can be used to determine the diameter of a bell. Tables that facilitate the selection of pier and bell diameters for drilled piers subjected to axial compression and FRPELQHGD[LDOFRPSUHVVLRQDQGÀH[XUHDUHJLYHQLQ5HIHUHQFH

10-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

10.4.5 Reinforcement Details

Recommended reinforcement details for drilled piers subjected to primarily axial compression loads are given LQ)LJXUH7KHUHFRPPHQGHGORQJLWXGLQDOUHLQIRUFHPHQWLQWKH¿JXUHLVEDVHGRQDPLQLPXPORQJLWXGLQDO UHLQIRUFHPHQWUDWLRRIZKLFKFRUUHVSRQGVWRFROXPQVZLWKFURVVVHFWLRQVODUJHUWKDQUHTXLUHGIRUWKHDSSOLHG or ORDGV$&, 7KHPLQLPXPHPEHGPHQWOHQJWKRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLVWKHORQJHURI IW ݀௣௜௘௥  ‫ܣ‬௦ ൒ ͲǤͲͲͷ‫ܣ‬௚ 

൒ ͵݀௣௜௘௥ ǡ ͳͲԢǦͲԣ

‘™‡Ž•

‹‡•

Ž‡ƒ”…‘˜‡”൒ ͵ԣሺ–›’Ǥ ሻ

ͳԢǦͲԣ

‡ŽŽ‡‹‰Š–

‡‰–Š൑ ͵Ͳ݀௣௜௘௥ 

‹‡Žƒ’•’Ž‹…‡Ž‡‰–Šൌ Ͷͺ݀௧௜௘ 

൒ ͸Ͳι

‹‡” ‘‰‹–—†‹ƒŽ †‹ƒ‡–‡” ”‡‹ˆ‘”…‡‡– ͳᇱ Ǧ͸ԣ ͸Ǧ͓ͷ ʹᇱ ǦͲԣ ͸Ǧ͓͸ ʹᇱ Ǧ͸ԣ ͸Ǧ͓͹ ͵ᇱ ǦͲԣ ͹Ǧ͓ͺ ͵ᇱ Ǧ͸ԣ ͹Ǧ͓ͻ Ͷᇱ ǦͲԣ ͺǦ͓ͳͲ Ͷᇱ Ǧ͸ԣ ͻǦ͓ͳͲ ͷᇱ ǦͲԣ ͳͲǦ͓ͳͳ ͷᇱ Ǧ͸ԣ ͳͳǦ͓ͳͳ ͸ᇱ ǦͲԣ ͳͶǦ͓ͳͳ ͸ᇱ Ǧ͸ԣ ͳ͸Ǧ͓ͳͳ ͹ᇱ ǦͲԣ ͳͺǦ͓ͳͳ

‹‡• ͓͵̷ͳͲԣ ͓͵̷ͳʹԣ ͓͵̷ͳͶԣ ͓͵̷ͳ͸ԣ ͓͵̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ ͓Ͷ̷ͳͺԣ

݀௕௘௟௟  Figure 10.19 Recommended reinforcement details for drilled piers subjected to axial compression loads.

$VQRWHGLQ6HFWLRQORQJLWXGLQDOUHLQIRUFHPHQWPXVWEHSURYLGHGWRUHVLVWWKHWHQVLOHIRUFHVLQGULOOHGSLHUV unless the VXEMHFWHGWRXSOLIWRUWRDIDFWRUHGEHQGLQJPRPHQWH[FHHGLQJSHUFHQWRIWKHFUDFNLQJPRPHQW GULOOHGSLHULVHQFORVHGE\DVWUXFWXUDOVWHHOSLSHRUWXEH7KHORQJLWXGLQDOEDUVPXVWH[WHQGLQWRWKHSLHUDVXI¿FLHQW distance to fully develop the bars in tension. 9HUWLFDODQGKRUL]RQWDOORDGVPXVWEHWUDQVIHUUHGIURPWKHVXSSRUWHGPHPEHUWRWKHWRSRIWKHSLHU7KHIRUFHWUDQVIHUUHTXLUHPHQWVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQDUHDSSOLFDEOHWRGULOOHGSLHUV

10-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

10.5 Examples 10.5.1 Example 10.1 – Design of a Wall Footing Subjected to Axial Compression: Building #2

'HVLJQDIRRWLQJIRUWKHLQWKLFNLQWHULRUUHLQIRUFHGFRQFUHWHZDOORI%XLOGLQJLQ([DPSOHVHH)LJXUH  7KHZDOOLVUHLQIRUFHGZLWKRQHOD\HURIORQJLWXGLQDOEDUVVSDFHGDWLQRQFHQWHU$VVXPH*UDGHUHLQIRUFHfor the footing and for the wall. Also assume a ment and normalweight concrete with  SHUPLVVLEOHVRLOSUHVVXUHRIOEIW . 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the width of the footing

From Example 8.1, the axial service dead and live loads are equal to the following:

8VHWKHEDVLFORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,WRGHWHUPLQHWKHPD[LPXPD[LDO force transferred to the footing: 1.  Assuming the axial forces act at the centroid of the wall, the required area of the footing is equal to the following: Eq. (10.1)

$IWLQZLGHZDOOIRRWLQJLVDGHTXDWHIRUVRLOEHDULQJFDSDFLW\+RZHYHULWLVXQOLNHO\WKDWZLGWKLVDGHTXDWHWR GHYHORSWKHÀH[XUDOUHLQIRUFHPHQW7KHUHIRUHWU\DIWZLGHIRRWLQJ Step 2 – Determine the thickness of the footing

8VHWKHORDGFRPELQDWLRQVLQ$&,7DEOHWRGHWHUPLQHWKHWKLFNQHVVRIWKHIRRWLQJ ACI Eq. (5.3.1a) ACI Eq. (5.3.1b)

The maximum factored pressure at the base of the footing is equal to the following:

0LQLPXPHIIHFWLYHGHSWKIRUÀH[XUH Eq. (10.15)

10-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Minimum effective depth for one-way shear:

Eq. (10.18)

7KHPLQLPXPHIIHFWLYHGHSWKIRUÀH[XUHJRYHUQVLQWKLVFDVH+RZHYHUDPLQLPXPGHSWKRILQLVUHTXLUHGDERYH WKHERWWRPUHLQIRUFHPHQWRIWKHIRRWLQJ$&, 7KHUHIRUHDWOHDVWDLQWKLFNIRRWLQJLVUHTXLUHG,WLV XQOLNHO\DLQIRRWLQJWKLFNQHVVLVDGHTXDWHWRSURSHUO\GHYHORSWKHGRZHOEDUVIURPWKHIRRWLQJWRWKHZDOO Try a 16-in.-thick footing Step 3 – Determine the required flexural reinforcement

%HFDXVHWKHSURYLGHGHIIHFWLYHGHSWKRILQLVJUHDWHUWKDQWKHLQPLQLPXPHIIHFWLYHGHSWKFDOFXODWHGIRU ÀH[XUHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWH ACI 7.6.1.1

7U\EDUVVSDFHGDWLQRQFHQWHU $&,  ing of

7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDF-

Step 4 – Check if the flexural reinforcement can be developed for tension

The tension development length,

RIWKHEDUVPXVWEHOHVVWKDQRUHTXDOWRWKHDYDLODEOHGHYHORSPHQWOHQJWK Eq. (10.21)

Determine

IURP$&,(TD  Eq. (10.22)

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26)

)RUOHVVWKDQLQRIFRQFUHWHSODFHGEHORZWKHÀH[XUDOUHLQIRUFHPHQW

Eq. (10.27)

Eq. (10.28)

Because there is no transverse reinforcement,

10-30

Eq. (10.29)

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

%HFDXVHWKHDYDLODEOHGHYHORSPHQWOHQJWKLVJUHDWHUWKDQWKHWHQVLRQGHYHORSPHQWOHQJWKWKHEDUVFDQEHIXOO\ GHYHORSHGIRUÀH[XUH Step 5 – Determine the shrinkage and temperature reinforcement

ACI 24.4.3.2

3URYLGHEDUV XQLIRUPO\VSDFHGZLWKLQWKHLQZLGWKWKLVUHLQIRUFHPHQWLVSHUSHQGLFXODUWRWKHPDLQÀH[XUDOUHLQIRUFHPHQW Step 6 – Determine the required dowel reinforcement

Only vertical forces need to be considered for force transfer between the wall and footing. • Check the bearing force on the wall: Eq. (10.30)

• Check the bearing force on the footing: Eq. (10.31)

8VLQJ)LJXUH

is determined as follows:

Thickness of footing +RUL]RQWDOSURMHFWLRQIRUDVORSH Projected length Therefore,

• Determine the required interface reinforcement Because the design bearing strength is adequate, provide the minimum area of reinforcement across the interface: Eq. (10.33)

7U\EDUVVSDFHGDWLQRQFHQWHU bars in the wall.

ZKLFKPDWFKHVWKHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDO

• Development of the dowel bars into the footing The dowel bars must extend into the footing at least a compression development length,

10-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (10.34)

where

IRUUHLQIRUFLQJEDUVZLWKRXWFRQ¿QHPHQW

Table 10.4

The minimum footing thickness for the development of the dowel bars in compression is the following: Eq. (10.36)

The provided footing thickness is equal to the required footing thickness, so the dowel bars can be fully developed for compression into the footing. • Development of the dowel bars into the wall The dowel bars are lap spliced to the longitudinal bars in the wall. The dowel bars and the longitudinal bars are LVHTXDOWRWKHIROORZLQJIRU*UDGHUHLQIRUFHPHQW WKHVDPHVL]HVRWKHFRPSUHVVLRQODSVSOLFHOHQJWK Eq. (10.37)

Provide a lap splice length equal to 1 ft-3 in. 5HLQIRUFHPHQWGHWDLOVIRUWKHZDOOIRRWLQJDUHJLYHQLQ)LJXUH

ͻԣ

͓Ͷ̷ͳͺԣ

–ƒ†ƒ”† Š‘‘

ͷǦ͓ͷ•Š”‹ƒ‰‡ƒ† –‡’‡”ƒ–—”‡”‡‹ˆ‘”…‡‡–

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͵ԣሺ–›’Ǥሻ

ͳԢǦͶԣ

ͳԢǦ͵ԣ

͓Ͷ†‘™‡Ž„ƒ”•̷ͳͺԣ

Figure 10.20 Reinforcement details for the wall footing in Example 10.1.

10-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

10.5.2 Example 10.2 – Design of a Square Isolated Spread Footing Subjected to Axial Compression: Building #1 (Framing Option B), SDC A

'HVLJQDVTXDUHLVRODWHGIRRWLQJIRUWKHLQVTXDUHLQWHULRUFROXPQRI%XLOGLQJ)UDPLQJ2SWLRQ%LQ([DPSOH VHH)LJXUH 7KHFROXPQLVUHLQIRUFHGZLWKEDUV$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKW IRUWKHFROXPQDQGWKHIRRWLQJ$OVRDVVXPHDSHUPLVVLEOHVRLOSUHVVXUHRIOEIW. concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the base dimensions of the footing

)URP([DPSOHWKHVHUYLFHD[LDOGHDGURRIOLYHDQGOLYHORDGVDUHHTXDOWRWKHIROORZLQJ

8VHWKHEDVLFORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,WRGHWHUPLQHWKHPD[LPXPD[LDO force transferred to the footing: 1.  3.  8VLQJ$6&(6(,(TWKHUHTXLUHGDUHDRIWKHIRRWLQJLVHTXDOWRWKHIROORZLQJ Eq. (10.1)

An 11 ft-6 in. square isolated footing is adequate for soil bearing capacity. Step 2 – Determine the thickness of the footing

8VHWKHORDGFRPELQDWLRQVLQ$&,7DEOHWRGHWHUPLQHWKHWKLFNQHVVRIWKHIRRWLQJ ACI Eq. (5.3.1a) ACI Eq. (5.3.1b) ACI Eq. (5.3.1c)

The maximum factored pressure at the base of the footing is equal to the following:

10-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• 0LQLPXPHIIHFWLYHGHSWKIRUÀH[XUH Eq. (10.15)

• Minimum effective depth for one-way shear:

Eq. (10.18)

• Minimum effective depth for two-way shear:

Eq. (10.20)

The minimum effective depth for one-way shear governs in this case. 7U\DLQWKLFNIRRWLQJ Step 3 – Determine the required flexural reinforcement

%HFDXVHWKHSURYLGHGHIIHFWLYHGHSWKRILQLVJUHDWHUWKDQWKHLQPLQLPXPHIIHFWLYHGHSWKFDOFXODWHGIRU ÀH[XUHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWH ACI 7.6.1.1

7U\EDUVVSDFHGDWLQRQFHQWHU ing of

7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDF-

Step 4 – Check if the flexural reinforcement can be developed for tension

The tension development length,

RIWKHEDUVPXVWEHOHVVWKDQRUHTXDOWRWKHDYDLODEOHGHYHORSPHQWOHQJWK Eq. (10.21)

Determine

IURP$&,(TD  Eq. (10.22)

10-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26)

)RUOHVVWKDQLQRIFRQFUHWHSODFHGEHORZWKHÀH[XUDOUHLQIRUFHPHQW

Eq. (10.27)

Eq. (10.28)

Because there is no transverse reinforcement,

Eq. (10.29)

Therefore,

%HFDXVHWKHDYDLODEOHGHYHORSPHQWOHQJWKLVJUHDWHUWKDQWKHWHQVLRQGHYHORSPHQWOHQJWKWKHEDUVFDQEHIXOO\ GHYHORSHGIRUÀH[XUH Step 5 – Determine the required dowel reinforcement

Only vertical forces need to be considered for force transfer between the column and footing. • Check the bearing force on the column: Eq. (10.30)

• Check the bearing force on the footing: Eq. (10.31)

8VLQJ)LJXUH

is determined as follows:

Thickness of footing +RUL]RQWDOSURMHFWLRQIRUDVORSH Projected length Therefore,

• Determine the required interface reinforcement Because the design bearing strength is adequate, provide the minimum area of reinforcement across the interface:

10-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Sect. 10.3.7

7U\GRZHOEDUV • Development of the dowel bars into the footing The dowel bars must extend into the footing at least a compression development length,

Eq. (10.34)

where

IRUUHLQIRUFLQJEDUVZLWKRXWFRQ¿QHPHQW

Table 10.4

The minimum footing thickness for the development of the dowel bars in compression is the following: Eq. (10.35)

The provided footing thickness is greater than the required footing thickness, so the dowel bars can be fully developed for compression into the footing. • Development of the dowel bars into the column The dowel bars are lap spliced to the longitudinal bars in the column. Because the dowel bars are smaller in must be greater than or equal to the diameter than the longitudinal bars, the compression lap splice length, larger of the following: Development length in compression,

RIWKHORQJLWXGLQDOEDUV

Eq. (10.34)

Compression lap splice length,

RIWKHGRZHOEDUV

Eq. (10.37)

3URYLGHDODSVSOLFHOHQJWKHTXDOWRIWLQ 5HLQIRUFHPHQWGHWDLOVIRUWKHIRRWLQJDUHJLYHQLQ)LJXUH

10-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳԢǦ͸ԣ ͳԢǦ͸ԣ



ʹԢǦͲԣ

ͶǦ͓ͻ

͓͹ˆ‘‘–‹‰„ƒ”•̷ͳͲԣ ͳͳԢǦ͸ԣ

͵ԣ…Ž‡ƒ” …‘˜‡”ሺ–›’Ǥሻ

ʹԢǦ͸ԣ

ͶǦ͓͸†‘™‡Ž„ƒ”•

Figure 10.21 Reinforcement details for the footing in Example 10.2.

10.5.3 Example 10.3 – Design of a Rectangular Isolated Spread Footing Subjected to Axial Compression: Building #1 (Framing Option B), SDC A

'HVLJQDUHFWDQJXODULVRODWHGIRRWLQJIRUWKHLQVTXDUHLQWHULRUFROXPQRI%XLOGLQJ)UDPLQJ2SWLRQ%LQ ([DPSOHVHH)LJXUH %HFDXVHRISURSHUW\OLQHUHVWULFWLRQVRQHGLPHQVLRQRIWKHIRRWLQJLVOLPLWHGWR IW7KHFROXPQLVUHLQIRUFHGZLWKEDUV$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKWFRQFUHWHZLWK IRUWKHFROXPQDQGWKHIRRWLQJ$OVRDVVXPHDSHUPLVVLEOHVRLOSUHVVXUHRIOEIW. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the base dimensions of the footing

)URP([DPSOHWKHVHUYLFHD[LDOGHDGURRIOLYHDQGOLYHORDGVDUHHTXDOWRWKHIROORZLQJ

10-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8VHWKHEDVLFORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,WRGHWHUPLQHWKHPD[LPXPD[LDO force transferred to the footing: 1.  3.  8VLQJ$6&(6(,(TWKHUHTXLUHGDUHDRIWKHIRRWLQJLVHTXDOWRWKHIROORZLQJ Eq. (10.1)

Given $QIWLQE\IWLQLVRODWHGIRRWLQJLVDGHTXDWHIRUVRLOEHDULQJFDSDFLW\ Step 2 – Determine the thickness of the footing

8VHWKHORDGFRPELQDWLRQVLQ$&,7DEOHWRGHWHUPLQHWKHWKLFNQHVVRIWKHIRRWLQJ ACI Eq. (5.3.1a) ACI Eq. (5.3.1b) ACI Eq. (5.3.1c)

The maximum factored pressure at the base of the footing is equal to the following:

The minimum effective depth of the footing is determined using the longest cantilever length, 0LQLPXPHIIHFWLYHGHSWKIRUÀH[XUH Eq. (10.15)

Minimum effective depth for one-way shear:

Eq. 10.18)

Minimum effective depth for two-way shear:

10-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (10.20)

The minimum effective depth for one-way shear governs in this case. 7U\DLQWKLFNIRRWLQJ Step 3 – Determine the required flexural reinforcement

%HFDXVHWKHSURYLGHGHIIHFWLYHGHSWKRILQLVJUHDWHUWKDQWKHLQPLQLPXPHIIHFWLYHGHSWKFDOFXODWHGIRU ÀH[XUHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWH ACI 7.6.1.1

5HLQIRUFHPHQWLQWKHORQJGLUHFWLRQLVXQLIRUPO\GLVWULEXWHGDFURVVWKHIWZLGWK7KXVWU\EDUVVSDFHGDW 7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDFLQJRI RULQ in. on center In the short direction, the portion of the total reinforcement, FHQWHUHGRQWKHFROXPQVHH)LJXUH  to

is uniformly distributed over a band width equal

PXVWEHGLVWULEXWHGZLWKLQDQIWZLGWK

Therefore, Within the

width, provide

$OWHUQDWLYHO\SURYLGHWKHIROORZLQJDPRXQWRIUHLQIRUFHPHQWZKLFKFDQEHXQLIRUPO\GLVWULEXWHGDFURVVWKHIW width: Sect. 10.3.5

7U\EDUVVSDFHGDWLQRQFHQWHU Step 4 – Check if the flexural reinforcement can be developed for tension

The tension development length,

RIWKHEDUVPXVWEHOHVVWKDQRUHTXDOWRWKHDYDLODEOHGHYHORSPHQWOHQJWK Eq. (10.21)

Determine

IURP$&,(TD 

Eq. (10.22)

For normalweight concrete,

Eq. (10.23)

10-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26)

)RUOHVVWKDQLQRIFRQFUHWHSODFHGEHORZWKHÀH[XUDOUHLQIRUFHPHQW

Eq. (10.27)

Eq. (10.28)

Because there is no transverse reinforcement,

Eq. (10.29)

Therefore,

%HFDXVHWKHDYDLODEOHGHYHORSPHQWOHQJWKLVJUHDWHUWKDQWKHWHQVLRQGHYHORSPHQWOHQJWKWKHEDUVFDQEHIXOO\ GHYHORSHGIRUÀH[XUH,WLVHYLGHQWWKDWWKHEDUVFDQEHIXOO\GHYHORSHGIRUWHQVLRQLQWKHSHUSHQGLFXODUGLUHFWLRQ as well. Step 5 – Determine the required dowel reinforcement

Only vertical forces need to be considered for force transfer between the column and footing. • Check the bearing force on the column: Eq. (10.30)

• Check the bearing force on the footing: Eq. (10.31)

8VLQJ)LJXUH

is determined as follows:

Thickness of footing +RUL]RQWDOSURMHFWLRQIRUDVORSH Projected length Therefore,

s • Determine the required interface reinforcement Because the design bearing strength is adequate, provide the minimum area of reinforcement across the interface:

10-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Sect. 10.3.7

7U\GRZHOEDUV • Development of the dowel bars into the footing The dowel bars must extend into the footing at least a compression development length,

Eq. (10.34)

where

IRUUHLQIRUFLQJEDUVZLWKRXWFRQ¿QHPHQW

Table 10.4

The minimum footing thickness for the development of the dowel bars in compression is the following: Eq. (10.35)

The provided footing thickness is greater than the required footing thickness, so the dowel bars can be fully developed for compression into the footing. • Development of the dowel bars into the column The dowel bars are lap spliced to the longitudinal bars in the column. Because the dowel bars are smaller in must be greater than or equal to the diameter than the longitudinal bars, the compression lap splice length, larger of the following: Development length in compression,

RIWKHORQJLWXGLQDOEDUV

Eq. (10.34)

Compression lap splice length,

RIWKHGRZHOEDUV

Eq. (10.37)

3URYLGHDODSVSOLFHOHQJWKHTXDOWRIWLQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVIRRWLQJDUHVLPLODUWRWKRVHLQ)LJXUH 10.5.4 Example 10.4 – Design of a Square Isolated Spread Footing Subjected to Axial Compression and Flexure: Building #1 (Framing Option C), SDC A

'HVLJQDVTXDUHLVRODWHGIRRWLQJIRUWKHLQVTXDUHHGJHFROXPQRI%XLOGLQJ)UDPLQJ2SWLRQ&LQ([DPSOH VHH)LJXUH 7KHFROXPQLVUHLQIRUFHGZLWKEDUV$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKW IRUWKHFROXPQDQGWKHIRRWLQJ$OVRDVVXPHDSHUPLVVLEOHVRLOSUHVVXUHRIOEIW. concrete with

10-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the base dimensions of the footing

7KHDOORZDEOHVWUHVVDQGVWUHQJWKGHVLJQORDGFRPELQDWLRQVDUHJLYHQLQ7DEOHVHH([DPSOH 7KHDOORZDEOHVWUHVVGHVLJQORDGFRPELQDWLRQVDUHIURP$6&(6(,7KHVWUHQJWKGHVLJQORDGFRPELQDWLRQVDUHIURP$&, 7DEOH7KHEHQGLQJPRPHQWLQ7DEOHGXHWRZLQGLVDWWKHERWWRPRIWKHFROXPQ Table 10.11 Summary of Axial Forces and Bending Moments for Column C1 Load Case

Axial Force (kips)

Bending Moment (ft-kips)

Dead





Roof live



ņ

Live





Wind



Allowable Stress Load Combinations $6&(6(,(T





$6&(6(,(T





$6&(6(,(T





$6&(6(,(T





$6&(6(,(T



$6&(6(,(T



$6&(6(,(T



Strength Design Load Combinations $&,(TD





$&,(TE









$&,(TF



$&,(TG



$&,(TI



'HWHUPLQHWKHUHTXLUHGEDVHGLPHQVLRQVRIWKHIRRWLQJEDVHGRQD[LDOIRUFHRQO\$6&(6(,(T DQGWKHQFKHFN if the maximum combined pressure due to axial force and bending moment is less than the permissible soil pressure. Eq. (10.1)

7U\DIWLQVTXDUHLVRODWHGIRRWLQJ 10-42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

&KHFNWKHPD[LPXPFRPELQHGSUHVVXUHEDVHGRQ$6&(6(,(T Pressure due to axial force

Pressure due to moment Total pressure &KHFNWKHPD[LPXPFRPELQHGSUHVVXUHEDVHGRQ$6&(6(,(T Pressure due to axial force

Pressure due to moment Total pressure $IWLQLVRODWHGVTXDUHIRRWLQJLVDGHTXDWHIRUVRLOEHDULQJFDSDFLW\ Note that there is no net uplift (tension) at the base of the footing for any of the allowable stress load combinations LQ7DEOH Step 2 – Determine the thickness of the footing

%DVHGRQWKHVWUHQJWKGHVLJQORDGFRPELQDWLRQVLQ7DEOHWKHPD[LPXPIDFWRUHGSUHVVXUHDWWKHEDVHRIWKH IRRWLQJEDVHGRQRQO\D[LDOIRUFHLVREWDLQHGIURP$&,(TE ZKLFKLVDXQLIRUPVWUHVVRYHUWKHEDVHDUHDRI the footing:

0LQLPXPHIIHFWLYHGHSWKIRUÀH[XUH Eq. (10.15)

Minimum effective depth for one-way shear:

Eq. (10.18)

Minimum effective depth for two-way shear:

10-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (10.20)

The minimum effective depths determined above are based on a uniform factored pressure at the base of the footing GXHWRD[LDOIRUFHRQO\&KHFNWZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVEDVHGRQ$&,(TG IRUFRPELQHGD[LDO force and bending assuming

7KHFULWLFDOVHFWLRQDURXQGWKHLQFROXPQLVDWDGLVWDQFHRI from the face of the column (see )LJXUH 7KHWRWDOVKHDUVWUHVVLVHTXDOWRWKHVKHDUVWUHVVGXHWRWKHIDFWRUHGD[LDOIRUFHSOXVWKHVKHDUVWUHVV due to the transferred bending moment. The magnitude of the factored pressure along the length of the footing is equal to the following:

where

is measured from the edge of the footing with minimum factored pressure.

The boundaries of the critical sections are at the following locations from the edge of the footing with minimum factored pressure:

The factored pressures at these locations are equal to the following:

The factored shear force at the critical section, is equal to the factored axial force from the column minus the IDFWRUHGVRLOSUHVVXUHERXQGHGE\WKHFULWLFDOVHFWLRQVHH)LJXUH 

10-44

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

͵ʹͳǤͳ‹’•

       

͸͵Ǥͻˆ–Ǧ‹’•

ʹͶԣ

ͳͲԣ ͳͲԣ ͳͲԣ ͳͲԣ

ʹǤͺͳ ‹’•Τˆ– ʹ 

͵ǤͲʹ ‹’•Τˆ– ʹ  ʹͲǤʹԣ ͵ʹǤͶ‹’•

Figure 10.22 Free-body diagram of the critical section for two-way shear in Example 10.4.

The moment transferred between the column and the footing, troid of the footing:

The factor

is obtained by summing moments about the cen-

LVGHWHUPLQHGE\$&,(T 

7KHVHFWLRQSURSHUWLHVRIWKHFULWLFDOVHFWLRQDUHGHWHUPLQHGXVLQJ&DVHLQ7DEOHRIWKLVSXEOLFDWLRQIRUDQ interior column:

The total factored shear stress at the critical section is equal to the following:

7KHUHIRUHDPLQLPXPHIIHFWLYHGHSWKRILQLVDOVRDGHTXDWHIRUWZRZD\VKHDUEDVHGRQFRPELQHGD[LDOIRUFH and bending. 7KHPLQLPXPHIIHFWLYHGHSWKIRUÀH[XUHDQGIRURQHZD\VKHDUDUHQRWGHWHUPLQHGXVLQJ$&,(TG EHFDXVH WKHPD[LPXPIDFWRUHGSUHVVXUHDWWKHEDVHRIWKHIRRWLQJEDVHGRQ$&,(TE LVJUHDWHUWKDQWKDWEDVHGRQ $&,(TG  7U\DLQWKLFNIRRWLQJ 10-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the required flexural reinforcement

%HFDXVHWKHSURYLGHGHIIHFWLYHGHSWKRILQLVJUHDWHUWKDQWKHLQPLQLPXPHIIHFWLYHGHSWKFDOFXODWHGIRU ÀH[XUHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWH ACI 7.6.1.1

7U\EDUVVSDFHGDWLQRQFHQWHU ing of

7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDF-

Step 4 – Check if the flexural reinforcement can be developed for tension

The tension development length,

RIWKHEDUVPXVWEHOHVVWKDQRUHTXDOWRWKHDYDLODEOHGHYHORSPHQWOHQJWK Eq. (10.21)

Determine

IURP$&,(TD 

Eq. (10.22)

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26)

)RUOHVVWKDQLQRIFRQFUHWHSODFHGEHORZWKHÀH[XUDOUHLQIRUFHPHQW

Eq. (10.27)

Eq. (10.28)

Because there is no transverse reinforcement,

Eq. (10.29)

Therefore,

%HFDXVHWKHDYDLODEOHGHYHORSPHQWOHQJWKLVJUHDWHUWKDQWKHWHQVLRQGHYHORSPHQWOHQJWKWKHEDUVFDQEHIXOO\ GHYHORSHGIRUÀH[XUH

10-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the required dowel reinforcement

• Bearing strength of the column is determined for compression using the factored axial force determined by ACI Factored bearing stress, (TVE DQGIRUFRPSUHVVLRQSOXVEHQGLQJXVLQJWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHG E\$&,(TVG DQGI >VHH7DEOH@ $&,(TE 

$&,(TG 

$&,(TI 

Design bearing strength:

Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match WKHVL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQsion forces are adequately transferred from the column into the footing. • Bearing strength of the footing There is no need to check the bearing strength of the footing because interface reinforcement must be provided GXHWRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(TI  • Determine the required interface reinforcement 7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHVWKDW both the compression and tension forces are adequately transferred through the interface. Check minimum area of reinforcement across the interface:

10-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Development of the dowel bars into the footing $VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSRIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN ment length,

Eq. (10.38)

Table 10.5

Eq. (10.39)

The minimum LVJUHDWHUWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQ7KXVLQFUHDVHWKHIRRWLQJ WKLFNQHVVWRLQWRDFFRPPRGDWHGHYHORSPHQWRIWKHGRZHOEDUV 'HWHUPLQHPLQLPXPÀH[XUHUHLQIRUFHPHQWLQWKHIRRWLQJEDVHGRQWKHLQFUHDVHGWKLFNQHVV ACI 7.6.1.1

8VHEDUVVSDFHGDWLQRQFHQWHU 7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXP 1RWHWKDWWKHGRZHOEDUVFDQEHIXOO\GHYHORSHGLQWKHLQWKLFNIRRWLQJ spacing of ZLWKWKHEDUVIRUÀH[XUH$OVRWKHEDUVFDQEHIXOO\GHYHORSHGIRUÀH[XUHEDVHGRQWKHZLGWKRIWKHIRRWLQJ • Development of the dowel bars into the column The dowel bars must be lap spliced to the longitudinal reinforcement in the column using a tension lap splice. Because all the dowel bars are spliced at the same location, a Class B tension lap splice is required (ACI Table   Development length in tension,

RIWKHEDUV Eq. (10.22)

10-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26) Eq. (10.27)

Eq. (10.28)

Set

ACI 25.4.2.4

Therefore,

3URYLGHDODSVSOLFHOHQJWKRIIWLQ • &KHFNKRUL]RQWDOIRUFHWUDQVIHU LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGIRRWLQJ7KHUHTXLUHG )URP([DPSOH area of shear-friction reinforcement is determined as follows assuming the surface between the column and footing has not been intentionally roughened: Table 10.6, Eq. (10.40)

7KHGRZHOEDUVSURYLGH

across the interface, which is greater than the required amount of

Check the upper shear limit:

Table 10.7

5HLQIRUFHPHQWGHWDLOVIRUWKLVIRRWLQJDUHVLPLODUWRWKRVHLQ)LJXUH 10.5.5 Example 10.5 – Design of a Square Isolated Spread Footing Subjected to Axial Compression and Flexure: Building #1 (Framing Option B), SDC A

'HVLJQDVTXDUHLVRODWHGIRRWLQJIRUWKHLQVTXDUHHGJHFROXPQRI%XLOGLQJ)UDPLQJ2SWLRQ%LQ([DPSOH VHH)LJXUH 7KHFROXPQLVUHLQIRUFHGZLWKEDUV$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKW IRUWKHFROXPQDQGWKHIRRWLQJ$OVRDVVXPHDSHUPLVVLEOHVRLOSUHVVXUHRIOEIW. concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW 10-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the base dimensions of the footing

7KHDOORZDEOHVWUHVVDQGVWUHQJWKGHVLJQORDGFRPELQDWLRQVDUHJLYHQLQ7DEOHVHH([DPSOH 7KHDOORZDEOHVWUHVVGHVLJQORDGFRPELQDWLRQVDUHIURP$6&(6(,7KHVWUHQJWKGHVLJQORDGFRPELQDWLRQVDUHIURP$&, DQG$6&(6(,ZKLFKDUHWKHVDPH7KHEHQGLQJPRPHQWVLQ7DEOHGXHWRJUDYLW\DQGZLQGDUHDW WKHERWWRPRIWKHFROXPQWKHODWWHURIZKLFKDUHPDJQL¿HGWRDFFRXQWIRUWKHHIIHFWVRIVOHQGHUQHVVVHH7DEOHV DQGRIWKLVSXEOLFDWLRQ  Table 10.12 Summary of Axial Forces and Bending Moments for Column D1 Axial Force (kips)

Bending Moment (ft-kips)

Dead





Roof live





Live





$6&(6(,(T





$6&(6(,(T





$6&(6(,(T





$6&(6(,(T





Load Case

Wind Allowable Stress Load Combinations

SSR



SSL



SSR



SSL



SSR



SSL





$&,(TD



3.9

$&,(TE





316.9



$6&(6(,(T $6&(6(,(T $6&(6(,(T

 

Strength Design Load Combinations

$&,(TF

$&,(TG $&,(TI

10-50

SSR



SSL



SSR



SSL



SSR



SSL



  

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HWHUPLQHWKHUHTXLUHGEDVHGLPHQVLRQVRIWKHIRRWLQJEDVHGRQD[LDOIRUFHRQO\$6&(6(,(T DQGWKHQFKHFN if the maximum combined pressure due to axial force and bending moment is less than the permissible soil pressure. Eq. (10.1)

Try a 9 ft-6 in. square isolated footing. &KHFNWKHPD[LPXPFRPELQHGSUHVVXUHEDVHGRQ$6&(6(,(T Pressure due to axial force

Pressure due to moment Total pressure &KHFNWKHPD[LPXPFRPELQHGSUHVVXUHEDVHGRQ$6&(6(,(T Pressure due to axial force

Pressure due to moment Total pressure Therefore, a 9 ft-6 in. square isolated footing is not adequate for soil bearing capacity based on combined axial force and bending. ,WFDQEHGHWHUPLQHGWKDWDIWLQLVRODWHGVTXDUHIRRWLQJLVDGHTXDWH Note that there is no net uplift (tension) at the base of the footing for any of the allowable stress load combinations LQ7DEOHEDVHGRQDIWLQIRRWLQJ Step 2 – Determine the thickness of the footing

7KHIDFWRUHGSUHVVXUHGHWHUPLQHGE\$&,(TE LVWUDSH]RLGDORYHUWKHEDVHDUHDRIWKHIRRWLQJDQGWKZPD[Lmum pressure is equal to the following:

7KHIDFWRUHGSUHVVXUHGHWHUPLQHGE\$&,(TG LVWUDSH]RLGDORYHUWKHEDVHDUHDRIWKHIRRWLQJDQGWKHPD[Lmum pressure is equal to the following: Maximum

10-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Conservatively assume this pressure is uniformly distributed across the entire width of the footing and determine the minimum effective depths based on this assumption. 0LQLPXPHIIHFWLYHGHSWKIRUÀH[XUH Eq. (10.15)

Minimum effective depth for one-way shear:

Eq. (10.18)

Minimum effective depth for two-way shear:

Eq. (10.20)

The minimum effective depth for one-way shear governs in this case. 7U\DLQWKLFNIRRWLQJ Step 3 – Determine the required flexural reinforcement

%HFDXVHWKHSURYLGHGHIIHFWLYHGHSWKRILQLVJUHDWHUWKDQWKHLQPLQLPXPHIIHFWLYHGHSWKFDOFXODWHGIRU ÀH[XUHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWH ACI 7.6.1.1

7U\EDUVVSDFHGDWLQRQFHQWHU ing of

7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDF-

Step 4 – Check if the flexural reinforcement can be developed for tension

The tension development length,

RIWKHEDUVPXVWEHOHVVWKDQRUHTXDOWRWKHDYDLODEOHGHYHORSPHQWOHQJWK Eq. (10.21)

10-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Determine

IURP$&,(TD 

Eq. (10.22)

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26)

)RUOHVVWKDQLQRIFRQFUHWHSODFHGEHORZWKHÀH[XUDOUHLQIRUFHPHQW

Eq. (10.27)

Eq. (10.28)

Because there is no transverse reinforcement,

Eq. (10.29)

Therefore,

%HFDXVHWKHDYDLODEOHGHYHORSPHQWOHQJWKLVJUHDWHUWKDQWKHWHQVLRQGHYHORSPHQWOHQJWKWKHEDUVFDQEHIXOO\ GHYHORSHGIRUÀH[XUH Step 5 – Determine the required dowel reinforcement

• Bearing strength of the column is determined for compression plus bending using the factored axial forces and Factored bearing stress, EHQGLQJPRPHQWVGHWHUPLQHGE\$&,(TVE DQGG >VHH7DEOH@ $&,(TE 

10-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$&,(TG 

Design bearing strength:

Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match WKHVL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQsion forces are adequately transferred from the column into the footing. • Bearing strength of the footing There is no need to check the bearing strength of the footing because interface reinforcement must be provided GXHWRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(TG  • Determine the required interface reinforcement 7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHV that both the compression and tension forces are adequately transferred through the interface. Check minimum area of reinforcement across the interface:

• Development of the dowel bars into the footing $VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSRIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN ment length,

Eq. (10.38)

Table 10.5

10-54

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (10.39)

The minimum LVJUHDWHUWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQ7KXVLQFUHDVHWKHIRRWLQJ WKLFNQHVVWRLQWRDFFRPPRGDWHGHYHORSPHQWRIWKHGRZHOEDUV 'HWHUPLQHPLQLPXPÀH[XUHUHLQIRUFHPHQWLQWKHIRRWLQJEDVHGRQWKHLQFUHDVHGWKLFNQHVV ACI 7.6.1.1

8VHEDUVVSDFHGDWLQRQFHQWHU 7KHLQVSDFLQJLVOHVVWKDQWKHPD[LPXP 1RWHWKDWWKHGRZHOEDUVFDQEHIXOO\GHYHORSHGLQWKHLQWKLFNIRRWLQJ spacing of ZLWKWKHEDUVIRUÀH[XUH • Development of the dowel bars into the column RIWKHEDUV Development length in tension, Eq. (10.22)

For normalweight concrete,

Eq. (10.23)

)RU*UDGHUHLQIRUFHPHQW

Eq. (10.24)

For uncoated reinforcing bars,

Eq. (10.25)

)RUEDUV

Eq. (10.26) Eq. (10.27)

Eq. (10.28)

Set

ACI 25.4.2.4

Therefore,

3URYLGHDODSVSOLFHOHQJWKRIIWLQ

10-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• &KHFNKRUL]RQWDOIRUFHWUDQVIHU LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGWKHIRRWLQJ7KHUH)URP([DPSOH quired area of shear-friction reinforcement is determined as follows assuming the surface between the column and the footing has not been intentionally roughened: Table 10.6, Eq. (10.40)

7KHGRZHOEDUVSURYLGHV

across the interface, which is greater than the required amount of

Check the upper shear limit:

Table 10.7

5HLQIRUFHPHQWGHWDLOVIRUWKLVIRRWLQJDUHVLPLODUWRWKRVHLQ)LJXUH Comments.7KHWKLFNQHVVRIWKHIRRWLQJDQGWKHDPRXQWRIÀH[XUDOUHLQIRUFHPHQWLQWKHIRRWLQJFDQEHUHGXFHGE\ XVLQJVPDOOHUORQJLWXGLQDOEDUVLQWKHFROXPQDQGVPDOOHUGRZHOEDUVLQWKHIRRWLQJ,WFDQEHGHWHUPLQHGWKDW ORQJLWXGLQDOEDUVDUHDGHTXDWHIRUWKHLQVTXDUHFROXPQ(LJKWGRZHOEDUVZLWKDVWDQGDUGGHJUHHKRRN FDQEHIXOO\GHYHORSHGLQDLQWKLFNIRRWLQJ7KHFRUUHVSRQGLQJPLQLPXPÀH[XUDOUHLQIRUFHPHQWLQWKHIRRWLQJ LVEDUVVSDFHGDWLQRQFHQWHU 10.5.6 Example 10.6 – Design of a Combined Rectangular Spread Footing Subjected to Axial Compression: Building #1 (Framing Option B), SDC A

'HVLJQDFRPELQHGUHFWDQJXODUVSUHDGIRRWLQJVXSSRUWLQJFROXPQV$DQG%LQ%XLOGLQJ)UDPLQJ2SWLRQ %VHH)LJXUH 7KHFROXPQVZKLFKDUHLQE\LQDQGDUHUHLQIRUFHGZLWKEDUVDUHQRWSDUWRIWKH for the columns and the /)56$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKWFRQFUHWHZLWK IRRWLQJ$OVRDVVXPHDSHUPLVVLEOHVRLOSUHVVXUHRIOEIW. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the base dimensions of the footing

The service axial dead, roof live, and live loads are equal to the following: • &ROXPQ$

• &ROXPQ%

10-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

8VHWKHEDVLFORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,WRGHWHUPLQHWKHPD[LPXPD[LDO force transferred to the footing: • &ROXPQ$ 1.  3.  • &ROXPQ% 1.  3.  $6&(6(,(TJRYHUQVIRUERWKFROXPQV The base dimension of the footing is determined so that the pressure at the base of the footing is uniform (see FigXUH 

Eq. (10.6)

Use ܲଵ ൌ 263.6 kips

ܲ ൌ ܲଵ ൅ ܲଶ ൌ 759.1 kips

Column B2

Column A2 ‫ݔ‬ଵ ൌ 9ԣ

ܲଶ ൌ 495.5 kips

‫ ݔ‬ൌ 16.1Ԣ ‫ݔ‬ଶ ൌ 23.5 ൅ 0.75 ൌ 24.25Ԣ

‫ ܮ‬ൌ 2‫ ݔ‬ൌ 32.2ᇱ ሺuse ‫ ܮ‬ൌ 32.25ᇱ ሻ

‫ݍ‬ൌ

ܲ 759.1 ൈ 1,000 ൌ ൌ 2,942 lbΤft 2 ൏ ‫ݍ‬௣ ൌ 3,000 lbΤft 2  ‫ܤܮ‬ 32.25 ൈ 8.0

Figure 10.23 Combined footing in Example 10.6

10-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Eq. (10.7)

Use Step 2 – Determine the thickness of the footing

7KHIDFWRUHGD[LDOIRUFHVRQHDFKFROXPQEDVHGRQWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,7DEOHDUHWKHIROlowing: • &ROXPQ$

• &ROXPQ%

7KHODUJHVWD[LDOIRUFHVDQGSUHVVXUHDWWKHEDVHRIWKHIRRWLQJDUHEDVHGRQ$&,(TE  The factored pressure,

at the base of the footing is equal to the following:

The factored uniformly distributed load, width of the footing,

along the length of the footing is obtained by multiplying

by the

7KHWKLFNQHVVRIWKHIRRWLQJLVGHWHUPLQHGXVLQJWKHIDFWRUHGGLVWULEXWHGORDGFRQVLGHULQJERWKÀH[XUHDQGVKHDU 7KHVKHDUDQGPRPHQWGLDJUDPVIRUWKHIRRWLQJDUHJLYHQLQ)LJXUH1RWHWKDWLQRUGHUIRUWKHPRPHQWGLDJUDPWRFORVHWKHIRRWLQJOHQJWKLVWDNHQDVIWLQVWHDGRIIW • One-way shear RFFXUVDWWKHLQWHULRUFROXPQDQGLVHTXDOWRNLSV7KHIDFWRUHG The maximum factored shear force, shear force at the critical section located a distance from the face of the column is equal to the following (see )LJXUH 

Assuming a reinforcement ratio in the footing at this section, the design shear strength is equal to the following for members without shear reinforcement: ACI Table 22.5.5.1

10-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



352.0 kips

661.6 kips

Column A2

Column B2

‫ݓ‬௨ ൌ 31.44 kipsΤft

23.5Ԣ

9ԣ

411.9

23.6 ‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ

249.7 328.4

9.7Ԣ

990.0 811.6 689.9

8.9 ‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬--‫ܛܘܑܓ‬ሻ 228.6

1.9Ԣ

1,702.7

Figure 10.24 Shear and moment diagrams for the combined footing in Example 10.6.

Set the required shear strength equal to the design shear strength and solve for

7KXVDQHIIHFWLYHGHSWKHTXDOWRLQLVDGHTXDWHIRURQHZD\VKHDU • Two-way shear Two-way shear requirements are checked at each column assuming

10-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.5Ԣ

8.0Ԣ

1.5Ԣ

Column A2

Critical section for one-way shear

݀ Column B2

23.5Ԣ 32.25Ԣ Figure 10.25 Critical section for one-way shear.

(GJHFROXPQ±&ROXPQ$ The edge column has a three-sided critical section located GLUHFWLRQVVHH)LJXUH 

from the face of the column in both

The maximum factored shear force at the face of the critical section is equal to the factored column load minus the factored soil pressure in the area bounded by the critical section:

'HWHUPLQHWKHVHFWLRQSURSHUWLHVRIWKHFULWLFDOVHFWLRQXVLQJ&DVHLQ7DEOHRIWKLVSXEOLFDWLRQIRUDQHGJH column bending perpendicular to the edge:

The load from the column and the load from the factored soil pressure in the area bounded by the critical section occurs at the critical section. Determine at do not act through the same point; thus, a transfer moment, WKHFHQWURLGRIWKHFULWLFDOVHFWLRQVHH)LJXUH 

7KHWRWDOIDFWRUHGVKHDUVWUHVVDWWKHIDFHRIFULWLFDOVHFWLRQ$%LVGHWHUPLQHGE\(T RIWKLVSXEOLFDWLRQ

10-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾଵ ൌ 18 ൅ 21 ൌ 39ԣ

B

D

       

18ԣ

ܾଶ ൌ 18 ൅ 42 ൌ 60ԣ

18ԣ

C

A ܿ஼஽ ൌ 28.0ԣ

ܿ஺஻ ൌ 11.0ԣ

352.0 kips 9ԣ

46ԣ

9ԣ

3.93 kipsΤft 2 19.5ԣ 63.9 kips

Figure 10.26 Critical section for two-way shear at the edge column.

The two-way shear design strength for members without shear reinforcement, is equal to the least of the YDOXHVGHWHUPLQHGE\WKHWKUHHHTXDWLRQVLQ$&,7DEOH)RUVTXDUHFROXPQVWKH¿UVWRIWKHVHHTXDWLRQV governs:

,QWHULRUFROXPQ±&ROXPQ% The interior column has a four-sided critical section located directions.

from the face of the column in both

The maximum factored shear force at the face of the critical section is equal to the factored column load minus the factored soil pressure in the area bounded by the critical section:

10-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The load from the column and the load from the factored soil pressure in the area bounded by the critical section act through the centroid of the critical section; thus, no transfer moment occurs at this location. 'HWHUPLQHWKHVHFWLRQSURSHUWLHVRIWKHFULWLFDOVHFWLRQXVLQJ&DVHLQ7DEOHRIWKLVSXEOLFDWLRQIRUDQ interior column:

The total factored shear stress at the critical section is equal to the following:

7KHUHIRUHDQHIIHFWLYHGHSWKHTXDOWRLQLVDGHTXDWHIRUWZRZD\VKHDU 8VHDLQWKLFNIRRWLQJ Step 3 – Determine the required flexural reinforcement

• Moment in the longitudinal direction near midspan )URP)LJXUHWKHPRPHQWQHDUPLGVSDQLVHTXDOWRIWNLSV7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWWR resist this moment must be placed at the top of the footing section; it is analogous to positive reinforcement.

$WWKLVVHFWLRQWKHUHTXLUHGVKHDUVWUHQJWKLV]HURVRDPLQLPXP strength requirements. Thus,

is not needed to satisfy one-way shear

8VHEDUVVSDFHGDWLQRQFHQWHU • Moment in the longitudinal direction at the interior column )URP)LJXUHWKHPD[LPXPPRPHQWRFFXUVDWWKHULJKWIDFHRIWKHLQWHULRUFROXPQDQGLVHTXDOWR IWNLSV7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWWRUHVLVWWKLVPRPHQWPXVWEHSODFHGDWWKHERWWRPRIWKHIRRWLQJ section; it is analogous to negative reinforcement.

10-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$WWKHOHIWIDFHRIWKHLQWHULRUFROXPQWKHPRPHQWLVHTXDOWRIWNLSVZKLFKZRXOGUHTXLUHPLQLPXPÀH[XUDOUHLQIRUFHPHQW+RZHYHUWKHIDFWRUHGVKHDUIRUFHDWWKLVORFDWLRQLVPD[LPXPDQGWKHGHVLJQRQHZD\VKHDU 7KHUHIRUHSURYLGHWKHIROORZLQJÀH[XUDOUHLQIRUFHPHQW VWUHQJWKZDVGHWHUPLQHGLQ6WHSXVLQJ at the interior column:

8VHEDUVVSDFHGDWLQRQFHQWHU • Moment in the longitudinal direction at the edge column )URP)LJXUHWKHPRPHQWDWWKHLQWHULRUIDFHRIWKHHGJHFROXPQLVHTXDOWRIWNLSVZKLFKUHTXLUHV minimum reinforcement:

Note that this moment is positive, which causes tension on the top of the footing. Therefore, reinforcement must be provided at the top of the footing at this location. &KHFNWKDWWKHPLQLPXPÀH[XUDOUHLQIRUFHPHQWLVDGHTXDWHWRVDWLVI\WKHPRPHQWWUDQVIHUUHTXLUHPHQWVLQ$&,  The transfer moment

ZDVGHWHUPLQHGLQ6WHSDQGLVHTXDOWRIWNLSV$IUDFWLRQRIWKLVPRPHQW is transferred over the following effective width:

Therefore, the entire width of the footing is available to resist this transfer moment. 'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQW

&KHFNWKHPLQLPXPUHLQIRUFHPHQWUHTXLUHGE\$&,

7KHUHIRUHWKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,QHHGQRWEHVDWLV¿HG ([WHQGLQJWKHWRSEDUVVSDFHGDWLQZKLFKZHUHGHWHUPLQHGIRUWKHPRPHQWDWPLGVSDQ WKHIXOOOHQJWK RIWKHIRRWLQJVDWLV¿HVWKHÀH[XUDOUHTXLUHPHQWVDWWKHIDFHRIWKHHGJHFROXPQ 3URYLGHPLQLPXPÀH[XUDOUHLQIRUFHPHQWDWWKHERWWRPRIWKHIRRWLQJDWWKLVORFDWLRQ7KXVXVHEDUVVSDFHG DWLQRQFHQWHU • Moment in the transverse direction at the interior column The bending moment in the transverse direction is determined assuming the factored load from the column is distributed over a width on each side of the column. The actual width is not important at this stage because the moments are independent of it.

10-63

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The factored distributed load and factored bending moment at the interior column in the transverse direction are HTXDOWRWKHIROORZLQJVHH)LJXUH 

ͳͺԣ

͵ǤʹͷԢ

Ͷ͸ԣ

͵ǤʹͷԢ

ͺʹǤ͹‹’•Ȁˆ–

Figure 10.27 Factored distributed load in the transverse direction at the interior column.

Assuming the effective width of the transverse member is equal to the width of the column plus the effective RQHDFKVLGHRIWKHFROXPQ WKHUHTXLUHGÀH[XUDOUHLQIRUFHdepth, (that is, the effective width extends ment to resist this moment is equal to the following:

8VHEDUVVSDFHGDWLQRQFHQWHU • Moment in the transverse direction at the edge column The factored distributed load and factored bending moment at the interior column in the transverse direction are equal to the following:

10-64

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Assuming the effective width of the transverse member is equal to the width of the column plus TXLUHGÀH[XUDOUHLQIRUFHPHQWWRUHVLVWWKLVPRPHQWLVHTXDOWRWKHIROORZLQJ

the re-

8VHEDUVVSDFHGDWLQRQFHQWHU $WDOOFULWLFDOVHFWLRQVIRUÀH[XUHLQERWKGLUHFWLRQVWKHSURYLGHGVSDFLQJLVOHVVWKDQWKHPD[LPXPVSDFLQJRI

Step 4 – Check if the flexural reinforcement can be developed for tension

• Reinforcement in the longitudinal direction near midspan 7KHEDUVDWLQRQFHQWHUPXVWEHGHYHORSHGRQERWKVLGHVRIWKHFULWLFDOVHFWLRQZKLFKLVORFDWHGIW IURPWKHLQWHULRUIDFHRIWKHHGJHFROXPQVHH)LJXUH ,WLVHYLGHQWIURPWKHPRPHQWGLDJUDPWKDWWKH SRLQWVRILQÀHFWLRQRFFXUZLWKLQDQGYHU\FORVHWRWKHHGJHDQGLQWHULRUFROXPQVUHVSHFWLYHO\7KHUHIRUHWKHVH top bars are extended over the entire footing length. • Reinforcement in the longitudinal direction at the columns The bottom bars at the edge column are extended to the left face of the interior column. The bottom bars at the LQWHULRUFROXPQPXVWEHGHYHORSHGSDVVHGWKHSRLQWRILQÀHFWLRQZKLFKLVORFDWHGIWIURPWKHOHIWIDFHRIWKH LQWHULRUFROXPQVHH)LJXUH LQDFFRUGDQFHZLWK$&, • Reinforcement in the transverse direction ,WFDQEHGHWHUPLQHGWKDWWKHERWWRPEDUVFDQEHIXOO\GHYHORSHGIRUWHQVLRQXVLQJVWUDLJKWEDUVZLWKRXW hooks. Step 5 – Determine the required dowel reinforcement

Only vertical forces need to be considered for force transfer between the columns and the footing. Calculations are given below for the interior column. Similar calculations can be performed for the edge column. • Check the bearing force on the column: Eq. (10.30)

• Check the bearing force on the footing: Eq. (10.31)

8VLQJ)LJXUH

is determined as follows:

Thickness of footing +RUL]RQWDOSURMHFWLRQIRUDVORSH Projected length Therefore,

10-65

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Determine the required interface reinforcement Because the design bearing strength is adequate, provide the minimum area of reinforcement across the interface: Sect. 10.3.7

7U\GRZHOEDUV • Development of the dowel bars into the footing The dowel bars must extend into the footing at least a compression development length,

Eq. (10.34)

where

IRUUHLQIRUFLQJEDUVZLWKRXWFRQ¿QHPHQW

Table 10.4

The minimum footing thickness for the development of the dowel bars in compression is the following: Eq. (10.35)

The provided footing thickness is greater than the required footing thickness, so the dowel bars can be fully developed for compression into the footing. • Development of the dowel bars into the column The dowel bars are lap spliced to the longitudinal bars in the column. Because the dowel bars are smaller in must be greater than or equal to the diameter than the longitudinal bars, the compression lap splice length, larger of the following: 1. Development length in compression,

RIWKHORQJLWXGLQDOEDUV

Eq. (10.34)

 Compression lap splice length,

RIWKHGRZHOEDUV Eq. (10.37)

3URYLGHDODSVSOLFHOHQJWKHTXDOWRIWLQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVIRRWLQJDUHJLYHQLQ)LJXUH

10-66

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳԢǦ͸ԣ

ͳԢǦ͸ԣ

ʹʹԢǦͲԣ

ʹԢǦͲԣ ሺ–›’Ǥሻ



͵ԢǦͳͲԣ

͓ͻ̷ͳͲԣ

ͶǦ‘Ǥ͸†‘™‡Ž„ƒ”•ሺ–›’Ǥሻ

Ͳԣ

͓ͻ̷ͳʹԣ

ͷԢǦ͸ԣ

͓ͳͳ̷͸ԣ

͵ʹԢǦ͵ԣ Figure 10.28 Reinforcement details for the combined rectangular footing in Example 10.6.

10.5.7 Example 10.7 – Design of a Drilled Pier Subjected to Axial Compression: Building #1 (Framing Option B), SDC A

'HVLJQDGULOOHGSLHUVXSSRUWLQJDW\SLFDOLQWHULRUFROXPQLQ%XLOGLQJ)UDPLQJ2SWLRQ%VHH)LJXUH 7KH FROXPQZKLFKLVLQE\LQDQGLVUHLQIRUFHGZLWKEDUVLVQRWSDUWRIWKH/)56$VVXPH*UDGHUHfor the columns and the drilled pier. Also assume the inforcement and normalweight concrete with IROORZLQJIRUWKHGULOOHGSLHU ODWHUDOVXSSRUWLVSURYLGHGRYHUWKHHQWLUHKHLJKW EHQGLQJPRPHQWVGXHWRWKH HIIHFWVRIJUDYLW\ORDGVDUHQHJOLJLEOH QRFDVLQJLVSURYLGHGDQG WKHDOORZDEOHEHDULQJFDSDFLW\RIWKHURFNLV OEIW. 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the pier size

%HFDXVHWKHWZRFRQGLWLRQVLQ$&,DUHVDWLV¿HGLWLVSHUPLWWHGWRGHVLJQWKHGULOOHGSLHUXVLQJDOORZDEOH stress design. The service axial dead, roof live, and live loads on a typical interior column are equal to the following:

8VHWKHEDVLFORDGFRPELQDWLRQVIRUDOORZDEOHVWUHVVGHVLJQLQ$6&(6(,WRGHWHUPLQHWKHPD[LPXPD[LDO force transferred to the drilled pier: 1.  3.  $6&(6(,(TJRYHUQV The diameter of the pier can be determined by the following equation, which does not consider the contribution of the longitudinal reinforcement:

10-67

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 10.10

$SLHUZLWKDGLDPHWHURIIWLQLVDGHTXDWHKRZHYHUDIWLQGLDPHWHULVXVHGEDVHGRQWKHVL]HRIWKHIW square column. Step 2 – Determine the bell size

The diameter of the bell can be determined by the following equation: Eq. (10.19)

8VHDEHOOZLWKDGLDPHWHURIIWLQ Step 3 – Determine the longitudinal and transverse reinforcement

Because the drilled pier is subjected to primarily axial compression loads, provide a minimum longitudinal reinIRUFHPHQWUDWLRRI

8VHORQJLWXGLQDOEDUV

Figure 10.19

Length of longitudinal bars $OVRXVHWLHVVSDFHGDWLQRQFHQWHU Step 4 – Determine the required dowel reinforcement

Only vertical forces need to be considered for force transfer between the column and the pier. The maximum factored axial force transferred from the column to the pier is equal to the following:

• Check the bearing force on the column: Eq. (10.30)

• Check the bearing force on the pier: Eq. (10.31)

The largest square that can be inscribed in the 3-ft-diameter pier has sides equal to Therefore,

10-68

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Determine the required interface reinforcement Because the design bearing strength is adequate, provide the minimum area of reinforcement across the interface: Sect. 10.3.7

8VHGRZHOEDUV • Development of the dowel bars into the pier The dowel bars must extend into the footing at least a compression development length,

Eq. (10.34)

where

IRUUHLQIRUFLQJEDUVZLWKRXWFRQ¿QHPHQW

Table 10.4

8VHGRZHOEDUVIWLQLQOHQJWK • Development of the dowel bars into the column The dowel bars are lap spliced to the longitudinal bars in the column. Because the dowel bars are smaller in must be greater than or equal to the diameter than the longitudinal bars, the compression lap splice length, larger of the following: (1) Development length in compression,

RIWKHORQJLWXGLQDOEDUV

Eq. (10.34)

  Compression lap splice length,

RIWKHGRZHOEDUV

Eq. (10.37)

3URYLGHDODSVSOLFHOHQJWKHTXDOWRIWLQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVSLHUDUHJLYHQLQ)LJXUH

10-69

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͳԢǦ͸ԣ ͳԢǦ͸ԣ



ʹԢǦͲԣ

ͶǦ͓ͻ

ͳԢǦͶԣ

ͶǦ͓͸†‘™‡Ž„ƒ”•

͹Ǧ͓ͺ

ͺԢǦͺԣ

͓͵̷ͳ͸ԣ

ͳԢǦͲԣ

ͶԢǦͶԣ

͵ԢǦͲԣ

ͺԢǦͲԣ Figure 10.29 Reinforcement details for the drilled pier in Example 10.7.

10-70

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 11 BEAM-COLUMN AND SLAB-COLUMN JOINTS 11.1 Overview -RLQWVLQDUHLQIRUFHGFRQFUHWHVWUXFWXUHWUDQVIHUD[LDOIRUFHVIURPWKHFROXPQDERYHDQGPXVWUHVLVWWKHKRUL]RQWDO VKHDUIRUFHVIURPWKHFRQQHFWHGEHDPVDQGRUVODEVIUDPLQJLQWRWKHMRLQW$FFRUGLQJWR$&,5DMRLQWLVWKH portion of a structure that is common to intersecting members whereas a connection consists of a joint and portions RIDGMRLQLQJPHPEHUV,QRUGHUIRUDMRLQWWRDGHTXDWHO\WUDQVIHUWKHDSSOLFDEOHIRUFHVLWPXVWEHFRQ¿QHGE\VWUXFWXUDOPHPEHUVEHDPVDQGRUVODEV E\WUDQVYHUVHUHLQIRUFHPHQWRUE\ERWK 3URYLVLRQVIRUEHDPFROXPQDQGVODEFROXPQMRLQWVDUHJLYHQLQ$&,&KDSWHUZKLFKDUHDSSOLFDEOHWREXLOGLQJV assigned to Seismic Design Category (SDC) A and B.

11.2 Design Criteria 7KHJHQHUDOUHTXLUHPHQWVLQ$&,IRUWKHGHVLJQDQGGHWDLOLQJRIEHDPFROXPQDQGVODEFROXPQMRLQWVLQFDVW LQSODFHUHLQIRUFHGFRQFUHWHEXLOGLQJVDUHVXPPDUL]HGLQ7DEOH Table 11.1 General Requirements for Beam-Column and Slab-Column Joints in Accordance with ACI 15.2 Requirement

ACI Section No.

%HDPFROXPQDQGVODEFROXPQMRLQWVPXVWVDWLVI\WKHGHWDLOLQJSURYLVLRQVRI$&,DQG WKHVWUHQJWKUHTXLUHPHQWVRI$&,



%HDPFROXPQDQGVODEFROXPQMRLQWVPXVWVDWLVI\WKHUHTXLUHPHQWVLQ$&,IRUWUDQVIHU RID[LDOIRUFHWKURXJKWKHÀRRUV\VWHP



Beam-column joints must be designed for the shear forces resulting from moment transfer at the joint due to gravity, wind, earthquake, and any other lateral force.



At corner joints between two members (such as a roof-level exterior joint in a building), the effects of moment reversals within the joint must be considered in the design of the joint.



7KHVWUXWDQGWLHPHWKRGLQ$&,&KDSWHUPXVWEHXVHGWRDQDO\]HDQGGHVLJQDMRLQW where the depth of the beam framing into the joint that generates the joint shear is more than twice the depth of the column in the direction of analysis. In addition to the requirePHQWVRI$&,&KDSWHUWKHMRLQWVKHDUVWUHQJWKUHTXLUHPHQWVRI$&,DQGWKHMRLQW GHWDLOLQJUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG



Column and beam extensions are assumed to provide continuity through a beam-column MRLQWSURYLGHGWKHUHTXLUHPHQWVLQ$&,DQGDUHVDWLV¿HGUHVSHFWLYHO\VHH)LJXUHVDQG ([WHQVLRQVGRQRWFRQWULEXWHWRMRLQWVKHDUIRUFHLIWKH\GRQRWVXSSRUW externally applied loads.



)RUDJLYHQGLUHFWLRQRIDQDO\VLVDMRLQWLVFRQVLGHUHGWREHODWHUDOO\FRQ¿QHGZKHUHWZR transverse beams (that is, two beams framing into the column in the direction perpendicular WRWKHGLUHFWLRQRIDQDO\VLV VDWLVI\WKHWKUHHUHTXLUHPHQWVLQ$&,VHH)LJXUH 



Strength and detailing requirements in accordance with the applicable provisions in ACI &KDSWHU$&,DQG$&,PXVWEHVDWLV¿HGIRUVODEFROXPQFRQQHFWLRQVWKDW transfer moment.



11-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

൒ ݄

Column extension

Longitudinal and transverse reinforcement from  the column below continued through the extension

 Direction of analysis

Beam Column All other reinforcement not shown for clarity

݄

Figure 11.1 A column extension that provides continuity through a beam-column joint in accordance with ACI 15.2.6.

Longitudinal and transverse reinforcementfrom WKHEHDP RQWhe opposite side of the MRLQW continued through the extension



݄

Direction of analysis

Beam extension Beam Column All other reinforcement not shown for clarity ൒ ݄

Figure 11.2 A beam extension that provides continuity through a beam-column joint in accordance with ACI 15.2.7.

11.3 Detailing of Joints 11.3.1 Beam-Column Joint Transverse Reinforcement

%HDPFROXPQMRLQWVPXVWEHGHWDLOHGLQDFFRUGDQFHZLWK$&,WKURXJKXQOHVVDOOWKHIROORZLQJDUH VDWLV¿HG$&,  (a) 7  KHMRLQWLVFRQ¿QHGE\WUDQVYHUVHEHDPVLQDFFRUGDQFHZLWK$&,IRUDOOVKHDUGLUHFWLRQVFRQVLGHUHG (see Figure 11.3). (b) The joint is not part of a designated seismic-force-resisting system (SFRS). (c) The joint is not part of a structure assigned to SDC D, E, or F.

11-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



݄

Direction of analysis

Transverse beams not shown for clarity

Elevation Transverse beam ሺtyp.ሻ* 

ܿଶ 

* Instead of continuous transverse beams, a beam extension satisfying ACI 15.2.7 may be used to provide continuity ሺsee Figure 11.2ሻ

At least 2 continuous top and bottom bars satisfying ACI 9.6.1.2

#3 or larger stirrups satisfying ACI 9.6.3.4 and 9.7.6.2.2

൒ 3ܿଵ /4 ሺtyp.ሻ ܿଵ  Plan

Figure 11.3 Requirements for joint confinement by transverse beams in accordance with ACI 15.2.8.

7KHGHWDLOLQJUHTXLUHPHQWVRI$&,WKURXJKDUHDSSOLFDEOHWREHDPFROXPQMRLQWVWKDWDUHQRWFRQ¿QHGVXFKDVDWDMRLQWIRUDQHGJHRUDFRUQHUFROXPQ 7KHVHGHWDLOLQJUHTXLUHPHQWVDUHLOOXVWUDWHGLQ)LJXUH 11.3.2 Slab-Column Joint Transverse Reinforcement

$FFRUGLQJWR$&,FROXPQWUDQVYHUVHUHLQIRUFHPHQWPXVWEHFRQWLQXHGWKURXJKDQ\VODEFROXPQMRLQWWKDW is not laterally supported on four sides by the slab (that is, it must be continued through the joints at edge and corner columns). Such reinforcement must also be provided through column capitals, drop panels, and shear caps that are present. 11.3.3 Longitudinal Reinforcement

Longitudinal reinforcement in a column or beam that is terminated in the joint or within a column or beam extenVLRQPXVWVDWLVI\WKHDSSOLFDEOHGHYHORSPHQWUHTXLUHPHQWVLQ$&,$&, :KHUHVWDQGDUGKRRNV are used to develop the longitudinal reinforcement, the hook must be turned toward the mid-depth of the beam or FROXPQ$&,  An example of an edge or corner column where the longitudinal reinforcement from the beam is terminated in the MRLQWZLWKDVWDQGDUGGHJUHHKRRNLVLOOXVWUDWHGLQ)LJXUH,QDFFRUGDQFHZLWK$&,WKHKRRNVDUH turned toward the mid-depth of the beam. The critical section for development of the hooked bars is taken at the face of the column.

11-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



݄ଵ 

At least 2 layers of horizontal transverse reinforcement within ݄ଵ 

݄ଶ 

­Ties in accordance with ACI 25.7.2 ° °° ®Spirals in accordance with ACI 25.7.3  ° ° °¯Hoops in accordance with ACI 25.7.4

‫ ݏ‬൑ 8ᇳ within ݄ଶ 

Other reinforcement not shown for clarity

ܿଵ 

ܿଶ 

Transverse beam

Direction of analysis

Figure 11.4 Detailing requirements for beam-column joints in accordance with ACI 15.3.1.2 through 15.3.1.4.

The development length of a deformed reinforcing bar in tension with a standard hook, 

, is given in ACI

(11.1)

This development length is measured from the critical section (which is taken at the face of the column) to the outVLGHIDFHRIWKHKRRN7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH 

11-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ  κௗ௛ 

–ƒ†ƒ”†ͻͲǦ†‡‰Š‘‘ሺ–›’Ǥሻ

݂߰ ߰ ߰ ߰ ‫ۓ‬ቆ ௬ ௘ ௥ ௢ ௖ ቇ ݀௕ଵǤହ ͷͷߣඥ݂௖ᇱ ۖ ۖ κௗ௛ ൌ Žƒ”‰‡”‘ˆ κௗ௛ 

‫۔‬ͺ݀௕  ۖ ۖ ‫ە‬͸ᇳ 



Figure 11.5 Longitudinal beam reinforcement terminated in a beam-column joint with a standard 90-degree hook.

Table 11.2 Modification Factors for Development of Hooked Bars in Tension Modification Factor

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



Lightweight,

Epoxy, &RQ¿QLQJ reinforcement,

Location,

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK



or Other

)RUDQGVPDOOHUKRRNHGEDUV 1. terminating inside a column core with side cover normal to the plane of the hook or  with side cover normal to the plane of the hook Other

Concrete strength,

1.6







11-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term LVWKHWRWDOFURVVVHFWLRQDODUHDRIWLHVRUVWLUUXSVFRQ¿QLQJWKHKRRNHGEDUVDQG is the total crosssectional area of the hooked bars being developed at the same critical section. The term is the center-to-center are given in ACI spacing of the hooked bars, which have a nominal diameter . Detailing requirements for ZKLFKDUHLOOXVWUDWHGLQ$&,)LJXUHV5DDQG5EIRUFRQ¿QLQJUHLQIRUFHPHQWSODFHGSDUDOOHO and perpendicular to hooked bars being developed, respectively. An example of an edge or corner column where the longitudinal reinforcement from the beam is terminated in the MRLQWZLWKDVWDQGDUGKHDGLVLOOXVWUDWHGLQ)LJXUH+HDGHGGHIRUPHGEDUVDUHSHUPLWWHGWREHXVHGRQO\ZKHQ WKHFRQGLWLRQVRI$&,DUHVDWLV¿HG (a) (b) (c) (d) (e) (f)

%DUPXVWFRQIRUPWR$&, %DUVL]HPXVWEHRUVPDOOHU , must be at least where is the area of the bar Net bearing area of head, Concrete must be normalweight where is the nominal diameter of the bar Clear cover to the bar must be greater than or equal to Center-to-center spacing of the bars must be greater than or equal to ܿଵ  κௗ௧ 

‡ƒ†‡††‡ˆ‘”‡†„ƒ”ሺ–›’Ǥሻ

݂߰ ߰ ߰ ߰ ‫ۓ‬ቆ ௬ ௘ ௣ ௢ ௖ ቇ ݀௕ଵǤହ ͹ͷඥ݂௖ᇱ ۖ ۖ κௗ௧ ൌ Žƒ”‰‡”‘ˆ

‫ ۔‬ͺ݀௕  ۖ ۖ ‫ە‬͸ᇳ 



Figure 11.6 Longitudinal beam reinforcement terminated in a beam-column joint with a standard head.

The development length of a headed deformed reinforcing bar in tension,

LVJLYHQLQ$&,

(11.2)

11-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

This development length is measured from the critical section (which is taken at the face of the column) to the bearLQJIDFHRIWKHKHDGVHH$&,)LJXUHV5DDQG5E 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUH JLYHQLQ7DEOHVHH$&,7DEOH  Where beam negative moment reinforcement is provided by headed deformed bars that terminate in a joint, the FROXPQPXVWH[WHQGDERYHWKHWRSRIWKHMRLQWDGLVWDQFHRIDWOHDVWWKHKRUL]RQWDOGHSWKRIWKHMRLQWLQWKHGLUHFWLRQ RIDQDO\VLV$&, ,IVXFKDQH[WHQVLRQLVQRWSRVVLEOHRUIHDVLEOHWKHEHDPUHLQIRUFHPHQWPXVWEHHQFORVHG E\DGGLWLRQDOYHUWLFDOMRLQWUHLQIRUFHPHQWWKDWSURYLGHVHTXLYDOHQWFRQ¿QHPHQWWRWKHWRSIDFHRIWKHMRLQW7KLVVLWXation is likely to occur for the joints in the top story of the structure; for joints below the top story where the columns are continuous above the joint, this requirement is not applicable. Table 11.3 Modification Factors for Development of Headed Bars in Tension Modification Factor

Epoxy, Parallel tie reinforcement,

Location,

Condition

Value of Factor

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK

or



Other

1.6

For headed bars (1) terminating inside a column core with side RU ZLWKVLGHFRYHUWREDU cover to bar



Other



Concrete strength,



The term DFFRXQWVIRUFRQ¿QLQJHIIHFWVSURYLGHGE\VWLUUXSVRUWLHVWKDWDUHRULHQWHGSDUDOOHOWRWKHGHYHORSPHQW where tie reinforcement with a total cross-sectional area length of headed bars. At beam-column joints, LVSURYLGHGWKDWVDWLV¿HVWKHUHTXLUHPHQWVLQ$&,VHH$&,)LJXUH5 :KHUHVXFKUHLQIRUFHment is not provided in beam-column joints,

11.4 Strength Requirements for Beam-Column Joints 11.4.1 Required Shear Strength Overview

7KHUHTXLUHGIDFWRUHGKRUL]RQWDOMRLQWVKHDU is determined at the mid-height of a joint in the direction of analyVLVIURPVWDWLFV$FFRUGLQJWR$&, LVFDOFXODWHGXVLQJÀH[XUDOWHQVLOHDQGFRPSUHVVLYHEHDPIRUFHVDQG column shear consistent with (a) or (b): (a) The maximum moment transferred between the beam and the column based on a factored load analysis for beam-column joints with continuous beams in the direction of analysis. and (b) 7KHQHJDWLYHDQGSRVLWLYHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHEHDPDWWKHIDFHRIWKHMRLQW Joints in Moment Frames Subjected to Gravity Loads Only

A free-body diagram of an interior column or an edge column with the direction of analysis parallel to the edge in a PRPHQWIUDPHWKDWLVVXEMHFWHGWRWKHHIIHFWVIURPJUDYLW\ORDGVRQO\LVJLYHQLQ)LJXUH7KHEHQGLQJPRPHQWV and and the shear forces and WKDWDUHWUDQVIHUUHGIURPWKHEHDPWRWKHMRLQWDUHVKRZQLQWKH¿JXUH ,WLVDOVRDVVXPHGWKDWWKHSRLQWRILQÀHFWLRQLVDWWKHPLG It is assumed for purposes of discussion that KHLJKWRIWKHFROXPQZKLFKLVDUHDVRQDEOHDVVXPSWLRQIRUFROXPQVDERYHWKH¿UVWVWRU\DQGEHORZWKHWRSVWRU\

11-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ  ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦ ܸଵ 

κ௖ 

ܸଶ  ‫ܯ‬ଵି 

‫ܯ‬ଶି  ܸଶ 

ܸଵ  ‫ܣ‬ା ௦

ܸ௖௢௟ 

Figure 11.7 Free-body diagram of an interior column or an edge column with the direction of analysis parallel to the edge in a moment frame subjected to gravity loads. ܿଵ  ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦

κ௖ 

ܸଵ  ‫ܯ‬௡ି  ܸଵ  ‫ܣ‬ା ௦

ܸ௖௢௟ 

Figure 11.8 Free-body diagram of an edge column with the direction of analysis perpendicular to the edge or a corner column in a moment frame subjected to gravity loads.

11-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The length

is equal to the depths of the beams plus one-half the clear story height above and below the joint.

Where the negative and positive reinforcing bars in the beams framing into an interior joint are continuous through the joint, the forces are transmitted through the joint by the reinforcing bars, which means there is no need to check the shear strength of the joint. The free-body diagram of an edge column with the direction of analysis perpendicular to the edge or a corner column in a moment frame that is subjected to the effects from gravity loads only is given in Figure 11.8. The shear force in the column is determined from equilibrium by summing moments about the center of the joint: (11.3)

The loads that are transferred from the beam and column to the joint are shown on the free-body diagram of the and are the axial forces and and are the bending moments in the coljoint in Figure 11.9(a) where umn. It is assumed that the axial force in the beam is negligible. The internal forces in the joint due to these loads DUHJLYHQLQ)LJXUHE 7HQVLRQDQGFRPSUHVVLRQIRUFHVDUHGHQRWHG³7´DQG³&´UHVSHFWLYHO\,QRUGHUWR must be equal to the compression VDWLVI\KRUL]RQWDOHTXLOLEULXPWKHWHQVLRQIRUFHLQWKHQHJDWLYHUHLQIRUFHPHQW force resisted by the concrete, ܲଶ 

ܶଶ  ‫ܯ‬ଶ 

‫ܥ‬ଶ 

ܸ௖௢௟ 

ܸ௖௢௟ 

ܶଵ 

‫ିܣ‬ ௦

ܸଵ 

ܸ௖௢௟ 

ܸଵ 

‫ିܣ‬ ௦

ܶଵ 

‫ܯ‬ଵି  ܸ௨ ൌ ܶଵ െ ܸ௖௢௟ 

‫ܣ‬ା ௦ ܸ௖௢௟ 

ܸ௖௢௟  ‫ܯ‬ଷ  ܲଷ  ሺࢇሻ

‫ܥ‬ଵ 

ሺࢉሻ

‫ܥ‬ଷ 

ܶଷ  ሺ࢈ሻ

Figure 11.9 Free-body diagram of a joint in a moment frame subjected to gravity loads. (a) Load transfer to the joint. (b) Internal forces in the joint. (c) Horizontal shear force in the joint.

