Design Guide for Reinforced Concrete Diaphragms 978-1943961467

Table of contents :
Foreword......Page 4
Contents......Page 5
1.3 Scope......Page 7
1.4 Organization of this Publication......Page 8
2.3 Nonprestressed Reinforcement......Page 11
3.2.2 Two-way Slabs......Page 13
3.3.3 Shear......Page 14
3.4.2 One-way Shear......Page 16
3.4.3 Two-way Shear......Page 17
3.6 Minimum Diaphragm Thickness for SDC D, E, and F......Page 21
4.2.1 Wind Forces......Page 23
4.2.3 Seismic Forces......Page 28
4.2.4 Soil Lateral Forces......Page 33
4.2.5 Flood and Tsunami Forces......Page 34
4.3 Transfer Forces......Page 35
4.4.2 Wind Forces......Page 36
4.4.3 Seismic Forces......Page 37
4.5 Column Bracing Forces......Page 38
4.7.2 Collector Design Forces for Buildings Assigned to SDC C through F......Page 39
5.2 Strength Design Load Combinations......Page 41
5.3 Seismic Load Combinations for SDC B......Page 42
5.4 Seismic Load Combinations for SDC C......Page 43
5.5 Seismic Load Combinations for SDC D, E, and F......Page 44
6.2 In-plane Stiffness Modeling......Page 47
6.3.2 Load Paths for Lateral Forces......Page 48
6.3.3 Horizontal Distribution of Lateral Forces......Page 49
6.3.4 ACI 318 Analysis Methods......Page 52
6.4.2 Equivalent Beam Model with Rigid Supports......Page 53
6.4.3 Corrected Equivalent Beam Model withSpring Supports......Page 57
6.7 Alternative Models......Page 61
7.3.3 Shear......Page 63
7.3.5 Collectors......Page 65
8.3 Reinforcement Detailing Requirements......Page 67
8.4.2 Location of Chord Reinforcement......Page 68
8.4.3 Compression Chords......Page 69
8.6.2 Shear Transfer in a Diaphragm......Page 70
8.6.3 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS– Construction Method A......Page 71
8.6.4 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS– Construction Method B......Page 75
8.6.5 Required Shear Transfer Reinforcement Betweenthe Diaphragm and the Collector Elements......Page 76
8.8.3 Wind Forces......Page 77
8.8.4 Seismic Forces......Page 78
8.9.2 Slabs......Page 79
8.9.3 Beams......Page 81
8.10 Summary of Design and Detailing Requirements......Page 83
9.3 Step 2 – Determine the Diaphragm Thickness......Page 86
9.6 Step 5 – Determine the Combined Load Effects......Page 87
9.12 Step 11 – Determine the Collector Reinforcement......Page 88
10.2 Example 10.1 – One-story Retail Building (SDC A)......Page 103
10.3 Example 10.2 – Seven-story Office Building (SDC B)......Page 118
10.4 Example 10.3 – Eighteen-story Residential Building (SDC C)......Page 140
10.5 Example 10.4 – Thirty-story Office Building (SDC D)......Page 165
10.6 Example 10.5 – Five-story Residential Building (SDC D)......Page 187
Chapter 11 References......Page 212
Notations......Page 215

Citation preview

Design Guide for Reinforced Concrete Diaphragms

A guide to assist design professionals in efficiently designing and detailing reinforced concrete diaphragms. First Edition

2019

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.

Cover photos courtesy of Magnusson Klemencic Associates.

Design Guide for Reinforced Concrete Diaphragms

Publicaton No: 10-DG-RC-DIAPHRAGMS-2019 ISBN: 978-1-943961-46-7 Copyright © 2019 By Concrete Reinforcing Steel Institute First Edition Printed 2019 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.

i

Design Guide for Reinforced Concrete Diaphragms

Foreword In the 2014 edition of ACI 318, a new chapter devoted to the design and detailing of reinforced concrete diaphragms was included for the first time. The purpose of this Design Guide is to provide explanatory material based on the provisions of that chapter for diaphragms in buildings assigned to any Seismic Design Category. Included are many design aids and worked-out example problems that illustrate the proper application of the code requirements. Since the first CRSI Design Handbook in 1952, users of CRSI publications have been cooperative in suggesting to the Design Aids Committee and CRSI Staff, many improvements, clarifications and additional design short-cuts. 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 Mike Mota, Ph.D., P.E., SECB, F.SEI, F.ASCE, F.ACI, CRSI Vice President of Engineering.

ii

Design Guide for Reinforced Concrete Diaphragms

Contents Foreword Chapter 1 Introduction

ii 1-1

1.1 Overview

1-1

1.2 Definitions

1-1

1.3 Scope

1-2

1.4 Organization of this Publication

1-2

Chapter 2 Materials

2-1

2.1 Overview

2-1

2.2 Concrete

2-1

2.3 Nonprestressed Reinforcement

2-1

Chapter 3 Determining the Diaphragm Thickness

3-1

3.1 Overview

3-1

3.2 Serviceability Requirements

3-1

3.2.1 One-way Slabs

3-1

3.2.2 Two-way Slabs

3-1

4.3 Transfer Forces

4-13

4.4 Anchorage and Connection Forces

4-14

4.4.1 Overview

4-14

4.4.2 Wind Forces

4-14

4.4.3 Seismic Forces

4-15

4.5 Column Bracing Forces

4-16

4.6 Out-of-plane Forces

4-17

4.7 Collector Design Forces

4-17

4.7.1 Overview

4-17

4.7.2 Collector Design Forces for Buildings Assigned to SDC C through F

4-17

Chapter 5 Load Combinations

5-1

5.1 Overview

5-1

5.2 Strength Design Load Combinations

5-1

5.3 Seismic Load Combinations for SDC B

5-3

5.4 Seismic Load Combinations for SDC C

5-3

5.5 Seismic Load Combinations for SDC D, E, and F

5-4

Chapter 6 Diaphragm Modeling and Analysis

6-1

3-2

6.1 Overview

6-1

3.3.1 Overview

3-2

6.2 In-plane Stiffness Modeling

6-1

3.3.2 Flexure

3-2

6.3 Analysis Methods

6-2

3.3.3 Shear

3-2

6.3.1 Overview

6-2

3.3 In-plane Strength Requirements

3-4

6.3.2 Load Paths for Lateral Forces

6-2

3.4.1 Flexure

3-4

6.3.3 Horizontal Distribution of Lateral Forces

6-3

3.4.2 One-way Shear

3-4

6.3.4 ACI 318 Analysis Methods

6-7

3.4.3 Two-way Shear

3-5

3.4 Out-of-plane Strength Requirements

3.5 Minimum Diaphragm Thickness for SDC A, B, and C

3-9

3.6 Minimum Diaphragm Thickness for SDC D, E, and F

3-9

Chapter 4 Diaphragm Design Forces

4-1

4.1 Overview

4-1

4.2 In-plane Forces

4-1

4.2.1 Wind Forces

4-1

4.2.2 General Structural Integrity Forces

4-6

4.2.3 Seismic Forces

4-6

6.4 Beam Models

6-8

6.4.1 Overview

6-8

6.4.2 Equivalent Beam Model with Rigid Supports

6-8

6.4.3 Corrected Equivalent Beam Model with Spring Supports

6-11

6.5 Strut-and-Tie Models

6-15

6.6 Finite Element Models

6-15

6.7 Alternative Models

6-16

Chapter 7 Design Strength

7-1

7.1 Overview

7-1 7-1 7-2

4.2.4 Soil Lateral Forces

4-11

7.2 Strength Reduction Factors

4.2.5 Flood and Tsunami Forces

4-12

7.3 Nominal Strength

iii

Design Guide for Reinforced Concrete Diaphragms 7.3.1 Overview

7-2

9.4 Step 3 – Determine the Diaphragm Design Forces 9-1

7.3.2 Moment and Axial Forces

7-2

7.3.3 Shear

7-2

7.3.4 Shear Transfer

7-3

9.5 Step 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine the Diaphragm Internal Forces

9-2

7.3.5 Collectors

7-3

9.6 Step 5 – Determine the Combined Load Effects

9-2

9.7 Step 6 – Determine the Chord Reinforcement

9-2

9.8 Step 7 – Determine the Diaphragm Shear Reinforcement

9-2

9.9 Step 8 – Determine the Shear Transfer Reinforcement

9-2

9.10 Step 9 – Deterimine the Reinforcement Due to Eccentricity of Collector Forces

9-2 9-2

Chapter 8 Determining and Detailing the Required Reinforcement 8.1 Overview

8-1

8.2 Diaphragm Reinforcement Limits

8-1

8.3 Reinforcement Detailing Requirements

8-1

9.11 Step 10 – Determine the Anchorage and Connection Reinforcement

8.4 Chord Reinforcement

8-2

9.12 Step 11 – Determine the Collector Reinforcement 9-3

8.4.1 Required Area of Chord Reinforcement

8-2

8.4.2 Location of Chord Reinforcement

8-2

8.4.3 Compression Chords

8-3

8.5 Diaphragm Shear Reinforcement

8-4

8.6 Shear Transfer Reinforcement

8-5

8.6.1 Overview

8-5

8.6.2 Shear Transfer in a Diaphragm

8-5

8.6.3 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS – Construction Method A 8.6.4 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS – Construction Method B 8.6.5 Required Shear Transfer Reinforcement Between the Diaphragm and the Collector Elements

