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&D