$IUHHERG\GLDJUDPRIWKHXSSHUKDOIRIWKHMRLQWZLWKRQO\KRUL]RQWDOIRUFHVLVJLYHQLQ)LJXUHF 7KHKRUL]RQFDQEHGHWHUPLQHGIURPHTXLOLEULXPE\VXPPLQJIRUFHVLQWKHKRUL]RQWDOGLUHFWLRQ tal shear force in the joint, (11.4)

Where a factored load analysis is used to determine ZKLFKLVWKH¿UVWRSWLRQLQ$&, WKHWHQVLRQIRUFH is determined based on (1) which is the factored bending moment at in the negative reinforcement, which is the corresponding stress in the WKHIDFHRIWKHMRLQWGHWHUPLQHGIURPWKHIDFWRUHGORDGDQDO\VLVDQG  negative reinforcement; that is,

11-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In lieu of using a factored load analysis, FDQEHGHWHUPLQHGEDVHGRQWKHQHJDWLYHQRPLQDOÀH[XUDOVWUHQJWKRI ZKLFKLVWKHVHFRQGRSWLRQLQ$&, ,QVXFKFDVHV and the shear force in the the beam, joint is equal to the following: (11.5)

where

is determined by the following equation: (11.6)

The shear force cluded when determining based on

is usually much smaller than the shear force , and, thus, is often not inE\(TXDWLRQ 7KLVVLPSOL¿FDWLRQLVFRQVHUYDWLYHEHFDXVHWKHFDOFXODWHGYDOXHRI is greater than the calculated value where this shear force is included.

Joints in Moment Frames Subjected to Gravity and Lateral Loads

The free-body diagram of an interior column or an edge column with the direction of analysis parallel to the edge in DPRPHQWIUDPHVXEMHFWHGWRJUDYLW\DQGODWHUDOIRUFHVLVJLYHQLQ)LJXUHIRUVLGHVZD\WRWKHOHIWZKHUHQRPLQDOÀH[XUDOVWUHQJWKVDUHXVHGWRGHWHUPLQH ܿଵ  ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦ ܸଵ 

κ௖ 

ܸଶ  ‫ܯ‬௡ି 

‫ܯ‬௡ା  ܸଶ 

ܸଵ 

‫ܣ‬ା ௦

ܸ௖௢௟ 

Figure 11.10 Free-body diagram of an interior column or an edge column with the direction of analysis parallel to the edge in a moment frame subjected to gravity and lateral loads.

11-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The shear force in the column, joint:

can be obtained from equilibrium by summing moments about the center of the

(11.7)

where

is determined by Equation (11.6) and

is determined by the following equation: (11.8)

The loads that are transferred from the beam and column to the joint are shown on the free-body diagram of the joint in Figure 11.11(a) and the internal forces in the joint due to these loads are given in Figure 11.11(b). In order WRVDWLVI\KRUL]RQWDOHTXLOLEULXPWKHWHQVLRQIRUFHVLQWKHUHLQIRUFHPHQWPXVWEHHTXDOWRWKHFRPSUHVVLRQIRUFHV and resisted by the concrete, that is, ܲଷ 

ሺ࢈ሻ

ܶଷ  ‫ܯ‬ଷ 

‫ܥ‬ଷ 

ܸ௖௢௟ 

ܸ௖௢௟  ‫ܥ‬ଶ 

ܶଵ 

‫ିܣ‬ ௦

ܸଵ 

ܸ௖௢௟ 

ܸଵ 

‫ܥ‬ଶ ൌ ܶଶ 

‫ିܣ‬ ௦

ܶଵ 

‫ܯ‬௡ି 

‫ܯ‬௡ା 

ܸଶ 

ܸଶ  ܶଶ  ܸ௖௢௟ 

ܸ௖௢௟  ‫ܯ‬ସ  ܲସ 

ܸ௨ ൌ ܶଵ ൅ ܶଶ െ ܸ௖௢௟ 

‫ܣ‬ା ௦

‫ܥ‬ଵ 

ሺࢉሻ

‫ܥ‬ସ 

ܶସ 

ሺࢇሻ

ሺ࢈ሻ

Figure 11.11 Free-body diagram of a joint in a moment frame subjected to gravity and lateral loads. (a) Load transfer to the joint. (b) Internal forces in the joint. (c) Horizontal shear force in a joint.

$IUHHERG\GLDJUDPRIWKHXSSHUKDOIRIWKHMRLQWZLWKRQO\KRUL]RQWDOIRUFHVLVVKRZQLQ)LJXUHF ZKHUHLW is obtained LVDVVXPHGWKDWWKHD[LDOIRUFHVRQWKHEHDPVDUHQHJOLJLEOH7KHKRUL]RQWDOVKHDUIRUFHLQWKHMRLQW IURPHTXLOLEULXPE\VXPPLQJIRUFHVLQWKHKRUL]RQWDOGLUHFWLRQ (11.9)

where

and

Equation (11.9) can be rewritten in the following form: (11.10)

7KHKRUL]RQWDOVKHDUIRUFHLQWKHMRLQWIRUVLGHVZD\WRWKHULJKWLVWKHVDPHDVWKDWIRUVLGHVZD\WRWKHOHIWZKHUH WKHEHDPVIUDPLQJLQWRWKHMRLQWKDYHWKHVDPHQHJDWLYHDQGSRVLWLYHQRPLQDOÀH[XUDOVWUHQJWKVDWERWKHQGVRIWKH beams. Otherwise, PXVWEHGHWHUPLQHGE\(TXDWLRQ IRUVLGHVZD\WRWKHULJKWDQGWKHODUJHURIWKHWZR factored shear forces must be used to check shear strength requirements.

11-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For edge columns with the direction of analysis perpendicular to the edge and for corner columns in a moment IUDPHWKDWLVVXEMHFWHGWRERWKJUDYLW\DQGODWHUDOORDGVWKHKRUL]RQWDOVKHDUIRUFHPXVWEHGHWHUPLQHGIRUERWKWKH SRVLWLYHDQGQHJDWLYHQRPLQDOÀH[XUDOVWUHQJWKVWKDWLVVLGHVZD\WRWKHULJKWDQGWRWKHOHIW DQGWKHODUJHURIWKH two must be used to check shear strength requirements. For sidesway to the right: (11.11)

(11.12)

For sidesway to the left: (11.13)

(11.14)

It is assumed in all the above methods for determining that the beams in the direction of analysis are as wide as RUQDUURZHUWKDQWKHFROXPQVWKDWWKH\IUDPHLQWR)RUEHDPVWKDWDUHVLJQL¿FDQWO\ZLGHUWKDQWKHFROXPQVORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDPVSDVVHVRXWVLGHRIWKHFRQ¿QHGFRUHRIWKHFROXPQDQGWKHIRUFHJHQHUDWHGE\WKDW reinforcement cannot be equilibrated by the compression strut in the joint. Guidelines on how to address this situaWLRQDUHJLYHQLQ5HIHUHQFH,WLVSURSRVHGWKDWMRLQWVKHDUVWUHQJWKIRUEHDPVWKDWDUHDWPRVWLQZLGHUWKDQWKH columns can be determined using the methods presented above. )RUFROXPQVLQWKH¿UVWRUWRSVWRU\RIDPRPHQWIUDPHRUIRUPRPHQWIUDPHVZKHUHWKHDERYHDVVXPSWLRQUHJDUGLQJWKHORFDWLRQRIWKHLQÀHFWLRQSRLQWLVQRWVXLWDEOHVLPLODUDQDO\VHVFDQEHSHUIRUPHGWRGHWHUPLQH using appropriate assumptions for the case at hand. The results from the overall analysis of the moment frame can be used as DJXLGHWRORFDWHSRLQWVRILQÀHFWLRQLQWKHFROXPQV$OWHUQDWLYHO\ can be conservatively determined by setting HTXDOWR]HUR It is important to note that joint shear strength is evaluated in each direction of analysis separately; it is not required to consider interaction in two principal directions simultaneously. 11.4.2 Design Shear Strength

The design shear strength, of a cast-in-place beam-column joint must be greater than or equal to the required shear strength, $&, 7KHVWUHQJWKUHGXFWLRQIDFWRU LVHTXDOWRIRUVKHDULQDFFRUGDQFHZLWK $&,7DEOH LVGHWHUPLQHGXVLQJ$&,7DEOHDQGGHSHQGVRQWKHFRQWLQXLW\ The nominal shear strength of the joint, RIWKHFROXPQVDQGEHDPVIUDPLQJLQWRWKHMRLQWDQGZKHWKHUWKHMRLQWLVFRQ¿QHGRUQRWE\WUDQVYHUVHEHDPVVHH 7DEOH 

11-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 11.4 Nominal Joint Shear Strength, Vn Column

Continuous or a column extension is provided that VDWLV¿HV$&,

Beam in Direction of Analysis

Continuous or a beam extension is SURYLGHGWKDWVDWLV¿HV$&,

Confinement by Transverse Beams*

Vn (lbs)**

&RQ¿QHG 1RWFRQ¿QHG &RQ¿QHG

Other 1RWFRQ¿QHG Continuous or a beam extension is SURYLGHGWKDWVDWLV¿HV$&,

&RQ¿QHG 1RWFRQ¿QHG

Other &RQ¿QHG Other 1RWFRQ¿QHG * Transverse beams that satisfy the requirements of ACI 15.2.8 are considered to provide confinement (see Figure 11.3). Examples of the various joint types in this table are given in Figure 11.12. ** The modification factor, , that reflects the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength is equal to 0.75 for lightweight concrete and 1.0 for normalweight concrete.

([DPSOHVRIWKHYDULRXVMRLQWW\SHVLQ7DEOHDQGWKHFRUUHVSRQGLQJ DUHJLYHQLQ)LJXUH-RLQWVZLWK continuous columns (that is, columns frame into the top and bottom of the joint) or with a column extension that satLV¿HVWKHUHTXLUHPHQWVRI$&,DUHGHVLJQDWHG7\SH,-RLQWVZLWKQRQFRQWLQXRXVFROXPQVWKDWLVMRLQWVZKHUH columns frame into the bottom of the joints only, such as those in the top story of a building), are designated Type ,,7\SH$DQG7\SH%GHVLJQDWLRQVLQ)LJXUHFRUUHVSRQGWRMRLQWVZLWKFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRI DQDO\VLVRUZLWKEHDPH[WHQVLRQVWKDWVDWLVI\WKHUHTXLUHPHQWVLQ$&,7\SH&DQG7\SH'GHVLJQDWLRQVFRUUHVSRQGWRMRLQWVZLWKQRQFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRIDQDO\VLV$GHVLJQDWLRQRI³&´DWWKHIDFHRIDMRLQW PHDQVDWUDQVYHUVHEHDPSURYLGHVFRQ¿QHPHQWDWWKDWIDFHLQDFFRUGDQFHZLWK$&,$Q³1&´GHVLJQDWLRQ PHDQVWKDWQRWUDQVYHUVHEHDPLVSUHVHQWDWWKDWIDFHRUDWUDQVYHUVHEHDPLVSUHVHQWEXWGRHVQRWVDWLVI\WKHFRQ¿QHPHQWUHTXLUHPHQWVLQ$&, FDQEHREWDLQHGIURP)LJXUHIRUWKHYDULRXVFRPELQDWLRQVRIMRLQWW\SHV)RU The nominal shear strengths, example, a Type I-A joint consists of a continuous column and continuous beams in the direction of analysis with This joint type corresponds FRQ¿QHPHQWSURYLGHGE\WUDQVYHUVHEHDPVRQERWKIDFHVLQWKLVFDVH to an interior column located in the structure other than the top story. Similarly, a corner column in the top story of a structure corresponds to a Type II-D joint, and The term LVWKHHIIHFWLYHFURVVVHFWLRQDODUHDZLWKLQDMRLQWZKLFKLVJLYHQLQ$&,%\GH¿QLWLRQ is equal to the product of the joint depth and the effective joint width. The depth of the joint is always equal to the is equal depth of the column parallel to the direction of analysis, (see Figure 11.13). The effective joint width, to the width of the column perpendicular to the direction of analysis where the beams in the direction of analysis are For beams not as wide as the column, is equal as wide as or wider than the column. In such cases, and where is the beam width and is the perpendicular distance from the edge of to the lesser of the beam to the nearest side face of the column.

11-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Note 1 Transverse beam

C

C or NC

C

NC

Type A

Type B

C

C or NC

C

NC

Type C

Type D

Note 2

Note 2



Type I

Transverse beam

Type II

Plan

Elevation Direction of analysis

Joint Type

ܸ௡

I-A

24ߣඥ݂௖ᇱ ‫ܣ‬௝

I-B

20ߣඥ݂௖ᇱ ‫ܣ‬௝

I-C

20ߣඥ݂௖ᇱ ‫ܣ‬௝

I-D

15ߣඥ݂௖ᇱ ‫ܣ‬௝

II-A

20ߣඥ݂௖ᇱ ‫ܣ‬௝

II-B

15ߣඥ݂௖ᇱ ‫ܣ‬௝

II-C

15ߣඥ݂௖ᇱ ‫ܣ‬௝

II-D

12ߣඥ݂௖ᇱ ‫ܣ‬௝

Notes 1.Column is continuous or a column extension satisfying ACI 15.2.6 is provided ሺsee Figure 11.1ሻ. 2.Beam in direction of analysis is continuous or a beam extension satisfying ACI 15.2.7 is provided ሺsee Figure 11.2ሻ. 3.C – Transverse beam provides confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3ሻ. 4.NC – Transverse beam is not provided or transverse beam does not provide confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3ሻ.

Figure 11.12 Examples of joint types and the corresponding nominal joint shear strength, Vn.

11-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



‫ܣ‬௝ ൌ ܿଵ ‫ݓ‬

ܿଵ 

‫ ݔ‬

ܾ ‫ݓ‬

‫ݔ‬

ܿଶ  ‹”‡…–‹‘‘ˆƒƒŽ›•‹• ܾ ൅ ܿଵ   ˆˆ‡…–‹˜‡Œ‘‹–™‹†–Š‫ ݓ‬ൌ Ž‡••‡”‘ˆ ൝ ܾ ൅ ʹ‫ݔ‬

Figure 11.13 Effective cross-sectional area within a joint, Aj .

11.5 Transfer of Column Axial Force Through the Floor System 7KHUHTXLUHPHQWVLQ$&,FRQVLGHUWKHHIIHFWRIWKHVWUHQJWKRIWKHFRQFUHWHLQDÀRRUV\VWHPRQWKHD[LDO VWUHQJWKRIDFROXPQ,IWKHVWUHQJWKRIWKHFRQFUHWHLQWKHÀRRUV\VWHPLVVLJQL¿FDQWO\OHVVWKDQWKDWLQWKHFROXPQ FRQ¿QHPHQWRIWKHMRLQWE\WKHÀRRUV\VWHPPD\QRWEHDGHTXDWHZKLFKPD\KDYHDQLPSDFWRQORDGWUDQVIHU WKURXJKWKHÀRRUV\VWHP 7KHSURYLVLRQVLQ$&,PXVWEHVDWLV¿HGZKHUHWKHFRPSUHVVLYHVWUHQJWKRIWKHÀRRUV\VWHPLVOHVVWKDQ percent of the compressive strength of a column. For edge and corner columns, one of the following requirements PXVWEHVDWLV¿HG 1. &  RQFUHWHVSHFL¿HGIRUWKHFROXPQPXVWEHSODFHGLQWKHÀRRUV\VWHPDWWKHFROXPQORFDWLRQDQGPXVWH[WHQG DWOHDVWIWLQWRWKHÀRRUV\VWHPIURPWKHIDFHVRIWKHFROXPQIRUWKHIXOOGHSWKRIWKHÀRRUV\VWHP7KHFROXPQFRQFUHWHZLWKLQWKLV]RQHPXVWEHIXOO\LQWHJUDWHGZLWKWKHFRQFUHWHIRUWKHÀRRUV\VWHP  7KHGHVLJQVWUHQJWKRIDFROXPQWKURXJKDÀRRUV\VWHPPXVWEHFDOFXODWHGXVLQJWKHVSHFL¿HGFRQFUHWH FRPSUHVVLYHVWUHQJWKRIWKHÀRRUV\VWHP9HUWLFDOGRZHOVDQGWUDQVYHUVHUHLQIRUFHPHQWPXVWEHSURYLGHGWR achieve the required design strength, if required. ,QWKH¿UVWRSWLRQZKLFKLVVRPHWLPHVUHIHUUHGWRDV³SXGGOLQJ´ WZRGLIIHUHQWFRQFUHWHPL[WXUHVPXVWEHSURYLGHG LQWKHÀRRUV\VWHP7KHFROXPQDQGÀRRUV\VWHPFRQFUHWHPL[WXUHVPXVWUHPDLQSODVWLFORQJHQRXJKVRWKDWWKHWZR can be vibrated and well-integrated within the designated area.

11-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In the second option, the design strength of the column is checked using the concrete compressive strength of the ÀRRUV\VWHP,QFDVHVZKHUHWKHVWUHQJWKUHTXLUHPHQWVRIWKHFROXPQDUHQRWVDWLV¿HGGRZHOEDUVDQGWUDQVYHUVH reinforcement can be used to increase the design strength of the column. In addition to the two options described above, one additional option is available for interior columns: For interior columns laterally supported by beams of approximately equal depth that satisfy the continuity requirements in ACI DQGIRUVODEFROXPQMRLQWVVXSSRUWHGRQIRXUVLGHVE\WKHVODELWLVSHUPLWWHGWRFDOFXODWHWKHGHVLJQVWUHQJWK RIWKHFROXPQDWWKHMRLQWXVLQJDFRQFUHWHFRPSUHVVLYHVWUHQJWKHTXDOWRSHUFHQWRIWKHVSHFL¿HGFRPSUHVVLYH VWUHQJWKRIWKHFROXPQFRQFUHWHSOXVSHUFHQWRIWKHVSHFL¿HGFRPSUHVVLYHVWUHQJWKRIWKHÀRRUV\VWHPFRQFUHWH :KHQXWLOL]LQJWKLVRSWLRQWKHUDWLRRIWKHFROXPQFRQFUHWHVWUHQJWKWRWKHÀRRUV\VWHPFRQFUHWHVWUHQJWKPXVWQRW H[FHHGLQWKHGHVLJQFDOFXODWLRQV7KHUHIRUHWKHGHVLJQVWUHQJWKRIWKHFROXPQLQDQLQWHULRUMRLQWPXVWEHGHWHUmined using the following concrete strength, (11.15)

11.6 Examples 11.6.1 Example 11.1 – Check of Joint Shear Strength, Edge Column is Not Part of the LFRS: Building #1 (Framing Option C), SDC A

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHQRUWKVRXWKGLUHFWLRQIRUHGJHFROXPQ(RI%XLOGLQJ)UDPLQJ2SWLRQ& DWWKHVHFRQGÀRRUOHYHOVHH)LJXUH $VVXPHWKHIROORZLQJ WKHFROXPQLVQRWSDUWRIWKH/)56 WKH FROXPQLVLQVTXDUH WKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGWLHV EHDPVIUDPHLQWR WKHMRLQWRQWKUHHIDFHV WKHEHDPVDUHLQZLGHDQGLQGHHSDQG WKHEHDPVDUHUHLQIRUFHGZLWK EDUVDWWKHWRSRIWKHVHFWLRQDQGEDUVDWWKHERWWRPRIWKHVHFWLRQ$OVRDVVXPH*UDGHUHLQIRUFHPHQWDQG normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Check if the beam longitudinal reinforcement can be developed for tension in the edge column

'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGGHJUHHKRRNVDUHSURYLGHGDWWKH ends of the bars: Eq. (11.1)

For uncoated reinforcing bars,

Table 11.2

)RUKRRNHGEDUVZLWKRXWFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&, )RUKRRNHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUQRUPDOWRWKHSODQHRIWKHKRRNJUHDWHUWKDQ LQ For For normalweight concrete, Therefore,

Available development length

11-16

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHIRUHWKHEDUVFDQEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGGHJUHHKRRNVDWWKHHQGVRIWKH bars. 'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGKHDGVDUHSURYLGHGDWWKHHQGVRIWKHEDUV Eq. (11.2)

For uncoated reinforcing bars,

Table 11.3

)RUKHDGHGEDUVZLWKRXWSDUDOOHOWLHUHLQIRUFHPHQWVDWLVI\LQJWKHFRQGLWLRQVLQ$&,7DEOH )RUKHDGHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUWRWKHEDUJUHDWHUWKDQLQ For Therefore,

Available development length 7KHUHIRUHWKHEDUVFDQEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGKHDGVDWWKHHQGVRIWKHEDUV Step 2 – Determine the factored horizontal joint shear force

ACI 15.4.1.1

Because the joint is not part of the LFRS (that is, it is subjected to the effects from gravity loads only), and because WKHDUHDRIWKHQHJDWLYHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPLVJUHDWHUWKDQWKHDUHDRIWKHSRVLWLYHUHLQIRUFHPHQWWKH conservatively assuming following equation for sidesway to the left can be used to determine Eq. (11.14) Step 3 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of DQDO\VLVLVQRWFRQWLQXRXV7KHWUDQVYHUVHEHDPVSURYLGHFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&,EHFDXVHWKH (see Figure 11.3). Therefore, the joint type is I-C (see LQZLGWKVDUHJUHDWHUWKDQ )LJXUH  Because the beam in the direction of analysis is the same width as the column, the effective cross-sectional area within the joint is equal to the area of the column: Figure 11.13

The design shear strength of the joint is equal to the following for a Type I-C joint: Figure 11.12 11.6.2 Example 11.2 – Check of Joint Shear Strength, Edge Column is Part of the LFRS: Building #1 (Framing Option C), SDC A

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHQRUWKVRXWKGLUHFWLRQIRUHGJHFROXPQ&RI%XLOGLQJ)UDPLQJ2SWLRQ& DWWKHVHFRQGÀRRUOHYHOVHH)LJXUH $VVXPHWKHIROORZLQJ WKHFROXPQLVSDUWRIWKH/)56 WKHFROXPQ LVLQVTXDUH WKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGWLHV EHDPVIUDPHLQWRWKHMRLQW RQWKUHHIDFHV WKHEHDPVDUHLQZLGHDQGLQGHHSDQGDUHFHQWHUHGRQWKHFROXPQDQG WKHEHDPVDUH

11-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

UHLQIRUFHGZLWKEDUVDWWKHWRSRIWKHVHFWLRQDQGEDUVDWWKHERWWRPRIWKHVHFWLRQ$OVRDVVXPH*UDGH reinforcement and normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Check if the beam longitudinal reinforcement can be developed for tension in the edge column

'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGGHJUHHKRRNVDUHSURYLGHGDWWKH ends of the bars: Eq. (11.1)

For uncoated reinforcing bars,

Table 11.2

)RUKRRNHGEDUVZLWKRXWFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&, )RUKRRNHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUQRUPDOWRWKHSODQHRIWKHKRRNJUHDWHUWKDQ LQ For For normalweight concrete, Therefore,

Available development length 7KHUHIRUHWKHEDUVFDQQRWEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGGHJUHHKRRNVDWWKHHQGVRIWKH bars. $VVXPLQJWKHVL]HRIWKHFROXPQFDQQRWEHLQFUHDVHGDQGQRFRQ¿QLQJUHLQIRUFHPHQWLVSURYLGHGGHFUHDVHWKHVL]H RIWKHORQJLWXGLQDOEDUV,WFDQEHGHWHUPLQHGWKDWWRSEDUVDQGERWWRPEDUVFDQEHIXOO\GHYHORSHGLQWHQVLRQZLWKLQWKHFROXPQXVLQJVWDQGDUGGHJUHHKRRNVDWWKHHQGVRIWKHEDUV 'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGKHDGVDUHSURYLGHGDWWKHHQGVRIWKHEDUV Eq. (11.2)

For uncoated reinforcing bars, )RUKHDGHGEDUVZLWKRXWSDUDOOHOWLHUHLQIRUFHPHQWVDWLVI\LQJWKHFRQGLWLRQVLQ$&,7DEOH )RUKHDGHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUWRWKHEDUJUHDWHUWKDQLQ For Therefore,

Available development length 11-18

Table 11.3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHIRUHWKHEDUVFDQQRWEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGKHDGVDWWKHHQGVRIWKHEDUV ,WFDQEHGHWHUPLQHGWKDWWRSEDUVDQGERWWRPEDUVFDQEHIXOO\GHYHORSHGLQWHQVLRQZLWKLQWKHFROXPQ using standard heads at the ends of the bars. The headed bars are used in the remainder of this example. Step 2 – Determine the factored horizontal joint shear force

The following equation can be used to determine

ACI 15.4.1.1

conservatively assuming Eq. (11.10)

Step 3 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of DQDO\VLVLVFRQWLQXRXV7KHWUDQVYHUVHEHDPSURYLGHVFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&,EHFDXVHWKHLQ (see Figure 11.3). Therefore, the joint type is I-B (see Figure width is greater than   Because the beams in the direction of analysis are narrower than the column, the effective joint width, the following:

is equal to

Figure 11.13

The design shear strength of the joint is equal to the following for a Type I-B joint: Figure 11.12 11.6.3 Example 11.3 – Check of Joint Shear Strength, Corner Column is Part of the LFRS: Building #1 (Framing Option B), SDC A

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHQRUWKVRXWKGLUHFWLRQIRUFRUQHUFROXPQ(RI%XLOGLQJ)UDPLQJ2SWLRQ% DWWKHVHFRQGÀRRUOHYHOVHH)LJXUH $VVXPHWKHIROORZLQJ WKHFROXPQLVSDUWRIWKH/)56 WKHFROXPQ LVLQVTXDUH WKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGWLHV EHDPVIUDPHLQWRWKHMRLQW RQWZRIDFHV WKHEHDPVDUHLQZLGHDQGLQGHHSDQGDUHÀXVKZLWKWKHRXWVLGHHGJHRIWKHFROXPQDQG  WKHEHDPVDUHUHLQIRUFHGZLWKEDUVDWWKHWRSRIWKHVHFWLRQDQGEDUVDWWKHERWWRPRIWKHVHFWLRQ$VVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKWFRQFUHWHZLWK 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Check if the beam longitudinal reinforcement can be developed for tension in the corner column

'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGGHJUHHKRRNVDUHSURYLGHGDWWKH ends of the bars: Eq. (11.1)

For uncoated reinforcing bars,

Table 11.2

11-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUKRRNHGEDUVZLWKRXWFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&, )RUKRRNHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUQRUPDOWRWKHSODQHRIWKHKRRNJUHDWHUWKDQ LQ For For normalweight concrete, Therefore,

Available development length 7KHUHIRUHWKHEDUVFDQEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGGHJUHHKRRNVDWWKHHQGVRIWKHEDUV 'HWHUPLQHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVDVVXPLQJVWDQGDUGKHDGVDUHSURYLGHGDWWKHHQGVRIWKHEDUV Eq. (11.2)

For uncoated reinforcing bars,

Table 11.3

)RUKHDGHGEDUVZLWKRXWSDUDOOHOWLHUHLQIRUFHPHQWVDWLVI\LQJWKHFRQGLWLRQVLQ$&,7DEOH )RUKHDGHGEDUVWHUPLQDWLQJZLWKLQWKHFROXPQFRUHZLWKVLGHFRYHUWRWKHEDUJUHDWHUWKDQLQ For Therefore,

Available development length 7KHUHIRUHWKHEDUVFDQEHIXOO\GHYHORSHGZLWKLQWKHFROXPQZLWKVWDQGDUGKHDGVDWWKHHQGVRIWKHEDUV Step 2 – Determine the factored horizontal joint shear force

ACI 15.4.1.1

%HFDXVHWKHDUHDRIWKHQHJDWLYHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPLVJUHDWHUWKDQWKHDUHDRIWKHSRVLWLYHUHLQconservatively assuming forcement, the following equation for sidesway to the left can be used to determine Eq. (11.14) Step 3 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of DQDO\VLVLVQRWFRQWLQXRXV7KHWUDQVYHUVHEHDPSURYLGHVFRQ¿QHPHQWLQDFFRUGDQFHZLWK$&,EHFDXVHWKH (see Figure 11.3). Therefore, the joint type is I-D (see 18-in. width is greater than )LJXUH 

11-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Because the beam in the direction of analysis is narrower than the column, the effective joint width, the following:

is equal to

Figure 11.13

The design shear strength of the joint is equal to the following for a Type I-D joint: Figure 11.12 11.6.4 Example 11.4 – Check of Joint Shear Strength, Edge Column is Part of the SFRS: Building #1 (Framing Option B), SDC B

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHQRUWKVRXWKGLUHFWLRQIRUHGJHFROXPQ'RI%XLOGLQJ)UDPLQJ2SWLRQ%DW WKHVHFRQGÀRRUOHYHOVHH)LJXUH $VVXPHWKHIROORZLQJ WKHFROXPQLVSDUWRIWKH/)56 WKHFROXPQLV LQVTXDUH WKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGWLHV EHDPVIUDPHLQWRWKHMRLQWRQ WZRIDFHV WKHEHDPVDUHLQZLGHDQGLQGHHSDQG WKHEHDPVDUHUHLQIRUFHGZLWKEDUVDWWKHWRS RIWKHVHFWLRQDQGEDUVDWWKHERWWRPRIWKHVHFWLRQ$OVRDVVXPH*UDGHUHLQIRUFHPHQWDQGQRUPDOZHLJKW concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Determine the factored horizontal joint shear force

Determine the shear force in the column,

conservatively neglecting

ACI 15.4.1.1

and Eq. (11.7)

Eq. (11.6)

Eq. (11.8)

)URPDQDO\VLVRIWKHVWUXFWXUHWKHSRLQWRILQÀHFWLRQLQWKHFROXPQDERYHWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURP WKHERWWRPRIWKHFROXPQDQGWKHSRLQWRILQÀHFWLRQLQWKHFROXPQEHORZWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURP the top of the column. Therefore,

The factored shear force in the joint is equal to the following: Eq. (11.10)

11-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of analysis is continuous. There are no beams in the transverse direction. Therefore, the joint type is I-B (see Figure 11.12). Because the beam in the direction of analysis is the same width as the column, the effective cross-sectional area within the joint is equal to the area of the column: Figure 11.13

The design shear strength of the joint is equal to the following for a Type I-B joint: Figure 11.12 Step 3 – Determine the detailing for the joint

%HFDXVHWKHMRLQWLVSDUWRIWKH6)56WKHGHWDLOLQJUHTXLUHPHQWVLQ$&,WKURXJKPXVWEHVDWLV¿HG The details for this joint are given in Figure 11.14 (see Figure 11.4). ʹᇱ ǦͲԣ



ͺǦ͓ͺ

͓Ͷ–‹‡•̷ͳ͸ԣሺ–›’Ǥሻ

ʹᇱ ǦͲԣ

͵Ǧ͓Ͷ–‹‡•̷ͺԣ

–Š‡””‡‹ˆ‘”…‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 11.14 Reinforcement details for the joint in Example 11.4.

11-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

11.6.5 Example 11.5 – Adequacy of Transfer of Column Axial Force, Interior Column: Building #1 (Framing Option B), SDC A

'HWHUPLQHWKHDGHTXDF\RIWKHWUDQVIHURIFROXPQD[LDOIRUFHIRUWKHLQWHULRUFROXPQ&RI%XLOGLQJ)UDPLQJ 2SWLRQ%VHH)LJXUH $VVXPHWKHFROXPQLVLQVTXDUH$OVRDVVXPHWKHFRQFUHWHIRUWKHFROXPQLVQRUPDODQGWKHFRQFUHWHIRUWKHÀRRUV\VWHPLVQRUPDOZHLJKWFRQFUHWHZLWK weight with 'HVLJQGDWDDUHJLYHQLQ6HFW Solution

7KHSURYLVLRQVRI$&,PXVWEHVDWLV¿HGZKHUHWKHFRPSUHVVLYHVWUHQJWKRIWKHÀRRUV\VWHPLVOHVVWKDQ percent of the compressive strength of the column. In this example,

VRRQHRIWKHUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG

Because the column is an interior column that is laterally supported on all four faces, any one of three requirements LQ$&,FDQEHXVHGWRVDWLVI\WUDQVIHURIFROXPQD[LDOIRUFHWKURXJKWKHMRLQW ,QWKH¿UVWUHTXLUHPHQWWKHSVLFROXPQFRQFUHWHPXVWEHIXOO\LQWHJUDWHGZLWKWKHSVLÀRRUV\VWHPFRQFUHWHZLWKLQD]RQHH[WHQGLQJLQWRWKHÀRRUV\VWHPDWOHDVWIWIURPWKHIRXUIDFHVRIWKHFROXPQIRUWKHIXOOGHSWK RIWKHÀRRUV\VWHP ,QWKHVHFRQGUHTXLUHPHQWWKHLQFROXPQPXVWEHGHVLJQHGDWWKHMRLQWXVLQJWKHSVLÀRRUV\VWHPFRQFUHWH Vertical dowels and transverse reinforcement must be provided to achieve the design strength, if required. ,QWKHWKLUGUHTXLUHPHQWWKHLQFROXPQPXVWEHGHVLJQHGDWWKHMRLQWXVLQJWKHIROORZLQJFRQFUHWHFRPSUHVVLYH strength:

Eq. (11.15)

11-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

11-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 12 EARTHQUAKE-RESISTANT STRUCTURES — OVERVIEW 12.1 Overview Design and detailing requirements for structures assigned to Seismic Design Category (SDC) B through F are given in ACI Chapter 18, including those for members designated as part of the seismic-force-resisting system (SFRS) in buildings assigned to SDC B and C. For buildings assigned to SDC D through F, requirements are given for members designated as part of the SFRS and for members not designated as part of the SFRS. The provisions in ACI Chapter 18 are considered to be the minimum required in order for cast-in-place concrete structures to respond in the nonlinear range when subjected to design-level ground motions without critical strength GHFD\7RDFKLHYHDQLQHODVWLFUHVSRQVHSDUWLFXODUGHWDLOLQJUHTXLUHPHQWVPXVWEHVDWLV¿HGZKLFKDUHUHODWHGWRWKH SDC. The purpose of the detailing is to dissipate energy caused by the ground motion; even though a decrease in structure stiffness is likely to occur during a design earthquake event, the design and detailing requirements ensure that the strength of the structure is not compromised. This chapter provides an overview of the design and detailing requirements in ACI Chapter 18. Included is information on the various structural systems available to resist the effects from earthquakes based on SDC.

12.2 Seismic Design Category In order to apply the applicable design and detailing requirements in ACI Chapter 18, all structures must be assigned WRD6'&LQDFFRUGDQFHZLWK$&,$&,  The SDC is a trigger mechanism for many seismic requirements, including the following: • • • •

Permissible SFRS Limitations on building height Consideration of structural irregularities The need for additional special inspections, structural testing, and structural observation for seismic resistance

$PHWKRGWRGHWHUPLQHWKH6'&LQDFFRUGDQFHZLWK$6&(6(,LVJLYHQLQ6HFWLRQDQG)LJXUHRIWKLV publication.

12.3 Design and Detailing Requirements Structures assigned to SDC A need not satisfy the requirements in ACI Chapter 18. The requirements that must be VDWLV¿HGIRUVWUXFWXUHVDVVLJQHGWR6'&%WKURXJK)DUHJLYHQLQ$&,WKURXJKVHH7DEOH  Table 12.1 Requirements to be Satisfied Based on SDC SDC

ACI Section No.

B



C

DQG

D, E, or F

WKURXJK WKURXJK

7KHVHFWLRQVRI$&,&KDSWHUWKDWPXVWEHVDWLV¿HGLQW\SLFDODSSOLFDWLRQVEDVHGRQ6'&DUHJLYHQLQ7DEOH VHH$&,7DEOH5 7KHDSSOLFDEOHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVLQ$&,&KDSWHUVWKURXJKDQG WKURXJKPXVWDOVREHVDWLV¿HGDGDVKLQ7DEOHVLJQL¿HVWKHDSSOLFDWLRQLVJRYHUQHGE\WKHUHTXLUHPHQWVRI those chapters for a given SDC (that is, no additional requirements are given in ACI Chapter 18).

12-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 12.2 Sections of ACI Chapter 18 That Must be Satisfied in Typical Applications Based on SDC* ACI Chapter 18 Section No. Application SDC B

SDC C

SDC D, E, or F

Analysis and design requirements





 

Material requirements

ņ

ņ

WKURXJK

Design and detailing of frame members

18.3



18.6 through 18.9

Design and detailing of structural walls and coupling beams

ņ

ņ



Design and detailing of diaphragms and trusses

ņ





Design and detailing of foundations

ņ

18.13

18.13

Design and detailing of frames members not designated as part of the SFRS

ņ

ņ



* The applicable design and detailing requirements in ACI Chapters 1 through 17 and 19 through 26 must also be satisfied. Where the requirements in ACI Chapter 18 conflict with other chapters, the requirements of ACI Chapter 18 govern (ACI 18.2.1.2). ** The requirements of ACI 18.12 must be satisfied for buildings assigned to SDC C only where precast concrete diaphragms are used as part of the SFRS (ACI 18.12.1.2).

12.4 Structural Systems 12.4.1 Overview

For jurisdictions that have adopted the IBC, structural systems designated to part of the SFRS are restricted to those JLYHQLQ$6&(6(,7DEOHRWKHUV\VWHPVDUHSHUPLWWHGSURYLGHGWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG $&, 7KHFDVWLQSODFHUHLQIRUFHGFRQFUHWHV\VWHPVLQ$6&(6(,7DEOHDUHJLYHQLQ7DEOH 1RWHWKDWWKHQXPEHULQ7DEOHSUHFHGLQJHDFKV\VWHPLVWKHQXPEHUIURP$6&(6(,7DEOH6HLVPLF structural system limitations, and building height limits are given in GHVLJQFRHI¿FLHQWDQGIDFWRUV WKHWDEOHEDVHGRQ6'&'HVFULSWLRQVRIHDFK6)56DUHJLYHQLQ7DEOHDQGDGGLWLRQDOLQIRUPDWLRQLVSURYLGHG in the following sections (Note: Cantilevered column systems are not covered in this publication). Information on WKHGHVLJQFRHI¿FLHQWDQGIDFWRUVLQ7DEOHLVJLYHQLQ7DEOH Table 12.3 Design Coefficient and Factors for Seismic-Force-Resisting Systems of Reinforced Concrete Structural System Limitations Including Structural Height Limits (ft) Seismic-Force-Resisting System

R

ȍo

Cd

Seismic Design Category B

C

D

E

F

A. Bearing Wall Systems 1. Special reinforced concrete shear walls



1



NL

NL







2UGLQDU\UHLQIRUFHGFRQFUHWHVKHDUZDOOV



1



NL

NL

NP

NP

NP

6SHFLDOUHLQIRUFHGFRQFUHWHVKHDUZDOOV

6

1



NL

NL







2UGLQDU\UHLQIRUFHGFRQFUHWHVKHDUZDOOV



1 1

NL

NL

NP

NP

NP

B. Building Frame Systems

(table continued on next page)

12-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 12.3 Design Coefficient and Factors for Seismic-Force-Resisting Systems of Reinforced Concrete (cont.) Structural System Limitations Including Structural Height Limits (ft) Seismic-Force-Resisting System

R

ȍo

Cd

Seismic Design Category B

C

D

E

F

C. Moment-Resisting Frame Systems 3. Special reinforced concrete moment frames

8

3

1

NL

NL

NL

NL

NL

,QWHUPHGLDWHUHLQIRUFHGFRQFUHWHPRPHQWIUDPHV



3

1

NL

NL

NP

NP

NP

2UGLQDU\UHLQIRUFHGFRQFUHWHPRPHQWIUDPHV

3

3

1

NL

NP

NP

NP

NP

''XDO6\VWHPVZLWK6SHFLDO0RPHQW)UDPHV&DSDEOHRI5HVLVWLQJDW/HDVWRI3UHVFULEHG6HLVPLF)RUFHV 3. Special reinforced concrete shear walls



1 1

NL

NL

NL

NL

NL

2UGLQDU\UHLQIRUFHGFRQFUHWHVKHDUZDOOV

6

1

NL

NL

NP

NP

NP



('XDO6\VWHPVZLWK,QWHUPHGLDWH0RPHQW)UDPHV&DSDEOHRI5HVLVWLQJDW/HDVWRI3UHVFULEHG Seismic Forces 6SHFLDOUHLQIRUFHGFRQFUHWHVKHDUZDOOV

61 1

8. Ordinary reinforced concrete shear walls

1 1 1



NL

NL







NL

NL

NP

NP

NP

NP

NP

F. Shear Wall-Frame Interactive System with Ordinary Reinforced Concrete Moment Frames and Ordinary Reinforced Concrete Shear Walls —

1 1



NL

NP

NP

NL = No height limit NP = Not permitted A “shear wall” is defined in ACI 2.3 as a “structural wall”.

7KHWHUPV³RUGLQDU\´³LQWHUPHGLDWH´DQG³VSHFLDO´LQ7DEOHDUHDVVRFLDWHGZLWKWKHOHYHOVRILQHODVWLFUHVSRQVH assumed in the determination of the design earthquake forces. A higher level of inelastic response means more stringent requirements for member proportioning and detailing, which results in an accompanying increase in deformation capacity. Table 12.4 Seismic-Force-Resisting Systems of Reinforced Concrete System

Description

Bearing wall

Bearing walls provide support for most or all the gravity loads, and resistance to seismic forces is provided by the same bearing walls acting as shear walls. These systems do not have an essentially complete space frame that provides support for gravity loads.

Building frame

A structural system with an essentially complete space frame that supports the gravity loads and shear walls that resist the seismic forces. It is assumed that the seismic forces are allocated to the shear walls only; no interaction is considered between the shear walls and the frames. (table continued on next page)

12-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 12.4 Seismic-Force-Resisting Systems of Reinforced Concrete (cont.) System

Description

Moment-resisting frame

Gravity loads are supported by an essentially complete space frame and seismic forces DUHUHVLVWHGSULPDULO\E\ÀH[XUDODFWLRQRIGHVLJQDWHGIUDPHPHPEHUVWKHHQWLUHVSDFH frame or selected portions of the space frame may be designated as the SFRS).

Dual

An essentially complete space frame provides support for gravity loads and resistance to seismic forces is provided by moment-resisting frames and shear walls. The frames and shear walls are designed to resist seismic forces in proportion to their relative rigidities. An additional requirement is that the moment frames alone must be capable RIUHVLVWLQJDWOHDVWSHUFHQWRIWKHVHLVPLFIRUFHVLHWKHIUDPHVDFWDVDEDFNXSWR the shear walls).

Shear wall-frame interactive

An essentially complete space frame provides support for gravity loads and resistance to seismic forces is provided by moment-resisting frames and shear walls, which are designed to resist seismic forces in proportion to their relative rigidities. The shear VWUHQJWKRIWKHVKHDUZDOOVPXVWEHDWOHDVWSHUFHQWRIWKHGHVLJQVWRU\VKHDUDWHDFK VWRU\DQGWKHPRPHQWUHVLVWLQJIUDPHVPXVWEHFDSDEOHRIUHVLVWLQJDWOHDVWSHUFHQW of the design story shear in every story.

Table 12.5 Design Coefficient and Factors Related to Seismic-Force-Resisting Systems Design Coefficient and Factors

Description

7KLVFRHI¿FLHQWDFFRXQWVIRUWKHDELOLW\RID6)56WRUHVSRQGWRJURXQGVKDNLQJLQ a ductile manner without loss of load-carrying capacity. It basically represents the UDWLRRIWKHIRUFHVWKDWZRXOGGHYHORSXQGHUWKHJURXQGPRWLRQVSHFL¿HGLQ$6&( 6(,LIWKHVWUXFWXUHKDGUHVSRQGHGWRWKHJURXQGPRWLRQLQDOLQHDUHODVWLFPDQQHU Response PRGL¿FDWLRQ FRHI¿FLHQW

A system with no ability to respond in a ductile manner has an while systems capable of a highly ductile response have an

equal to 1, equal to 8.

7KHVHLVPLFIRUFHHIIHFWVRQDEXLOGLQJDUHHVVHQWLDOO\UHGXFHGE\WKLVFRHI¿FLHQW Although the required design strength decreases as increases, the design and detailing requirements substantially increase with increasing

Overstrength factor,

'HÀHFWLRQ DPSOL¿FDWLRQ factor,

12-4

The purpose of this factor is to ensure that critical elements in the SFRS that are not within anticipated hinge regions remain essentially elastic during the design-level earthquake. In the design and detailing of certain members, such as collector elements, the reactions obtained from an analysis based on the code-prescribed forces to ensure that such elements remain elastic for the duramust be multiplied by tion of ground shaking. This factor is used to adjust the lateral displacements determined for a structure using the code-prescribed seismic forces to the actual anticipated lateral displacements during the design-level earthquake. are equal to or slightly less than the corresponding The more Values of ), the greater the difference is ductile a system is (that is, the greater the between the values of and

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

12.4.2 Bearing Wall Systems Overview

In a bearing wall system, bearing walls provide support for most or all of the gravity loads, and resistance to seismic IRUFHVLVSURYLGHGE\WKHVDPHEHDULQJZDOOVDFWLQJDVVKHDUZDOOVVHH)LJXUH  

‡ƒ”‹‰™ƒŽŽሺ–›’Ǥሻ

Figure 12.1 Example of a building with a bearing wall system.

SDC B

Ordinary reinforced concrete shear walls are permitted to be used in structures assigned to SDC B without any limiWDWLRQV6XFKZDOOVPXVWVDWLVI\WKHDSSOLFDEOHUHTXLUHPHQWVRI$&,&KDSWHUVWRDQGWRWKHSURYLVLRQV RI$&,&KDSWHUQHHGQRWEHVDWLV¿HG$QRUGLQDU\UHLQIRUFHGFRQFUHWHVKHDUZDOOV\VWHPLVWKHPLQLPXPV\Vtem type that needs to be provided for SDC B; a system listed for a higher SDC can always be provided if desired (which in this case would be a system with special reinforced concrete shear walls), as long as all the applicable GHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWKDWV\VWHPDUHVDWLV¿HGWKLVLVWUXHIRUDQ\W\SHRIV\VWHPFRYHUHGLQWKHIROlowing sections). SDC C

Ordinary reinforced concrete shear walls are permitted to be used in structures assigned to SDC C without any limitations. It is assumed the design and detailing requirements in ACI Chapter 11 are compatible with the anticipated level of inelastic response when the walls are subjected to the effects from moderately strong ground motion. SDC D, E, or F

Special reinforced concrete shear walls are required in buildings assigned to SDC D, E, or F. The building height is OLPLWHGWRIWIRUVWUXFWXUHVDVVLJQHGWR6'&'DQG(DQGWRIWIRUVWUXFWXUHVDVVLJQHGWR6'&)7KHGHVLJQ DQGGHWDLOLQJUHTXLUHPHQWVRI$&,WKURXJKDQGPXVWEHVDWLV¿HGIRUZDOOVLQVWUXFWXUHVDVVLJQHG to SDC D, E, or F.

12-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

12.4.3 Building Frame Systems Overview

A building frame system is a structural system with an essentially complete space frame that supports the gravity ORDGVDQGVKHDUZDOOVWKDWUHVLVWWKHVHLVPLFIRUFHVVHHIRUH[DPSOH%XLOGLQJLQ)LJXUHRIWKLVSXEOLFDWLRQ  It is assumed the seismic forces are allocated to the shear walls only; no interaction is considered between the shear walls and the frames. SDC B

Ordinary reinforced concrete shear walls are permitted to be used in buildings assigned to SDC B with no limitaWLRQV7KHDSSOLFDEOHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVRI$&,&KDSWHUVWRDQGWRPXVWEHVDWLV¿HG SDC C

6WUXFWXUHVDVVLJQHGWR6'&&DUHSHUPLWWHGWRXWLOL]HRUGLQDU\UHLQIRUFHGFRQFUHWHVKHDUZDOOVZLWKQROLPLWDWLRQV Like in the case of bearing wall systems, it is assumed the design and detailing requirements in ACI Chapter 11 are compatible with the anticipated level of inelastic response. SDC D, E, or F

Special reinforced concrete shear walls are required to be used in a building frame system (along with the applicable building height limits) in structures assigned to SDC D and higher. The deformational compatibility requirements of $&,PXVWDOVREHVDWLV¿HG7KHEHDPFROXPQIUDPHVPXVWEHGHVLJQHGWRUHVLVWWKHHIIHFWVFDXVHGE\WKHODWHUDOGHÀHFWLRQVGXHWRWKHVHLVPLFIRUFHVEHFDXVHWKH\DUHFRQQHFWHGWRWKHZDOOVE\WKHGLDSKUDJPDWHDFKOHYHO,Q other words, the frame members, which are not designated as part of the SFRS, must be able to support the tributary gravity loads when subjected to the design displacements caused by the seismic forces. 12.4.4 Moment-Resisting Frame Systems Overview

In a moment-resisting frame system, gravity loads are supported by an essentially complete space frame and seismic IRUFHVDUHUHVLVWHGSULPDULO\E\ÀH[XUDODFWLRQRIGHVLJQDWHGIUDPHPHPEHUVVHHIRUH[DPSOH%XLOGLQJ)UDPing Option C in Figure 1.1 of this publication). The entire space frame or selected portions of the space frame may be designated as the SFRS. SDC B

An ordinary reinforced concrete moment is permitted to be used in buildings assigned to SDC B with no limitations. ,QDGGLWLRQWRWKHUHTXLUHPHQWVRI$&,&KDSWHUVWRDQGWRWKHUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG SDC C

6WUXFWXUHVDVVLJQHGWR6'&&DUHSHUPLWWHGWRXWLOL]HLQWHUPHGLDWHUHLQIRUFHGFRQFUHWHPRPHQWIUDPHVZLWKQROLPLWDWLRQV7KHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG SDC D, E, or F

Special reinforced concrete moment frames are required in structures assigned to SDC D, E, or F. This system can EHXVHGZLWKRXWDQ\OLPLWDWLRQVDQGPXVWEHGHVLJQHGDQGGHWDLOHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&, WKURXJKDQG$&,WKURXJK 12.4.5 Dual Systems Overview

In a dual system, an essentially complete space frame provides support for gravity loads and resistance to seismic IRUFHVLVSURYLGHGE\PRPHQWUHVLVWLQJIUDPHVDQGE\VKHDUZDOOVVHHIRUH[DPSOH%XLOGLQJLQ)LJXUHRI this publication). The frames and shear walls are designed to resist seismic forces in proportion to their relative

12-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ULJLGLWLHV$QDGGLWLRQDOUHTXLUHPHQWLVWKDWWKHPRPHQWIUDPHVDORQHPXVWEHFDSDEOHRIUHVLVWLQJDWOHDVWSHUFHQW of the seismic forces (i.e., the frames act as a backup to the shear walls). SDC B

$Q\RIWKHGXDOV\VWHPVOLVWHGLQ$6&(6(,7DEOHDUHSHUPLWWHGWREHXVHGLQVWUXFWXUHVDVVLJQHGWR6'&% ,WLVFRPPRQIRUVKHDUZDOOIUDPHLQWHUDFWLYHV\VWHPVWREHVSHFL¿HGLQVXFKFDVHV SDC C

A dual system with intermediate moment frames and ordinary reinforced concrete shear walls is permitted as a PLQLPXPIRUVWUXFWXUHVDVVLJQHGWR6'&&7KHPRPHQWIUDPHVZKLFKPXVWEHGHVLJQHGWRLQGHSHQGHQWO\UHVLVW SHUFHQWRIWKHFRGHSUHVFULEHGVHLVPLFIRUFHVPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,7KHUHTXLUHPHQWVRI$&, &KDSWHUVWRDQGWRPXVWEHVDWLV¿HGIRUWKHZDOOV SDC D, E, or F

Dual systems with special moment frames and special reinforced concrete shear walls are required for structures assigned to SDC D, E, or F without limitations. Each component must be designed and detailed in accordance with WKHDSSOLFDEOHSURYLVLRQVRI$&,$GXDOV\VWHPZLWKLQWHUPHGLDWHPRPHQWIUDPHVDQGVSHFLDOUHLQIRUFHG FRQFUHWHVKHDUZDOOVLVDOVRSHUPLWWHGLQ$6&(6(,7DEOHZLWKWKHIROORZLQJOLPLWDWLRQV IRU6'&'WKH EXLOGLQJKHLJKWLVOLPLWHGWRIWDQG IRU6'&(DQG)WKHEXLOGLQJKHLJKWLVOLPLWHGWRIW7KHXVHRILQWHUPHGLDWHPRPHQWIUDPHVDVSDUWRIDGXDOV\VWHPLQ6'&'(RU)LVQRWUHFRPPHQGHGVHH$&,5  12.4.6 Shear Wall-Frame Interactive Systems

Shear wall-frame interactive systems are similar to dual systems in that an essentially complete space frame provides support for gravity loads and resistance to seismic forces is provided by moment-resisting frames and by shear walls, which are designed to resist lateral forces in proportion to their relative rigidities. Shear wall-frame interactive systems with ordinary reinforced concrete moment frames and ordinary reinforced concrete shear walls are permitted to be used in structures assigned to SDC B with no limitations; dual systems are UHTXLUHGLQVWUXFWXUHVDVVLJQHGWR6'&&DQGDERYH7KHVKHDUVWUHQJWKRIWKHVKHDUZDOOVPXVWEHDWOHDVWSHUFHQWRIWKHGHVLJQVWRU\VKHDUDWHDFKVWRU\DQGWKHPRPHQWUHVLVWLQJIUDPHVPXVWEHFDSDEOHRIUHVLVWLQJDWOHDVW SHUFHQWRIWKHGHVLJQVWRU\VKHDULQHYHU\VWRU\$6&(6(, 2WKHUWKDQWKHUHTXLUHPHQWVLQ$&,IRU ordinary moment frames, no special seismic design and detailing requirements are needed for this system.

12-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

12-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 13 EARTHQUAKE-RESISTANT STRUCTURES — SDC B and C 13.1 Overview This chapter covers the design and detailing requirements in ACI Chapter 18 for the following: • )UDPHPHPEHUV$&,DQG • Foundations (ACI 18.13) Ordinary moment frames are permitted to be used as a minimum for structures assigned to Seismic Design Category 6'& %ZLWKQROLPLWDWLRQVVHH7DEOHRIWKLVSXEOLFDWLRQ 5HTXLUHPHQWVIRURUGLQDU\PRPHQWIUDPHVDUH given in ACI 18.3. For structures assigned to SDC C, intermediate moment frames are permitted to be used with no OLPLWDWLRQV5HTXLUHPHQWVIRULQWHUPHGLDWHPRPHQWIUDPHVDUHJLYHQLQ$&, 7KHUHTXLUHPHQWVLQ$&,&KDSWHUPXVWEHVDWLV¿HGIRUIRXQGDWLRQVVXSSRUWLQJVWUXFWXUHVDVVLJQHGWR6'&%7KH DSSOLFDEOHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWKLVFDVHDUHFRYHUHGLQ&KDSWHURIWKLVSXEOLFDWLRQ5HTXLUHPHQWVIRUIRXQGDWLRQVVXSSRUWLQJVWUXFWXUHVDVVLJQHGWR6'&&DUHJLYHQLQ6HFWLRQEHORZ No requirements are given in ACI Chapter 18 for the design and detailing of cast-in-place reinforced concrete diaSKUDJPVLQEXLOGLQJVDVVLJQHGWR6'&%RU&7KHUHTXLUHPHQWVLQ$&,&KDSWHUDUHDSSOLFDEOHLQVXFKFDVHVVHH Chapter 9 of this publication). Collector elements must be designed for the applicable forces determined in accorGDQFHZLWK$6&(6(, Ordinary reinforced concrete shear walls are permitted to be used in bearing wall, building frame, and shear wallIUDPHLQWHUDFWLYHVHLVPLFIRUFHUHVLVWLQJV\VWHPV6)56 LQVWUXFWXUHVDVVLJQHGWR6'&%RU&VHH7DEOHRI WKLVSXEOLFDWLRQ $VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQLWLVDVVXPHGWKHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWV given in ACI Chapter 11 for ordinary shear walls are compatible with the anticipated level of inelastic response when the walls are subjected to the effects from moderately strong ground motion associated with SDC C. Design and detailing requirements for ordinary walls are given in Chapter 8 of this publication. It is important to note that the applicable seismic design and detailing requirements in ACI Chapter 18 must be satis¿HGHYHQLQFDVHVZKHUHVWUHQJWKGHVLJQORDGFRPELQDWLRQVLQFOXGLQJHDUWKTXDNHHIIHFWVGRQRWJRYHUQWKHGHVLJQRI the structural member. For example, if the combined effects due to gravity and wind forces govern with respect to WKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWLQDEHDPWKDWLVSDUWRIDPRPHQWUHVLVWLQJIUDPHLQ6'&%WKHVHLVPLF ÀH[XUDOGHWDLOLQJUHTXLUHPHQWVIRU6'&%LQ$&,PXVWEHVDWLV¿HGHYHQWKRXJKZLQGORDGHIIHFWVJRYHUQWKH design.

13.2 Ordinary Moment Frames (SDC B) 13.2.1 Overview

Design and detailing requirements for beams, columns, and beam-column joints in ordinary moment frames are given in ACI 18.3. The provisions in ACI 18.3 are not applicable to moment frames consisting of slabs and columns; the requirements in ACI Chapter 8 must be used to design and detail two-way slab-column moment frames in VWUXFWXUHVDVVLJQHGWR6'&%VHH&KDSWHURIWKLVSXEOLFDWLRQ  Gravity and lateral load effects on members in an ordinary moment frame are obtained from an analysis method in $&,'HVLJQVWUHQJWKORDGFRPELQDWLRQVIRUVWUXFWXUHVDVVLJQHGWR6'&%DUHJLYHQLQ7DEOHIRUJUDYLW\ ZLQGDQGVHLVPLFORDGHIIHFWVVHH6HFWLRQRIWKLVSXEOLFDWLRQ 

13-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.1 Strength Design Load Combinations for Structures Assigned to SDC B ACI Equation Number

ASCE/SEI 7 Load Combination

D

1

E



F

3

G



H

6

I



J



Load Combination

13.2.2 Beams

7KHUHTXLUHGEHDPVL]HIRUVWUHQJWKDQGVHUYLFHDELOLW\DQGWKHUHTXLUHGÀH[XUDOVKHDUDQGWRUVLRQUHLQIRUFHPHQW are determined using the provisions in ACI Chapter 9 for the combined effects due to gravity and lateral forces (see Chapter 6 of this publication). ,QDGGLWLRQWRWKHDSSOLFDEOHGHWDLOLQJUHTXLUHPHQWVLQ$&,&KDSWHUWKHGHWDLOLQJUHTXLUHPHQWVLQ$&,PXVW EHVDWLV¿HGIRUEHDPVLQRUGLQDU\PRPHQWIUDPHVWKDWDUHSDUWRIWKH6)56VHH)LJXUH  • 7  KHÀH[XUDOUHLQIRUFHPHQWDWWKHWRSDQGERWWRPIDFHVRIWKHVHFWLRQPXVWLQFOXGHWZREDUVFRQWLQXRXVDORQJ the span. • 7  KHDUHDRIWKHFRQWLQXRXVERWWRPEDUVLQWKHVHFWLRQPXVWKDYHDQDUHDRIDWOHDVWSHUFHQWRIWKHPD[LPXP DUHDRIWKHERWWRPÀH[XUDOUHLQIRUFHPHQWDORQJWKHVSDQ • 7KHÀH[XUDOUHLQIRUFHPHQWPXVWEHDQFKRUHGWRGHYHORS in tension at the face of the support.         

Anchored to develop ݂௬ in tension ሺtyp.ሻ ‫ିܣ‬ ௦ 

‫ܣ‬ା ௦

൒ 2 bars continuous

൒ 2 bars continuous with ‫ܣ‬௦ ൒ ሺ‫ܣ‬ା ௦ ሻ௠௔௫ Τ4

Other reinforcement not shown for clarity

Figure 13.1 Requirements for beams in ordinary moment frames in structures assigned to SDC B.

7KHUHTXLUHPHQWVIRUÀH[XUDOUHLQIRUFHPHQWDUHLQWHQGHGWRLPSURYHVHLVPLFIRUFHUHVLVWDQFHDQGVWUXFWXUDOLQWHJULW\ for structures not expected to be subjected to strong ground motion. 13.2.3 Columns

7KHUHTXLUHGFROXPQVL]HDQGWKHUHTXLUHGORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQVLQRUGLQDU\PRment frames that are part of the SFRS in structures assigned to SDC B are determined using the provisions in ACI &KDSWHUIRUWKHFRPELQHGHIIHFWVGXHWRJUDYLW\DQGODWHUDOIRUFHVVHH&KDSWHURIWKLVSXEOLFDWLRQ 

13-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Shear strength requirements for certain columns in ordinary moment frames that are part of the SFRS are given in LVOHVVWKDQRUHTXDOWR¿YHWLPHVWKHSODQGLPHQVLRQRI ACI 18.3.3. Where the unsupported length of a column, in the direction of analysis, the design shear strength of the column must be taken as the lesser of the column, the following: DVVRFLDWHGZLWKWKHGHYHORSPHQWRIQRPLQDOÀH[XUDOVWUHQJWKV of the column at each • The shear force, UHVWUDLQHGHQGRIWKHXQVXSSRUWHGOHQJWKGXHWRUHYHUVHFXUYDWXUHEHQGLQJVHH)LJXUH ,QWKH¿JXUH DUHWKHQRPLQDOÀH[XUDOVWUHQJWKVDWWKHWRSDQGERWWRPRIWKHFROXPQUHVSHFWLYHO\LQWKHGLUHFWLRQRI and consistent with the direcDQDO\VLV7KHVHÀH[XUDOVWUHQJWKVPXVWEHFDOFXODWHGIRUWKHIDFWRUHGD[LDOIRUFH WLRQRIDQDO\VLVUHVXOWLQJLQWKHKLJKHVWÀH[XUDOVWUHQJWK6LGHVZD\WRWKHULJKWDQGVLGHVZD\WRWKHOHIWPXVW both be considered. • 7KHPD[LPXPVKHDUIRUFHREWDLQHGIURPWKHGHVLJQORDGFRPELQDWLRQVRI$&,&KDSWHUWKDWLQFOXGHWKHHDUWKwith substituted for in the load combinations. The term is the overstrength factor quake effect, IRURUGLQDU\PRPHQWIUDPHVRIUHLQIRUFHGFRQFUHWHZKLFKLVHTXDOWRVHH$6&(6(,7DEOHRU7DEOH LQWKLVSXEOLFDWLRQ  ‫ܯ‬௡௧  ܿଵ 



ܸ௨ 

ܸ௨ ൌ

‫ܯ‬௡௧ ൅ ‫ܯ‬௡௕  κ௨

κ௨ ൑ ͷܿଵ 

ܸ௨  ‫ܯ‬௡௕  Figure 13.2 Design shear strength of columns in ordinary moment frames that are part of the SFRS in accordance with ACI 18.3.3.

The provisions for the design shear strength are intended to provide additional shear capacity in relatively squat columns, which are vulnerable to shear failure under earthquake loading. 13.2.4 Beam-Column Joints

Beam-column joints in ordinary moment frames that are part of the SFRS in structures assigned to SDC B must EHGHVLJQHGDQGGHWDLOHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&,&KDSWHU$&,VHH&KDSWHURIWKLV publication). is determined at the mid-height of the $FFRUGLQJWR$&,WKHUHTXLUHGIDFWRUHGKRUL]RQWDOMRLQWVKHDU joint in the direction of analysis from statics using tensile and compressive beam forces and column shear consistent and ZLWKWKHQHJDWLYHDQGSRVLWLYHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHEHDPDWWKHIDFHRIWKHMRLQW 7KHLQIRUPDWLRQLQ6HFWLRQRIWKLVSXEOLFDWLRQFDQEHXVHGWRFDOFXODWH IRULQWHULRUDQGHGJHFRUQHUMRLQWV can be determined using Figure subjected to gravity and lateral forces. The design shear strength of joints, LQ6HFWLRQRIWKLVSXEOLFDWLRQ)RUMRLQWVLQEXLOGLQJVDVVLJQHGWR6'&%WKHVWUHQJWKUHGXFWLRQIDFWRU LVHTXDOWR

13-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

13.3 Intermediate Moment Frames (SDC C) 13.3.1 Overview

Design and detailing requirements for beams, columns, beam-column joints, slab-column joints, and two-way slabs ZLWKRXWEHDPVGHVLJQDWHGSDUWRIWKH6)56LQLQWHUPHGLDWHPRPHQWIUDPHVDUHJLYHQLQ$&, Gravity and lateral load effects on members in an intermediate moment frame are obtained from an analysis method LQ$&,'HVLJQVWUHQJWKORDGFRPELQDWLRQVIRUVWUXFWXUHVDVVLJQHGWR6'&&DUHJLYHQLQ7DEOHIRUJUDYLW\ ZLQGDQGVHLVPLFORDGHIIHFWVVHH6HFWLRQRIWKLVSXEOLFDWLRQ  Table 13.2 Strength Design Load Combinations for Structures Assigned to SDC C ACI Equation Number

ASCE/SEI 7 Load Combination

D

1

E



F

3

G



H

6

I



J



Load Combination

13.3.2 Beams Overview

7KHUHTXLUHGEHDPVL]HIRUVWUHQJWKDQGVHUYLFHDELOLW\DQGWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWDUHGHWHUPLQHGXVLQJ the provisions in ACI Chapter 9 for the combined effects due to gravity and lateral forces (see Chapter 6 of this pubOLFDWLRQ 7KHVKHDUVWUHQJWKUHTXLUHPHQWVLQ$&,DQGWKHGHWDLOLQJUHTXLUHPHQWVLQ$&,WKURXJK PXVWEHVDWLV¿HGLQDGGLWLRQWRDQ\UHOHYDQWSURYLVLRQVLQ$&,&KDSWHU Flexural Strength Requirements

5HTXLUHPHQWVKDYLQJDSRVVLEOHLPSDFWRQWKHDPRXQWRISRVLWLYHÀH[XUDOUHLQIRUFHPHQWLQEHDPVWKDWDUHSDUWRIDQ LQWHUPHGLDWHPRPHQWIUDPHDUHJLYHQLQ$&,$WWKHIDFHVRIWKHMRLQWVWKHSRVLWLYHPRPHQWVWUHQJWK at that joint. The area of must be at least equal to 33 percent of the corresponding negative moment strength, WKHSRVLWLYHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPDWDMRLQWPD\QHHGWREHLQFUHDVHGWRVDWLVI\WKLVUHTXLUHPHQW$OVR DWDQ\VHFWLRQDORQJWKHOHQJWKRIDEHDPWKHQHJDWLYHDQGSRVLWLYHPRPHQWVWUHQJWKPXVWEHHTXDOWRDWOHDVW percent of the maximum moment strength provided at the face of either joint. $VXPPDU\RIWKHÀH[XUDOUHTXLUHPHQWVLQ$&,&KDSWHUDQG$&,IRUEHDPVLQDQLQWHUPHGLDWHPRPHQW frame is given in Figure 13.3. 7KHUHDUHQRUHVWULFWLRQVZKHUHODSVSOLFHVRIWKHÀH[XUDOUHLQIRUFHPHQWFDQEHORFDWHGDORQJWKHVSDQ7KHWRS reinforcement is usually spliced near midspan and the bottom reinforcement is spliced near the supports. Because plastic hinges have the potential to form at the ends of a beam, it would be appropriate to locate the splices of the bottom reinforcement somewhere between the end of the anticipated plastic hinge region and midspan. The assumed from each end of a beam where is the overall depth of length of the anticipated plastic hinge region is equal to WKHEHDP$&, 

13-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ܣ‬௦,௠௜௡ ൌ 3ඥ݂௖ᇱ ܾ௪ ݀/݂௬ ൒ 200ܾ௪ ݀/݂௬ , unless ACI 9.6.1.3 is satisfied Minimum 2 bars continuous at top Minimum 2 bars continuous at bottom with ‫ܣ‬௦ ൒ ሺ‫ܣ‬ା ௦,௠௔௫ ሻΤ4 Structural integrity reinforcement in accordance with ACI 9.7.7

ି ‫ܯ‬௡,௥

ି ‫ܯ‬௡,κ

ା ൒ ‫ܯ‬௡,κ

ି ‫ܯ‬௡,κ 3

ା ‫ܯ‬௡,௥ ൒

ି ‫ܯ‬௡,௥ 3

‫ܯ‬௡ି or ‫ܯ‬௡ା ൒

Maximum ‫ܯ‬௡ at either joint 5

Note: Transverse reinforcement not shown for clarity

Figure 13.3 Flexural requirements for beams in intermediate moment frames.

Shear Strength Requirements

Required shear reinforcement in beams of intermediate moment frames is calculated differently than in beams of ordinary moment frames, the latter of which is determined solely on the effects from factored load combinations VHH6HFWLRQRIWKLVSXEOLFDWLRQ  $FFRUGLQJWR$&,WKHGHVLJQVKHDUVWUHQJWKRIDEHDP is equal to the lesser of the following: required shear strength,

which must be greater than or equal to the

• 7  KHVXPRIWKHVKHDUIRUFHVDVVRFLDWHGZLWKGHYHORSPHQWRIWKHQHJDWLYHDQGSRVLWLYHQRPLQDOÀH[XUDO and at each end of the beam and the factored gravity and vertical earthquake loads acting strengths, DORQJWKHVSDQ>$&,D @ • The maximum shear force obtained from the factored load combinations that include earthquake load effects, where LVWDNHQWREHWZLFHWKDWSUHVFULEHGE\WKHJRYHUQLQJEXLOGLQJFRGH>$&,E @ 7KHUHTXLUHPHQWLQ$&,D LVLOOXVWUDWHGLQ)LJXUHIRUDEHDPZLWKDIDFWRUHGXQLIRUPJUDYLW\ORDG along its span. Because earthquake effects can act in either direction, both sidesway to the right and sidesway to the which must include the vertical OHIWPXVWFRQVLGHUHG$&,(TXDWLRQH LVDSSOLFDEOHZKHQGHWHUPLQLQJ VHH7DEOH 7KXV is combined with and where it earthquake load effect is assumed the load factor on FDQEHWDNHQDVLQDFFRUGDQFHZLWK$&,7KHWRWDOIDFWRUHGVKHDUIRUFHDW is determined from statics. In cases where the top reinforcement is the same at both ends of each end of a beam, and the bottom reinforcement is continuous over the entire span, maximum is the same for a beam both sidesway to the right and to the left. $FFRUGLQJWR$&,E WKHPD[LPXP is determined from analysis of the structure using the factored load where Thus, the following equation can be used to combination determine (13.1)

13-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ିܣ‬ ௦ǡκ 

‫ିܣ‬ ௦ǡ௥ 

‫ܣ‬ା ௦ κ௡ 



‫ݓ‬௨ ൌ ሺͳǤʹ ൅ ͲǤʹܵ஽ௌ ሻ‫ ܦ‬൅ ͲǤͷ‫ ܮ‬൅ ͲǤʹܵ

”ƒ˜‹–› ‫ݓ‬௨ κ௡  ʹ ‫ܯ‬௡ା ൌ ‫ܣ‬ା ௦ ݂௬ ቆ݀ െ

‫ܣ‬ା ௦ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

‫ݓ‬௨ κ௡  ʹ ି ൌ ‫ିܣ‬ ‫ܯ‬௡௥ ௦ǡ௥ ݂௬ ቆ݀ െ

‫ିܣ‬ ௦ǡ௥ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪



‹†‡•™ƒ›”‹‰Š– ି ‫ܯ‬௡ା ൅ ‫ܯ‬௡௥  κ௡ 

‫ܯ‬௡ା 

”ƒ˜‹–›൅ ‹†‡•™ƒ›”‹‰Š–

ܸ௨ ൌ

ି ‫ܯ‬௡κ

ି ‫ܯ‬௡ା ൅ ‫ܯ‬௡௥  κ௡ 

ൌ

ି ‫ܯ‬௡௥  

ି ‫ݓ‬௨ κ௡ ‫ܯ‬௡ା ൅ ‫ܯ‬௡௥ െ  ʹ κ௡

‫ିܣ‬ ௦ǡκ ݂௬

‫ିܣ‬ ௦ǡκ ݂௬ ቆ݀ െ ቇ ͳǤ͹݂௖ᇱ ܾ௪

ܸ௨ ൌ

ି ‫ݓ‬௨ κ௡ ‫ܯ‬௡ା ൅ ‫ܯ‬௡௥ ൅  ʹ κ௡

‫ܯ‬௡ା ൌ ‫ܣ‬ା ௦ ݂௬ ቆ݀ െ

‫ܣ‬ା ௦ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

‹†‡•™ƒ›Ž‡ˆ–  ି ‫ܯ‬௡ା ൅ ‫ܯ‬௡κ  κ௡

”ƒ˜‹–›൅ ‹†‡•™ƒ›Ž‡ˆ–

ି ‫ܯ‬௡ା ൅ ‫ܯ‬௡κ  κ௡

ି ‫ܯ‬௡κ 

ܸ௨ ൌ

ି ‫ݓ‬௨ κ௡ ‫ܯ‬௡ା ൅ ‫ܯ‬௡κ ൅  ʹ κ௡

‫ܯ‬௡ା  ܸ௨ ൌ

ି ‫ݓ‬௨ κ௡ ‫ܯ‬௡ା ൅ ‫ܯ‬௡κ െ  ʹ κ௡

Figure 13.4 Required shear forces for beams in intermediate moment frames in accordance with ACI 18.4.2.3(a).

13-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHOHVVHURIWKHIDFWRUHGVKHDUIRUFHVGHWHUPLQHGE\$&,D DQGE LVXVHGLQGHWHUPLQLQJWKH UHTXLUHGVL]HDQGVSDFLQJRIWKHVKHDUUHLQIRUFHPHQW As noted previously, plastic hinges can form at the ends of a beam during a design-level seismic event and the FRQFUHWHPXVWEHSURSHUO\FRQ¿QHGDWWKHVHORFDWLRQV+RRSVPXVWEHSURYLGHGZLWKLQDGLVWDQFHRIDWOHDVW from HDFKHQGRIDEHDPZKLFKLVWKHDVVXPHGOHQJWKRIWKHDQWLFLSDWHGSODVWLFKLQJHUHJLRQ+RRSVDUHGH¿QHGDVD closed tie or continuously wound tie made up of one or several reinforcement elements, each having seismic hooks DWHDFKHQGFRQIRUPLQJWR$&,$&,VHH)LJXUH 7KHHQGVRIWKHUHLQIRUFHPHQWHOHPHQWVPXVW HQJDJHDORQJLWXGLQDOEDULQWKHEHDPDQGWKHH[WHQVLRQVPXVWSURMHFWLQWRWKHLQWHULRURIWKHKRRS+RRSVIRUPHG E\VWLUUXSVZLWKVHLVPLFKRRNVDQGFURVVWLHV'HWDLOV%DQG&LQ)LJXUH DUHSUHIHUUHGRYHUWKRVHIRUPHGE\ closed stirrups with seismic hooks (Detail A) because the longitudinal bars in the beam can be placed more easily DQGHI¿FLHQWO\&URVVWLHVPXVWFRQIRUPWR$&,1RWHWKDWLQWHUORFNLQJKHDGHGGHIRUPHGEDUVDUHQRWSHUPLWWHGEHDFORVHGWLH$&, 

Figure 13.5 Examples of hoops and overlapping hoops.

7KHVL]HDQGVSDFLQJRIWKHKRRSVZLWKLQ

FDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ$&,  (13.2)

where IRUVKHDUVHH$&,7DEOH DQG LVGHWHUPLQHGE\$&,:KHUH the folORZLQJHTXDWLRQIURP$&,7DEOHFDQEHXVHGWRGHWHUPLQH for beams with negligible axial force, (13.3)

7KHPRGL¿FDWLRQIDFWRUWKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHWRQRUPDOZHLJKW concrete of the same compressive strength, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, 2QFHWKHVL]HDQGVSDFLQJRIWKHKRRSVDUHGHWHUPLQHGEDVHGRQWKHJRYHUQLQJ the spacing must be checked must be less DJDLQVWWKHPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,7KHFDOFXODWHGKRRSVSDFLQJ within than or equal to the smallest of the following (see Figure 13.6): 13-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(13.4)

݀

݄

൑ 2ԣ

݀ Τ4 ‫ۓ‬ ۖ ۖ8݀௕ ሺlongitudinal barሻ ‫ ݏ‬൑ smallest of  ‫۔‬24݀ ሺhoop barሻ ௕ ۖ ۖ ‫ە‬12ᇳ

Hoops

൒ 2݄

Stirrups

‫ ݏ‬൑ ݀/2 Hoop reinforcement determined in accordance with ACI 18.4.2.3 at both ends

Figure 13.6 Transverse reinforcement requirements for beams in intermediate moment frames.