8-6

8-9

Chapter 10 Examples

10-1

10.2 Examples 10.1 – One-story Retail Building (SDC A)

10-1

10.3 Example 10.2 – Seven-story Office Building (SDC B)

10-16

10.4 Example 10.3 – Eighteen-story Residential Building (SDC C)

10-38

10.5 Example 10.4 – Thirty-story Office Building (SDC D)

10-63

10.6 Example 10.5 – Five-story Residential Building (SDC D)

10-85

8-10

8.7 Required Reinforcement Due to Eccentricity of Collector Forces

8-11

Notations

8.8 Anchorage Reinforcement

8-11

8.8.1 Overview

8-11

8.8.2 General Structural Integrity Forces

8-11

8.8.3 Wind Forces

8-12

8.8.4 Seismic Forces

8-12

8.9 Collector Reinforcement

8-13

8.9.1 Overview

8-13

8.9.2 Slabs

8-13

8.9.3 Beams

8-15

8.9.4 Subdiaphragms

8-17

Chapter 9 Design Procedure

8-17

9-1

9.1 Overview

9-1

9.2 Step 1 – Select the Materials

9-1

9.3 Step 2 – Determine the Diaphragm Thickness

9-1

10-1

10.1 Overview

Chapter 11 References

8.10 Summary of Design and Detailing Requirements

iv

8-1

11-1 N-1

Design Guide for Reinforced Concrete Diaphragms

Chapter 1 Introduction 1.1 Overview The purpose of this publication is to assist in the analysis, design, and detailing of reinforced concrete diaphragms in accordance with the 2014 edition of Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14) [Reference 1]. One of the main goals is to provide step-by-step design procedures and design aids that make designing and detailing reinforced concrete diaphragms simpler and faster. The procedures and design aids contained in this publication can be used in the design and detailing of reinforced concrete diaphragms in buildings of any size that are assigned to Seismic Design Categories A through F. The 2018 edition of the International Building Code (IBC) [Reference 2] references ACI 318-14, and Section 1901.2 of the IBC requires that structural concrete be designed and constructed in accordance with the provisions of Chapter 19 of the IBC and the 2014 edition of ACI 318 as amended in IBC Section 1905. Thus, ACI 318 is part of the IBC and its applicable provisions must be satisfied when the IBC is adopted in a jurisdiction. It is important to note that ACI 318 provides minimum requirements for the materials, design, construction, and strength evaluation of structural concrete members and systems in any structure designed and constructed under the requirements of the general building code, such as the IBC. The purpose and applicability of the requirements in ACI 318-14 can be found in Sections 1.3 and 1.4 of that document, respectively. The requirements of the 2016 edition of ASCE/SEI 7 Minimum Design Loads and Associated Criteria for Buildings and Other Structures (Reference 3) are also used throughout this publication, including several important provisions pertaining to diaphragm loading and Seismic Design Category (SDC).

1.2 Definitions Diaphragms have an essential role in both the gravity and lateral force-resisting systems in building structures. Reference 2 provides the following definition of a diaphragm: A horizontal or sloped system acting to transmit lateral forces to vertical elements of the lateral force-resisting system. When the term “diaphragm” is used, it shall include horizontal bracing systems. A similar definition is given in Reference 3. Basically, diaphragms are roof and floor systems within a structure that must resist and transfer gravity and lateral forces. Diaphragms must support and transfer gravity loads to columns, walls, and other supporting elements in a building. The weight of the structure, superimposed dead loads, and live loads are common out-of-plane gravity loads that are applied to the surface of a diaphragm. Roof diaphragms must also be able to support effects due to rain, snow, and wind, to name a few. Minimum gravity and vertical environmental loads for floors and roofs are determined accordance with References 2 and 3. Lateral forces from wind, earthquakes, soil pressure, and other types of forces are transferred from diaphragms to walls, moment-resisting frames, and other types of lateral force-resisting systems in a building. The load path from the application of the force on the diaphragm to the vertical elements of the lateral force-resisting system (LFRS) depends on several factors, including the type of applied force and the rigidity (or, flexibility) of the diaphragm. The in-plane forces, which are determined mainly in accordance with the provisions in References 2 and 3, generate in-plane shear forces, bending moments, and axial forces in a diaphragm. The connections between the diaphragm and the vertical elements of the LFRS are very important; these connections must be properly designed and detailed otherwise there is no mechanism for proper load transfer to the LFRS. Collectors, which are part of a diaphragm and are sometimes referred to as drag struts, are required where the LFRS does not extend the full depth of a diaphragm or where there is a discontinuity in the LFRS, for example. The following definition of a collector is given in Reference 2: A horizontal diaphragm element parallel and in line with the applied force that collects and transfers diaphragm shear forces to the vertical elements of the lateral force-resisting system or distributes forces within the diaphragm, or both. A specific definition related to diaphragms that transfer forces to the vertical elements of a seismic force-resisting system (SFRS) is given in Reference 3. These elements are also an essential part of the overall load path for lateral forces, and References 1 through 3 contain design and detailing requirements based on SDC. Figure R12.1.1 in Reference 1 depicts typical diaphragm actions in a reinforced concrete building. In general, diaphragms must be designed and detailed for the combined effects due to in-plane and out-of-plane load effects in accordance with the factored load combinations in Chapter 5 of Reference 1.

1-1

Design Guide for Reinforced Concrete Diaphragms 1.3 Scope The design and detailing of diaphragms that are cast-in-place concrete slabs utilizing nonprestressed, steel reinforcement in building structures are covered in this publication. Slabs-on-ground that are part of the LFRS and transmit lateral forces from other portions of the structure to the soil are also covered (see Sections 1.4.7 and 13.2.4 of Reference 1). Slabs-on-ground that resist earthquake forces from walls or columns that are part of the SFRS must be designed as diaphragms in accordance with Section 18.12 of ACI 318-14 (see Section 18.13.3.4 of that document). Because in-plane load effects from wind, earthquakes, and other types of forces play an important part in the design and detailing process, in-depth information from References 2 and 3 is presented on how to calculate these lateral forces for buildings assigned to SDC A through F. ACI 318-14 provisions related to diaphragms are organized in tables, figures, and flowcharts throughout this publication. These provide a roadmap that guides the reader through the requirements, and are especially useful because ACI 318 underwent a major reorganization going from the 2011 to the 2014 edition, the first such reorganization since 1971. Requirements in ACI 31814 are organized mainly by member type, which is a major change of how the document was organized in earlier editions. New Chapter 12 and Chapter 18 in ACI 318-14 contain the provisions for reinforced concrete diaphragms, and supplementary requirements in other chapters of ACI 318 are covered as needed. Throughout this publication, section numbers from ACI 318-14 are referenced as illustrated by the following: Section 12.4 of ACI 318-14 is denoted as ACI 12.4. Section numbers from the 2018 IBC and ASCE/SEI 7-16 are referenced as follows: Section 1613 from the 2018 IBC and Section 12.3 of ASCE/SEI 7-16 are denoted as IBC 1613 and ASCE/SEI 12.3, respectively.

1.4 Organization of this Publication Chapter 2 contains limits on the specified compressive strength of concrete and the types of nonprestressed reinforcement that are permitted to be specified for reinforced concrete diaphragms. Information is provided based on whether the diaphragm is in a building that contains special seismic systems or not. Methods on how to determine diaphragm thickness based on serviceability, in-plane, and out-of-plane requirements are given in Chapter 3. Included are minimum thickness provisions for diaphragms in buildings assigned to SDC A through C and SDC D through F. Chapter 4 contains comprehensive information on how to determine the following forces on a diaphragm: (1) in-plane forces (wind, general structural integrity, seismic, soil, flood, tsunami, transfer, connection, and column bracing), (2) out-of-plane forces, and (3) collector forces. Step-by-step procedures are given on how to calculate these diaphragm forces for buildings assigned to SDC A through F. Strength design load combinations from References 1 through 3 that are to be used to design diaphragms are given in Chapter 5. Specific load combinations are provided for buildings assigned to SDC A, SDC B, SDC C, and SDC D through F. Included is information on in-plane inertial and transfer forces and direction of loading requirements that must be satisfied based on SDC. Chapter 6 covers the general modeling and analysis requirements for diaphragms that are given in ACI 12.4.2. Included is information on how to determine whether a diaphragm is rigid or not and horizontal distribution of lateral forces based on the in-plane stiffness of a diagram. A method is provided on how to determine the location of the center of rigidity for a rigid diaphragm. Also included are approximate methods to determine in-plane stiffnesses of moment frames and shear walls, which can be used in equivalent beam models of diaphragms. This chapter also contains the ACI 318 analysis methods, which can be used to determine in-plane design moments and shears in diaphragms. Finally, beam models are presented; these models are utilized to determine in-plane internal forces in rigid diaphragms with and without openings in buildings without major irregularities or transfer forces. Equations are provided on how to calculate chord, shear-transfer, and collector forces for the cases where the collector is the same width as or is wider than the vertical element of the LFRS it frames in to. The general requirements for strength design pertaining to reinforced concrete diaphragms are given in Chapter 7. Included are basic performance requirements, strength reduction factors, and nominal strength of diaphragms and collectors (moment and axial force, shear, and shear transfer). Chapter 8 contains comprehensive procedures on how to determine the required reinforcement for diaphragms based on two different types of common construction methods. Numerous design aids and figures are provided that summarize the pertinent design and detailing requirements based on SDC. A step-by-step design procedure that can be used to design and detail reinforced concrete diaphragms in buildings assigned to SDC A through F based on the information in Chapters 2 through 8 is given in Chapter 9. The following design steps are provided that reference the equations, tables, and figures in the chapters (flowcharts are given for steps 2 through 11):

1-2

Design Guide for Reinforced Concrete Diaphragms 1. Selecting the materials 2. Determining the diaphragm thickness 3. Determining the design forces 4. Determining the diaphragm classification, selecting the diaphragm model, and determining the diaphragm internal forces 5. Determining the combined load effects 6. Determining the chord reinforcement 7. Determining the diaphragm shear reinforcement 8. Determining the shear transfer reinforcement 9. Determining the reinforcement due to eccentricity of collector forces 10. Determining the anchorage reinforcement 11. Determining the collector reinforcement Completely worked-out examples are given in Chapters 10 that illustrate the proper application of the code requirements for buildings assigned to SDC A through D. These examples follow the design procedures in the chapters and the flowcharts in Chapter 9 and utilize the design aids wherever possible. Examples are included for diaphragms with relatively large openings and for buildings with horizontal irregularities.

1-3

Design Guide for Reinforced Concrete Diaphragms

1-4

Design Guide for Reinforced Concrete Diaphragms

Chapter 2 Materials 2.1 Overview Limits on the specified compressive strength of concrete and the types of nonprestressed reinforcement that are permitted to be specified for reinforced concrete diaphragms and collectors are covered in this chapter. Information is provided for all the Seismic Design Categories.