Outside of the anticipated plastic hinge regions, it is permitted to use stirrups spaced no greater than on center is calculated at the location where the hoops are terminated and the $&, 7KHIDFWRUHGVKHDUIRUFH VL]HDQGVSDFLQJRIWKHVWLUUXSVDUHGHWHUPLQHGXVLQJ(TXDWLRQ  transverse reinforcement conformFor beams subjected to a factored axial compressive force greater than LQJWRWKHWLHUHTXLUHPHQWVLQ$&,DQGHLWKHU$&,RUPXVWEHSURYLGHGLQWKHUHJLRQRIWKH EHDPRXWVLGHRIWKHDQWLFLSDWHGSODVWLFKLQJHUHJLRQV$&, 7KHVHDGGLWLRQDOUHTXLUHPHQWVDUHLQWHQGHGWR provide lateral support for the longitudinal reinforcement subjected to factored axial compressive forces greater than the prescribed limit. 13.3.3 Columns Overview

7KHUHTXLUHGFROXPQVL]HDQGWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQVLQLQWHUPHGLDWHPRPHQWIUDPHV WKDWDUHSDUWRIWKH6)56LQVWUXFWXUHVDVVLJQHGWR6'&&DUHGHWHUPLQHGXVLQJWKHSURYLVLRQVLQ$&,&KDSWHUIRU WKHFRPELQHGHIIHFWVGXHWRJUDYLW\DQGODWHUDOIRUFHVVHH&KDSWHURIWKLVSXEOLFDWLRQ 7KHVKHDUVWUHQJWKUHTXLUHPHQWVLQ$&,DQGWKHGHWDLOLQJUHTXLUHPHQWVLQ$&,WKURXJKPXVWEHVDWLV¿HGLQDGGLWLRQ WRDQ\UHOHYDQWSURYLVLRQVLQ$&,&KDSWHU

13-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Shear Strength Requirements

Plastic hinges can form at the ends of a column during a design-level seismic event and the concrete must be properO\FRQ¿QHGZLWKLQWKHVHUHJLRQV&ROXPQVPXVWEHVSLUDOO\UHLQIRUFHGLQDFFRUGDQFHZLWK$&,&KDSWHURUWUDQVfrom each end of a column where verse reinforcement in the form of hoops must be provided over a distance of LVWKHDVVXPHGOHQJWKRIWKHDQWLFLSDWHGSODVWLFKLQJHUHJLRQ$&, 

(13.5)

6  KHDUVWUHQJWKUHTXLUHPHQWVIRUFROXPQVLQLQWHUPHGLDWHPRPHQWIUDPHVDUHJLYHQLQ$&,WKHVHUHTXLUHments are similar to those for beams in intermediate moment frames. must be greater than or equal to the required shear strength, which The design shear strength of a column, is equal to the lesser of the following: • 7  KHVKHDUIRUFHVDVVRFLDWHGZLWKGHYHORSPHQWRIWKHQRPLQDOÀH[XUDOVWUHQJWKV and at the top and ERWWRPRIWKHFROXPQGXHWRUHYHUVHFXUYDWXUHEHQGLQJ7KHQRPLQDOÀH[XUDOVWUHQJWKLVFDOFXODWHGIRUWKHIDFWRUHGD[LDOIRUFHFRQVLVWHQWZLWKWKHGLUHFWLRQRIDQDO\VLVUHVXOWLQJLQWKHODUJHVWQRPLQDOÀH[XUDOVWUHQJWK • The maximum shear force obtained from the factored load combinations that include earthquake effects, where is substituted for 7KHUHTXLUHPHQWLQ$&,D LVLOOXVWUDWHGLQ)LJXUH7KHQRPLQDOÀH[XUDOVWUHQJWKVDWWKHWRSDQG and are applied in reverse curvature bending and the factored shear force, is bottom of the column, obtained from statics: (13.6)

‫ܯ‬௡௧  ܲ௨ 



ܸ௨ 

κ௨ 

ܸ௨ ൌ

ܲ௨ 

‫ܯ‬௡௧ ൅ ‫ܯ‬௡௕  κ௨

ܸ௨ 

‫ܯ‬௡௕ 

Figure 13.7 Required shear forces for columns in intermediate moment frames in accordance with ACI 18.4.3.1(a).

13-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Axial Force Axial Force

Range of Axial Forces

ACI Eq. ሺ5.3.1eሻ: 1.2‫ ܦ‬൅ 1.0‫ ܧ‬൅ 1.0‫ ܮ‬൅ 0.2ܵ ACI Eq. ሺ5.3.1gሻ: 0.9‫ ܦ‬൅ 1.0‫ܧ‬

‫ܯ‬௡  Bending Moment ሺࢇሻ Bending Moment

   Axial Force Axial Force

Range of Axial Forces

ACI Eq. ሺ5.3.1eሻ: 1.2‫ ܦ‬൅ 1.0‫ ܧ‬൅ 1.0‫ ܮ‬൅ 0.2ܵ ACI Eq. ሺ5.3.1gሻ: 0.9‫ ܦ‬൅ 1.0‫ܧ‬ Bending Moment ‫ܯ‬௡  Bending Moment ሺ࢈ሻ Figure 13.8 Nominal strength interactions diagrams for columns in intermediate moment frames.

13-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

:KHQGHWHUPLQLQJWKHQRPLQDOÀH[XUDOVWUHQJWKVWKHIDFWRUHGD[LDOIRUFH within the range of axial forces inFOXGLQJHDUWKTXDNHHIIHFWVPXVWEHVHOHFWHGWKDWUHVXOWVLQWKHODUJHVWQRPLQDOÀH[XUDOVWUHQJWK1RPLQDOVWUHQJWKLQteraction diagrams for a column are given in Figure 13.8. For the range of axial forces shown in Figure 13.8(a), the RFFXUVDWWKHEDODQFHGSRLQWHYHQWKRXJKWKHIDFWRUHGD[LDOIRUFHFRUUHVSRQGLQJWRWKLVQRPLQDOÀH[XUDO largest VWUHQJWKLVQRWREWDLQHGIURPHLWKHURIWKHWZRDSSOLFDEOHORDGFRPELQDWLRQVZKLFKDUH$&,(TXDWLRQVH DQG at the balanced point must be used in the calculation of For the range of axial forces J ,QWKLVFDVH LVHTXDOWRWKHQRPLQDOÀH[XUDOVWUHQJWKDVVRFLDWHGZLWKWKHIDFWRUHGD[LDOIRUFH in Figure 13.8(b), the largest IURP$&,(TXDWLRQH EHFDXVHWKDWQRPLQDOÀH[XUDOVWUHQJWKLVJUHDWHUWKDQWKHQRPLQDOÀH[XUDOVWUHQJWKVDVsociated with any of the other factored axial forces in that range. 7KHUHTXLUHPHQWLQ$&,E LVVLPLODUWRWKDWIRUEHDPVLQLQWHUPHGLDWHPRPHQWIUDPHVH[FHSWLQVWHDGRI which is equal to 3 for intermediate increasing E\DIDFWRURILWLVLQFUHDVHGE\WKHRYHUVWUHQJWKIDFWRU PRPHQWIUDPHVVHH$6&(6(,7DEOHRU7DEOHLQWKLVSXEOLFDWLRQ 7KLVKLJKHUIDFWRUUHFRJQL]HVWKH importance of columns compared to beams in regard to shear design. 7KHOHVVHURIWKHIDFWRUHGVKHDUIRUFHVIURPWKHVHWZRUHTXLUHPHQWVLVXVHGLQGHWHUPLQLQJWKHUHTXLUHGVL]HDQG VSDFLQJRIWKHVKHDUUHLQIRUFHPHQW,QJHQHUDOVKHDUVWUHQJWKLQWKHUHJLRQVRIÀH[XUDO\LHOGLQJLVSURYLGHGE\ERWK and transverse reinforcement 7KHVL]HDQGVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWFDQEHGHWHUconcrete PLQHGE\WKHIROORZLQJHTXDWLRQ$&,  (13.7)

where IRUVKHDUVHH$&,7DEOH DQG LVGHWHUPLQHGE\$&,)RUUHFWLOLQHDUKRRSV is equal to the area of all hoop legs or all hoop plus crosstie legs in the direction of analysis within the longituis equal to two times the area of the tie bar within (ACI dinal center-to-center spacing, For circular ties, is equal to two times the area of the spiral bar within the spiral pitch,  6LPLODUO\IRUVSLUDOV 2QFHWKHVL]HDQGVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWKDVEHHQGHWHUPLQHGEDVHGRQWKHJRYHUQLQJ the VSDFLQJPXVWEHFKHFNHGDJDLQVWWKHPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,IRUKRRSVDQG$&,IRU spirals. Within

the calculated hoop spacing based on

must be less than or equal to

(see Figure 13.9):

(13.8)

For columns with spiral reinforcement, the calculated spiral pitch based on must be less than or equal to where LVWKHGLDPHWHURIWKHVSLUDOEDU$&, 7KHVSLUDOSLWFKPXVWEHJUHDWHU and where LVWKHQRPLQDOPD[LPXPVL]H than or equal to the greater of of coarse aggregate in the concrete mix. Outside of the anticipated plastic hinge regions, the spacing of the transverse reinforcement must conform to the latHUDOUHLQIRUFHPHQWSURYLVLRQVLQ$&,IRUWLHVDQG$&,IRUVSLUDOVDQGWRWKHPD[LPXPVSDFLQJUHTXLUHPHQWVIRUVKHDUUHLQIRUFHPHQWLQ$&,$&, 7KHOHDVWVSDFLQJREWDLQHGIURPWKHVHUHTXLUHPHQWV must be used in the center region of a column.

13-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Joint reinforcement in accordance with ACI 18.4.4



A

A

Spacing of transverse reinforcement in accordance with ACI 18.4.3.5

κ௢ ൒ longest of

‫ۓ‬ ۖ

Clear spanΤ6

See note below on splice location

Larger of ܿଵ and ܿଶ  ‫۔‬ ۖ ᇳ ‫ ە‬18

Transverse reinforcement in accordance with ACI 18.4.3.1

‫ݏ‬௢ ൑ least of ൑ ‫ݏ‬௢ /2

‫ۓ‬ ۖ

Smaller of 8݀௕ሺ௟௢௡௚.ሻ and 8ᇳ for Grade 60 bars

Smaller of 6݀௕ሺ௟௢௡௚.ሻ and 6ᇳ for Grade 80 bars ‫۔‬ ۖ ‫ە‬ሺSmaller of ܿଵ and ܿଶ ሻΤ2

ܿଶ 

Note: There is no restriction on the location of longitudinal bar splices for intermediate moment frames. The splice may be located away from the potential hinge regions as shown above, or it may be located immediately above the floor slab. ܿଵ Section A-A

Figure 13.9 Transverse reinforcement requirements for columns with hoops in intermediate moment frames.

Columns Supporting Reactions from Discontinuous Stiff Members

Where a stiff member, such as a wall, is discontinued and supported on columns instead of on a foundation, the columns must contain transverse reinforcement at a spacing of LQDFFRUGDQFHZLWK$&,IRUFROXPQV in intermediate moment frames over the full height at all levels beneath the level at which the discontinuity oc(ACI curs where the factored axial compressive force related to earthquake effects in the columns exceeds pertains to columns designed using load combinations that include the overstrength  7KHOLPLWRI ZKLFKDUHJLYHQLQ$6&(6(,VHH7DEOHRIWKLVSXEOLFDWLRQ 2WKHUZLVHWKHOLPLWLV factor 7KHWUDQVYHUVHUHLQIRUFHPHQWPXVWH[WHQGDERYHDQGEHORZWKHFROXPQLQDFFRUGDQFHZLWK$&,E ZKLFK is the requirement applicable to columns supporting discontinued stiff members in special moment frames (see )LJXUH  As noted previously, the required reinforcement details presented above must be provided even where the effects from wind loads govern the design of a column.

13-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



Reinforcement in wall not shown for clarity Wall

൒ κௗ of largest longitudinal column bar where κௗ is determined by ACI 18.8.5 ሺtyp.ሻ

Transverse reinforcement spaced at ‫ݏ‬௢ ሺACI 18.4.3.3ሻ

Footing or mat

Wall

൒ 12ԣ

൒ κௗ of largest longitudinal column bar where κௗ is determined by ACI 18.8.5 Figure 13.10 Transverse reinforcement requirements for columns supporting discontinuous stiff members in intermediate moment frames.

13.3.4 Joints Beam-Column Joints

%HDPFROXPQMRLQWVLQLQWHUPHGLDWHPRPHQWIUDPHVPXVWVDWLVI\WKHUHTXLUHPHQWVLQ$&,DQG WKURXJK$&, 6KHDUVWUHQJWKUHTXLUHPHQWVIRUEHDPFROXPQMRLQWVDUHJLYHQLQ$&,  7KHVWUXWDQGWLHPHWKRGLQ$&,&KDSWHUPXVWEHXVHGWRDQDO\]HDQGGHVLJQDMRLQWZKHUHWKHGHSWKRIWKHEHDP framing into the joint generating the joint shear is more than twice the depth of the column in the direction of analyVLV$&, ,QDGGLWLRQWRWKHUHTXLUHPHQWVRI$&,&KDSWHUWKHMRLQWVKHDUVWUHQJWKUHTXLUHPHQWVRI$&, DQGWKHMRLQWGHWDLOLQJUHTXLUHPHQWVRI$&,WKURXJKPXVWEHVDWLV¿HG Beam longitudinal reinforcement terminated in a joint must extend to the far face of the joint core and must be GHYHORSHGLQWHQVLRQLQDFFRUGDQFHZLWKWKHWHQVLRQGHYHORSPHQWOHQJWKUHTXLUHPHQWVRI$&,IRUMRLQWVLQ VSHFLDOPRPHQWIUDPHVDQGLQFRPSUHVVLRQLQDFFRUGDQFHZLWK$&,$&,  An example of an edge or corner column where the longitudinal reinforcement from the beam is terminated in a VWDQGDUGGHJUHHKRRNLVJLYHQLQ)LJXUH,QDFFRUGDQFHZLWK$&,WKHKRRNPXVWEHORFDWHGZLWKLQ WKHFRQ¿QHGFRUHRIWKHFROXPQDQGLWPXVWEHEHQWLQWRWKHMRLQW7KHFULWLFDOVHFWLRQIRUGHYHORSPHQWRIWKHKRRNHG bars is taken at the outside edge of the joint transverse reinforcement and not at the face of column because the conFUHWHRXWVLGHRIWKHFRQ¿QHGFRUHPD\VSDOOGXULQJDGHVLJQOHYHOVHLVPLFHYHQW is 7KHGHYHORSPHQWOHQJWKRIDGHIRUPHGWKURXJKUHLQIRUFLQJEDUZLWKDVWDQGDUGKRRNLQWHQVLRQ GHWHUPLQHGE\$&,(TXDWLRQ 7KHPRGL¿FDWLRQIDFWRU LVHTXDOWRIRUFRQFUHWHZLWKOLJKWZHLJKW based on the requirements in DJJUHJDWHDQGRWKHUZLVH7KHIROORZLQJHTXDWLRQVFDQEHXVHGWRGHWHUPLQH $&,

13-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ  κௗ௛ 

ܿ௖ 

݀௕ሺ௛௢௢௣ሻ 

‘‘’•ሺ–›’Ǥሻ

݀௕ ሺ–›’Ǥሻ –ƒ†ƒ”†ͻͲǦ†‡‰ Š‘‘ሺ–›’Ǥሻ

݀௕ሺ௖௢௟Ǥሻ 

ƒŽŽ‡”‘ˆͺ݀௕ሺ௖௢௟Ǥሻ ƒ†ͺᇳ ˆ‘” ”ƒ†‡͸Ͳ„ƒ”•  ‫ݏ‬௢ ൑ Ž‡ƒ•–‘ˆ ƒŽŽ‡”‘ˆ͸݀௕ሺ௖௢௟Ǥሻ ƒ†͸ᇳ ˆ‘” ”ƒ†‡ͺͲ„ƒ”• ‫۔‬  ۖ ‫ە‬ሺƒŽŽ‡”‘ˆܿଵ ƒ†ܿଶ ሻΤʹ ‫ۓ‬ ۖ

κௗ௛ 

κௗ௛

‘”ƒŽ™‡‹‰Š–…‘…”‡–‡

ᇱ ‫݂ ۓ‬௬ ݀௕ Τ͸ͷඥ݂௖ ۖ  ൌ ‰”‡ƒ–‡”‘ˆ ͺ݀௕   ‫۔‬  ۖ ‫ە‬͸ᇳ  ᇱ

 ‹‰Š–™‡‹‰Š–…‘…”‡–‡ 

κௗ௛

‫݂ۓ‬௬ ݀௕ Τሺ͸ͷ ൈ ͲǤ͹ͷሻඥ݂௖ ۖ  ൌ ‰”‡ƒ–‡”‘ˆ ͳͲ݀௕   ‫۔‬  ۖ ‫ە‬͹Ǥͷᇳ 

‘”‡’‘š›Ǧ…‘ƒ–‡†‘”œ‹…ƒ†‡’‘š›†—ƒŽǦ…‘ƒ–‡†„ƒ”•ǡ—Ž–‹’Ž›κௗ௛ „›ͳǤʹǤ

 Figure 13.11 Development of flexural reinforcement with standard hooks in beams of intermediate moment frames.

For normalweight concrete:

(13.9)

13-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For lightweight concrete:

(13.10)

These development lengths are measured from the outside edge of the joint transverse reinforcement to the outside face of the hook (see Figure 13.11). In order for the hooked longitudinal bars to be fully developed in the joint for tension, the available development length must be greater than or equal to (13.11)

In this equation, is the clear cover to the hoop reinforcement in the joint, is the diameter of the hoop reinis the diameter of the longitudinal reinforcement in the column. forcement in the joint, and ,QOLHXRIKRRNHGEDUVWKURXJKVWUDLJKWEDUVPD\EHXVHGSURYLGHGWKHEDUVDUHSURSHUO\GHYHORSHGLQDFFRUGDQFHZLWK$&,DQG7KHUHTXLUHGGHYHORSPHQWOHQJWKVRIWKURXJKVWUDLJKWEDUVDUHPXOWLSOHVRIWKHKRRNHGEDUGHYHORSPHQWOHQJWKVLQ$&,IRUWRSEDUVWKHIDFWRULVDQGIRUERWWRPEDUVLWLV 1RWHWKDWDQGEDUVDUHQRWLQFOXGHGEHFDXVHRIWKHODFNRILQIRUPDWLRQSHUWDLQLQJWRDQFKRUDJHRIWKHVH EDUVL]HVZKHQVXEMHFWHGWRORDGUHYHUVDOV $QH[DPSOHRIDQHGJHRUFRUQHUFROXPQZLWKKHDGHGGHIRUPHGEDUVLVJLYHQLQ)LJXUH6XFKEDUVDUHSHUPLWWHGWREHXVHGZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG (a) (b) (c) (d) (e) (f)

%DUPXVWFRQIRUPWR$&, %DUVL]HPXVWEHRUVPDOOHU , must be at least where is the area of the bar Net bearing area of head, Concrete must be normalweight where is the nominal diameter of the bar Clear cover to the bar must be greater than or equal to Center-to-center spacing of the bars must be greater than or equal to

$FFRUGLQJWR$&,WKHGHYHORSPHQWOHQJWKRIDKHDGHGGHIRUPHGUHLQIRUFLQJEDULQWHQVLRQ is substituted for $&,ZKHUH

, is given in

(13.12)

>1RWH7KHGHYHORSPHQWOHQJWKVRIKHDGHGEDUVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,DUHJUHDWHUWKDQWKH GHYHORSPHQWOHQJWKVRIKRRNHGEDUVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,ZKLFKLVQRWFRUUHFW$&,&RPPLWWHHLVDZDUHRIWKLVDQGLVZRUNLQJWRPRGLI\WKHSURYLVLRQVLQ$&,DFFRUGLQJO\@

13-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ  κௗ௧ 

ܿ௖ 

݀௕ሺ௛௢௢௣ሻ 

‘‘’•ሺ–›’Ǥሻ

‡ƒ†‡††‡ˆ‘”‡†„ƒ”ሺ–›’Ǥሻ

‫ݐ‬௛௘௔ௗ 

݀௕ሺ௖௢௟Ǥሻ 

ƒŽŽ‡”‘ˆͺ݀௕ሺ௖௢௟Ǥሻ ƒ†ͺᇳ ˆ‘” ”ƒ†‡͸Ͳ„ƒ”•  ‫ݏ‬௢ ൑ Ž‡ƒ•–‘ˆ ƒŽŽ‡”‘ˆ͸݀௕ሺ௖௢௟Ǥሻ ƒ†͸ᇳ ˆ‘” ”ƒ†‡ͺͲ„ƒ”• ‫۔‬  ۖ ‫ە‬ሺƒŽŽ‡”‘ˆܿଵ ƒ†ܿଶ ሻΤʹ ‫ۓ‬ ۖ

κௗ௧

ͳǤʹͷ݂௬ ߰௘ ߰௣ ߰௢ ߰௖

ቇ ݀௕ଵǤହ ͹ͷඥ݂௖ᇱ  ൌ ‰”‡ƒ–‡”‘ˆ  ‫۔‬ͺ݀௕  ۖ  ‫ ە‬͸ᇳ  ‫ۓ‬ቆ ۖ

Figure 13.12 Development of flexural reinforcement with headed bars in beams of intermediate moment frames.

7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  Table 13.3 Modification Factors for Development of Headed Bars in Tension Modification Factor

Epoxy,

Parallel tie reinforcement,

Condition

Value of Factor

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK

or

Other

Location,

For headed bars (1) terminating inside a column core with RU ZLWKVLGHFRYHUWREDU side cover to bar Other

Concrete strength,

13-16

 1.6  



Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term DFFRXQWVIRUFRQ¿QLQJHIIHFWVSURYLGHGE\WUDQVYHUVHUHLQIRUFHPHQWRULHQWHGSDUDOOHOWRWKHGHYHORSment length of headed bars. At beam-column joints, the total cross-sectional area of transverse reinforcement, of the centerline of the headed bar toward the middle of the joint where is the nomimust be located within is greater than or equal to or where the QDOGLDPHWHURIWKHKHDGHGEDUVHH$&,)LJXUH5 :KHUH The term is the total crosscenter-to center spacing of the headed bars is greater than or equal to sectional area of the headed bars being developed. In order for the headed longitudinal bars to be fully developed in the joint for tension, the available development length must be greater than or equal to (13.13)

where

is the thickness of the head.

The development length,

IRUGHIRUPHGEDUVLQFRPSUHVVLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,

(13.14)

7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  Table 13.4 Modification Factors for Development of Deformed Bars in Compression Modification Factor

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



Lightweight,

&RQ¿QLQJ reinforcement,

Reinforcement enclosed within the following: • A spiral • A circular continuously wound tie with RILQ • EDUWLHVLQDFFRUGDQFHZLWK$&,VSDFHG • +RRSVLQDFFRUGDQFHZLWK$&,VSDFHG Other

and pitch



on center on center 

This development length corresponds to the straight portion of a hooked or headed bar measured from the critical section (in this case, the outside edge of the joint transverse reinforcement) to the onset of the bend for hooked bars and from the critical section to the head for headed bars. Therefore, in order for the hooked longitudinal bars to be fully developed in the joint for compression, the available development length must be greater than or equal to (13.15)

In this equation, is the clear cover to the hoop reinforcement in the column, is the diameter of the hoop reis the diameter of the longitudinal reinforcement in the column, is the diameter inforcement in the column, of the longitudinal reinforcement in the beam, and is the bend radius of the beam longitudinal bar determined in DFFRUGDQFHZLWK$&,7DEOHIRUVWDQGDUGGHJUHHKRRNV

13-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUKHDGHGEDUVWREHIXOO\GHYHORSHGLQWKHMRLQWIRUFRPSUHVVLRQWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG (13.16)

$FFRUGLQJWR$&,WKHVL]HDQGVSDFLQJRIWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWDWWKHHQGVRIDFROXPQLQ an intermediate moment frame must be carried through a joint within the height of the deepest beam framing into WKHMRLQWVHH)LJXUHVDQG  +HDGHGEDUVXVHGDVQHJDWLYHUHLQIRUFHPHQWLQEHDPVWHUPLQDWHGLQDMRLQWZKHUHWKHUHLVQRFROXPQDERYHVXFKDV LQWKHWRSVWRU\RIDEXLOGLQJ UHTXLUHFRQ¿QHPHQWDORQJWKHWRSIDFHRIWKHMRLQW7ZRFRQ¿QHPHQWRSWLRQVDUHSURYLGHGLQ$&,D WKHFROXPQEHORZWKHMRLQWPXVWEHH[WHQGHGDERYHWKHWRSRIWKHMRLQWDGLVWDQFHHTXDO

Column extension

൒ ܿଵ 



Headed bars

All other reinforcment not shown for clarity

ܿଵ  ሺࢇሻ

Headed bars

൒ κௗ 

A

A

Vertical transverse reinforcement

‫ ܖܗܑܜ܋܍܁‬A-A

All other reinforcment not shown for clarity

ܿଵ  ሺ࢈ሻ

Figure 13.13 Confinement for headed bars in joints of intermediate moment frames. (a) Column extension. (b) Vertical transverse reinforcement.

13-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

WRDWOHDVWWKHGHSWKRIWKHMRLQWLQWKHGLUHFWLRQRIDQDO\VLV>VHH)LJXUHD @RUE YHUWLFDOMRLQWUHLQIRUFHPHQW must be provided that hooks around the headed bars and extends downward into the joint in addition to the column ORQJLWXGLQDOUHLQIRUFHPHQW>VHH)LJXUHE @'HVLJQUHFRPPHQGDWLRQVIRUWKHYHUWLFDOMRLQWUHLQIRUFHPHQWDUH JLYHQLQ5HIHUHQFH Slab-Column Joints

$FFRUGLQJWR$&,VODEFROXPQMRLQWVLQLQWHUPHGLDWHPRPHQWIUDPHVPXVWVDWLVI\WKHWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,&ROXPQWUDQVYHUVHUHLQIRUFHPHQWPXVWEHFRQWLQXHGWKURXJKDQ\VODEFROXPQ joint not laterally supported on four sides by the slab (that is, it must be continued through the joints at edge and corner columns). Such reinforcement must also be provided through column capitals, drop panels, and shear caps. Additionally, at least one layer of joint transverse reinforcement must be provided between the top and bottom layHUVRIÀH[XUDOUHLQIRUFHPHQWLQWKHVODE Shear Strength Requirements for Beam-Column Joints

The design shear strength of cast-in-place beam-column joints in intermediate moment frames must satisfy the folORZLQJHTXDWLRQ$&,  (13.17)

7KHKRUL]RQWDOMRLQWVKHDUIRUFH LVGHWHUPLQHGXVLQJWKHUHTXLUHPHQWVLQ$&,IRUMRLQWVLQRUGLQDU\PRPHQWIUDPHV$&, WKDWLV is determined at the mid-height of the joint in the direction of analysis from statics using tensile and compressive beam forces and column shear consistent with the negative and positive and VHH6HFWLRQRIWKLVSXEOLFDWLRQ  QRPLQDOÀH[XUDOVWUHQJWKVRIWKHEHDPDWWKHIDFHRIWKHMRLQW 7KHLQIRUPDWLRQLQ6HFWLRQRIWKLVSXEOLFDWLRQFDQEHXVHGWRFDOFXODWH IRULQWHULRUDQGHGJHFRUQHUMRLQWV and subjected to gravity and lateral forces. For convenience, equations for the shear force in the column, DUHJLYHQLQ7DEOHIRUVLGHVZD\WRWKHOHIWDVVXPLQJWKHSRLQWRILQÀHFWLRQLVDWWKHPLGKHLJKWRIWKHFROXPQ ZKLFKLVDUHDVRQDEOHDVVXPSWLRQIRUFROXPQVDERYHWKH¿UVWVWRU\DQGEHORZWKHWRSVWRU\VHH)LJXUH  it is conservative to disregard the terms associated with the beam shear forces, and When calculating when calculating Similarly, it is conservative to disregard Table 13.5 Horizontal Joint Shear Forces for Interior, Edge, and Corner Columns Column Location Shear Force

Interior Columns and Edge Columns with the Direction of Analysis Parallel to the Edge

,QDFFRUGDQFHZLWK$&,7DEOHWKHVWUHQJWKUHGXFWLRQIDFWRU

Edge Columns with the Direction of Analysis Perpendicular to the Edge and Corner Columns

LVHTXDOWRIRUVKHDU$&, 

The nominal joint shear strength, in an intermediate moment frame is determined using the requirements of ACI IRUMRLQWVLQVSHFLDOPRPHQWIUDPHV$&, 7KHFRQWLQXLW\RIWKHFROXPQVDQGEHDPVIUDPLQJ Continuous colLQWRWKHMRLQWDQGZKHWKHUWKHMRLQWLVFRQ¿QHGRUQRWE\WUDQVYHUVHEHDPVKDYHDQLQÀXHQFHRQ umns and beams refer to members present on both sides of a joint. Alternatively, column and beam extensions are DVVXPHGWRSURYLGHFRQWLQXLW\WKURXJKDEHDPFROXPQMRLQWSURYLGHGWKHUHTXLUHPHQWVLQ$&,DQGDUH VDWLV¿HGUHVSHFWLYHO\VHH)LJXUHVDQGRIWKLVSXEOLFDWLRQ $MRLQWLVFRQVLGHUHGWREHODWHUDOO\FRQ¿QHG where two transverse beams (that is, two beams framing into the column in the direction perpendicular to the direcWLRQRIDQDO\VLV VDWLVI\WKHWKUHHUHTXLUHPHQWVLQ$&,VHH)LJXUHRIWKLVSXEOLFDWLRQ 7KHQRPLQDOMRLQW VKHDUVWUHQJWKVLQ$&,7DEOHDUHJLYHQLQ7DEOH

13-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 



ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦ ܸଵ 

κ௖ 

ܸଶ  ‫ܯ‬௡ା 

‫ܯ‬௡ି  ܸଶ 

ܸଵ 

‫ܣ‬ା ௦

–‡”‹‘”…‘Ž—ƒ†‡†‰‡…‘Ž—™‹–Š–Š‡ †‹”‡…–‹‘‘ˆƒƒŽ›•‹•’ƒ”ƒŽŽ‡Ž–‘–Š‡‡†‰‡

ܸ௖௢௟  ܿଵ  ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦

κ௖ 

ܸଵ  ‫ܯ‬௡ି  ܸଵ 

†‰‡…‘Ž—™‹–Š–Š‡†‹”‡…–‹‘‘ˆƒƒŽ›•‹• ’‡”’‡†‹…—Žƒ”–‘–Š‡‡†‰‡ƒ†…‘”‡”…‘Ž—

ܸ௖௢௟ 

Figure 13.14 Free-body diagrams of columns in an intermediate moment frame subjected to gravity and lateral loads.

13-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.6 Nominal Joint Shear Strength, Vn Column

Continuous or a column extension is provided WKDWVDWLV¿HV$&,

Beam in Direction of Analysis

Continuous or a beam extension is provided WKDWVDWLV¿HV$&,

Confinement by Transverse Beams*

Vn (lbs)**

&RQ¿QHG 1RWFRQ¿QHG &RQ¿QHG

Other 1RWFRQ¿QHG Continuous or a beam extension is provided WKDWVDWLV¿HV$&,

&RQ¿QHG 1RWFRQ¿QHG

Other &RQ¿QHG Other 1RWFRQ¿QHG *Transverse beams that satisfy the requirements of ACI 15.2.8 are considered to provide confinement (see Figure 11.3 of this publication). Examples of the various joint types in this table are given in Figure 13.15. **The modification factor, , that reflects the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength is equal to 0.75 for lightweight concrete and 1.0 for normalweight concrete.

Examples of the various joint types in Table 13.6 and the corresponding DUHJLYHQLQ)LJXUH-RLQWVZLWK continuous columns (that is, columns frame into the top and bottom of the joint) or with a column extension that satLV¿HVWKHUHTXLUHPHQWVRI$&,DUHGHVLJQDWHG7\SH,-RLQWVZLWKQRQFRQWLQXRXVFROXPQVWKDWLVMRLQWVZKHUH columns frame into the bottom of the joints only, such as those in the top story of a building), are designated Type ,,7\SH$DQG7\SH%GHVLJQDWLRQVLQ)LJXUHFRUUHVSRQGWRMRLQWVZLWKFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRI DQDO\VLVRUZLWKEHDPH[WHQVLRQVWKDWVDWLVI\WKHUHTXLUHPHQWVLQ$&,7\SH&DQG7\SH'GHVLJQDWLRQVFRUUHVSRQGWRMRLQWVZLWKQRQFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRIDQDO\VLV$GHVLJQDWLRQRI³&´DWWKHIDFHRIDMRLQW PHDQVDWUDQVYHUVHEHDPSURYLGHVFRQ¿QHPHQWDWWKDWIDFHLQDFFRUGDQFHZLWK$&,$Q³1&´GHVLJQDWLRQ PHDQVWKDWQRWUDQVYHUVHEHDPLVSUHVHQWDWWKDWIDFHRUDWUDQVYHUVHEHDPLVSUHVHQWEXWGRHVQRWVDWLVI\WKHFRQ¿QHPHQWUHTXLUHPHQWVLQ$&, FDQEHREWDLQHGIURP)LJXUHIRUWKHYDULRXVFRPELQDWLRQVRIMRLQWW\SHV)RU The nominal shear strengths, example, a Type I-A joint consists of a continuous column and continuous beams in the direction of analysis with This joint type corresponds FRQ¿QHPHQWSURYLGHGE\WUDQVYHUVHEHDPVRQERWKIDFHVLQWKLVFDVH to an interior column located in the structure other than the top story. Similarly, a corner column in the top story of a structure corresponds to a Type II-D joint, and The term LVWKHHIIHFWLYHFURVVVHFWLRQDODUHDZLWKLQDMRLQWZKLFKLVJLYHQLQ$&,%\GH¿QLWLRQ is equal to the product of the joint depth and the effective joint width. The depth of the joint is always equal to the depth of the column parallel to the direction of analysis, (see Figure 11.13 of this publication). The effective is equal to the width of the column perpendicular to the direction of analysis where the beams in the joint width, For beams not as wide direction of analysis are as wide as or wider than the column. In such cases, and where is the beam width and is the perpendicular as the column, is equal to the lesser of distance from the edge of the beam to the nearest side face of the column.

13-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Note 1 Transverse beam

C

C or NC

C

NC

Type A

Type B

C

C or NC

C

NC

Type C

Type D

Note 2

Note 2



Type I

Transverse beam

Type II

Plan

Elevation Direction of analysis

Joint Type

ܸ௡

I-A

20ߣඥ݂௖ᇱ ‫ܣ‬௝

I-B

15ߣඥ݂௖ᇱ ‫ܣ‬௝

I-C

15ߣඥ݂௖ᇱ ‫ܣ‬௝

I-D

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-A

15ߣඥ݂௖ᇱ ‫ܣ‬௝

II-B

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-C

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-D

8ߣඥ݂௖ᇱ ‫ܣ‬௝

Notes 1.Column is continuous or a column extension satisfying ACI 15.2.6 is provided ሺsee Figure 11.1 of this publicationሻ. 2.Beam in direction of analysis is continuous or a beam extension satisfying ACI 15.2.7 is provided ሺsee Figure 11.2 of this publicationሻ. 3.C – Transverse beam provides confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3 of this publicationሻ. 4.NC – Transverse beam is not provided or transverse beam does not provide confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3 of this publicationሻ.

Figure 13.15 Examples of joint types and the corresponding nominal joint shear strength, Vn.

13-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

13.3.5 Two-way Slabs Without Beams Overview

Flat plate systems, which are two-way slabs without beams, are permitted to be part of the SFRS in intermediate PRPHQWIUDPHV$VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHWKLFNQHVVRIDÀDWSODWHV\VWHPLVXVXDOO\GHWHUPLQHG¿UVWRQWKHEDVLVRIWKHDSSURSULDWHGHÀHFWLRQUHTXLUHPHQWV7ZRZD\VKHDUUHTXLUHPHQWVPD\FRQWUROWKH WKLFNQHVVRIWKHVODELQVWHDGRIVHUYLFHDELOLW\UHTXLUHPHQWV+HDGHGVKHDUVWXGUHLQIRUFHPHQWSURYLGHVDQHFRQRPLcal way to increase the two-way shear resistance of a slab. Analysis Methods

)ODWSODWHÀRRUDQGURRIV\VWHPVFDQEHPRGHOHGIRUODWHUDOORDGUHVLVWDQFHXVLQJDQ\PHWKRGVDWLVI\LQJERWKHTXLlibrium and compatibility. It is common to assume the intermediate moment frames consist of column strips and columns. This essentially means all the lateral load effects get assigned to the column strips. In lieu of that method, an effective beam width can be determined that is a fraction of the transverse width of the slab based on a model that produces results in reasonable agreement with test data. The combined effects due to gravity and lateral load effects are obtained using the strength design load combinaWLRQVLQ7DEOH,WLVSHUPLWWHGWRFRPELQHWKHUHVXOWVRIWKHJUDYLW\ORDGDQDO\VLVZLWKWKHUHVXOWVIURPWKHODWHUDO ORDGDQDO\VLV$&, ,QRWKHUZRUGVWKHVODEFDQEHDQDO\]HGIRUJUDYLW\ORDGVXVLQJDQ\UDWLRQDOPHWKRG LQFOXGLQJWKH'LUHFW'HVLJQ0HWKRGLQ6HFWLRQRIWKLVSXEOLFDWLRQZKHUHDSSOLFDEOH DQGWKHUHVXOWVIURPWKDW method can be combined with the results from the lateral load analysis assuming, for example, the column strips alone resist the lateral load effects. Required Flexural Reinforcement

2QFHWKHWKLFNQHVVRIWKHVODEKDVEHHQGHWHUPLQHGRQWKHEDVLVRIGHÀHFWLRQDQGWZRZD\VKHDUFULWHULDDQGWKHVODE KDVEHHQDQDO\]HGIRUWKHHIIHFWVGXHWRJUDYLW\DQGODWHUDOORDGVWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWFDQEHGHWHUVHH6HFWLRQRIWKLVSXEOLFDWLRQ  mined in the column strips and middle strips using $VLQWKHFDVHIRUVODEFROXPQMRLQWVLQEXLOGLQJVDVVLJQHGWR6'&$DQG%VSHFLDOÀH[XUDOUHTXLUHPHQWVPXVWEH VDWLV¿HGDWVODEFROXPQMRLQWVLQEXLOGLQJVDVVLJQHGWR6'&&ZKHUHLQWHUPHGLDWHPRPHQWIUDPHVFRQVLVWLQJRI WZRZD\VODEVZLWKRXWEHDPVDUHXWLOL]HG,WLVDVVXPHGWKHVODEPRPHQWUHVLVWHGE\WKHFROXPQDWDVODEFROXPQ GXHWRJUDYLW\DQGHDUWKTXDNHHIIHFWVLVWUDQVIHUUHGE\DFRPELQDWLRQRIÀH[XUHDQGHFFHQWULFLW\RIVKHDU joint, WUDQVIHUUHGE\ÀH[XUHLVHTXDOWR where is determined by ACI Equa$&, 7KHSRUWLRQRI WLRQ  (13.18)

In this equation, and are the dimensions of the critical section for shear measured parallel and perpendicular to the direction of analysis, respectively. at edge and corner columns for intermediate moment The requirements for the effective slab width resisting IUDPHVDUHJLYHQLQ$&,DQGDUHLOOXVWUDWHGLQ$&,)LJXUH5VHH)LJXUH )OH[XUDOUHLQIRUFHunless it is placed within the efment perpendicular to the edge is not considered fully effective to resist fective slab widths indicated in Figure 13.16. The portion of the effective slab width extending beyond the face of or where is the distance from the interior face of the column to the slab edge the column is the lesser of LQWKHGLUHFWLRQRIDQDO\VLV7KHGH¿QLWLRQRIWKHHIIHFWLYHVODEZLGWKLVEDVHGRQ\LHOGOLQHVWKDWH[WHQGIURPWKH LQWHULRUFRUQHUVRUFRUQHURIWKHFROXPQWRWKHHGJHRIWKHVODEDWDGHJUHHDQJOH7KHHIIHFWLYHVODEZLGWKVIRU interior columns and columns bending parallel to the edge are the same as those for ordinary moment frames, which must be placed DUHJLYHQLQ$&,7DEOH7KHDPRXQWRIÀH[XUDOUHLQIRUFHPHQWUHTXLUHGWRUHVLVW ZLWKLQWKHHIIHFWLYHZLGWKGH¿QHGLQ)LJXUHDQGPXVWEHJUHDWHUWKDQRUHTXDOWRRQHKDOIRIWKHUHLQIRUFHPHQW LQWKHFROXPQVWULSDWWKHVXSSRUW$&, $GGLWLRQDOÀH[XUDOUHLQIRUFHPHQWPD\EHUHTXLUHGLQWKHVODEWR satisfy this strength requirement.

13-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is possible for a reversal of moments to occur at the joints. In such cases, bottom reinforcement is required in addition to the top reinforcement within the effective slab width. ܿ௧ 

 ͳǤͷ݄ ൑ ܿ௧ 

‹‡Ž†Ž‹‡ሺ–›’Ǥሻ

ܿଶ  ͳǤͷ݄ ൑ ܿ௧ 

൑ Ͷͷιሺ–›’Ǥሻ

ˆˆ‡…–‹˜‡•Žƒ„™‹†–Š

ܿଵ  ۳‫܍܏܌‬۱‫ܖܕܝܔܗ‬

ܿଵ  ܿଶ  ͳǤͷ݄ ൑ ܿ௧ 

ˆˆ‡…–‹˜‡•Žƒ„™‹†–Š

ܿ௧  ۱‫ܚ܍ܖܚܗ‬۱‫ܖܕܝܔܗ‬

‹”‡…–‹‘‘ˆƒƒŽ›•‹•

Figure 13.16 Effective slab width in edge and corner connections of intermediate moment frames consisting of two-way slabs without beams.

Detailing the Flexural Reinforcement

7KHGHWDLOLQJUHTXLUHPHQWVLQ$&,WKURXJKIRUÀH[XUDOUHLQIRUFHPHQWPXVWEHVDWLV¿HGIRUDOOWZRZD\ VODEV\VWHPVLQFOXGLQJWKRVHLQEXLOGLQJVDVVLJQHGWR6'&&VHH6HFWLRQRIWKLVSXEOLFDWLRQ )RUWZRZD\ VODEVWKDWDUHSDUWRIDQLQWHUPHGLDWHPRPHQWIUDPHWKHGHWDLOLQJSURYLVLRQVRI$&,WKURXJKPXVW DOVREHVDWLV¿HG As noted above, at least one-half of the reinforcement provided in the column strip at the support must be placed within WKHHIIHFWLYHVODEZLGWKJLYHQLQ$&,7DEOH$&, 7KLVUHTXLUHPHQWLVLOOXVWUDWHGLQ)LJXUH 13-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHGHWDLOLQJUHTXLUHPHQWVLQ$&,WKURXJKDUHJLYHQLQ)LJXUH

½ Middle Strip

ܿଶ 

ܾ௦௟௔௕ ൌ ܿଶ ൅ 3݄

Column Strip All reinforcement to resist ‫ܯ‬௦௖ to be placed in column strip ሺACI 18.4.5.1ሻ

½ Middle Strip

Reinforcement required to resist ߛ௙ ‫ܯ‬௦௖ (ACI 18.4.5.2) ‫ܣ‬௦ ൒larger of Reinforcement in column strip/2 (ACI 18.4.5.3)

Note: Detailing requirements apply to both top and bottom reinforcement

Figure 13.17 Reinforcement details at supports of two-way slabs without beams.

Shear Strength Requirements

7KHRQHZD\DQGWZRZD\VKHDUUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\PXVWEHVDWLV¿HGIRUVKHDUGHVLJQLQWZRZD\VODEVVHH&KDSWHURIWKLVSXEOLFDWLRQ  For two-way systems without column-line beams, it has been shown that slab-column connections are susceptible to punching shear failure and reduced lateral displacement ductility during seismic events where shear stresses due on shear caused by factored gravity loads without moment to gravity loads are relatively large. A limit of is determined WUDQVIHULVSUHVFULEHGLQ$&,ZKHUHWKHQRPLQDOWZRZD\VKHDUVWUHQJWKRIWKHFRQFUHWH LQDFFRUGDQFHZLWK$&,7KLVUHTXLUHPHQWQHHGQRWEHVDWLV¿HGZKHUHWKHSURYLVLRQVRI$&,DUHVDWLV¿HGZKLFKDUHDSSOLFDEOHWRVODEFROXPQFRQQHFWLRQVLQIUDPHVWKDWDUHQRWSDUWRIWKH6)56

13.4 Foundations No design and detailing requirements other than those in ACI Chapter 13 are needed for foundations supporting VWUXFWXUHVDVVLJQHGWR6'&%7KHLQIRUPDWLRQLQ&KDSWHURIWKLVSXEOLFDWLRQFDQEHXVHGLQVXFKFDVHV Design and detailing requirements for piles, pile caps, and deep foundations (including drilled piers) supporting structures assigned to SDC C are given in ACI 18.13 (there are no requirements in ACI 18.13 for spread footings supporting buildings assigned to SDC C). Requirements for slabs-on-ground resisting in-plane earthquake forces IURPZDOOVRUFROXPQVWKDWDUHSDUWRIWKH6)56DUHJLYHQLQ$&,7KHUHTXLUHPHQWVWKDWPXVWEHVDWLV¿HG IRUGULOOHGSLHUVFDLVVRQV DUHJLYHQLQ7DEOH

13-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

                

             

‫ିܣ‬ ௦,௖௢௡௧௜௡௨௢௨௦ ൒ larger of ൝

‫ିܣ‬ ௦,κ 

‫ିܣ‬ ௦,κ Τ4 ‫ିܣ‬ ௦,௥ Τ4

 ACI 18.4.5.4 

‫ିܣ‬ ௦,௥ 

‫ܣ‬ା ௦

‫ିܣ‬ ௦,κ Τ3 ൒ larger of ൝

‫ିܣ‬ ௦,௥ Τ3

 ACI 18.4.5.5 

ା ‫ିܣ‬ ௦,κ and ‫ܣ‬௦ to be developed ACI 18.4.5.6 and 18.4.5.7 

‘Ž—–”‹’

‫ିܣ‬ ௦,κ 

‫ିܣ‬ ௦,௥ 

ା ‫ܣ‬ା ௦,௖௢௡௧௜௡௨௢௨௦ ൒ ‫ܣ‬௦ /2

‫ܣ‬ା ௦

ACI 18.4.5.6  ା ‫ିܣ‬ ௦,κ and ‫ܣ‬௦ to be developed ACI 18.4.5.6 and 18.4.5.7 

‹††Ž‡–”‹’ Figure 13.18 Reinforcement details for two-way slabs without beams in intermediate moment frames.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.7 Requirements for Drilled Piers Supporting Structures Assigned to SDC C Requirement

ACI Section No.

Caissons must be interconnected by foundation seismic ties in orthogonal directions, unless it can be demonstrated that equivalent restraint is provided by other means.



Foundation seismic ties must have a design strength in tension and compression at least times the greater of (1) the factored dead load plus factored live load equal to VXSSRUWHGE\WKHSLOHFDSRU WKHIDFWRUHGGHDGORDGSOXVIDFWRUHGOLYHORDGVXSSRUWHG by the column. This requirement is permitted to be waived if it can be demonstrated that another means of equivalent restraint will be provided.



Caissons resisting design tension forces must have continuous longitudinal reinforcement over their length.



0LQLPXPORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHGLQ$&,PXVWEH extended over the entire unsupported length of the portion of a caisson in air or water, or in soil not capable of providing adequate lateral restraint to prevent buckling throughout this length.



+RRSVVSLUDOVDQGWLHVLQFDLVVRQVPXVWEHWHUPLQDWHGZLWKVHLVPLFKRRNV



Longitudinal and transverse reinforcement in uncased cast-in-place caissons must be proYLGHGLQDFFRUGDQFHZLWK$&,7DEOHDQGZLWK$&,DQG



Longitudinal reinforcement in caissons resisting tension forces must be detailed to transfer the tension forces within the pile cap to the supported structural members.

18.13.6.1

Caisson longitudinal reinforcement must be embedded in the pile cap a distance equal to the development length in compression or tension. Alternatively, it may be embedded by WKHXVHRI¿HOGSODFHGGRZHOVDQFKRUHGLQWKHFDLVVRQ



7KHSXUSRVHRIWKHIRXQGDWLRQVHLVPLFWLHVVSHFL¿HGLQ$&,LVWRPLQLPL]HWKHUHODWLYHPRYHPHQWRIWKHVXSported member relative to the movement of the caisson. A grade beam can be used as a seismic tie (see Figure 13.19). ‘Ž—‘”™ƒŽŽ

ƒ‹••‘ሺ’‹‡”ሻ…ƒ’

‡‹•‹…–‹‡„‡ƒ

ƒ‹••‘ሺ’‹‡”ሻ

Figure 13.19 Foundation tie for caissons supporting structures in SDC C.

13-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHIRXQGDWLRQVHLVPLFWLHGHVLJQVWUHQJWKVSHFL¿HGLQ$&,LVPHDQWWRSURYLGHDPLQLPXPFRQQHFWLRQ times the larger between adjacent caissons. The required design strength for the tie beam must be at least force from the column or pile cap, whichever applies, on either end of the tie beam. Longitudinal reinforcement in the tie beam, which is designed for the applicable factored tension and compression forces, must be anchored in the caisson (see Figure 13.19). 5HTXLUHPHQWVIRUXQFDVHGFDVWLQSODFHRUDXJHUHGFRQFUHWHSLOHVRUSLHUVDUHLOOXVWUDWHGLQ)LJXUHIRUDFLUcular pile or pier with spiral reinforcement where it is assumed a hard or stiff layer of soil is present over the entire length of the pile or pier. Pile cap or mat foundation

݀௣ 

൒ κௗ 

‫ܦ‬௖௛ 

Min. #3 spiral ‫ܣ‬௦ ൒ 0.0025‫ܣ‬௚  Min. 6 long. bars

4ԣ

Section A-A

A 8݀௕

6ᇳ ‫ ݏ‬൑ lesser of ൝

Zone 1

κଶ 

κଵ ൒ 3݀௣ 



A

‫ ݏ‬൑ 16݀௕ 

κ௙ ൌ distance from bottom of pile cap to where 0.4‫ܯ‬௖௥ ൐ ‫ܯ‬௨

Zone 2

κ௣ 

κଵ ൌ length of transverse confinement reinforcement zone κ௣ Τ 3 ‫ۓ‬ ۖ ۖ 10ᇱ κଶ ൌ minimum reinforced pier/pile length ൒ longest of ‫ ۔‬3݀௣ ۖ ۖ ‫ ە‬κ௙

Longitudinal reinforcement must extend at least κௗ beyond the distance from the bottom of the pile cap to where 0.4‫ܯ‬௖௥ ൐ ‫ܯ‬௨ . 

Figure 13.20 Requirements for uncased cast-in-place or augered concrete piles or piers supporting structures assigned to SDC C – Hard or stiff soil over entire length.

13-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QWKH¿JXUH LVWKHOHQJWKRIWKHUHTXLUHGWUDQVYHUVHFRQ¿QHPHQWUHLQIRUFHPHQW]RQHDQG is the minimum OHQJWKZKHUHUHLQIRUFHPHQWLVUHTXLUHG7KHVHOHQJWKVDUHGH¿QHGLQ$&,7DEOHDQGDUHWKHVDPHIRUDQ\ 6LWH&ODVV=RQHLQ)LJXUHFRUUHVSRQGVWRWKHVHJPHQWZKHUHPLQLPXPWUDQVYHUVHFRQ¿QHPHQWUHLQIRUFH=RQHFRUUHVSRQGVWRWKHVHJPHQWZKHUHPLQLPXPWUDQVYHUVHUHLQIRUFHPHQW ment is required over the length is required in the remainder of the required reinforcement length (that is, in the region outside of Zone 1 where the 7KHGDVKHGORQJLWXGLQDOUHLQIRUFHPHQWFRUUHVSRQGVWRWKHFDVHZKHUHWKHSLOHSLHUPXVW length is equal to UHVLVWGHVLJQWHQVLRQIRUFHV$&,  Where a portion of a pile or pier is in air or water or is in soil not capable of providing adequate lateral restrain to prevent buckling, Zone 1 longitudinal and transverse reinforcement must be provided over the entire unsupported OHQJWK$&,  7KHSXUSRVHRIWKHUHTXLUHPHQWVLQ$&,DQGLVWRHQVXUHWKDWDFRPSOHWHORDGSDWKLVSUHVHQWWKDW transfers the force from the longitudinal reinforcement in the supported member through the pile (caisson) cap to the longitudinal reinforcement in the caisson.

13.5 Examples 13.5.1 Example 13.1 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option B), Beam is Part of the SFRS (Intermediate Moment Frame), SDC C

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHEHDPDORQJFROXPQOLQHEHWZHHQFROXPQOLQHV&DQG'LQ %XLOGLQJ)UDPLQJ2SWLRQ%DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQV RIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ DQG*UDGHUHLQIRUFHPHQW Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& where Step 1 – Determine the factored bending moments along the span

A summary of the service bending moments due to the dead and live loads at the critical sections of the beam is given in Table 13.8. Table 13.8 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Beam in Example 13.1 Negative – Line C

Positive

Negative – Line D

7KHEHQGLQJPRPHQWVLQWKHEHDPGXHWRHDUWKTXDNHORDGVDUHJLYHQLQ7DEOH7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHQRUWKGLUHFWLRQDQGWKHVRXWKGLUHFWLRQ (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). The bending moments due to wind loads are smaller than those due to the earthquake loads and are not considered in this example. Table 13.9 Bending Moments (ft-kips) due to Earthquake Loads at the Second-Floor Level for the Beam in Example 13.1 Negative – Line C

Positive

Negative – Line D

ņ

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOHVHH7DEOH  Table 13.10 Design Bending Moments (ft-kips) at the Second-Floor Level for the Beam in Example 13.1 Load Combination

Negative – Line C

Positive

$&,(TD



$&,(TE

 SSR

$&,(TH



SSR $&,(TJ

 

SSL 



 

SSL

Negative – Line D



Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR ZKHUHDVQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHUHGXQGDQF\IDFWRU is SHUPLWWHGWREHWDNHQDVIRUEXLOGLQJVDVVLJQHGWR6'&&$6&(6(, 7KHUHIRUHWKLVORDGFRPELQDWLRQ becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 2 – Determine the required flexural reinforcement at the critical sections

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW at the critical sections is determined by the following equations, ZKLFKDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUVHFWLRQVZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFH(an effective slab width at positive moment critical sections is not considered ment where in this example):

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWDOOVHFWLRQVDUHWHQVLRQ controlled because Table 13.11 Required Flexural Reinforcement for the Beam in Example 13.1 Location

SSR 1HJDWLYH±/LQH& SSL

Mu (ft-kips)

Rn (psi)

As (in.2)*









3.69 (table continued on next page)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.11 Required Flexural Reinforcement for the Beam in Example 13.1 (cont.) Location

Mu (ft-kips)

Rn (psi)

As (in.2)*

Positive















SSR 1HJDWLYH±/LQH' SSL



*Min. Max.

Step 3 – Select the flexural reinforcement

Because the beam is subjected to torsional loading from the dead and live loads, additional longitudinal reinforcePHQWPD\EHUHTXLUHGRQDOOIRXUIDFHVRIWKHEHDPLQDFFRUGDQFHZLWK$&,VHH([DPSOH ,QRUGHU WRGHWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWWKHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHEHDPPXVWEHNQRZQZKLFK GHSHQGRQWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW7KHUHIRUHDWWKLVVWDJHLQWKHGHVLJQPRUHUHLQIRUFHPHQWWKDQUHTXLUHGIRUÀH[XUHDORQHLVSURYLGHGEHFDXVHRIWRUVLRQDOORDGLQJ,WZLOOEHFKHFNHGLQ([DPSOHZKHWKHUDOOWKH DSSOLFDEOHUHTXLUHPHQWVDUHVDWLV¿HGRQFHWKHWRWDODUHDRIORQJLWXGLQDOUHLQIRUFHPHQWLVGHWHUPLQHGEDVHGRQÀH[XUH and torsion. 6HOHFWWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVEDVHGRQWKHPD[LPXPDQGPLQLPXPVSDFLQJUHTXLUHPHQWVLQ$&, DQGUHVSHFWLYHO\ 1HJDWLYHUHLQIRUFHPHQW±/LQH&: maximum and minimum number of longitudinal bars in a single 8VHEDUV OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  Positive reinforcement: maximum and minimum number of longitudinal bars in a single 8VHEDUV OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  DUHDOVRDGHTXDWHIRUWKHIWNLSDQGIWNLSSRVLWLYHPRPHQWVDWWKH 7KHEDUV faces of the supports at column lines C and D, respectively (see Table 13.11). 1HJDWLYHUHLQIRUFHPHQW±/LQH': maximum and minimum number of longitudinal bars in a single 8VHEDUV OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  Step 4 – Check the nominal strength requirements of ACI 18.4.2.2

The positive moment strength at the face of a joint must be greater than or equal to one-third the negative moment VWUHQJWKSURYLGHGDWWKDWORFDWLRQ$&, 7KLVUHTXLUHPHQWLVVDWLV¿HGDWERWKMRLQWVDVVXPLQJWKHERWtom bars are continuous over the span:

13-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Also, the negative or positive moment strength at any section of the beam must be greater than or equal to RQH¿IWKWKHPD[LPXPPRPHQWVWUHQJWKSURYLGHGDWWKHIDFHRIHLWKHUMRLQWZKLFKLQWKLVH[DPSOHLVHTXDOWR 3URYLGLQJFRQWLQXRXVERWWRPEDUVDQGFRQWLQXRXVWRSEDUVVDWLV¿HVWKLVUHquirement. 13.5.2 Example 13.2 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option B), Beam is Part of the SFRS (Intermediate Moment Frame), SDC C

Determine the required shear reinforcement for the beam along column line 1 between column lines C and D in %XLOGLQJ)UDPLQJ2SWLRQ%DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQV RIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ DQG*UDGHUHLQIRUFHPHQW Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& See Example 13.1. where Step 1 – Determine the factored shear forces

'HVLJQVKHDUVWUHQJWKPXVWEHJUHDWHUWKDQRUHTXDOWRWKHOHVVHURIWKHWZRYDOXHVFDOFXODWHGE\$&,D DQG E  $FFRUGLQJWR$&,D PD[LPXP is obtained by adding the shear forces associated with the development RIWKHQRPLQDOÀH[XUDOVWUHQJWKVDWHDFKHQGRIWKHEHDPDQGWKHVKHDUIRUFHVIURPWKHIDFWRUHGJUDYLW\DQGYHUWLFDO earthquake loads. Sidesway to the right and to the left must both be investigated to obtain the maximum 7KHODUJHVWVKHDUIRUFHDVVRFLDWHGZLWKVHLVPLFHIIHFWVLVREWDLQHGIURP$&,(TXDWLRQH 7KHIDFWRUHGORDG that is uniformly distributed over the length of the beam is obtained by substituting LQWR$&,(TXDWLRQH 

6KHDUIRUFHVGXHWRWKHFRPELQDWLRQRIJUDYLW\ORDGVDQGQRPLQDOÀH[XUDOVWUHQJWKIRUVLGHVZD\WRWKHULJKWDUH JLYHQLQ)LJXUHDVVXPLQJWKHERWWRPEDUVDUHFRQWLQXRXVRYHUWKHVSDQVHH)LJXUH 7KHPD[LPXP LVHTXDOWRNLSV%HFDXVHWKHQHJDWLYHÀH[XUDOUHLQIRUFHPHQWLVWKHVDPHDWERWKVXSSRUWVWKHPD[LPXP is also equal to 66.6 kips for sidesway to the left. ‫ݓ‬௨ ൌ ͵Ǥͳ‹’•Ȁˆ– ‹†‡•™ƒ›”‹‰Š–

‫ܯ‬௡ି ൌ ͶͶʹǤͷˆ–Ǧ‹’•

‫ܯ‬௡ା ൌ ʹ͹ʹǤͺˆ–Ǧ‹’• ʹͳԢǦ͸ԣ ͲǤͳ‹’•

͸͸Ǥ͸‹’•

Figure 13.21 Factored shear forces in accordance with ACI 18.4.2.3(a).

$FFRUGLQJWR$&,E PD[LPXP where combination mine maximum

is determined from analysis of the structure using the factored load Thus, the following equation can be used to deterEq. (13.1)

From analysis,

13-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore, use

ACI 18.4.2.3

Step 2 – Determine the required shear reinforcement

Assuming ACI Table 22.5.5.1

Transverse reinforcement required for shear: ACI 22.5.8.5.3

Because the beam is subjected to torsional loading, additional transverse reinforcement may be required (see Example 13.3). 13.5.3 Example 13.3 – Determination of Torsion Reinforcement: Beam in Building #1 (Framing Option B), Beam is Part of the SFRS (Intermediate Moment Frame), SDC C

Determine the required torsion reinforcement for the beam along column line 1 between column lines C and D in %XLOGLQJ)UDPLQJ2SWLRQ%DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQV RIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ DQG*UDGHUHLQIRUFHPHQW Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& 6HH([DPSOHVDQG where Step 1 – Determine the maximum factored torsional moment

From analysis, the maximum factored torsional moment,

LVHTXDOWRIWNLSV

Step 2 – Determine if torsional effects must be considered

Torsion can be neglected where the factored torsional moment from analysis is less than the threshold torsion: ACI 22.7.1.1

'HWHUPLQHWKHRYHUKDQJLQJÀDQJHZLGWK , permitted to be used in the calculation of

and

:

ACI 9.2.4.4

7RUVLRQDOVHFWLRQSURSHUWLHVLQFOXGLQJWKHRYHUKDQJLQJÀDQJH Figure 6.3

13-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7RUVLRQDOVHFWLRQSURSHUWLHVZLWKRXWWKHRYHUKDQJLQJÀDQJH

Because LQJÀDQJHXVH

IRUWKHEHDPZLWKWKHRYHUKDQJLQJÀDQJHLVJUHDWHUWKDQWKDWIRUWKHEHDPZLWKRXWWKHRYHUKDQJ>$&,E @

Therefore, torsional effects must be considered. Because the edge beam is part of an indeterminate system in which redistribution of internal forces can occur following torsional cracking, the maximum factored torsional moment need not exceed the cracking torsional moment: ACI 22.7.3.2

This is equivalent to a factored uniformly distributed torsional loading,

equal to the following:

Step 3 – Check the adequacy of the cross-sectional dimensions

:LWKLQFOHDUFRYHUWRKRRSV Figure 6.17

)URP6WHS )URP6WHSLQ([DPSOH )URP6WHSLQ([DPSOH

ACI 22.7.7.1

Therefore, the cross-sectional dimensions are adequate.

13-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the required transverse reinforcement for torsion ACI 22.7.6.1

Step 5 – Determine the required longitudinal reinforcement for torsion

ACI 22.7.6.1

ACI 9.6.4.3

Use The transverse and longitudinal reinforcement for torsion are combined with the transverse reinforcement required IRUVKHDUDQGWKHORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHGIRUÀH[XUHVHH([DPSOH  13.5.4 Example 13.4 – Design for Combined Flexure, Shear, and Torsion: Beam in Building #1 (Framing Option B), Beam is Part of the SFRS (Intermediate Moment Frame), SDC C

'HVLJQWKHEHDPDORQJFROXPQOLQHEHWZHHQFROXPQOLQHV&DQG'LQ%XLOGLQJ)UDPLQJ2SWLRQ%DWWKH VHFRQGÀRRUOHYHOIRUWKHFRPELQHGHIIHFWVGXHWRÀH[XUHVKHDUDQGWRUVLRQDVVXPLQJWKHEHDPLVSDUWRIWKH6)56 DQGWKHGLPHQVLRQVRIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUH DQG*UDGHUHLQIRUFHPHQW LQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& 6HH([DPSOHVDQG where Step 1 – Determine the total transverse reinforcement

7KHWRWDOUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWSHUVWLUUXSOHJLVHTXDOWRWKDWUHTXLUHGIRUVKHDU([DPSOH  plus that required for torsion (Example 13.3):

ACI 9.6.4.2

13-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The maximum spacing of the hoops over the length of the beam is the smallest of the following:

from the face of the support at each end

•

ACI 18.4.2.4, Figure 13.6

• • • LQ The maximum spacing of the legs of the transverse reinforcement across the width of the member is determined LQDFFRUGDQFHZLWK$&,7DEOHZKHUH Maximum spacing across width $VVXPLQJWUDQVYHUVHEDUVZLWKOHJVFHQWHUWRFHQWHUVSDFLQJ Thus, 3 legs must be provided. 7U\KRRSVOHJV DWHDFKHQGRIWKHEHDPVSDFHGDWLQRQFHQWHUZLWKWKH¿UVWKRRSORFDWHGLQIURPWKH face of the support. Provided transverse reinforcement (per leg) 7KHUHIRUHXVHKRRSVOHJV DWHDFKHQGRIWKHEHDPVSDFHGDWLQRQFHQWHUSURYLGHGWUDQVYHUVHUHLQIRUFH). ment per leg 7KHSURYLGHGKRRSVSDFLQJRILQLVOHVVWKDQWKHOHVVHURIWKHIROORZLQJZKLFKLVWKHPD[LPXPSHUPLWWHGVSDFLQJ for the transverse torsional reinforcement:

ACI 9.7.6.3.3

For the remainder of the beam, the maximum closed stirrup spacing is

ACI 18.4.2.5

7U\FORVHGVWLUUXSVOHJV DWLQRQFHQWHUIRUWKHUHPDLQGHURIWKHEHDP $WLQIURPWKHIDFHRIWKHVXSSRUW

ACI 22.5.8.5.3

ACI 22.7.6.1

13-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Provided transverse reinforcement (per leg) 7KHUHIRUHXVHFORVHGVWLUUXSVVSDFHGDWLQIRUWKHUHPDLQGHURIWKHEHDP Step 2 – Determine the total longitudinal reinforcement

)URP6WHSLQ([DPSOH This reinforcement must be distributed around the perimeter of the EHDPZLWKDPD[LPXPVSDFLQJRILQDQGPXVWEHFRPELQHGZLWKWKDWUHTXLUHGIRUÀH[XUH$&,DQG to each face of the beam.  7KXVDVVLJQ 8VHEDUVRQHDFKVLGHIDFHRIWKHEHDP Check spacing of side bars: :LWKWRSDQGERWWRPORQJLWXGLQDOEDUVKRRSVDQGFORVHGVWLUUXSVDQGLQFOHDUFRYHUWKHFOHDUVSDFLQJ between the bars on each side face &HQWHUWRFHQWHUVSDFLQJRIWKHEDUV &HQWHUWRFHQWHUVSDFLQJRIWKHDQGEDUV &KHFNUHTXLUHPHQWRI$&,

where the largest spacing of the transverse reinforcement,

is used to check this requirement.

The remaining longitudinal reinforcement for torsion is distributed equally between the top and bottom of the section: • 1HJDWLYH±/LQH& Total longitudinal reinforcement

Table 13.11

7KHEDUVVHOHFWHGLQ6WHSRI([DPSOHDUHDGHTXDWH • Positive Total longitudinal reinforcement 7KHEDUVVHOHFWHGLQ6WHSRI([DPSOHDUHDGHTXDWH • 1HJDWLYH±/LQH' Total longitudinal reinforcement 7KHEDUVVHOHFWHGLQ6WHSRI([DPSOHDUHDGHTXDWH %HFDXVHWKHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQWWKDWZDVVHOHFWHGLQ([DPSOHDUHDGHTXDWHIRUWKH FRPELQHGHIIHFWVRIÀH[XUHDQGWRUVLRQWKHQHJDWLYHDQGSRVLWLYHQRPLQDOÀH[XUDOVWUHQJWKVDUHWKHVDPHDVGHWHUPLQHGSUHYLRXVO\ZKLFKPHDQVWKHQRPLQDOÀH[XUDOVWUHQJWKUHTXLUHPHQWVRI$&,DUHDXWRPDWLFDOO\ VDWLV¿HGVHH6WHSLQ([DPSOH DQGWKHVKHDUIRUFHVEDVHGRQWKHQRPLQDOÀH[XUDOVWUHQJWKVGRQRWQHHGWREH UHYLVHGVHH6WHSRI([DPSOH  Step 3 – Detail the longitudinal reinforcement

$VVXPHWKHWRSEDUVDQGERWWRPEDUVDUHDOVRUHTXLUHGLQDQHQGVSDQ7KHVHEDUVPXVWH[WHQGWRWKHIDU end of the joint core (that is, the core of the edge column) and must be fully developed in tension in accordance with $&,DQGLQFRPSUHVVLRQLQDFFRUGDQFHZLWK$&,$&, 

13-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPLQJVWDQGDUGGHJUHHKRRNVDUHSURYLGHGDWWKHHQGVRIWKHEDUVWKHWHQVLRQGHYHORSPHQWOHQJWKRID is the greater of the following for normalweight concrete: deformed hooked bar,

Eq. (13.9)

$VVXPLQJORQJLWXGLQDOEDUVDQGKRRSVLQWKHHGJHFROXPQWKHDYDLODEOHGHYHORSPHQWOHQJWKLVHTXDOWRWKH following (see Figure 13.11): Eq. (13.11)

7KHUHIRUHWKHORQJLWXGLQDOEDUVLQWKHEHDPFDQEHIXOO\GHYHORSHGLQWHQVLRQLQDQHGJHFROXPQXVLQJVWDQGDUG GHJUHHKRRNV The development length of a deformed bar in compression,

is the greater of the following:

Eq. (13.14)

$VVXPLQJORQJLWXGLQDOEDUVDQGKRRSVLQWKHHGJHFROXPQWKHDYDLODEOHGHYHORSPHQWOHQJWKLVHTXDOWRWKH IROORZLQJIRUWKHVWUDLJKWSRUWLRQRIWKHKRRNHGEDU>(T @

Eq. (13.15)

where is the bend radius of the beam longitudinal bar, which is equal to  

IRUEDUVVHH$&,7DEOH

7KHUHIRUHWKHORQJLWXGLQDOEDUVLQWKHEHDPFDQEHIXOO\GHYHORSHGLQFRPSUHVVLRQLQDQHGJHFROXPQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVEHDPDWDQLQWHULRUVSDQDUHJLYHQLQ)LJXUH

13-38

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

CC

DC 23ᇱ -6ᇱᇱ 

        

2ᇱ -0ᇱᇱ 

2ᇱ -0ᇱᇱ 

A

A

2ᇱᇱ 

11-#4 hoops  @ 5ԣ

10-#6 

6-#6

2-#5 11-#4 hoops 

15-#4 closed stirrups

2ᇱᇱ 

@ 5ԣ

21ᇱ -6ᇱᇱ 

8.5ᇱᇱ 

Other reinforcement not shown for clarity

10-#6 

15.5ᇱᇱ 

2-#5 ሺtyp.ሻ #4 hoops and crossties 6-#6  2ᇱ -4ᇱᇱ  ‫ۯ ܖܗܑܜ܋܍܁‬-‫ۯ‬

Figure 13.22 Reinforcement details for the beam in Examples 13.1 through 13.4.

13.5.5 Example 13.5 – Determination of Longitudinal Reinforcement: Column in Building #1 (Framing Option B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& where Step 1 – Determine the factored axial forces and bending moments

Table 13.2

$VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVDWWKHERWWRPRIFROXPQ'LQWKH¿UVWVWRU\LVJLYHQLQ7DEOH IRUVLGHVZD\WRWKHULJKW665 DQGVLGHVZD\WRWKHOHIW66/ )DFWRUHGZLQGORDGHIIHFWVDUHVPDOOHUWKDQ WKRVHGXHWRIDFWRUHGVHLVPLFORDGHIIHFWVDQGDUHQRWFRQVLGHUHGLQWKLVH[DPSOH,WFDQEHGHWHUPLQHGWKDWWKH¿UVW VWRU\LVQRQVZD\DQGVOHQGHUQHVVHIIHFWVQHHGQRWEHFRQVLGHUHGVHH6HFWRIWKLVSXEOLFDWLRQ 

13-39

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.12 Summary of Axial Forces and Bending Moments for Column D1 Load Case

Axial Force (kips)

Dead



Roof live



Live



Bending Moment (ft-kips)

ņ

Seismic Load Combination $&,(TD



$&,(TE



$&,(TF

 SSR



SSL



SSR



SSL



$&,(TH

$&,(TJ





Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR ZKHUHDVQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHUHGXQGDQF\IDFWRU is SHUPLWWHGWREHWDNHQDVIRUEXLOGLQJVDVVLJQHGWR6'&&$6&(6(, 7KHUHIRUHWKLVORDGFRPELQDWLRQ becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 2 – Determine the required area of longitudinal reinforcement

%DVHGRQDLQVTXDUHWLHGFROXPQWKHUHTXLUHGUHLQIRUFHPHQWUDWLR is equal to the following for the largest IDFWRUHGD[LDOIRUFHLQ7DEOHZKLFKRFFXUVZLWKRXWDQ\DSSUHFLDEOHEHQGLQJPRPHQWV Eq. (7.32)

Therefore, use the minimum required

6HOHFWEDUV

13-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Check if the selected longitudinal reinforcement is adequate for all load combinations

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUVLVJLYHQLQ)LJXUH$OVR VKRZQLQWKH¿JXUHDUHORDGFRPELQDWLRQSRLQWVIRUWKHIDFWRUHGD[LDOIRUFHVDQGEHQGLQJPRPHQWVLQ7DEOH  1,500

1,250

Axial Force ሺkipsሻ

݂௦ ൌ 0 1,000

݂௦ ൌ 0.5݂௬  750

500

 

 

250

0 0

100

200

300

400

500

600

Bending Moment ሺft-kipsሻ Figure 13.23 Design strength interaction diagram for column D1.

,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHFROXPQLVDGHTXDWHIRUDOOIDFWRUHGORDGFRPELQDWLRQV Step 3 – Check the minimum number of longitudinal bars and the minimum face dimension of the column

For rectangular, tied columns: Minimum number of longitudinal bars with rectangular ties Minimum face dimension

IRUEDUVSHUIDFHDQGQRUPDOODSVSOLFHV

Table 7.16 Table 7.17

8VHEDUV 13.5.6 Example 13.6 – Determination of Transverse Reinforcement: Column in Building #1 (Framing Option B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ% DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KH VODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& 6HH([DPSOH where

13-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the factored shear forces

'HVLJQVKHDUVWUHQJWKPXVWEHJUHDWHUWKDQRUHTXDOWRWKHOHVVHURIWKHWZRYDOXHVGHWHUPLQHGE\$&,D  DQGE  $FFRUGLQJWR$&,D PD[LPXP LVREWDLQHGIURPWKHGHYHORSPHQWRIWKHQRPLQDOÀH[XUDOVWUHQJWKVDW each end of the column for the factored axial force consistent with the direction of analysis resulting in the highest moment strength. Sidesway to the right and to the left must both be investigated to obtain the maximum The nominal strength diagram and the factored load combinations that include seismic load effects are given in )LJXUHIRUWKHORDGHIIHFWVDWWKHERWWRPRIWKHFROXPQ,QWKLVH[DPSOHWKHD[LDOIRUFHDQGWKHFRUUHVSRQGLQJQRPLQDOÀH[XUDOVWUHQJWKDWWKHWRSRIWKHFROXPQLVVPDOOHUWKDQWKHD[LDOIRUFHDQGWKHFRUUHVSRQGLQJQRPLQDO ÀH[XUDOVWUHQJWKDWWKHERWWRPRIWKHFROXPQ&RQVHUYDWLYHO\XVHWKHQRPLQDOÀH[XUDOVWUHQJWKDWWKHERWWRPRIWKH column at both the top and bottom of the column to determine  2,500 2,250 2,000

Axial Force ሺkipsሻ

1,750 1,500 1,250 1,000 750 ACI Eq. 5.3.1ሺeሻ

500

 

ACI Eq. 5.3.1ሺgሻ

250 0 0

100

200

300

400

500

Bending Moment ሺft-kipsሻ

600

700

800

‫ܯ‬௡௕ ൌ 598.7 ft-kips ‫ܯ‬௡௕ ൌ 509.7 ft-kips

Figure 13.24 Nominal strength interaction diagram for column D1.

,WLVHYLGHQWIURP)LJXUHWKDWWKHODUJHVW (TH IRUVLGHVZD\WRWKHULJKW For

13-42

within the range of factored axial forces is obtained from ACI

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$FFRUGLQJWR$&,E PD[LPXP is determined from analysis of the structure using the factored load where Thus, the following equation can be used to detercombination where IRULQWHUPHGLDWHPRPHQWIUDPHVVHH7DEOHRIWKLVSXEOLFDWLRQ  mine maximum

From analysis,

Therefore, use

ACI 18.4.3.1

Step 2 – Determine the required shear reinforcement

Assuming

ACI Table 22.5.5.1

where ZKLFKLVWKHD[LDOIRUFHGHWHUPLQHGE\$&,(TH FRUUHVSRQGLQJWRWKHQRPLQDO is determined from ÀH[XUDOVWUHQJWKXVHGWRGHWHUPLQHWKHPD[LPXPVKHDUIRUFHVHH7DEOH DQG for this case, which means two layers of a strain compatibility analysis of the section. Neutral axis depth longitudinal reinforcement are in tension.