2.2 Concrete ACI Table 19.2.1.1 contains limits on the specified compressive strength of concrete, (see Table 2.1). For structural members in buildings—including diaphragms and collectors—that are not cast monolithically with portions of special moment frames and/or special structural walls, the minimum compressive strength is 2,500 psi for both normalweight and lightweight concrete. There is no upper limit in such cases. The durability requirements in ACI Table 19.3.2.1 must be satisfied as well.

Table 2.1 Limits for Concrete Compressive Strength Application General Special seismic systems(1)

Concrete

f 'c (psi) Minimum

Maximum

Normalweight and lightweight

2,500

None

Normalweight

3,000

None

Lightweight

3,000

5,000(2)

(1) Applicable to special moment frames and special structural walls.

(2) Limit is permitted to be exceeded where demonstrated by experimental evidence The minimum value of for diaphragms and collectors that members with lightweight concrete provide strength and toughness that are cast monolithically with portions of special moequal to or exceeding those of comparable members made with normalweight concrete of the same strength. ment frames or special structural walls is 3,000 psi for both normalweight and lightweight concrete. There is no upper limit on for normalweight concrete, but for lightweight concrete, the upper limit is 5,000 psi. This limit is imposed because of the very limited amount of experimental and field data that are available on the behavior and performance of members made with lightweight concrete. However, values of greater than 5,000 psi for lightweight mixes are permitted if it can be shown that members made from the lightweight mix provide strength and toughness levels that equal or exceed those of comparable members made from normalweight concrete of the same compressive strength.

Typical concrete compressive strengths used in reinforced concrete floor and roof systems with nonprestressed reinforcement are 4,000 psi and 5,000 psi. Using a compressive strength greater than 5,000 psi is usually not warranted.

2.3 Nonprestressed Reinforcement ACI Table 20.2.2.4a contains maximum values of specified yield strength, , and permissible bar types for deformed reinforcing bars based on usage and application. The following types of reinforcement in diaphragms and collectors are limited to for design calculations: 1. Shear reinforcement in diaphragms 2. S  hear-friction reinforcement used to transfer forces between the diaphragm and the vertical elements of the lateral forceresisting system 3. Shear-friction reinforcement used to transfer forces between the diaphragm and collectors 4. Collector transverse reinforcement, including that required by ACI 18.12.7.5 The specified yield strength of collector and chord reinforcement used for flexure and axial force in diaphragms and collectors that are not cast monolithically with portions of special moment frames and/or special structural walls is limited to 80,000 psi; otherwise, is limited to 60,000 psi. Using reinforcing bars with different specified yield strengths in different parts of a slab system is not recommended. That is why Grade 60 reinforcement is commonly specified for the entire slab system. Any of the following types of reinforcement may be used in diaphragms and collectors that are not cast monolithically with portions of special moment frames and/or special structural walls: 1. Carbon-steel (ASTM A615) 2. Low-alloy steel (ASTM A706) 3. Stainless-steel (ASTM A955) 4. Rail-steel and Axle-steel (ASTM A996)

2-1

Design Guide for Reinforced Concrete Diaphragms According to ACI 20.2.2.5, ASTM A706 and A615 reinforcement that conforms to ACI 20.2.2.5(b) are required to be used in diaphragms and collectors that are cast monolithically with portions of special moment frames and/or special structural walls. Chord or collector reinforcement that is placed within beams of special moment frames, including the effective slab width of the beam, must be ASTM A706 or equivalent.

2-2

Design Guide for Reinforced Concrete Diaphragms

Chapter 3 Determining the Diaphragm Thickness 3.1 Overview According to ACI 12.3.1, diaphragms must have sufficient thickness so that all applicable strength and serviceability requirements are satisfied. The following load effects must be investigated for strength: (1) in-plane moments, in-plane shear forces, and axial forces due to wind, seismic, and other applicable lateral forces and (2) out-of-plane moments and shears due to gravity forces or combinations of gravity and lateral forces. ACI 318 serviceability requirements are covered in Section 3.2 below, and in-plane and out-of-plane strength requirements are given in Sections 3.3 and 3.4, respectively. Information on how to determine minimum diaphragm thickness for buildings assigned to Seismic Design Category (SDC) A, B, or C is given in Section 3.5. Section 3.6 contains similar information for buildings assigned to SDC D, E, or F.

3.2 Serviceability Requirements 3.2.1 One-way Slabs

Table 3.1 Minimum Thickness, h , of Solid, Nonprestressed One-way Slabs

Minimum h(1)

Support Condition Simply supported

l/20

One end continuous

l/24

Both ends continuous

l/28

Cantilever

l/10

(1) Expressions for minimum h are applicable to Grade 60 reinforcement and normalweight concrete. These expressions are to be modified in accordance with ACI 7.3.1.1.1 for reinforcement other than Grade 60 and ACI 7.3.1.1.2 for lightweight concrete.

Expressions for minimum thickness of solid, nonprestressed one-way slabs that are not supporting or attached to partitions or other construction likely to be damaged by large deflections are given in ACI Table 7.3.1.1 (see Table 3.1). These expressions are based on span length, , and are applicable to Grade 60 reinforcement and normalweight concrete. For reinforcement with a specified yield strength other than 60,000 psi, the expressions in the table are to be multiplied by where the specified yield strength of the reinforcement, , is in pounds per square inch (ACI 7.3.1.1.1). For lightweight concrete with an equilibrium density, , in the range of 90 to 115 lb/ft3, the expressions in the table are to be multiplied by the greater of and 1.09 (ACI 7.3.1.1.2).

3.2.2 Two-way Slabs Minimum thickness of nonprestressed two-way slabs without interior beams is determined by the expressions in ACI Table 8.3.1.1 (see Table 3.2), which are based on the clear span length, , in the long direction. For slab systems without drop panels (drop panels are defined in ACI 8.2.4), the minimum thickness is the larger of the value obtained from the expression in Table 3.2 and 5 in. (ACI 8.3.1.1). Similarly, for slab systems with drop panels, the minimum thickness is the larger of the tabulated value and 4 in. According to ACI 8.3.1.1, exterior panels are considered to be without edge beams where . The term is defined in ACI 8.10.2.7 as the ratio of the flexural stiffness of the beam to the flexural stiffness of a slab width bounded laterally by the centerlines of adjacent panels, if any, on each side of the beam . In typical situations where the concrete mix for the beams and slab are the same, the modulus of elasticity of the beam, , and the modulus of elasticity of the slab, , are equal, so where and are the moments of inertia of the beam and slab, respectively. Table 3.2 Minimum Thickness, h, of Nonprestressed Two-way Slabs Without Interior Beams(1)

Without Drop Panels(3) fy (psi)(2)

Exterior Panels Without With Edge Edge Beams Beams(4)

With Drop Panels(3) Exterior Panels

Interior Panels

Without Edge Beams

With Edge Beams(4)

Interior Panels

40,000

ln/33

ln/36

ln/36

ln/36

ln/40

ln/40

60,000

ln/30

ln/33

ln/33

ln/33

ln/36

ln/36

75,000

ln/28

ln/31

ln/31

ln/31

ln/34

ln/34

(1) ln is the clear span in the long direction, measured face-to-face of supports (in.). (2) For fy between the values given in the table, minimum thickness is to be calculated by linear interpolation. (3) Drop panels are defined in ACI 8.2.4. (4) Exterior panels with edge beams are defined as slabs with beams between columns along exterior edges. Exterior panels are considered to be without edge beams where af < 0.8. The value of af for an edge beam is calculated in accordance with ACI 8.10.2.7.

3-1

Design Guide for Reinforced Concrete Diaphragms Expressions for minimum thickness of nonprestressed two-way slabs with beams spanning between supports on all sides are given in ACI Table 8.3.1.2 (see Table 3.3). For panels without sufficiently stiff beams around the edges (that is, where ), the expressions for minimum thickness in Table 3.2 for two-way slabs without interior beams must be used. The term is the average value of for all beams on the edges of a panel. Table 3.3 Minimum Thickness, h, of Nonprestressed Two-way Slabs With Beams Spanning Between Supports on all Sides

αfm (1)

Minimum h (in.)

αfm ≤ 0.2

ACI 8.3.1.1 applies (see Table 3.2)

0.2 < αfm ≤ 2.0

Greater of(2),(3): 5.0

αfm > 2.0

Greater of(2),(3): 3.5

(1) αfm is the average value of αf for all beams on the edges of a panel where αf is determined in accordance with ACI 8.10.2.7. (2) n is the clear span in the long direction, measured face-to-face of supports (in.). (3) β is the ratio of clear spans in the long to short directions of the slab.

Figure 3.1 can be used to determine minimum slab thickness based on the requirements in ACI 8.3.1.1 and 8.3.1.2 for various types of two-way slab systems assuming Grade 60 reinforcement.

3.3 In-plane Strength Requirements 3.3.1 Overview Cast-in-place reinforced concrete slabs must have adequate strength to resist in-plane effects, including those due to wind, seismic, fluid, or lateral earth pressure. In general, both flexure and shear must be considered when determining a required slab thickness. A diaphragm thickness based on serviceability requirements is usually adequate to satisfy in-plane strength requirements.

3.3.2 Flexure In-plane moments, which are determined by one of the models in Chapter 6 of this publication, are typically resisted by chord reinforcement placed perpendicular to the direction of the lateral force. In most cases, this reinforcement is determined using a slab thickness that is sufficient for serviceability or other strength requirements; there is usually no need to increase the thickness of a slab based on in-plane flexural requirements. Chord reinforcement is normally concentrated near the edges of the slab and around any openings in the slab; however, it is permitted to distribute this reinforcement within 25 percent of the diaphragm depth from the tension edge of the diaphragm (ACI 12.5.2.3; see ACI Figure R12.5.2.3 and Section 8.4 of this publication).

3.3.3 Shear In-plane shear strength requirements for diaphragms are given in ACI 12.5.3 for buildings assigned to SDC A, B, or C. The nominal shear strength, , of a cast-in-place reinforced concrete diaphragm is determined by ACI Equation (12.5.3.3), which is a function of the area of the diaphragm, , the modification factor, , that accounts for the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength, (which is limited to 10,000 psi), and the amount of distributed shear reinforcement in the slab, . The following equation must be satisfied for in-plane shear strength: (3.1)

3-2

Design Guide for Reinforced Concrete Diaphragms 12.0

�at

11.0

�bt �ct

�dt �et

.

10.0

��t

n

9.0

Minimum slab thickness h �i

i

�gt

8.0 7.0 6.0 5.0 4.0 3.0

10

12

14

16

18

20

22

Longer clear span n �� t f

24

26

28

30

�at Flat plate, external panels without edge beams �bt Flat plate, internal panels and external panels with edge beams and Flat slab, external panels without edge beams �ct Two-way beam-supported slab,    1 fm 0.5,

�dt Flat slab, internal panels and external panels with edge beams

�et Two-way beam-supported slab,  fm  1.0,  1 ��t Two-way beam-supported slab,  fm  1.5,  1

�gt Two-way beam-supported slab,  fm  2.0,  1

Figure 3.1 Minimum Slab Thickness of Two-Way Slab Systems for Grade 60 Reinforcement

Equation (3.1) is also applicable to diaphragms in buildings assigned to SDC D, E, or F [see ACI Equation (18.12.9.1) for , which is the same as that in ACI Equation (12.5.3.3)]. Required shear strength, , due to in-plane forces is determined using one of the models in Chapter 6 of this publication. Strength reduction factor, , is equal to 0.60 or 0.75, and information on the correct value to use is given in Section 7.2 of this publication. The gross area of the diaphragm, , is usually determined using a one-foot diaphragm length in the direction of analysis, that is, where is the thickness of the reinforced concrete slab. The shear reinforcement ratio, , is equal to the area of the uniformly distributed slab reinforcement that is oriented parallel to the shear force divided by the gross area of the slab perpendicular to that reinforcement. It is not uncommon for the design shear strength of the concrete, , alone to satisfy in-plane shear strength requirements. Assuming , Equation (3.1) can be solved for the minimum slab thickness, , based on a one-foot diaphragm length: (3.2)

3-3

Design Guide for Reinforced Concrete Diaphragms In this equation, is the maximum factored in-plane shear force in the diaphragm divided by the overall depth of the diaphragm in the direction of analysis (minus any opening lengths). For normalweight concrete with , minimum can be determined by the following equations: • For

: (3.3)

• For

: (3.4)

In these equations,

has the units of kips per linear foot.

The cross-sectional dimensions of a diaphragm must also be selected so that ACI Equation (12.5.3.4) is satisfied (the same requirement is also given in ACI 18.12.9.2 for buildings assigned to SDC D, E, or F): (3.5) This equation essentially establishes a maximum design shear strength for the diaphragm, or equivalently, a lower limit on the diaphragm thickness for a maximum factored shear force, . Assuming a one-foot diaphragm length, Equation (3.5) can be solved for : (3.6)

For normalweight concrete with units of kips per linear foot: • For

, minimum

can be determined by the following equations where

has the

: (3.7)

• For

: (3.8)

Comparing Equations (3.2) and (3.6), it is evident that the minimum required slab thickness based on the design shear strength of the concrete alone is always larger than the minimum required slab thickness based on the maximum permitted design shear strength where .

3.4 Out-of-plane Strength Requirements 3.4.1 Flexure Flexural reinforcement is provided in the slab based on the factored moments, , at the critical sections along the span. Where applicable, the simplified method of analysis in ACI 6.5 for one-way slabs or the Direct Design Method in ACI 8.10 for two-way slabs can be used to determine for gravity loads. For slab systems that are part of the lateral force-resisting system (LFRS), these factored moments are due to the combined effects from gravity and lateral forces. Reasonable amounts of flexural reinforcement can usually be determined at the critical sections using a slab thickness based on serviceability or twoway shear requirements; out-of-plane flexural strength requirements rarely have an impact on the required slab thickness.

3.4.2 One-way Shear The one-way shear strength requirements in ACI 22.5 must be satisfied for both one-way and two-way slab systems at the critical sections defined in ACI 7.4.3 and 8.4.3, respectively. Because shear reinforcement is rarely, if ever, used in one-way or two-way slabs to supplement one-way shear strength, the design shear strength of the concrete, , must be greater than or equal to the required shear strength, One-way slabs are usually not part of the LFRS, so the required shear strength, . , is equal to the factored distributed gravity load, , times the shaded area in Figure 3.2. In this figure, the critical section is located a distance from the face of the support, which means the requirements of ACI 7.4.3.2 are satisfied; otherwise, the critical section must be located at the face of the support (ACI 7.4.3.1).

3-4

Design Guide for Reinforced Concrete Diaphragms With the nominal shear strength of the concrete, , determined by ACI Equation (22.5.5.1) [that is, ] and assuming , the minimum to satisfy one-way shear requirements in a one-way slab can be determined by the following equation:

ℓ�

ℓ � ⁄2

12��

(3.9)

The critical section for one-way shear in a two-way slab system is defined in ACI 8.4.3 and is illustrated in Figure 3.3 for a flat plate where the requirements of ACI 8.4.3.2 are satisfied. The minimum slab thickness to satisfy one-way shear strength requirements in this case can be determined by the following equation:

𝑑𝑑

Critical section

(3.10)

where is the width of the slab that resists the shear force, which is equal to for the case shown in Figure 3.3. For gravity loads, is equal to the factored distributed gravity load, , times the tributary area indicated in Figure 3.3, that is, .

Figure 3.2 Critical Section for One-way Shear in a One-way Slab System

ℓ�  

One-way shear requirements must also be checked in the perpendicular direction to that shown in Figure 3.3. It is rare for one-way shear to govern in a two-way slab system.

Critical section

3.4.3 Two-way Shear

(3.11)

where









𝑐𝑐�  

ℓ�  

The two-way shear requirements in ACI 8.4.4 must be satisfied for two-way slab systems. Where column-line beams are present, these requirements rarely govern. For systems without beams, such as flat plates, the thickness of the slab may have to be greater than that required for serviceability to satisfy these requirements. For slabs without shear reinforcement, factored shear stresses, , at the critical sections defined in ACI 22.6.4 must be less than or equal to the design two-way shear strength, , of the concrete where is defined in ACI Table 22.6.5.2:

𝑑𝑑 

𝑐𝑐�  

Tributary area

 

  Figure 3.3 Critical Section for One-way Shear in a Flat Plate System       properties of lightweight concrete relative to normalmodification factor that reflects the reduced mechanical   weight concrete of the same compressive strength (see ACI Table 19.2.4.2)   ratio of long to short dimensions of the sides of a  column, concentrated load, or reaction area 40 for interior columns (four-sided critical section); 30 for edge columns (three-sided critical section); and 20 for  corner columns (two-sided critical section) perimeter of critical section for two-way shear (see ACI 22.6.4)

3-5

Design Guide for Reinforced Concrete Diaphragms The critical section for two-way shear at an interior column in a flat plate system is illustrated in Figure 3.4.

ℓ�  

The assumed distribution of two-way shear stresses due to direct shear and eccentricity of shear in accordance with ACI 8.4.4 is illustrated in Figure 3.5. The maximum factored shear stresses, , on faces and of the critical section can be determined by the following equations: ℓ�  

𝑏𝑏�   𝑐𝑐�  

Critical section

𝑐𝑐�   𝑏𝑏�  

𝑑𝑑𝑑𝑑 �t���� 

(3.12)





















direct shear force on the column due to  gravity and lateral forces, where applicable

Figure 3.4 Critical Section for Two-way Shear in a Flat Plate System

area of critical section factor used to determine the fraction of  transferred by eccentricity of shear

𝑏𝑏�   𝑐𝑐�  

D 

A 

(see ACI 8.4.4.2)

dimension of critical section measured paral lel to the direction of analysis dimension of critical section measured per pendicular to the direction of analysis slab moment resisted by the column due  to gravity and lateral forces, where applicable, transferred by a combination of flexure and eccentricity of shear distance from the centroid of the critical  section to face AB distance from the centroid of the critical  section to face CD

C  𝑐𝑐   𝑐𝑐   B  �� ��

𝑏𝑏�  

𝑐𝑐�  

D 

C 

In buildings assigned to SDC A, B, or C, two-way slab systems without beams are permitted to resist the effects from wind and seismic forces. Load combinations must include gravity and lateral force effects, where applicable, when determining the maximum shear stress, in such cases. Because two-way slab systems without beams are not permitted to be part of the seismic force-resisting system (SFRS) in buildings assigned to SDC D, E, or F, includes

A 

𝑐𝑐��   𝑐𝑐��  B 

Critical section

𝑐𝑐�   𝑏𝑏�  

Interior Column

Critical section

𝑐𝑐�   𝑏𝑏�  

Edge Column

𝑣𝑣����

𝑣𝑣����

𝑐𝑐��  

𝑐𝑐��  

𝑣𝑣����

𝑣𝑣����

𝑐𝑐��   𝑐𝑐��  

Figure 3.5 Assumed Distribution of Shear Stresses Due to Direct Shear and Eccentricity of Shear 𝐷𝐷

property of the critical section analogous to  the polar moment of inertia

Section properties of the critical section for two-way shear are given in Table 3.4 for rectangular columns and in Figure 3.6 for interior circular columns. Shear strength requirements are generally more critical at edge and corner columns where relatively large unbalanced moments cause relatively large factored shear stresses.