Therefore,

is equal to the following:

3URYLGHDWOHDVWWKHPLQLPXPVKHDUUHLQIRUFHPHQWLQ$&,FRQVLGHULQJWKHUHTXLUHPHQWVLQ$&, $FFRUGLQJWR$&,KRRSVPXVWEHSURYLGHGDWHDFKHQGRIWKHFROXPQRYHUWKHOHQJWK

Eq. (13.5)

13-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ThePD[LPXPVSDFLQJRIWKHKRRSVLVWKHOHDVWRIWKHIROORZLQJIRU*UDGHUHLQIRUFHPHQW

Eq. (13.8)

$VVXPHKRRSVZKLFKDUHSHUPLWWHGWREHXVHGZLWKORQJLWXGLQDOEDUVVHH$&,DQG)LJXUHRI WKLVSXEOLFDWLRQ &KHFNWKHVSDFLQJUHTXLUHPHQWVRI$&,E 

7KHUHIRUHFURVVWLHVPXVWEHSURYLGHGDURXQGWKHLQWHULRUORQJLWXGLQDOEDUV 'HWHUPLQHWKHUHTXLUHGVSDFLQJRIWKHKRRSVEDVHGRQ ACI 22.5.8.5.3

&KHFNWKHPLQLPXPVKHDUUHLQIRUFHPHQWUHTXLUHPHQWVXVLQJDQLQKRRSVSDFLQJ

ACI 10.6.2.2

Provided 8VHKRRSVZLWKFURVVWLHVVSDFHGDWLQRYHUDOHQJWKRIDWOHDVWLQIURPHDFKHQGRIWKHFROXPQ7KLV WUDQVYHUVHUHLQIRUFHPHQWLVDOVRSURYLGHGZLWKLQWKHMRLQWZKLFKVDWLV¿HVWKHUHTXLUHPHQWVRI$&, /RFDWHWKH¿UVWKRRSQRPRUHWKDQ

from the face of the joint.

ACI 18.4.3.4

)RUWKHUHPDLQGHURIWKHFROXPQLWLVSHUPLWWHGWRXVHWLHVDQGFURVVWLHVDWDVSDFLQJRIQRPRUHWKDQWKHOHVVHURI DQGLQEHFDXVH $&,DQG  8VHWLHVDQGFURVVWLHVDWDVSDFLQJRILQZKLFKPDWFKHVWKHVSDFLQJRIWKHKRRSVLQWKHSODVWLF hinge region. 13.5.7 Example 13.7 – Determination of Lap Splice Length: Column in Building #1 (Framing Option B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C

'HWHUPLQHWKHUHTXLUHGODSVSOLFHOHQJWKIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KHVODE and LVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& 6HH([DPSOHVDQG where

13-44

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the tension development length

The tension development length, PHQWVLQ$&,

RIWKHORQJLWXGLQDOEDUVIRUFROXPQ'LVGHWHUPLQHGXVLQJWKHUHTXLUH-

ACI Eq. (25.4.2.4a)

ACI Table 25.4.2.5

ACI Eq. (25.4.2.4b)

Therefore,

Step 2 – Determine the required lap splice length

,WLVHYLGHQWIURP)LJXUHWKDWVRPHRIWKHIDFWRUHGORDGFRPELQDWLRQVIDOOZLWKLQWKHUHJLRQRIWKHLQWHUDFWLRQ Therefore, a Class B tension lap splice must be provided in accordance with ACI Table diagram where LVGHWHUPLQHGIURP$&,7DEOH 7KH&ODVV%WHQVLRQODSVSOLFHOHQJWK

3URYLGHDLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUFROXPQ'DUHJLYHQLQ)LJXUH7KHODSVSOLFHLVORFDWHGQHDUWKHPLGKHLJKWRIWKH FROXPQDOWKRXJKORFDWLQJLWLPPHGLDWHO\DERYHWKHÀRRUVODELVSHUPLWWHG

13-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

#3 hoops @ 8ԣ



2Ԣ-0ԣ  2ԣ 2Ԣ-0ԣ 

2Ԣ-0ԣ 

A

A

8-#8  #3 hoops and crossties 

2Ԣ-0ԣ 

#3 ties @ 8ԣ

3Ԣ-2ԣ 

2Ԣ-0ԣ 

8-#8

#3 hoops @ 8ԣ

2ԣ

2Ԣ-0ԣ 

Section A-A

Other reinforcement not shown for clarity

Figure 13.25 Reinforcement details for column D1 in Examples 13.5 through 13.7.

13.5.8 Example 13.8 – Check of Joint Shear Strength: Column in Building #1 (Framing Option B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHQRUWKVRXWKGLUHFWLRQIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH )LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW,WLVGHWHUPLQHGLQ([DPSOH3DUWF WKDWWKHEXLOGLQJLVDVVLJQHGWR6'&& where Step 1 – Determine the factored horizontal joint shear force

ACI 15.4.1.1

$IUHHERG\GLDJUDPRIWKHMRLQWLVJLYHQLQ)LJXUH7KHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQWLQWKH EHDPVRQERWKVLGHVRIWKHMRLQWDUHWKHVDPHWKDWLVWRSEDUVDQGERWWRPEDUVDUHXVHGLQERWKEHDPV VHH([DPSOH 

13-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ ൌ ʹͶԣ ܸ௖௢௟ ൌ ͷͳǤͷ‹’•

‘‹– ͳͲǦ͓͸

κ௖ ൌ ͳͷǤʹԢ

‫ܯ‬௡ି ൌ ͶͶʹǤͷˆ–Ǧ‹’•

ܸଵ 

ܸଵ ൌ ͸͸Ǥ͸‹’•

‫ܯ‬௡ା ൌ ʹ͹ʹǤͺˆ–Ǧ‹’•

ܸଶ ൌ ͲǤͳ‹’•

ܸଶ  ͸Ǧ͓͸

ܸ௖௢௟ ൌ ͷͳǤͷ‹’• Figure 13.26 Free-body diagram of the joint at column D1.

Determine the shear force in the column,

for an edge column with the direction of analysis parallel to the edge: Table 13.5

)URPDQDO\VLVRIWKHVWUXFWXUHWKHSRLQWRILQÀHFWLRQLQWKHFROXPQDERYHWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURP WKHFHQWHURIWKHMRLQWDQGWKHSRLQWRILQÀHFWLRQLQWKHFROXPQEHORZWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURPWKH center of the joint. Therefore, Table 13.5

The factored shear force in the joint,

is equal to the following:

13-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of analysis is continuous. There are no beams in the transverse direction. Therefore, the joint type is I-B (see Figure   Because the beam in the direction of analysis is slightly larger than the width of the column, the effective crosssectional area within the joint is equal to the area of the column: Figure 11.13

The design shear strength of the joint is equal to the following for a Type I-B joint: Figure 13.15 Step 3 – Determine the detailing for the joint

%HFDXVHWKHMRLQWLVSDUWRIWKH6)56WKHGHWDLOLQJUHTXLUHPHQWVLQ$&,DQGWKURXJK PXVWEHVDWLV¿HG$&,  3URYLGLQJWLHVDQGFURVVWLHVDWDVSDFLQJRILQLQWKHMRLQWZKLFKPDWFKHVWKHVSDFLQJRIWKHKRRSVLQWKH SODVWLFKLQJHUHJLRQVDWLV¿HV$&,VHH)LJXUH  13.5.9 Example 13.9 – Determination of Flexural Reinforcement: Two-way Slab in Building #1 (Framing Option A), Two-way Slab is Part of the SFRS (Intermediate Moment Frame), SDC C

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQHQGVSDQRIDQLQWHULRUGHVLJQVWULSDORQJFROXPQOLQHLQ %XLOGLQJ)UDPLQJ2SWLRQ$DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHWZRZD\VODELVSDUWRIWKH6)56VHH)LJXUH  7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&&ZKHUH Step 1 – Determine the factored bending moments along the span

A summary of the service bending moments due to the dead and live loads at the critical sections in the column strip DQGPLGGOHVWULSLQDQHQGVSDQLVJLYHQLQ7DEOHVHH7DEOHLQ([DPSOH  Table 13.13 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Flat Plate System in Example 13.9 Design Strip

Exterior Negative

Positive

First Interior Negative

Column strip

Middle strip



7KHEHQGLQJPRPHQWVGXHWRHDUWKTXDNHORDGVDUHDVVLJQHGWRWKHFROXPQVWULSDQGDUHJLYHQLQ7DEOHDVVXPLQJDOOIUDPHVLQWKHGLUHFWLRQRIDQDO\VLVDUHSDUWRIWKH6)567KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHG YDOXHVVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHQRUWKGLUHFWLRQDQGWKHVRXWKGLUHFWLRQWKDWLVVLGHVZD\WR the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). The bending moments due to wind loads are smaller than those due to the earthquake loads and are not considered in this example.

13-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 13.14 Bending Moments (ft-kips) due to Earthquake Loads at the Second-Floor Level for the Flat Plate System in Example 13.9 Exterior Negative

Positive

First Interior Negative

ņ 7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOHVHH7DEOH  Table 13.15 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in Example 13.9 Load Combination

Exterior Negative

Location

Positive

First Interior Negative



Column strip $&,(TD



Middle strip

 

Column strip $&,(TE



Middle strip



SSR

86.1

SSL

86.1

Column strip $&,(TH



Middle strip



SSR



SSL



Column strip $&,(TJ Middle strip



33.6

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR ZKHUHDVQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHUHGXQGDQF\IDFWRU is SHUPLWWHGWREHWDNHQDVIRUEXLOGLQJVDVVLJQHGWR6'&&$6&(6(, 7KHUHIRUHWKLVORDGFRPELQDWLRQ becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 2 – Determine the required flexural reinforcement at the critical sections

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW at the critical sections is determined by the following equations, ZKLFKDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUVHFWLRQVZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHment where is the width of the column strip or middle strip and

13-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A summary of the required reinforcement is given in Table 13.16. It is evident that all sections are tension-controlled because Table 13.16 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in Example 13.9

Mu (ft-kips)

Location

Exterior Negative Column strip



Positive First Interior Negative

Middle strip

b (in.)

Rn (psi)

As (in.2)*

Reinforcement*











161













Exterior Negative











Positive



















First Interior Negative * Min. for the column strip Min. for the middle strip Max. spacing For For For the column strip: For the middle strip:

Step 3 – Check that the flexural reinforcement is adequate for moment transfer requirements

ACI 8.4.2.2

• Edge columns Determine the required reinforcement based on the largest transfer moment at the edge columns. Table 13.16

ACI Eq. (8.4.2.2.2)

where Table 5.11, Case 3

Determining whether

13-50

can be increased toWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The moment

must be transferred over the effective slab width ACI 18.4.5.2

Determine the required

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQVHH7DEOH )RUV\PPHWU\DGGEDUDQG check the bar spacing: )RUZLWKLQWKHLQZLGWK )RUZLWKLQWKH 7KHUHIRUHSURYLGHDGGLWLRQDOEDUVZLWKLQWKHLQZLGWKWRVDWLVI\PD[LPXPVSDFLQJUHTXLUHPHQWV$WRWDO RIEDUVLVUHTXLUHGDWWKHHGJHFROXPQVZLWKLQWKHFROXPQVWULSZLWKRIWKHEDUVFRQFHQWUDWHG ZLWKLQDZLGWKRILQ Check

within

ACI 8.6.1.2

Table 5.11, Case 3

Factored shear force at the critical section:

Therefore, ACI 8.6.1.2

where ACI Eq. (22.5.5.1.3)

and

13-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Thus,

ACI 8.6.1.2

Provided

within

Table 13.14

Therefore,

ACI 8.6.1.2

Provided

within

$FFRUGLQJWR$&,DWOHDVWRQHKDOIRIWKHUHLQIRUFHPHQWLQWKHFROXPQVWULSDWWKHVXSSRUWPXVWEH SODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK3URYLGLQJRIWKHEDUVZLWKLQWKHHIIHFWLYHVODEZLGWK VDWLV¿HVWKLVUHTXLUHPHQW 5HLQIRUFHPHQWGHWDLOVIRUWKHWRSEDUVDWWKHHGJHFROXPQVDUHJLYHQLQ)LJXUH        

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Figure 13.27 Reinforcement details for the top reinforcing bars at the edge column in the flat plate system in Example 13.9.

13-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Interior columns Determine reinforcement based on the largest transfer moment at the interior columns, which occurs for the load combination In lieu of a more exact analysis, the factored slab moment, transferred to an interior column due to the gravity load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal: Eq. (5.20)

$WWKH¿UVWLQWHULRUFROXPQ

due to seismic load effects.

Therefore, total ACI Eq. (8.4.2.2.2)

where Table 5.11, Case 1

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH

The moment

must be transferred over the effective slab width ACI Table 8.4.2.2.3

Determine the required

7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKDXQLIRUPEDUVSDFLQJLQWKHFROXPQVWULSRIWKH EDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHment is required to satisfy moment transfer requirements at these locations. Check

within

ACI 8.6.1.2

Table 5.11, Case 1

Factored shear force at the critical section:

13-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

where ACI Eq. (22.5.5.1.3)

and

Thus,

ACI 8.6.1.2

$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG

within

$WWKH¿UVWLQWHULRUFROXPQV Therefore, maximum shear stress is equal to the following:

ACI 8.6.1.2

Thus,

It is evident that the

13-54

load combination governs for the determination of minimum reinforcement.

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

To satisfy minimum reinforcement requirements within EDUVPXVWEHSODFHGZLWKLQWKHLQHIDWWKH¿UVWLQWHULRUFROXPQVZLWKWKHUHfective slab width width of the column strip. For symmetry, add PDLQLQJEDUVWREHSODFHGZLWKLQWKH EDUWRWKHLQZLGWK7KXVDWOHDVWEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH¿UVWLQWHULRUFROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK$FFRUGLQJWR$&,, DWOHDVWRQHKDOIRIWKHUHLQIRUFHPHQWLQWKHFROXPQVWULSPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK 3URYLGLQJRIWKHEDUVZLWKLQWKHHIIHFWLYHVODEZLGWKGRHVQRWVDWLVI\WKLVUHTXLUHPHQW7KHUHIRUHSURYLGHZLWKLQWKHLQHIIHFWLYHZLGWKDQGEDUVZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS7KH UHLQIRUFHPHQWGHWDLOIRUWKLVFDVHLVVKRZQLQ)LJXUH

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Figure 13.28 Reinforcement details for the top reinforcing bars at the first interior columns in the flat plate system in Example 13.9.

Step 4 – Determine the reinforcement details

7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQ$&,)LJXUHFDQQRWEHXVHGLQWKHFROXPQVWULSEHFDXVHRIWKH HIIHFWVGXHWRHDUWKTXDNHORDGV$FFRUGLQJWR$&,DWOHDVWSHUFHQWRIWKHWRSEDUVLQWKHFROXPQVWULSV PXVWEHPDGHFRQWLQXRXVRYHUWKHVSDQV,QWKLVFDVHWKHPD[LPXPDPRXQWRIWRSUHLQIRUFHPHQWRFFXUVDWWKH¿UVW LQWHULRUVXSSRUWVEDUV VREDUVDUHPDGHFRQWLQXRXV 7KHUHPDLQLQJEDUVLQWKHFROXPQVWULSDQGWKHEDUVLQWKHPLGGOHVWULSFDQEHWHUPLQDWHGDWWKHORFDWLRQVLGHQWL¿HG LQ$&,)LJXUHWKLV¿JXUHDOVRLQFOXGHVWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&, A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap splice length is determined as follows:

ACI Eq. (25.4.2.4a)

13-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

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ͻǤͷԣ

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–ƒ†ƒ”†Š‘‘ሺ–›’Ǥሻ

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ʹᇱ Ǧʹԣ

ʹͳԢǦ͸ԣ

ʹͳԢǦ͸ԣ ‘Ž—–”‹’

ʹԢǦͲԣ ͶԢǦͻԣ

ͶԢǦͻԣ ʹԢǦͲԣ ͶԢǦͻԣ

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ͻǦ͓ͷ

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Figure 13.29 Reinforcement details for the flat plate system in Example 13.9.

13-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Table 25.5.2.1

3URYLGHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSLQDFFRUGDQFHZLWKWKHSURYLVLRQVRI$&, WKURXJKDUHJLYHQLQ)LJXUHVHH)LJXUH  13.5.10 Example 13.10 – Check of Two-way Shear Strength Requirements: Two-way Slab in Building #1 (Framing Option A), Two-way Slab is Part of the SFRS (Intermediate Moment Frame), SDC C

Check the two-way shear strength requirements at the edge columns in the interior design strip along column line LQ%XLOGLQJ)UDPLQJ2SWLRQ$DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHWZRZD\VODELVSDUWRIWKH6)56VHH )LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&&ZKHUH Step 1 – Check two-way shear strength requirements

The critical section is located a distance

ACI 8.4.4

from the face of the column.

Factored shear force at the critical section:

For the gravity load combination where the Direct Design Method (DDM) is used to determine design bending VHH6HFWRIWKLVSXEOLFDWLRQ  moments, Dead load:

Live load:

13-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 5.11, Case 3

Determining whether

FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH ACI Eq. (8.4.4.2.2)

Table 5.11, Case 3

Total factored shear stress:

ACI Table 22.6.5.2

where The factored two-way shear stress caused by factored gravity loads without moment transfer must be less than $&,  or equal to

,QOLHXRILQFUHDVLQJWKHFROXPQVL]HDWWKLVSRLQWGHWHUPLQHLIWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG 7KLVFKHFNLVJLYHQLQ6WHSEHORZ

Table 13.14

Total factored shear force:

13-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For other than gravity load combinations,

is equal to that from the actual load combination: Table 13.15

Total factored shear stress:

The factored two-way shear stress caused by factored gravity loads without moment transfer must be less than $&,  or equal to

Step 2 – Check two-way shear strength requirements of ACI 18.14.5

Because

GHWHUPLQHLIWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG

ACI 18.4.5.8

)URPDQDO\VLVODWHUDOGLVSODFHPHQWDWWKHVHFRQGÀRRUOHYHOGXHWRWKHFRGHSUHVFULEHGVHLVPLFIRUFHVLV Design displacement

ASCE/SEI Eq. (12.8-15)

where LVWKHGHÀHFWLRQDPSOL¿FDWLRQIDFWRUZKLFKLVHTXDOWRIRULQWHUPHGLDWHPRPHQWIUDPHVVHH7DEOH RIWKLVSXEOLFDWLRQ 

7KHUHIRUHSURYLGHVKHDUUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&, Required shear reinforcement must provide Try ¾-in. diameter headed shear studs

with

In the direction parallel to the column face, the maximum spacing between adjacent lines of headed shear studs $&,7DEOH  )RUDLQZLGHFROXPQXVHOLQHVRIKHDGHGVKHDUVWXGVRQHDFKIDFHRIWKHFROXPQ Maximum spacing

In the direction perpendicular to the column face, maximum spacing

$&,7DEOH 

Therefore,

13-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QDFFRUGDQFHZLWK$&,WKHKHDGHGVKHDUVWXGVPXVWH[WHQGDWOHDVW FROXPQ3URYLGHKHDGHGVKHDUVWXGVLQHDFKOLQHRQHDFKIDFHRIWKHFROXPQ

from the face of the

+HDGHGVKHDUVWXGUHLQIRUFHPHQWGHWDLOVIRUWKHHGJHFROXPQVDUHJLYHQLQ)LJXUH Ͷᇱᇱ 

͸̷͸ᇱᇱ ൌ ͵͸ԣ

ʹͺǤͳ͵ᇱᇱ 

ʹͶᇱᇱ 

ʹͶᇱᇱ 

”‹–‹…ƒŽ •‡…–‹‘

͵ʹǤʹͷԣ

͵ȀͶǦ‹Ǥ†‹ƒǤŠ‡ƒ†‡† •Š‡ƒ”•–—†ሺ–›’Ǥሻ

Figure 13.30 Headed shear stud reinforcement at the edge columns in Example 13.10.

13.5.11 Example 13.11 – Design of Foundation Seismic Tie: Column in Building #1 (Framing Option B), Column is Part of the SFRS (Intermediate Moment Frame), SDC C

'HVLJQDIRXQGDWLRQVHLVPLFWLHIRUWKHFDLVVRQVXSSRUWLQJFROXPQ'LQ%XLOGLQJ)UDPLQJ2SWLRQ%VHH)LJXUH DQG*UDGHUHLQIRUFHPHQW 1.1). Assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&&ZKHUH Step 1 – Determine the seismic tie force

ACI 18.13.4.3

The seismic tie must have a design strength in tension and compression of at least times the factored dead load plus factored live load in accordance with the load combination on column D1 that includes earthquake effects (assuming the axial forces on the adjacent columns are less than or equal to that on column D1). )URP7DEOHLQ([DPSOHWKHODUJHVWD[LDOIRUFHFRUUHVSRQGVWRWKHORDGFRPELQDWLRQLQ$&,(TH  for sidesway to the right:

Therefore, the required axial tension and compression force in the seismic tie Step 2 – Determine the required reinforcement in the seismic tie

ACI 18.13.4.3

$VVXPHWKHVHLVPLFWLHLVDLQE\LQUHLQIRUFHGFRQFUHWHEHDPDQGWKHWLHEHDPLVQRWVXEMHFWHGWRDQ\JUDYLW\ or seismic load effects transmitted by the column or any other structural members.

13-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Also assume the tie beam has a longitudinal reinforcement ratio Thus, 7U\ORQJLWXGLQDOEDUV Required longitudinal reinforcement for the axial tension force in the seismic tie $VVXPLQJWKHWLHEHDPKDVWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIWLHVFRQIRUPLQJWR$&,WKHGHVLJQD[LDO RIWKHVHFWLRQLVGHWHUPLQHGXVLQJ$&,7DEOHDQG$&,(T  strength,

8VHORQJLWXGLQDOEDUVZLWKWLHVVSDFHGLQRQFHQWHURYHUWKHOHQJWKRIWKHVHLVPLFWLHEHDP 5HLQIRUFHPHQWGHWDLOVIRUWKHVHLVPLFWLHEHDPLQWKHHDVWZHVWGLUHFWLRQDUHVLPLODUWRWKRVHLQ)LJXUH7KH longitudinal bars are developed for tension and compression in the caisson cap.

13-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

13-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Chapter 14 EARTHQUAKE-RESISTANT STRUCTURES — SDC D, E and F 14.1 Overview This chapter covers the design and detailing requirements in ACI Chapter 18 for the following: • • • • •

Frame members (ACI 18.6 through 18.9) 6WUXFWXUDOZDOOVDQGFRXSOLQJEHDPV$&, 'LDSKUDJPVDQGWUXVVHV$&, Foundations (ACI 18.13) )UDPHPHPEHUVQRWGHVLJQDWHGDVSDUWRIWKHVHLVPLFIRUFHUHVLVWLQJV\VWHP6)56 >$&,@

*UDYLW\DQGODWHUDOORDGHIIHFWVRQVWUXFWXUDOPHPEHUVDUHREWDLQHGIURPDQDQDO\VLVPHWKRGLQ$&,FRQVLGHULQJ WKHDQDO\VLVDQGSURSRUWLRQLQJUHTXLUHPHQWVLQ$&,'HVLJQVWUHQJWKORDGFRPELQDWLRQVIRUVWUXFWXUHVDVVLJQHGWR6'&'(RU)DUHJLYHQLQ7DEOHIRUJUDYLW\ZLQGDQGVHLVPLFORDGHIIHFWVVHH6HFWLRQRIWKLV publication). Table 14.1 Strength Design Load Combinations for Structures Assigned to SDC D, E, or F ACI Equation Number

ASCE/SEI 7 Load Combination

D

1

E



F

3

G



H

6

I



J



Load Combination

The seismic load combinations for the design of structural members where seismic load effects including overVWUHQJWKDUHUHTXLUHGDUHJLYHQLQ7DEOH Table 14.2 Seismic Load Combinations Where Seismic Loads Including Overstrength are Required ACI Equation Number

ASCE/SEI 7 Load Combination

H

6

J



Load Combination

6WUHQJWKUHGXFWLRQIDFWRUV$&, DQGUHTXLUHPHQWVIRUFRQFUHWH$&, DQGUHLQIRUFHPHQW$&,  LQVSHFLDOPRPHQWIUDPHVDQGVSHFLDOVWUXFWXUDOZDOOVDUHJLYHQLQ&KDSWHURIWKLVSXEOLFDWLRQ5HTXLUHPHQWVIRU PHFKDQLFDODQGZHOGHGVSOLFHVLQVSHFLDOPRPHQWIUDPHVDQGVSHFLDOVWUXFWXUDOZDOOVDUHJLYHQLQ$&,DQG UHVSHFWLYHO\

14-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.2 Beams of Special Moment Frames 14.2.1 Overview

Design and detailing requirements for beams designated part of the SFRS in special moment frames are given in ACI 18.6. The provisions in this section have been developed based on the assumption that beams in a special moPHQWIUDPHDUHGHVLJQHGSULPDULO\IRUÀH[XUHDQGVKHDU$&,  ,WLVDOVRDVVXPHGWKDWDVSHFLDOPRPHQWIUDPHFRQVLVWVRIKRUL]RQWDOEHDPVIUDPLQJLQWRYHUWLFDOFROXPQVZKHUH WKHFROXPQVVDWLVI\WKHUHTXLUHPHQWVRI$&,$&, ,WLVDFFHSWDEOHIRUWKHEHDPVDQGFROXPQVWR EHLQFOLQHGIURPWKHKRUL]RQWDODQGYHUWLFDOUHVSHFWLYHO\SURYLGHGWKHV\VWHPEHKDYHVDVDIUDPHWKDWLVODWHUDO resistance is provided primarily by moment transfer between beams and columns rather than by other means, such as strut or brace action). It is also acceptable for beams of special moment frames to (1) be designed as chords or FROOHFWRUVRIDGLDSKUDJPWKDWLVEHGHVLJQHGWRUHVLVWFRPELQHGPRPHQWDQGD[LDOIRUFHV DQG IUDPHLQWRD boundary element of a wall provided the boundary is reinforced as a column in a special moment frame in accorGDQFHZLWK$&,%HDPVWKDWFDQWLOHYHUEH\RQGFROXPQVRIVSHFLDOPRPHQWIUDPHVDUHQRWFRQVLGHUHGWREHSDUW of the SFRS. 14.2.2 Dimensional Limits

7KHUHTXLUHGVL]HRIDEHDPLQDVSHFLDOPRPHQWIUDPHIRUVWUHQJWKDQGVHUYLFHDELOLW\DUHGHWHUPLQHGXVLQJWKH provisions in ACI Chapter 9 for the combined effects due to gravity and lateral forces (see Chapter 6 of this publicaWLRQ DQGWKHGLPHQVLRQDOOLPLWVLQ$&,7KHUHTXLUHPHQWVLQ$&,DUHJLYHQLQ)LJXUH ܿଵ  Column ܿଵ ൈ ܿଶ ሺtyp.ሻ

 ݄

A

A

݀

κ௡ ൒ 4݀

ܾ௪  ൒ lesser of 0.3݄ and 10ᇳ ൝  ൑ lesser of 3ܿଶ and ܿଶ ൅ 1.5ܿଵ Section A-A Figure 14.1 Dimensional limits for beams in special moment frames.

14-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

For beams wider than the columns that they frame into, the maximum beam width that can effectively transfer and VHH$&,F DQG$&,)LJXUH5  forces into a beam-column joint is the lesser of Transverse reinforcement is required in the portions of the beam on either side of the column and through the FROXPQWRFRQ¿QHWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDPSDVVLQJRXWVLGHRIWKHFROXPQFRUH 14.2.3 Longitudinal Reinforcement Determining the Required Flexural Reinforcement

2QFHWKHFURVVVHFWLRQDOGLPHQVLRQVRIWKHEHDPKDYHEHHQGHWHUPLQHGWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW can be calculated at the critical sections for the combined effects due to gravity and lateral forces using the proviVLRQVLQ$&,&KDSWHUVHH&KDSWHURIWKLVSXEOLFDWLRQ ,QDGGLWLRQWRWKHÀH[XUDOUHTXLUHPHQWVLQ$&,&KDSWHU WKHUHTXLUHPHQWVLQ$&,DQGPXVWDOVREHVDWLV¿HGVHH)LJXUH  ͳʹᇳ  ‫ۓ‬  ۖ ۖʹͲ݀௕ Τߣˆ‘”–Š‡Žƒ”‰‡•– ”ƒ†‡͸Ͳ„ƒ”   ܿଵ ൒ ‰”‡ƒ–‡”‘ˆ ‫ʹ۔‬͸݀ ˆ‘”–Š‡Žƒ”‰‡•– ”ƒ†‡ͺͲ„ƒ” ௕ ۖ ۖ  ‫݄ ە‬Τʹ



ͲǤͲʹͷܾ௪ ݀ˆ‘” ”ƒ†‡͸Ͳ„ƒ”• ͵ඥ݂௖ᇱ ܾ௪ ݀ ൗ݂௬ ‫ۍ‬ ି ା 

”‡ƒ–‡”‘ˆ ቐ ൑ ‫ܣ‬ ‘”‫ܣ‬ ൑ ൝   ௦ ௦ ‫ێ‬ ‫ێ‬ ͲǤͲʹͲܾ௪ ݀ˆ‘” ”ƒ†‡ͺͲ„ƒ”•  ʹͲͲܾ௪ ݀ Τ݂௬ ‫ێ‬ ‫ێ‬ ‫‘––‘„†ƒ’‘–•—‘—‹–‘…•”ƒ„ʹ—‹‹ۏ‬

ି ‫ܯ‬௡ǡ௥ 

݀

݄

ି ‫ܯ‬௡ǡκ 

ା ‫ܯ‬௡ǡκ ൒

ି ‫ܯ‬௡ǡκ  ʹ

ା ൒ ‫ܯ‬௡ǡ௥

ି ‫ܯ‬௡ǡ௥  ʹ

‫ܯ‬௡ି ‘”‫ܯ‬௡ା ൒

ƒš‹—‫ܯ‬௡ ƒ–‡‹–Š‡”Œ‘‹–  Ͷ

”ƒ•˜‡”•‡”‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 14.2 Flexural requirements for beams in special moment frames.

7KHPLQLPXPUHLQIRUFHPHQWPXVWEHLQDFFRUGDQFHZLWK$&,DQGWKHXSSHUOLPLWIRUWKHDPRXQWRIÀH[XUDO IRU*UDGHUHLQIRUFHPHQWDQG IRU*UDGHUHLQIRUFHPHQW$&,  reinforcement is The purpose of the maximum reinforcement ratio is to help reduce congestion and to limit shear stresses. A practical UHLQIRUFHPHQWUDWLRIRUEHDPVLQVSHFLDOPRPHQWIUDPHVLV 5HTXLUHPHQWVKDYLQJDSRVVLEOHLPSDFWRQWKHDPRXQWRISRVLWLYHÀH[XUDOUHLQIRUFHPHQWLQEHDPVWKDWDUHSDUWRI DVSHFLDOPRPHQWIUDPHDUHJLYHQLQ$&,$WWKHIDFHVRIWKHMRLQWVWKHSRVLWLYHPRPHQWVWUHQJWK at that joint. The area of PXVWEHDWOHDVWHTXDOWRSHUFHQWRIWKHFRUUHVSRQGLQJQHJDWLYHPRPHQWVWUHQJWK SRVLWLYHÀH[XUDOUHLQIRUFHPHQWPD\QHHGWREHLQFUHDVHGWRVDWLVI\WKLVUHTXLUHPHQW$OVRDWDQ\VHFWLRQDORQJWKH OHQJWKRIDEHDPWKHQHJDWLYHDQGSRVLWLYHPRPHQWVWUHQJWKPXVWEHHTXDOWRDWOHDVWSHUFHQWRIWKHPD[LPXP 14-3

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

moment strength provided at the face of either joint. This requirement ensures strength and ductility under large lateral displacements. 2QFHWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWKDVEHHQGHWHUPLQHGWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVFDQ be selected using Tables 6.8 and 6.9 of this publication. At least two continuous bars must be provided at both the top and bottom faces of a beam (ACI 18.6.3.1). 7KHUHTXLUHPHQWVRI$&,PXVWDOVREHVDWLV¿HGZKHQVHOHFWLQJWKHVL]HRIWKHORQJLWXGLQDOEDUVLQDEHDP H[WHQGLQJWKURXJKDQLQWHULRUMRLQWLQDVSHFLDOPRPHQWIUDPHVHH)LJXUH 7KHGLPHQVLRQRIWKHMRLQWFROtimes the diameter of the largest umn) parallel to the longitudinal reinforcement in a beam must be at least ORQJLWXGLQDOEHDPEDUIRU*UDGHUHLQIRUFHPHQWZKHUH LVHTXDOWRIRUOLJKWZHLJKWFRQFUHWHDQGIRU DOORWKHUFDVHV)RU*UDGHUHLQIRUFHPHQWWKHPLQLPXPMRLQWGLPHQVLRQLVWLPHVWKHGLDPHWHURIWKHODUJHVW ORQJLWXGLQDOEDUQRUPDOZHLJKWFRQFUHWHPXVWEHXVHGLQMRLQWVZLWK*UDGHUHLQIRUFHPHQWVHH$&,  IRU*UDGHUHLQ7KLVPHDQVWKHVL]HRIWKHORQJLWXGLQDOEDUVLQDEHDPPXVWEHVHOHFWHGVXFKWKDW IRU*UDGHUHLQIRUFHPHQW7KHSXUSRVHRIWKHVHUHTXLUHPHQWVLVWROLPLWWKHDPRXQW forcement and of potential slip of the longitudinal bars in a beam-column joint during a design-level earthquake event, thereby SUHVFULEHGLQ$&,DQGLQ ensuring adequate development of the bars. The minimum dimensions of SUHVFULEHGLQ$&,D IRUFROXPQVRIVSHFLDOPRPHQWIUDPHVDUHDOVRJLYHQLQ)LJXUH Detailing the Flexural Reinforcement

Development of Flexural Reinforcement Beam longitudinal reinforcement terminated in a joint must extend to the far face of the joint core and must be deYHORSHGLQWHQVLRQLQDFFRUGDQFHZLWKWKHWHQVLRQGHYHORSPHQWOHQJWKUHTXLUHPHQWVRI$&,DQGLQFRPSUHVVLRQLQDFFRUGDQFHZLWK$&,$&,  Hooked Bars in Tension. An example of an edge or corner column where the longitudinal reinforcement from the EHDPLVWHUPLQDWHGLQDVWDQGDUGGHJUHHKRRNLVLOOXVWUDWHGLQ)LJXUH,QDFFRUGDQFHZLWK$&,WKH KRRNPXVWEHORFDWHGZLWKLQWKHFRQ¿QHGFRUHRIWKHFROXPQDQGLWPXVWEHEHQWLQWRWKHMRLQW7KHFULWLFDOVHFWLRQ for development of the hooked bars is taken at the outside edge of the joint transverse reinforcement and not at the IDFHRIWKHFROXPQEHFDXVHWKHFRQFUHWHRXWVLGHRIWKHFRQ¿QHGFRUHPD\VSDOOGXULQJDGHVLJQOHYHOHDUWKTXDNH event. is 7KHGHYHORSPHQWOHQJWKRIDGHIRUPHGWKURXJKUHLQIRUFLQJEDUZLWKDVWDQGDUGKRRNLQWHQVLRQ GHWHUPLQHGE\$&,(TXDWLRQ 7KHPRGL¿FDWLRQIDFWRU LVHTXDOWRIRUOLJKWZHLJKWFRQFUHWHDQG EDVHGRQWKHUHTXLUHPHQWVLQ$&, otherwise. The following equations can be used to determine For normalweight concrete:

(14.1)

)RUOLJKWZHLJKWFRQFUHWH*UDGHUHLQIRUFHPHQWRQO\ 

(14.2)

14-4

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ κௗ௛

ܿ௖

݀௕ሺ௛௢௢௣ሻ

‘‘’•ሺ–›’Ǥሻ

݀௕ ሺ–›’Ǥሻ –ƒ†ƒ”†ͻͲǦ†‡‰ Š‘‘ሺ–›’Ǥሻ

݀௕ሺ௖௢௟Ǥሻ

κௗ௛

ᇱ

‫݂ۓ‬௬ ݀௕ Τ͸ͷඥ݂௖ ۖ ൌ ‰”‡ƒ–‡”‘ˆ ͺ݀   ‫ ۔‬௕ ۖ ‫ە‬͸ᇳ 

‘”ƒŽ™‡‹‰Š–…‘…”‡–‡

κௗ௛

 ‹‰Š–™‡‹‰Š–…‘…”‡–‡ȗ 

‫݂ۓ‬௬ ݀௕ Τሺ͸ͷ ൈ ͲǤ͹ͷሻඥ݂௖ ۖ ൌ ‰”‡ƒ–‡”‘ˆ ͳͲ݀  ௕  ‫۔‬ ۖ ‫ە‬͹Ǥͷᇳ 

ᇱ

κௗ௛

‘”‡’‘š›Ǧ…‘ƒ–‡†‘”œ‹…ƒ†‡’‘š›†—ƒŽǦ…‘ƒ–‡†„ƒ”•ǡ—Ž–‹’Ž›κௗ௛ „›ͳǤʹǤ * Applicable to Grade 60 reinforcement only Figure 14.3 Development of flexural reinforcement with standard hooks in beams of special moment frames.

These development lengths are measured from the outside edge of the joint transverse reinforcement to the outside IDFHRIWKHKRRNVHH)LJXUH ,QRUGHUIRUWKHKRRNHGORQJLWXGLQDOEDUVWREHIXOO\GHYHORSHGLQWKHMRLQWIRU tension, the available development length must be greater than or equal to (14.3)

In this equation, is the clear cover to the hoop reinforcement in the joint, is the diameter of the hoop reinis the diameter of the longitudinal reinforcement in the column. forcement in the joint, and Straight Bars in Tension.,QOLHXRIKRRNHGEDUVWKURXJKVWUDLJKWEDUVPD\EHXVHGSURYLGHGWKHEDUVDUH SURSHUO\GHYHORSHGLQDFFRUGDQFHZLWK$&,DQG

14-5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHTXLUHGGHYHORSPHQWOHQJWKVRIWKURXJKVWUDLJKWEDUVDUHPXOWLSOHVRIWKHKRRNHGEDUGHYHORSPHQW OHQJWKVLQ$&, where the depth of the concrete cast in one lift beneath the bar is less • Minimum development length WKDQRUHTXDOWRLQW\SLFDOO\ERWWRPEDUV  where the depth of the concrete cast in one lift beneath the bar is • Minimum development length JUHDWHUWKDQLQW\SLFDOO\WRSEDUV  1RWHWKDWDQGEDUVDUHQRWLQFOXGHGEHFDXVHRIWKHODFNRILQIRUPDWLRQSHUWDLQLQJWRDQFKRUDJHRIWKHVHEDU VL]HVZKHQVXEMHFWHGWRORDGUHYHUVDOV :KHUHWKHUHTXLUHGVWUDLJKWGHYHORSPHQWOHQJWKRIWKHORQJLWXGLQDOEHDPEDUVH[WHQGVEH\RQGWKHFRQ¿QHGFRUHRI the column, the required development length must be increased because the limiting bond stress outside the con¿QHGUHJLRQLVOHVVWKDQWKDWLQVLGHWKHFRQ¿QHGUHJLRQ7KHOHQJWKRIWKHEDUVRXWVLGHRIWKHFRQ¿QHGFRUHPXVWEH LQFUHDVHGE\$&, 7KHHTXDWLRQVLQ$&,5FDQEHXVHGWRGHWHUPLQHWKHUHTXLUHGGHYHORSPHQW length in such cases. Headed Bars in Tension. An example of an edge or corner column with headed deformed bars is illustrated in FigXUH6XFKEDUVDUHSHUPLWWHGWREHXVHGZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG (a) (b) (c) (d) (e) (f)

%DUVPXVWFRQIRUPWR$&, %DUVL]HPXVWEHRUVPDOOHU , must be at least where is the area of the bar Net bearing area of the head, Concrete must be normalweight where is the nominal diameter of the bar Clear cover to the bar must be greater than or equal to Center-to-center spacing of the bars must be greater than or equal to

$FFRUGLQJWR$&,WKHGHYHORSPHQWOHQJWKRIDKHDGHGGHIRUPHGUHLQIRUFLQJEDULQWHQVLRQ is substituted for FDOFXODWHGLQDFFRUGDQFHZLWK$&,ZKHUH

, is to be

(14.4)

>1RWH7KHGHYHORSPHQWOHQJWKVRIKHDGHGEDUVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,DUHJUHDWHUWKDQWKH GHYHORSPHQWOHQJWKVRIKRRNHGEDUVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,ZKLFKLVQRWFRUUHFW$&,&RPPLWWHHLVDZDUHRIWKLVDQGLVZRUNLQJWRPRGLI\WKHSURYLVLRQVLQ$&,DFFRUGLQJO\@ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  DFFRXQWVIRUFRQ¿QLQJHIIHFWVSURYLGHGE\WUDQVYHUVHUHLQIRUFHPHQWRULHQWHGSDUDOOHOWRWKHGHYHORSThe term ment length of headed bars. At beam-column joints, the total cross-sectional area of transverse reinforcement, of the centerline of the headed bar toward the middle of the joint where is the nomimust be located within is greater than or equal to or where the QDOGLDPHWHURIWKHKHDGHGEDUVHH$&,)LJXUH5 :KHUH The term is the total crosscenter-to center spacing of the headed bars is greater than or equal to sectional area of the headed bars being developed.

14-6

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.3 Modification Factors for Development of Headed Bars in Tension Modification Factor

Epoxy, Parallel tie reinforcement,

Location,

Condition

Value of Factor

(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW



8QFRDWHGRU]LQFFRDWHGJDOYDQL]HG UHLQIRUFHPHQW



)RUDQGVPDOOHUEDUVZLWK



or

Other

1.6

For headed bars (1) terminating inside a column core with side RU ZLWKVLGHFRYHUWREDU cover to bar

 

Other Concrete strength,



+HDGHGEDUVXVHGDVQHJDWLYHUHLQIRUFHPHQWLQEHDPVWHUPLQDWHGLQDMRLQWZKHUHWKHUHLVQRFROXPQDERYHVXFKDV LQWKHWRSVWRU\RIDEXLOGLQJ UHTXLUHFRQ¿QHPHQWDORQJWKHWRSIDFHRIWKHMRLQW7ZRFRQ¿QHPHQWRSWLRQVDUHSURYLGHLQ$&,D WKHFROXPQEHORZWKHMRLQWPXVWEHH[WHQGHGDERYHWKHWRSRIWKHMRLQWDGLVWDQFHHTXDOWR DWOHDVWWKHGHSWKRIWKHMRLQWLQWKHGLUHFWLRQRIDQDO\VLV>VHH)LJXUHD RIWKLVSXEOLFDWLRQ@RUE YHUWLFDOMRLQW reinforcement must be provided that hooks around the headed bars and extends downward into the joint in addition WRWKHFROXPQORQJLWXGLQDOUHLQIRUFHPHQW>VHH)LJXUHE RIWKLVSXEOLFDWLRQ@'HVLJQUHFRPPHQGDWLRQVIRUWKH YHUWLFDOMRLQWUHLQIRUFHPHQWDUHJLYHQLQ5HIHUHQFH>VHHDOVR$&,)LJXUH5E @ ܿଵ  ܿ௖ 

κௗ௧  ݀௕ሺ௛௢௢௣ሻ 

‘‘’•ሺ–›’Ǥሻ

‫ݐ‬௛௘௔ௗ 

݀௕ሺ௖௢௟Ǥሻ 

‡ƒ†‡††‡ˆ‘”‡†„ƒ”ሺ–›’Ǥሻ

ͳǤʹͷ݂௬ ߰௘ ߰௣ ߰௢ ߰௖ ଵǤହ ‫ۓ‬ቆ ቇ ݀௕ ͹ͷඥ݂௖ᇱ ۖ ۖ κௗ௧ ൌ ‰”‡ƒ–‡”‘ˆ

‫۔‬ͺ݀௕ ۖ ۖ ‫ ە‬͸ᇳ 



Figure 14.4 Development of flexural reinforcement with headed bars in beams of special moment frames.

14-7

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In order for the headed longitudinal bars to be fully developed in the joint for tension, the available development length must be greater than or equal to (14.5)

where

is the thickness of the head.

Deformed Bars in Compression. The development length, DFFRUGDQFHZLWK$&,

for deformed bars in compression is determined in

(14.6)

7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOHVHH$&,7DEOH  Table 14.4 Modification Factors for Development of Deformed Bars in Compression Modification Factor

Lightweight,

&RQ¿QLQJ reinforcement,

Condition

Value of Factor

Lightweight concrete



Normalweight concrete



Reinforcement enclosed within the following: • A spiral • A circular continuously wound tie with SLWFKRILQ • EDUWLHVLQDFFRUGDQFHZLWK$&,VSDFHG on center • +RRSVLQDFFRUGDQFHZLWK$&,VSDFHG on center Other

and 



This development length corresponds to the straight portion of a hooked or headed bar measured from the critical section (in this case, the outside edge of the joint transverse reinforcement) to the onset of the bend for hooked bars and from the critical section to the head for headed bars. For hooked longitudinal bars to be fully developed in the joint for compression, the available development length must be greater than or equal to (14.7)

In this equation, is the clear cover to the hoop reinforcement in the joint, is the diameter of the hoop is the diameter of the longitudinal reinforcement in the column, is the diameter reinforcement in the joint, of the longitudinal reinforcement in the beam, and is the bend radius of the beam longitudinal bar determined in DFFRUGDQFHZLWK$&,7DEOHIRUVWDQGDUGGHJUHHKRRNV

14-8

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUKHDGHGEDUVWREHIXOO\GHYHORSHGLQWKHMRLQWIRUFRPSUHVVLRQWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG (14.8)

Cutoff Points of Flexural Reinforcement. Flexural bars may be terminated along the span in accordance with the in combiUHTXLUHPHQWVLQ$&,7KHIDFWRUHGORDGXVHGWRGHWHUPLQHWKHFXWRIISRLQWVLV and at the ends of the beam because this QDWLRQZLWKWKHQHJDWLYHDQGSRVLWLYHSUREDEOHÀH[XUDOVWUHQJWKV FRPELQDWLRQSURGXFHVWKHORQJHVWEDUOHQJWKVVHH)LJXUH 



൒ κௗ  ݀ ൒ ൝   ͳʹ݀௕

‫ݔ‬

‫ିܣ‬ ௦ǡ௥ 

‫ିܣ‬ ௦ǡκ 

‫ܣ‬ା ௦

κ௡ 

ା ‫ܯ‬௣௥ ൌ ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ ቆ݀ െ

ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪



‫ݓ‬௨ ൌ ሺͲǤͻ െ ͲǤʹܵ஽ௌ ሻ‫ݓ‬஽ 

ି ൌ ͳǤʹͷ‫ିܣ‬ ‫ܯ‬௣௥ ௦ǡ௥ ݂௬ ቆ݀ െ

”ƒ˜‹–›൅ ‡‹•‹… ܸ௨ ൌ 

ͳǤʹͷ‫ିܣ‬ ௦ǡ௥ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

 ା ି ൅ ‫ܯ‬௣௥ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ െ  ʹ κ௡

ܸ௨ ൌ 

ା ି ൅ ‫ܯ‬௣௥ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ ൅  ʹ κ௡

ା  ‫ܯ‬௣௥

‫ݔ‬ ‘‡– ‹ƒ‰”ƒ ߶‫ܯ‬௡ ൌ ߶‫ିܣ‬ ௦ǡκ ݂௬ ቆ݀ െ

‫ିܣ‬ ௦ǡκ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

ି  ‫ܯ‬௣௥

Figure 14.5 Cutoff point of negative flexural reinforcement for beams in special moment frames.

14-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHSUREDEOHÀH[XUDOVWUHQJWK ZKLFKLVDVVRFLDWHGZLWKSODVWLFKLQJLQJLQDEHDPLVGH¿QHGDVWKHVWUHQJWKRI DÀH[XUDOPHPEHUEDVHGRQWKHSURSHUWLHVRIWKHPHPEHUDWWKHMRLQWIDFHVDVVXPLQJWKHWHQVLOHVWUHVVLQWKHORQJLWXand the strength reduction factor, LVHTXDOWR$&,  dinal reinforcement is equal to (14.9)

where The reasons for using are the following: (1) the actual yield strength of the reinforcement may exceed the VSHFL¿HG\LHOGVWUHQJWKDQG VWUDLQKDUGHQLQJRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLVOLNHO\WRRFFXUDWDMRLQWXQGHUgoing large rotations, which would be expected during a design-level earthquake event. The following equation can be used to calculate the theoretical cutoff point QHJDWLYHUHLQIRUFLQJEDUVVHH)LJXUH 

from the face of the support for the

(14.10)

where LVWKHGHVLJQÀH[XUDOVWUHQJWKRIWKHEHDPEDVHGRQWKHDUHDRIQHJDWLYHUHLQIRUFHPHQW at the left at the right support is no support. The distance is the location where the total area of negative reinforcement DORQHLVVXI¿FLHQWWRUHVLVWWKHUHTXLUHGEHQGLQJPRPHQW longer needed, that is, it is the point where Once a theoretical cutoff point is determined, the bars must extend a distance equal to the larger of or be\RQGWKDWSRLQW$&,VHH)LJXUH $GGLWLRQDOO\WKHWRWDOOHQJWKRIWKHEDUVIURPWKHIDFHRIWKHVXSSRUW GHWHUPLQHGE\$&, must be at least equal to the development length, )OH[XUDOUHLQIRUFHPHQWLVQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRUPRUHRIWKHFRQGLWLRQVRI $&,LVVDWLV¿HG Lap Splices Requirements for lap splices in beams of special moment frames are given in ACI 18.6.3.3: 1. +  RRSRUVSLUDOUHLQIRUFHPHQWVSDFHGRQFHQWHUQRPRUHWKDQWKHOHVVHURI the lap splice length /DSVSOLFHVPXVWQRWEHORFDWHGZLWKLQWKHIROORZLQJUHJLRQVVHH)LJXUH 

RULQPXVWEHSURYLGHGRYHU

(a) Within the joints from the face of the joint where is the overall depth of the beam (b) Within a distance of IURPFULWLFDOVHFWLRQVZKHUHÀH[XUDO\LHOGLQJLVOLNHO\WRRFFXUDVDUHVXOW (c) Within a distance of of lateral displacements beyond the elastic range of behavior Like intermediate moment frames, the plastic hinge regions in beams of special moment frames are expected to ocat both ends of a beam. Yielding will typically take place at these locations for relatively cur over a distance of short beams where bending moments due to gravity loads are low compared to those from the earthquake load effects. Where gravity load bending moments are large compared to the bending moments from earthquake load effects, it is possible for a plastic hinge to form away from the faces of the columns. For a beam subjected to a factored uniformWKDWKDVDFRQVWDQWÀH[XUDOVWUHQJWKDORQJWKHHQWLUHVSDQOHQJWKSODVWLFKLQJHVPD\ ly distributed gravity load form away from the ends of beam at locations where

14-10

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 

൒ ʹ݄ ሺ–›’Ǥሻ

‘‘’‘”•’‹”ƒŽ ”‡‹ˆ‘”…‡‡–

݀

݄

݀௕ 

ƒ’•’Ž‹…‡…‘ˆ‹‡†ƒ†Ž‘…ƒ–‡†‘—–•‹†‡ ‘ˆ’‘–‡–‹ƒŽŠ‹‰‡”‡‰‹‘•

݀ ΤͶ ‫ݏ‬൑൝  Ͷᇳ 

Figure 14.6 Lap splice requirements for beams in special moment frames.

Mechanical and Welded Splices Mechanical and welded splices of the longitudinal reinforcement in beams of special moment frames must conform WRWKHUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\ 7\SHPHFKDQLFDOVSOLFHVPXVWFRQIRUPWR$&,DQGPXVWEHDEOHWRGHYHORSLQWHQVLRQRUFRPSUHVVLRQDW RIWKHEDU7KHVSHFL¿HGWHQVLOHVWUHQJWKRIWKHVSOLFHGEDUVPXVWEHGHYHORSHGE\7\SHPHFKDQLFDO least VSOLFHV([FHSWIRU7\SHPHFKDQLFDOVSOLFHVRQ*UDGHUHLQIRUFHPHQWPHFKDQLFDOVSOLFHVDUHQRWSHUPLWWHGWREH from the face of the column or from critical sections where yielding of the reinforcement is likely located within WRRFFXUDVDUHVXOWRIODWHUDOGLVSODFHPHQWVEH\RQGWKHOLQHDUUDQJHRIEHKDYLRU$7\SHPHFKDQLFDOVSOLFHRQ *UDGHUHLQIRUFHPHQWLVSHUPLWWHGDWDQ\ORFDWLRQLQDFDVWLQSODFHUHLQIRUFHGFRQFUHWHEHDP of :HOGHGVSOLFHVPXVWFRQIRUPWR$&,DQGPXVWEHDEOHWRGHYHORSLQWHQVLRQRUFRPSUHVVLRQDWOHDVW from the face of the column or from critical sections the bar. Such splices are not permitted to be located within where yielding of the reinforcement is likely to occur as a result of lateral displacements beyond the linear range of behavior. $FFRUGLQJWR$&,LWLVQRWSHUPLWWHGWRZHOGVWLUUXSVWLHVLQVHUWVRURWKHUVLPLODUHOHPHQWVWRWKHORQJLWXdinal reinforcement in a beam that is part of a special moment frame. Such welding can lead to local embrittlement of the reinforcing steel. 14.2.4 Transverse Reinforcement Determining the Required Transverse Reinforcement

,QRUGHUWRDVVXUHWKDWDEHDPLQDVSHFLDOPRPHQWIUDPH\LHOGVLQÀH[XUHEHIRUHLWIDLOVLQVKHDUWKHEHDPLVVXEjected to a required shear force based on the maximum shear that can develop in the beam. Instead of designing the required shear reinforcement based solely on the results from a structural analysis using factored load combinations, the design shear forces must be determined using the factored gravity and vertical earthquake loads plus the shear and at IRUFHVDVVRFLDWHGZLWKWKHGHYHORSPHQWRIWKHQHJDWLYHDQGSRVLWLYHSUREDEOHÀH[XUDOVWUHQJWKV WKHHQGVRIWKHEHDPZKLFKDUHGHWHUPLQHGE\(TXDWLRQ >$&,@7KHGHWHUPLQDWLRQRIWKHIDFWRUHG EDVHGRQWKLVSURYLVLRQLVLOOXVWUDWHGLQ)LJXUHIRUDEHDPVXEMHFWHGWRDIDFWRUHGXQLIRUP shear forces, $&,(TXDWLRQH LVXVHGWRGHWHUPLQH is combined with and where it load, is assumed the load factor on FDQEHWDNHQDVLQDFFRUGDQFHZLWK$&, Because earthquake effects can act in either direction, both sidesway to the right and sidesway to the left must be due to gravity and earthquake load effects are determined from considered. The total factored shear forces, ) and the statics. In cases where the top reinforcement is the same at both ends of a beam (that is, where is the same for both sidesway to the right bottom reinforcement is continuous along the entire span, maximum and to the left.

14-11

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ିܣ‬ ௦ǡκ 

‫ିܣ‬ ௦ǡ௥ 

‫ܣ‬ା ௦

κ௡ 

ା  ‫ܯ‬௣௥

‫ݓ‬௨ ൌ ሺͳǤʹ ൅ ͲǤʹܵ஽ௌ ሻ‫ݓ‬஽ ൅ ͲǤͷ‫ݓ‬௅ ൅ ͲǤʹ‫ݓ‬ௌ 

ି ‫ܯ‬௣௥ଶ 

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›”‹‰Š– ܸ௨ ൌ

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ି ା ൅ ‫ܯ‬௣௥ଶ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ െ ʹ κ௡

ି  ‫ܯ‬௣௥ଵ

ܸ௨ ൌ

ܸ௨ ൌ

‫ݓ‬௨ ൌ ሺͳǤʹ ൅ ͲǤʹܵ஽ௌ ሻ‫ݓ‬஽ ൅ ͲǤͷ‫ݓ‬௅ ൅ ͲǤʹ‫ݓ‬ௌ 

ି ା ൅ ‫ܯ‬௣௥ଵ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ ൅ ʹ κ௡

ܸ௨ ൌ ା ൌ ͳǤʹͷ‫ܣ‬ା ‫ܯ‬௣௥ ௦ ݂௬ ቆ݀ െ

ି ‫ܯ‬௣௥ଵ ൌ ͳǤʹͷ‫ିܣ‬ ௦ǡκ ݂௬ ቆ݀ െ

ͳǤʹͷ‫ିܣ‬ ௦ǡκ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

ି ା ൅ ‫ܯ‬௣௥ଶ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ ൅ ʹ κ௡

ା ‫ܯ‬௣௥ 

ି ା ൅ ‫ܯ‬௣௥ଵ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ െ ʹ κ௡

ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ ቇ ᇱ ͳǤ͹݂௖ ܾ௪ ି ൌ ͳǤʹͷ‫ିܣ‬ ‫ܯ‬௣௥ଶ ௦ǡ௥ ݂௬ ቆ݀ െ

ͳǤʹͷ‫ିܣ‬ ௦ǡ௥ ݂௬ ቇ ͳǤ͹݂௖ᇱ ܾ௪

Figure 14.7 Design shear forces for beams in special moment frames.

+RRSVDUHUHTXLUHGLQUHJLRQVZKHUHÀH[XUDO\LHOGLQJLVH[SHFWHG$&,  • Over a length equal to measured from the face of the supporting column toward midspan, at both ends of the beam. RQERWKVLGHVRIDVHFWLRQZKHUHÀH[XUH\LHOGLQJLVOLNHO\WRRFFXUDVDUHVXOWRIODWHUDO • Over lengths equal to displacements beyond the elastic range of behavior. +RRSVDUHGH¿QHGLQ$&,DVFORVHGWLHVRUFRQWLQXRXVO\ZRXQGWLHVPDGHXSRIRQHRUVHYHUDOUHLQIRUFHPHQW HOHPHQWVHDFKKDYLQJVHLVPLFKRRNVDWHDFKHQGFRQIRUPLQJWR$&,$&,VHH)LJXUH 7KHHQGV of the reinforcement elements must engage a longitudinal bar in the beam and the extensions must project into the LQWHULRURIWKHKRRS+RRSVIRUPHGE\VWLUUXSVZLWKVHLVPLFKRRNVDQGFURVVWLHV'HWDLOV%DQG&LQ)LJXUH DUH preferred over those formed by closed stirrups with seismic hooks (Detail A) because the longitudinal bars in the EHDPFDQEHSODFHGPRUHHDVLO\DQGHI¿FLHQWO\&URVVWLHVPXVWFRQIRUPWR$&,1RWHWKDWKRRSVPDGHXSRI LQWHUORFNLQJKHDGHGGHIRUPHGEDUVDUHQRWSHUPLWWHG$&, 

14-12

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Figure 14.8 Examples of hoops and overlapping hoops.

,QJHQHUDOVKHDUVWUHQJWKLQWKHUHJLRQVRIÀH[XUDO\LHOGLQJLVSURYLGHGE\ERWKFRQFUHWH and transverse reinLQWKHIRUPRIKRRSV7KHVL]HDQGVSDFLQJRIWKHKRRSVFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ forcement $&,  (14.11)

where IRUVKHDUVHH$&,7DEOHDQG$&, 7KHQRPLQDOVKHDUVWUHQJWKRIWKHFRQFUHWH is GHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VVXPLQJWKHSURYLGHGWUDQVYHUVHUHLQIRUFHPHQWLVJUHDWHUWKDQRUHTXDO JLYHQLQ$&,7DEOHDQGWKHEHDPLVQRWVXEMHFWHG to the minimum required transverse reinforcement, to an axial force, LVGHWHUPLQHGDVIROORZVVHH$&,7DEOH 

(14.12)

where and is the area of longitudinal reinforcement in the beam located more than IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU

away

$FFRUGLQJWR$&, PXVWEHWDNHQHTXDOWR]HURLQ(TXDWLRQ RYHUWKHOHQJWKVLGHQWL¿HGLQ$&, ZKHQERWKRIWKHIROORZLQJWZRFRQGLWLRQVRFFXU 1. The earthquake-induced shear force required shear force,  The factored axial compressive force,

is greater than or equal to one-half of the maximum on the beam, which includes earthquake effects, is less than

14-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHVHFRQGLWLRQVPXVWEHFKHFNHGRYHUWKHHQWLUHOHQJWKVLGHQWL¿HGLQ$&,%HFDXVHWKHVKHDUIRUFHGXH to earthquake effects is constant along the span, it is possible that at the faces of the supports where shear forces which means LVQRWHTXDOWR]HUR+RZHYHU due to gravity loads are relatively large, and away from the faces of the supports where shear forces due to gravity loads are smaller, at sections within which means LVHTXDOWR]HUR,QVXFKFDVHV must be set equal it is possible that regardless if is WR]HURRYHUWKHHQWLUHOHQJWKVLGHQWL¿HGLQ$&,1RWHWKDW is limited to LQFOXGHGRUQRW$&,  Detailing the Transverse Reinforcement

2QFHWKHVL]HDQGVSDFLQJRIWKHKRRSVDUHGHWHUPLQHGEDVHGRQWKHJRYHUQLQJ the required spacing must be FKHFNHGDJDLQVWWKHPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,7KHFDOFXODWHGKRRSVSDFLQJ within PXVWEHOHVVWKDQRUHTXDOWRWKHVPDOOHVWRIWKHIROORZLQJVHH)LJXUH  ݀ Τ4 ‫ۓ‬ ۖ ᇳ ۖ6 ‫ ݏ‬൑ smaller of

‫۔‬6݀௕ሺ௟௢௡௚.ሻ for Grade 60 bars ۖ ۖ ‫ە‬5݀௕ሺ௟௢௡௚.ሻ for Grade 80 bars



൑ 2ԣ

݀

݄

A

A Stirrups with seismic hooks

Hoops ൒ 2݄

‫ ݏ‬൑ ݀/2 Hoop reinforcement determined in accordance with ACI 18.6.5 at both ends

Crosstie ሺDetail Bሻ

Stirrup with seismic hooks ሺDetail Cሻ ൑ 6ᇳ ሺtyp.ሻ

൑ 14ԣ Section A-A Figure 14.9 Transverse reinforcement requirements for beams in special moment frames.

14-14

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

(14.13)

/RQJLWXGLQDOVNLQUHLQIRUFHPHQWUHTXLUHGE\$&,LVQRWLQFOXGHGZKHQFDOFXODWLQJ based on the smallest longitudinal bar in the section. In the regions where hoops are required, the primary longitudinal reinforcing bars in the beam closest to the tension DQGFRPSUHVVLRQIDFHVRIWKHPHPEHUPXVWKDYHWKHVDPHODWHUDOVXSSRUWUHTXLUHGLQ$&,DQGIRU WKHORQJLWXGLQDOEDUVLQFROXPQVZLWKUHFWLOLQHDUWLHVRUFLUFXODUWLHVUHVSHFWLYHO\$&,VHH)LJXUHIRU a beam with rectilinear hoops). Additionally, the center-to-center spacing of the transversely supported longitudinal EDUVLQWKHEHDPPXVWEHOHVVWKDQRUHTXDOWRLQDGGLWLRQDOYHUWLFDOFURVVWLHVDUHUHTXLUHGZKHUHWKLVUHTXLUHPHQWLVQRWVDWLV¿HG1RWHWKDWVNLQUHLQIRUFHPHQWUHTXLUHGE\$&,QHHGQRWEHODWHUDOO\VXSSRUWHG Where hoops are not required, it is permitted to use stirrups with seismic hooks at both ends spaced no greater than RQFHQWHU$&, 7KHIDFWRUHGVKHDUIRUFH is calculated at the location where the hoops are termiQDWHGDQGWKHVL]HDQGVSDFLQJRIWKHVWLUUXSVDUHGHWHUPLQHGXVLQJ(TXDWLRQ 7KHUHDUHQRUHVWULFWLRQVRQWKH is included in nominal shear strength of the concrete outside of the plastic hinge regions, so the contribution of WKHQRPLQDOVKHDUVWUHQJWKDQGLVGHWHUPLQHGE\(TXDWLRQ  the transverse reinforcement For beams subjected to a factored axial compressive force greater than UHTXLUHPHQWVLQ$&,WKURXJKIRUFROXPQVLQVSHFLDOPRPHQWIUDPHVPXVWEHVDWLV¿HGDORQJWKH OHQJWKVRIWKHDQWLFLSDWHGSODVWLFKLQJHUHJLRQVJLYHQLQ$&,$&, 2XWVLGHRIWKHVHUHJLRQV of the smallKRRSVFRQIRUPLQJWR$&,PXVWEHSURYLGHGDWDVSDFLQJQRWWRH[FHHGWKHOHDVWRILQ HVW*UDGHHQFORVHGORQJLWXGLQDOEHDPEDUDQG RIWKHVPDOOHVW*UDGHHQFORVHGORQJLWXGLQDOEHDPEDU,Q EHDPVZKHUHWKHFRQFUHWHFRYHURYHUWKHWUDQVYHUVHUHLQIRUFHPHQWH[FHHGVLQDGGLWLRQDOWUDQVYHUVHUHLQIRUFHPHQW KDYLQJDFRYHURILQRUOHVVDQGDVSDFLQJRILQRUOHVVPXVWEHSURYLGHG7KHVHDGGLWLRQDOUHTXLUHPHQWVDUH intended to provide lateral support for the longitudinal reinforcement subjected to factored axial compressive forces greater than the prescribed limit.

14.3 Columns of Special Moment Frames 14.3.1 Overview

Design and detailing requirements for columns designated part of a SFRS in special moment frames are given in $&,7KHVHUHTXLUHPHQWVDUHDSSOLFDEOHWRDQ\FROXPQWKDWLVSDUWRIDVSHFLDOPRPHQWIUDPHUHJDUGOHVVRIWKH magnitude of the axial force on the column. $GGLWLRQDODQDO\VLVUHTXLUHPHQWVPXVWEHVDWLV¿HGIRUFROXPQVWKDWDUHSDUWRIVSHFLDOPRPHQWIUDPHVLQEXLOGLQJV DVVLJQHGWR6'&'WKURXJK)$Q\FROXPQ SDUWRIWZRRUPRUHLQWHUVHFWLQJVSHFLDOPRPHQWIUDPHVDQG VXEMHFWHGWRDQD[LDOIRUFHGXHWRVHLVPLFIRUFHVDFWLQJDORQJHLWKHUSULQFLSDOSODQD[LVJUHDWHUWKDQRUHTXDOWRSHUcent of the axial design strength of the column must be designed for the most critical load effects due to application RIWKHVHLVPLFIRUFHVLQDQ\GLUHFWLRQ$6&(6(, (LWKHURIWKHSURFHGXUHVLQ$6&(6(,DUHSHUPLWWHGWREHXVHGWRVDWLVI\WKLVUHTXLUHPHQW,QWKH¿UVWSURFHGXUHWKHFROXPQVDQGWKHLUIRXQGDWLRQV DUHGHVLJQHGIRU SHUFHQWRIWKHVHLVPLFIRUFHVLQRQHGLUHFWLRQSOXVSHUFHQWRIWKHVHLVPLFIRUFHVLQWKHSHUSHQGLFXODUGLUHFWLRQ XVLQJDQ\RIWKHDQDO\VLVSURFHGXUHVLQ$6&(6(,D,QWKHVHFRQGSURFHGXUHWKHVWUXFWXUHLVDQDO\]HGXVLQJWKHDQDO\VLVPHWKRGVLQ$6&(6(,EZKHUHRUWKRJRQDOSDLUVRIJURXQGPRWLRQDFFHOHUDWLRQKLVWRULHVDUH applied simultaneously to the structure.

14-15

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHTXLUHGFROXPQVL]HDQGWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQVLQVSHFLDOPRPHQWIUDPHVDUH GHWHUPLQHGXVLQJWKHSURYLVLRQVLQ$&,&KDSWHUIRUWKHFRPELQHGHIIHFWVGXHWRJUDYLW\DQGODWHUDOIRUFHVVHH &KDSWHURIWKLVSXEOLFDWLRQ 7KHIROORZLQJUHTXLUHPHQWVPXVWDOVREHVDWLV¿HG 1.  3.  

'LPHQVLRQDOOLPLWV$&, 0LQLPXPÀH[XUDOVWUHQJWKRIFROXPQV$&, 'HWDLOLQJRIORQJLWXGLQDOUHLQIRUFHPHQW$&, 'HWDLOLQJRIWUDQVYHUVHUHLQIRUFHPHQW$&, 6KHDUVWUHQJWK$&,

14.3.2 Dimensional Limits

7KHGLPHQVLRQDOOLPLWVLQ$&,PXVWEHVDWLV¿HGIRUFROXPQVLQVSHFLDOPRPHQWIUDPHVVHH)LJXUH  7KHOLPLWRQWKHFURVVVHFWLRQDOGLPHQVLRQVRIDFROXPQUHVXOWVLQDPRUHFRPSDFWVHFWLRQDVRSSRVHGWRDORQJ UHFWDQJXODUVHFWLRQOLNHDZDOO7KHSUHVFULEHGLQPLQLPXPFROXPQGLPHQVLRQLVQRWSUDFWLFDOLQPRVWVSHFLDO PRPHQWIUDPHVDODUJHUFROXPQVL]HLVXVXDOO\UHTXLUHGEDVHGRQVWUHQJWKUHTXLUHPHQWVIRUWKHFROXPQDQGRU strength requirements for the joint. The dimensional limits for beam-column joints where longitudinal beam reinIRUFHPHQWH[WHQGVWKURXJKWKHMRLQWDUHDOVRJLYHQLQ)LJXUHVHH$&,  12ᇳ ‫ۓ‬ ۖ ۖ20݀௕ Τߣ for the largest Grade 60 beam bar ܿଵ ൒ greater of

‫۔‬26݀௕ for the largest Grade 80 beam bar ۖ ۖ



݄

‫݄ ە‬Τ2

A

For ܿଵ ൏ ܿଶ : ܿଶ ൑ 2.5ܿଵ 

A

ܿଵ Section A-A

Figure 14.10 Dimensional limits of columns in special moment frames.

14-16



Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QDGGLWLRQWR$&,WKHPLQLPXPÀH[XUDOVWUHQJWKSURYLVLRQVIRUFROXPQVLQVSHFLDOPRPHQWIUDPHV$&,  DQGWKHVKHDUVWUHQJWKUHTXLUHPHQWVIRUMRLQWVLQVSHFLDOPRPHQWIUDPHV$&, PD\KDYHDQLPSDFW RQFROXPQVL]H7KHVHUHTXLUHPHQWVDUHFRYHUHGEHORZ 7KHJHQHUDOJXLGHOLQHVWRDFKLHYHRYHUDOOHFRQRP\JLYHQLQ&KDSWHURIWKLVSXEOLFDWLRQDUHDOVRYDOLGIRUFROXPQV LQVSHFLDOPRPHQWIUDPHV8VLQJDODUJHUFROXPQVL]HZLWKDORQJLWXGLQDOUHLQIRUFHPHQWUDWLREHWZHHQDQGSHUcent helps in alleviating congestion problems and in satisfying shear strength requirements at beam-column joints. 6SHFLI\LQJODUJHUORQJLWXGLQDOEDUVL]HVDQGRU*UDGHORQJLWXGLQDOEDUVFDQDOVRKHOSZLWKFRQJHVWLRQDQGUHGXFHV the number of pieces that must be handled on site. 14.3.3 Minimum Flexural Strength of Columns

7KHUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HGDWMRLQWVLQVSHFLDOPRPHQWIUDPHVH[FHSWDWFRQQHFWLRQVZKHUH  WKHFROXPQLVGLVFRQWLQXRXVDERYHWKHFRQQHFWLRQOLNHLQWKHWRSVWRU\RIDEXLOGLQJ DQG WKHFROXPQIDFWRUHG under load combinations including earthquake effect, is less than axial compressive force, ,QOLHXRIVDWLVI\LQJ$&,WKHÀH[XUDOVWUHQJWKRIWKHFROXPQVDWDMRLQWLQDVSHFLDOPRPHQWIUDPHPXVW VDWLVI\$&,(TXDWLRQ  (14.14)

In this equation, LVWKHVXPRIWKHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHFROXPQVIUDPLQJLQWRWKHMRLQWDQG LVWKHVXPRIWKHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHEHDPVIUDPLQJLQWRWKHMRLQW%RWKWKHFROXPQDQGEHDPQRPLQDO ÀH[XUDOVWUHQJWKVDUHHYDOXDWHGDWWKHIDFHVRIWKHMRLQW7KLVUHTXLUHPHQWLVLOOXVWUDWHGLQ)LJXUHIRUVLGHVZD\ to the right (sidesway to the left must also be checked in this direction, and sidesway to the right and left must be FKHFNHGLQGHSHQGHQWO\LQWKHGLUHFWLRQSHUSHQGLFXODUWRWKHRQHVKRZQLQ)LJXUH 

‫ܯ‬௡௖ሺ௧௢௣ሻ 

‫ିܣ‬ ௦ ି ‫ܯ‬௡௕ 

ା ‫ܯ‬௡௕ 

‫ܣ‬ା ௦

‫ܯ‬௡௖ሺ௕௢௧ሻ  ା ሻ ି ‫ܯ‬௡௖ሺ௧௢௣ሻ ൅ ‫ܯ‬௡௖ሺ௕௢௧ሻ ൒ ሺ͸Τͷሻሺ‫ܯ‬௡௕ ൅ ‫ܯ‬௡௕

Figure 14.11 Minimum flexural strength requirements for columns in special moment frames.

The main intent of this requirement is to reduce the possibility of yielding in the columns during a design earthquake event; that is, yielding is more likely to occur at the ends of the beams. This is a more desirable situation than the one where yielding would occur in the columns: If yielding takes place at both ends of all the columns in a story of a building, a story mechanism can occur, which could lead to collapse.

14-17

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Axial Force

7KHQRPLQDOÀH[XUDOVWUHQJWKRIDFROXPQ is dependent on the magnitude of the axial force in the column. is to be calculated for the factored axial force consistent with the direction of $FFRUGLQJWR$&, DQDO\VLVUHVXOWLQJLQWKHORZHVWÀH[XUDOVWUHQJWK7KHIDFWRUHGD[LDOIRUFHVIURP$&,(TXDWLRQVH DQGJ  which include earthquake effects, DQGWKHFRUUHVSRQGLQJQRPLQDOÀH[XUDOVWUHQJWKVDUHLQGLFDWHGRQWKHQRPLQDO VWUHQJWKLQWHUDFWLRQGLDJUDPLQ)LJXUH,QWKLVFDVHWKHORZHVWQRPLQDOÀH[XUDOVWUHQJWKLVHTXDOWR ZKLFKFRUUHVSRQGVWRWKHD[LDOIRUFHDVVRFLDWHGZLWK$&,(TXDWLRQJ WKLVQRPLQDOÀH[XUDOVWUHQJWKLVWREH XVHGLQ(TXDWLRQ 

ACI Eq. ሺ5.3.1eሻ: 1.2‫ ܦ‬൅ 1.0‫ ܧ‬൅ 1.0‫ ܮ‬൅ 0.2ܵ ACI Eq. ሺ5.3.1gሻ: 0.9‫ ܦ‬൅ 1.0‫ܧ‬

‫ܯ‬௡௖|ହ.ଷ.ଵ௚ 

‫ܯ‬௡௖|ହ.ଷ.ଵ௘ 

Bending Moment

Figure 14.12 Determination of nominal flexural strength for a column in a special moment frame.

7KHVODEUHLQIRUFHPHQWZLWKLQWKHHIIHFWLYHZLGWKGH¿QHGLQ$&,PXVWEHLQFOXGHGZKHQFDOFXODWLQJWKHQRPL$&,VHH)LJXUHIRUDEHDPZLWKDVODERQHDFKVLGHRIWKH QDOÀH[XUDOVWUHQJWKVRIWKHEHDPV even where the web and a beam with a slab on one side of the web). The contribution of this reinforcement to minimum required amount is present, can be considerable and must be included in all cases. ,I(TXDWLRQ LVQRWVDWLV¿HGDWDMRLQWHLWKHU RU PXVWEHVDWLV¿HG (1) 7  KHQRPLQDOÀH[XUDOVWUHQJWKRIWKHFROXPQPXVWEHLQFUHDVHGE\LQFUHDVLQJWKHD VL]HRIWKHFROXPQE  DPRXQWRIÀH[XUDOUHLQIRUFHPHQWDQGRUF FRQFUHWHFRPSUHVVLYHVWUHQJWK   Ignore any contribution the column has to the lateral strength and stiffness of the structure, and design and GHWDLOWKHFROXPQLQDFFRUGDQFHZLWKWKHUHTXLUHPHQWVRI$&,IRUPHPEHUVWKDWDUHQRWSDUWRIWKH6)56 $&,  7KHFROXPQVXQGHULWHPDERYHPXVWQRWEHLJQRUHGZKHQFDOFXODWLQJWKHVHLVPLFEDVHVKHDUDQGWKHRYHUDOOWRUsional effects must not be reduced because the stiffness of these columns have been disregarded in the overall model of the structure.