3-6

Tributary area 

𝛾𝛾� 𝑀𝑀��

𝑑𝑑

where

𝐴𝐴� � ��𝐷𝐷 � 𝑑𝑑�𝑑𝑑 𝑐𝑐 � 𝑐𝑐 � �

𝐷𝐷 � 𝑑𝑑 2

𝐽𝐽 𝐽𝐽 𝐷𝐷 � 𝑑𝑑 � 𝑑𝑑 � � � �𝑑𝑑 � � � 𝑐𝑐 𝑐𝑐 � 3 2

Figure 3.6 Section Properties of the Critical Section for Interior Circular Columns

Design Guide for Reinforced Concrete Diaphragms Table 3.4 Section Properties of the Critical Section for Rectangular Columns Case 1

2

𝑐𝑐�   𝑐𝑐�  

Section Property

𝑐𝑐�  

𝛾𝛾� 𝑀𝑀��   𝐴𝐴 

𝑏𝑏� � 𝑐𝑐� � � 

𝑐𝑐�  

𝐷𝐷 

𝐶𝐶  𝑐𝑐��  

𝐵𝐵 

𝑐𝑐��  

𝛾𝛾� 𝑀𝑀��  

𝑏𝑏� � 𝑐𝑐� � � 

𝑑𝑑 𝑏𝑏� � 𝑐𝑐� �   2

 

Table 3.4 –

𝑐𝑐��  

𝐵𝐵 

𝐴𝐴 

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

𝐷𝐷 

𝐶𝐶 

𝑐𝑐��  

𝑏𝑏� � 𝑐𝑐� � 𝑑𝑑 

 

             

        Continued  

Case 3

4

𝑐𝑐�   𝑐𝑐�  

Section Property

𝛾𝛾� 𝑀𝑀��   𝐴𝐴 

𝑐𝑐�  

𝐷𝐷 

𝐶𝐶  𝐵𝐵 

𝑐𝑐��  

𝑐𝑐��  

𝑐𝑐�   𝛾𝛾� 𝑀𝑀��  

𝑑𝑑 𝑏𝑏� � 𝑐𝑐� �   2

𝐴𝐴 

𝑑𝑑 𝑏𝑏� � 𝑐𝑐� �   2

𝑏𝑏� � 𝑐𝑐� � 𝑑𝑑 

Case 3: Case 4:

          Edge rectangular column bending perpendicular to the edge   Corner rectangular column bending perpendicular to the edge  

               

𝐷𝐷 

𝐶𝐶  𝐵𝐵 

𝑐𝑐��  

𝑐𝑐��  

𝑑𝑑 𝑏𝑏� � 𝑐𝑐� �   2

 

3-7

Design Guide for Reinforced Concrete Diaphragms only gravity load effects (wind force effects in this case are resisted by the SFRS); however, the slab-column connection requirements of ACI 18.14.5 must be satisfied. Figure 3.7 can be used to determine a preliminary slab thickness, , for flat plate systems subjected to gravity loads without shear reinforcement where slab thickness is controlled by two-way shear requirements. The information in the figure is based on the following assumptions: • Square edge column of size

bending perpendicular to the slab edge with a three-sided critical section

• Column supports a tributary area • G  ravity load moment transferred between the slab and the edge column in accordance with the Direct Design Method of ACI 8.10   • Normalweight concrete with a compressive strength of 4,000 psi  

The term is the factored distributed gravity load, which must include the slab weight; this weight can be estimated using a   slab thickness based on is obtained from Figure 3.7 as a function of and the area   serviceability requirements. The ratio   ratio . A preliminary slab thickness, , is obtained by adding 1.25 in. to .      

0.90

A/c12 ��250

0.80

225 200

0.70

175

0.60

d/c1

150

0.50

125

0.40

100

0.30

75

0.20

50

0.10

200

250

300

qu ��psf

350

400

Figure 3.7 Preliminary Slab Thickness for Flat Plate Systems

In general, determining the thickness of a flat plate system based on two-way shear requirements for out-of-plane loads is an iterative process: the requirements are initially checked using the slab thickness determined for serviceability requirements, and if shear strength requirements are not satisfied, the slab thickness is increased until such requirements are satisfied. It may be possible to satisfy two-way shear requirements using a slab thickness based on serviceability requirements if shear reinforcement in accordance with ACI 22.6.7 (single- or multiple-leg stirrups) or ACI 22.6.8 (headed shear stud reinforcement) is provided around the columns. Shear strength may also be increased by drop panels and shear caps but using these types of elements may not always result in the most economical solution.

3-8

Design Guide for Reinforced Concrete Diaphragms 3.5 Minimum Diaphragm Thickness for SDC A, B, and C For buildings assigned to SDC A, B, or C, the minimum diaphragm thickness must be the largest of the following: 1. T  hickness based on serviceability requirements of ACI 7.3.1.1 for one-way slabs or ACI 8.3.1.1 for two-way slabs (Section 3.2) 2. Thickness based on in-plane strength requirements (Section 3.3) 3. Thickness based on out-of-plane strength requirements (Section 3.4) Fire-resistance requirements of the governing building code must also be considered when selecting a minimum slab thickness. For local jurisdictions that have adopted the IBC (Reference 2), minimum slab thickness for various fire-resistance ratings based on concrete type is given in IBC Table 722.2.2.1. It is common for fire-resistance requirements to be satisfied using a slab thickness that satisfies serviceability requirements, strength requirements, or both.

3.6 Minimum Diaphragm Thickness for SDC D, E, and F In addition to the requirements outlined in Section 3.5, diaphragms in buildings assigned to SDC D, E, or F must satisfy the minimum thickness requirements in ACI 18.12.6.1. According to that section, a minimum 2-in.-thick diaphragm must be provided to transmit seismic forces. This minimum thickness reflects the current practice in joist and waffle systems in cast-in-place construction. Like in the case for buildings assigned to SDC A, B, or C, fire-resistance requirements of the governing building code must also be considered when selecting a minimum diaphragm thickness in buildings assigned to SDC D, E, or F. According to IBC Table 722.2.2.1, a slab thickness greater than 2 in. must be provided to achieve a 2-hour fire-resistance rating for all concrete types.

3-9

Design Guide for Reinforced Concrete Diaphragms

3-10

Design Guide for Reinforced Concrete Diaphragms

Chapter 4 Diaphragm Design Forces 4.1 Overview Lateral forces from wind, earthquakes, soil pressure, fluid pressure, floods, or tsunamis generate in-plane shear forces, bending moments, and axial forces in diaphragms. Methods to determine these in-plane forces in accordance with ASCE/SEI 7 (Reference 3) are given in Section 4.2. Diaphragm transfer forces, connection forces, column bracing forces, and out-of-plane forces are covered in Sections 4.3, 4.4, 4.5, and 4.6, respectively. Procedures on how to calculate collector design forces are given in Section 4.7.

4.2 In-Plane Forces 4.2.1 Wind Forces Wind Load Provisions in ASCE/SEI Chapters 27 and 28 Wind pressures on the main wind force-resisting system (MWFRS) of buildings can be determined using the provisions in ASCE/ SEI Chapters 27 and 28 subject to the conditions and limitations in those chapters. The directional procedures in Chapter 27 and the envelope procedures in Chapter 28 are summarized in Table 4.1 (see next page). Because of its broader applicability than the other procedures in Chapters 27 and 28, the steps to determine wind pressures in accordance with Part 1 of Chapter 27 are given below. Determination of Wind Pressures in Accordance with Part 1 of ASCE/SEI Chapter 27 The steps to determine wind pressures on the MWFRS of enclosed, partially enclosed, and open buildings of all heights are given in ASCE/SEI Table 27.2-1. These steps are covered in detail below with emphasis on the determination of wind pressures needed for diaphragm design. • Step 1: Determine the Risk Category of the Building The Risk Category of a building is a function of its use or occupancy. IBC Table 1604.5 or ASCE/SEI Table 1.5-1 is to be used to establish the Risk Category based on the nature of the use or occupancy defined in the tables. • Step 2: Determine Basic Wind Speed, The basic wind speed, , at a site is determined using the wind hazard maps in IBC 1609.3 or ASCE/SEI 26.5 based on Risk Category. A summary of the wind hazard maps and the corresponding return periods in years is given in Table 4.2. Reference 5 can also be used to determine for a given Risk Category. Table 4.2 Wind Hazard Maps in IBC 1609.3 and ASCE/SEI 26.5 Location

Conterminous U.S. Alaska Puerto Rico Guam Virgin Islands American Samoa

Hawaii

Figure Number IBC

ASCE/SEI 7

Risk Category*

Return Period (years)

1609.3(4)

26.5-1A

I

300

1609.3(1)

26.5-1B

II

700

1609.3(2)

26.5-1C

III

1,700

1609.3(3)

26.5-1D

IV

3,000

1609.3(8)

26.5-2A

I

300

1609.3(5)

26.5-2B

II

700

1609.3(6)

26.5-2C

III

1,700

1609.3(7)

26.5-2D

IV

3,000

*See IBC Table 1604.5 or ASCE/SEI Table 1.5-1 for definitions of Risk Categories.

4-1

Design Guide for Reinforced Concrete Diaphragms Table 4.1 Summary of Wind Load Provisions in Chapters 27 and Chapter 28 of ASCE/SEI 7-16 for MWFRSs Chapter

Applicability

Part

Building Type(1)

Enclosed 1

Conditions / Limitations

Height Limit

Partially enclosed

None

Open

• Regular-shaped building(2) • Building does not have response characteristics making it subject to across-wind loading, vortex shedding, or instability due to galloping or flutter • Building is not located on a site where channeling effects or buffeting in the wake of upwind obstructions warrant special consideration • Same conditions/limitations as in Part 1 • Building must meet the conditions for either a Class 1 or Class 2 building(4): 1.

27

Class 1 building a. b.

2

Enclosed, simple diaphragm(3)

160 ft

2.

Mean roof height



Class 2 building a.



b.



c.  Fundamental natural frequency ≥ ( is in ft)

• Building must have either a rigid or flexible diaphragm • Regular-shaped building(2) • Building does not have response characteristics making it subject to across-wind loading, vortex shedding, or instability due to galloping or flutter

Enclosed, low-rise(5) 1

Partially enclosed, low-rise Open, low-rise

Least horizontal dimension of building

28

2

(1)

60 ft

Enclosed, simple diaphragm low-rise

• Building is not located on a site where channeling effects or buffeting in the wake of upwind obstructions warrant special consideration • Same conditions as in Part 1 • Fundamental natural frequency ≥ 1 Hz • Building has an approximately symmetrical cross-section in each direction with either a flat roof or a gable or hip roof with an angle with respect to the horizontal less than or equal to 45 degrees • Building is exempted from torsional load cases indicated in Note 5 of ASCE/SEI Figure 28.3-1, or the torsional load cases defined in Note 5 do not control the design of any of the MWFRS of the building

Enclosure classifications are defined in ASCE/SEI 26.2. A regular-shaped building is defined as a building that has no unusual geometrical irregularity in spatial form (ASCE/SEI 26.2). (3) A simple diaphragm building is defined as a building in which both windward and leeward wind forces are transmitted by roof and vertically-spanning wall assemblies, through continuous roof and floor diaphragms, to the MWFRS (ASCE/SEI 26.2). (4) See ASCE/SEI Figure 27.4-1 for definitions of Class 1 and Class 2 buildings. (5) A low-rise building is defined as a building with a mean roof height less than or equal to (a) 60 ft and (b) the least horizontal dimension of the building (ASCE/SEI 26.2). (2)

4-2

Design Guide for Reinforced Concrete Diaphragms • Step 3: Determine Wind Pressure Parameters (a) Determine Wind Directionality Factor, The wind directionality factor, , which accounts for statistical nature of wind flow and the probability of the maximum effects occurring at any time for any given wind direction, is given in ASCE/SEI Table 26.6-1. For the MWFRS of building structures, . (b) Determine Exposure Categories Exposure categories are based on the surface roughness categories defined in IBC 1609.4.2 and ASCE/SEI 26.7.2 (see Table 4.3). These definitions are descriptive and have been purposely expressed this way so that they can be applied easily—while still being sufficiently precise—in most practical applications. Table 4.3 Surface Roughness Categories Surface Roughness Category

Description

B

Urban and suburban areas, wooded areas, or other terrain with numerous, closely spaced obstructions having the size of single-family dwellings or larger.

C

Open terrain with scattered obstructions having heights generally less than 30 feet. This category includes flat, open country and grasslands.

D

Flat, unobstructed areas and water surfaces. This category includes smooth mud flats, salt flats, and unbroken ice.

Definitions of the three exposure categories given in IBC 1609.4.3 and ASCE/SEI 26.7.3 are given in Table 4.4. Exposure B and Exposure D are illustrated in ASCE/SEI Figures C26.7-1 and C26.7-2, respectively. Table 4.4 Exposure Categories Exposure Category

Description • Mean roof height

Surface Roughness B prevails in the upwind direction for a distance >1,500 ft B

C

• Mean roof height Surface Roughness B prevails in the upwind direction for a distance > 2,600 ft or 20 times the height of the building, whichever is greater Applies for all cases where Exposures B and D do not apply

D

• Surface Roughness D prevails in the upwind direction for a distance > 5,000 ft or 20 times the height of the building, whichever is greater • Surface roughness immediately upwind of the site is B or C, and the site is within a distance of 600 ft or 20 times the building height, whichever is greater, from an Exposure D condition as defined above

According to IBC 1609.4 and ASCE/SEI 26.7, an exposure category must be determined upwind of a building for each wind direction that is considered in design. Wind must be assumed to come from any horizontal direction when determining wind pressures (IBC 1609.1.1 and ASCE/SEI 26.5.1). One rational way of satisfying this requirement is to assume that there are eight wind directions: four that are perpendicular to the main axes of the building and four that are at 45-degree angles to the main axes. Shown in ASCE/SEI Figure C26.7-8 are the sectors that are to be used to determine the exposure for a selected wind direction (IBC 1609.4.1 and ASCE/SEI 26.7.1). (c) Determine Topographic Factor, Buildings that are sited on the upper half of an isolated hill, ridge, or escarpment can experience significantly higher wind velocities than those sited on relatively level ground. The topographic factor, , in ASCE/SEI 26.8 accounts for this increase in wind speed, which is commonly referred to as wind speed-up. When all the conditions of ASCE/SEI 26.8.1 are met, is determined in accordance with ASCE/SEI Equation (26.8-1) and must be included in the calculation of design wind pressures. In all other cases, .

4-3

Design Guide for Reinforced Concrete Diaphragms (d) Determine Ground Elevation Factor, The ground elevation factor, , adjusts the velocity pressure, , determined in accordance with ASCE/SEI 26.10 based on the reduced mass density of air at elevations above sea level. ASCE/SEI Table 26.9-1 contains values of , which can be calculated using the equation in Note 2 of the table where is the ground elevation above sea level. A more complete version of ASCE/SEI Table 26.9-1 that includes air density values is provided in ASCE/SEI Table C26.9-1. It is permitted to take

for all elevations (see Note 3 in ASCE/SEI Table 26.9-1).

(e) Determine Gust-Effect Factor,

or

The gust effect factor, which accounts for both atmospheric and aerodynamic effects in the along-wind direction, is defined in ASCE/SEI 26.11 and depends on the natural frequency, , of a building. For rigid buildings ( ), is to be taken as 0.85 or may be calculated by ASCE/SEI Equation (26.11-6). For flexible buildings, ( ), is to be calculated by ASCE/SEI Equation (26.11-10). ASCE/SEI 26.11.3 contains approximate methods to determine for buildings that meet the limitations in ASCE/SEI 26.11.2.1. (f) Determine Enclosure Classification Any building or other structure must be classified as enclosed, partially enclosed, partially open, or open based on the definitions in ASCE/SEI 26.2. A summary of these definitions is given in Table 4.5. Table 4.5 Enclosure Classifications Classification

Definition

A building that complies with the following condition for each wall: Enclosed Building

A building that complies with the following conditions for each wall:

Partially enclosed building

1. 2.

Partially open building

A building that does not comply with the requirements for open, partially enclosed, or enclosed buildings

Open building

For each wall of the building,

total area of openings in a wall that receives positive external pressure gross area of wall in which

is identified

sum of the areas of openings in the building envelope (walls and roof) not including sum of the gross surface areas of the building envelope (walls and roof) not including

Special requirements are given in IBC 1609.2 and ASCE/SEI 26.12.3.2 for the protection of glazed openings in windborne debris regions. (g) Determine Internal Pressure Coefficients, Internal pressure coefficients tions defined in ASCE/SEI 26.12.

are given in ASCE/SEI Table 26.13-1 and are based on the enclosure classifica-

• Step 4: Determine Velocity Pressure Exposure Coefficients, This coefficient modifies wind velocity (or pressure) with respect to exposure and height above ground. Values of for Exposures B, C, and D at various heights above ground level are given in ASCE/SEI Table 26.10-1. In lieu of linear interpolation for heights that are not tabulated and for heights greater than 500 ft above the ground level, may be calculated at any height using the equations in Note 1 at the bottom of the table:

4-4

Design Guide for Reinforced Concrete Diaphragms

(4.1)

Values of and are given in ASCE/SEI Table 26.11-1 based on exposure. The above discussion on the determination of is valid for the case of a single roughness category (that is, uniform terrain). Procedures on how to determine for a single roughness change or multiple roughness changes are given in ASCE/SEI C26.10.1. • Step 5: Determine Velocity Pressures, The velocity pressure at height above the ground level is determined by ASCE/SEI Equation (26.10-1); this equation converts the basic wind speed, , to a velocity pressure: (4.2) where all terms in this equation have been defined previously. • Step 6: Determine External Pressure Coefficients, Values of are given in ASCE/SEI Figure 27.3-1 for windward walls, leeward walls, side walls, and roofs for enclosed and partially enclosed buildings with gable and hip roofs, monoslope roofs, and mansard roofs. Wall pressure coefficients are constant on windward and side walls; on leeward walls, the coefficients vary with the plan dimensions of the building (that is, the aspect ratio of the building ). The table in the upper part of ASCE/SEI Figure 27.3-1 also designates which velocity pressure to use— or —on a wall surface. Roof pressure coefficients vary with the ratio of the mean roof height to the plan dimension of the building and with the roof angle for a given wind direction (normal to ridge or parallel to ridge). All these roof pressure coefficients are intended to be used with , and the parallel-to-ridge wind direction is applicable for flat roofs. Values of or for other roof configurations and for open buildings with various roof configurations are given in ASCE Figures 27.3-2 through 27.3-7. , On

𝐿𝐿

This equation is used to calculate wind pressures on each surface of the building: windward wall, leeward wall, side walls, and roof. The pressures are applied simultaneously on the walls and roof, as depicted in ASCE/SEI Figure 27.3-1. Velocity pressure, , varies with respect to height on windward walls . For leeward walls, sidewalls, and roofs, the velocity pressure is constant and is evaluated at the mean roof height, . The effects from internal pressure cancel out when evaluating the total horizontal wind pressure on the MWFRS of a building. Thus, the total horizontal wind pressure at any height, , above ground in the direction of wind is equal to the external pressure on the windward face at height

𝑝𝑝�

Level 𝑥𝑥 � �

Level 𝑥𝑥

ℎ�

(4.3)

ℎ� � ℎ��� 2

Design wind pressures, , are determined by ASCE/SEI Equation (27.3-1) for enclosed and partially enclosed rigid and flexible buildings:

𝐵𝐵

ℎ���

• S  tep 7: Determine Wind Pressures, Each Building Surface

Wind

Level 𝑥𝑥 � �

Figure 4.1 Wind Pressure Uniformly Distributed Over Tributary Floor Height

4-5

Design Guide for Reinforced Concrete Diaphragms plus the external pressure on the leeward face

𝐵𝐵

.

For multistory buildings, it is common practice to calculate wind pressures at the roof and floor levels of a building and to assume that the total pressure at a level is uniformly distributed over the tributary height of that level. Illustrated in Figure 4.1 is the wind pressure on the MWFRS at level , which is located at distance from the base of a multistory building where is the sum of the windward and leeward pressures determined by Equation (4.3), that is, . It is assumed that is uniformly distributed over the tributary height

𝐿𝐿

Determination of Wind Forces on Diaphragms

.