14-18

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Žƒ„”‡‹ˆ‘”…‡‡–

݄

ܾ௘ 

‫ିܣ‬ ௦

‫ܣ‬ା ௦

ͳ͸݄ ൅ ܾ௪ ‫ۓ‬ ۖ ܾ௘ ൌŽ‡ƒ•–‘ˆ ൫‫ݏ‬௪ǡκ Τʹ൯ ൅ ൫‫ݏ‬௪ǡ௥ Τʹ൯ ൅ ܾ௪  ‫۔‬ ۖ ‫ ە‬ሺκ௡ ΤͶሻ ൅ ܾ௪  ‫ݏ‬௪ǡκ ൌ …Ž‡ƒ”†‹•–ƒ…‡–‘–Š‡™‡„‘–Š‡Ž‡ˆ–

ܾ௪ 

‫ݏ‬௪ǡ௥ ൌ …Ž‡ƒ”†‹•–ƒ…‡–‘–Š‡™‡„‘–Š‡”‹‰Š–

Žƒ„”‡‹ˆ‘”…‡‡–

݄

ܾ௘ 

‫ିܣ‬ ௦

‫ܣ‬ା ௦

͸݄ ൅ ܾ௪  ‫ۓ‬ ۖ ܾ௘ ൌŽ‡ƒ•–‘ˆ ሺ‫ݏ‬௪ Τʹሻ ൅ ܾ௪  ‫۔‬ ۖ ‫ ە‬ሺκ௡ Τͳʹሻ ൅ ܾ௪  ‫ݏ‬௪ ൌ …Ž‡ƒ”†‹•–ƒ…‡–‘–Š‡ƒ†Œƒ…‡–™‡„

ܾ௪ 

Figure 14.13 Determination of nominal flexural strength for a beam in a special moment frame.

14.3.4 Longitudinal Reinforcement Determining the Required Longitudinal Reinforcement

7KHPHWKRGVJLYHQLQ&KDSWHURIWKLVSXEOLFDWLRQFDQEHXVHGLQGHWHUPLQLQJWKHUHTXLUHGDUHDRIORQJLWXGLQDO UHLQIRUFHPHQWIRUUHLQIRUFHGFRQFUHWHFROXPQVLQVSHFLDOPRPHQWIUDPHVVXEMHFWHGWRFRPELQHGÀH[XUHDQGD[LDO instead of $&, WKHSXUSRVHRI forces. The area of longitudinal reinforcement is limited to this is to help control congestion and the development of high shear stresses. As noted above, it is good practice to to Providing larger areas of reinforcement is usually limit the area of longitudinal reinforcement to not practical or economical, and can result in very congested joints and splice locations. 7KHPLQLPXPÀH[XUDOVWUHQJWKSURYLVLRQVLQ$&,PD\KDYHDQLPSDFWRQWKHDUHDRIORQJLWXGLQDOUHLQIRUFHment required in a column that is part of a special moment frame. 5HIHUHQFHFRQWDLQVGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHIROORZLQJ • • • • •

7LHGUHFWDQJXODUFROXPQVUDQJLQJLQVL]HIURPWRLQLQFOXVLYH 7LHGDQGVSLUDOFLUFXODUFROXPQVUDQJLQJLQGLDPHWHUIURPWRLQLQFOXVLYH *UDGHDQG*UDGHORQJLWXGLQDOUHLQIRUFHPHQW &RQFUHWHFRPSUHVVLYHVWUHQJWKVIURPSVLWRSVLLQFOXVLYHDQG /RQJLWXGLQDOUHLQIRUFHPHQWUDWLRVIURPSHUFHQWWROHVVWKDQSHUFHQW

7KHVHLQWHUDFWLRQGLDJUDPVFDQEHXVHGWRIDFLOLWDWHVHOHFWLRQRIWKHFROXPQVL]HDQGDUHDRIORQJLWXGLQDOUHLQIRUFHPHQW 14-19

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Detailing the Longitudinal Reinforcement

7KHORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,DUHJLYHQLQ)LJXUH/DSVSOLFHVDUHUHTXLUHGWREH located within the center half of the column length, away from the ends of the column where spalling of the conFUHWHVKHOOVXUURXQGLQJWKHWUDQVYHUVHUHLQIRUFHPHQWLVOLNHO\WRRFFXUGXHWRKLJKEHQGLQJPRPHQWV$&,  7KHVHVSOLFHVPXVWEHGHVLJQHGDVWHQVLRQODSVSOLFHVLQDFFRUGDQFHZLWK$&,7UDQVYHUVHUHLQIRUFHPHQW FRQIRUPLQJWR$&,DQGZKLFKLVDSSOLFDEOHDWWKHHQGVRIWKHFROXPQPXVWEHSURYLGHGRYHUWKH lap splice length mainly to ensure that the splice performs as intended when subjected to stress reversals.



݄

ܿଵ 

A

A

ܿଶ 

splice*

Tension lap

κ௨ ൒ 2.5κௗ 

0.01‫ܣ‬௚ ൑ ‫ܣ‬௦௧ ൑ 0.06‫ܣ‬௚

ܿଵ Section A-A

* Tension lap splices must be located within the center half of the column and must be enclosed by transverse reinforcement in accordance with ACI 18.7.5.2 and 18.7.5.3

Figure 14.14 Longitudinal reinforcement requirements for columns in special moment frames.

is greater than or The longitudinal reinforcement must be selected such that the clear height of the column, HTXDOWRWLPHVWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUV GHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,  7KHSXUSRVHRIWKLVUHTXLUHPHQWLVWRUHGXFHWKHOLNHOLKRRGRIERQGVSOLWWLQJIDLOXUHDORQJWKHORQJLWXGLQDO EDUVZLWKLQWKHFOHDUFROXPQKHLJKW:KHUHWKLVUHTXLUHPHQWLVQRWVDWLV¿HG can be reduced by using smaller longitudinal bars, increasing the amount of transverse reinforcement, increasing the compressive strength of the concrete, or a combination of one or more of the above. 0HFKDQLFDODQGZHOGHGVSOLFHVFRQIRUPLQJWR$&,DQGUHVSHFWLYHO\PD\EHXVHGLQVWHDGRIODS VSOLFHV$&,  $WOHDVWVL[ORQJLWXGLQDOEDUVDUHUHTXLUHGLQFROXPQVRIVSHFLDOPRPHQWIUDPHVXWLOL]LQJFLUFXODUKRRSVDVWKHWUDQVYHUVHUHLQIRUFHPHQW$&, 

14-20

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.3.5 Transverse Reinforcement Determining the Required Transverse Reinforcement

Plastic hinges can form at the ends of a column in a special moment frame during a design-level earthquake event DQGWKHFRQFUHWHPXVWEHSURSHUO\FRQ¿QHGZLWKLQWKHVHUHJLRQV$FFRUGLQJWR$&,WUDQVYHUVHUHLQIRUFHment in the form of spirals, circular hoops, or rectilinear hoops with or without crossties satisfying the requirements RI$&,WKURXJKPXVWEHSURYLGHGRYHUDOHQJWK from each joint face and on both sides of any VHFWLRQZKHUHÀH[XUDO\LHOGLQJLVOLNHO\WRRFFXUZKHUH is the assumed length of the anticipated plastic hinge region:

(14.15)

Similar to beams, columns in special moment frames must be designed for the shear forces generated by the probat the ends of the column, which are associated with the range of factored axial forces, DEOHÀH[XUDOVWUHQJWKV that include earthquake effects, The determination of the factored shear force, based on this requirement LVJLYHQLQ)LJXUHIRUDFROXPQVXEMHFWHGWRDIDFWRUHGD[LDOIRUFH ,WLVHYLGHQWIURPWKH¿JXUHWKDW is constant over the height of the column. In order to obtain the maximum shear force, sidesway to the right and sidesway to left must both be considered. ܲ௨ 

‫ܯ‬௣௥ሺ௧௢௣ሻ 

κ௨ 

ܸ௨ 

ܸ௨ ൌ

‫ܯ‬௣௥ሺ௧௢௣ሻ ൅ ‫ܯ‬௣௥ሺ௕௢௧ሻ  κ௨

ܸ௨ 

‫ܯ‬௣௥ሺ௕௢௧ሻ 

ܲ௨ 

Figure 14.15 Design shear force for columns in special moment frames.

14-21

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Interaction diagrams for a column where the tensile stress in the longitudinal reinforcement is equal to and the strength reduction factor LVHTXDOWRDUHJLYHQLQ)LJXUHWKHVHGLDJUDPVDUHDUHSUHVHQWDWLRQRIWKH of a column as a function of factored axial forces. For the range of axial forces SUREDEOHÀH[XUDOVWUHQJWKV occurs at the balanced point even though the factored axial force corVKRZQLQ)LJXUHD WKHODUJHVW UHVSRQGLQJWRWKLVSUREDEOHÀH[XUDOVWUHQJWKLVQRWREWDLQHGIURPHLWKHURIWKHWZRDSSOLFDEOHORDGFRPELQDWLRQVWKDW at the balanced point must be used in the contain >WKDWLV$&,(TXDWLRQVH DQGJ @,QWKLVFDVH )RUWKHUDQJHRID[LDOIRUFHVLQ)LJXUHE WKHODUJHVW LVHTXDOWRWKHSUREDEOHÀH[XUDO calculation of VWUHQJWKDVVRFLDWHGZLWKWKHIDFWRUHGD[LDOIRUFHIURP$&,(TXDWLRQH EHFDXVHWKDWSUREDEOHÀH[XUDOVWUHQJWK LVJUHDWHUWKDQWKHSUREDEOHÀH[XUDOVWUHQJWKVDVVRFLDWHGZLWKDQ\RIWKHRWKHUIDFWRUHGD[LDOIRUFHVLQWKDWUDQJH

Axial Force

Range of Axial Forces

ACI Eq. ሺ5.3.1eሻ: 1.2‫ ܦ‬൅ 1.0‫ ܧ‬൅ 1.0‫ ܮ‬൅ 0.2ܵ ACI Eq. ሺ5.3.1gሻ: 0.9‫ ܦ‬൅ 1.0‫ܧ‬

‫ܯ‬௣௥  Bending Moment ሺࢇሻ

Axial Force

Range of Axial Forces

ACI Eq. ሺ5.3.1eሻ: 1.2‫ ܦ‬൅ 1.0‫ ܧ‬൅ 1.0‫ ܮ‬൅ 0.2ܵ ACI Eq. ሺ5.3.1gሻ: 0.9‫ ܦ‬൅ 1.0‫ܧ‬

‫ܯ‬௣௥  Bending Moment ሺ࢈ሻ

Figure 14.16 Nominal strength interaction diagrams for columns with fs = 1.25fy and ݊ = 1.0.

14-22

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

It is conservative to assume that GHYHORSVDWERWKHQGVVLPXOWDQHRXVO\LQDFROXPQLQDVWRU\DERYHWKH¿UVW VWRU\HVSHFLDOO\LQFDVHVZKHUH(TXDWLRQ LVVDWLV¿HG,WLVIRUWKLVUHDVRQWKDWWKHHDUWKTXDNHLQGXFHGVKHDU IRUFHLVSHUPLWWHGWREHWDNHQDVWKHVKHDUIRUFHFDOFXODWHGIURPMRLQWVKHDUVWUHQJWKVEDVHGRQWKHSUREDEOHÀH[XUDO VWUHQJWKVRIWKHEHDPVIUDPLQJLQWRWKHMRLQW$&, 7KHGHWHUPLQDWLRQRIWKHVKHDUIRUFHEDVHGRQ of the beams is indeterminate and an analysis must be performed to obtain it (see, for example, the analysis method of the column at its base and the earthLQ5HIHUHQFH )RUFROXPQVLQWKH¿UVWVWRU\LWLVSRVVLEOHWRGHYHORS quake-induced shear force must be determined based on that possibility. In any case, the design shear force must not be taken less than that determined from analysis of the building subjected to the code-prescribed seismic forces. and transverse rein,QJHQHUDOVKHDUVWUHQJWKLQWKHUHJLRQVRIÀH[XUDO\LHOGLQJLVSURYLGHGE\ERWKFRQFUHWH 7KHVL]HDQGVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ forcement $&,  (14.16)

where IRUVKHDUVHH$&,7DEOHDQG$&, 7KHQRPLQDOVKHDUVWUHQJWKRIWKHFRQFUHWH is GHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VVXPLQJWKHSURYLGHGWUDQVYHUVHUHLQIRUFHPHQWLVJUHDWHUWKDQRUHTXDO JLYHQLQ$&, is determined as follows (see to the minimum required transverse reinforcement, $&,7DEOH 

(14.17)

where is the area of longitudinal reinforcement in the column located more than away is the factored axial force on the column corresponding to the load IURPWKHH[WUHPHFRPSUHVVLRQ¿EHUDQG Note that is positive for compression and negative for tension, and combination used to determine $&,  LQ(TXDWLRQ PXVWQRWEHWDNHQJUHDWHUWKDQ For rectilinear hoops, is equal to the area of all hoop legs or all hoop legs plus crosstie legs in the direction of is equal to two times the area of analysis within the longitudinal center-to-center spacing, For circular ties, is equal to two times the area of the spiral bar within the tie bar within $&, 6LPLODUO\IRUVSLUDOV the spiral pitch, Like in the case of beams in special moment frames, PXVWEHWDNHQHTXDOWR]HURLQ(TXDWLRQ RYHUWKH lengths ZKHQERWKRIWKHIROORZLQJWZRFRQGLWLRQVRFFXU$&,  1. The earthquake-induced shear force mum required shear force,  The factored axial compressive force,

is greater than or equal to one-half of the maxion the column, which includes earthquake effects, is less than

It is typical for the earthquake-induced shear force to be greater than the shear forces from gravity loads (especially DWLQWHULRUFROXPQV VRWKH¿UVWFRQGLWLRQLVXVXDOO\PHW7KHPLQLPXPIDFWRUHGD[LDOIRUFHREWDLQHGIURP$&, (TXDWLRQVH DQGJ LVXVHGWRGHWHUPLQHLIWKHVHFRQGFRQGLWLRQLVPHWRUQRW ,IWKHVKHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGEDVHGRQWKHSUREDEOHÀH[XUDOVWUHQJWKVDWWKHHQGVRIDFROXPQD PRUHGHWDLOHGDQDO\VLVWRGHWHUPLQHWKHIDFWRUHGVKHDUIRUFHVEDVHGRQWKHSUREDEOHÀH[XUDOVWUHQJWKVWKDWFDQGHvelop at the ends of the beams need not be performed.

14-23

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Beyond the length the nominal shear strength of the concrete, E\(TXDWLRQ 

is permitted to be used and can be determined

The usual procedure for determining column transverse reinforcement is to calculate the area and spacing based on WKHWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,VHHEHORZ DQGWKHQFKHFNWKDWWKHVKHDUVWUHQJWKUHTXLUHPHQWVRI$&,RXWOLQHGDERYHDUHVDWLV¿HGXVLQJWKDWDUHDDQGVSDFLQJ Detailing the Transverse Reinforcement

Hoop Reinforcement Transverse reinforcement in the form of rectilinear hoops (with or without crossties) or circular hoops in accorGDQFHZLWK$&,PXVWEHSURYLGHGDORQJWKHHQWLUHOHQJWKRIFROXPQVLQVSHFLDOPRPHQWIUDPHV:LWKLQD distance IURPHDFKHQGRIDFROXPQWKHKRRSVPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,WKURXJK $&, %H\RQGWKHOHQJWK KRRSVVDWLVI\LQJ$&,DQGDUHSHUPLWWHGDWDVSDFLQJOHVVWKDQ RUHTXDOWRWKDWLQ$&,$VXPPDU\RIWKHVHUHTXLUHPHQWVLVJLYHQLQ7DEOH7UDQVYHUVHUHLQIRUFHPHQW and DUHJLYHQLQ)LJXUH requirements for columns with rectilinear hoops where DQGRU DUHJLYHQLQ)LJXUH Requirements where

Within ℓo

Table 14.5 Design and Detailing Requirements for Hoop Reinforcement in Columns of Special Moment Frames Requirement

ACI Section No.

Bends of rectilinear hoops and crossties must engage peripheral longitudinal reinforcing bars.

E

&URVVWLHVRIWKHVDPHRUVPDOOHUEDUVL]HDVWKHKRRSVDUHSHUPLWWHGVXEMHFWWRWKHOLPLWDWLRQRI$&,&RQVHFXWLYHFURVVWLHVPXVWEHDOWHUQDWHGHQGIRUHQGDORQJWKHORQJLtudinal reinforcement and around the perimeter of the cross-section.

F

Rectilinear hoops or crossties must provide lateral support to longitudinal reinforcement in DFFRUGDQFHZLWK$&,DQG

G

Center-to-center spacing, of longitudinal bars laterally supported by the corner of a FURVVWLHRUKRRSOHJPXVWEHOHVVWKDQRUHTXDOWRLQDURXQGWKHSHULPHWHURIWKHFROXPQ

H

In columns with rectilinear hoops where the factored axial compressive force every longitudinal bar or bundle of bars around the perimeter of or where the column core must have lateral support provided by the corner of a hoop or by a seismic in such cases must be less than or equal to 8 in. The factored axial hook. The value of is the largest value obtained from the load combinations that include the earthforce quake effect,

I

The center-to-center spacing of the hoops must not exceed the least of the following: (a) 0LQLPXPFROXPQGLPHQVLRQ of the smallest longitudinal bar (b) )RU*UDGHEDUV of the smallest longitudinal bar (c) )RU*UDGHEDUV where (d)



(table continued on next page)

14-24

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.5 Design and Detailing Requirements for Hoop Reinforcement in Columns of Special Moment Frames (cont.) Requirement

ACI Section No.

0LQLPXPWUDQVYHUVHUHLQIRUFHPHQWLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,7DEOH +RRS7\SH

Within ℓo

Rectilinear



Circular

Outside ℓo

+RRSUHLQIRUFHPHQWVDWLVI\LQJWKHSURYLVLRQVRI$&,DQGPXVWEHSURYLGHG The spacing of the transverse reinforcement, must not exceed the lesser of the following: (a) 6 in. of the smallest longitudinal bar (b) )RU*UDGHEDUV of the smallest longitudinal bar (c) )RU*UDGHEDUV



A larger amount of transverse reinforcement may be required based on the provisions of $&,RU that must be provided within the length is equal to the The minimum area of transverse reinforcement, JUHDWHURIWKDWGHWHUPLQHGE\(TXDWLRQVD DQGE LQ)LJXUHRUE\(TXDWLRQVD E DQGF LQ)LJXUH in these equations is equal to the area of the section measured to the outside of the transverse reinThe term IRUFHPHQWZKLFKLVWKHDUHDRIWKHFRQ¿QHGFRUHRIWKHFROXPQ7KHWHUP LVWKHGLPHQVLRQRIWKHFRQ¿QHGFRUH ,GHQWL¿HGLQ)LJXUHDUHWKHDSSURperpendicular to the legs of the transverse reinforcement that make up IRUWKHFROXPQGHSLFWHGLQWKH¿JXUH priate to determine The term in Equation (c) is equal to This term increases the minimum required area of transverse reinforcement where H[FHHGVSVLDQGLVPHDQWWRKHOSLQSUHYHQWLQJEULWWOHIDLOXUHRIWKH FROXPQ$&,5UHFRPPHQGVFRQFUHWHVWUHQJWKVJUHDWHUWKDQSVLVKRXOGEHXVHGZLWKFDXWLRQLQ columns of special moment frames because of the limited test data available on their overall performance. The term in Equation (c) is equal to where is the number of longitudinal bars around the perimeter of the column core laterally supported by the corner of hoops or by seismic hooks.

14-25

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Hoops ሺtyp.ሻ

 A

‫ۓ‬ ۖ

A ‫ ݏ‬൑ smaller of

κ௢ ൒ larger of

‫ۓ‬ ۖ

Larger of ܿଵ and ܿଶ

6ᇳ

6݀௕ሺ௟௢௡௚.ሻ for Grade 60 bars ‫۔‬ ۖ ‫ە‬5݀௕ሺ௟௢௡௚.ሻ for Grade 80 bars

Note 1 Clear spanΤ6

‫۔‬ ۖ ‫ە‬18ᇳ

 ሺSmaller of ܿଵ and ܿଶ ሻΤ4 ‫ۓ‬ ۖ ۖ6݀௕ሺ௟௢௡௚.ሻ for Grade 60 bars ‫ ݏ‬൑ smaller of

‫۔‬5݀ ௕ሺ௟௢௡௚.ሻ for Grade 80 bars ۖ ۖ ‫ݏ ە‬௢ ሺsee Notes 2 and 3ሻ

‫ܣ‬௚ ݂௖ᇱ ‫ۓ‬0.3‫ܾݏ‬௖ ൬ െ 1൰ ‫ܣ‬௖௛ ݂௬௧ ۖ ‫ܣ‬௦௛ ൒ greater of Note 6

‫۔‬ ݂ᇱ ۖ0.09‫ܾݏ‬௖ ௖ ݂௬௧ ‫ە‬

ሺaሻ ሺsee Note 5ሻ  ሺbሻ

‫ݔ‬௜  ܿଶ  ‫ݔ‬௜ 

Alternate 90-deg hooks ‫ݔ‬௜ 

‫ݔ‬௜ 

‫ݔ‬௜ 

Note 4

ܿଵ  Section A-A Notes 1. Transverse reinforcement in accordance with ACI 18.7.5.2 and 18.7.5.3 over the lap splice length. 2. 4ԣ ൑ ‫ݏ‬௢ ൌ 4 ൅ ሾሺ14 െ ݄௫ ሻΤ3ሿ ൑ 6ԣ 3. ݄௫ ൌ largest value of ‫ݔ‬௜ 4. ‫ݔ‬௜ ൑ 14ԣ 5. In determining ‫ܣ‬௦௛ , provisions of ACI 18.7.6 must also be satisfied. 6. Where the concrete cover ൐ 4ԣ, provide additional transverse reinforcement having cover ൑ 4 in. and spacing no greater than 12ᇳ . Figure 14.17 Requirement for rectilinear hoops in columns of special moment frames where Pu”0.3Ag Iࠢc and Iࠢc ”10,000 psi.

14-26



Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Hoops ሺtyp.ሻ

A

‫ۓ‬ ۖ

A ‫ ݏ‬൑ smaller of

‫ۓ‬ ۖ

Larger of ܿଵ and ܿଶ

κ௢ ൒ larger of Clear spanΤ6 ‫۔‬ ۖ ‫ ە‬18ᇳ

6ᇳ

6݀௕ሺ௟௢௡௚.ሻ for Grade 60 bars ‫۔‬ ۖ ‫ە‬5݀௕ሺ௟௢௡௚.ሻ for Grade 80 bars

Note 1  ሺSmaller of ܿଵ and ܿଶ ሻΤ4 ‫ۓ‬ ۖ ۖ 6݀௕ሺ௟௢௡௚.ሻ for Grade 60 bars ‫ ݏ‬൑ smaller of

‫ܣ‬௦௛ Note 8

‫ ۔‬5݀ ௕ሺ௟௢௡௚.ሻ for Grade 80 bars ۖ ۖ ‫ݏە‬௢ ሺsee Notes 2 and 3ሻ

‫ܣ‬௚ ݂௖ᇱ ‫ۓ‬0.3‫ܾݏ‬௖ ൬ െ 1൰ ‫ܣ‬௖௛ ݂௬௧ ۖ ۖ ۖ ݂௖ᇱ ൒ greater of 0.09‫ܾݏ‬௖ ݂௬௧ ‫۔‬ ۖ ۖ ܲ௨ ۖ0.2‫݇ ݇ ܾݏ‬ ௖ ௙ ௡ ݂௬௧ ‫ܣ‬௖௛ ‫ە‬



ሺaሻ

ሺbሻ ሺsee Notes 5 െ 7ሻ 

ሺcሻ

‫ݔ‬௜  ܿଶ  ‫ݔ‬௜  ‫ݔ‬௜  ‫ݔ‬௜  ‫ݔ‬௜  ‫ݔ‬௜  ‫ݔ‬௜  ‫ݔ‬௜ 

Note 4

ܿଵ  Section A-A Notes 1. Transverse reinforcement in accordance with ACI 18.7.5.2 and 18.7.5.3 over the lap splice length. 2. 4 in. ൑ ‫ݏ‬௢ ൌ 4 ൅ ሾሺ14 െ ݄௫ ሻΤ3ሿ ൑ 6 in. 3. ݄௫ ൌ largest value of ‫ݔ‬௜ 4. ‫ݔ‬௜ ൑ 8 in. 5. In determining ‫ܣ‬௦௛ , provisions of ACI 18.7.6 must also be satisfied. 6. ݇௙ ൌ ሺ݂௖ᇱ Τ25,000ሻ ൅ 0.6 ൒ 1.0 7. ݇௡ ൌ ݊௟ Τሺ݊௟ െ 2ሻ 8. Where the concrete cover ൐ 4 in., provide additional transverse reinforcement having cover ൑ 4 in. and spacing no greater than 12 in. Figure 14.18 Requirement for rectilinear hoops in columns of special moment frames where Pu > 0.3Ag Iࠢc and/or Iࠢc > 10,000 psi.

14-27

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ௖ǡଶ 

‫ܣ‬௦௛ǡଶ 

‫ܣ‬௦௛ǡଵ 

ܾ௖ǡଵ  ‫ܣ‬௖௛ ൌ ܾ௖ǡଵ ൈ ܾ௖ǡଶ 

Figure 14.19 Confined core dimensions for columns with rectilinear hoop reinforcement in special moment frames.

Spiral Reinforcement 7UDQVYHUVHUHLQIRUFHPHQWUHTXLUHPHQWVIRUFROXPQVZLWKVSLUDOUHLQIRUFHPHQWDUHJLYHQLQ$&,VHH)LJXUH  6SLUDOUHLQIRUFHPHQWLVJHQHUDOO\WKHPRVWHI¿FLHQWW\SHRIFRQ¿QHPHQWUHLQIRUFHPHQWKRZHYHUWKHVSLUDOV within the beam-column joints may cause constructability issues, especially when placing longitudinal bars from the beams through the joint.

 ‫ۓ‬ ۖ

൑ 3ᇳ

Clear space ൒ 1ᇳ ‫۔‬ ۖ ‫ە‬൒ ሺ4Τ3ሻ݀௔௚௚ A

A

Min. 6 longitudinal bars

Min. 3/8-in. diameter spiral bar Section A-A

‫ܣ‬௚ ݂௖ᇱ ‫ۓ‬0.45 ൬ െ 1൰ ‫ܣ‬௖௛ ݂௬௧ ۖ ۖ ۖ ݂௖ᇱ ߩ௦ ൒ greater of 0.12 ݂௬௧ ‫۔‬ ۖ ۖ ۖ 0.35݇ ܲ௨ ௙ ݂௬௧ ‫ܣ‬௖௛ ‫ە‬

ሺdሻ

ሺeሻ 

ሺfሻ

Notes 1. Equation (f) for ߩ௦ is applicable only where ܲ௨ ൐ 0.3‫ܣ‬௚ ݂௖ᇱ and/or ݂௖ᇱ ൐ 10,000 psi. 2. In determining ‫ܣ‬௦௛ , provisions of ACI 18.7.6 must also be satisfied. 3. ݇௙ ൌ ሺ݂௖ᇱ Τ25,000ሻ ൅ 0.6 ൒ 1.0

Figure 14.20 Requirements for spiral reinforcement in columns of special moment frames.

14-28

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Columns Supporting Reactions from Discontinuous Stiff Members Where a stiff member, such as a wall, is discontinued and supported on columns instead of on a foundation, the FROXPQVPXVWFRQWDLQWUDQVYHUVHUHLQIRUFHPHQWFRQIRUPLQJWRWKHUHTXLUHPHQWVLQ$&,WKURXJK for columns in special moment frames over the full height at all levels beneath the discontinuity where the factored $&,VHH)LJXUH axial compressive force related to earthquake effects in the columns exceeds pertains to columns designed using load combinations that include the overstrength  7KHOLPLWRI ZKLFKDUHJLYHQLQ$6&(6(,VHH7DEOHRIWKLVSXEOLFDWLRQ 2WKHUZLVHWKHOLPLWLV factor 7KHWUDQVYHUVHUHLQIRUFHPHQWPXVWH[WHQGDERYHDQGEHORZWKHFROXPQLQDFFRUGDQFHZLWK$&,E 



Reinforcement in wall not shown for clarity Wall

൒ κௗ of largest longitudinal column bar where κௗ is determined by ACI 18.8.5 ሺtyp.ሻ

Transverse reinforcement in accordance with ACI 18.7.5.2 through 18.7.5.4 ሺACI 18.7.5.6ሻ

Footing or mat

Wall

൒ 12ԣ

൒ κௗ of largest longitudinal column bar where κௗ is determined by ACI 18.8.5

Figure 14.21 Transverse reinforcement requirements for columns supporting discontinuous stiff members in special moment frames.

As noted previously, the required reinforcement details presented above must be provided even where the effects from wind loads govern the design of a column.

14.4 Joints of Special Moment Frames 14.4.1 Overview

Requirements for beam-column joints in special moment frames are given in ACI 18.8. The overall integrity and performance of a special moment frame is dependent on the behavior of the beam-column joints in the frames. Inelastic rotations at the faces of the joints produce strains in the beam longitudinal reinforcement well in excess of the strain corresponding to the yield strength of the reinforcement. As such, joint shear forces in the beam longitudinal reinforcement passing through or anmust be calculated using a stress equal to FKRUHGLQWKHMRLQW$&,  Slippage of the beam longitudinal reinforcement through a joint can lead to an increase in joint rotation. At edge and corner columns where the reinforcement does not continue through the joint, the terminated bars must extend to the IDUIDFHRIWKHFRQ¿QHGFROXPQFRUHDQGPXVWEHGHYHORSHGLQWHQVLRQLQDFFRUGDQFHZLWK$&,DQGLQFRP14-29

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

SUHVVLRQLQDFFRUGDQFHZLWK$&,$&, $WLQWHULRUMRLQWVWKHPLQLPXPMRLQWFROXPQ GLPHQVLRQ UHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGVHH)LJXUH  %HFDXVHWHVWGDWDIRUWKHFRPELQDWLRQRIOLJKWZHLJKWFRQFUHWHDQG*UDGHORQJLWXGLQDOUHLQIRUFHPHQWLQMRLQWVDUH QRWDYDLODEOHQRUPDOZHLJKWFRQFUHWHPXVWEHXVHGLQMRLQWVZKHUH*UDGHORQJLWXGLQDOUHLQIRUFHPHQWLVVSHFL¿HG $&,  14.4.2 Transverse Reinforcement

7UDQVYHUVHUHLQIRUFHPHQWLQDEHDPFROXPQMRLQWLVUHTXLUHGWRFRQ¿QHWKHFRQFUHWHVRWKDWLWEHKDYHVLQDGXFWLOH manner and maintains it ability to carry vertical loads even after the outer concrete shell were to spall off. The transYHUVHUHLQIRUFHPHQWVSHFL¿HGLQ$&,DQGLVUHTXLUHGDVDPLQLPXPLQDMRLQW regardless of the magnitude of the shear force (ACI 18.8.3.1); this is the same amount of transverse reinforcement at the ends of columns in special moment frames. An exception to this requirement ocrequired over the length FXUVZKHUHDMRLQWLVFRQ¿QHGE\EHDPVRQDOOIRXUVLGHVRIWKHMRLQWEHDPVDUHGH¿QHGDVSURYLGLQJFRQ¿QHPHQW to a joint where the width of the beam is at least three-fourths the width of the column): the amount of transverse UHLQIRUFHPHQWUHTXLUHGE\$&,LVSHUPLWWHGWREHUHGXFHGE\SHUFHQWDQGWKHVSDFLQJUHTXLUHGE\$&, LVSHUPLWWHGWREHLQFUHDVHGWRLQZLWKLQWKHRYHUDOOGHSWKRIWKHVKDOORZHVWEHDPIUDPLQJLQWRWKHMRLQW $&, )RUVLPSOHUGHWDLOLQJLWLVFRPPRQIRUWKHWUDQVYHUVHUHLQIRUFHPHQWDWWKHHQGVRIWKHFROXPQWR FRQWLQXHWKURXJKWKHMRLQWUHJDUGOHVVRIDQ\FRQ¿QHPHQWSURYLGHGE\EHDPV Where beams are wider than the joint that they frame into, the beam longitudinal reinforcement located outside of WKHFRQ¿QHGFRUHRIWKHFROXPQPXVWEHFRQ¿QHGE\WUDQVYHUVHUHLQIRUFHPHQWSDVVLQJWKURXJKWKHFROXPQVDWLVI\LQJWKHVSDFLQJUHTXLUHPHQWVRI$&,DQGZKHUHWUDQVYHUVHEHDPVGRQRWSURYLGHFRQ¿QHment in the joint (ACI 18.8.3.3). Where transverse beams with widths at least three-fourths the column width frame LQWRDMRLQWRQO\WKHVSDFLQJUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG$QH[DPSOHRIVXFKWUDQVYHUVHUHLQIRUFHPHQWLVVKRZQLQ$&,)LJXUH5 14.4.3 Shear Strength

The design shear strength of cast-in-place beam-column joints in special moment frames must satisfy the following HTXDWLRQ$&,  (14.18)

7KHKRUL]RQWDOMRLQWVKHDUIRUFH is determined at the mid-height of the joint in the direction of analysis from VWDWLFVXVLQJWHQVLOHDQGFRPSUHVVLYHEHDPIRUFHVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,DQGFROXPQVKHDU and FRQVLVWHQWZLWKWKHQHJDWLYHDQGSRVLWLYHSUREDEOHÀH[XUDOVWUHQJWKVRIWKHEHDPDWWKHIDFHRIWKHMRLQW where LVGHWHUPLQHGE\(TXDWLRQ  Free-body diagrams of (1) an interior column or an edge column with the direction of analysis parallel to the edge DQG DQHGJHFROXPQZLWKWKHGLUHFWLRQRIDQDO\VLVSHUSHQGLFXODUWRWKHHGJHRUDFRUQHUFROXPQLQDPRPHQW IUDPHVXEMHFWHGWRJUDYLW\DQGHDUWKTXDNHORDGVDUHJLYHQLQ)LJXUHIRUVLGHVZD\WRWKHOHIWZKHUHSUREDEOH ÀH[XUDOVWUHQJWKVRIWKHEHDPVDUHXVHGWRGHWHUPLQH For an interior column or an edge column with the direction of analysis parallel to the edge, the shear force in the can be obtained from equilibrium by summing moments about the center of the joint assuming the column, SRLQWRILQÀHFWLRQLVDWWKHPLGKHLJKWRIWKHFROXPQZKLFKLVDUHDVRQDEOHDVVXPSWLRQIRUFROXPQVDERYHWKH¿UVW VWRU\DQGEHORZWKHWRSVWRU\VHH)LJXUH  (14.19)

14-30

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܿଵ 



ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦ ܸଵ 

κ௖ 

ܸଶ  ା ‫ܯ‬௣௥ 

ି ‫ܯ‬௣௥ 

ܸଶ 

ܸଵ 

‫ܣ‬ା ௦

–‡”‹‘”…‘Ž—ƒ†‡†‰‡…‘Ž—™‹–Š–Š‡ †‹”‡…–‹‘‘ˆƒƒŽ›•‹•’ƒ”ƒŽŽ‡Ž–‘–Š‡‡†‰‡

ܸ௖௢௟  ܿଵ  ܸ௖௢௟ 

‘‹– ‫ିܣ‬ ௦

κ௖ 

ܸଵ  ି ‫ܯ‬௣௥ 

ܸଵ 

†‰‡…‘Ž—™‹–Š–Š‡†‹”‡…–‹‘‘ˆƒƒŽ›•‹• ’‡”’‡†‹…—Žƒ”–‘–Š‡‡†‰‡ƒ†…‘”‡”…‘Ž—

ܸ௖௢௟ 

Figure 14.22 Free-body diagrams of columns in a special moment frame subjected to gravity and earthquake loads.

14-31

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

where

and

are determined by the following equations: (14.20)

(14.21)

Similarly, for an edge column with the direction of analysis perpendicular to the edge or a corner column, the shear FDQEHREWDLQHGIURPWKHIROORZLQJHTXDWLRQVHH)LJXUH  force in the column, (14.22)

)UHHERG\GLDJUDPVRIWKHMRLQWVIRUWKHVHWZRFDVHVZLWKRQO\KRUL]RQWDOIRUFHVDUHJLYHQLQ)LJXUHZKHUHLW is assumed the axial forces on the beams are negligible. )RUDQLQWHULRUFROXPQRUDQHGJHFROXPQZLWKWKHGLUHFWLRQRIDQDO\VLVSDUDOOHOWRWKHHGJHWKHKRUL]RQWDOVKHDU LVREWDLQHGIURPHTXLOLEULXPE\VXPPLQJIRUFHVLQWKHKRUL]RQWDOGLUHFWLRQ force in the joint, (14.23)

ZKHUHWKHIRUFHLQWKHUHLQIRUFHPHQWLVLQFUHDVHGE\LQDFFRUGDQFHZLWK$&, Similarly, for an edge column with the direction of analysis perpendicular to the edge or a corner column, the horiis equal to the following: ]RQWDOVKHDUIRUFHLQWKHMRLQW (14.24)

When calculating it is conservative to disregard the term(s) associated with the beam shear force(s). Similarly, when calculating it is conservative to disregard

ܸ௖௢௟  ‫ܥ‬ଶ ൌ ܶଶ ൌ ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ 

‫ିܣ‬ ௦

ܸ௖௢௟  ܶଵ ൌ ͳǤʹͷ‫ିܣ‬ ௦ ݂௬ 

ܶଵ ൌ ͳǤʹͷ‫ିܣ‬ ௦ ݂௬ 

‫ିܣ‬ ௦

ା ܸ௨ ൌ ܶଵ ൅ ܶଶ െ ܸ௖௢௟ ൌ ͳǤʹͷሺ‫ିܣ‬ ௦ ൅ ‫ܣ‬௦ ሻ݂௬ െ ܸ௖௢௟ 

ܸ௨ ൌ ܶଵ െ ܸ௖௢௟ ൌ ͳǤʹͷ‫ିܣ‬ ௦ ݂௬ െ ܸ௖௢௟ 

‫ܣ‬ା ௦

‫ܣ‬ା ௦

ܶଶ ൌ ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ 

‫ܥ‬ଵ ൌ ܶଵ ൌ ͳǤʹͷ‫ିܣ‬ ௦ ݂௬ 

ܸ௖௢௟  –‡”‹‘”…‘Ž—ƒ†‡†‰‡…‘Ž—™‹–Š–Š‡ †‹”‡…–‹‘‘ˆƒƒŽ›•‹•’ƒ”ƒŽŽ‡Ž–‘–Š‡‡†‰‡

‫ܥ‬ଵ ൌ ܶଵ ൌ ͳǤʹͷ‫ିܣ‬ ௦ ݂௬ 

ܸ௖௢௟  †‰‡…‘Ž—™‹–Š–Š‡†‹”‡…–‹‘‘ˆƒƒŽ›•‹• ’‡”’‡†‹…—Žƒ”–‘–Š‡‡†‰‡ƒ†…‘”‡”…‘Ž—

Figure 14.23 Free-body diagrams of the joints in special moment frames.

14-32

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The strength reduction factor, $&,DQG 

LVHTXDOWRIRUVKHDULQEHDPFROXPQMRLQWVRIVSHFLDOPRPHQWIUDPHVVHH

in a special moment frame is determined using the requirements of ACI The nominal joint shear strength, 7KHFRQWLQXLW\RIWKHFROXPQVDQGEHDPVIUDPLQJLQWRWKHMRLQWDQGZKHWKHUWKHMRLQWLVFRQ¿QHGRUQRWE\ Continuous columns and beams refer to members present on both sides WUDQVYHUVHEHDPVKDYHDQLQÀXHQFHRQ of a joint. Alternatively, column and beam extensions are assumed to provide continuity through a beam-column MRLQWSURYLGHGWKHUHTXLUHPHQWVLQ$&,DUHDUHVDWLV¿HGUHVSHFWLYHO\VHH)LJXUHVDQGRIWKLV SXEOLFDWLRQ $MRLQWLVFRQVLGHUHGWREHODWHUDOO\FRQ¿QHGZKHUHWZRWUDQVYHUVHEHDPVWKDWLVWZREHDPVIUDPing into the column in the direction perpendicular to the direction of analysis) satisfy the three requirements in ACI VHH)LJXUHRIWKLVSXEOLFDWLRQ 7KHQRPLQDOMRLQWVKHDUVWUHQJWKVLQ$&,7DEOHDUHJLYHQLQ 7DEOH Table 14.6 Nominal Joint Shear Strength, Vn Column

Continuous or a column extension is provided that VDWLV¿HV$&,

Beam in Direction of Analysis

Continuous or a beam extension is SURYLGHGWKDWVDWLV¿HV$&,

Confinement by Transverse Beams*

Vn (lbs)**

&RQ¿QHG 1RWFRQ¿QHG &RQ¿QHG

Other 1RWFRQ¿QHG Continuous or a beam extension is SURYLGHGWKDWVDWLV¿HV$&,

&RQ¿QHG 1RWFRQ¿QHG

Other &RQ¿QHG Other 1RWFRQ¿QHG * Transverse beams that satisfy the requirements of ACI 15.2.8 are considered to provide confinement (see Figure 11.3 of this publication). Examples of the various joint types in this table are given in Figure 14.24. ** The modification factor, , that reflects the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength is equal to 0.75 for lightweight concrete and 1.0 for normalweight concrete.

([DPSOHVRIWKHYDULRXVMRLQWW\SHVLQ7DEOHDQGWKHFRUUHVSRQGLQJ DUHJLYHQLQ)LJXUH-RLQWVZLWK continuous columns (that is, columns frame into the top and bottom of the joint) or column extensions satisfying the UHTXLUHPHQWVRI$&,DUHGHVLJQDWHG7\SH,-RLQWVZLWKQRQFRQWLQXRXVFROXPQVWKDWLVMRLQWVZKHUHFROXPQV frame into the bottom of the joints only, such as those in the top story of a building), are designated Type II. Type $DQG7\SH%GHVLJQDWLRQVLQ)LJXUHFRUUHVSRQGWRMRLQWVZLWKFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRIDQDO\VLV RUEHDPH[WHQVLRQVVDWLVI\LQJWKHUHTXLUHPHQWVLQ$&,7\SH&DQG7\SH'GHVLJQDWLRQVFRUUHVSRQGWRMRLQWV ZLWKQRQFRQWLQXRXVEHDPVLQWKHGLUHFWLRQRIDQDO\VLV$GHVLJQDWLRQRI³&´DWWKHIDFHRIDMRLQWPHDQVDWUDQVYHUVH EHDPSURYLGHVFRQ¿QHPHQWDWWKDWIDFHLQDFFRUGDQFHZLWK$&,$Q³1&´GHVLJQDWLRQPHDQVQRWUDQVYHUVH EHDPLVSUHVHQWDWWKDWIDFHRUDWUDQVYHUVHEHDPLVSUHVHQWEXWGRHVQRWVDWLVI\WKHFRQ¿QHPHQWUHTXLUHPHQWVLQ$&,  FDQEHREWDLQHGIURP)LJXUHIRUWKHYDULRXVFRPELQDWLRQVRIMRLQWW\SHV)RU The nominal shear strengths, example, a Type I-A joint consists of a continuous column and continuous beams in the direction of analysis with This joint type corresponds FRQ¿QHPHQWSURYLGHGE\WUDQVYHUVHEHDPVRQERWKIDFHVLQWKLVFDVH to an interior column located in the structure other than the top story. Similarly, a corner column in the top story of a structure corresponds to a Type II-D joint, and

14-33

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Note 1 Transverse beam

C

C or NC

C

NC

Type A

Type B

C

C or NC

C

NC

Type C

Type D

Note 2

Note 2



Type I

Transverse beam

Type II

Plan

Elevation Direction of analysis

Joint Type

ܸ௡

I-A

20ߣඥ݂௖ᇱ ‫ܣ‬௝

I-B

15ߣඥ݂௖ᇱ ‫ܣ‬௝

I-C

15ߣඥ݂௖ᇱ ‫ܣ‬௝

I-D

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-A

15ߣඥ݂௖ᇱ ‫ܣ‬௝

II-B

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-C

12ߣඥ݂௖ᇱ ‫ܣ‬௝

II-D

8ߣඥ݂௖ᇱ ‫ܣ‬௝

Notes 1.Column is continuous or a column extension satisfying ACI 15.2.6 is provided ሺsee Figure 11.1 of this publicationሻ. 2.Beam in direction of analysis is continuous or a beam extension satisfying ACI 15.2.7 is provided ሺsee Figure 11.2 of this publicationሻ. 3.C – Transverse beam provides confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3 of this publicationሻ. 4.NC – Transverse beam is not provided or transverse beam does not provide confinement in accordance with ACI 15.2.8 ሺsee Figure 11.3 of this publicationሻ.

Figure 14.24 Examples of joint types and the corresponding nominal joint shear strength, Vn.

14-34

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term LVWKHHIIHFWLYHFURVVVHFWLRQDODUHDZLWKLQDMRLQWZKLFKLVJLYHQLQ$&,%\GH¿QLWLRQ is equal to the product of the joint depth and the effective joint width. The depth of the joint is always equal to the depth of the column parallel to the direction of analysis, VHH)LJXUH 7KHHIIHFWLYHMRLQWZLGWK is equal to the width of the column perpendicular to the direction of analysis where the beams in the direction of analysis are For beams not as wide as the column, is equal as wide as or wider than the column. In such cases, and where is the beam width and is the perpendicular distance from the edge of to the lesser of the beam to the nearest side face of the column.



‫ܣ‬௝ ൌ ܿଵ ‫ݓ‬

ܿଵ 

‫ ݔ‬

ܾ ‫ݓ‬

‫ݔ‬

ܿଶ  ‹”‡…–‹‘‘ˆƒƒŽ›•‹• ܾ ൅ ܿଵ   ˆˆ‡…–‹˜‡Œ‘‹–™‹†–Š‫ ݓ‬ൌ Ž‡••‡”‘ˆ ൝ ܾ ൅ ʹ‫ݔ‬

Figure 14.25 Effective cross-sectional area within a joint, Aj.

14.4.4 Development Length of Bars in Tension

Requirements for the development length of hooked bars and headed bars terminating in a joint and the developPHQWOHQJWKRIVWUDLJKWEDUVWHUPLQDWHGDWDMRLQWDUHJLYHQLQ$&, 'HWDLOHGLQIRUPDWLRQRQWKHVHUHTXLUHPHQWVLVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ

14-35

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.5 Special Structural Walls 14.5.1 Overview

Design and detailing requirements for special structural walls, including ductile coupled walls and all components of special structural walls (including coupling beams and wall piers that are part of the SFRS) are given in ACI $FFRUGLQJWR$&,DZDOOLVDYHUWLFDOVWUXFWXUDOHOHPHQWGHVLJQHGWRUHVLVWD[LDOORDGODWHUDOORDGRUERWK ZLWKDQRYHUDOOKRUL]RQWDOOHQJWKWRWKLFNQHVVUDWLRJUHDWHUWKDQ 6WUXFWXUDOZDOOVFDQEHFRQ¿JXUDWHGLQDQXPEHURIZD\VLQFOXGLQJWKHIROORZLQJVHH)LJXUH D UHFWDQJXODU ZDOOVE UHFWDQJXODUZDOOVZLWKFROXPQVLQFRUSRUDWHGDWWKHHQGVVRPHWLPHVUHIHUUHGWRDVD³EDUEHOO´ZDOOFRQ¿JXUDWLRQ F LQWHUVHFWLQJUHFWDQJXODUZDOOVDQGG FRUHZDOOVZLWKFRXSOLQJEHDPV

ሺƒሻ



ሺ„ሻ

ሺ…ሻ

‘—’Ž‹‰„‡ƒ

ሺ†ሻ Figure 14.26 Structural wall configurations. (a) Rectangular wall. (b) Rectangular wall with columns. (c) Intersecting rectangular walls. (d) Core walls.

14-36

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

:DOOVZLWKRSHQLQJVFRQVLVWRIYHUWLFDODQGKRUL]RQWDOZDOOVHJPHQWV$YHUWLFDOZDOOVHJPHQWLVDVHJPHQWRID VWUXFWXUDOZDOOERXQGHGKRUL]RQWDOO\E\WZRRSHQLQJVRUE\DQRSHQLQJDQGDQHGJHVHH)LJXUH :DOOSLHUV DUHGH¿QHGLQ$&,DVYHUWLFDOZDOOVHJPHQWVVDWLVI\LQJWKHIROORZLQJGLPHQVLRQDOUDWLRV KRUL]RQWDOOHQJWKWR DQG FOHDUKHLJKWWRKRUL]RQWDOOHQJWK wall thickness

ሺ݄௪ ሻ஺ 

$KRUL]RQWDOZDOOVHJPHQWLVDVHJPHQWRIDVWUXFWXUDOZDOOERXQGHGYHUWLFDOO\E\WZRRSHQLQJVRUE\DQRSHQLQJ DQGDQHGJHVHH)LJXUH  ሺκ௪ ሻଵ 

ሺκ௪ ሻଶ 

ᬅ

ᬆ

:DOOSLHU

ሺ݄௪ ሻ஻ 

x κ௪ Τܾ௪ ൑ ͸ǤͲƒ† ሺκ௪ ሻଷ 

ሺκ௪ ሻସ 

ሺκ௪ ሻହ 

ᬇ

ᬈ

ᬉ

x ݄௪ Τκ௪ ൒ ʹǤͲ

‡”–‹…ƒŽ™ƒŽŽ•‡‰‡–•ȋ•Šƒ†‡†Ȍ

‘”‹œ‘–ƒŽ™ƒŽŽ•‡‰‡–•ȋ•Šƒ†‡†Ȍ Figure 14.27 Vertical and horizontal wall segments in a structural wall.

14.5.2 Reinforcement Minimum Reinforcement Requirements

Special structural walls must have reinforcement in two orthogonal directions in the plane of the wall. The miniPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,DUHVXPPDUL]HGLQ)LJXUHIRUUHLQIRUFHPHQWZLWK where is the ratio of the area of the distributed longitudinal reinforcement in the wall to the gross is the ratio of the area of the distributed transverse reinconcrete area perpendicular to that reinforcement and forcement in the wall to the gross concrete area perpendicular to that reinforcement.

14-37

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

           

‫ ݏ‬൑ 18ԣ  ߩκ 

‫ ݏ‬൑ 18ԣ 

ߩ௧ 

ܾ௪ 

All other reinforcement not shown for clarity

‫ܣ‬௖௩ ൌ ܾ௪ κ௪ 

κ௪ 

ࢂ࢛ ‫כ‬ ൑ ߣඥ݂௖ᇱ ‫ܣ‬௖௩

࢚࣋ #5 and smaller

#6 and larger

൒ 0.0020

൒ 0.0025

൐ ߣඥ݂௖ᇱ ‫ܣ‬௖௩

൒ 0.0025

࣋र  ൒ 0.0025

* For ܸ௨ ൐ 2ߣඥ݂௖ᇱ ‫ܣ‬௖௩ or ݄௪ Τκ௪ ൒ 2.0, provide two curtains of reinforcement where ݄௪ ൌ height of entire wall.

Figure 14.28 Web reinforcement requirements for special structural walls.

Tension Development and Splice Requirements

/RQJLWXGLQDO%DU7HUPLQDWLRQ 7KHWHQVLRQGHYHORSPHQWDQGVSOLFHUHTXLUHPHQWVLQ$&,DUHDSSOLFDEOHWRDQ\VSHFLDOVWUXFWXUDOZDOO /RQJLWXGLQDOEDUFXWRIISRLQWVDUHJLYHQLQ$&,D :KHUHORQJLWXGLQDOUHLQIRUFHPHQWLVQRORQJHUUHTXLUHG WRUHVLVWWKHFRPELQHGHIIHFWVRIÀH[XUHDQGD[LDOIRUFHVLWPXVWH[WHQGDWOHDVWIWDERYHWKHWKHRUHWLFDOFXWRII GHWHUPLQHGLQDFFRUGDQFHZLWK$&, point but need not extend more than the tension development length, DERYHWKHQH[WÀRRUOHYHO7KLVUHTXLUHPHQWLVLOOXVWUDWHGLQ)LJXUHZKHUHLWKDVEHHQGHWHUPLQHGE\DQDO\VLV from the base of the wall. In WKDWWKHWKHRUHWLFDOFXWRIISRLQWIRUWKHEDUVODEHOOHG³$´LQWKH¿JXUHLVORFDWHG RUGHUWRVDWLVI\WKHUHTXLUHPHQWVLQ$&,D WKHOHQJWKRIWKH³$´EDUVPXVWEHHTXDOWRWKHIROORZLQJ (14.25)

where

14-38

LVWKHGLVWDQFHIURPWKHEDVHRIWKHZDOOWRWKHWRSRIWKHÀRRUOHYHODERYHWKHWKHRUHWLFDOFXWRIISRLQW

൒ 12Ԣ

κௗ 

݄ 

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Floor ‫ݔ‬

݄௫ 

݄௖௨௧௢௙௙ 

Floor ‫ ݔ‬െ 1

Theoretical cutoff point for “A” bars

“A” bars

κ௪ 

‫ܯ‬௨ 

Figure 14.29 Longitudinal bar cutoff points for special structural walls.

7KHORQJLWXGLQDOUHLQIRUFHPHQWDWWKHWRSRIDZDOOQHHGQRWVDWLVI\$&,D ,QJHQHUDOORQJLWXGLQDOUHLQforcement should be terminated gradually over the height of a wall and should not occur near critical sections where yielding of the longitudinal reinforcement is expected (such as at the base of a wall where plastic hinges are likely to form). When determining tension development lengths at locations where yielding of the longitudinal reinforcement is likely to occur, PXVWEHPXOWLSOLHGE\>$&,E @7KLVPXOWLSOLHUDFFRXQWVIRU WKHOLNHOLKRRG WKDWWKHDFWXDO\LHOGVWUHQJWKH[FHHGVWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHEDUV VWUDLQKDUGHQLQJDQG F\FOLFORDG reversals. Lap Splices Requirements where lap splices can occur within boundary regions of special structural walls are given in ACI F ,QJHQHUDOERXQGDU\UHJLRQVDUHWKRVHUHJLRQVDWWKHHQGVRIDZDOODQGDWWKHHGJHVDGMDFHQWWRRSHQings subjected to large compressive forces when the wall undergoes cyclic deformations caused by a design-level HDUWKTXDNHHYHQW6XFKUHJLRQVW\SLFDOO\UHTXLUHVSHFLDOWUDQVYHUVHUHLQIRUFHPHQWWRFRQ¿QHWKHFRQFUHWHDQGWR UHVWUDLQWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHZDOOVREXFNOLQJRIWKHEDUVGRHVQRWRFFXU7KHKRUL]RQWDOOHQJWKRID LVGH¿QHGLQ$&,D >VHHEHORZ@ boundary element, Lap splices of longitudinal wall reinforcement within boundary regions are not permitted over the following reRUIWZKLFKHYHULVOHVVDERYHDFULWLFDOVHFWLRQZKHUH\LHOGLQJLVOLNHO\WRRFFXUDVD gions: (1) a story height below a critical section where yielding is UHVXOWRIODWHUDOGLVSODFHPHQWVDQG DWHQVLRQGHYHORSPHQWOHQJWK likely to occur as a result of lateral displacements. $FULWLFDOVHFWLRQORFDWHGDWWKHWRSRIDÀRRUVODELVGHSLFWHGLQ)LJXUH7KHKRUL]RQWDOOHQJWKRIDERXQGDU\ at the ends of walls and around openings is equal to the greater of and which is mearegion, VXUHGIURPWKHH[WUHPHFRPSUHVVLRQ¿EHURUIURPWKHHGJHRIDQRSHQLQJ7KHWHUP is the largest neutral axis depth FDOFXODWHGIURPDVWUDLQFRPSDWLELOLW\DQDO\VLVIRUWKHIDFWRUHGD[LDOIRUFHDQGQRPLQDOÀH[XUDOVWUHQJWKFRQVLVWHQW at the top of the wall. At wall intersections, boundary regions occur within a length with the design displacement, HTXDOWRWKHZDOOWKLFNQHVVPHDVXUHGEH\RQGWKHLQWHUHVWLQJUHJLRQV RIWKHFRQQHFWHGZDOOVVHH)LJXUH 

14-39

݄௦௫ 

൒ lesser of ሺ݄௦௫ , 20Ԣሻ

Lap splice length ሺtyp.ሻ

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

T/Floor

൒ κௗ 

ሺCritical sectionሻ

Other reinforcement not shown for clarity

Wall Elevation



3ܾ௪ሺଵሻ 

κ௕௘ 

κ௕௘ 

ܾ௪ሺଶሻ 

κ௕௘ 

κ௕௘ 

ܿ െ 0.1κ௪ κ௕௘ ൌ greater of ൝  ܿ Τ2

κ௕௘ 

Regions in plan and elevation where lap splices of longitudinal Ueinforcementare not permitted ሾACI 18.10.2.3ሺcሻሿ

ܾ௪ሺଵሻ 

Wall Plan

 Figure 14.30 Regions where lap splices of longitudinal reinforcement are not permitted in special structural walls.

14-40

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Mechanical and Welded Splices Mechanical and welded splices of the reinforcement in special structural walls must conform to the requirements in $&,DQGUHVSHFWLYHO\>$&,G @ RIWKHEDU7KHVSHFL¿HG Type 1 mechanical splices must be able to develop in tension or compression at least WHQVLOHVWUHQJWKRIWKHVSOLFHGEDUVPXVWEHGHYHORSHGE\7\SHPHFKDQLFDOVSOLFHV([FHSWIRU7\SHPHFKDQLFDO VSOLFHVRQ*UDGHUHLQIRUFHPHQWPHFKDQLFDOVSOLFHVDUHQRWSHUPLWWHGWREHORFDWHGLQUHJLRQVZKHUH\LHOGLQJRI WKHUHLQIRUFHPHQWLVOLNHO\WRRFFXUDVDUHVXOWRIODWHUDOGLVSODFHPHQWVEH\RQGWKHOLQHDUUDQJHRIEHKDYLRU$7\SH PHFKDQLFDOVSOLFHRQ*UDGHUHLQIRUFHPHQWLVSHUPLWWHGDWDQ\ORFDWLRQLQDFDVWLQSODFHVSHFLDOVWUXFWXUDOZDOO of the bar; these types of splices Welded splices must be able to develop in tension or compression at least are not permitted to be located within regions where yielding of the reinforcement is likely to occur as a result of lateral displacements beyond the linear range of behavior. $FFRUGLQJWR$&,LWLVQRWSHUPLWWHGWRZHOGVWLUUXSVWLHVLQVHUWVRURWKHUVLPLODUHOHPHQWVWRWKHORQJLtudinal reinforcement in a special structural wall. Such welding can lead to local embrittlement of the reinforcing steel. Minimum End Reinforcement for Walls or Piers with a Single Critical Section for Flexure and Axial Loads A minimum amount of longitudinal reinforcement is required at the ends of walls or walls piers that meet the following conditions: • • The wall or pier is essentially continuous from the base of the structure to the top of the wall or pier • 7KHZDOORUSLHULVGHVLJQHGWRKDYHDVLQJOHFULWLFDOVHFWLRQIRUÀH[XUHDQGD[LDOORDGVVXFKDVDFDQWLOHYHU wall) 7KHIROORZLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HGLQVXFKFDVHV from the end of a vertical wall segment and over a width 1. The longitudinal reinforcement ratio within must be greater than or equal to equal to the wall thickness,  The minimum longitudinal reinforcement prescribed in item 1 above must extend vertically above and below and (which is the length over which the critical section a length equal to at least the greater of yielding is expected). 3. 1RPRUHWKDQSHUFHQWRIWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWSUHVFULEHGLQLWHPDERYHLVSHUPLWWHG to be terminated at any one section. 7KHUHTXLUHPHQWVRI$&,DUHLOOXVWUDWHGLQ)LJXUHIRUDUHFWDQJXODUFDQWLOHYHUZDOOWKDWVDWLV¿HVWKH within from the ends of conditions noted above. The minimum area of longitudinal reinforcement, WKHZDOOLVHTXDOWRWKHIROORZLQJEDVHGRQWKHUHTXLUHPHQWLQ$&,D  (14.26)

/RFDWLRQVZKHUHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWLVUHTXLUHGE\$&,D IRUGLIIHUHQWZDOOFRQ¿JXUDWLRQVDUHJLYHQLQ$&,)LJXUH5 'HYHORSPHQWRI5HLQIRUFHPHQWLQ&RXSOLQJ%HDPV ,QDGGLWLRQWRWKHGHYHORSPHQWUHTXLUHPHQWVLQ$&,DQGWKHUHLQIRUFHPHQWLQFRXSOLQJEHDPVPXVWEH GHYHORSHGLQWHQVLRQLQDFFRUGDQFHZLWKWKHIROORZLQJUHTXLUHPHQWVLQ$&,

14-41

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ܣ‬κ ൒ 0.5‫ܣ‬κሺ௘௡ௗሻ ሺtyp.ሻ

κ௪ greater of ൝  ‫ܯ‬௨ Τ3ܸ௨

Other reinforcement not shown for clarity

Critical section

κ௪  0.15κ௪ 

ܾ௪ 

0.15κ௪ 

‫ܣ‬κሺ௘௡ௗሻ ൒ 0.9ܾ௪ κ௪ ඥ݂௖ᇱ ൗ݂௬  Regions in plan and elevation where end longitudinal Ueinforcementin accordance with ACI 18.10.2.4 is required

Figure 14.31 End longitudinal reinforcement requirements for walls and piers in accordance with ACI 18.10.2.4.

1. For coupling beams reinforced with longitudinal reinforcement in accordance with the requirements in ACI 18.6.3.1 for beams in special moment frames, the tension development length of the longitudinal reinforcement must be determined using  For coupling beams reinforced with two intersecting groups of diagonally placed bars symmetrical about the PLGVSDQRIWKHEHDPLQDFFRUGDQFHZLWK$&,WKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHGLDJRQDOUHLQforcement must be determined using 14.5.3 Design Shear Force

)RUUHODWLYHO\VOHQGHUZDOOVWKDWLVZDOOVZLWKDQRYHUDOOKHLJKWWROHQJWKUDWLRJUHDWHUWKDQ ZLWKRXWVLJQL¿FDQW RSHQLQJVWKHEHKDYLRULVPXFKOLNHDÀH[XUDOFDQWLOHYHUZKHUHWKHFULWLFDOVHFWLRQIRUÀH[XUHDQGD[LDOIRUFHVLV DWWKHEDVHRIWKHZDOOZKHUHÀH[XUDO\LHOGLQJFDQRFFXU'XHWRPDWHULDORYHUVWUHQJWKDQGVWUDLQKDUGHQLQJRIWKH UHLQIRUFHPHQWDZDOO\LHOGLQJLQÀH[XUHZLOOOLNHO\GHYHORSDPRPHQWDWWKHFULWLFDOVHFWLRQHTXDOWRWKHSUREDEOH 7KXVWKHVKHDUFDSDFLW\RIWKHZDOOPXVWEHVXI¿FLHQWVRWKDW can develop at the critical ÀH[XUDOVWUHQJWK VHFWLRQSULRUWRVKHDUIDLOXUH)RUQRQVOHQGHUZDOOVÀH[XUDOVWUHQJWKLVUHODWLYHO\KLJKDQGLQHODVWLFUHVSRQVHRFFXUV LQVKHDUUDWKHUWKDQÀH[XUDO\LHOGLQJ &RQVLGHUWKHZDOOLQ)LJXUHZKHUHLWLVDVVXPHGWKDWWKHKHLJKWRIWKHHQWLUHZDOODERYHWKHFULWLFDOVHFWLRQ LVDWOHDVWWLPHVWKHOHQJWKRIWKHZDOO For purposes of discussion, assume the wall is subjected to a triangular lateral load distribution, which approximates the distribution of code-prescribed, static earthquake forces

14-42

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

over the height of the wall (the following discussion is equally valid where a parabolic distribution of earthquake IRUFHVLVUHTXLUHG 7KHIDFWRUHGVKHDUDQGPRPHQWGLDJUDPVDUHDOVRVKRZQLQWKH¿JXUHDQGDUHREWDLQHGIURPWKH DSSURSULDWHORDGFRPELQDWLRQV%HFDXVHPXOWLSOHORDGFRPELQDWLRQVPXVWEHFRQVLGHUHGÀH[XUDORYHUVWUHQJWKRFis greater than at the base of the wall for one or both of the load combicurs at the critical section, that is, nations that include earthquake effects, 7KHUHIRUHLQRUGHUIRUWKHZDOOWR\LHOGLQÀH[XUHDWWKHFULWLFDOVHFWLRQ WKHODWHUDOIRUFHVDSSOLHGWRWKHZDOOPXVWEHJUHDWHUWKDQWKHFRGHSUHVFULEHGIRUFHVVHH)LJXUH 7KLVPHDQV the shear forces in the wall must be greater as well.

݄௪௖௦ 

Code-prescribed  earthquake forces

Additional earthquake forces to develop ‫ܯ‬௣௥ at the critical section ܸ௨ 

ܸ௨ 

ܸ௘ 

‫ܯ‬௨ 

‫ܯ‬௣௥ 

Critical section ‫ܯ‬௨  Figure 14.32 Determination of shear demand in a slender wall in accordance with ACI 18.10.3.

LVDPHDVXUHRIÀH[XUDORYHUVWUHQJWKEXLOWLQWRWKHZDOO,QJHQHUDOWKHVKHDUIRUFHFRUUHVSRQGThe ratio at the critical section is equal to the design shear force, due to the code-preing to the development of and a factor to account for the dynamic response of a building. scribed forces multiplied by $FFRUGLQJWR$&,VSHFLDOVWUXFWXUDOZDOOVPXVWEHGHVLJQHGIRUVKHDUXVLQJWKHGHVLJQVKHDUIRUFH GHWHUPLQHGE\$&,(TXDWLRQ  (14.27)

The overstrength factor, LVREWDLQHGIURP$&,7DEOHDQGLVHTXDOWRZKHUH is equal to the following: is, for nonslender walls). In cases where

(that

(14.28)

14-43

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Because depends on the magnitude of the axial force at the critical section, which is different for different load resulting in the largest must be used. combinations, the ratio The dynamic response of a multistory building produces changing patterns of lateral inertial forces. One or more patterns can shift the centroid of the lateral forces downward, thereby increasing the shear force at the critiLQ(TXDWLRQ DFFRXQWVIRUWKLVG\QDPLFVKHDUDPSOL¿FDWLRQ)RUZDOOVZKHUH cal section. The term $&, RWKHUZLVH LVGHWHUPLQHGE\$&,(TXDWLRQ 

(14.29)

In this equation, is the number of stories above the critical section, which must be taken greater than or equal to This limit is imposed on to account for buildings with relatively large story heights. It is permitted to use when determining the design shear strength of a slender special structural wall where LVGHWHUPLQHGE\(TXDWLRQ >VHH$&,@)RUVKRUWVTXDWZDOOVZKHUHLWLVLPSUDFWLFDOWRGHVLJQIRU the shear force corresponding to the development of 14.5.4 Shear Strength

$FFRUGLQJWR$&,WKHQRPLQDOVKHDUVWUHQJWKRIVSHFLDOVWUXFWXUDOZDOOV WLRQ 

is determined by ACI Equa(14.30)

where

is equal to the following:

(14.31)

ܾ௪ 

κ௪ 

‫ܣ‬௖௩ ൌ ܾ௪ κ௪  κ௪ 

ܾ௪ 

κ௢௣௘௡ 

‫ܣ‬௖௩ ൌ ܾ௪ ሺκ௪ െ κ௢௣௘௡  Figure 14.33 Area of wall resisting shear force.

14-44

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The term is the gross area of the cross-section of the wall resisting the shear force. For a rectangular wall with is equal to the product of the thickness of the web and the length of the web in the direction of no openings, The areas to be analysis. If a wall section has openings, the area of the openings must not be included in XVHGLQ(TXDWLRQ ZKHUHWKHHQGVRIDZDOODUHHQODUJHGDUHJLYHQLQ)LJXUH Transverse and longitudinal shear reinforcement must be appropriately distributed along the height and length of a wall to effectively restrain inclined cracks. Wherever practical, shear reinforcement should be uniformly distribmust be greater than or equal to uted and at a relatively small spacing. The longitudinal reinforcement ratio, where the overall height-to-length ratio $&, $Q\ the transverse reinforcement ratio, concentrated reinforcement near the ends of the wall provided primarily for resisting bending moments is not to be and included when determining 1RPLQDOVKHDUVWUHQJWKUHTXLUHPHQWVIRUZDOOVZLWKRSHQLQJVDUHJLYHQLQ$&,DQG:KHUHVHYthe nominal shear strength, is limited eral vertical wall segments all resist a common factored shear force, $&,  to When designing an individual vertical wall segment, LVFDOFXODWHGE\(TXDWLRQ ZKHUH is determined is obtained using the larger of the following two ratios: using the transverse reinforcement in that segment and EDVHGRQWKHGLPHQVLRQVRIWKHHQWLUHZDOODQG  based on the dimensions of the wall segment. (1) The intent of this requirement is to ensure that a vertical wall segment will not have a unit shear strength greater of the than that of the entire wall; however, the vertical segment may have a lower unit shear strength if is limited to for any vertical segment is greater than that of the entire wall. The nominal shear strength, is the area of the concrete section of the individual vertical one of the vertical wall segments in the group where resisted by an individual vertical wall segment is based on wall segment. The portion of the total shear force WKHÀH[XUDODQGVKHDUULJLGLWLHVRIWKDWVHJPHQWDQGWKDWVKHDUIRUFHPXVWEHOHVVWKDQRUHTXDOWRWKHGHVLJQVKHDU IRUWKDWVHJPHQWVHH)LJXUH  strength, ܸ௨  ሺܸ௨ ሻଵ 

ሺܸ௨ ሻଶ 

ሺܸ௨ ሻଷ 

༃

༄

༅

ܸ௨ ൌ ሺܸ௨ ሻଵ ൅ ሺܸ௨ ሻଶ ൅ ሺܸ௨ ሻଷ  ሺܸ௨ ሻଵ ൑ ߶ ቂሺߙ௖ ሻଵ ߣඥ݂௖ᇱ ൅ ሺߩ௧ ሻଵ ݂௬௧ ቃ ሺ‫ܣ‬௖௩ ሻଵ ൑ ߶ͳͲඥ݂௖ᇱ ሺ‫ܣ‬௖௪ ሻଵ  ሺܸ௨ ሻଶ ൑ ߶ ቂሺߙ௖ ሻଶ ߣඥ݂௖ᇱ ൅ ሺߩ௧ ሻଶ ݂௬௧ ቃ ሺ‫ܣ‬௖௩ ሻଶ ൑ ߶ͳͲඥ݂௖ᇱ ሺ‫ܣ‬௖௪ ሻଶ  ሺܸ௨ ሻଷ ൑ ߶ ቂሺߙ௖ ሻଷ ߣඥ݂௖ᇱ ൅ ሺߩ௧ ሻଷ ݂௬௧ ቃ ሺ‫ܣ‬௖௩ ሻଷ ൑ ߶ͳͲඥ݂௖ᇱ ሺ‫ܣ‬௖௪ ሻଷ  ‘”…‘„‹‡†˜‡”–‹…ƒŽ™ƒŽŽ•‡‰‡–•ǣ߶ܸ௡ ൑ ߶ͺඥ݂௖ᇱ ‫ܣ‬௖௩  ™Š‡”‡‫ܣ‬௖௩ ൌ ሺ‫ܣ‬௖௪ ሻଵ ൅ ሺ‫ܣ‬௖௪ ሻଶ ൅ ሺ‫ܣ‬௖௪ ሻଷ  Figure 14.34 Shear strength requirements for vertical wall segments.

14-45

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QWKHFDVHRIKRUL]RQWDOZDOOVHJPHQWVDQGFRXSOLQJEHDPV is limited to where is the area of the FRQFUHWHVHFWLRQRIWKHKRUL]RQWDOZDOOVHJPHQWRUWKHDUHDRIWKHFRXSOLQJEHDP$&,  for shear are not applicable to walls or wall piers designed in 7KHUHTXLUHPHQWVLQ$&,ZKHUH DFFRUGDQFHZLWK$&,EHFDXVHWKHVHPHPEHUVDUHFRQWUROOHGE\ÀH[XUDO\LHOGLQJDQGWKHFRGHSUHVFULEHG in such cases. VKHDUIRUFHVKDYHEHHQDPSOL¿HG$&, $VQRWHGSUHYLRXVO\ 14.5.5 Design for Flexure and Axial Force

6WUXFWXUDOZDOOVDUHGHVLJQHGIRUFRPELQHGÀH[XUHDQGD[LDOIRUFHVLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&, $&, $QLQWHUDFWLRQGLDJUDPIRUDZDOOFDQEHFRQVWUXFWHGXVLQJDVWUDLQFRPSDWLELOLW\DQDO\VLVEDVHGRQ WKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHVHFWLRQ-XVWOLNHLQWKHGHVLJQRIUHLQIRUFHGFRQFUHWHFROXPQVDOOWKHFRPbined factored axial force and bending moment points obtained from the applicable load combinations in ACI ChapWHUPXVWIDOOZLWKLQRURQWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPLQRUGHUIRUVWUHQJWKUHTXLUHPHQWVWREHVDWLV¿HG Openings in walls must be considered when determining the strength requirements of a special structural wall. Relatively large openings can dramatically change the overall behavior and it is important to understand how a wall will perform when one or more openings are introduced. Additional reinforcement around the openings is typically provided to counteract large tensile forces that may develop, especially at the corners. )RUZDOOVZLWKÀDQJHVOLNHWKRVHGHSLFWHGLQ)LJXUHDSRUWLRQRIWKHÀDQJHLVFRQVLGHUHGWREHHIIHFWLYHLQ UHVLVWLQJWKHHIIHFWVIURPFRPELQHGD[LDOIRUFHDQGEHQGLQJPRPHQW,QJHQHUDOWKHHIIHFWLYHZLGWKRIDÀDQJH GHSHQGVRQWKHPDJQLWXGHRIWKHD[LDOIRUFHRQWKHZDOOWKHDPRXQWRIODWHUDOGULIWHIIHFWLYHÀDQJHZLGWKLQFUHDVHV ZLWKLQFUHDVLQJGULIW DQGZKHWKHUWKHÀDQJHLVLQWHQVLRQRUFRPSUHVVLRQ,QOLHXRIDUDWLRQDODQDO\VLVWKDWWDNHV WKHVHYDULDEOHVLQWRDFFRXQW$&,FRQWDLQVDVLPSOHPHWKRGWRGHWHUPLQHWKHHIIHFWLYHÀDQJHZLGWKZKLFK is the total wall height above the section under consideration. Because LVLOOXVWUDWHGLQ)LJXUHZKHUH WKHHIIHFWLYHFRPSUHVVLRQÀDQJHZLGWKKDVOLWWOHHIIHFWRQWKHVWUHQJWKDQGGHIRUPDWLRQFDSDFLW\RIDZDOOWKHSURYLVLRQVLQ$&,DUHEDVHGRQWKHHIIHFWLYHWHQVLRQÀDQJHZLGWKRQO\,WLVDVVXPHGWKHHIIHFWLYHÀDQJHDQGWKH reinforcement in it fully contribute to the nominal strength of the wall section.

‫ݏ‬ଶ Τʹ

‫ݏ‬ଵ Τʹ ‡••‡”‘ˆ ቐ

 ݄௪ǡ௔௕௢௩௘ ΤͶ

‫ݏ‬ଵ 

‡••‡”‘ˆ ቐ

 ݄௪ǡ௔௕௢௩௘ ΤͶ

‫ݏ‬ଶ 

݄௪ǡ௔௕௢௩௘ ൌ –‘–ƒŽ™ƒŽŽŠ‡‹‰Š–ƒ„‘˜‡–Š‡•‡…–‹‘—†‡”…‘•‹†‡”ƒ–‹‘ Figure 14.35 Effective flange width for special structural walls.

14-46

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.5.6 Boundary Elements of Special Structural Walls Overview

The ends of a structural wall and the edges adjacent to openings in structural walls can be subjected to large compressive forces when the wall undergoes cyclic deformations caused by a design-level earthquake event. Special WUDQVYHUVHUHLQIRUFHPHQWPD\EHUHTXLUHGDWWKHVHORFDWLRQVWRFRQ¿QHWKHFRQFUHWHDQGWRUHVWUDLQWKHORQJLWXGLQDO reinforcement in the wall so buckling of the bars does not occur. Two design approaches for evaluating the need of special elements are given in ACI 18.10.6: (1) displacementEDVHGDSSURDFKDQG FRPSUHVVLYHVWUHVVDSSURDFK7KH¿UVWDSSURDFKLVDSSOLFDEOHWRVOHQGHUZDOOVRUZDOOSLHUV effectively continuous from the base of the structure to the top of the wall and are designed to have RQHFULWLFDOVHFWLRQIRUÀH[XUHDQGD[LDOIRUFHVZKLFKW\SLFDOO\RFFXUVDWWKHEDVHRIWKHZDOO 7KHFRPSUHVVLYH stress approach can be used to assess the need for special boundary elements for any structural wall. Displacement-Based Approach (ACI 18.10.6.2)

In the displacement-based approach, special transverse reinforcement must be provided in compression zones of VSHFLDOVWUXFWXUDOZDOOVZKHUHWKHVWUDLQDWWKHH[WUHPHFRPSUHVVLRQ¿EHUH[FHHGVDFULWLFDOYDOXHZKHQWKHZDOOLV which occurs at the top of the wall. According to ASCE/SEI subjected to 1.5 times the design displacement, where LVWKHGHÀHFWLRQDPSOL¿FDWLRQIDFWRUJLYHQLQ$6&(6(,7DEOH is 12.8.6, WKHGHÀHFWLRQDWWKHWRSRIWKHZDOOGHWHUPLQHGIURPDQDO\VLVZKHUHWKHFRGHSUHVFULEHGODWHUDOHDUWKTXDNHIRUFHV are applied over the height of the structure, and is the earthquake importance factor determined in accordance with ASCE/SEI 11.5.1. &RPSUHVVLRQ]RQHVPXVWEHUHLQIRUFHGZLWKVSHFLDOERXQGDU\HOHPHQWVZKHUH$&,(TXDWLRQD LVVDWLV¿HG (14.32)

where must be taken greater than or equal to 0.005. The lower limit of 0.005 on the ratio ¿HGWRUHTXLUHPRGHUDWHZDOOGHIRUPDWLRQFDSDFLW\IRUVWLIIEXLOGLQJV

is speci-

7KHQHXWUDOD[LVGHSWK LQ(TXDWLRQ FRUUHVSRQGVWRWKHODUJHVWQHXWUDOD[LVGHSWKFDOFXODWHGIRUWKH IDFWRUHGD[LDOIRUFHDQGQRPLQDOPRPHQWVWUHQJWKRIWKHZDOOZKHQLWLVGLVSODFHGLQWKHVDPHGLUHFWLRQDV In general, can be obtained from a strain compatibility analysis for each load combination that includes earthquake effects, considering sidesway to the left and to the right. The wall dimensions, the material properties, and the amount and distribution of the longitudinal reinforcement in the wall are required to perform such an analysis. at Equation (14.32) is derived from the relationships shown in Figure 14.36. It is assumed the displacement which is centered on the critical section of the wall at its the top of the wall is due entirely to the curvature, base. It is also assumed the length of the plastic hinge that forms at the base is equal to one-half the length of the Assuming small curvatures, Therefore, Rearranging wall, terms and rounding down the numerical constant in the denominator results in Equation (14.32). Where special boundary elements are required by Equation (14.32), the requirements in (1) and either (2) or (3) PXVWEHVDWLV¿HG 1. 6  SHFLDOERXQGDU\WUDQVYHUVHUHLQIRUFHPHQWPXVWH[WHQGYHUWLFDOO\DERYHDQGEHORZWKHFULWLFDOVHFWLRQDW and WKDWLVWKHDQWLFLSDWHGSODVWLFKLQJHOHQJWK H[FHSWDVSHUPLWWHGE\$&, least the greater of 18.10.6.4(j). 2. :LGWKRIWKHÀH[XUDOFRPSUHVVLRQ]RQH 3.

where

14-47

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

1.5ߜ௨ 

Special boundary elements

݄௪௖௦ 

κ௪ greater of ൝  ‫ܯ‬௨ Τ4ܸ௨

0.5κ௪ 

߶௨ 

ܸ௨  ‫ܯ‬௨  κ௪ 

ܾ

ܿ

ܿ െ 0.1κ௪ greater of ൝  ܿ Τ2

߶௨  ߝ௖௨ ൌ 0.003

Special boundary elements are required where

ܾ ൒ greater of

‫ۓ‬ ۖ

1.5ߜ௨ κ௪ ൒ ݄௖௦ 600ܿ

݄௨ Τ16 ‫כ‬

 ඥ0.025ܿκ௪ ‫۔‬ ۖ ‫ە‬12ᇳ where ݄௪ Τκ௪ ൒ 2.0 and ܿ Τκ௪ ൒ 3Τ8

* This requirement need not be satisfied if ACI 18.10.6.2ሺbሻሺiiiሻ is satisfied. Figure 14.36 Special boundary element requirements in accordance with ACI 18.10.6.2.

The term LQ LVWKHZDOOGLVSODFHPHQWFDSDFLW\DWWKHWRSRIWKHZDOODWDSHUFHQWORVVRIODWHUDOVWUHQJWK 7KXV UHTXLUHVWKHGULIWFDSDFLW\RIDZDOOWRH[FHHGWLPHVWKHGULIWGHPDQG7KHH[SUHVVLRQIRU LQ LV derived from the equation in (3) assuming Compressive Stress Approach (ACI 18.10.6.3)

7KHFRPSUHVVLYHVWUHVVDSSURDFKLQ$&,FDQEHXVHGWRDVVHVVWKHQHHGIRUVSHFLDOERXQGDU\HOHPHQWVIRU any structural wall. In this method, the wall is subjected to gravity loads and the maximum bending moments and VKHDUIRUFHVGXHWRHDUWKTXDNHHIIHFWVLQDJLYHQGLUHFWLRQVHH)LJXUH 7KHFRPELQHGFRPSUHVVLYHVWUHVVGXH to gravity loads and bending moments is calculated assuming a linearly elastic model and gross section properties

14-48

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

݄௪௖௦ 

’‡…‹ƒŽ„‘—†ƒ”›‡Ž‡‡–•

’‡…‹ƒŽ„‘—†ƒ”›‡Ž‡‡–•ƒ›„‡ †‹•…‘–‹—‡†™Š‡”‡݂௖௨ ൏ ͲǤͳͷ݂௖ᇱ 

ܲ௨  ܸ௨  ‫ܯ‬௨  κ௪ 

ܾ

ܿ

ܿ െ ͲǤͳκ௪ ‰”‡ƒ–‡”‘ˆ ൝  ܿ Τʹ

ܲ௨  ‫ܣ‬௚

‫ܯ‬௨ κ௪  ʹ‫ܫ‬௚

’‡…‹ƒŽ„‘—†ƒ”›‡Ž‡‡–•ƒ”‡”‡“—‹”‡†™Š‡”‡݂௖௨ ൌ

ܲ௨ ‫ܯ‬௨ κ௪ ൅ ൐ ͲǤʹ݂௖ᇱ  ‫ܣ‬௚ ʹ‫ܫ‬௚

݄௨ Τͳ͸  ܾ ൒ ‰”‡ƒ–‡”‘ˆ ቐ ͳʹᇳ ™Š‡”‡ ݄௪ Τκ௪ ൒ ʹǤͲƒ† ܿ Τκ௪ ൒ ͵Τͺ Figure 14.37 Special boundary element requirements in accordance with ACI 18.10.6.3.

RIWKHZDOOFRQVLGHULQJWKHHIIHFWLYHÀDQJHZLGWKUHTXLUHPHQWVLQ$&,IRUZDOOVZLWKÀDQJHVVHH)LJXUH  6SHFLDOERXQGDU\HOHPHQWVDUHUHTXLUHGDWWKHHQGVRIWKHZDOORUDWHGJHVDURXQGRSHQLQJVZKHUHWKHPD[Lexceeds mum compressive stress, (14.33)

The special boundary elements may be discontinued where the combined stress is less than

14-49

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Design and Detailing Requirements for Special Boundary Elements

7KHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGZKHUHVSHFLDOERXQGDU\HOHPHQWVDUH UHTXLUHGE\$&,RU$VXPPDU\RIWKHVHUHTXLUHPHQWVLVJLYHQLQ7DEOHDQGLQ)LJXUH for the case of a rectangular wall with (1) a perimeter rectilinear hoop and crossties where the web reinforcement LVGHYHORSHGE\VWDQGDUGKRRNVDWWKHHQGVDQG RYHUODSSLQJKRRSVDQGFURVVWLHVZKHUHWKHZHEUHLQIRUFHPHQW are the lengths of the hoop legs in the direction of analysis. It is LVGHYHORSHGE\KHDGV,QWKH¿JXUH and DVVXPHGWKHZDOOVHFWLRQVLQ)LJXUHDUHZLWKLQWKHSODVWLFKLQJHUHJLRQGH¿QHGLQ$&,E VRWKDWWKH web vertical reinforcement outside of the special boundary element must have lateral support in accordance with $&,L  Table 14.7 Design and Detailing Requirements for Special Boundary Elements ACI Section Number

Figure Number(s)

+RUL]RQWDOOHQJWKRIDVSHFLDOERXQGDU\HOHPHQW at each end of a wall or from the edge of an opening must be greater than or equal to the greater of and

D



LQFOXGLQJDÀDQJHLISUHV:LGWKRIWKHÀH[XUDOFRPSUHVVLRQ]RQH over where is the laterally unsupent, must be greater than or equal to SRUWHGKHLJKWRIDZDOORUSLHUDWWKHH[WUHPHFRPSUHVVLRQ¿EHU

E



F



)RUZDOOVZLWKÀDQJHVWKHERXQGDU\HOHPHQWPXVWLQFOXGHWKHHIIHFWLYHÀDQJH ZLGWKGH¿QHGLQ$&,DQGPXVWH[WHQGLQWRWKHZHEDWOHDVWLQ

G



Transverse reinforcement in the boundary element must satisfy the detailing UHTXLUHPHQWVRI$&,D WKURXJKG DQG$&,ZKLFKDUHDSSOLcable to columns in special moment frames, except the vertical spacing limit in $&,D PXVWEHRQHWKLUGRIWKHOHDVWGLPHQVLRQRIWKHERXQGDU\HOHPHQW7KHPD[LPXPYHUWLFDOVSDFLQJUHTXLUHPHQWVLQ$&,7DEOHE  for transverse reinforcement where special boundary elements are not required PXVWDOVREHVDWLV¿HG

H



of the laterally supported longiThe maximum center-to-center spacing, tudinal bars around the perimeter of the boundary element must be less than RUHTXDOWRWKHOHVVHURILQDQGWZRWKLUGVWKHWKLFNQHVVRIWKHERXQGDU\ element. Lateral support must be provided by a seismic hook of a crosstie or DFRUQHURIDKRRS7KHOHQJWKRIWKHKRRSOHJPXVWEHOHVVWKDQRUHTXDOWR WLPHVWKHWKLFNQHVVRIWKHFRQ¿QHGFRUH$GMDFHQWKRRSVPXVWRYHUODSDWOHDVW the lesser of 6 in. and two-thirds of the boundary element thickness.

I



Requirement

must be greater than or :LGWKRIWKHÀH[XUDOFRPSUHVVLRQ]RQH over HTXDOWRLQZKHUHDOOWKHIROORZLQJDUHVDWLV¿HG • the wall or wall pier is effectively continuous from the base of the structure to the top of the wall • WKHZDOORUZDOOSLHULVGHVLJQHGWRKDYHDVLQJOHFULWLFDOVHFWLRQIRUÀH[XUH and axial loads • •

(table continued on next page)

14-50

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.7 Design and Detailing Requirements for Special Boundary Elements (cont.) ACI Section Number

Figure Number(s)

Minimum transverse reinforcement must be in accordance with ACI Table J 

J



7KHVSHFL¿HGFRPSUHVVLYHVWUHQJWKRIWKHFRQFUHWHZLWKLQWKHWKLFNQHVVRIWKH ÀRRUV\VWHPPXVWEHDWOHDVWSHUFHQWRI of the wall.

K

ņ

Web vertical reinforcement must have lateral support provided by the corner of hoop or by a crosstie with seismic hooks at each end for a distance above and EHORZWKHFULWLFDOVHFWLRQVSHFL¿HGLQ$&,E 7KHKRRSVRUFURVVWLHV PXVWKDYHDGLDPHWHURIDWOHDVWWKDWVSHFL¿HGLQ$&,IRUWLHVDQGWKH YHUWLFDOVSDFLQJPXVWEHOHVVWKDQRUHTXDOWRLQ

L



M



N



Requirement

Where the critical section occurs at the base of a wall, special boundary eleof the ment transverse reinforcement must extend into the support at least is largest longitudinal reinforcement in the special boundary element where GHWHUPLQHGLQDFFRUGDQFHZLWK$&, 6SHFLDOERXQGDU\HOHPHQWWUDQVYHUVHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQ into footings, mats, or piles caps, unless a greater extension is required by ACI  +RUL]RQWDOUHLQIRUFHPHQWLQDZDOOZHEPXVWH[WHQGWRZLWKLQLQRIWKHHQG of the wall and it must be anchored to develop ZLWKLQWKHFRQ¿QHGFRUHRI the boundary element using standard hooks or heads. ,WLVSHUPLWWHGWRWHUPLQDWHWKHKRUL]RQWDOZHEUHLQIRUFHPHQWZLWKRXWDVWDQGDUG KRRNRUKHDGZKHUH WKHFRQ¿QHGERXQGDU\HOHPHQWKDVVXI¿FLHQWOHQJWK RIWKHKRUL]RQWDO WRGHYHORSWKHKRUL]RQWDOZHEUHLQIRUFHPHQWDQG  of the boundary element web reinforcement is less than or equal to WUDQVYHUVHUHLQIRUFHPHQWSDUDOOHOWRWKHKRUL]RQWDOZHEUHLQIRUFHPHQW

Design and Detailing Requirements Where Special Boundary Elements Are Not Required

(YHQWKRXJKVSHFLDOERXQGDU\HOHPHQWVPD\QRWEHUHTXLUHGE\$&,RUWUDQVYHUVHUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,PD\QHHGWREHSURYLGHGDWWKHHQGVRIDZDOO,QSDUWLFXODUWUDQVYHUVH UHLQIRUFHPHQWDSSOLFDEOHWRFROXPQVLQVSHFLDOPRPHQWIUDPHV>$&,D WKURXJKH @PXVWEHSURYLGHGRYHU WKHGLVWDQFHFDOFXODWHGLQDFFRUGDQFHZLWK$&,D DWDQ\ORFDWLRQRYHUWKHKHLJKWRIWKHZDOOZKHUHWKH Methods to determine for two longitudinal reinforcement ratio at the wall boundary, is greater than FDVHVDUHJLYHQLQ)LJXUH ORQJLWXGLQDOEDUVDWWKHHQGVRIWKHZDOODUHODUJHUWKDQWKHXQLIRUPO\GLVWULEXWHG ZHEORQJLWXGLQDOEDUVOHIWSDUWRIWKH¿JXUH DQG XQLIRUPO\GLVWULEXWHGZHEORQJLWXGLQDOEDUVRIWKHVDPHVL]H DQGVSDFLQJDUHSURYLGHGWKURXJKRXWWKHOHQJWKRIWKHZDOOULJKWSDUWRIWKH¿JXUH ,WKDVEHHQVKRZQWKDWZDOOV with relatively large amounts of longitudinal reinforcement at their ends can attract relatively large compressive forces; thus, the purpose of these provisions is to help prevent these bars from buckling.

14-51

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ

ܾ

ܾ௖ଶ 

ܾ௖ଶ 

Perimeter hoop 

൑6ԣ κଶ ൑ 2ܾ௖ଶ 

‫ܣ‬௦௛ 

൒κௗ௧ 

‫ݏ‬௢௩௘௥  ݄௫ 

κଵ ൑ 2ܾ௖ଶ 

ܾ௖ଵ 

‫ܣ‬௦௛ 

κ௕௘ 

൒κௗ௛ 

݄௫ 

κଵ ൌ ܾ௖ଵ ൑ 2ܾ௖ଶ 

κ௕௘ 

൑6ԣ





Overlapping hoops 

ܿ െ 0.1κ௪

Length of boundary element, κ௕௘

Width of flexural compression zone, ܾ

Horizontal spacing, ݄௫

Vertical spacing, ‫ݏ‬ Transverse reinforcement

κ௕௘ ൒ greater of ൝ ܿ Τ2

ܾ ൒ greater of

‫ۓ‬ ۖ

݄௨ Τ16

ඥ0.025ܿκ௪ where ACI 18.10.6.2 is used ‫۔‬ ۖ ᇳ ‫ە‬12 where ݄௪ Τκ௪ ൒ 2.0 and ܿ Τκ௪ ൒ 3Τ8 14ᇳ ݄௫ ൑ lesser of ൝ 2ܾΤ3

lesser of ܾΤ3 and κ௕௘ Τ3 ‫ۓ‬ ۖ ۖ lesser of 6݀௕ of the smallest long. bar and 6ԣ for Grade 60 bars ۖ ۖ lesser of 5݀௕ of the smallest long. bar and 6ԣ for Grade 80 bars ‫ ݏ‬൑ lesser of ‫۔‬ ۖlesser of 4݀௕ of the smallest long. bar and 6ԣ for Grade 100 bars ۖ ۖ 14 െ ݄௫ ۖ ᇳ ᇳ ‫ ە‬4 ൑ ‫ݏ‬௢ ൌ 4 ൅ ൬ 3 ൰ ൑ 6 ܾκ௕௘ ݂௖ᇱ ‫ۓ‬0.3‫ܾݏ‬௖ଶ ൬ െ 1൰ ܾ ܾ ݂ ௖ଵ ௖ଶ ௬௧ ۖ

Area, ‫ܣ‬௦௛

‫ܣ‬௦௛ ൌ greater of

‫۔‬ ݂ᇱ ۖ 0.09‫ܾݏ‬௖ଶ ௖ ݂௬௧ ‫ە‬ 6ᇳ

Hoop overlap, ‫ݏ‬௢௩௘௥

‫ݏ‬௢௩௘௥ ൒ lesser of ൝

 2ܾΤ3

Figure 14.38 Detailing requirements for special boundary elements with rectilinear hoops and crossties.

14-52

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ܾ

൑ 14ԣ

‫ܣ‬௦௕ 400 ൐ ‫ܣ‬௖௕ ݂௬

‫ݔ‬

ߩൌ

ܽ

‫ܣ‬௦௕ ൌ area of longitudinal reinforcement in ‫ܣ‬௖௕

൑ 14ԣ

‫ݏ‬௕ 

‫ݔ‬

ܾ

‫ܣ‬௖௕ ൌ ܾ‫ݏ‬௕ 

‫ܣ‬௖௕ ൌ ܾሺ2‫ ݔ‬൅ ܽሻ Notes:

1. Where ܸ௨ ൒ ߣඥ݂௖ᇱ ‫ܣ‬௖௩ , provide either of the following: a.Standard hooks at the ends of the horizontal reinforcement engaging the end edge reinforcement. b.U-stirrups spliced to the horizontal reinforcement with the same size and spacing as the horizontal reinforcement. 2. Size of transverse reinforcement must satisfy the requirements of ACI 25.7.2.2: ሺaሻ #3 bars for #10 and smaller longitudinal bars and ሺbሻ #4 bars for #11 and larger longitudinal bars. 3. Vertical spacing, ‫ݏ‬, of transverse reinforcement at the wall boundary must be in accordance with ACI Table 18.10.6.5ሺbሻ: Maximum Vertical Spacing of Transverse Reinforcement, ‫ݏ‬ Location

Within the greater of κ௪ and ‫ܯ‬௨ Τ4ܸ௨ above and below critical sections

Grade of Primary Flexural Reinforcing Bar 60

80

100

6݀௕ Lesser of ൝ 6ᇳ

5݀௕ Lesser of ൝ 6ᇳ

4݀௕ Lesser of ൝ 6ᇳ

6݀௕ 6݀௕ 8݀௕ Lesser of ൝ Lesser of ൝ Lesser of ൝ 8ᇳ 6ᇳ 6ᇳ ݀௕ ൌ diameter of the smallest primary flexural reinforcing bar Other locations

Figure 14.39 Transverse reinforcement requirements where special boundary elements are not required.

7UDQVYHUVHUHLQIRUFHPHQWLQWKLVFDVHPXVWH[WHQGRYHUWKHVDPHGLVWDQFHSUHVFULEHGLQ$&,D IRUZDOOV where special boundary elements are required. The maximum vertical spacing for the transverse reinforcement at WKHZDOOERXQGDU\DUHJLYHQLQ$&,7DEOHE VHH)LJXUH 

14-53

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Where in-plane shear forces are relatively large (that is, where ), standard hooks must be provided at WKHHQGVRIWKHKRUL]RQWDOZDOOUHLQIRUFHPHQWVRWKDWWKH\FDQEHHIIHFWLYHLQUHVLVWLQJWKHUHTXLUHGVKHDUIRUFH,Q lieu of providing hooks on these bars, U-stirrups can be provided, which enclose the edge reinforcement. The UVWLUUXSVPXVWKDYHWKHVDPHVL]HDQGVSDFLQJDVWKHKRUL]RQWDOUHLQIRUFHPHQWLQWKHZDOODQGPXVWEHVSOLFHGWRWKH KRUL]RQWDOZDOOEDUV Summary of Boundary Element Requirements for Special Structural Walls

A summary of the requirements for special structural walls with that are effectively continuous from WKHEDVHRIWKHVWUXFWXUHWRWKHWRSRIWKHZDOODQGDUHGHVLJQHGWRKDYHDVLQJOHFULWLFDOVHFWLRQIRUÀH[XUHDQGD[LDO ORDGVLVJLYHQLQ)LJXUH$&,DQG ,QFOXGHGDUHWKHWUDQVYHUVHUHLQIRUFHPHQW UHTXLUHPHQWVWKDWPXVWVDWLV¿HGDWIRXQGDWLRQVLQDFFRUGDQFHZLWK$&,M DQG

Ties not required Ties in accordance with ACI 18.10.6.5 ሺFigure 14.39ሻ

1.25݂௬ ሺtyp.ሻ

൑ ݄Τ2 ൒ κௗ based on 

Critical section

൒ 12ԣ

݄

൐ ݄Τ2

Special boundary elements ሺACI 18.10.6.2, 18.10.6.4; Figure 14.38ሻ

൒ greater of κ௪ and

‫ܯ‬௨  4ܸ௨

݄௪௖௦ 

ɏ൐

400  ݂௬

ɏ൑

400  ݂௬



Figure 14.40 Summary of requirements for special structural walls designed in accordance with ACI 18.10.6.2.

14-54

݂௖௨ ൐ ൑ ݄Τ2

Ties not required Ties in accordance with ACI 18.10.6.5 ሺFigure 14.39ሻ

1.25݂௬ ሺtyp.ሻ

݄

൒ 12ԣ

Critical section

൒ κௗ based on 

൐ ݄Τ2

Special boundary elements ሺACI 18.10.6.3, 18.10.6.4; Figure 14.38ሻ

400  ݂௬ ɏ൑ ݂௖௨ ൏ 0.15݂௖ᇱ and ɏ ൐ 0.2݂௖ᇱ 

See Note

൒ κௗ ሺtyp.ሻ

݄௪௖௦ 

400  ݂௬

݂௖௨ ൏ 0.15݂௖ᇱ and

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Note: For ݂௖௨ ൐ 0.2݂௖ᇱ : provide special boundary elements For ݂௖௨ ൏ 0.15݂௖ᇱ and ߩ ൐ 400Τ݂௬ : provide ties in accordance with ACI 18.10.6.5

Figure 14.41 Summary requirements for special structural walls designed in accordance with ACI 18.10.6.3.

6LPLODUO\DVXPPDU\RIWKHUHTXLUHPHQWVIRUVSHFLDOVWUXFWXUDOZDOOVGHVLJQHGLQDFFRUGDQFHZLWK$&,LV JLYHQLQ)LJXUH 14.5.7 Coupling Beams Overview

Coupling beams are beams connecting two structural walls together and are typically aligned vertically above openLQJVLQWKHZDOORYHULWVKHLJKWVHH)LJXUH :KHQSURSHUO\GHVLJQHGDQGGHWDLOHGVXFKEHDPVFDQSURYLGHDQ HI¿FLHQWPHDQVRIHQHUJ\GLVVLSDWLRQZKHQWKHVWUXFWXUHLVVXEMHFWHGWRHDUWKTXDNHJURXQGPRWLRQ

14-55

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

–”—…–—”ƒŽ™ƒŽŽ



݄

‘—’Ž‹‰ „‡ƒሺ–›’Ǥሻ

κ௡ 

Figure 14.42 Structural wall with coupling beams.

Because of geometric constraints, coupling beams usually have relatively large depth to span ratios. As such, the HQGVRIWKHEHDPVDUHW\SLFDOO\VXEMHFWHGWRODUJHLQHODVWLFURWDWLRQVDQGODUJHVKHDUIRUFHV6XI¿FLHQWVKHDUUHLQIRUFHPHQWDQGSURSHUGHWDLOLQJDQGFRQ¿QHPHQWRIDOOWKHUHLQIRUFHPHQWLQWKHEHDPDUHQHHGHGWRSUHYHQWVKHDU failure and to ensure ductility and energy dissipation. Tests have shown that adequate resistance can be achieved by XVLQJFRQ¿QHGGLDJRQDOUHLQIRUFHPHQWLQGHHSFRXSOLQJEHDPV Design and Detailing Requirements

7KHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVLQ$&,GHSHQGRQWKHFOHDUVSDQOHQJWKWRGHSWKUDWLR EHDPDQGWKHVKHDUGHPDQG$VXPPDU\RIWKHUHTXLUHPHQWVLVJLYHQLQ7DEOH

of the

Table 14.8 Design and Detailing Requirements for Coupling Beams Case

$&,

$&,

Requirements

• &  RXSOLQJEHDPVDUHGHVLJQHGDQGGHWDLOHGIRUÀH[XUHDQGVKHDULQDFFRUGDQFHZLWKWKH requirements in ACI 18.6 for beams in special moment frames. • Wall boundaries take the place of columns. • 7KHGLPHQVLRQDOOLPLWVLQ$&,E DQGF QHHGQRWEHVDWLV¿HGZKHUHDQDO\VLV shows that beams have adequate lateral stability. • The use of diagonal reinforcement in this case is not required because the beams are conVLGHUHGWREHWRRVKDOORZIRUWKHGLDJRQDOEDUVWREHHI¿FLHQW • The longitudinal reinforcement must be embedded into the wall boundaries in accordance ZLWKWKHSURYLVLRQVLQ$&,IRUGHYHORSPHQWRIVWUDLJKWEDUV

• Reinforcement must consist of two intersecting groups of diagonally placed bars symmetrical about the midspan of the beam unless it can be demonstrated that safety and stability are not compromised. • 7KHGLDJRQDOO\SODFHGEDUVDUHLQWHQGHGWRSURYLGHERWKWKHVKHDUDQGÀH[XUDOVWUHQJWKRI the beam. • 1RPLQDOVKHDUVWUHQJWKDQGUHLQIRUFHPHQWGHWDLOVDUHJLYHQLQ$&, (table continued on next page)

14-56

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.8 Design and Detailing Requirements for Coupling Beams (cont.) Case

Requirements

All other cases $&,

Coupling beams that are not in the other cases may be reinforced with either two intersecting groups of diagonally placed bars symmetrical about the midspan of the beam or in accordance with the longitudinal and transverse reinforcement requirements of ACI 18.6.3 WKURXJKIRUEHDPVLQVSHFLDOPRPHQWIUDPHVZLWKWKHZDOOERXQGDULHVDFWLQJDVWKH FROXPQV7KHFKRLFHEHWZHHQWKHVHUHLQIRUFHPHQWRSWLRQVOHDGLQJWRWKHPRVWHI¿FLHQW design typically depends on geometry and the magnitude of the factored shear force.

Coupling beams reinforced with two intersecting groups of diagonally placed bars symmetrical about the midspan RIWKHEHDPPXVWVDWLVI\WKHUHTXLUHPHQWVLQ$&,D DQGE DQGHLWKHU$&,F RUG 7KHUHTXLUHPHQWVLQ$&,IRUGHHSEHDPVQHHGQRWEHVDWLV¿HG The nominal shear strength of a coupling beam reinforced with two intersecting groups of diagonally placed bars is GHWHUPLQHGE\$&,(TXDWLRQ  (14.34) ƒ”•†‡˜‡Ž‘’‡†‹–‡•‹‘ˆ‘”ͳǤʹͷ݂௬  ‹‡‘ˆ•›‡–”› ‫ܣ‬௩ௗ ൒ ƒ”•‘–†‡˜‡Ž‘’‡† ‹–‡•‹‘ˆ‘”݂௬ 

ܸ௨ Τ߶ ʹ݂௬ •‹ ߙ

݀௕ 



݄

ߙ

‫ݏ‬



κ௡ 

ƒŽŽ„‘—†ƒ”› ”‡‹ˆ‘”…‡‡–

‘–‡ǣ Ž›’ƒ”–‘ˆ–Š‡”‡‹ˆ‘”…‡‡–‹••Š‘™ ‘‡ƒ…Š•‹†‡‘ˆ–Š‡Ž‹‡‘ˆ•›‡–”›ˆ‘”…Žƒ”‹–›

൑ ͳͶԣ

൑ ͳͶԣ

ͲǤͲͻ‫ܾݏ‬௖ ݂௖ᇱ Τ݂௬௧  ‫ܣ‬௦௛ ൒ ‰”‡ƒ–‡”‘ˆ ቐ ͲǤ͵‫ܾݏ‬௖ ൣ൫‫ܣ‬௚ Τ‫ܣ‬௖௛ ൯ െ ͳ൧ ݂௖ᇱ Τ݂௬௧ ܾଵ ൒ ܾ௪ Τͷ Ͷԣ ൑ ‫ݏ‬௢ ൌ Ͷ ൅ ൬

‘‰‹–—†‹ƒŽƒ†–”ƒ•˜‡”•‡”‡‹ˆ‘”…‡‡– ൒ ͲǤͲͲʹܾ௪ ‫ ݏ‡”‡Š™ݏ‬൑ ͳʹԣ ‹‹—Ͷ„ƒ”•‹ʹ‘”‘”‡Žƒ›‡”•

‫ ݏ‬൑ Ž‡••‡”‘ˆ ൞ ܾଶ ൒ ܾ௪ Τʹ ܾ௪ 

ͳͶ െ ݄௫ ൰ ൑ ͸ԣ ͵

͸݀௕  ܿ௖ 

‫ܣ‬௖௛ ൌ ܾଵ ܾଶ  ‫ܣ‬௚ ൌ ሺܾଵ ൅ ʹܿ௖ ሻ ൈ ሺܾଶ ൅ ʹܿ௖ ሻ

Ǧ ‫ܖܗܑܜ܋܍܁‬

Figure 14.43 Detailing requirements for coupling beams with confinement of individual diagonals.

14-57

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

where is the total area of reinforcement in each group of diagonals bars, is the angle between the diagonal is the gross area of the coupling beam. Each group bars and the longitudinal axis of the coupling beam, and of diagonal bars must have a minimum of four bars provided in two or more layers and the diagonal bars must be +HDGHGEDUVFDQEHXVHGWRVKRUWHQGHYHORSPHQWOHQJWKVDQG developed in tension in the wall boundary for to help alleviate congestion, if required. must be less than or equal to the design shear strength, where for The required shear strength, GLDJRQDOO\UHLQIRUFHGFRXSOLQJEHDPV$&, 0RPHQWUHVLVWDQFHLVDXWRPDWLFDOO\SURYLGHGE\WKHGLDJRQDO bars as well. 7ZRRSWLRQVDUHJLYHQUHJDUGLQJFRQ¿QHPHQWRIWKHGLDJRQDOEDUV,QWKH¿UVWRSWLRQHDFKJURXSRIGLDJRQDOEDUV LVFRQ¿QHGE\UHFWLOLQHDUWUDQVYHUVHUHLQIRUFHPHQWVDWLVI\LQJWKHUHTXLUHPHQWVRI$&,F 7KHVHUHTXLUHPHQWVDUHLOOXVWUDWHGLQ)LJXUH7KHWUDQVYHUVHUHLQIRUFHPHQWDURXQGWKHGLDJRQDOEDUVPXVWKDYHRXWWRRXW in the direction parallel to and along the other sides where is the web dimensions of at least must be determined in each direcwidth of the coupling beam. The required area of transverse reinforcement, Equations to tion where LVWKHRXWWRRXWFURVVVHFWLRQDOGLPHQVLRQRIWKHFRQ¿QHGFRUHSHUSHQGLFXODUWR and DUHJLYHQLQ)LJXUHEDVHGRQWKHUHTXLUHPHQWVLQ$&,F $PLQLPXPDPRXQW determine of transverse and longitudinal reinforcement must be provided around the beam section. The longitudinal reinforcePHQWVKRXOGQRWEHIXOO\GHYHORSHGLQWRWKHZDOOVRWKDWLWGRHVQRWGHYHORSVLJQL¿FDQWWHQVLOHVWUHVVHV ƒ”•†‡˜‡Ž‘’‡†‹–‡•‹‘ˆ‘”ͳǤʹͷ݂௬  ‹‡‘ˆ•›‡–”› ‫ܣ‬௩ௗ ൒ ƒ”•‘–†‡˜‡Ž‘’‡† ‹–‡•‹‘ˆ‘”݂௬ 

ܸ௨ Τ߶ ʹ݂௬ •‹ ߙ

݀௕ 



݄

ߙ



͸ᇳ  ‫ ݏ‬൑ Ž‡••‡”‘ˆ ൝ ͸݀௕ κ௡ 

ƒŽŽ„‘—†ƒ”› ”‡‹ˆ‘”…‡‡–

‘–‡ǣ Ž›’ƒ”–‘ˆ–Š‡”‡‹ˆ‘”…‡‡–‹••Š‘™ ‘‡ƒ…Š•‹†‡‘ˆ–Š‡Ž‹‡‘ˆ•›‡–”›ˆ‘”…Žƒ”‹–›

ܾ௖ଵ 

൑ ͺᇳ ሺ–›’Ǥሻ

൑ ͺᇳ ሺ–›’Ǥሻ

ͲǤͲͻ‫ܾݏ‬௖ ݂௖ᇱ Τ݂௬௧  ‫ܣ‬௦௛ ൒ ‰”‡ƒ–‡”‘ˆ ቐ ͲǤ͵‫ܾݏ‬௖ ൣ൫‫ܣ‬௚ Τ‫ܣ‬௖௛ ൯ െ ͳ൧ ݂௖ᇱ Τ݂௬௧ ‫ܣ‬௖௛ ൌ ܾ௖ଵ ܾ௖ଶ  ‫ܣ‬௚ ൌ ܾ௪ ݄

ܾ௖ଶ 

‹‹—Ͷ„ƒ”•‹ʹ‘”‘”‡Žƒ›‡”•

ܾ௪  ‫ܖܗܑܜ܋܍܁‬ Ǧ

Figure 14.44 Detailing requirements for coupling beams with confinement of the entire cross-section.

14-58

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QWKHVHFRQGRSWLRQWUDQVYHUVHUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,G LVSURYLGHGIRUWKHHQWLUHFURVV VHFWLRQRIWKHFRXSOLQJEHDPVHH)LJXUH 7KLVRSWLRQKHOSVIDFLOLWDWHSODFHPHQWRIWKHUHLQIRUFLQJEDUVLQ must be WKH¿HOGHVSHFLDOO\DWWKHORFDWLRQZKHUHWKHGLDJRQDOEDUVLQWHUVHFWHDFKRWKHU/LNHLQWKH¿UVWRSWLRQ and RIWKHFRQ¿QHGFRUH determined in each direction using the out-to-out dimensions 5HJDUGOHVVRIWKHW\SHRIFRXSOLQJEHDPXWLOL]HGDVSDUWRIWKH6)56LWLVLPSRUWDQWWRHQVXUHWKDWDOOWKHUHTXLUHG reinforcement can be accommodated within the section and the wall boundary elements. In the case of diagonally UHLQIRUFHGFRXSOLQJEHDPVDPLQLPXPZLGWKRILQLVVXJJHVWHG 14.5.8 Wall Piers $VQRWHGSUHYLRXVO\DZDOOSLHULVDYHUWLFDOZDOOVHJPHQWZLWKLQDVWUXFWXUDOZDOOERXQGHGKRUL]RQWDOO\E\WZR RSHQLQJVRUE\DQRSHQLQJDQGDQHGJHDQGVDWLVI\LQJWKHIROORZLQJ KRUL]RQWDOOHQJWKWRZDOOWKLFNQHVV DQG FOHDUKHLJKWWRKRUL]RQWDOOHQJWK VHH)LJXUH 7KHVHPHPEHUVDUHHVVHQtially columns; however, the dimensions do not satisfy the requirements of columns in special moment frames. :DOOSLHUVDUHGHVLJQHGXVLQJWKHUHTXLUHPHQWVIRUYHUWLFDOZDOOVHJPHQWVLQ6HFWLRQRIWKLVSXEOLFDWLRQ and the following additional design and detailing requirements applicable for columns in special moment frames:  $&,ORQJLWXGLQDOUHLQIRUFHPHQW  $&,WUDQVYHUVHUHLQIRUFHPHQW DQG $&,VKHDU strength). The joint faces are taken at the top and bottom of the clear height of the wall pier, Alternatively, wall piers with $&,D WKURXJKI 

are permitted to be designed and detailed according to the provisions of

is equal to the lesser of the following: (1) the shear force corresponding to the • The factored shear force, DWERWKHQGVRIWKHSLHUDQG WKHVKHDUIRUFHGHWHUPLQHG GHYHORSPHQWRIWKHSUREDEOHÀH[XUDOVWUHQJWK from analysis using code-prescribed earthquake forces multiplied by the overstrength factor, • The design shear strength, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,IRUVWUXFWXUDOZDOOVDQGPXVWEH greater than or equal to • 7UDQVYHUVHUHLQIRUFHPHQWLVUHTXLUHGWREHKRRSV+RZHYHUZKHUHRQHFXUWDLQRIVKHDUUHLQIRUFHPHQWLVSURYLGVLQJOHOHJKRUL]RQWDOVKHDUUHLQIRUFHHGZKLFKDFFRUGLQJWR$&,LVSHUPLWWHGRQO\LI ZLWKGHJUHHEHQGVDWHDFKHQGHQJDJLQJWKHZDOOSLHUERXQGDU\ORQJLWXGLQDOUHLQIRUFHment parallel to ment may be used. • The vertical spacing of the transverse reinforcement is limited to 6 in. • 7KHWUDQVYHUVHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQDERYHDQGEHORZWKHFOHDUKHLJKWRIWKHZDOOSLHU • 6SHFLDOERXQGDU\HOHPHQWVDWWKHHQGVRIWKHSLHUPXVWEHSURYLGHGLIUHTXLUHGE\$&, )RUZDOOSLHUVORFDWHGDWWKHHGJHRIZDOODVLOOXVWUDWHGLQ$&,)LJXUH5KRUL]RQWDOUHLQIRUFHPHQWLVUHTXLUHG in the adjacent wall segments above and below the wall pier to transfer the design shear force from the wall pier into the adjacent wall segments. The required length of the reinforcement into the adjacent wall is equal to the greater of the development length of the bars in tension and the shear strength of the wall segment. In the latter case, the total in the wall pier divided by the design unit shear strength embedment length is equal to the factored shear force in the adjacent wall. 14.5.9 Ductile Coupled Structural Walls

Ductile coupled structural walls is a type of SFRS where limits are given for the aspect ratios of the structural walls and coupling beams and for the development lengths so as to induce an energy dissipation mechanism associated with inelastic deformation reversal of the coupling beams. To achieve this intended behavior, the following requirePHQWVPXVWEHVDWLV¿HG$&,  PXVWEHJUHDWHUWKDQRUHTXDOWR • The height-to-length aspect ratio of an individual wall, • $QLQGLYLGXDOZDOOPXVWVDWLVI\WKHDSSOLFDEOHUHTXLUHPHQWVRI$&,IRUVSHFLDOVWUXFWXUDOZDOOV

14-59

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• At all levels of the building, the length-to-height aspect ratios of the coupling beams, must be greater WKDQRUHTXDOWR LQDWOHDVWSHUFHQWRIWKHOHYHOVRIWKHEXLOGLQJ • $OOFRXSOLQJEHDPVDWDÀRRUOHYHOPXVWKDYH • 7KHGHYHORSPHQWOHQJWKUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HGDWERWKHQGVRIDOOFRXSOLQJEHDPV 14.5.10 Construction Joints

$FFRUGLQJWR$&,FRQVWUXFWLRQMRLQWVLQVSHFLDOVWUXFWXUDOZDOOVDUHWREHVSHFL¿HGLQDFFRUGDQFHZLWK $&,7KHKDUGHQHGFRQFUHWHDWWKHMRLQWPXVWEHFOHDQIUHHRIODLWDQFHDQGLQWHQWLRQDOO\URXJKHQHGWRDIXOO DPSOLWXGHRIóLQ>$&,7DEOHE @ 14.5.11 Discontinuous Walls

For special structural walls supported on columns instead of a foundation, the columns must be designed and GHWDLOHGLQDFFRUGDQFHZLWK$&,$&, 5HTXLUHPHQWVIRUWKHVHFROXPQVDUHJLYHQLQ6HFWLRQ RIWKLVSXEOLFDWLRQ

14.6 Diaphragms 14.6.1 Overview

Diaphragms and collectors that are part of the SFRS in buildings assigned to SDC D through F must be designed DQGGHWDLOHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&,'LDSKUDJPGHVLJQIRUFHVDUHSUHVFULEHGLQ$6&(6(, DQGFROOHFWRUVPXVWEHGHVLJQHGWRUHVLVWWKHPD[LPXPRIWKHWKUHHIRUFHVJLYHQLQ$6&(6(, VHH5HIHUHQFH 7KHGLDSKUDJPPRGHOLQJDQGDQDO\VLVPHWKRGVSUHVHQWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ can be used to determine the in-plane design bending moments, shear forces, and axial forces in a diaphragm. Information on how to determine the required chord, shear, and shear transfer reinforcement in a diaphragm is given LQ6HFWLRQRIWKLVSXEOLFDWLRQDQGLQ5HIHUHQFH 14.6.2 Minimum Thickness

'LDSKUDJPVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRWKDWDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG0HWKRGVWRGHWHUPLQHPLQLPXPVODEWKLFNQHVVEDVHGRQVHUYLFHDELOLW\UHTXLUHPHQWVRXWRISODQHVWUHQJWK UHTXLUHPHQWVDQGLQSODQHVWUHQJWKUHTXLUHPHQWVDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ $LQPLQLPXPWKLFNQHVVLVSUHVFULEHGLQ$&,IRUFRQFUHWHVODEVVHUYLQJDVGLDSKUDJPV,WLVLPSRUWDQWWR QRWHWKDW¿UHUHVLVWDQFHUHTXLUHPHQWVPXVWDOVREHFRQVLGHUHGZKHQVHOHFWLQJDPLQLPXPVODEWKLFNQHVV7\SLFDOO\ WKHVODEWKLFNQHVVUHTXLUHGWRVDWLVI\VWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVLVDGHTXDWHIRU¿UHUHVLVWDQFH 14.6.3 Reinforcement Minimum Reinforcement

The minimum reinforcement that must be provided in diaphragms corresponds to the minimum shrinkage and temSHUDWXUHUHLQIRUFHPHQWLQ$&,ZKLFKLVHTXDOWRWLPHVWKHJURVVFRQFUHWHDUHD$&, 7KH maximum spacing of the reinforcement is 18 in., which is intended to control the width of inclined cracks that may form in the diaphragm. Where reinforcement is required for shear strength, the reinforcement must be continuous and must be distributed uniformly across the shear plane. Development and Splices

Tension development lengths and lap splice lengths in diaphragms and collectors are to be determined in accordance ZLWK$&,&KDSWHU5HGXFWLRQLQGHYHORSPHQWRUODSVSOLFHOHQJWKVEDVHGRQH[FHVVUHLQIRUFHPHQWLQDFFRUGDQFH ZLWK$&,LVQRWSHUPLWWHGIRUPHPEHUVRIWKH6)56LQEXLOGLQJVDVVLJQHGWR6'&&'(RU)>$&, H @

14-60

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7\SHVSOLFHVLQDFFRUGDQFHZLWK$&,DUHUHTXLUHGZKHUHPHFKDQLFDOVSOLFHVRQ*UDGHUHLQIRUFHPHQW DUHXVHGWRWUDQVIHUIRUFHVEHWZHHQWKHGLDSKUDJPDQGWKHYHUWLFDOHOHPHQWVRIWKH6)56$&, ,WLVQRW SHUPLWWHGWRPHFKDQLFDOO\VSOLFH*UDGHDQG*UDGHUHLQIRUFHPHQWIRUWKLVDSSOLFDWLRQ Collectors

&ROOHFWRUVPXVWEHGHVLJQHGWRUHVLVWWKHFRPELQHGHIIHFWVRIÀH[XUHVKHDUDQGD[LDOFRPSUHVVLRQDQGWHQVLRQ forces caused by the combination of gravity and earthquake load effects. Tension and compression requirements DUHJLYHQLQ$&,DQGUHVSHFWLYHO\7RKHOSFRQWUROFUDFNLQJRYHUWKHOHQJWKRIDFROOHFWRUWKH longitudinal reinforcement must be proportioned such that the average tensile stress over the following lengths in where LVOLPLWHGWRSVLD WKHOHQJWKEHWZHHQWKHHQGRIDFROOHFWRUDQG (a) or (b) does not exceed the location where transfer of load to the vertical element of the SFRS begins or (b) the length between two vertical HOHPHQWVRIWKH6)56$&, 1RWHWKDWWKHFDOFXODWLRQRIWKHDYHUDJHWHQVLOHVWUHVVDORQJWKHOHQJWKLVQRW QHFHVVDU\LIWKHFROOHFWRULVGHVLJQHGXVLQJSVLIRU HYHQLI*UDGHUHLQIRUFHPHQWLVVSHFL¿HG>ZKLFKLV SHUPLWWHGLQDFFRUGDQFHZLWK$&,7DEOHD @ transverse reinforcement conAt any section in a collector where the combined compressive stress exceeds IRUPLQJWR$&,D WKURXJKH DQG$&,IRUFROXPQVRIVSHFLDOPRPHQWIUDPHVPXVWEHSURYLGHG ZLWKWKHH[FHSWLRQWKDWWKHVSDFLQJOLPLWRI$&,D PXVWEHRQHWKLUGWKHOHDVWGLPHQVLRQRIWKHFROOHFWRU $&, 7KHFRPELQHGFRPSUHVVLYHVWUHVVLVFDOFXODWHGXVLQJWKHORDGFRPELQDWLRQVLQ$&,&KDSWHUVHH 6HFWLRQRIWKLVSXEOLFDWLRQ DQGDOLQHDUO\HODVWLFPRGHOEDVHGRQJURVVVHFWLRQSURSHUWLHV7KHUHTXLUHGDPRXQW RIWUDQVYHUVHUHLQIRUFHPHQWLVJLYHQLQ$&,7DEOHLWLVSHUPLWWHGWRGLVFRQWLQXHWKLVUHLQIRUFHPHQWZKHUH ,QFDVHVZKHUHWKHIRUFHVKDYHEHHQDPSOL¿HGE\WKHRYHUthe combined compressive stress is less than the limits of and are to be increased to and respectively. To avoid trigstrength factor, JHULQJWKHUHTXLUHPHQWVIRUWUDQVYHUVHUHLQIRUFHPHQWWKHFROOHFWRUFDQEHVL]HGVRWKDWWKHFRPSUHVVLYHVWUHVVLVOHVV that is, where is the gross area of the collector and is the maximum factored than has been used to determine the forces, axial compressive force in the collector. Where 'HWDLOLQJUHTXLUHPHQWVDWVSOLFHDQGDQFKRUDJH]RQHORFDWLRQVIRUWKHORQJLWXGLQDOUHLQIRUFHPHQWLQFROOHFWRUV DUHJLYHQLQ$&,7KHUHTXLUHPHQWVLQHLWKHU$&,D RUE QHHGWREHVDWLV¿HGQRWERWK7KH purpose of these requirements is to reduce the possibility of longitudinal bar buckling and to provide adequate bar GHYHORSPHQWLQWKHVHUHJLRQV7KHPLQLPXPDUHDRIWUDQVYHUVHUHLQIRUFHPHQWLQ$&,E FRUUHVSRQGVWR WKDWIRUEHDPVLQ$&,7DEOHDWVHFWLRQVZKHUHWKHSURYLVLRQVRI$&,GRQRWJRYHUQ Detailing requirements for diaphragms and collectors with rectilinear transverse reinforcement are given in Figure  Elements of a structural diaphragm system subjected primarily to axial forces and used to transfer diaphragm shear RUÀH[XUDOIRUFHVDURXQGRSHQLQJVRURWKHUGLVFRQWLQXLWLHVPXVWVDWLVI\WKHUHTXLUHPHQWVLQ$&,DQG IRUFROOHFWRUV$&, $QH[DPSOHRIDQHOHPHQWDGMDFHQWWRDQRSHQLQJZKHUHWKHVHUHTXLUHPHQWVPXVWEHVDWLV¿HGLVJLYHQLQ$&,)LJXUH5 14.6.4 Flexural Strength

'LDSKUDJPVDUHGHVLJQHGIRUÀH[XUHXVLQJWKHVDPHDVVXPSWLRQVXVHGLQWKHGHVLJQRIEHDPVFROXPQVDQGZDOOV LQFOXGLQJWKHDVVXPSWLRQWKDWVWUDLQVYDU\OLQHDUO\DORQJWKHGHSWKRIWKHGLDSKUDJP$&, ,QSDUWLFXODU WKHÀH[XUDOVWUHQJWKLVFDOFXODWHGXVLQJWKHSURYLVLRQVLQ$&,&KDSWHUDQGWKHUHTXLUHGVWUHQJWKLVGHWHUPLQHG XVLQJWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHU$GGLWLRQDOLQIRUPDWLRQFDQEHIRXQGLQ&KDSWHURIWKLV publication. 14.6.5 Shear Strength

The nominal shear strength, $&,(TXDWLRQ 

RIDGLDSKUDJPLVGHWHUPLQHGE\$&,(TXDWLRQ ZKLFKLVWKHVDPHDV (14.35)

14-61

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Minimum reinforcement in accordance with ACI 24.4 ൒ κௗ in accordance with ACI 25.4.2 Collector element ሺtyp.ሻ

‫ ݏ‬൑ 18ԣ

൒ κௗ in accordance with ACI 25.5.2 A

A

‫ ݏ‬൑ 18ԣ Average tensile stress ൑ ߶݂௬ (typ.)

spacing ൒ greater of ൝

3݀௕ 1.5ԣ 2.5݀௕ 2ԣ

‫ݔ‬௜

൒ 2ԣ

Clear cover ൒ greater of ൝

݄

spacing ൒ greater of ൝

3݀௕

‫ݔ‬௜

1.5ԣ Transverse reinforcement, ‫ܣ‬௦௛

Other reinforcement not shown for clarity

‫ݔ‬௜ ܾ௪  Section A-A

Compressive stress*

Transverse reinforcement, ࡭࢙ࢎ

Spacing, ࢙ ሺlesser of ݄ and ܾ௪ ሻΤ3 ‫ۓ‬ ۖ ۖ 6݀௕ for Grade 60 reinforcement

൐ 0.2݂௖ᇱ

‫ܣ‬௦௛ ൒ 0.09‫ܾݏ‬௖ ݂௖ᇱ Τ݂௬௧ 

‫ ݏ‬൑ lesser of

‫ ۔‬5݀ for Grade 80 reinforcement ۖ ௕ ۖ ‫ ە‬4ԣ ൑ ‫ݏ‬௢ ൌ 4 ൅ ሾሺ14 െ ݄௫ ሻΤ3ሿ ൑ 6ԣ

݄௫ ൌ maximum of ‫ݔ‬௜ ൑ 14ԣ

൏ 0.15݂௖ᇱ

0.75ඥ݂௖ᇱ ൫ܾ௪ ‫ݏ‬Τ݂௬௧ ൯ ‫ܣ‬௦௛ ൒ greater of ቐ 50 ܾ௪ ‫ݏ‬Τ݂௬௧

‫ ݏ‬determined in accordance with ACI 22.5 ൑ maximum spacing in ACI Table 9.7.6.2.2

*Where design forces have been amplified by ȳ௢, limits of 0.2݂௖ᇱ and 0.15݂௖ᇱ are increased to 0.5݂௖ᇱ and 0.4݂௖ᇱ,”‡•’‡…–‹˜‡Ž›Ǥ

Figure 14.45 Detailing requirements for diaphragms and collectors.

In this equation, is the gross area of the diaphragm and is limited to the thickness of the diaphragm times the width of the diaphragm in the direction of analysis; this corresponds to the gross area of the deep beam that forms that contributes to is the reinforcement parallel to the in-plane shear. the diaphragm. The reinforcement ratio, If required for shear strength, shear reinforcement for diaphragms must be in addition to reinforcement in the slab UHTXLUHGWRUHVLVWRWKHUORDGHIIHFWV$&, 5HLQIRUFHPHQWGHVLJQHGWRUHVLVWVKULQNDJHDQGWHPSHUDWXUHORDG effects is permitted to also resist in-plane shear forces in the diaphragm. 14-62

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$FFRUGLQJWR$&,WKHVWUHQJWKUHGXFWLRQIDFWRU for shear in diaphragms must not exceed the minimum used in the shear design of the vertical elements of the SFRS connected to the diaphragm. For example, in a reinforced concrete building with nonslender walls where the shear strength of the walls has been determined using the shear strength of the diaphragm must be determined using 14.6.6 Construction Joints

$FFRUGLQJWR$&,FRQVWUXFWLRQMRLQWVLQGLDSKUDJPVDUHWREHVSHFL¿HGLQDFFRUGDQFHZLWK$&, The hardened concrete at the joint must be clean, free of laitance, and intentionally roughened to a full amplitude of óLQ>$&,7DEOHE @

14.7 Foundations 14.7.1 Overview

Requirements for foundations supporting buildings assigned to SDC D, E, or F are given in ACI 18.13. Provisions are provided for footings, foundation mats, pile caps, grade beams, slabs-on-ground, foundation seismic ties, piles, SLHUVDQGFDLVVRQV,WLVGHVLUDEOHIRUIRXQGDWLRQVQRWWRXQGHUJRDQ\VLJQL¿FDQWLQHODVWLFUHVSRQVHZKHQVXEMHFWHG WRVWURQJJURXQGPRWLRQEHFDXVHUHSDLURIIRXQGDWLRQVLVHLWKHUGLI¿FXOWRULPSRVVLEOH$&,5GLVFXVVHV methods that can be implemented to achieve this goal. is permit$FFRUGLQJWRWKHVHFRQGH[FHSWLRQLQ$6&(6(,WKHYHUWLFDOVHLVPLFORDGHIIHFW WHGWREHWDNHQDV]HURLQ$6&(6(,(TXDWLRQ ZKHQGHWHUPLQLQJGHPDQGVRQWKHVRLOVWUXFWXUHLQWHUIDFHRI foundations. This equation is applicable when gravity and earthquake effects counteract, and is used when checking overturning resistance of a footing. $6&(6(,SHUPLWVDVWUXFWXUHWREH¿[HGDWLWVEDVHIRUSXUSRVHVRIGHWHUPLQLQJHDUWKTXDNHIRUFHV7KLV assumption is generally conservative because considering soil-structure interaction may reduce the effects of the ground motion, increase the period of the building, or increase damping, any of which can reduce the earthquake effects on the structure except, possibly, for the lateral displacements. A geotechnical report provides important information on allowable soil bearing capacities under gravity (static) and JUDYLW\SOXVODWHUDOVWDWLFSOXVWUDQVLHQW ORDGLQJFRHI¿FLHQWRIIULFWLRQIRUVOLGLQJUHVLVWDQFHSDVVLYHUHVLVWDQFH SUHVVXUHYHUWLFDODQGKRUL]RQWDOPRGXOXVRIVXEJUDGHUHDFWLRQDQWLFLSDWHGGLIIHUHQWLDODQGWRWDOVHWWOHPHQWVDQG DQ\SRWHQWLDOJHRORJLFDODQGVHLVPLFKD]DUGVVXFKDVVRLOOLTXHIDFWLRQ WRQDPHDIHZ$FOHDUXQGHUVWDQGLQJRIWKH dynamic properties of the soil is essential in the design of any foundation system for buildings assigned to SDC D through F. 14.7.2 Footings, Foundation Mats, and Pile Caps

Longitudinal reinforcement of columns and structural walls that are part of the SFRS must be fully developed for WHQVLRQLQWRWKHIRRWLQJVPDWVDQGSLOHFDSVVXSSRUWLQJWKHP$&, :KHUHVXSSRUWHGPHPEHUVFROXPQV DQGZDOOV DUHDVVXPHGWREH¿[HGDWWKHWRSRIWKHIRXQGDWLRQWHVWVKDYHGHPRQVWUDWHGWKDWORQJLWXGLQDOUHLQIRUFHPHQWGHYHORSHGLQWRWKHIRXQGDWLRQIURPWKHVXSSRUWHGPHPEHUPXVWKDYHGHJUHHKRRNVWXUQHGLQZDUGVWRZDUGV WKHD[LVRIWKHPHPEHULQRUGHUIRUWKHMRLQWWREHDEOHWRUHVLVWÀH[XUHDWWKLVORFDWLRQ$&,  Requirements for columns or boundary elements of special structural walls located near the edge of the foundation DUHJLYHQLQ$&,7KHVHUHTXLUHPHQWVDUHLOOXVWUDWHGLQ)LJXUHVDQGIRUDIRXQGDWLRQVXSSRUWLQJDVWUXFWXUDOZDOODQGLQ)LJXUH Flexural reinforcement must be provided in the top of the foundation where uplift forces are generated in columns DQGERXQGDU\HOHPHQWVRIVSHFLDOVWUXFWXUDOZDOOV$&, 7KHUHLQIRUFHPHQWPXVWEHGHVLJQHGIRUWKHDSSOLFDEOHORDGFRPELQDWLRQVLQ$&,&KDSWHUDQGPXVWEHJUHDWHUWKDQRUHTXDOWRWKHDSSOLFDEOHPLQLPXPUHLQIRUFHPHQWIRURQHZD\VODEVLQ$&,RUIRUEHDPVLQ$&, $VXPPDU\RIWKHSURYLVLRQVLQWKLVVHFWLRQLVLOOXVWUDWHGLQ)LJXUH 14-63

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete



Longitudinal reinforcement fully developed for tension

Column or boundary element of special structural wall

ܿ

Top reinforcement to resist tension uplift forces ‫ܣ‬௦ ൒ minimum reinforcement in accordance with ACI 7.6.1 or 9.6.1 Footing, foundation mat, or pile cap ሺother reinforcement not shown for clarityሻ

݄

For ܿ ൑ ݄/2, provide transverse reinforcement in accordance with ACI 18.7.5.2 through 18.7.5.4. Transverse reinforcement must extend into the footing, mat, or pile cap a length equal to the development length calculated for ݂௬ in tension.

90-deg hooks with free ends of bars oriented towards the center of the column or boundary element for fixed-end condition Figure 14.46 Requirements for footings, foundation mats, and pile caps

14.7.3 Grade Beams and Slabs-on-ground

Grade beams and beams that are part of a mat foundation supporting columns that are part of the SFRS must be designed and detailed in accordance with ACI 18.6 for beams in special moment frames (ACI 18.13.3.1; see Section RIWKLVSXEOLFDWLRQ  A slab-on-ground not subjected to earthquake effects is generally considered nonstructural, and ACI 318 does not govern its design and construction unless the slab transmits vertical load or lateral forces from other portions of the VWUXFWXUHWRWKHVRLO$&, +RZHYHUIRUVWUXFWXUHVDVVLJQHGWR6'&'(RU)DVODERQJURXQGLVRIWHQSDUW RIWKH6)56DFWLQJDVDGLDSKUDJPWKDWKROGVWKHVWUXFWXUHWRJHWKHUDWWKHJURXQGOHYHOWKHUHE\PLQLPL]LQJWKHHIfects of out-of-phase ground motion that may occur over the footprint of the structure. In such cases, it must be deVLJQHGDQGGHWDLOHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVRI$&,IRUGLDSKUDJPV$&,VHH6HFWLRQ of this publication). Construction documents must clearly indicate the slabs-on-ground that are part of the SFRS so as to prohibit saw cutting of the slab. 14.7.4 Foundation Seismic Ties Overview

Individual pile caps, piers, or caissons must be interconnected by foundation seismic ties in orthogonal directions ZKHUHHTXLYDOHQWUHVWUDLQWLVQRWSURYLGHGE\RWKHUPHDQV$&, 7KHSXUSRVHRIWKLVUHTXLUHPHQWLVWR PLQLPL]HWKHPRYHPHQWRIWKHVXSSRUWHGPHPEHUUHODWLYHWRWKHPRYHPHQWRIWKHIRXQGDWLRQ ,QGLYLGXDOVSUHDGIRRWLQJVIRXQGHGRQ6LWH&ODVV(RU)VRLOVHH$6&(6(,&KDSWHU PXVWEHLQWHUFRQQHFWHG E\IRXQGDWLRQVHLVPLFWLHV$&, 7KHWLHVLQWHUFRQQHFWWKHIRRWLQJVVRWKDWWKH\FDQDFWDVDXQLWWKHUHE\ PLQLPL]LQJWKHPRYHPHQWRIWKHLQGLYLGXDOO\VXSSRUWHGPHPEHUVUHODWLYHWRWKHPRYHPHQWRIWKHIRRWLQJV Design and Detailing Requirements

'HVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUIRXQGDWLRQVHLVPLFWLHVDUHJLYHQLQ$&,DQG7LHVDUH times the greater of the required to have a design strength in compression or tension greater than or equal to pile cap or column factored dead load plus factored live load occurring on either end of the tie. This requirement QHHGQRWEHVDWLV¿HGLILWFDQEHGHPRQVWUDWHGWKDWHTXLYDOHQWUHVWUDLQWLVSURYLGHGE\DWOHDVWRQHRIWKHZD\VLQ$&, D WKURXJKG 

14-64

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'LPHQVLRQDODQGPLQLPXPFORVHGWLHUHTXLUHPHQWVIRUJUDGHEHDPVGHVLJQHGDVKRUL]RQWDOIRXQGDWLRQVHLVPLFWLHV EHWZHHQSLOHFDSVRUIRRWLQJVDUHJLYHQLQ$&,VHH)LJXUHIRUWKHFDVHRIFROXPQVVXSSRUWHGRQ footings with a grade beam at the same top elevation as the footings). Longitudinal reinforcement in the grade beam must be continuous, and at discontinuous locations (for example, at perimeter pile caps or footings), the longitudinal reinforcement must be either developed within or beyond the supported column or boundary element or anchored within the pile cap or footing (straight or hooked bar development). Footing ሺtyp.ሻ

Grade beam

Column ሺtyp.ሻ

A

A κ௡  Centerline of column Slab-on-ground

൒ κௗ௛ 

‫ۓ‬ ۖ

݄Τ2 Other reinforcement not shown for clarity

‫ ݏ‬d lesser of ܾ௪ Τ2 ‫۔‬ ۖ ‫ە‬12ᇳ

Closed ties @ •

݄

κ௡ Τ20 Lesser of ܾ௪ and ݄ ൒ lesser of ൝ 18ᇳ ܾ௪  Section A-A

Figure 14.47 Dimensional and detailing requirements for grade beams designed as foundation seismic ties.

14.7.5 Deep Foundations

5HTXLUHPHQWVIRUGHHSIRXQGDWLRQVVXSSRUWLQJVWUXFWXUHVDVVLJQHGWR6'&'(RU)DUHJLYHQLQ$&,DQG are applicable to uncased cast-in-place concrete drilled piles (caissons), uncased augered piles, metal cased concrete SLOHVFRQFUHWH¿OOHGSLSHSLOHVDQGSUHFDVWFRQFUHWHSLOHV$VXPPDU\RIWKHJHQHUDOUHTXLUHPHQWVLVJLYHQLQ7DEOH ZKLFKDOVRLQFOXGHVWKHVSHFL¿FUHTXLUHPHQWVIRUXQFDVHGGULOOHGSLHUVFDLVVRQV SLOHVDQGDXJHUHGSLOHVLQ $&, 14-65

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.9 Requirements for Deep Foundations Supporting Structures Assigned to SDC D, E, or F Requirement

ACI Section No.

Piles, piers, or caissons resisting design tension forces must have continuous longitudinal reinforcement over their length.



0LQLPXPORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHGLQ$&,WKURXJK PXVWEHH[WHQGHGRYHUWKHHQWLUHXQVXSSRUWHGOHQJWKIRUWKHSRUWLRQRIWKHGHHS foundation member in air or water, or in soil not capable of providing adequate lateral restraint to prevent buckling throughout this length.



+RRSVVSLUDOVDQGWLHVLQGHHSIRXQGDWLRQPHPEHUVPXVWEHWHUPLQDWHGZLWKVHLVPLFKRRNV



Deep foundation members supporting members structures assigned to SDC D, E, or F or located in Site Class E or F must have transverse reinforcement in accordance with ACI DQG$&,7DEOH,WHPH ZLWKLQVHYHQPHPEHUGLDPHWHUVDERYHDQG EHORZWKHLQWHUIDFHVEHWZHHQVWUDWDWKDWDUHKDUGRUVWLIIDQGVWUDWDWKDWDUHOLTXH¿DEOHRUVRIW



Longitudinal and transverse reinforcement in uncased cast-in-place caissons or augered conFUHWHSLOHVRUSLHUVPXVWEHSURYLGHGLQDFFRUGDQFHZLWK$&,7DEOH



Minimum longitudinal and transverse reinforcement must be provided along minimum reinforced lengths measured from the top of the pile (caisson) in accordance with ACI Table 



beyond Longitudinal reinforcement must extend at least the tension development length, WKHÀH[XUDOOHQJWKRIWKHSLOHZKLFKLVGH¿QHGLQ$&,7DEOHDVWKHGLVWDQFHIURP is greater than WKHERWWRPRIWKHSLOHFDSWRZKHUHSHUFHQWRIWKHFUDFNLQJPRPHQW the factored moment,



Longitudinal reinforcement in piles, piers, or caissons resisting tension forces must be detailed to transfer the tension forces within the pile cap to the supported structural members.

18.13.6.1

&RQFUHWHSLOHVDQGFRQFUHWH¿OOHGSLSHSLOHVPXVWEHFRQQHFWHGWRWKHSLOHFDSE\ HPEHGding the pile reinforcement in pile cap a distance equal to the tension or compression develRSPHQWOHQJWKIRUSLOHVVXEMHFWHGWRXSOLIWDQGFRPSUHVVLRQUHVSHFWLYHO\RU WKHXVHRI ¿HOGSODFHGGRZHOVDQFKRUHGLQWKHFRQFUHWHSLOH



Where tension forces induced by earthquake effects are transferred between a pile cap or mat foundation and a precast pile by reinforcing bars grouted or post-installed in the top of the of the bar, which must be demonstrated pile, the grouting system must develop at least by testing.

18.13.6.3

5HTXLUHPHQWVIRUXQFDVHGFDVWLQSODFHRUDXJHUHGFRQFUHWHSLOHVRUSLHUVDUHLOOXVWUDWHGLQ)LJXUHIRUVWUXFtures located in Site Classes A through D and in Site Class E and F for a circular pile or pier with spiral reinforcement where it is assumed a hard or stiff layer of soil is present over the entire length of the pile or pier. Similar GHWDLOVDUHJLYHQLQ)LJXUHZKHUHDOLTXH¿DEOHRUVRIWOD\HURIVRLODQGDKDUGRUVWLIIOD\HURIVRLODUHSUHVHQW ,QWKH¿JXUHV LVWKHOHQJWKRIWKHUHTXLUHGWUDQVYHUVHFRQ¿QHPHQWUHLQIRUFHPHQW]RQHDQG is the minimum OHQJWKZKHUHUHLQIRUFHPHQWLVUHTXLUHG7KHVHOHQJWKVDUHGH¿QHGLQ$&,7DEOHEDVHGRQ6LWH&ODVV =RQHLQWKH¿JXUHVFRUUHVSRQGVWRWKHVHJPHQWZKHUHPLQLPXPWUDQVYHUVHFRQ¿QHPHQWUHLQIRUFHPHQWLVUHTXLUHG ,Q)LJXUH=RQHFRUUHVSRQGVWRWKHVHJPHQWZKHUHPLQLPXPWUDQVYHUVHUHLQIRUFHPHQW over the length is required in the remainder of the required reinforcement length (that is, in the region outside of Zone 1 where the 7KHGDVKHGSRUWLRQRI=RQHFRUUHVSRQGVWRWKHFDVHIRU6LWH&ODVVHV(DQG)ZKHUH length is equal to reinforcement is required over the full length of the member.

14-66

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Pile cap or mat foundation

݀௣  ‫ܦ‬௖௛ 

‫ܣ‬௕௦  ‫ܣ‬௦ ൒ 0.005‫ܣ‬௚  Min. 6 long. bars

4ԣ

Section A-A

κଵ

Site Class

κଶ κ௣ Τ 2 ‫ۓ‬ ۖ ᇱ ۖ 10

‫ݏ‬

Zone 1

A

A–D

൒ 3݀௣

൒ longest of

E and F

൒ 7݀௣

Full length of pileሺ2ሻ

‫ ۔‬3݀௣ ۖ ۖ ‫ ە‬κ௙

Zone 2

ሺ1ሻ κଵ ൌ length of transverse confinement reinforcement zone κଶ ൌ minimum reinforced pier/pile length κ௙ ൌ distance from bottom of pile cap to where 0.4‫ܯ‬௖௥ ൐ ‫ܯ‬௨ Longitudinal reinforcement must extend at least κௗ beyond the distance from the bottom of the pile cap to where 0.4‫ܯ‬௖௥ ൐ ‫ܯ‬௨ . ሺ2ሻ Minimum reinforced pier/pile length need not extend the full length where ሾ1ሿ or ሾ2ሿ in ACI Table 18.13.5.7.1 are satisfied. 

ሺwhere pile/pier resists axial tension loadsሻ 

Pitch, ‫ݏ‬

Zone 2

κ௣ 

κଶ 

κଵ 

A

Zone

Site Class A–D

1

2

൑

E and F

66.7‫ܣ‬௕௦ ݂௬௧ ቆ ᇱቇ ‫ܦ‬௖௛ ݂௖

൑ least of

൑ ‫ۓ‬ ۖ

33.3‫ܣ‬௕௦ ݂௬௧ ቆ ᇱቇ ‫ܦ‬௖௛ ݂௖

12݀௕ሺ௟௢௡௚.ሻ

݀௣ Τ 2 ‫۔‬ ۖ ‫ە‬12ᇳ

In Zones 1 and 2: x Minimum #3 spiral where ݀௣ ൑ 20ԣ x Minimum #4 spiral where ݀௣ ൐ 20ԣ

Figure 14.48 Requirements for uncased cast-in-place or augered concrete piles or piers supporting structures assigned to SDC D, E, or F – Hard or stiff soil over entire length.

14-67

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Pile cap or mat foundation

݀௣  ‫ܦ‬௖௛ 

‫ܣ‬௕௦  ‫ܣ‬௦ ൒ 0.005‫ܣ‬௚  Min. 6 long. bars

4ԣ

Section A-A

A

‫ݏ‬

Zone 1

Zone 2

κଵ 

A

Site Class

κଵ

A–D

൒ 3݀௣

E and F

൒ 7݀௣

Liquefiable or  soft soil

Zone 1

7݀௣ 

κ௣ 

κଵ ൌ length of transverse confinement reinforcement zone

Pitch, ‫ݏ‬ Zone A–D 1

Hard or  stiff soil

൑

E and F

66.7‫ܣ‬௕௦ ݂௬௧ ቆ ᇱቇ ‫ܦ‬௖௛ ݂௖

൑ least of

൑ ‫ۓ‬ ۖ

33.3‫ܣ‬௕௦ ݂௬௧ ቆ ᇱቇ ‫ܦ‬௖௛ ݂௖

12݀ሺ௟௢௡௚.ሻ

݀௣ Τ 2 ‫۔‬ ۖ ‫ە‬12ᇳ

In Zones 1 and 2: x Minimum #3 spiral where ݀௣ ൑ 20ԣ x Minimum #4 spiral where ݀௣ ൐ 20ԣ

Zone 2

Zone 1

2

7݀௣ 

Site Class

Figure 14.49 Requirements for uncased cast-in-place or augered concrete piles or piers supporting structures assigned to SDC D, E, or F – Liquefiable or soft soil layer adjacent to hard or stiff soil layer.

14-68

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The maximum spiral pitch, ZLWKLQWKHWUDQVYHUVHFRQ¿QHPHQWUHLQIRUFHPHQW]RQH=RQH LVEDVHGRQRQHKDOI WKHUHTXLUHPHQWRI$&,7DEOHH IRU6LWH&ODVV$WKURXJK'DQGLVGHWHUPLQHGE\HTXDWLQJRQHKDOIRI from the equation in that table with the general expression for the volume of spiral reinforcement within (14.36)

In this equation, is the area of the spiral bar and of the spiral reinforcement.

LVWKHGLDPHWHURIWKHSLOHSLHUFRUHPHDVXUHGWRWKHRXWVLGH

Thus, the maximum pitch must be less than or equal to the following for Site Classes A through D in Zone 1: (14.37)

Similarly, for Site Classes E and F where LVEDVHGRQWKHUHTXLUHPHQWRI$&,7DEOHH WKHPD[LPXP pitch must be less than or equal to the following in Zone 1: (14.38)

For piers or piles with rectilinear transverse reinforcement, the transverse reinforcement requirements of ACI PXVWDOVREHVDWLV¿HGLQ=RQH

14.8 Members Not Designated as Part of the SFRS 14.8.1 Overview

Not all the structural members in a building are usually assigned to the SFRS. In a building frame system, it is assumed the effects due to the applied earthquake forces are resisted solely by special structural walls. In structures XWLOL]LQJVSHFLDOPRPHQWIUDPHVLWLVW\SLFDOWKDWRQO\VRPHRIWKHIUDPHVLQWKHEXLOGLQJDUHGHVLJQDWHGWREHSDUW of the SFRS in both orthogonal directions. 5HJDUGOHVVRIWKHW\SHRIWKH6)56XWLOL]HGDQGWKHPHPEHUVVHOHFWHGWREHSDUWRILWDOOPHPEHUVLQDVWUXFWXUH are subjected to the effects of an earthquake because all are tied together by the diaphragm at each level. During an earthquake event, the entire structure undergoes displacements and all members are subjected to the effects of these displacements, including those that have not been designated to be part of the SFRS. 0HPEHUVLQDVWUXFWXUHWKDWDUHQRWSDUWRI6)56PXVWVDWLVI\WKHGHIRUPDWLRQFRPSDWLELOLW\UHTXLUHPHQWVLQ$6&( 6(,7KHSXUSRVHRIWKHVHUHTXLUHPHQWVLVWRHQVXUHWKDWWKHVHPHPEHUVFDQVXSSRUWWKHLUUHVSHFWLYHJUDYLW\ caused by the design-level earthquake, which are calculated loads when subjected to the design displacements, LQDFFRUGDQFHZLWK$6&(6(, (14.39)

In this equation, LVWKHGHÀHFWLRQDPSOL¿FDWLRQIDFWRUJLYHQLQ$6&(6(,7DEOH DUHWKHGHÀHFWLRQV DWWKHÀRRUOHYHOVZKHUHWKHFRGHSUHVFULEHGODWHUDOHDUWKTXDNHIRUFHVDUHDSSOLHGRYHUWKHKHLJKWRIWKHVWUXFWXUH and LVWKHHDUWKTXDNHLPSRUWDQFHIDFWRUGHWHUPLQHGLQDFFRUGDQFHZLWK$6&(6(,,Q$6&(6(, 7KHSURYLVLRQVRI$&,PXVWEHVDWLV¿HGIRUUHLQIRUFHGFRQFUHWHPHPEHUVWKDWKDYHQRWEHHQGHVLJQDWHGDV part of the SFRS. Requirements are provided for beams, columns, joints, slab-column connections, and wall piers. 7KHSXUSRVHRIWKHVHRIWKHVHUHTXLUHPHQWVLVWRHQDEOHGXFWLOHÀH[XUDO\LHOGLQJRIWKHVHPHPEHUVE\SURYLGLQJ VXI¿FLHQWFRQ¿QHPHQWDQGVKHDUVWUHQJWK ZKHQVXEMHFWHGWRWKHHIIHFWVFDXVHGE\WKHGHVLJQGLVSODFHPHQWV 14-69

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A structural analysis is performed to determine axial forces, bending moments, and shear forces in the members that ZKLFKDUHDSSOLHGDWHDFKÀRRUOHYHO2QO\WKHPHPare not part of the SFRS due to the design displacements EHUVWKDWDUHQRWSDUWRIWKH6)56DUHLQFOXGHGLQWKHVWUXFWXUDOPRGHOLQWKLVDQDO\VLV$FFRUGLQJWR$&, and the vertical ground motion are to be combined with factored gravity load effects deterthe effects caused by PLQHGE\WKHORDGFRPELQDWLRQVLQ$&,7KHUHIRUHHDFKPHPEHUWKDWLVQRWSDUWRIWKH6)56PXVWEHGHVLJQHG for the following load combinations. (14.40) (14.41)

In these equations,

are the effects in the members due to

The design and detailing requirements for beams and columns depend on the magnitude of the bending moments One set of requirements is applicable and shear forces induced in those members when they are subjected to where the induced bending moments and shear forces do not exceed the corresponding design moment and shear VWUHQJWKVDQGDQRWKHUVHWPXVWEHVDWLV¿HGZKHUHWKHGHVLJQVWUHQJWKVDUHH[FHHGHG,QFDVHVZKHUHLWKDVEHHQ are not going to be calculated, the set of requirements decided that the induced moments and shears caused by ZKHUHGHVLJQVWUHQJWKVDUHH[FHHGHGPXVWEHVDWLV¿HG,WLVDVVXPHGWKDWEHDPVDQGFROXPQV\LHOGLQFDVHVZKHUH the combined effects of factored gravity loads and design displacements exceed the design strengths, or where the effects of design displacements are not calculated. As expected, the requirements where the members yield are more VWULQJHQWWKDQWKRVHZKHUH\LHOGLQJLVDVVXPHGQRWWRRFFXUWKDWLVZKHUHWKHGHVLJQVWUHQJWKVDUHVDWLV¿HG  14.8.2 Beams

$VXPPDU\RIWKHUHTXLUHPHQWVIRUEHDPVWKDWDUHQRWSDUWRIWKH6)56LVJLYHQLQ7DEOH:KHUHWKHLQGXFHG PRPHQWVDQGVKHDUVGRQRWH[FHHGWKHGHVLJQPRPHQWDQGVKHDUVWUHQJWKVWKHUHTXLUHPHQWVRI$&,D  PXVWEHVDWLV¿HGZKLFKDUHLOOXVWUDWHGLQ)LJXUH Table 14.10 Detailing Requirements for Beams That are Not Part of the SFRS

Mu”݊Mn and Vu”݊Vn

Longitudinal Reinforcement

Transverse Reinforcement

Mu and Vu not calculated

5HTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG

5HTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG

No restrictions on mechanical and welded splices

Mechanical and welded splices must satisfy the UHTXLUHPHQWVRI$&,WKURXJK

No restrictions on location of lap splices

No restrictions on location of lap splices

6L]HDQGVSDFLQJRI stirrups determined in accordance with $&,

Spacing throughout the length of the beam

14-70

Mu > ݊Mn or Vu > ݊Vn or

6L]HDQGVSDFLQJRI hoops and crossties satisfying ACI GHWHUPLQHG in accordance with $&, and 6 in. throughout the length of the beam

6L]HDQGVSDFLQJRI stirrups with seismic hooks determined in accordance with $&, Spacing throughout the length of the beam

6L]HDQGVSDFLQJRI hoops and crossties satisfying ACI GHWHUPLQHG in accordance with $&, and 6 in. throughout the length of the beam

݀

݄

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ݏ‬

Longitudinal reinforcement  0.025ܾ௪ ݀ for Grade 60 bars 3ඥ݂௖ᇱ ܾ௪ ݀ൗ݂௬ ା x Greater of ቐ ൑ ‫ିܣ‬ ௦ or ‫ܣ‬௦ ൑ ቐ 0.020ܾ௪ ݀ for Grade 80 bars 200ܾ௪ ݀ Τ݂௬ x Minimum 2 bars continuous top and bottom x No restrictions on lap splice locations x No restrictions on mechanical and welded splices Transverse reinforcement  x For ܲ௨ ൑ ‫ܣ‬௚ ݂௖ᇱ Τ10: Stirrups determined by ACI 22.5.8 with ‫ ݏ‬൑ ݀ Τ2 x For ܲ௨ ൐ ‫ܣ‬௚ ݂௖ᇱ Τ10: Hoops satisfying ACI 18.7.5.2 determined by ACI 22.5.8 with ‫ ݏ‬൑ lesser of ൝

6݀௕ሺ௟௢௡௚.ሻ 6ᇳ

ࡹ࢛ ൑ ࣘࡹ࢔ and ࢂ࢛ ൑ ࣘࢂ࢔ 

Figure 14.50 Requirements for beams that are not part of the SFRS in accordance with ACI 18.14.3.2(a).