A plan view of level is given in Figure 4.2. The resultant   wind force, at this level is equal to the total pressure, , , times the tributary , times the width of the building, height and acts at the centroid of the windward face. For wind pressures determined by Part 1 (or Part 2) of ASCE/SEI Chapter 27, this represents Case 1 in ASCE/SEI Figure 27.3-8. The diaphragm resists the in-plane effects caused by and transfers this force to the vertical elements of the MWFRS. Cases 2, 3, and 4 in ASCE/SEI Figure 27.3-8 must also be considered where Part 1 of Chapter 27 is used to determine wind pressures. These cases account for torsion and the wind force acting along the diagonal of a building.

4.2.2 General Structural Integrity Forces Buildings assigned to SDC A need only comply with the general structural integrity requirements of ASCE/SEI 1.4 (ASCE/SEI 11.7). To ensure general structural integrity, the LFRS must be proportioned to resist a lateral force at each floor level equal to 1 percent of , which is the portion of the total dead load of the building, , located or assigned to level (see Figure 4.3). The diaphragm at level

must be designed to resist the force

𝐵𝐵⁄𝐵

𝑝𝑝�

𝑊𝑊� � 𝑝𝑝� 𝐵𝐵 �ℎ� � ℎ��� �⁄2

Figure 4.2 Resultant Wind Force Acting on Diaphragm

0.01𝑤𝑤�

0.01𝑤𝑤� 0.01𝑤𝑤� 0.01𝑤𝑤�

𝑤𝑤�

𝑤𝑤�

𝑤𝑤� 𝑤𝑤�

� � 0.01�𝑤𝑤� � 𝑤𝑤� � 𝑤𝑤� � 𝑤𝑤� �

Figure 4.3 Structural Integrity Lateral Forces for Buildings Assigned to SDC A

plus any other applicable in-plane forces.

4.2.3 Seismic Forces Seismic Force Provisions Diaphragms are required to be designed as part of the seismic force-resisting system (SFRS) for buildings assigned to SDC B through F. The seismic forces that are applied to the SFRS over the height of a building are needed to determine diaphragm forces at each level. The Equivalent Lateral Force (ELF) Procedure of ASCE/SEI 12.8 can be used to determine seismic forces on the SFRS provided the building possesses the applicable structural characteristics in ASCE/SEI Table 12.6-1. The steps to determine design seismic forces in accordance with the ELF Procedure are given below. In cases where the ELF Procedure cannot be used, design seismic forces can be determined using one of the following methods (see ASCE/SEI Table 12.6-1): (1) Modal Response Spectrum Analysis (ASCE/SEI 12.9.1), (2) Linear Response History Analysis (ASCE/SEI 12.9.2), and (3) Nonlinear Response History Procedures (ASCE/SEI Chapter 16). Equivalent Lateral Force (ELF) Procedure • Step 1: Determine Seismic Ground Motion Values IBC Figures 1613.2.1(1) and 1613.2.1(2) and ASCE/SEI Figures 22-1 and 22-2 contain mapped risk-targeted maximum considered earthquake (MCER) spectral response acceleration parameters and at periods of 0.2 second and 1.0

4-6

Design Guide for Reinforced Concrete Diaphragms

second, respectively, for conterminous U.S. sites that have an effective average small-strain shear wave velocity of 2,500 ft/s and 5-percent damping. IBC Figures 1613.2.1(3) through 1613.2.1(8) and ASCE/SEI Figures 22-3 through 22-8 contain similar contour maps for Hawaii, Alaska, Puerto Rico, U.S. Virgin Islands, Guam, North Mariana Islands, and American Samoa. The mapped spectral accelerations are the smaller of the probabilistic risk-based and deterministic ground motion values obtained at a site. In lieu of the maps, MCER spectral response accelerations can be obtained from Reference 4 or Reference 5. Six site classes are defined in ASCE/SEI Table 20.3-1. A site is to be classified as one of these six based on one of three soil properties measured over the top 100 ft of the site. Once the mapped MCER spectral acceleration parameters and site class have been established, the risk-targeted MCER spectral response acceleration parameters for short periods, , and at a 1-second period, , adjusted for site class effects, are determined by IBC Equations (16-36) and (16-37), respectively, or ASCE/SEI Equations (11.4-1) and (11.4-2), respectively: (4.4) (4.5) where short-period site coefficient determined from IBC Table 1613.2.3(1) or ASCE/SEI Table 11.4-1 and longperiod site coefficient determined from IBC Table 1613.2.3(2) or ASCE/SEI Table 11.4-2. Mapped accelerations must be adjusted by these site coefficients for other than reference site conditions (a reference site, as noted previously, is a site where the shear-wave velocity is equal to 2,500 ft/s). ASCE/SEI C11.4.4 contains information on how the site coefficients were derived. It is evident from IBC Table 1613.2.3(1) or ASCE/SEI Table 11.4-1 that the short-period site coefficient for Site Class C is greater than that for Site Class D where the short-period acceleration is greater than or equal to 1.0. Thus, in cases where Site Class D is selected as the default site class in accordance with IBC 1613.2.2 or ASCE/SEI 11.4.3, the value of must be taken greater than or equal to 1.2. This essentially makes Site Class C the default class in cases where . Design earthquake spectral response acceleration parameters at short periods, , and at a 1-second period, , are determined by IBC Equations (16-38) and (16-39), respectively, or ASCE/SEI Equations (11.4-3) and (11.4-4), respectively: (4.6) (4.7) • Step 2: Determine the Risk Category of the Building The Risk Category of a building is a function of its use or occupancy. IBC Table 1604.5 or ASCE/SEI Table 1.5-1 is to be used to establish the Risk Category based on the four use/occupancy types defined in the tables. • Step 3: Determine the Seismic Design Category (SDC) All buildings must be assigned to a SDC in accordance with IBC 1613.2.5 or ASCE/SEI 11.6. In general, the SDC is a function of the Risk Category and the design spectral acceleration parameters at the site. Six SDCs are defined ranging from A (minimal seismic risk) to F (highest seismic risk). As the SDC of a structure increases, so do the strength and detailing requirements. The SDC of a building is assigned as follows where : • Buildings classified as Risk Category I, II or III are assigned to SDC E • Buildings classified as Risk Category IV are assigned to SDC F Where , the SDC is determined twice: first as a function of by IBC Table 1613.2.5(1) or ASCE/SEI Table 11.6-1 and second as a function of by IBC Table 1613.2.5(2) or ASCE/SEI Table 11.6-2. The more severe SDC of the two governs. The SDC may be determined by IBC Table 1613.2.5(1) or ASCE/SEI Table 11.6-1 based solely on in cases where is less than 0.75 provided that all the conditions listed under IBC 1613.2.5.1 or ASCE/SEI 11.6 are satisfied. This exception is usually applicable to stiff, low-rise buildings.

4-7

Design Guide for Reinforced Concrete Diaphragms • Step 4: Determine the Seismic Response Coefficient, The seismic response coefficient,

, is determined by ASCE/SEI Equation (12.8-2):



(4.8)

where is the response modification coefficient in ASCE/SE Table 12.2-1 for the SFRS that is required based on the SDC and is the seismic importance factor in ASCE/SEI Table 1.5-2 based on the Risk Category. The value of • For

need not exceed the following: : (4.9)

• For

: (4.10)

where is the long-period transition period determined by ASCE/SEI 11.4.6 and is the fundamental period of the building determined by ASCE/SEI 12.8.2. In lieu of performing an analysis to obtain , an approximate building period, , determined by ASCE/SEI Equation (12.8-7) may be used in Equations (4.9) and (4.10) instead of , that is, where the parameters and are given in ASCE/SEI Table 12.8-2 based on structure type. The value of

must not be less than the following: (4.11)

In addition to Equation (4.11),

must not be less than the following for buildings located where

:

(4.12)

The seismic response coefficient, , (and the vertical seismic load effect, ) are permitted to be calculated using equal to the greater of 1.0 or provided all the criteria in ASCE/SEI 12.8.1.3 are satisfied. This cap on the maximum value of reflects engineering judgment on the performance of regular, low-rise buildings in past earthquakes that comply to the design and detailing requirements prescribed in the codes. • Step 5: Determine the Effective Seismic Weight, The effective seismic weight, , includes the dead load of the building above its base (as defined in ASCE/SEI 3.1) plus the following applicable loads (ASCE/SEI 12.7.2): 1. In areas used for storage, a minimum of 25 percent of the floor live load, with the following two exceptions: (a) Storage loads need not be included where the inclusion of such loads adds no more than 5 percent to the effective seismic weight at that level. (b) Floor live loads in public garages and open parking structures need not be included. 2. Where partitions are required by ASCE/SEI 4.3.2, the actual partition weight or a minimum weight of 10 psf of floor area, whichever is greater. 3. Total operating weight of permanent equipment. 4. Where the flat roof snow load, , determined by ASCE/SEI 7.3 exceeds 30 psf, 20 percent of the uniform design snow load, regardless of actual roof slope. 5. Weight of landscaping and other materials at roof gardens and similar areas.

4-8

Design Guide for Reinforced Concrete Diaphragms • Step 6: Determine Seismic Base Shear, The seismic base shear,

, in the direction of analysis is determined by ASCE/SEI Equation (12.8-1):

(4.13) • Step 7: Distribute the Seismic Base Shear, The lateral seismic force,

, Over the Height of the Building

, induced at any level of a building is determined by ASCE/SEI Equations (12.8-11) and (12.8-12):

(4.14) (4.15) where

seismic base shear determined by ASCE/SEI Equation (12.8-1) [see Equation (4.13)]



and

portion of the total effective seismic weight of the building,

and

height from the base to level



, located or assigned to level

or

or

exponent related to the building period:



•

for buildings that have a period less than or equal to 0.5 seconds

•

for buildings that have a period greater than or equal to 2.5 seconds





•

. is to be determined by linear interpolation between 1 and 2 for buildings that have a period between 0.5 and 2.5 seconds or can be taken equal to 2 

Depicted in Figure 4.4(a) is the linear distribution of the seismic base shear, , over the height of a building with a fundamental period less than or equal to 0.5 seconds . A parabolic distribution is depicted in Figure 4.4(b) for a building with a fundamental period greater than 2.5 seconds . For periods between 0.5 and 2.5 seconds, a linear interpolation between a linear and parabolic distribution is permitted, or a parabolic distribution may be used.