Where the induced moments or shears exceed the respective design strengths, or where the induced moments and shear due to DUHQRWFDOFXODWHGWKHUHTXLUHPHQWVRI$&,D DQGE PXVWEHVDWLV¿HGVHH)LJXUH  14.8.3 Columns

$VXPPDU\RIWKHUHTXLUHPHQWVIRUFROXPQVWKDWDUHQRWSDUWRIWKH6)56LVJLYHQLQ7DEOH:KHUHWKH induced moments and shears do not exceed the design moment and shear strengths, the requirements of ACI E DQGF PXVWEHVDWLV¿HG7KHUHTXLUHPHQWVIRUWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIUHFWLOLQHDUKRRSV DUHLOOXVWUDWHGLQ)LJXUH

14-71

݄

݀

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

‫ݏ‬

Longitudinal reinforcement 0.025ܾ௪ ݀ for Grade 60 bars 3ඥ݂௖ᇱ ܾ௪ ݀ൗ݂௬ ା x Greater of ቐ ൑ ‫ିܣ‬ ௦ or ‫ܣ‬௦ ൑ ቐ 0.020ܾ௪ ݀ for Grade 80 bars 200ܾ௪ ݀ Τ݂௬ x Minimum 2 bars continuous top and bottom x No restrictions on lap splice locations x Type 1 mechanical splices must be located outside of 2݄ from column face x Type 2 mechanical splices on Grade 60 bars are permitted at any location x Welded splices must be located outside of 2݄ from column face Transverse reinforcement x For ܲ௨ ൑ ‫ܣ‬௚ ݂௖ᇱ Τ10: Stirrups with seismic hooks determined by ACI 18.6.5 with ‫ ݏ‬൑ ݀ Τ2 x For ܲ௨ ൐ ‫ܣ‬௚ ݂௖ᇱ Τ10: Hoops satisfying ACI 18.7.5.2 determined by ACI 18.6.5 with ‫ ݏ‬൑ lesser of ൝

6݀௕ሺ௟௢௡௚.ሻ 6ᇳ

ࡹ࢛ ൐ ࣘࡹ࢔ or ࢂ࢛ ൐ ࣘࢂ࢔ RU ,QGXFHGPRPHQWVDQGVKHDUVQRWFDOFXODWHG

Figure 14.51 Requirements for beams that are not part of the SFRS in accordance with ACI 18.14.3.3(a) and (b).

Table 14.11 Detailing Requirements for Columns That are Not Part of the SFRS

Mu”݊Mn and Vu”݊Vn

Longitudinal Reinforcement

Mu > ݊Mn or Vu > ݊Vn or Mu and Vu not calculated

5HTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG

5HTXLUHPHQWVRI$&,IRUFROXPQVLQ VSHFLDOPRPHQWIUDPHVPXVWEHVDWLV¿HG

No restrictions on mechanical and welded splices

Mechanical and welded splices must satisfy WKHUHTXLUHPHQWVRI$&,

No restrictions on location of tension lap splices

Tension lap splices are permitted only within the center half of the member length (table continued on next page)

14-72

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.11 Detailing Requirements for Columns That are Not Part of the SFRS (cont.)

Mu”݊Mn and Vu”݊Vn

Mu > ݊Mn or Vu > ݊Vn or Mu and Vu not calculated

6L]HDQGVSDFLQJRIWUDQVYHUVHUHLQIRUFHPHQW GHWHUPLQHGLQDFFRUGDQFHZLWK$&, 6SLUDOUHLQIRUFHPHQWVDWLVI\LQJ$&, RUKRRSUHLQIRUFHPHQWVDWLVI\LQJ$&, must be provided over the full length of the column and 6 in. Transverse reinforcement satisfying ACI D WKURXJKH PXVWEHSURYLGHGRYHU the length GH¿QHGLQ$&,IURP each joint face Transverse Reinforcement

In addition, where • Transverse reinforcement requirements RI$&,PXVWEHVDWLV¿HG • For columns with rectilinear hoops, minimum transverse reinforcement must be at least one-half the greater of (a) and E LQ$&,7DEOHRYHUWKHOHQJWK GH¿QHGLQ$&,IURPHDFK joint face • For columns with spirals or circular hoops, minimum transverse reinforcement must be at least one-half the greater RIG DQGH LQ$&,7DEOHRYHU the length GH¿QHGLQ$&, from each joint face

6L]HDQGVSDFLQJRIWUDQVYHUVHUHLQIRUFHPHQW must be determined in accordance with the UHTXLUHPHQWVRI$&,DQGIRU columns in special moment frames

Where the induced moments or shears exceed the respective design strengths, or where the induced moments and shear due to DUHQRWFDOFXODWHGWKHUHTXLUHPHQWVRI$&,F PXVWEHVDWLV¿HGZKLFKDUHWKHVDPHDV WKRVHIRUFROXPQVLQVSHFLDOPRPHQWIUDPHV7KHVHUHTXLUHPHQWVDUHLOOXVWUDWHGLQ)LJXUHVDQG IRUWKHFDVHRIUHFWLOLQHDUKRRSVDQGFURVVWLHVDQGLQ)LJXUHIRUVSLUDOV 14.8.4 Joints

For joints that are not part of the SFRS where the induced moments and shears do not exceed the design moment DQGVKHDUVWUHQJWKVWKHUHTXLUHPHQWVLQ$&,&KDSWHUPXVWEHVDWLV¿HG>$&,G @7KHVHUHTXLUHPHQWV are covered in Chapter 11 of this publication. Where the induced moments or shears exceed the respective design strengths, or where the induced moments and shear due to DUHQRWFDOFXODWHGWKHUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG>$&,G @7KHVH UHTXLUHPHQWVDUHFRYHUHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ

14-73

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

A

κ௢ ൒ larger of

‫ۓ‬ ۖ

Larger of ܿଵ and ܿଶ ‫ ݏ‬൑ lesser of Clear spanΤ6

‫۔‬ ۖ ‫ە‬18ᇳ

‫ۓ‬ ۖ

6݀௕ሺ௟௢௡௚.ሻ

6ᇳ ‫۔‬ ۖ ‫ ە‬Spacing determined in accordance with ACI 18.7.6

No restrictions on location of tension lap splices

Transverse reinforcement

0.01‫ܣ‬௚ ൑ ‫ܣ‬௦௧ ൑ 0.06‫ܣ‬௚ 

x ܲ௨ ൑ 0.35ܲ௢

See Note 3 under ܲ௨ ൐ 0.35ܲ௢

1. Over length κ௢ : Hoops satisfying ACI 18.7.5.2ሺaሻ through ሺeሻ

‫ݔ‬௜  ܿଶ  ‫ݔ‬௜ 

Alternate 90-deg hooks ‫ݔ‬௜ 

‫ݔ‬௜ 

‫ݔ‬௜ 

ܿଵ 

2. Remainder of column: Hoops satisfying ACI 25.7.4 x ܲ௨ ൐ 0.35ܲ௢ 1. Over length κ௢ : Hoops satisfying ACI 18.7.5.2ሺaሻ through ሺeሻ that are determined by the following:

Section A-A ݄௫ ൌ maximum of ‫ݔ‬௜ ൑ 14ԣ

‫ܣ‬௦௛ ൒ larger of

‫ܣ‬௚ ݂௖ᇱ ‫ۓ‬0.15‫ܾݏ‬௖ ൬ െ 1൰ ‫ܣ‬ ݂ ۖ ௖௛ ௬௧

ሺaሻ

‫۔‬ ݂ᇱ ۖ0.045‫ܾݏ‬௖ ௖ ݂௬௧ ‫ە‬

ሺbሻ

2. Remainder of column: Hoops satisfying ACI 25.7.4 3. Where concrete cover outside the transverse reinforcement ൐ 4ᇳ , provide additional transverse reinforcement in accordance with ACI 18.7.5.7 at ‫ ݏ‬൑ 12ԣ

ࡹ࢛ ൑ ࣘࡹ࢔ and ࢂ࢛ ൑ ࣘࢂ࢔ 

Figure 14.52 Requirements for columns that are not part of the SFRS in accordance with ACI 18.14.3.2(b) and (c).

14-74

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.8.5 Slab-Column Connections

3URYLVLRQVIRUVODEFROXPQFRQQHFWLRQVLQWZRZD\VODEV\VWHPVZLWKRXWEHDPVDUHJLYHQLQ$&,7KHSXUpose of these requirements is to reduce the likelihood of two-way shear failure when the structure is subjected to the caused by the design-level earthquake. design displacements Slab shear reinforcement must be provided at all slab-column connections of nonprestressed two-way slabs without EHDPVZKHUHWKHIROORZLQJHTXDWLRQLVVDWLV¿HG (14.42)

 UHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQGERWWRPRIDVWRU\VWRU\GULIW story height under consideration factored shear stress at the slab critical section due to gravity loads without moment transfer where the factored gravity shear force is determined using the load combinations that include  QRPLQDOWZRZD\VKHDUVWUHQJWKSURYLGHGE\WKHFRQFUHWHFDOFXODWHGLQDFFRUGDQFHZLWK$&, ‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘ ൑ ݀ Τʹ

൒ Ͷ݄ሺ–›’Ǥ ሻ

݀Ȁʹ

‫ݏ‬

݀Ȁʹሺ–›’Ǥሻ

൑ ʹ݀ –‡”‹‘”‘Ž—

‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ ”‹–‹…ƒŽ•‡…–‹‘ ‡ƒ†‡†•Š‡ƒ”•–—†ƒ••‡„Ž›ሺ–›’Ǥሻ

൑ ݀ Τʹ

൑ ݀ Τʹ

݀Ȁʹ

”‹–‹…ƒŽ•‡…–‹‘ ൑ ʹ݀

†‰‡‘Ž—

݀Ȁʹ ݀Ȁʹሺ–›’Ǥሻ ‫ݏ‬

൒ Ͷ݄ሺ–›’Ǥ ሻ

݀Ȁʹሺ–›’Ǥሻ

‫ݏ‬

൒ Ͷ݄ሺ–›’Ǥ ሻ

where

”‹–‹…ƒŽ•‡…–‹‘ ”‹–‹…ƒŽ•‡…–‹‘ ൑ ʹ݀ ‘”‡”‘Ž—

Figure 14.53 Headed shear stud reinforcement at slab-column connections that are not part of the SFRS.

14-75

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The value of WRXVHLQ(TXDWLRQ LVWKHJUHDWHURIWKHYDOXHVRIWKHDGMDFHQWVWRULHVDERYHDQGEHORZ the slab-column connection under consideration. Note that the shear reinforcement requirements of this section are >$&,D @7KHFRQGLWLRQVZKHUHVKHDUUHLQIRUFHPHQWLVUHTXLUHGDQG not applicable where QRWUHTXLUHGDUHJLYHQLQ$&,)LJXUH5 :KHUH(TXDWLRQ LVVDWLV¿HGVWLUUXSVFRQIRUPLQJWR$&,RUKHDGHGVKHDUVWXGVFRQIRUPLQJWR$&, greater than or equal to PXVWEHXWLOL]HGDWWKHVODEFULWLFDO SURYLGLQJDQRPLQDOVKHDUVWUHQJWK VHFWLRQ$&,DQG 7KLVUHLQIRUFHPHQWPXVWH[WHQGDPLQLPXPRIIRXUWLPHVWKHVODEWKLFNQHVV from the face of the column in both principal directions. Typical details for headed shear stud reinforcement at inteULRUHGJHDQGFRUQHUFROXPQVWKHGHWDLOVIRUVWLUUXSVDUHVLPLODUDUHJLYHQLQ)LJXUH 14.8.6 Wall Piers

:DOOSLHUVQRWGHVLJQDWHGSDUWRIWKH6)56PXVWVDWLVI\WKHUHTXLUHPHQWVLQ$&,$&, 7KHGHVLJQ with the joint faces taken at the top and VKHDUIRUFHPXVWEHGHWHUPLQHGXVLQJWKHSUREDEOHÀH[XUDOVWUHQJWK ERWWRPRIWKHFOHDUKHLJKWRIWKHZDOOSLHU$&,  ,QOLHXRIGHWHUPLQLQJWKHGHVLJQVKHDUIRUFHLQDFFRUGDQFHZLWK$&,LWLVSHUPLWWHGWRFDOFXODWH and the overstrength factor, product of the shear induced in the pier due to the design displacement

as the

14.9 Examples 14.9.1 Example 14.1 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C), Beam is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHEHDPDORQJFROXPQOLQH&EHWZHHQFROXPQOLQHVDQGLQ %XLOGLQJ)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQV RIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ DQG*UDGHUHLQIRUFHPHQW Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'EDVHGRQWKHIROORZLQJ • Site Class D (determined) • • Step 1 – Check the dimensional limits

ACI 18.6.2

Prior to determining the design bending moments, check to ensure theGLPHQVLRQDOOLPLWVRI$&,DUHVDWLV¿HG and where • •

•

7KHUHIRUHWKHGLPHQVLRQVRIWKHEHDPVDWLVI\WKHUHTXLUHPHQWVRI$&,

14-76

Figure 14.1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the factored bending moments along the span

A summary of the service bending moments due to the dead and live loads at the critical sections of the beam is JLYHQLQ7DEOH Table 14.12 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the Beam in Example 14.1 Negative – Line 1

Positive

Negative – Line 2

A three-dimensional analysis was performed for seismic forces in the east-west direction where it is assumed the frames along column lines A, C, and E are part of the SFRS. In the north-south direction, the frames along column OLQHVDQGDUHSDUWRIWKH6)565LJLGGLDSKUDJPVDUHDVVLJQHGDWHDFKOHYHODQGWRDFFRXQWIRUFUDFNLQJ WKHIROORZLQJUHGXFHGPRPHQWVRILQHUWLDDUHXVHGIRUWKHEHDPVDQGFROXPQV>VHH$&,7DEOHD @ • Beams: • Columns: 7KHEHQGLQJPRPHQWVLQWKHEHDPGXHWRHDUWKTXDNHORDGVDUHJLYHQLQ7DEOH7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQGWKHZHVWGLUHFWLRQWKDW is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the south elevation of the building). The bending moments due to wind loads are smaller than those due to the earthquake loads and are not considered in this example. Table 14.13 Bending Moments (ft-kips) due to Earthquake Loads at the Second-Floor Level for the Beam in Example 14.1 Negative – Line 1

Positive

Negative – Line 2

ņ 7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOHVHH7DEOH ,W can be determined that the redundancy factor, LVHTXDOWRLQDFFRUGDQFHZLWK$6&(6(, Table 14.14 Design Bending Moments (ft-kips) at the Second-Floor Level for the Beam in Example 14.1 Load Combination

Negative – Line 1

Positive

Negative – Line 2

$&,(TD

í



í

$&,(TE

í



í

SSR





í

SSL

í



í

SSR

119.8



í

SSL

í





$&,(TH

$&,(TJ

14-77

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR Therefore, this load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWso , and this load combination becomes the following:

Step 3 – Determine the required flexural reinforcement at the critical sections

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW at the critical sections is determined by the following equations, ZKLFKDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUVHFWLRQVZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHment (an effective slab width at positive moment critical sections is not considered in this example):

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWDOOVHFWLRQVDUHWHQVLRQ Also, the required reinforcement is less than the maximum amount prescribed in ACI controlled because IRU*UDGHUHLQIRUFHPHQWZKLFKLVEDVHGRQ Table 14.15 Required Flexural Reinforcement for the Beam in Example 14.1

Mu (ft-kips)

Rn (psi)

As (in.2)*

SSR

119.8





SSL

í

336









SSR

í





SSL







Location

1HJDWLYH±/LQH Positive 1HJDWLYH±/LQH * Min. Max.

Step 4 – Select the flexural reinforcement

6HOHFWWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVEDVHGRQWKHPD[LPXPDQGPLQLPXPVSDFLQJUHTXLUHPHQWVLQ$&, DQGUHVSHFWLYHO\ 1HJDWLYHUHLQIRUFHPHQW±/LQH: maximum and minimum number of longitudinal bars in a single 8VHEDUV OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  14-78

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Positive reinforcement: 8VHEDUV maximum and minimum number of longitudinal bars in a single OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  DUHDOVRDGHTXDWHIRUWKHIWNLSDQGIWNLSSRVLWLYHPRPHQWVDWWKH 7KHEDUV IDFHVRIWKHVXSSRUWVDWFROXPQOLQHVDQGUHVSHFWLYHO\VHH7DEOH  1HJDWLYHUHLQIRUFHPHQW±/LQH: maximum and minimum number of longitudinal bars in a single 8VHEDUV OD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\  Step 5 – Check the dimensional requirements of ACI 18.8.2.3

&KHFNLIWKHEDUVVDWLVI\WKHGLPHQVLRQDOUHTXLUHPHQWVLQ$&, Figure 14.2

Step 6 – Check the nominal strength requirements of ACI 18.6.3.2

The positive moment strength at the face of a joint be greater than or equal to one-half the negative moment strength SURYLGHGDWWKDWORFDWLRQ$&,  At the interior joint:

7KLVUHTXLUHPHQWLVDOVRVDWLV¿HGDWWKHH[WHULRUMRLQWZKHUHWKHQHJDWLYHUHLQIRUFHPHQWLVEDUV Also, the negative or positive moment strength at any section of the beam must be greater than or equal to onefourth the maximum moment strength provided at the face of either joint, which in this example is equal to 3URYLGLQJFRQWLQXRXVWRSDQGERWWRPEDUV VDWLV¿HVWKLVUHTXLUHment. Step 7 – Check if the flexural reinforcement can be fully developed in the column at column line 1 ACI 18.8.2.2

7KHEDUVPXVWEHGHYHORSHGLQWHQVLRQLQDFFRUGDQFHZLWK$&,DQGLQFRPSUHVVLRQLQDFFRUGDQFHZLWK $&, The required development length of the hooked bars in tension is equal to the following for normalweight concrete:

Eq. (14.1)

14-79

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHDYDLODEOHGHYHORSPHQWOHQJWKLVHTXDOWRWKHIROORZLQJDVVXPLQJKRRSVDQGORQJLWXGLQDOEDUVLQWKH column: Eq. (14.3)

7KHUHTXLUHGGHYHORSPHQWOHQJWKRIWKHEDUVLQFRPSUHVVLRQLVHTXDOWRWKHIROORZLQJ

Eq. (14.6)

7KHDYDLODEOHGHYHORSPHQWOHQJWKLVHTXDOWRWKHIROORZLQJDVVXPLQJKRRSVDQGORQJLWXGLQDOEDUVLQWKH column:

Eq. (14.7)

7KHUHIRUHWKHEDUVFDQEHDGHTXDWHO\GHYHORSHGLQWKHFROXPQDWFROXPQOLQH 14.9.2 Example 14.2 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C), Beam is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUWKHEHDPDORQJFROXPQOLQH&EHWZHHQFROXPQOLQHVDQGLQ %XLOGLQJ)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQV RIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHFROXPQVDUHLQE\LQ DQG*UDGHUHLQIRUFHPHQW Also assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the factored shear forces

ACI 18.6.5

Design shear forces are determined from statics assuming the beam is loaded with the factored gravity and vertical HDUWKTXDNHORDGVDORQJWKHVSDQDQGPRPHQWVRIRSSRVLWHVLJQFRUUHVSRQGLQJWRWKHSUREDEOHÀH[XUDOVWUHQJWK acting at the joint faces. 7KHODUJHVWVKHDUIRUFHDVVRFLDWHGZLWKHDUWKTXDNHHIIHFWVLVREWDLQHGIURP$&,(TH 7KHGLVWULEXWHGORDG DORQJWKHOHQJWKRIWKHEHDPLVWUDSH]RLGDOVHH)LJXUHRIWKLVSXEOLFDWLRQ DQGWKHPD[LPXPIDFWRUHGGLVWULEXWHGORDGLQWKHFHQWHUIWVHJPHQWRIWKHEHDPLVGHWHUPLQHGDVIROORZV

14-80

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore,

Shear forces due to the combination of and for sidesway to the right and to the left are given in Figure ZKHUHWKHSRVLWLYHDQGQHJDWLYHSUREDEOHÀH[XUDOVWUHQJWKVDUHGHWHUPLQHGE\(T 

ͳA

ʹA

‫ݓ‬௨ ൌ ͶǤ͸Ͷ ‹’•Τˆ–

ି ‫ܯ‬௣௥ ൌ ͷʹͳǤͲˆ–Ǧ‹’•

ା ‫ܯ‬௣௥ ൌ ͵ͲͺǤ͵ˆ–Ǧ‹’•

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›”‹‰Š–

ͳͲǤͷͺԢ

ͳǤͷԢ

ͳͲǤͷͺԢ

ͺǤ͸‹’•

͸ͶǤ͸‹’•

ͳA

ʹA

‫ݓ‬௨ ൌ ͶǤ͸Ͷ ‹’•Τˆ–

ି ା ି ൅ ‫ܯ‬௣௥ଵ ‫ݓ‬௨ κ௡ ‫ܯ‬௣௥ ‫ܯ‬௣௥ ൌ ൌ ൅ ܸ௨ ͶͷͳǤͺˆ–Ǧ‹’• ʹ κ௡

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›Ž‡ˆ–

ା ‫ܯ‬௣௥ ൌ ͳǤʹͷ‫ܣ‬ା ͳͲǤͷͺԢ ௦ ݂ͳǤͷԢ ௬ ቆ݀ െ

ܸ௨ ൌ

ି ା ‫ܯ‬ା௣௥ ൅ ͵ͲͺǤ͵ˆ–Ǧ‹’• ‫ܯ‬௣௥ଵ ‫ݓ‬௨ κ௡ ‫ܯ‬ ൌ െ ௣௥ ʹ κ

ͳǤʹͷ‫ܣ‬ା ௦ ݂௬ ቇ ͳͲǤͷͺԢ ᇱ ͳǤ͹݂௖ ܾ௪

͸ͳǤͷ‹’• ି ൌ ͳǤʹͷ‫ିܣ‬ ‫ܯ‬௣௥ଵ ௦ǡκ ݂௬ ቆ݀ െ

ͷǤͷ‹’• ି ‫ܯ‬௣௥ଶ ൌ ͳǤʹͷ‫ିܣ‬ ௦ǡ௥ ݂௬ ቆ݀ െ

Figure 14.54 Factored shear forces on the beam in Example 14.2.

The maximum

LVHTXDOWRNLSVZKLFKLVJUHDWHUWKDQWKHPD[LPXP

from analysis.

Step 2 – Determine the required transverse reinforcement

$FFRUGLQJWR$&,KRRSVDUHUHTXLUHGRYHUDOHQJWKHTXDOWR both ends of the beam.

from the face of the columns at

14-81

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHVL]HDQGVSDFLQJRIWKHKRRSVFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ Eq. (14.11)

Check if

PXVWEHWDNHQHTXDOWR]HUR

ACI 18.6.5.2

At the faces of the supports, the maximum earthquake-induced shear force One-half of the maximum shear force Factored axial compressive force on the beam including earthquake effects Therefore,

PXVWEHWDNHQHTXDOWR]HUR

$VVXPLQJKRRSVZLWKFURVVWLHVWKHUHTXLUHGKRRSVSDFLQJLVHTXDOWRWKHIROORZLQJFURVVWLHVPXVWEHXVHG EHFDXVHWKHFOHDUVSDFLQJRIWKHERWWRPEDUVLVJUHDWHUWKDQLQVHH$&,DQG 

Check maximum hoop spacing along the length:

ACI Table 9.7.6.2.2

Therefore, maximum Also,

ACI 18.6.4.4, Figure 14.9

8VHKRRSVZLWKFURVVWLHVDWHDFKHQGRIWKHEHDPVSDFHGLQRQFHQWHUZLWKWKH¿UVWVHWORFDWHGLQ IURPWKHIDFHRIWKHVXSSRUW$&,  Where hoops are no longer required, stirrups with seismic hooks at both ends may be used instead of hoops $&, $WDGLVWDQFHRILQIURPWKHIDFHRIWKHVXSSRUWDWFROXPQOLQHIRUVLGHVZD\WRWKHULJKW 7KHUHTXLUHGVSDFLQJRIVWLUUXSVLVHTXDOWRWKHIROORZLQJZKHUHLWLVSHUPLWWHGWRXVH in this , so 3 stirrup region of the beam (the maximum spacing of stirrup legs across the width of the beam OHJVDUHUHTXLUHGVHH$&,7DEOH 

ACI Table 9.6.3.4

14-82

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Also, ACI 18.6.4.6

8VHVWLUUXSVZLWKRQHFURVVWLHVSDFHGDWLQRQFHQWHULQWKHUHJLRQRIWKHEHDPRXWVLGHRIWKHDQWLFLSDWHG plastic hinge regions. 14.9.3 Example 14.3 – Determination of Cutoff Points of Flexural Reinforcement: Beam in Building #1 (Framing Option C), Beam is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHORFDWLRQZKHUHRIWKHWRSEDUVDWWKHLQWHULRUVXSSRUWFDQEHWHUPLQDWHGIRUWKHEHDPDORQJ FROXPQOLQH&EHWZHHQFROXPQOLQHVDQGLQ%XLOGLQJ)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJ WKHEHDPLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHEHDPDUHLQE\LQVHH)LJXUH 7KHVODELVLQ DQG*UDGH WKLFNDQGWKHFROXPQVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the load combination to determine the cutoff point

The load combination used to determine the cutoff point is in combinaWLRQZLWKWKHSUREDEOHÀH[XUDOVWUHQJWKVDWWKHHQGVRIWKHEHDPIRUVLGHVZD\WRWKHULJKWEHFDXVHWKLVFRPELQDWLRQ produces the longest bar lengths. Step 2 – Determine the theoretical cutoff point

7KUHHRIWKHWRSEDUVFDQEHWKHRUHWLFDOO\FXWRIIDWWKHORFDWLRQZKHUHWKHIDFWRUHGEHQGLQJPRPHQW ZLWKEDUVZKLFKLVHTXDOWRIWNLSV equal to the design moment strength of the section,

is

The distance IURPWKHIDFHRIWKHVXSSRUWDWFROXPQOLQHWRWKHORFDWLRQZKHUHWKHPRPHQWLVHTXDOWRIW LQ)LJXUH kips can be determined by summing moments about section

ͳA

ʹA

‫ݓ‬௨ ൌ ͳǤͻͺ ‹’•Τˆ– ܽ

ି ‫ܯ‬௣௥ ൌ ͷʹͳǤͲˆ–Ǧ‹’•

ା ‫ܯ‬௣௥ ൌ ͵ͲͺǤ͵ˆ–Ǧ‹’•

‫ݔ‬

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›”‹‰Š–

ͳͲǤͷͺԢ ʹͶǤ͸‹’•

ͳǤͷԢ

ܽ

ͳͲǤͷͺԢ ͶͺǤͷ‹’•

Figure 14.55 Cutoff location for the top bars at the interior support for the beam in Example 14.3.

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›Ž‡ˆ–

”ƒ˜‹–›൅‡‹•‹…ǡ ‹†‡•™ƒ›Ž‡ˆ–

14-83

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Solving for

gives a distance of 6.3 ft from the face of the support.

7KHEDUVPXVWH[WHQGDGLVWDQFHHTXDOWRWKHJUHDWHURIWKHIROORZLQJEH\RQG

ACI 9.7.3.3

Thus, the total bar length from the face of the support must be at least equal to Additionally, the bars must extend at least

beyond the face of the support:

ACI 9.7.3.4

ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV )RUPRUHWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHQHJDWLYHUHLQIRUFHPHQW

Set

ACI 25.4.2.4

Therefore,

$FFRUGLQJWR$&,ÀH[XUDOUHLQIRUFHPHQWLVQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRU PRUHRIWKHFRQGLWLRQVLQWKDWVHFWLRQDUHVDWLV¿HG,QWKLVH[DPSOHWKHSRLQWRILQÀHFWLRQLQWKHEHQGLQJPRPHQW GLDJUDPRFFXUVDSSUR[LPDWHO\IWIURPWKHIDFHRIWKHVXSSRUWZKLFKLVJUHDWHUWKDQIWZKLFKPHDQVWKH EDUVFDQQRWEHWHUPLQDWHGDWWKDWORFDWLRQEHFDXVHWKH\DUHLQDWHQVLRQ]RQH &KHFNLI$&,D LVVDWLV¿HGDWWKHLQÀHFWLRQSRLQWZKLFKZRXOGSHUPLWWKHEDUVWREHWHUPLQDWHGDWWKLV location: At 8.1 ft from the face of the support,

:LWKLQWKLVUHJLRQRIWKHEHDPVWLUUXSVZLWKRQHFURVVWLHVSDFHGDWLQRQFHQWHUDUHSURYLGHGWKXV

14-84

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHIRUHWKHEDUVDUHSHUPLWWHGWREHWHUPLQDWHGIWIURPWKHIDFHRIWKHVXSSRUW 6LPLODUFDOFXODWLRQVFDQEHSHUIRUPHGIRUWKHFXWRIISRLQWIRUWZRRIWKHWRSEDUVDWWKHH[WHULRUVXSSRUWDWFROXPQOLQH7ZRRIWKHEDUVFDQEHWHUPLQDWHGIWIURPWKHIDFHRIWKDWVXSSRUW 5HLQIRUFHPHQWGHWDLOVIRUWKLVEHDPDUHJLYHQLQ)LJXUH 2Ԣ-4ԣ

7Ԣ-0ԣ

2Ԣ-4ԣ

8Ԣ-2ԣ

2-#7

4-#7

3-#7

2Ԣ-0ԣ

A

A

4-#7 2ԣ

#3 stirrups and crosstie @ 10ԣ 11-#3 hoops and crossties @ 5ԣ

7.0ԣ

22Ԣ-8ԣ

7-#7

17.0ԣ

#3 hoops and crossties

4-#7

28ԣ Section A-A Other reinforcement not shown for clarity

Figure 14.56 Reinforcement details for the beam in Examples 14.1, 14.2, and 14.3.

14.9.4 Example 14.4 – Determination of Longitudinal Reinforcement: Interior Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQ%XLOGLQJ)UDPLQJ2SWLRQ&LQWKH¿UVW VWRU\DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW

14-85

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'EDVHGRQWKHIROORZLQJ • Site Class D (determined) • • Step 1 – Check the dimensional limits

ACI 18.7.2

Shortest cross-sectional dimension

Figure 14.10

Ratio of shortest cross-sectional dimension to the perpendicular dimension Therefore, the provided cross-sectional dimensions are adequate. Step 2 – Determine the factored load combinations

A three-dimensional analysis was performed for gravity and earthquake effects in the east-west direction where it is assumed the frames along column lines A, C, and E are part of the SFRS. In the north-south direction, the frames DORQJFROXPQOLQHVDQGDUHSDUWRIWKH6)565LJLGGLDSKUDJPVDUHDVVLJQHGDWHDFKOHYHODQGWRDFcount for cracking, the following reduced moments of inertia are used for the beams and columns [see ACI Table D @ • Beams: • Columns: This column is part of the SFRS in the east-west direction only, which means it need not be designed for orthogoQDOORDGHIIHFWVLQDFFRUGDQFHZLWK$6&(6(,$OVR3GHOWDHIIHFWVZHUHDXWRPDWLFDOO\DFFRXQWHGIRULQWKH analysis and were found to be negligible. 7KHJUDYLW\DQGHDUWKTXDNHD[LDOIRUFHVDQGEHQGLQJPRPHQWVDWWKHWRSDQGERWWRPRIFROXPQ$DUHJLYHQLQ 7DEOH%HQGLQJPRPHQWVGXHWRJUDYLW\ORDGHIIHFWVDUHQHJOLJLEOHFRPSDUHGWRWKRVHIURPHDUWKTXDNHHIIHFWV DQGDUHWDNHQHTXDOWR]HUR7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGVHLVPLFD[LDOIRUFHEHQGLQJPRPHQWV DQGVKHDUIRUFHVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQGWKHZHVWGLUHFWLRQWKDWLV sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the south elevation of the building). The effects due to wind loads are smaller than those due to the earthquake loads and are not considered in this example. Table 14.16 Design Axial Forces, Bending Moments, and Shear Forces for Column C2 in Example 14.4 Load Case

Axial Force (kips)

Bending Moment (ft-kips) Bottom

Top

Shear Force (kips)

Dead









Live









Roof live load

11.8







Seismic Load Combination $&,(TD



(table continued on next page)

14-86

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.16 Design Axial Forces, Bending Moments, and Shear Forces for Column C2 in Example 14.4 (cont.) Axial Force (kips)

Load Case

Bending Moment (ft-kips) Bottom

Top

Shear Force (kips)

$&,(TE









$&,(TF









SSR



í



í

SSL





í



SSR



í



í

SSL





í



$&,(TH

$&,(TJ SSR = sidesway to the right, SSL = sidesway to the left

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR Therefore, this load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 3 – Determine the required longitudinal reinforcement

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUV LVJLYHQLQ)LJXUH$OOORDGFRPELQDWLRQVIDOOZLWKLQWKHERXQGDU\RIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDP for this column. 7U\EDUV Step 4 – Check the minimum flexural strength requirements

ACI 18.7.3

7KHPLQLPXPÀH[XUDOVWUHQJWKUHTXLUHPHQWVRI$&,DUHFKHFNHGIRUWKLVFROXPQLQWKHHDVWZHVWGLUHFWLRQ 7KHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHFROXPQVDQGEHDPVDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,2QO\ ORDGFRPELQDWLRQVWKDWLQFOXGHHDUWKTXDNHHIIHFWVQHHGWREHFRQVLGHUHGZKHQFKHFNLQJ$&,(T  7KHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPVIUDPLQJLQWRWKHMRLQWLVGHWHUPLQHGLQ([DPSOHVHH)LJXUH  7KHQHJDWLYHÀH[XUDOVWUHQJWKRIWKHEHDPPXVWLQFOXGHWKHFRQWULEXWLRQRIWKHUHLQIRUFHPHQWLQWKHVODEZLWKLQWKH HIIHFWLYHVODEZLGWKGH¿QHGLQ$&,WKHVODELVLQWHQVLRQ ,WFDQEHGHWHUPLQHGWKDWEDUVDUHUHTXLUHG IRUERWKWKHSRVLWLYHDQGQHJDWLYHUHLQIRUFHPHQWZLWKLQWKHIWFROXPQVWULS7KHHIIHFWLYHVODEZLGWKLQDFFRUGDQFHZLWK$&,LVHTXDOWRWKHIROORZLQJ

14-87

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

2,000 1,750

Axial Force ሺkipsሻሻ

1,500 1,250 1,000 750

 



500



250 0 0

200

400

600

800

1,000

1,200

Bending Moment ሺft-kipsሻ Figure 14.57 Design strength interaction diagram for column C2 in Example 14.4.

Figure 14.13

7KHEHDPXVHGLQGHWHUPLQLQJWKHQHJDWLYHÀH[XUDOVWUHQJWKLVJLYHQLQ)LJXUH 96ԣ

7.0ԣ

#4 ሺtyp.ሻ

17.0ԣ

7-#7

4-#7

28ԣ Figure 14.58 Beam framing into the interior joint at column C2 in Example 14.4.

14-88

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VWUDLQFRPSDWLELOLW\DQDO\VLVRIWKHVHFWLRQZLWKWKHLQHIIHFWLYHZLGWKWKHWRSEDUVLQWKHEHDPDQGWKH Also, EDUVLQWKHVODE\LHOGV 7KHQRPLQDOÀH[XUDOVWUHQJWKRIWKHFROXPQLVGHWHUPLQHGIRUWKHIDFWRUHGD[LDOFRPSUHVVLYHIRUFHUHVXOWLQJLQWKH ORZHVWÀH[XUDOVWUHQJWKFRQVLVWHQWZLWKWKHGLUHFWLRQRIDQDO\VLV)RUWKHFROXPQEHORZWKHMRLQWWKHD[LDOIRUFHVLQ 7DEOHDUHDSSOLFDEOH7KHVHIRUFHVDUHLQGLFDWHGLQ)LJXUHZKLFKFRQWDLQVWKHQRPLQDOVWUHQJWKLQWHUDFtion diagram for this column. 

4,000 ;

3,500

Column above

 Column below

Axial Force ሺkipsሻሻ

3,000 2,500 2,000 1,500 1,000

 

;

500

; 0 0

200

400

600

800

1,000

1,200

1,400

Bending Moment ሺft-kipsሻ Figure 14.59 Nominal strength interaction diagram for column C2 in Example 14.4.

The factored axial compressive forces and corresponding bending moments for the load combinations with earthTXDNHHIIHFWVDUHDOVRLQGLFDWHGLQ)LJXUHIRUWKHFROXPQDERYHWKHMRLQWZKHUHLWLVDVVXPHGWKHFROXPQ above the joint has the same cross-section, concrete compressive strength, and longitudinal reinforcement as the column below the joint. )URP)LJXUHWKHPLQLPXPÀH[XUDOVWUHQJWKVIRUWKHFROXPQVDERYHDQGEHORZWKHMRLQWDUHHTXDOWRWKHIROlowing:

&KHFN$&,(T 

Figure 14.11

7KHUHIRUHWKHPLQLPXPÀH[XUDOVWUHQJWKUHTXLUHPHQWVRI$&,DUHVDWLV¿HG

14-89

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Check the development length requirements of ACI 18.7.4.3

Unsupported length of column: Determine the tension development length,

RIWKHORQJLWXGLQDOEDUV ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV

$VVXPLQJKRRSV

Set

ACI 25.4.2.4

Therefore,

Figure 14.14

7KHUHIRUHWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG 8VHEDUV 14.9.5 Example 14.5 – Determination of Transverse Reinforcement: Interior Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQ%XLOGLQJ)UDPLQJ2SWLRQ&LQWKH¿UVWVWRU\ DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KH VODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH

14-90

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the length over which flexural yielding is likely to occur

ACI 18.7.5.1

Eq. (14.015)

Step 2 – Check if the requirements of ACI 18.7.5.2(f) must be satisfied

7KHODUJHVWIDFWRUHGD[LDOIRUFHLQFOXGLQJHDUWKTXDNHHIIHFWVLVHTXDOWRNLSV

Table 14.16

Also, 7KHUHIRUHWKHUHTXLUHPHQWVRI$&,I QHHGQRWEHVDWLV¿HG Step 3 – Determine the maximum hoop spacing based on the requirements of ACI 18.7.5.3

The maximum hoop spacing within

from each end of the column is equal to the following:

Figure 14.17

In the equation for the maximum center-to-center spacing of the laterally supported longitudinal bars, GHWHUPLQHGDVIROORZVDVVXPLQJKRRSVZLWKFURVVWLHVDURXQGDOOWKHORQJLWXGLQDOEDUV

is

Step 4 – Determine the minimum area of hoop reinforcement based on the requirements of ACI 18.7.5.4

In both directions,

Figure 14.17

KRRSVZLWKWZRFURVVWLHVVSDFHGDWLQSURYLGHV KRRSVZLWKWZRFURVVWLHVVSDFHGDWLQSURYLGHV 7U\KRRSVDQGFURVVWLHVVSDFHGDWLQRQFHQWHUZLWKLQ (note: recalculating LQVWHDGRIKRRSVDQGFURVVWLHVUHVXOWVLQWKHVDPHYDOXHIRU ).

ZLWKKRRSVDQGFURVVWLHV

14-91

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Check if the size and spacing of the hoop reinforcement determined in Step 4 are adequate for shear

Design shear force, LVGHWHUPLQHGXVLQJWKHSUREDEOHÀH[XUDOVWUHQJWKV at the ends of the column associDWHGZLWKWKHUDQJHRIIDFWRUHGD[LDOIRUFHVDFWLQJRQWKHFROXPQ$&, 7KHGHVLJQVWUHQJWKLQWHUDFWLRQ and LVJLYHQLQ)LJXUH,WLVHYLGHQWIURPWKH diagram for this column with LVHTXDOWRIWNLSVZKLFKFRUUHVSRQGVWRDQD[LDOIRUFHRINLSVIRUVLGHVZD\ ¿JXUHWKDWWKHODUJHVW to the left. 4,000 3,500

Axial Force ሺkipsሻሻ

3,000 2,500 2,000 1,500 1,000

 

500 0 0

300

600

900

1,200

1,500

Bending Moment ሺft-kipsሻ

Figure 14.60 Nominal strength interaction diagram for column C2 with fy = 75 ksi and ݊ = 1.0.

Therefore, Figure 14.15

7KLVVKHDUIRUFHLVJUHDWHUWKDQWKHPD[LPXPVKHDUIRUFHRINLSVGHWHUPLQHGIURPDQDO\VLVVHH7DEOH  The earthquake-induced shear force is the total shear force on the column. Also, the minimum factored axial force Therefore, can LQFOXGLQJHDUWKTXDNHHIIHFWVLVHTXDOWRNLSVZKLFKLVJUHDWHUWKDQ EHLQFOXGHGZKHQGHWHUPLQLQJWKHUHTXLUHGVKHDUUHLQIRUFHPHQW$&, 

Eq. (14.17)

where

14-92

is determined from a strain compatibility analysis of the section for the axial compressive force which corresponds to used to determine

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHUHTXLUHGVSDFLQJRIWKHKRRSVDQGFURVVWLHVLQWKHGLUHFWLRQRIDQDO\VLVLVHTXDOWRWKHIROORZLQJ

7KXVWKHVL]HDQGVSDFLQJRIWKHKRRSVGHWHUPLQHGLQ6WHSDUHDGHTXDWHIRUVKHDU 8VHKRRSVZLWKWZRFURVVWLHVVSDFHGDWLQRQFHQWHURYHUDGLVWDQFHRIDWOHDVW the column.

from each end of

Step 6 – Determine the size and spacing of the hoop reinforcement in the remainder of the column

Outside of

the spacing of the hoops is equal to the following: Figure 14.17

The same transverse reinforcement required within segment over the lap splice length.

is used over the entire length of the column, including the VHH([DPSOHIRUWKH

A Class B lap splice must be used. The length of the lap splice calculation of ).

8VHDIWLQODSVSOLFHOHQJWKZKLFKPXVWEHORFDWHGQHDUWKHPLGKHLJKWRIWKHFROXPQ 5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH

24ԣ

28ԣ 

28ԣ 

12-#10 

4'- 9ԣ

28ԣ 



Section A-A #5 hoops and crossties @ 6ԣ

A

A

Other reinforcement not shown for clarity

Figure 14.61 Reinforcement details for column A2 in Examples 14.4 and 14.5.

14-93

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.9.6 Example 14.6 – Check of Joint Shear Strength: Interior Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHHDVWZHVWGLUHFWLRQIRUFROXPQ&LQ%XLOGLQJ)UDPLQJ2SWLRQ&DWWKH VHFRQGÀRRUOHYHODVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQ VHH)LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK Step 1 – Determine the factored horizontal joint shear force

ACI 18.8.4.1

)UHHERG\GLDJUDPVRIWKHMRLQWDUHJLYHQLQ)LJXUHIRUVLGHVZD\WRWKHULJKWDQGVLGHVZD\WRWKHOHIW7KH QHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPVRQERWKVLGHVRIWKHMRLQWDUHWKHVDPHWKDWLVWRS EDUVDQGERWWRPEDUVDUHXVHGLQERWKEHDPVVHH([DPSOH  and ,WLVGHWHUPLQHGLQ([DPSOHWKDW WKHHQGVRIWKHEHDPVDUHJLYHQLQ)LJXUHIRUVLGHVZD\WRWKHULJKWDQGWRWKHOHIW

The shear forces at

)URPDQDO\VLVRIWKHVWUXFWXUHWKHSRLQWRILQÀHFWLRQLQWKHFROXPQDERYHWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURP WKHFHQWHURIWKHMRLQWDQGWKHSRLQWRILQÀHFWLRQLQWKHFROXPQEHORZWKHMRLQWRFFXUVDSSUR[LPDWHO\IWIURPWKH center of the joint. Therefore, The shear force in the column is obtained by summing moments about the center of the joint for sidesway to the right and to the left. Sidesway to the right:

Eq. (14.19)

Sidesway to the left:

The shear force in the column for sidesway to the left will be used in determining the joint shear force because this results in a larger joint shear force, Eq. (14.23) Step 2 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of analysis is continuous. There are beams in the transverse direction that have the same width as the joint, which SURYLGHFRQ¿QHPHQW7KHUHIRUHWKHMRLQWW\SHLV,$VHH)LJXUH 

14-94

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ʹͺԣ



ܸ௖௢௟ ൌ ͻ͹Ǥ͵‹’•

͹Ǧ͓͹ ܸଶ ൌ ͸ͶǤ͸‹’• ܸଶ ൌ ͸ͶǤ͸‹’•

ܸଵ ൌ ͺǤ͸‹’•

ͻǤͶԢ

ܸଵ ൌ ͺǤ͸‹’•

ା ‫ܯ‬௣௥ ൌ ͵ͲͺǤ͵ˆ–Ǧ‹’•

ି ‫ܯ‬௣௥ ൌ ͷʹͳǤͲˆ–Ǧ‹’•

ͶǦ͓͹

‹†‡•™ƒ›”‹‰Š–

ܸ௖௢௟ ൌ ͻ͹Ǥ͵‹’• ʹͺԣ ܸ௖௢௟ ൌ ͻ͸Ǥͻ‹’•

͹Ǧ͓͹ ܸଶ ൌ ͷǤͷ‹’•

ܸଵ ൌ ͸ͶǤ͸‹’•

ܸଵ ൌ ͸ͶǤ͸‹’•

ͻǤͶԢ

ܸଶ ൌ ͷǤͷ‹’•

ା ‫ܯ‬௣௥ ൌ ͵ͲͺǤ͵ˆ–Ǧ‹’•

ି ‫ܯ‬௣௥ ൌ ͷʹͳǤͲˆ–Ǧ‹’•

ͶǦ͓͹

‹†‡•™ƒ›Ž‡ˆ–

ܸ௖௢௟ ൌ ͻ͸Ǥͻ‹’•

Figure 14.62 Free-body diagram of the joint at column C2.

14-95

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Because the beam in the direction of analysis is the same the width as the column, the effective cross-sectional area within the joint is equal to the area of the column: Figure 14.25

The design shear strength of the joint is equal to the following for a Type I-A joint: Figure 14.24 Step 3 – Determine the detailing for the joint

Because beams that have the same width as the column frame into all four sides of the joint, one-half the amount of WUDQVYHUVHUHLQIRUFHPHQWUHTXLUHGE\$&,LVSHUPLWWHGLQWKHMRLQW$&, )RUVLPSOHUGHWDLOLQJWKH VDPHWUDQVYHUVHUHLQIRUFHPHQWSURYLGHGDWWKHHQGVRIWKHFROXPQLVSURYLGHGLQWKHMRLQWVHH)LJXUH  14.9.7 Example 14.7 – Determination of Longitudinal Reinforcement: Corner Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ(LQ%XLOGLQJ)UDPLQJ2SWLRQ&LQWKH¿UVW VWRU\DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW Step 1 – Check the dimensional limits

Shortest cross-sectional dimension

ACI 18.7.2 Figure 14.10

Ratio of shortest cross-sectional dimension to the perpendicular dimension Therefore, the provided cross-sectional dimensions are adequate. Step 2 – Determine the factored load combinations

A three-dimensional analysis was performed for gravity and earthquake effects in the north-south and east-west diUHFWLRQVZKHUHLWLVDVVXPHGWKHIUDPHVDORQJFROXPQOLQHVDQGLQWKHQRUWKVRXWKGLUHFWLRQDQGWKHIUDPHV along column lines A, C, and E in the east-west direction are part of the SFRS. Rigid diaphragms are assigned at each level and to account for cracking, the following reduced moments of inertia are used for the beams and colXPQV>VHH$&,7DEOHD @ • Beams: • Columns: This column is part of the SFRS in both the north-south and east-west directions, which means it may need to be GHVLJQHGIRURUWKRJRQDOORDGHIIHFWVLQDFFRUGDQFHZLWK$6&(6(,7KHPD[LPXPD[LDOIRUFHGXHWRHDUWKTXDNHHIIHFWVLVHTXDOWRNLSVEDVHGRQHDUWKTXDNHIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQ$WWKLVVWDJHLQWKH design, the required area of longitudinal reinforcement is not known. Assume a 1 percent longitudinal reinforcement ratio and determine the design axial strength of the column:

ACI 22.4.2

14-96

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ThereforeLQDFFRUGDQFHZLWK$6&(6(,RUWKRJRQDOORDGHIIHFWVGXHWRHDUWKTXDNHIRUFHVQHHGQRWEHFRQsidered. This requirement will be checked again once the required area of longitudinal reinforcement is determined. Also, P-delta effects were automatically accounted for in the analysis and were found to be negligible. The gravity and earthquake axial forces and bending moments at the top and bottom of column E1 are given in 7DEOHIRUHDUWKTXDNHIRUFHVLQWKHHDVWZHVWGLUHFWLRQ7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGVHLVPLF D[LDOIRUFHEHQGLQJPRPHQWVDQGVKHDUIRUFHVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQG the west direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the south elevation of the building). The effects due to wind loads are smaller than those due to the earthquake loads and are not considered in this example. Table 14.17 Design Axial Forces, Bending Moments, and Shear Forces for Column E1 in Example 14.7 Axial Force (kips)

Load Case

Bending Moment (ft-kips) Bottom

Top

Shear Force (kips)

Dead





í

6.8

Live





í

3.1

Roof live load



Seismic Load Combination $&,(TD





í



$&,(TE





í

13.1

$&,(TF





í



SSR



í

í

í

SSL





í



SSR

66.6

í

í

í

SSL





í



$&,(TH

$&,(TJ SSR = sidesway to the right, SSL = sidesway to the left

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR Therefore, this load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

14-97

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the required longitudinal reinforcement

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUV is JLYHQLQ)LJXUH$OOORDGFRPELQDWLRQVIDOOZLWKLQWKHERXQGDU\RIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRU this column.  2,000 1,750

Axial Force ሺkipsሻሻ

1,500 1,250 1,000 750 500





250



 

0 0

200

400

600

800

1,000

Bending Moment ሺft-kipsሻ Figure 14.63 Design strength interaction diagram for column E1 in Example 14.7.

7U\EDUV 1RWHWKDWRUWKRJRQDOORDGHIIHFWVLQDFFRUGDQFHZLWK$6&(6(,QHHGQRWEHFRQVLGHUHGEDVHGRQWKH longitudinal bars in the column. Step 4 – Check the minimum flexural strength requirements

ACI 18.7.3

7KHPLQLPXPÀH[XUDOVWUHQJWKUHTXLUHPHQWVRI$&,DUHFKHFNHGIRUWKLVFROXPQLQWKHHDVWZHVWGLUHFWLRQ 7KHQRPLQDOÀH[XUDOVWUHQJWKVRIWKHFROXPQVDQGEHDPDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,2QO\ORDG FRPELQDWLRQVWKDWLQFOXGHHDUWKTXDNHHIIHFWVQHHGWREHFRQVLGHUHGZKHQFKHFNLQJ$&,(T  7KHÀH[XUDOUHLQIRUFHPHQWLQWKHEHDPIUDPLQJLQWRWKHMRLQWLQWKHGLUHFWLRQRIDQDO\VLVLVWKHIROORZLQJ $WFROXPQOLQHEDUV $WPLGVSDQEDUV $WFROXPQOLQHEDUV

14-98

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHQHJDWLYHÀH[XUDOVWUHQJWKRIWKHEHDPPXVWLQFOXGHWKHFRQWULEXWLRQRIWKHUHLQIRUFHPHQWLQWKHVODEZLWKLQWKH HIIHFWLYHVODEZLGWKGH¿QHGLQ$&,WKHVODELVLQWHQVLRQ 7KHHIIHFWLYHVODEZLGWKLQDFFRUGDQFHZLWK$&, LVHTXDOWRWKHIROORZLQJ

Figure 14.13

7KHEHDPXVHGLQGHWHUPLQLQJWKHQHJDWLYHÀH[XUDOVWUHQJWKLVJLYHQLQ)LJXUH 50.7ԣ

17.0ԣ

5-#7

7.0ԣ

#4 ሺtyp.ሻ

4-#7

28ԣ Figure 14.64 Beam framing into the corner joint at column E1 in Example 14.7.

$VWUDLQFRPSDWLELOLW\DQDO\VLVRIWKHVHFWLRQZLWKWKHLQHIIHFWLYHZLGWKWKHWRSEDUVLQWKHEHDPDQGWKH EDUVLQWKHVODE\LHOGV 7KHQRPLQDOÀH[XUDOVWUHQJWKRIWKHFROXPQLVGHWHUPLQHGIRUWKHIDFWRUHGD[LDOFRPSUHVVLYHIRUFHUHVXOWLQJLQWKH ORZHVWÀH[XUDOVWUHQJWKFRQVLVWHQWZLWKWKHGLUHFWLRQRIDQDO\VLV)RUWKHFROXPQEHORZWKHMRLQWWKHD[LDOIRUFHVLQ 7DEOHDUHDSSOLFDEOH7KHVHIRUFHVDUHLQGLFDWHGLQ)LJXUHZKLFKFRQWDLQVWKHQRPLQDOVWUHQJWKLQWHUDFtion diagram for this column. The factored axial compressive forces and corresponding bending moments for the load combinations with earthTXDNHHIIHFWVDUHDOVRLQGLFDWHGLQ)LJXUHIRUWKHFROXPQDERYHWKHMRLQWZKHUHLWLVDVVXPHGWKHFROXPQ above the joint has the same cross-section, concrete compressive strength, and longitudinal reinforcement as the column below the joint. )URP)LJXUHWKHPLQLPXPÀH[XUDOVWUHQJWKVIRUWKHFROXPQVDERYHDQGEHORZWKHMRLQWDUHHTXDOWRWKHIROlowing:

&KHFN$&,(T 

Figure 14.11

14-99

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3,500 ;

3,000

Column above

 Column below

Axial Force ሺkipsሻሻ

2,500

2,000

1,500

1,000

500 ; 0

;

0

;

;

200

    400

600

800

1,000

1,200

Bending Moment ሺft-kipsሻ

Figure 14.65 Nominal strength interaction diagram for column E1 in Example 14.7.

7KHUHIRUHWKHPLQLPXPÀH[XUDOVWUHQJWKUHTXLUHPHQWVRI$&,DUHVDWLV¿HG Step 5 – Check the development length requirements of ACI 18.7.4.3

Unsupported length of column: Determine the tension development length,

RIWKHORQJLWXGLQDOEDUV ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV

14-100

ACI Table 25.4.2.5

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

$VVXPLQJKRRSV

Set

ACI 25.4.2.4

Therefore,a

Figure 14.14

7KHUHIRUHWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG 8VHEDUV Comments.7KHORQJLWXGLQDOEDUVDUHDOVRDGHTXDWHIRUELD[LDOEHQGLQJRIWKHFROXPQGXHWRZLQGIRUFHV 14.9.8 Example 14.8 – Determination of Transverse Reinforcement: Corner Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ(LQ%XLOGLQJ)UDPLQJ2SWLRQ&LQWKH¿UVWVWRU\ DVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQVHH)LJXUH 7KH VODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH Step 1 – Determine the length over which flexural yielding is likely to occur

ACI 18.7.5.1

Eq. (14.15)

Step 2 – Check if the requirements of ACI 18.7.5.2(f) must be satisfied

7KHODUJHVWIDFWRUHGD[LDOIRUFHLQFOXGLQJHDUWKTXDNHHIIHFWVLVHTXDOWRNLSV

Table 14.17

Also, 7KHUHIRUHWKHUHTXLUHPHQWVRI$&,I QHHGQRWEHVDWLV¿HG

14-101

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the maximum hoop spacing based on the requirements of ACI 18.7.5.3

The maximum hoop spacing within

from each end of the column is equal to the following:

Figure 14.17

In the equation for the maximum center-to-center spacing of the laterally supported longitudinal bars, GHWHUPLQHGDVIROORZVDVVXPLQJKRRSVZLWKFURVVWLHVDURXQGDOOWKHORQJLWXGLQDOEDUV

is

Step 4 – Determine the minimum area of hoop reinforcement based on the requirements of ACI 18.7.5.4

In both directions,

Figure 14.17

KRRSVZLWKWZRFURVVWLHVVSDFHGDWLQSURYLGHV KRRSVZLWKWZRFURVVWLHVVSDFHGDWLQSURYLGHV 7U\KRRSVDQGFURVVWLHVVSDFHGDWLQRQFHQWHUZLWKLQ (note: recalculating LQVWHDGRIKRRSVDQGFURVVWLHVUHVXOWVLQWKHVDPHYDOXHIRU ).

ZLWKKRRSVDQGFURVVWLHV

Step 5 – Check if the size and spacing of the hoop reinforcement determined in Step 4 is adequate for shear

Design shear force, LVGHWHUPLQHGXVLQJWKHSUREDEOHÀH[XUDOVWUHQJWKV at the ends of the column associDWHGZLWKWKHUDQJHRIIDFWRUHGD[LDOIRUFHVDFWLQJRQWKHFROXPQ$&, 7KHGHVLJQVWUHQJWKLQWHUDFWLRQ and LVJLYHQLQ)LJXUH,WLVHYLGHQWIURPWKH diagram for this column with LVHTXDOWRIWNLSVZKLFKFRUUHVSRQGVWRDQD[LDOIRUFHRINLSVIRUVLGHVZD\ ¿JXUHWKDWWKHODUJHVW to the left. Therefore, Figure 14.15

7KLVVKHDUIRUFHLVJUHDWHUWKDQWKHPD[LPXPVKHDUIRUFHRINLSVGHWHUPLQHGIURPDQDO\VLVVHH7DEOH  The earthquake-induced shear force is the total shear force on the column. Also, the minimum factored axial force Therefore, must be including earthquake effects is equal to 66.6 kips, which is less than WDNHQHTXDOWR]HUR$&, 

14-102

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

3,500

3,000

Axial Force ሺkipsሻሻ

2,500

2,000

1,500

1,000

500

   

0 0

200

400

600

800

1,000

1,200

1,400

Bending Moment ሺft-kipsሻ

Figure 14.66 Nominal strength interaction diagram for column E1 with fy = 75 ksi and ݊ = 1.0.

7KHUHTXLUHGVSDFLQJRIWKHKRRSVDQGFURVVWLHVLQWKHGLUHFWLRQRIDQDO\VLVLVWKHIROORZLQJ

where LVGHWHUPLQHGIURPDVWUDLQFRPSDWLELOLW\DQDO\VLVXVLQJWKHD[LDOORDGHTXDOWRNLSVZKLFK is the axial load used in determining the shear force on the column. 7KXVWKHVL]HDQGVSDFLQJRIWKHKRRSVGHWHUPLQHGLQ6WHSDUHDGHTXDWHIRUVKHDU 8VHKRRSVZLWKWZRFURVVWLHVVSDFHGDWLQRQFHQWHURYHUDGLVWDQFHRIDWOHDVW the column.

from each end of

Step 6 – Determine the size and spacing of the hoop reinforcement in the remainder of the column

Outside of

the spacing of the hoops is equal to the following: Figure 14.17

The same transverse reinforcement required within segment over the lap splice length.

is used over the entire length of the column, including the

A Class B lap splice must be used. The length of the lap splice calculation of ).

VHH([DPSOHIRUWKH

8VHDIWLQODSVSOLFHOHQJWKZKLFKPXVWEHORFDWHGQHDUWKHPLGKHLJKWRIWKHFROXPQ

14-103

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHVLPLODUWRWKRVHLQ)LJXUH Comments. The transverse reinforcement determined above is also adequate for biaxial shear due to wind forces VHH$&,  14.9.9 Example 14.9 – Check of Joint Shear Strength: Corner Column in Building #1 (Framing Option C), Column is Part of the SFRS (Special Moment Frame), SDC D

&KHFNWKHMRLQWVKHDUVWUHQJWKLQWKHHDVWZHVWGLUHFWLRQIRUFROXPQ(LQ%XLOGLQJ)UDPLQJ2SWLRQ&DWWKH VHFRQGÀRRUOHYHODVVXPLQJWKHFROXPQLVSDUWRIWKH6)56DQGWKHGLPHQVLRQVRIWKHFROXPQDUHLQE\LQ VHH)LJXUH 7KHVODELVLQWKLFNDQGWKHEHDPVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWH DQG*UDGHUHLQIRUFHPHQW with 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG Step 1 – Determine the factored horizontal joint shear force

ACI 18.8.4.1

7KHMRLQWVKHDUIRUFHLVGHWHUPLQHGEDVHGRQQHJDWLYHUHLQIRUFLQJEDUVVHH([DPSOH DQGFRQVHUYDWLYHO\ assuming Eq. (14.24) Step 2 – Determine the design shear strength of the joint

Columns frame into the bottom and top of the joint (that is, the column is continuous). The beam in the direction of DQDO\VLVSURYLGHVFRQ¿QHPHQWRQWKDWIDFHRIWKHMRLQW7KHUHLVDEHDPLQWKHWUDQVYHUVHGLUHFWLRQWKDWLVWKHVDPH ZLGWKDVWKHMRLQWZKLFKSURYLGHVFRQ¿QHPHQWRQWKDWIDFHRIWKHMRLQW7KHUHIRUHWKHMRLQWW\SHLV,'VHH)LJXUH   Because the beam in the direction of analysis is the same the width as the column, the effective cross-sectional area within the joint is equal to the area of the column: Figure 14.25

The design shear strength of the joint is equal to the following for a Type I-D joint: Figure 14.24 Step 3 – Determine the detailing for the joint

%HFDXVHWKHMRLQWLVFRQ¿QHGRQRQO\WZRIDFHVWKHVDPHWUDQVYHUVHUHLQIRUFHPHQWSURYLGHGDWWKHHQGVRIWKHFROumn must be provided in the joint (ACI 18.8.3.1). 14.9.10 Example 14.10 – Design of Special Structural Wall: Building #3, Wall is Part of the SFRS (Building Frame System), SDC D, Displacement-Based Approach

'HVLJQWKHZDOORQFROXPQOLQHLQ%XLOGLQJLQWKH¿UVWVWRU\DVVXPLQJWKHZDOOLVSDUWRIWKH6)56DQGWKHZDOO WKLFNQHVVLVLQZLWKLQE\LQHQODUJHGVHJPHQWVDWHDFKHQGVHH)LJXUH 7KHVODELVLQWKLFN$OVR DQG$670$*UDGHUHLQIRUFHPHQW assume normalweight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'EDVHGRQWKHIROORZLQJ • Site Class D (determined) • •

14-104

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 1 – Determine the factored load combinations

A three-dimensional analysis was performed for seismic forces in the north-south direction where the walls on FROXPQOLQHVDQGDUHSDUWRIWKH6)56,QWKHHDVWZHVWGLUHFWLRQWKHZDOOVRQFROXPQOLQHV$DQG'DUHSDUW of the SFRS. Thus, in both directions the SFRS is a building frame system. Rigid diaphragms are assigned at each OHYHODQGWRDFFRXQWIRUFUDFNLQJSHUFHQWRIWKHJURVVPRPHQWRILQHUWLDLVXVHGIRUWKHZDOOV>VHH$&,7DEOH D @ The gravity and earthquake axial forces, bending moments, and shear forces at the base of the wall are given in 7DEOHWKHFULWLFDOVHFWLRQIRUWKLVZDOORFFXUVDWWKHEDVH 7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHG VHLVPLFEHQGLQJPRPHQWDQGVKHDUIRUFHVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHQRUWKGLUHFWLRQDQGWKH south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building). The effects due to wind loads are smaller than those due to the earthquake loads and are not considered in this example. Table 14.18 Design Axial Forces, Bending Moments, and Shear Forces for the Wall in Example 14.10 Axial Force (kips)

Bending Moment (ft-kips)

Shear Force (kips)

Dead







Live







Roof live load



Seismic



Load Case

Load Combination $&,(TD







$&,(TE







$&,(TF







SSR



í

í

SSL







SSR



SSL







$&,(TH

$&,(TJ SSR = sidesway to the right, SSL = sidesway to the left

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR where LQDFFRUGDQFHZLWK$6&(6(,7KHUHIRUHWKLV load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

14-105

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 2 – Determine the minimum reinforcement ratios for the web

ACI 18.10.2.1

Therefore, minimum Step 3 – Determine the number of curtains of reinforcement in the web

ACI 18.10.2.2

7KHUHIRUHFXUWDLQVRIORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWDUHUHTXLUHGLQWKHZHE Step 4 – Determine the required area of longitudinal reinforcement for flexure and axial forces

ACI 18.10.5

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVZDOOLVJLYHQLQ)LJXUH7KHORQJLWXGLQDOUHLQIRUFHPHQWLQWKH ZHELVEDUVVSDFHGDWLQRQFHQWHUDQGWKHORQJLWXGLQDOUHLQIRUFHPHQWLQHDFKLQE\LQVHJPHQWLV EDUV$OOORDGFRPELQDWLRQVIDOOZLWKLQWKHERXQGDU\RIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDP  35,000 35,000

30,000 30,000

AxialForce Forceሺkipsሻ ሺkipsሻሻ Axial

25,000 25,000

20,000 20,000

15,000 15,000

10,000 10,000

5,000 5,000

00 0 0

30,000 30,000

60,000 60,000

90,000 90,000

120,000 120,000

150,000 150,000

180,000 180,000

Bending Moment ሺft-kipsሻ Bending Moment ሺft-kipsሻ

Figure 14.67 Design strength interaction diagram for the wall in Example 14.10.

&KHFNWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,D  , which is equal to the depth of the 36 in. by 36 in. segments at the ends of the wall Longitudinal reinforcement ratio within the 36 in. by 36 in. segment is equal to the following: ACI Fig. R18.10.2.4

7KHUHIRUHWKHUHTXLUHPHQWVRI$&,D DUHVDWLV¿HG

14-106

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 5 – Determine the design shear force

The design shear force,

ACI 18.10.3

LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, Eq. (14.27)

Because

DQG

is equal to the greater of

ACI Table 18.10.3.1.2

The design strength interaction diagram for this wall with and is given in Figure LVHTXDOWRIWNLSVZKLFKFRUUHVSRQGVWRDQD[LDO ,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHODUJHVW IRUFHRINLSVVHH7DEOH   40,000

35,000

Axial Force ሺkipsሻሻ

30,000

25,000

20,000

15,000

10,000

5,000

0 0

40,000

80,000

120,000

160,000

200,000

Bending Moment ሺft-kipsሻ

Figure 14.68 Nominal strength interaction diagram for the wall in Example 14.10 with fy = 100 ksi and ݊ = 1.0.

)RUWKLVVWRU\EXLOGLQJWKHG\QDPLFVKHDUDPSOL¿FDWLRQIDFWRULVHTXDOWRWKHIROORZLQJ Eq. (14.29)

Therefore,

Step 6 – Check shear strength requirements

The design shear strength,

ACI 18.10.4

must be greater than or equal to

14-107

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWUDWLRXVLQJ(TV DQG 

where Therefore, required $VVXPLQJWZROD\HUVRIKRUL]RQWDOEDUVLQWKHZHEWKHUHTXLUHGVSDFLQJLVHTXDOWRWKHIROORZLQJ

&KHFNWKHXSSHUOLPLWRQVKHDUVWUHQJWKEDVHGRQEDUVZLWKDFHQWHUWRFHQWHUVSDFLQJRILQ

ACI 18.10.4.4

8VHKRUL]RQWDOEDUVVSDFHGDWLQRQFHQWHU Step 7 – Determine if special boundary elements are required

ACI 18.10.6.2

6SHFLDOERXQGDU\HOHPHQWVDUHUHTXLUHGZKHUHWKHIROORZLQJHTXDWLRQLVVDWLV¿HG Eq. (14.32)

From the analysis of the structure with the code-prescribed seismic forces applied over the height of the building, LVHTXDOWRLQ7KHUHIRUH the displacement at the top of the building, ASCE/SEI Eq. (12.8-15)

ZKHUHWKHGHÀHFWLRQDPSOL¿FDWLRQIDFWRU LVREWDLQHGIURP$6&(6(,7DEOHIRUDEXLOGLQJIUDPHV\Vtem with special reinforced concrete shear walls. ACI 18.10.6.2(a)

From a strain compatibility analysis of the section, the largest neutral axis depth, LVHTXDOWRLQZKLFKFRUUHVSRQGVWRWKHD[LDOIRUFHRINLSVREWDLQHGIURP$&,(TH >VHH7DEOH@ Thus,

Therefore, special boundary elements are required at the ends of the wall. Step 8 – Determine the vertical extent of the special boundary element transverse reinforcement

Provide special boundary element transverse reinforcement vertically over at least the greater of the following lengths from the base of the wall: ACI 18.10.6.2(b)(i)

14-108

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 9 – Check if ACI 18.10.6.2(b)(ii) or (iii) is satisfied

Because varies over an average or representative value of can be used to determine if these requirements are VDWLV¿HGRUQRW$OWHUQDWLYHO\FKHFNWKHVHUHTXLUHPHQWVXVLQJ HTXDOWRLQDQGLQ ACI 18.10.6.2(b)(ii)

ACI 18.10.6.2(b)(iii)

7KHUHIRUHERWK$&,E LL DQGLLL DUHVDWLV¿HGIRU It can also be determined that these reZKLFKPHDQVWKHUHTXLUHPHQWVDUHVDWLV¿HGIRUDQ\DYHUDJHYDOXHRI TXLUHPHQWVDUHVDWLV¿HGIRU Step 10 – Determine the horizontal length of the special boundary elements

ACI 18.10.6.4(a)

7KHVSHFLDOERXQGDU\HOHPHQWVPXVWH[WHQGKRUL]RQWDOO\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHUDGLVWDQFHHTXDOWRWKH following: Figure 14.38

For simpler detailing, provide special boundary element transverse reinforcement within the 36 in. by 36 in. segments at each end of the wall. Step 11 – Check the width of the flexural compression zone

ACI 18.10.6.4(b) and (c)

7KHUHIRUHWKHUHTXLUHPHQWRI$&,E LVVDWLV¿HGDQGWKHUHTXLUHPHQWRI$&,F LVQRWDSSOLFDEOH Step 12 – Determine the required special boundary element transverse reinforcement

ACI 18.10.6.4(e) and (f)

The maximum vertical spacing of the special boundary element transverse reinforcement is equal to the following:

Figure 14.38

14-109

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

In the equation for the maximum center-to-center spacing of the laterally supported longitudinal bars, GHWHUPLQHGDVIROORZVDVVXPLQJKRRSVZLWKFURVVWLHVDURXQGDOOWKHORQJLWXGLQDOEDUV

is

In both directions,

ACI Table 18.10.6.4(g)

KRRSVZLWKVL[FURVVWLHVVSDFHGDWLQSURYLGHV 8VHKRRSVDQGFURVVWLHVDURXQGDOOWKHORQJLWXGLQDOEDUVVSDFHGDWLQRQFHQWHU Step 13 – Determine if transverse reinforcement required by ACI 18.10.6.5 is needed at the ends of the wall

,WFDQEHGHWHUPLQHGWKDWORQJLWXGLQDOEDUVLQWKHLQE\LQVHJPHQWVDQGORQJLWXGLQDOEDUV VSDFHGDWLQLQWKHZHEDUHDGHTXDWHIRUÀH[XUHDQGD[LDOIRUFHVDWWKHVHFWLRQLPPHGLDWHO\DERYHWKHWRSRIWKH fourth story (which is above the required vertical extent of the special boundary elements; see Step 8). The reincalculated in accordance with ACI IRUFHPHQWUDWLRLQWKHLQE\LQVHJPHQWV>WKDWLVRYHUWKHGLVWDQFH D @LVHTXDOWRWKHIROORZLQJ Figure 14.39

7KHUHIRUHWUDQVYHUVHUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,LVUHTXLUHGDWWKHHQGVRIWKHZDOO7KHVL]H RIWKHWUDQVYHUVHUHLQIRUFHPHQWPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,WKXVXVHKRRSVDQGFURVVWLHV DURXQGHYHU\RWKHUORQJLWXGLQDOEDULQWKHLQE\LQVHJPHQWV Because the section immediately above the top of the fourth story is not expected to yield (that is, this section is see Step 8), the maximum spacing of the transverse reinforcement is equal to the lesser located above RIWKHIROORZLQJIRU*UDGHUHLQIRUFHPHQW Figure 14.39

$WWKHVHFWLRQLPPHGLDWHO\DERYHWKHWRSRIWKHHLJKWKVWRU\LWFDQEHGHWHUPLQHGWKDWORQJLWXGLQDOEDUVLQWKH HQODUJHGVHJPHQWVDQGORQJLWXGLQDOEDUVVSDFHGDWDLQLQWKHZHEDUHDGHTXDWHIRUÀH[XUHDQGD[LDOIRUFHV $VVXPLQJWKHHQODUJHGVHJPHQWVDWWKHHQGRIWKHZDOODUHLQE\LQLQVWRULHVWKURXJKWKHUHLQIRUFHPHQW ratio in the enlarged segments is equal to the following:

7KHUHIRUHXVHKRRSVDQGFURVVWLHVDURXQGHYHU\RWKHUORQJLWXGLQDOEDUDWDLQVSDFLQJLQVWRULHVWKURXJK >WKHQXPEHURISURYLGHGFURVVWLHVVDWLV¿HVWKHUHTXLUHPHQWLQ$&,E WKDWWKHKRRSVFRQIRUPWR$&, @ D WKURXJKH ZKHUH

14-110

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 14 – Detail the reinforcement

,WFDQEHGHWHUPLQHGWKDWWKHORQJLWXGLQDOEDUVLQWKHHQODUJHGVHJPHQWVDWWKHHQGVRIWKHZDOODQGWKH ORQJLWXGLQDOEDUVLQWKHZHEDUHQRORQJHUUHTXLUHGIRUÀH[XUHDQGD[LDOIRUFHVDWDVHFWLRQVOLJKWO\EHORZWKHWRSRI WKHWKLUGÀRRUOHYHO$FFRUGLQJWR$&,D ORQJLWXGLQDOUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWIWDERYHWKH above the base of the point it is no longer required, which in this example, is approximately ZDOORUHTXLYDOHQWO\IWDERYHWKHWRSRIWKHIRXUWKÀRRUOHYHO$OVRWKHORQJLWXGLQDOEDUVQHHGQRWH[WHQGPRUH than DERYHWKHQH[WÀRRUOHYHOZKLFKLVWKHIRXUWKÀRRUOHYHO'HWHUPLQH IRUWKHORQJLWXGLQDOEDUV ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV

Set

ACI 25.4.2.4

Therefore,

7KXVWKHORQJLWXGLQDOEDUVQHHGQRWH[WHQGPRUHWKDQIWDERYHWKHWRSRIWKHIRXUWKÀRRUOHYHO )RUVLPSOHUGHWDLOLQJH[WHQGWKHORQJLWXGLQDOEDUVWRWKHWRSRIWKH¿IWKÀRRUOHYHOZKHUHWKH\FDQEHVSOLFHG LPPHGLDWHO\DERYHWKH¿IWKÀRRUVODEZLWKWKHORQJLWXGLQDOEDUVUHTXLUHGLQWKHHQODUJHGVHJPHQWVDERYH VHH6WHS 7KHEDUVLQWKHZHEFDQEHVSOLFHGLQDVLPLODUIDVKLRQ $FFRUGLQJWR$&,F ODSVSOLFHVRIORQJLWXGLQDOUHLQIRUFHPHQWZLWKLQWKHVSHFLDOERXQGDU\UHJLRQVDUHQRW above the critical sections and below the critical sections permitted over a height equal to ZKHUH\LHOGLQJLVOLNHO\WRRFFXU,QWKLVH[DPSOHWKHFULWLFDOVHFWLRQRFFXUVDWWKHEDVHRIWKHZDOOVRWKHORQJLWXGLQDOEDUVDUHSHUPLWWHGWREHODSVSOLFHGWRWKHORQJLWXGLQDOEDUVLPPHGLDWHO\DERYHWKH¿IWKÀRRUVODE :KHUHEDUVRIGLIIHUHQWVL]HDUHODSVSOLFHGLQWHQVLRQWKHWHQVLRQODSVSOLFHOHQJWK RIWKHVPDOOHUEDU$&, )RUWKHEDUV of the larger bar and IRUWKHEDUV

is equal to the greater of (see above). Calculate

ACI Eq. (25.4.2.4a)

14-111

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV

Set

ACI 25.4.2.4

Therefore,

For a Class B tension lap splice: ACI Table 25.5.2.1

7KHUHIRUHSURYLGHDIWORQJODSVSOLFHLPPHGLDWHO\DERYHWKH¿IWKÀRRUVODE The special boundary element longitudinal reinforcement must extend into the support a distance of at least >$&,M @7KHH[WHQWRIWKHVSHFLDOERXQGDU\HOHPHQWWUDQVYHUVHUHLQIRUFHPHQWLQWRWKHVXSSRUWGHSHQGVRQ the type of support and the distance from the edge of the special boundary element to the edge of the support (see )LJXUH  5HLQIRUFHPHQWGHWDLOVIRUWKLVZDOODUHJLYHQLQ)LJXUH,QDFFRUGDQFHZLWK$&,L WKHZHEYHUWLcal reinforcement within the vertical extent of the special boundary elements must have lateral support provided by WKHFRUQHURIDKRRSRUE\DFURVVWLHZLWKVHLVPLFKRRNVDWHDFKHQG,QWKLVH[DPSOHFURVVWLHVDUHSURYLGHGDWD LQVSDFLQJZKLFKPDWFKHVWKHVSDFLQJRIWKHKRUL]RQWDOZHEUHLQIRUFHPHQW7KHFURVVWLHVVDWLVI\WKHUHTXLUHPHQWVRI$&,IRUWKHYHUWLFDOZHEEDUVDQGWKHSURYLGHGVSDFLQJLVOHVVWKDQWKHLQPD[LPXPVSDFLQJVSHFL¿HGLQ$&,L  Comments.)RUDFRQVWUXFWLRQVHTXHQFHZKHUH DZDOOLVFRQVWUXFWHGWRWKHXQGHUVLGHRIWKHVODE WKHVODELV constructed, and (3) the wall above the slab is constructed, the concrete within the thickness of the slab at the locaWLRQRIWKHVSHFLDOERXQGDU\HOHPHQWVPXVWKDYHDVSHFL¿HGFRPSUHVVLYHVWUHQJWKRIDWOHDVWWLPHVWKHVSHFL¿HG FRPSUHVVLYHVWUHQJWKRIWKHZDOO>$&,K @,QWKLVH[DPSOHWKHVODEPXVWKDYHDPLQLPXPFRPSUHVVLYH For a construction sequence where the wall is constructed ahead of the slab (for strength of example, where slip forms are used to construct the wall), this requirement is not applicable.

14-112

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

BA

CA 20ᇱ -6ԣ

R 16 3ᇱ -0ԣ

2-#7 @ 10ᇳ H

15

2-#7 @ 9ᇳ V

13

2ᇱ -0ԣ

3ᇱ -0ԣ

14

12

16 @ 9ᇱ -6ᇳ ൌ 152ᇱ -0ԣ

11

#3 crossties @ 10ԣ

#5 hoops and 28-#11 crossties @ 6ԣ

൑ 6ԣ

10 Section A-A 9 8

3ᇱ -0ԣ

2-#5 @ 10ᇳ H 2-#7 @ 9ᇳ V

7

4

A

#3 hoops and 28-#10 crossties @ 6ԣ Section B-B Level

Elevation boundary elements

2

Required extent of special

A

23.5Ԣ

3

2ᇱ -0ԣ

B boundary elements

5

Provided extent of special

B

3ᇱ -0ԣ

6

R 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

Height ሺftሻ 152.0 142.5 133.0 123.5 114.0 104.5 95.0 85.5 76.0 66.5 57.0 47.5 38.0 28.5 19.0 9.5

Story 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

݂௖ᇱ ሺpsiሻ

Web

Enlarged Segments

5,000

18ԣ 2-#5 @ 9ԣ V 2-#5 @ 10ԣ H

30ԣ ൈ 30ԣ 28-#8 #3 hoops and crossties @ 6ԣ

24ԣ 2-#7 @ 9ԣ V 2-#5 @ 10ԣ H

36ԣ ൈ 36ԣ 28-#10 #3 hoops and crossties @ 6ԣ

24ԣ 2-#7 @ 9ԣ V 2-#7 @ 10ԣ H #3 crossties @ 10ԣ

36ԣ ൈ 36ԣ 28-#11 #5 hoops and crossties @ 6ԣ

7,500

Figure 14.69 Reinforcement details for the wall in Example 14.10.

14-113

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.9.11 Example 14.11 – Design of Special Structural Wall: Building #4, Wall is Part of the SFRS (Dual System), SDC D, Compressive Stress Approach

'HVLJQWKHZDOORQFROXPQOLQH'LQ%XLOGLQJLQWKH¿UVWVWRU\DVVXPLQJWKHZDOOLVSDUWRIWKH6)56VHH)LJXUH  7KHZDOOWKLFNQHVVLVLQLQVWRULHVWKURXJKLQLQVWRULHVWKURXJKDQGLQLQVWRULHV LQVWRULHVWKURXJKDQG LQVWRULHV WKURXJK$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK  WKURXJK$OVRDVVXPH$670$*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'EDVHGRQWKHIROORZLQJ • 6LWH&ODVV'GHIDXOW • • Step 1 – Determine the factored load combinations

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ÀDQJHVDWWKHHQGVRIWKH ZDOOLQDFFRUGDQFHZLWK$&,WKHHIIHFWLYHÀDQJHZLGWKLVXVHGLQGHWHUPLQLQJWKHPRPHQWRILQHUWLDRI WKHVHFWLRQSHUPLWWHGWRUHVLVWWKHHDUWKTXDNHORDGHIIHFWVIRUGHWHUPLQDWLRQRIJUDYLW\ORDGHIIHFWVWKHIXOOZDOODUHD PXVWEHXVHG 7KHHIIHFWLYHÀDQJHZLGWKLVHTXDOWRWKHOHVVHURIWKHIROORZLQJ

Figure 14.35

7RWDOHIIHFWLYHÀDQJHZLGWK 7KHUHIRUHWKHWRWDOÀDQJHZLGWKLVHIIHFWLYHDQGPD\EHXVHGWRGHWHUPLQHWKHPRPHQWRILQHUWLDDQGWRUHVLVWODWHUDO HDUWKTXDNHHIIHFWV 7KHJUDYLW\DQGHDUWKTXDNHD[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVDWWKHEDVHRIWKHZDOODUHJLYHQLQ 7DEOHWKHFULWLFDOVHFWLRQIRUWKLVZDOORFFXUVDWWKHEDVH 7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGVHLV PLFEHQGLQJPRPHQWDQGVKHDUIRUFHVLJQL¿HVWKHHDUWKTXDNHORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQGWKHZHVW GLUHFWLRQWKDWLVVLGHVZD\WRWKHULJKW665 DQGVLGHVZD\WRWKHOHIW66/ ZKHQORRNLQJDWWKHVRXWKHOHYDWLRQRI WKHEXLOGLQJ 7KHHIIHFWVGXHWRZLQGORDGVDUHVPDOOHUWKDQWKRVHGXHWRWKHHDUWKTXDNHORDGVDQGDUHQRWFRQVLG HUHGLQWKLVH[DPSOH

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.19 Design Axial Forces, Bending Moments, and Shear Forces for the Wall in Example 14.11 Axial Force (kips)

Bending Moment (ft-kips)

Shear Force (kips)

Dead







Live







Roof live load



Seismic



Load Case

Load Combination $&,(TD







$&,(TE







$&,(TF







SSR



í

í

SSL







SSR



í

í

SSL







$&,(TH

$&,(TJ SSR = sidesway to the right, SSL = sidesway to the left

Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR where LQDFFRUGDQFHZLWK$6&(6(,7KHUHIRUHWKLV load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 2 – Determine the minimum reinforcement ratios for the web

ACI 18.10.2.1

Therefore, minimum Step 3 – Determine the number of curtains of reinforcement in the web

ACI 18.10.2.2

7KHUHIRUHFXUWDLQVRIORQJLWXGLQDODQGWUDQVYHUVHUHLQIRUFHPHQWDUHUHTXLUHGLQWKHZHE

14-115

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

60,000

Axial Force ሺkipsሻሻ

50,000

40,000

30,000

20,000

10,000

0 0

100,000

200,000

300,000

400,000

500,000

600,000

Bending Moment ሺft-kipsሻ

Figure 14.70 Design strength interaction diagram for the wall in Example 14.11.

Step 4 – Determine the required area of longitudinal reinforcement for flexure and axial force

ACI 18.10.5

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVZDOOLVJLYHQLQ)LJXUH7KHORQJLWXGLQDOUHLQIRUFHPHQW FRQVLVWVRIEDUVLQHDFKÀDQJHDQGEDUVVSDFHGDWLQRQFHQWHULQWKHZHE1RWHWKDWEDUVDUH UHTXLUHGDWWKHPLGGHSWKRIWKHÀDQJHWZREDUVHDFKHQG LQRUGHUWRVDWLVI\WKHPD[LPXPVSDFLQJUHTXLUHPHQWLQ $&,I IRUEDUVDURXQGWKHSHULPHWHURIWKHVSHFLDOERXQGDU\HOHPHQWWKHPD[LPXPVSDFLQJLVWKHOHVVHU RILQDQGWZRWKLUGVWKHERXQGDU\HOHPHQWWKLFNQHVVZKLFKLVHTXDOWRLQVHH6WHSVDQGEHORZ $OO load combinations fall within the boundary of the design strength interaction diagram. &KHFNWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,VHH)LJXUH  ACI Figure R18.10.2.4

7KHUHDUHEDUVZLWKLQWKHIWZLGWKDWHDFKHQGRIERWKÀDQJHVWKHUHIRUH

7KHUHDUHEDUVZLWKLQWKHIWZLGWKLQWKHZHEWKHUHIRUH

7KXVWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HGIRUWKLVZDOO

14-116

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ͲǤͳͷκ௪ ൌ ʹǤ͹Ԣ

ͲǤͳͷκ௪ ൌ ʹǤ͹Ԣ

ʹǤͲԢ

κ௪ ൌ ͳͺǤͲԢ

ʹǦ͓ͺ

ͲǤͳͷκ௪ ൌ ʹǤ͹Ԣ

͓ͺሺ–›’Ǥ ሻ

ʹǤͲԢ

–Š‡””‡‹ˆ‘”…‡‡–‘–•Š‘™ˆ‘”…Žƒ”‹–›

Figure 14.71 Minimum longitudinal reinforcement in accordance with ACI 18.10.2.4 (see Figure 14.71).

Step 5 – Determine the design shear force

The design shear force,

ACI 18.10.3

LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, Eq. (14.27)

Because

is equal to the greater of

DQG

ACI Table 18.10.3.1.2

The design strength interaction diagram for this wall with and is given in Figure LVHTXDOWRIWNLSVZKLFKFRUUHVSRQGVWRDQD[LDO ,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHODUJHVW IRUFHRINLSVVHH7DEOH 

)RUWKLVVWRU\EXLOGLQJWKHG\QDPLFVKHDUDPSOL¿FDWLRQIDFWRULVHTXDOWRWKHIROORZLQJ Eq. (14.29)

14-117

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

110,000 100,000 90,000

Axial Force ሺkipsሻሻ

80,000 70,000 60,000 50,000 40,000 30,000 20,000 10,000 0 0

100,000

200,000

300,000

400,000

500,000

600,000

700,000

Bending Moment ሺft-kipsሻ

Figure 14.72 Nominal strength interaction diagram for the wall in Example 14.11 with fy = 100 ksi and ݊ = 1.0.

Therefore,

Step 6 – Check shear strength requirements

The design shear strength,

ACI 18.10.4

must be greater than or equal to

'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWUDWLRXVLQJ(TV DQG 

where Therefore, required $VVXPLQJWZROD\HUVRIKRUL]RQWDOEDUVLQWKHZHEWKHUHTXLUHGVSDFLQJLVHTXDOWRWKHIROORZLQJ

&KHFNWKHXSSHUOLPLWRQVKHDUVWUHQJWKEDVHGRQEDUVZLWKDFHQWHUWRFHQWHUVSDFLQJRILQ

ACI 18.10.4.4

8VHKRUL]RQWDOEDUVVSDFHGDWLQRQFHQWHU

14-118

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 7 – Determine if special boundary elements are required

ACI 18.10.6.3

6SHFLDOERXQGDU\HOHPHQWVDUHUHTXLUHGZKHUHWKHIROORZLQJHTXDWLRQLVVDWLV¿HG Eq. (14.33)

)URP7DEOHWKHORDGFRPELQDWLRQWKDWLQFOXGHVHDUWKTXDNHHIIHFWVZKLFKUHVXOWVLQWKHODUJHVW H 

is ACI Eq.

Therefore, special boundary elements are required at the ends of the wall. Step 8 – Determine the vertical extent of the special boundary element transverse reinforcement

Special boundary elements are permitted to be discontinued where the calculated ,WFDQEHGHWHUPLQHGWKDWVSHFLDOERXQGDU\HOHPHQWVFDQEHWHUPLQDWHGDERYHWKHWZHQW\WKLUGÀRRUOHYHOWKHZDOO WKLFNQHVVLVLQDQGWKHFRPSUHVVLYHVWUHQJWKRIWKHFRQFUHWHLVHTXDOWRSVLDWWKLVHOHYDWLRQZKLFKPDNHV ). the stress limit equal to Step 9 – Determine the horizontal length of the special boundary elements

ACI 18.10.6.4(a)

7KHVSHFLDOERXQGDU\HOHPHQWVPXVWH[WHQGKRUL]RQWDOO\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHUDGLVWDQFHHTXDOWRWKH following: Figure 14.38

The largest neutral axis depth, used in the determination of is calculated from a strain compatibility analysis XVLQJWKHD[LDOIRUFHDQGQRPLQDOPRPHQWVWUHQJWKIURP$&,(TH LQ7DEOH ,WLVHYLGHQWWKDWWKHFDOFXODWHGKRUL]RQWDOOHQJWKRIWKHVSHFLDOERXQGDU\HOHPHQWLQFOXGHVWKHLQWKLFNÀDQJH LQWRWKHZHE$FFRUGLQJWR$&,G DVSHFLDOERXQGDU\HOHPHQWPXVW and extends H[WHQGDWOHDVWLQLQWRWKHZHEIRUÀDQJHGVHFWLRQV%DVHGRQWKHLQVSDFLQJRIWKHORQJLWXGLQDOEDUVLQ WKHZHEXVHDLQH[WHQVLRQLQWRWKHZHE Step 10 – Check the width of the flexural compression zone

ACI 18.10.6.4(b) and (c)

7KHUHIRUHWKHUHTXLUHPHQWRI$&,E LVVDWLV¿HGDQGWKHUHTXLUHPHQWRIACIF LVQRWDSSOLcable.

14-119

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 11 – Determine the required special boundary element transverse reinforcement

ACI 18.10.6.4(e) and (f)

• &RQ¿QHPHQWRIWKHÀDQJH 7KHPD[LPXPYHUWLFDOVSDFLQJRIWKHVSHFLDOERXQGDU\HOHPHQWWUDQVYHUVHUHLQIRUFHPHQWLQWKHÀDQJHLVHTXDO to the following:

Figure 14.38

In the equation for the maximum center-to-center spacing of the laterally supported longitudinal bars, is HTXDOWRDSSUR[LPDWHO\LQDVVXPLQJWUDQVYHUVHUHLQIRUFHPHQWFRQ¿QLQJDOOWKHORQJLWXGLQDOEDUVLQWKH ÀDQJH

)RUFRQ¿QHPHQWRIWKHÀDQJHLQWKHGLUHFWLRQRIDQDO\VLV

ACI Table 18.10.6.4(g)

:LWKRYHUODSSLQJKRRSVDQGFURVVWLHVHQJDJLQJDOOWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHÀDQJHSURYLGHG 8VHKRRSVDQGFURVVWLHVDURXQGDOOWKHORQJLWXGLQDOEDUVVSDFHGDWLQRQFHQWHULQWKHÀDQJHV • &RQ¿QHPHQWRIWKHZHE The maximum vertical spacing of the special boundary element transverse reinforcement in the web is equal to the following:

Figure 14.38

In the equation for HTXDOWRLQ

14-120

the maximum center-to-center spacing of the laterally supported longitudinal bars,

is

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUFRQ¿QHPHQWRIWKHZHELQWKHGLUHFWLRQRIDQDO\VLV ʹͶԣ

VHH)LJXUH 

ܾ௖ǡଶ ൌ ʹͶԣ

ʹͶԣ

ܾ௖ǡଵ ൌ ʹʹǤͷԣ



Figure 14.73 Special boundary transverse reinforcement in the web of the wall in Example 14.11.

ACI Table 18.10.6.4(g)

ZKHUHDYHUWLFDOVSDFLQJRILQLVXVHGLQWKHZHEWRPDWFKWKHYHUWLFDOVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWLQWKHÀDQJHV :LWKDKRRSDQGRQHFURVVWLHSURYLGHG In the direction perpendicular to the direction of analysis,

14-121

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

:LWKDKRRSDQGWZRFURVVWLHVSURYLGHG (one of the hoop legs is in WKHÀDQJHVRWKHDUHDRIWKDWOHJLVQRWLQFOXGHGLQWKHSURYLGHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQW  8VHDKRRSDQGFURVVWLHVVSDFHGDWLQRQFHQWHULQWKHZHE Step 12 – Determine if transverse reinforcement required by ACI 18.10.6.5 is needed at the ends of the wall

It can be determined that a reinforcement ratio greater than PXVWEHSURYLGHGLQWKHÀDQJHVLQVWRULHVDQGDERYH7KHUHIRUHWUDQVYHUVHUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,LVUHTXLUHGLQWKHÀDQJHV LQWKHVHVWRULHV7KHVL]HRIWKHWUDQVYHUVHUHLQIRUFHPHQWPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,

2Ԣ-0ԣ

Hoop overlap

2-#8 @ 10ᇳ V 2Ԣ-0ԣ

18Ԣ-0ԣ

2-#7 @ 11ᇳ H

2-#7 @ 11ᇳ H #5 hoop and crossties @ 4ԣ

#5 overlapping hoops and crossties @ 4ԣሺsee Detail Aሻ

۲‫ۯ ܔܑ܉ܜ܍‬

#8 bars Hoop overlap

#5 hoops 

4ԣ

ሺtyp. ሻ

‫ ܜܖ܍ܕ܍܋ܚܗ܎ܖܑ܍܀‬۳‫ܖܗܑܜ܉ܞ܍ܔ‬

Figure 14.74 Reinforcement details for the wall in Example 14.11.

14-122

ሺtyp. ሻ

44-#8 V

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

5HLQIRUFHPHQWGHWDLOVIRUWKLVZDOODUHJLYHQLQ)LJXUH7KHVSHFLDOWUDQVYHUVHUHLQIRUFHPHQWLQWKHÀDQJHV FRQVLVWVRIDVHULHVRIRYHUODSSLQJKRRSVDQGVXSSOHPHQWDOGHJUHHFURVVWLHVVLPLODUWR'HWDLOE GHSLFWHGLQ $&,)LJXUH5D7KHJHRPHWU\RIWKHRYHUODSSLQJKRRSVGHSLFWHGLQ'HWDLO$RI)LJXUHVDWLV¿HVWKH UHTXLUHPHQWVLQ$&,I  Comments.)RUDFRQVWUXFWLRQVHTXHQFHZKHUH DZDOOLVFRQVWUXFWHGWRWKHXQGHUVLGHRIWKHVODE WKHVODELV constructed, and (3) the wall above the slab is constructed, the concrete within the thickness of the slab at the locaWLRQRIWKHVSHFLDOERXQGDU\HOHPHQWVPXVWKDYHDVSHFL¿HGFRPSUHVVLYHVWUHQJWKRIDWOHDVWWLPHVWKHVSHFL¿HG FRPSUHVVLYHVWUHQJWKRIWKHZDOO>$&,K @,QWKLVH[DPSOHWKHVODELQÀRRUOHYHOVWKURXJKPXVW For a construction sequence where the wall is have a minimum compressive strength of constructed ahead of the slab (for example, where slip forms are used to construct the wall), this requirement is not applicable. This wall is also part of the SFRS for earthquake forces in the north-south direction. Because the axial force due to seismic forces acting along either of the principal plan axes is negligible, orthogonal load effects in accordance ZLWK$6&(6(,QHHGQRWEHFRQVLGHUHG7KHSURYLGHGUHLQIRUFHPHQWPXVWVDWLVI\WKHHIIHFWVIURPD[LDOIRUFHV bending moments, and shear forces in both orthogonal directions independently. 7KHVSHFLDOPRPHQWIUDPHVRQFROXPQOLQHV$DQG*PXVWEHGHVLJQHGWRLQGHSHQGHQWO\UHVLVWDWOHDVWSHUFHQW of the prescribed seismic forces. Design and detailing requirements for the beams, columns, and joints are given in 6HFWVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\ 14.9.12 Example 14.12 – Design of a Coupling Beam (Dual System): Building #4, SDC D

'HVLJQWKHFRXSOLQJEHDPRQFROXPQOLQHEHWZHHQFROXPQOLQHV&DQG'LQ%XLOGLQJDWWKHHOHYHQWKÀRRUOHYHO VHH)LJXUH 7KHZLGWKRIWKHEHDPLVHTXDOWRWKHZDOOWKLFNQHVVZKLFKLVLQDQGWKHGHSWKRIWKHEHDPLV DQG$670$*UDGHUHLQIRUFHPHQW LQ$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK Step 1 – Determine the factored load combinations

A three-dimensional analysis was performed for seismic forces in the north-south direction where the walls on colXPQOLQHVDQGDQGWKHPRPHQWIUDPHVRQFROXPQOLQHVDQGDUHSDUWRIWKH6)56,QWKHHDVWZHVWGLUHFWLRQ the walls on column lines C, D, and E and the moment frames on column lines A and G are part of the SFRS. Thus, in both directions the SFRS is a dual system with special structural walls and special moment frames capable of LQGHSHQGHQWO\UHVLVWLQJDWOHDVWSHUFHQWRIWKHSUHVFULEHGVHLVPLFIRUFHV5LJLGGLDSKUDJPVDUHDVVLJQHGDWHDFK OHYHODQGWRDFFRXQWIRUFUDFNLQJSHUFHQWRIWKHJURVVPRPHQWRILQHUWLDLVXVHGIRUWKHZDOOVDQGEHDPVDQG SHUFHQWIRUWKHFROXPQV>VHH$&,7DEOHD @ 7KHJUDYLW\DQGHDUWKTXDNHVKHDUIRUFHVLQWKHEHDPDWWKHHOHYHQWKÀRRUOHYHODUHJLYHQLQ7DEOHWKHPD[LPXPVKHDUIRUFHLQWKHFRXSOLQJEHDPRFFXUVDWWKLVÀRRUOHYHO 7KHHIIHFWVGXHWRZLQGORDGVDUHVPDOOHUWKDQ those due to the earthquake loads and are not considered in this example. Table 14.20 Design Shear Forces for the Coupling Beam in Example 14.12 Load Case

Shear Force (kips)

Dead



Live



Seismic (table continued on next page)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.20 Design Shear Forces for the Coupling Beam in Example 14.12 (cont.) Shear Force (kips)

Load Case

Load Combination 1





 SSR



SSL

í

SSR



SSL

í

3



Further explanation is needed on how the load factors in the load combinations that include are obtained. In $&,(TXDWLRQH WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQDUHDGGLWLYHVR where LQDFFRUGDQFHZLWK$6&(6(,7KHUHIRUHWKLV load combination becomes the following:

6LPLODUO\LQ$&,(TXDWLRQJ WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR , and this load combination becomes the following:

Step 2 – Determine the clear span to height ratio

Clear span +HLJKW

Because the coupling beam is permitted to be reinforced either with two intersecting groups of GLDJRQDOO\SODFHGEDUVV\PPHWULFDODERXWWKHPLGVSDQRIWKHEHDPRULQDFFRUGDQFHZLWK$&,WKURXJK IRUEHDPVLQVSHFLDOPRPHQWIUDPHV$&, 7KHIRUPHURIWKHWZRRSWLRQVLVVHOHFWHGLQWKLVH[DPSOH Step 3 – Determine the required area of the diagonal reinforcing bars

The required area of the diagonal bars,

where

ACI 18.10.7.4(a)

is determined by the following equation:

for diagonally reinforced coupling beams.

ACI 21.2.4.4

In order to determine the angle between the diagonal bars and the longitudinal axis of the coupling beam, the centroid of the diagonal bar bundles must be located with respect to the centroid of the beam. The minimum out-tofor the dimension parallel to and out dimensions of the transverse reinforcement are DORQJWKHRWKHUVLGHV$VVXPHLQE\LQRXWWRRXWGLPHQVLRQVRIWKHWUDQVYHUVH UHLQIRUFHPHQWDURXQGWKHGLDJRQDOEDUVDQGDLQFRYHUWRWKHWUDQVYHUVHUHLQIRUFHPHQWQHDUWKHHQGVRIWKHEHDP VHH)LJXUH 

14-124

ͳǤͷԣ

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Ͷʹԣ

ߙ

‡–”‘‹†‘ˆ †‹ƒ‰‘ƒŽ„ƒ”•

ͳ͵Ǥͷ ൰ ൌ ͳͷǤ͹ι ͶͺǤͲ

ߙ ൌ –ƒିଵ ൬

ͶԢǦͲԣ

ͺᇱ ǦͲԣ Figure 14.75 Determination of the angle between the diagonal bars and the longitudinal axis of the coupling beam in Example 14.12.

The distance from the top or bottom of the section to the centroid of the diagonal bar bundles at the face of the couwhich is approximately equal to for this beam. pling beam is equal to Thus, the distance from the centroid of the beam to the centroid of the diagonal bars is Therefore,

The required area of diagonal reinforcement is equal to the following:

3URYLGHEDUVLQHDFKJURXSRIGLDJRQDOEDUV Check the upper limit on the design shear strength:

ACI Eq. (18.10.7.4)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 4 – Determine the required area of transverse reinforcement

7KHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWLVGHWHUPLQHGXVLQJERWKRSWLRQVLQ$&, • 2SWLRQ&RQ¿QHPHQWRILQGLYLGXDOGLDJRQDOV Area LVGHWHUPLQHGDVVXPLQJDLQFRYHUDURXQGWKHGLDJRQDOEDUJURXS

ACI 18.10.7.4(c)

Also, $VVXPLQJEDUVIRUWUDQVYHUVHUHLQIRUFHPHQWDURXQGWKHGLDJRQDOEDUVWKHUHTXLUHGVSDFLQJLVHTXDOWRWKHIROlowing:

Figure 14.43

In the equation for equal to

the maximum center-to-center spacing of the laterally supported longitudinal bars,

is

7KHUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWLVHTXDOWRWKHIROORZLQJDVVXPLQJDLQVSDFLQJ

8VHKRRSVDQGFURVVWLHVDWDVSDFLQJRILQRQFHQWHU Additional longitudinal and transverse reinforcement must be provided around the perimeter of the beam with a VSDFLQJOHVVWKDQRUHTXDOWRLQ>$&,F LL @ Required area of reinforcement )RUWKHORQJLWXGLQDOUHLQIRUFHPHQWSURYLGHEDUV uniformly distributed around the SHULPHWHURIWKHEHDPDWDFHQWHUWRFHQWHUVSDFLQJOHVVWKDQLQ)RUWKHWUDQVYHUVHUHLQIRUFHPHQWXVHEDUV VSDFHGDWLQRQFHQWHU 5HLQIRUFHPHQWGHWDLOVIRUWKLVRSWLRQDUHJLYHQLQ)LJXUH • 2SWLRQ&RQ¿QHPHQWRIWKHHQWLUHFURVVVHFWLRQ The required spacing of the transverse reinforcement is equal to the following:

ACI 18.10.7.4(d)

Figure 14.44

$VVXPLQJDLQFRYHUWKHUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWSHUSHQGLFXODUWRWKHORQJIDFHRIWKH EHDPLVHTXDOWRWKHIROORZLQJDVVXPLQJDLQVSDFLQJWKHLQVSDFLQJLVXVHGLQVWHDGRIDLQVSDFLQJ to decrease the required area of transverse reinforcement):

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

A

ߙ ൌ 15.7°

A

8ᇱ -0ԣ

1ᇱ -0ԣ

8-#10

1ᇱ -0ԣ

#3 ሺtyp. ሻ #4 @ 12ԣ

1ᇱ -1½ԣ

2ᇱ -4½ԣ

#4 hoops and crossties @ 6ԣ

2ᇱ -0ԣ Section A-A

Figure 14.76 Reinforcement details for the coupling beam in Example 14.12 – Confinement of individual diagonals

,QRUGHUWRVDWLVI\WKHLQPD[LPXPVSDFLQJUHTXLUHPHQWLQ$&,G ORQJLWXGLQDOEDUVDUHUHquired on each of the long faces of the beam. :LWKDKRRSDQGFURVVWLHVSURYLGHG

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The required area of transverse reinforcement perpendicular to the short face of the beam is equal to the following:

,QRUGHUWRVDWLVI\WKHLQPD[LPXPVSDFLQJUHTXLUHPHQWLQ$&,G ORQJLWXGLQDOEDUVDUHUHquired on each of the short faces of the beam. :LWKDKRRSDQGFURVVWLHVSURYLGHG 5HLQIRUFHPHQWGHWDLOVIRUWKLVEHDPDUHJLYHQLQ)LJXUH

A

ߙ ൌ 15.7°

A

8ᇱ -0ԣ

8-#10per group

2ᇱ -4½ԣ

#4 hoops and crossties @ 4ԣ

1ᇱ -1½ԣ

#4 ሺtyp. ሻ

2ᇱ -0ԣ Section A-A

Figure 14.77 Reinforcement details for the coupling beam in Example 14.12 – Confinement of the entire cross-section

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.9.13 Example 14.13 – Determination of Diaphragm Reinforcement: Building #4, SDC D

'HVLJQWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWDWWKHVHFRQGÀRRUOHYHOLQ%XLOGLQJIRUVHLVPLFIRUFHVLQWKHHDVW ZHVWGLUHFWLRQVHH)LJXUH 7KHWKLFNQHVVRIWKHVODEZKLFKLVSDUWRIWKHZLGHPRGXOHMRLVWÀRRUV\VWHP LV DQG*UDGHUHLQIRUFHPHQW HTXDOWRLQ$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'EDVHGRQWKHIROORZLQJ • Site Class D (default) • • Step 1 – Determine the diaphragm in-plane forces

:LQGDQGVHLVPLFLQSODQHIRUFHVLQWKHHDVWZHVWGLUHFWLRQDUHJLYHQLQ([DPSOHVDQGUHVSHFWLYHO\,WLV HYLGHQWIURP7DEOHVDQGWKDWWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVDUHVLJQL¿FDQWO\ODUJHUWKDQWKRVHGXHWR wind. Therefore, wind forces are not considered in this example. Step 2 – Determine the diaphragm classification

7KHLQIRUPDWLRQLQ6HFWLRQRIWKLVSXEOLFDWLRQLVXVHGWRGHWHUPLQHWKHFODVVL¿FDWLRQRIWKLVGLDSKUDJP In the east-west direction, the maximum span-to-depth ratio is In the north-south direction, %HFDXVHWKHVSDQWRGHSWKUDWLRVDUHOHVVWKDQDQGWKH the maximum span-to-depth ratio is VWUXFWXUHGRHVQRWKDYHDQ\RIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOHWKHUHLQIRUFHGFRQFUHWHÀRRU V\VWHPFDQEHFODVVL¿HGDVDULJLGGLDSKUDJPZKHQVXEMHFWHGWRVHLVPLFIRUFHVLQERWKGLUHFWLRQVRIDQDO\VLV$6&( 6(,  Step 3 – Check the minimum diaphragm thickness

$FFRUGLQJWR$&,WKHPLQLPXPGLDSKUDJPWKLFNQHVVUHTXLUHGWRWUDQVPLWHDUWKTXDNHIRUFHVLVLQ,Q WKLVH[DPSOHDLQVODEWKLFNQHVVLVSURYLGHG Step 4 – Select the diaphragm model

An equivalent beam model is selected for this diaphragm. The forces in the walls and frames are determined using a three-dimensional model of the building (see Step 6 below where information is provided on how the diaphragm forces are obtained from the analysis). Step 5 – Determine the location of the center of mass and the center of rigidity

From symmetry,

and

There is no eccentricity between the CM and the CR in both directions, which means no inherent torsional moments are generated. Step 6 – Determine the seismic forces in the special structural walls and the special moment frames

$WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFH,QWKHHDVWZHVWGLUHFWLRQWKHZDOOV along column lines C, D, and E and the moment frames along column lines A and G are assumed to be part of the 6)56,QWKHPRGHOWKHFROXPQVDQGZDOOVDUH¿[HGDWWKHEDVH$6&(6(, ULJLGGLDSKUDJPVDUHDVVLJQHG at all levels in the building, and the following reduced moments of inertia are used, which account for the effects of FUDFNHGVHFWLRQV>VHH$6&(6(,DQG$&,7DEOHD @

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

• Columns: • Beams: • Walls: • Slabs (out-of-plane): • Slabs (in-plane): 6HLVPLFIRUFHVDUHDSSOLHGDWWKH&0LQERWKGLUHFWLRQV$FFLGHQWDOWRUVLRQLQDFFRUGDQFHZLWK$6&(6(, QHHGQRWEHDSSOLHGLQWKHDQDO\VLVIRUVWUHQJWKGHVLJQRUZKHQFKHFNLQJWKHVWRU\GULIWOLPLWVSUHVFULEHGLQ$6&( 6(,EHFDXVHWKHVWUXFWXUHZKLFKLVDVVLJQHGWR6'&'GRHVQRWKDYHD7\SHDRU7\SHEKRUL]RQWDOVWUXFDWWKHVHFRQGÀRRUOHYHOLVHTXDOWRNLSVLQERWKGLUHFWLRQVVHH tural irregularity. The diaphragm force, Table 3.33). %HFDXVHWKHEXLOGLQJGRHVQRWKDYHD7\SHKRUL]RQWDOVWUXFWXUDOLUUHJXODULW\DVGH¿QHGLQ$6&(6(,7DEOH the design seismic forces in the north-south and east-west directions are permitted to be applied independently in HDFKRIWKHWZRRUWKRJRQDOGLUHFWLRQVDQGRUWKRJRQDOLQWHUDFWLRQHIIHFWVDUHSHUPLWWHGWREHQHJOHFWHG$6&(6(,   The vertical seismic force distribution in a multistory building for use in the determination of diaphragm forces is JLYHQLQ)LJXUH7KHIRUFHV over the height of the building are the code-prescribed seismic forces, which FDQEHGHWHUPLQHGXVLQJWKH(/)3URFHGXUHLQ$6&(6(,$VVXPHWKHLQSODQHIRUFHVLQWKHGLDSKUDJPDUH at this level is replaced in the model with which is the required at level The code-prescribed force, GLDSKUDJPIRUFHGHWHUPLQHGLQDFFRUGDQFHZLWK$6&(6(,7KHUHDFWLRQVLQWKHYHUWLFDOHOHPHQWVRIWKH SFRS at this level are determined based on this vertical force distribution. ‫ܨ‬௡ 

‫ܨ‬௫ 

‫ܨ‬௣௜ 



‫ݔ‬

݅

ܸ Figure 14.78 Vertical seismic force distribution in a multistory building for use in the determination of diaphragm forces.

7KHGLDSKUDJPIRUFHVDWWKHVHFRQGÀRRUOHYHODUHREWDLQHGIURPWKHWKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJ from the ELF Procedure where DWWKHVHFRQGÀRRUOHYHOLVUHwhich is subjected to the forces, at this level (see Table 3.33 of this publication). Therefore, for purposes of analysis, placed with LVDSSOLHGDWWKH&0RIWKHVHFRQGÀRRUOHYHOLQWKH an additional force equal to model.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

%HFDXVHULJLGGLDSKUDJPVDUHVSHFL¿HGLQWKHPRGHOWKHIRUFH in the diaphragm is determined using the shear IRUFHVLQWKHYHUWLFDOPHPEHUVRIWKH6)56LPPHGLDWHO\DERYHDQGEHORZWKHVHFRQGÀRRUOHYHOVHH)LJXUH  For example, consider the special wall along column line D, which resists a portion of the east-west seismic force based on its relative rigidity. To determine the diaphragm force at this location, section cuts are made in the wall LPPHGLDWHO\DERYHDQGEHORZWKHVHFRQGÀRRUOHYHO7KHVKHDUIRUFHVREWDLQHGIURPDQDO\VLVDUHHTXDOWR NLSVDERYHWKHVHFRQGÀRRUOHYHODQGNLSVEHORZWKHVHFRQGÀRRUOHYHOVRWKHIRUFHLQWKHGLDSKUDJPDW 7KHNLSIRUFHLVHTXDOWRWKHUHDFWLRQDWWKLVVXSSRUW this location is equal to The forces in the walls along column lines C and E and in the moment frames along column lines A and G can be obtained in a similar fashion. The reactions at these supports are equal to the following: Reactions at column lines C and E Reactions at column lines A and G

ܸ௪ାଵ 

ܸ௨ ൌ ܸ௪ െ ܸ௪ାଵ  ܸ௪ 

Figure 14.79 Force transferred between a diaphragm and a vertical element of the SFRS.

7KHWRWDOIRUFHDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR matches the diaphragm force applied at this level.

which as expected,

Step 7 – Construct the shear and moment diagrams for the diaphragm

The equivalent in-plane distributed load for seismic forces in the east-west direction is equal to the sum of the reactions divided by the width of the diaphragm:

This load is uniformly distributed over the width of the diaphragm because there are no torsional moments. The corUHVSRQGLQJVKHDUDQGPRPHQWGLDJUDPVDUHJLYHQLQ)LJXUH Step 8 – Determine the chord forces

Chord forces must be determined in both directions considering the openings in the diaphragm. Based on the results from the analysis, these openings have a nominal effect on the overall behavior of the diaphragm. As such, chord forces are determined disregarding the openings, which means special chord and collector reinforcement at the edges of the openings need not be determined. 7\SLFDOO\WKHPRPHQWDUPXVHGWRGHWHUPLQHWKHWHQVLRQFKRUGIRUFHLVHTXDOWRSHUFHQWRIWKHGHSWKRIWKH diaphragm in the direction of analysis. If that depth is used in this example, the chord reinforcement will be located ZLWKLQWKHLQVODERXWVLGHRIWKHLQZLGWKRIWKHEHDPVLQWKHVSHFLDOPRPHQWIUDPHVDORQJFROXPQOLQHV DQGZKLFKLVDVXLWDEOHORFDWLRQIRUWKLVUHLQIRUFHPHQW

14-131

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

G

A

‫ݓ‬ௐ ൌ 2.81 kipsΤft E

50ᇱ -0ԣ

ܴீ ൌ 19.0 kips

25ᇱ -0ԣ

25ᇱ -0ԣ

50ᇱ -0ԣ

ܴா ൌ 124.0 kips ܴ஽ ൌ 135.5 kips ܴ஼ ൌ 124.0 kips

ܴ஺ ൌ 19.0 kips

19.0 2.5

Shear ሺkipsሻ

67.8

121.5

 Moment ሺft-kipsሻ

64.2

2,562.5

3,378.1

Figure 14.80 Equivalent distributed load, shear diagram, and moment diagram for seismic forces in the west direction.

7KHPD[LPXPPRPHQWLQWKHGLDSKUDJPLVHTXDOWRIWNLSVZKLFKRFFXUVDWFROXPQOLQH'VHH)LJXUH  Therefore, using Equation (9.1) of this publication, the maximum tension chord force is equal to the following:

Step 9 – Determine the unit shear forces, net shear forces, and collector forces

The unit shear forces in the diaphragm are equal to the following: • Along column lines A and G: • Along column lines C and E: • Along column line D:

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The unit shear forces in the walls along column lines C and E and in the wall along column line D are equal to the following: • Along column lines C and E: • Along column line D: 1 G

40� -0″

2

30� -0″

3

40� -0″

�typ.�

25� -0�

F

E

N

121.5 � 1.11 kipsft 110.0

2.5 � 0.03 kipsft 80.0

Unit shear forces

4

124.0 � 4.13 kipsft 30.0

4.13 kipsft

1.11 kipsft

1.14 kipsft

1.14 kipsft

3.02 kipsft

Net shear forces

1.14 kipsft

1.14 kipsft 45.6 kips

Collector forces 45.6 kips

Figure 14.81 Unit shear forces, net shear forces, and collector forces along column line E for the diaphragm in Example 14.13.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

The unit shear forces, net shear forces, and collector forces along column E are given in Figure 14.81 (the forces are the same along column line C). Forces along column line D can be obtained in a similar fashion. Collectors are not required along column lines A and G because the special moment frames extend the entire depth of the diaphragm in the east-west direction. Collectors are required along column lines C, D, and E. It is assumed the collector beams along these column lines transfer the entire axial force directly into the web of the walls. The beams must be designed for combined flexure due to gravity loads and axial tension and compression forces due to in-plane seismic forces (see Step 14 below). Step 10 – Determine combined load effects

Combined load effects are determined using the applicable strength design load combinations in ACI 5.3 (see Table 14.1 of this publication). The governing in-plane load effects are due to design seismic forces. In ACI Eq. (5.3.1e), the effects of gravity and seismic ground motion are additive, so where in accordance with ASCE/SEI 12.3.4.2. Therefore, this load combination becomes the following:

In buildings assigned to SDC D, collectors and their connections must be designed for the maximum of the three forces given in ASCE/SEI 12.10.2.1: 1. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces determined by the ELF Procedure of ASCE/SEI 12.8 or the modal response spectrum analysis procedure of ASCE/SEI 12.9.1. For dual systems with special reinforced concrete shear walls and special reinforced concrete moment frames is equal capable of resisting at least 25 percent of the prescribed seismic forces, the overstrength factor, to 2.5 from ASCE/SEI Table 12.2-1. The required in-plane diaphragm force at the second-floor level based on (see Table 3.33 of this publication). this requirement is equal to 2. Forces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces determined by ASCE/SEI Eq. (12.10-1). at the second-floor level determined by ASCE/ Using the information in Table 3.33, the diaphragm force, SEI Eq. (12.10-1) is equal to the following:



The required diaphragm in-plane force including overstrength is equal to based on this requirement. 3. Forces calculated using the load combinations of ASCE/SEI 2.3.6 with the seismic forces determined by ASCE/SEI Eq. (12.10-2). at the second-floor level determined by ASCE/SEI Eq. (12.10-2) is equal to 421.5 The diaphragm force, kips (see Table 3.33). The required diaphragm in-plane force based on this requirement is equal to times this value: Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the effects due to the 548.0-kip in-plane diaphragm force stipulated in the third requirement. Thus, the axial tension and compression forces determined in Step 9 above for the 421.5-kip diaphragm force must be increased by 1.3 (see . The collector beams must be designed for an 59.3-kip axial compression and Figure 14.81): tension forces in combination with the effects due to the tributary factored gravity loads along the length of the member (see Step 14 below).

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 11 – Determine the chord reinforcement

The required area of chord reinforcement for seismic forces in the east-west direction is equal to the following (see Step 8):

$WFROXPQOLQHVDQGSURYLGHEDUV these bars are located just outside the cross-secWLRQRIWKHEHDPVLQWKHVSHFLDOPRPHQWIUDPHVDORQJWKHVHOLQHVVHH)LJXUH   ͳ

ʹǦ͓ͷ…Š‘”†”‡‹ˆ‘”…‡‡–

ͶΦԣ

͓Ͷ̷ͳʹԣ

‡ƒ–”ƒ•˜‡”•‡ ”‡‹ˆ‘”…‡‡–

ʹԢǦͲԣ

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡–

‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘”…‡‡– ͵ԢǦͲԣ

Figure 14.82 Location of chord reinforcement in the diaphragm of Example 14.13 for seismic forces in the east-west direction.

Step 12 – Determine the diaphragm shear reinforcement

7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHLQWKHGLDSKUDJPLQWKHGLUHFWLRQRIDQDO\VLVLVHTXDOWRNLSVIWDORQJ for shear in a diaphragm must not exceed the column lines C and E (see Step 9). The strength reduction factor, least value of IRUVKHDUXVHGIRUWKHYHUWLFDOHOHPHQWVRIWKH6)56$&, )RUWKHVKHDUGHVLJQRIWKH beams and columns in the special moment frames, LVSHUPLWWHGWREHWDNHQDV$&, ,WLVGHWHUPLQHG for the shear design of the special structural walls. Therefore, use for the LQ([DPSOHWKDW shear design of the diaphragm. Determine

IURP(T DVVXPLQJWKHVKHDUUHLQIRUFHPHQW

Therefore, the shear strength of the diaphragm is adequate without shear reinforcement.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 13 – Determine the shear transfer reinforcement

• Special moment frames along column lines A and G The required shear-friction reinforcement between the diaphragm and the special moment frames along column lines A and G is equal to the following (see Step 9): Eq. (9.11)

where

EHFDXVHWKHVODEDQGPRPHQWIUDPHVDUHSODFHGPRQROLWKLFDOO\VHH$&,7DEOH 

Because the required area of shear-friction reinforcement is very small, it is safe to assume that the one layer RIÀH[XUDOUHLQIRUFHPHQWLQWKHLQVODEFDQDOVREHXVHGDVWKHVKHDUIULFWLRQUHLQIRUFHPHQWEHWZHHQWKH diaphragm and the special moment frames. 3URYLGHEDUVVSDFHGDWLQRQFHQWHURULHQWHGLQWKHQRUWKVRXWKGLUHFWLRQLQWKHLQVODERYHUWKHOHQJWK ACI of the special moment frames DQG)LJXUH $OWKRXJKEDUVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHEDUVDUHSURYLGHGWR match the required shear transfer reinforcement for the special structural walls (see below). • Special structural walls along column lines C and E: The required shear-friction reinforcement between the diaphragm and the walls along column line C and E is equal to the following (see Step 9):

where LQDFFRUGDQFHZLWK$&,>WKHZDOOVDQGVODEVDUHQRWFDVWPRQROLWKLFDOO\DQGWKH FRQWDFWVXUIDFHPXVWEHURXJKHQHGFRQVLVWHQWZLWKFRQGLWLRQE LQ$&,7DEOH@ 7KHWRWDODUHDRIUHLQIRUFHPHQWWKDWPXVWEHSURYLGHGIRUÀH[XUHDQGVKHDUWUDQVIHULVHTXDOWR 3URYLGHEDUVVSDFHGDWLQRQFHQWHURULHQWHGLQWKHQRUWKVRXWKGLUHFWLRQLQWKHLQVODERYHUWKH VHH)LJXUH  lengths of these walls • Special structural wall along column line D: The required shear-friction reinforcement between the diaphragm and the wall along column line D is equal to the following (see Step 9):

where LQDFFRUGDQFHZLWK$&,>WKHZDOOVDQGVODEVDUHQRWFDVWPRQROLWKLFDOO\DQGWKH FRQWDFWVXUIDFHPXVWEHURXJKHQHGFRQVLVWHQWZLWKFRQGLWLRQE LQ$&,7DEOH@ 7KHWRWDODUHDRIUHLQIRUFHPHQWWKDWPXVWEHSURYLGHGIRUÀH[XUHDQGVKHDUWUDQVIHU 3URYLGHEDUVVSDFHGDWLQRQFHQWHURULHQWHGLQWKHQRUWKVRXWKGLUHFWLRQLQWKHLQVODERYHUWKHOHQJWK of this wall

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

E

2′-0″

4½″

#4 @ 12″

2′-0″

Other reinforcement not shown for clarity

Figure 14.83 Shear transfer reinforcement between the diaphragm and the special structural wall along column line E in Example 14.13 for seismic forces in the east-west direction.

• Diaphragms and the collectors According to ASCE/SEI 12.10.2.1, the shear transfer reinforcement providing the connection between the diaphragm and the collectors must be based on the largest of the three forces determined in accordance with that section. From Step 10, the required diaphragm force is equal to 548.0 kips. The unit shear force between the diaphragm and the collector beams at column lines C and E is equal to 1.54 kips/ft based on a diaphragm force of 421.5 kips (see Figure 14.81). Therefore, the required shear-friction reinforcement at these locations is equal to the following:

The #4 bars spaced at 12 in. on center in the slab oriented in the north-south direction are adequate for combined flexure and shear transfer . Because the unit shear force along column line D is less than that along column lines C and E, #4 bars spaced at 12 in. are also adequate for combined flexure and shear transfer at this location. Step 14 – Determine the required collector reinforcement

Reinforcement is determined for the 36.0 in. by 28.5 in. collector beams along column lines C and E (determination of the reinforcement for the collector beam along column line D is similar). These beams must be designed for the combined effects from gravity loads (bending moments and shear forces) and seismic forces (axial compression and tension). A summary of the axial forces, bending moments, and shear forces is given in Table 14.21. The 59.3-kip axial compression and tension force in the collectors is determined in Step 10.

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.21 Summary of Design Axial Forces, Bending Moments, and Shear Forces for the Collector Beams Along Column Lines C and E in Example 14.13 Axial Force (kips)

Load Case

Bending Moment (ft-kips) Negative

Positive

Shear Force (kips)

Dead

0

−121.8

57.4

16.5

Live

0

−36.5

18.3

4.1

0

0

0

Seismic Load Combination ACI Eq. (5.3.1a)

0

−170.5

80.4

23.1

ACI Eq. (5.3.1b)

0

−204.6

98.2

26.4

ACI Eq. (5.3.1e)

−176.6

83.8

23.5

ACI Eq. (5.3.1g)

−97.4

45.9

13.2

The design strength interaction diagram for the collector beam with 5-#8 top bars, 5-#8 bottom bars, and 1-#5 bar on each side face is given in Figure 14.84. It is evident from the figure that the longitudinal reinforcement is adequate for the load combinations in Table 14.21. 2,250

2,000 1,750 Axial Force ሺkipsሻ

1,500 1,250 1,000

750 500 250 0

-250 -500

0

+++ +++ +200

400

600

800

1,000

1,200

1,400

Bending Moment ሺft-kipsሻ

Figure 14.84 Design strength interaction diagram for the collector beams along column lines C and E in Example 14.13.

Calculate the average tensile stress in the reinforcing bars over the length of the collector and check the requireit is ment in ACI 18.12.7.5 (note: this calculation is not necessary if the collector is designed using provided here for illustration purposes only):

14-138

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Check if confinement reinforcement in accordance with ACI 18.12.7.6 must be provided.

Thus, transverse reinforcement satisfying ACI 18.12.7.6 need not be provided. The limit of is used because the maximum axial force in the collector is based on ASCE/SEI Eq. (12.10-2) with the load combinations determined in accordance with ASCE/SEI 2.3.6 (see Step 10) and not on the overstrength factor, The maximum shear force in the collector is equal to 23.5 kips when the beam is subjected to combined gravity is determined from ACI Table and seismic effects (see Table 14.21). The design shear strength of the concrete, 22.5.5.1 assuming at least minimum shear reinforcement is provided in the section:

Thus,

The required spacing of #3 ties with one crosstie is equal to the following (the #3 crosstie is required to satisfy the maximum spacing of legs of shear reinforcement across the width of the member; see ACI Table 9.7.6.2.2):

Provide #3 ties and crossties spaced at 11.0 in. on center over the entire length of the collector beams. Note that this transverse reinforcement satisfies the detailing requirement of ACI 18.12.7.7(b) at splices and anchorage zones: Provided

Reinforcement details for the collector beams are given in Figure 14.85.

14-139

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

 

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Figure 14.85 Reinforcement details for the collector beams along column lines C and E in Example 14.13.

14.9.14 Example 14.14 – Design of Foundation Seismic Tie: Building #1 (Framing Option C), SDC D

'HVLJQDIRXQGDWLRQVHLVPLFWLHIRUWKHVSUHDGIRRWLQJVXSSRUWLQJFROXPQ&LQ%XLOGLQJ)UDPLQJ2SWLRQ&VHH and )LJXUH 7KHFROXPQGLPHQVLRQVDUHLQE\LQ$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK *UDGHUHLQIRUFHPHQW$OVRDVVXPHDJUDGHEHDPLVXVHGDVWKHVHLVPLFWLH 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'ZKHUH Step 1 – Determine the seismic tie force

ACI 18.13.4.3

The seismic tie must have a design strength in tension and compression of at least times the factored dead ORDGSOXVIDFWRUHGOLYHORDGLQDFFRUGDQFHZLWKWKHORDGFRPELQDWLRQRQFROXPQ&WKDWLQFOXGHVHDUWKTXDNHHIIHFWV DVVXPLQJWKHD[LDOIRUFHVRQWKHDGMDFHQWFROXPQVDUHOHVVWKDQRUHTXDOWRWKDWRQFROXPQ&  )URP7DEOHLQ([DPSOHWKHODUJHVWD[LDOIRUFHFRUUHVSRQGVWRWKHORDGFRPELQDWLRQLQ$&,(TH  for sidesway to the left:

Therefore, the required axial tension and compression force in the seismic tie Step 2 – Determine the required reinforcement in the seismic tie

ACI 18.13.4.3

$VVXPHWKHVHLVPLFWLHLVDQLQE\LQUHLQIRUFHGFRQFUHWHJUDGHEHDPDQGWKHJUDGHEHDPLVQRWVXEMHFWHGWR any gravity or seismic load effects transmitted by the column or any other structural members. &KHFNWKHGLPHQVLRQDOUHTXLUHPHQWVRI$&,D  0D[LPXPFOHDUVSDFLQJEHWZHHQFROXPQVDGMDFHQWWRFROXPQ&

14-140

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Smallest cross-sectional dimension Also assume the grade beam has a longitudinal reinforcement ratio Thus, 7U\ORQJLWXGLQDOEDUV Required longitudinal reinforcement for the axial tension force in the seismic tie $VVXPLQJWKHJUDGHEHDPKDVWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIWLHVFRQIRUPLQJWR$&,WKHGHVLJQ RIWKHVHFWLRQLVGHWHUPLQHGXVLQJ$&,7DEOHDQG$&,(T  axial strength,

Required tie spacing:

Figure 14.47

8VHORQJLWXGLQDOEDUVZLWKWLHVVSDFHGLQRQFHQWHURYHUWKHOHQJWKRIWKHJUDGHEHDP 5HLQIRUFHPHQWGHWDLOVIRUWKHJUDGHEHDPDUHVLPLODUWRWKRVHLQ)LJXUH7KHORQJLWXGLQDOEDUVDUHGHYHOoped for tension and compression in the footing. 14.9.15 Example 14.15 – Determination of Required Reinforcement: Beam in Building #4, Beam is Not Part of the SFRS, SDC D

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWIRUWKHEHDPRQFROXPQOLQH%EHWZHHQFROXPQOLQHVDQGDWWKHVHFRQG ÀRRUOHYHOLQ%XLOGLQJ7KHEHDPLVLQZLGHDQGLQGHHSDQGLVQRWSDUWRIWKH6)56$VVXPHQRUPDODQG*UDGHUHLQIRUFHPHQW weight concrete with 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'ZKHUH Step 1 – Determine the factored load combinations

$OOPHPEHUVWKDWDUHQRWSDUWRIWKH6)56PXVWVDWLVI\WKHGHIRUPDWLRQFRPSDWLELOLW\UHTXLUHPHQWVLQ$6&(6(, 7KLVEHDPPXVWEHGHVLJQHGIRUWKHFRPELQHGHIIHFWVIURPJUDYLW\DQGWKHGHVLJQGLVSODFHPHQWV FDXVHGE\WKHGHVLJQOHYHOHDUWKTXDNHLQWKHHDVWZHVWGLUHFWLRQ>VHH(T @ A structural analysis was performed to determine the bending moments and shear forces in this beam due to which are applied over the height of the building. Only the members that are not part of the SFRS were included in the analysis. $FFRUGLQJWR$&,WKHHIIHFWVFDXVHGE\ and the vertical ground motion are to be combined with the IDFWRUHGJUDYLW\ORDGHIIHFWVGHWHUPLQHGE\WKHORDGFRPELQDWLRQVLQ$&,$VXPPDU\RIWKHIDFWRUHGEHQGLQJ PRPHQWVDQGVKHDUIRUFHVIRUWKLVEHDPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ7DEOH

14-141

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table 14.22 Design Bending Moments and Shear Forces at the Second-Floor Level for the Beam in Example 14.15 Bending Moment (ft-kips) Negative

Positive

Shear Force (kips)

Dead

í





Live

í





Seismic

í





$&,(TD

í





$&,(TE

í





$&,(TH

í





$&,(TJ

í





Load Case

Load Combination

Step 2 – Determine the required flexural reinforcement at the critical sections

7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW at the critical sections is determined by the following equations, ZKLFKDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUVHFWLRQVZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHment (an effective slab width at positive moment critical sections is not considered in this example):

$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHEHDPLVGHVLJQHGIRU VRWKHUHTXLUHPHQWVRI$&,DUHDSSOLFDEOH,WLVHYLGHQWIURP7DEOHWKDWDOOVHFWLRQVDUHWHQVLRQ Also, the required reinforcement is less than the maximum amount prescribed in ACI controlled because >VHH$&,D DQG)LJXUH@ IRU*UDGHUHLQIRUFHPHQWZKLFKLVEDVHGRQ Table 14.23 Required Flexural Reinforcement for the Beam in Example 14.15 Location

Mu (ft-kips)

Rn (psi)

As (in.2)*

Negative

í





Positive



68



* Min. Max. Max.

6HOHFWWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVEDVHGRQWKHPD[LPXPDQGPLQLPXPVSDFLQJUHTXLUHPHQWVLQ$&, DQGUHVSHFWLYHO\ maximum and minimum 8VHEDUVIRUERWKWKHWRSDQGERWWRPUHLQIRUFHPHQW QXPEHURIORQJLWXGLQDOEDUVLQDVLQJOHOD\HUIRUDLQZLGHEHDPDUHHTXDOWRDQGIURP7DEOHVDQG of this publication, respectively). 14-142

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Step 3 – Determine the required transverse reinforcement

$FFRUGLQJWR$&,D VWLUUXSVDUHSHUPLWWHGWREHXVHGEHFDXVHWKHIDFWRUHGD[LDOIRUFHRQWKHEHDPLVOHVV VHH)LJXUH  than 7KHVL]HDQGVSDFLQJRIWKHVWLUUXSVFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ Eq. (14.11)

The design shear strength of the concrete, LVGHWHUPLQHGIURP$&,7DEOHDVVXPLQJDWOHDVWPLQLPXP shear reinforcement is provided in the section:

VHH7DEOH 

It is evident that

Therefore, determine the stirrup spacing based on minimum shear reinforcement requirements:

3URYLGHVWLUUXSVDQGFURVVWLHVVSDFHGDWLQRYHUWKHIXOOOHQJWKRIWKHEHDP 5HLQIRUFHPHQWGHWDLOVIRUWKLVEHDPDUHJLYHQLQ)LJXUH  

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Figure 14.86 Reinforcement details for the beam in Example 14.15.

14-143

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14.9.16 Example 14.16 – Determination of Required Reinforcement: Column in Building #4, Column is Not Part of the SFRS, SDC D

'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWIRUFROXPQ%LQWKH¿UVWVWRU\RI%XLOGLQJ$VVXPHWKHFROXPQLV DQG*UDGH LQE\LQDQGLVQRWSDUWRIWKH6)56$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK reinforcement. 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'ZKHUH Step 1 – Determine the factored load combinations

$OOPHPEHUVWKDWDUHQRWSDUWRIWKH6)56PXVWVDWLVI\WKHGHIRUPDWLRQFRPSDWLELOLW\UHTXLUHPHQWVLQ$6&(6(, 7KLVFROXPQPXVWEHGHVLJQHGIRUWKHFRPELQHGHIIHFWVIURPJUDYLW\DQGWKHGHVLJQGLVSODFHPHQWV FDXVHGE\WKHGHVLJQOHYHOHDUWKTXDNHLQWKHHDVWZHVWGLUHFWLRQ>VHH(T @ A structural analysis was performed to determine the axial forces, bending moments, and shear forces in this column which are applied over the height of the building. Only the members that are not part of the SFRS were due to included in the analysis. $FFRUGLQJWR$&,WKHHIIHFWVFDXVHGE\ and the vertical ground motion are to be combined with the IDFWRUHGJUDYLW\ORDGHIIHFWVGHWHUPLQHGE\WKHORDGFRPELQDWLRQVLQ$&,$VXPPDU\RIWKHIDFWRUHGD[LDO IRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVIRUWKLVFROXPQLQWKH¿UVWVWRU\LVJLYHQLQ7DEOH Table 14.24 Design Axial Forces, Bending Moments, and Shear Forces for Column B2 in Example 14.16 Axial Force (kips)

Bending Moment (ft-kips)

Shear Force (kips)

Dead







Live







Roof live load



Load Case

Seismic Load Combination $&,(TD







$&,(TE





13.3

$&,(TF

6,398.8





SSR





39.6

SSL



SSR







SSL



$&,(TH

$&,(TJ SSR = sidesway to the right, SSL = sidesway to the left

Step 2 – Determine the required longitudinal reinforcement

The column is designed so that the required strength is less than or equal to the design strength. Thus, the requirePHQWVRI$&,DUHDSSOLFDEOHVHH)LJXUH  14-144

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUV LVJLYHQLQ)LJXUH$OOORDGFRPELQDWLRQVIDOOZLWKLQWKHERXQGDU\RIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDP for this column.  8,000 7,000

Axial Force ሺkipsሻሻ

6,000

  

5,000

 

4,000 3,000 2,000 1,000 0 0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

Bending Moment ሺft-kipsሻ

Figure 14.87 Design strength interaction diagram for column B2 in Example 14.16.

8VHEDUV Step 3 – Determine the required transverse reinforcement

The required transverse reinforcement depends on the magnitude of the largest factored gravity axial force with ZKHUHWKHQRPLQDOD[LDOVWUHQJWKDW]HURHFFHQWULFLW\ is determined as follows: respect to ACI Eq. (22.4.2.2)

7KHODUJHVWIDFWRUHGJUDYLW\IRUFHLVHTXDOWRNLSVIURP7DEOHZKLFKLVJUHDWHUWKDQ 7KHUHIRUHWKHWUDQVYHUVHUHLQIRUFHPHQWPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,F >VHH)LJXUH @ 'HWHUPLQHWKHOHQJWKRYHUZKLFKÀH[XUDO\LHOGLQJLVOLNHO\WRRFFXU

Eq. (14.15)

14-145

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Maximum hoop spacing is equal to the following: Figure 14.52

In both directions,

KRRSVZLWKIRXUFURVVWLHVVSDFHGDWLQSURYLGHV Check if this transverse reinforcement is adequate for shear. $FFRUGLQJWR$&,E WKHGHVLJQVKHDUIRUFH PXVWEHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,7KH at the ends of the column associated with the range of factored axial forces are SUREDEOHÀH[XUDOVWUHQJWKV and VHH)LJXUH  obtained from the design strength interaction diagram with LVHTXDOWRIWNLSVZKLFKFRUUHVSRQGVWRDQD[LDOIRUFHRI ,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHODUJHVW NLSVIRUVLGHVZD\WRWKHOHIW 15,000

12,500



Axial Force ሺkipsሻሻ

10,000

7,500

 

5,000

  2,500

0 0

1,000

2,000

3,000

4,000

5,000

6,000

7,000

8,000

9,000

10,000

Bending Moment ሺft-kipsሻ

Figure 14.88 Nominal strength interaction diagram for column B2 with fy = 75 ksi and ݊ = 1.0.

14-146

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Therefore, Figure 14.15

7KLVVKHDUIRUFHLVJUHDWHUWKDQWKHPD[LPXPVKHDUIRUFHRINLSVGHWHUPLQHGIURPDQDO\VLVVHH7DEOH  The earthquake-induced shear force is the total shear force on the column. Also, the minimum factored axial force Therefore, is LQFOXGLQJHDUWKTXDNHHIIHFWVLVHTXDOWRNLSVZKLFKLVJUHDWHUWKDQ SHUPLWWHGWREHLQFOXGHGZKHQGHWHUPLQLQJWKHUHTXLUHGVKHDUUHLQIRUFHPHQW$&, 

Eq. (14.17)

In this equation,

is determined from a strain compatibility analysis of the section for the axial compressive force which corresponds to the used to determine

7KHUHTXLUHGVSDFLQJRIWKHKRRSVDQGFURVVWLHVLQWKHGLUHFWLRQRIDQDO\VLVLVHTXDOWRWKHIROORZLQJ

7KXVWKHVL]HDQGVSDFLQJRIWKHKRRSVGHWHUPLQHGSUHYLRXVO\DUHQRWDGHTXDWHIRUVKHDU 8VHKRRSVZLWKVL[FURVVWLHVVSDFHGDWLQRQFHQWHURYHUDGLVWDQFHRIDWOHDVW the column

from each end of

Outside of KRRSVZLWKIRXUFURVVWLHVVSDFHGDWLQRQFHQWHUDUHSHUPLWWHGWREHXVHG+RZHYHUJLYHQ from each end of the column (that is, only of the center portion of the column is the extent of XVHKRRSVDQGFURVVWLHVVSDFHGDWLQRQFHQWHURYHUWKHHQWLUHKHLJKWRIWKHFROXPQDQGZLWKLQ outside of the joints. Determine the tension development length,

RIWKHORQJLWXGLQDOEDUV ACI Eq. (25.4.2.4a)

7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV

ACI Table 25.4.2.5

For normalweight concrete, )RU*UDGHUHLQIRUFHPHQW For uncoated reinforcing bars, )RUUHLQIRUFLQJEDUV

14-147

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

)RUKRRSV

Set

ACI 25.4.2.4

Therefore,

Class B tension lap splice length 8VHDIWLQODSVSOLFHOHQJWK 5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH 4ᇱ -0ԣ

2ᇱ -4½ԣ



#6 hoops and crossties @ 4ԣ 4Ԣ-0ԣ 28-#11 A

4ᇱ -4ԣ

4Ԣ-0ԣ

A

Section A-A

Other reinforcement not shown for clarity

Figure 14.89 Reinforcement details for column B2 in Example 14.16.

14-148

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Comments. It may be possible to reduce the shear demand on the column (thereby reducing the required area of WUDQVYHUVHUHLQIRUFHPHQWDQGRULQFUHDVLQJWKHUHTXLUHGVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQW E\SHUIRUPLQJD PRUHUH¿QHGDQDO\VLVWRGHWHUPLQHWKHMRLQWVWUHQJWKVDWWKHVHFRQGÀRRUOHYHOWRSRIWKHFROXPQ EDVHGRQ of the RIWKHEHDPVIUDPLQJLQWRWKHMRLQWZKLFKLVSHUPLWWHGLQ$&,$WWKHERWWRPRIWKHFROXPQ column must still be used to determine the shear force. 14.9.17 Example 14.17 – Check of Slab-Column Connection: Column in Building #3, Column is Not Part of the SFRS, SDC D

&KHFNWKHVODEFROXPQFRQQHFWLRQDWFROXPQ%DWWKHVL[WHHQWKÀRRUOHYHORI%XLOGLQJ7KHFROXPQLVLQ E\LQDQGLVQRWSDUWRIWKH6)567KHVODELVLQWKLFN$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK DQG*UDGHUHLQIRUFHPHQW 'HVLJQGDWDDUHJLYHQLQ6HFW7KHEXLOGLQJLVDVVLJQHGWR6'&'ZKHUH Step 1 – Determine if slab shear reinforcement is required

6ODEVKHDUUHLQIRUFHPHQWLVUHTXLUHGZKHUHWKHIROORZLQJHTXDWLRQLVVDWLV¿HG Eq. (14.42)

,QWKHVL[WHHQWKDQG¿IWHHQWKVWRULHVWKHGHVLJQVWRU\GULIWV use

DUHHTXDOWRLQDQGLQUHVSHFWLYHO\7KXV

Slab weight Superimposed dead load Live load ,QDFFRUGDQFHZLWK$&,XVHRQO\WKHORDGFRPELQDWLRQVWKDWLQFOXGH

to determine

$&,(TH 

Therefore, (earthquake effects,

DUHHTXDOWR]HUREHFDXVHWKHWZRZD\VODE

and column are not part of the SFRS).

14-149

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Table 22.6.5.2

LQDFFRUGDQFHZLWK$&,DQG

where

for normalweight concrete.

7KHUHIRUHVKHDUUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,DQGHLWKHU$&,RUPXVWEHSURYLGHG Step 2 – Determine the required slab shear reinforcement

+HDGHGVKHDUVWXGUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&,DQGLVXVHGLQWKLVH[DPSOH Required shear reinforcement must provide Try ¾-in. diameter headed shear studs

with

In the direction parallel to the column face, the maximum spacing between adjacent lines of headed shear studs $&,7DEOH  )RUDLQZLGHFROXPQXVHOLQHVRIKHDGHGVKHDUVWXGVRQHDFKIDFHRIWKHFROXPQPD[LPXPVSDFLQJ $&,7DEOH 

In the direction perpendicular to the column face, maximum spacing Therefore,



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͵ͲǤ͹ͷԣ

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Figure 14.90 Headed shear stud reinforcement at column B3 in Example 14.17.

14-150

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

,QDFFRUGDQFHZLWK$&,WKHKHDGHGVKHDUVWXGVPXVWH[WHQGDWOHDVW from the face of the FROXPQ3URYLGHKHDGHGVKHDUVWXGVLQHDFKOLQHRQHDFKIDFHRIWKHFROXPQZLWKWKH¿UVWSHULSKHUDOOLQHRIKHDGHG IURPWKHIDFHRIWKHFROXPQVHH$&,7DEOH  shear studs located +HDGHGVKHDUVWXGUHLQIRUFHPHQWGHWDLOVIRUFROXPQ%LVJLYHQLQ)LJXUH

14-151

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

14-152

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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A-2

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Appendix B REINFORCING BAR DATA Table B.1 ASTM Standard Reinforcing Bars Bar Size

Nominal Diameter (in.)

Nominal Area (in.2)

Nominal weight (lb/ft)











































































1.693













B-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Table B.2 Overall Reinforcing Bar Diameters Nominal Diameter

A

A Overall Diameter

Section A–A

B-2

Bar Size

Approximate Diameter to the Outside Deformations (in.)



















1

























Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Appendix C SECTION INDEX

Structural System Requirements

Terminology

General

ACI Section Number

2QHZD\VODEV  %HDPV 

(DUWKTXDNHUHVLVWDQWVWUXFWXUHV6'&'(DQG)  :DOOV

 (DUWKTXDNHUHVLVWDQWVWUXFWXUHV6'&'(DQG)                    

Loads

Design Guide Chapter/Page Number(s)

       

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C-1

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number 6.2

6.2.3

6.2.4 6.2.4.1 6.2.5 6.2.5.1 6.2.5.3 6.2.6 6.3.2 6.3.2.1 6.5

Structural Analysis

6.5.1 6.5.2 6.5.3 6.5.4 6.6 6.6.1 6.6.2 6.6.3 6.6.3.1

6.6.3.1.1

6.6.3.1.2

6.6.3.1.3 6.6.3.2.2

6.6.4 6.6.4.3 6.6.4.4.4

C-2

Design Guide Chapter/Page Number(s) Earthquake-resistant structures (SDC B and C)/13-1, 13-4 Earthquake-resistant structures (SDC D, E, and F)/14-1 One-way slabs/4-2, 4-4 Two-way slabs/5-11 Beams/6-5 Columns/7-1 Walls/8-1 Two-way slabs/5-11 Two-way slabs/5-11, 5-26 Columns/7-5, 7-7 Walls/8-4, 8-5 Columns/7-5, 7-6, 7-8 Columns/7-1, 7-3 Columns/7-103 Beams/6-34 Earthquake-resistant structures (SDC D, E, and F)/14-18, 14-87, 14-99 Beams/6-14, 6-65, 6-80, 6-98, 6-112 One-way slabs/4-2, 4-22, 4-28 Beams/6-5, 6-9, 6-51 One-way slabs/4-2, 4-4, 4-25, 4-27 Beams/6-5, 6-63, 6-64, 6-73, 6-79, 6-97, 6-110, 6-111 One-way slabs/4-3 Beams/6-5 One-way slabs/4-4 Beams/6-5 One-way slabs/4-3, 4-25, 4-27 Beams/6-5 Two-way slabs/5-11, 5-35, 5-70, 5-92, 5-98, 5-106, 5-116 Diaphragms/9-6 Diaphragms/9-6 Diaphragms/9-6 Two-way slabs/5-36 Columns/7-4 Walls/8-3 Two-way slabs/5-35, 5-36 Columns/7-3 Walls/8-1 Two-way slabs/5-35, 5-36 Columns/7-4 Walls/8-3 Diaphragms/9-8 Two-way slabs/5-36 Two-way slabs/5-36 Columns/7-3 Walls/8-1 Columns/7-2, 7-10 Walls/8-5, 8-40 Columns/7-5, 7-6, 7-88, 7-106 Walls/8-3 Columns/7-2, 7-11, 7-13

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number

Structural Analysis (cont.)

             6.8 6.8.1.1  6.8.1.3 6.9

One-way Slabs

                      

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

One-way Slabs (cont.)

ACI Section Number             8.3.1.1

Two-way Slabs

  8.3.1.3    8.3.3.1                         

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number       8.6.1.1

Two-way Slabs (cont.)

                      8.8 8.8.1.1  8.8.1.3  8.8.1.6 8.8.1.8

Beams

 9.3.1 9.3.1.1 9.3.1.1.1  

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number  9.3.3.1                   

Beams (cont.)

9.6.1

 9.6.1.3 9.6.3                 

C-6

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number     

Beams (cont.)

          9.8  9.8.1.3   9.9.1.1 9.9.1.3

Columns

                     

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Columns (cont.)

ACI Section Number      

Diaphragms

Walls



C-8

  11.3.1.1                 11.6.1         11.8 11.8.1.1  11.8.3 11.8.3.1          

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Diaphragms (cont.)

ACI Section Number           

Beam-Column and Slab-Column Joints

Foundations

                 13.3.3.1  13.3.3.3                 

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C-9

Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number 

Beam-Column and Slab-Column Joints (cont.)

                  16.3

Connections Between Members







  

C-10

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

Connections (cont.)

ACI Section Number

&ROXPQV 

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Design Guide Chapter/Page Number(s)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number  

Earthquake-Resistant Structures (cont.)

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number         

Earthquake-Resistant Structures (cont.)

     18.8     18.8.3.1  18.8.3.3          18.9      

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number               

Earthquake-Resistant Structures (cont.)

                       18.13    

C-14

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number 18.13.3.1        

Earthquake-Resistant Structures (cont.)

      18.13.6.1  18.13.6.3       

Concrete Design and Durability

       

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

ACI Section Number  

Concrete Design and Durability (cont.)









    

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C-16

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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ACI Section Number



         

Strength Reduction Factors



 



    

Sectional Strength

 





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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete

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Description of Manual The purpose of this Design Guide is to assist in the proper application of the design and detailing requirements in Building Code Requirements for Structural Concrete (ACI 318-19) for cast-in-place concrete buildings with nonprestressed steel reinforcement. Many design aids and worked-out examples are provided to make designing and detailing reinforced concrete members simpler and faster. The goal is to acquire an understanding of the code requirements and to apply them properly and efficiently.

In addition to structural engineers, this Design Guide is a valuable resource for educators, students, individuals studying for licensing exams, and building officials.

ISBN 9781943961528

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933 North Plum Grove Road Schaumburg, IL 60173 Tel. 847.517.1200 www.crsi.org

9 781943 961528

DG-ACI-318-19