ℎ�  

𝐹𝐹�  

𝐹𝐹�  

 

𝑛𝑛  𝑥𝑥

𝑛𝑛 

𝐹𝐹�   ℎ�  

𝐹𝐹�  

ℎ�  

ℎ�  

                               

𝑤𝑤�   𝑉𝑉 

��� � � �����������

𝑥𝑥 

𝑤𝑤�  

𝑉𝑉 

��� � � �����������

Figure 4.4 Vertical Distribution of Seismic Base Shear, V

Determination of Seismic Forces on Diaphragms According to ASCE/SEI 12.10, diaphragms are to be designed for the larger of the forces in (1) and (2) below: (1) Design seismic force, the ELF Procedure]. (2) Diaphragm design force,

, determined from the structural analysis [see Equations (4.14) and (4.15) for

determined by

, determined by ASCE/SEI Equation (12.10-1): (4.16)

4-9

Design Guide for Reinforced Concrete Diaphragms where







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

[see Equations (4.14) and (4.15)]

, induced at level

portion of the effective seismic weight,

, that is located at or assigned to level

Minimum and maximum values of are determined by ASCE/SEI Equations (12.10-2) and (12.10-3), respectively, which are independent of the design base shear, (and, thus, of the selected SFRS and the corresponding response modification coefficient, ): Minimum Maximum For buildings with walls, the weights of the walls parallel to the direction of analysis are often not included in because these weights do not contribute to the diaphragm shear forces. However, they can be conservatively included in , which means at level . The design seismic forces, , determined by the ELF Procedure are used to design the SFRS of a building and represent mainly the first mode of vibration of the structure; the contributions of higher modes to the overall response are usually not substantial in buildings that are permitted to be analyzed by this procedure. However, significant forces can occur at any level in a building (especially in the lower levels) due to higher modes of vibration. The effects of higher modes must be considered in the design of diaphragms and are accounted for in the diaphragm design forces, .

R 8

7 6

5

4

The lower bound on typically applies to structures with a relatively low base shear due to a long period, a high response modification coefficient, or both. It essentially represents the effects of higher modes that generate relatively small base shears. The upper bound governs when there are relatively high base shears due to a short period, a low response modification coefficient, or both; it represents limited inelastic response or elastic response of a building.

A comparison of the forces and over the height of an 8-story building is given in Figure 4.5. These forces are equal in this example at the roof level, and at all other levels, is greater than . Note that the minimum governs in this example at levels 5 and below. Because it is an inertial force, is applied at the center of mass (CM) at each level (see Figure 4.6). For diaphragms that are rigid (see Section 6.2 for more information on diaphragm flexibility), the diaphragm translates and rotates about the center of rigidity (CR) due to ; the inherent torsional moment in this case is equal to times the eccentricity between the CM and the CR that is perpendicular to the direction of analysis (also see Section 6.3.3).

4-10

3

2

and

Figure 4.5 Seismic Forces Building

Over the Height of an 8-story

 

𝑒𝑒�  

CR 

CM 

𝐿𝐿 

Equation (4.16) is applicable to many common types of buildings with well-defined, discrete diaphragms where inertial forces and transfer forces are easily identified and separated. It is not meant to be used for unique or complex structures, such as stadiums or arenas, where identification and separation of inertial and transfer forces are not readily evident. In those cases, a more detailed analysis, such as a finite element analysis, is warranted.

𝐹𝐹��

𝐹𝐹�

𝐹𝐹��  

𝑒𝑒�  

𝐵𝐵 

Inherent torsional moment: 𝑀𝑀� � 𝐹𝐹�� 𝑒𝑒�  

Accidental torsional moment: 𝑀𝑀�� � 𝐹𝐹�� �0.05𝐵𝐵�∗   Total torsional moment: 𝑀𝑀� � 𝑀𝑀��  

*

𝑀𝑀�� to be included only when required by ASCE/SEI 12.8.4.2 

Figure 4.6 Inherent and Accidental Seismic Torsional Moments

Design Guide for Reinforced Concrete Diaphragms  

According to ASCE/SEI 12.8.4.2, an accidental torsional moment, , is to be included in the analysis of structures with diaphragms that are not flexible and that pos- Δ���   sess one or more of the horizontal irregularities given in ASCE/SEI Table 12.3-1. This accidental torsional moment is determined assuming that the CM is displaced each way from its actual location by a distance equal to 5 percent of the dimension of the structure perpendicular to the direction of analysis (see Figure 4.6). This is meant to account for any uncertainties in the actual locations of the CM and CR in structures with horizontal irregularities. Accidental torsion need not be included when determining the seismic forces, , in the design of the structure (including the diaphragms) and in the determination of the design story drifts in ASCE/SEI 12.8.6 or the story drift limits of ASCE/SEI 12.12.1, except for the following structures:

��� �

CM 

Δ��� � Δ���   2

Δ���  

CR 

𝐹𝐹��   Torsional irregularity �Ty�e �a�� Δ��� � ��2Δ���

Extreme torsional irregularity �Ty�e ���� ��� � ������

Figure 4.7 Type 1a and Type 1b Horizontal Torsional Irregularities • Structures assigned to SDC B with Type 1b (extreme torsional irregularity) horizontal structural irregularity (see Figure 4.7 for definitions of Type 1a and 1b horizontal structural irregularities). • Structures assigned to SDC C through F with Type 1a (torsional irregularity) or Type 1b (extreme torsional irregularity) horizontal structural irregularity. In such cases, at each level is to be multiplied by the torsional amplification factor, , determined by ASCE/SEI Equation (12.8-14) (also see ASCE/SEI Figure 12.8-1).

In cases where torsional moment,

must be included in accordance with ASCE/SEI 12.8.4.2, the total torsional moment is equal to the inherent . , plus the accidental torsional moment,

An alternative method to determine diaphragm design forces is given in ASCE/SEI 12.10.3, which is permitted to be used in lieu of the provisions in ASCE/SEI 12.10.1 for cast-in-place concrete diaphragms. Diaphragm acceleration coefficients are determined at the following locations: (1) at the base of the structure, (2) at a height equal to 80 percent of the total height of the structure, and (3) at the top of the structure. Linear interpolation is used to determine acceleration coefficients at intermediate levels. The diaphragm design force at a level is equal to the acceleration coefficient times the seismic mass tributary to that level divided by the diaphragm reduction factor, , which is obtained from ASCE/SEI Table 12.10-1. For cast-in-place concrete diaphragms, the diaphragm design forces determined by the alternative method are similar to those determined by the provisions of ASCE/SEI 12.10.1, except in the top 20 percent of the height of some buildings where the forces determined by the alternative method are greater than those determined by ASCE/SEI 12.10.1.

4.2.4 Soil Lateral Forces Static Forces For buildings with one or more basement levels, lateral forces are generated by soil pressure bearing against the basement walls. The magnitude of the soil pressure is usually obtained from a geotechnical investigation. In cases where the results of such an investigation are not available, the lateral soil loads in IBC Table 1610.1 are to be used (similar design lateral loads are provided in ASCE/SEI Table 3.2-1). The design lateral soil load, , depends on the type of soil and the boundary conditions at the top of the wall. Walls that are restricted to move at the top are to be designed for the at-rest pressures tabulated in IBC Table 1610.1 while walls that are free to deflect and rotate at the top are to be designed for the active pressures in that table. The distribution of at-rest soil pressure over the height of a reinforced concrete foundation wall is depicted in Figure 4.8 for a building with one basement level.

Reinforced concrete slab

Distribution of design lateral soil pressure, 𝐻𝐻

Reinforced concrete basement wall 𝐻

Reinforced concrete wall foundation

Figure 4.8 Distribution of At-Rest Soil Pressure on a Foundation Wall

4-11

Design Guide for Reinforced Concrete Diaphragms In addition to lateral pressures from soil, walls must be designed to resist the effects of hydrostatic pressure due to undrained backfill (unless a drainage system is installed) and to any surcharge loads that can result from sloping backfills or from driveways or parking spaces that are in proximity to a wall. Submerged or saturated soil pressures include the weight of the buoyant soil plus the hydrostatic pressure. It is evident from Figure 4.8 that a portion of the soil pressure is transferred to the foundation system and a portion is transferred to the reinforced concrete slab, which acts as a diaphragm. The diaphragm must be designed for the in-plane effects due to the soil pressure and must transfer the forces to the basement walls parallel to the direction of the soil pressure. Dynamic Forces IBC 1803.5.12 and ASCE/SEI 11.8.3 require that dynamic seismic lateral earth pressures on basement walls and retaining walls supporting more than 6 feet of backfill height due to design earthquake ground motions be included in the design of those members for structures assigned to SDC D, E, or F. However, no provisions or guidelines are provided on how to determine these dynamic pressures. Resource Paper 12 in Part 3 of Reference 6 contains several approaches for determining lateral seismic pressures on basement walls and retaining walls. Building basement walls restrained at the top and bottom are considered to be nonyielding walls, that is, the walls are assumed not to deform when subjected to seismic earth pressures. This assumption inherently implies a very stiff wall (deformations less than 0.2 percent of the height of the wall) in combination with a rigid base condition (wall foundation located on rock or very stiff soil). Two methods are given for nonyielding walls. In the first method, the dynamic thrust, , on a rigid, nonyielding wall retaining a homogeneous, linear elastic soil and connected to a rigid base is determined by the following equation: (4.17) where is the horizontal ground acceleration divided by the acceleration due to gravity, is the unit weight of the soil, and is the height of the wall. For purposes of analysis, is to be taken equal to the peak ground acceleration at the site. The point of application of is above the base of the wall. The second method, which is more involved than the first, considers kinematic soil-structure interaction and is based on dynamic soil properties and ground motion characteristics. Details on this five-step method to determine seismic soil pressure can be found in the Resource Paper. In lieu of modeling basement walls as nonyielding walls, references are provided in the Resource Paper for modeling such walls   using a soil-structure interaction approach. Regardless of the method used, the lateral seismic pressures must be multiplied by the seismic importance factor , which is the same as that of the building. In the design load combinations, the lateral seismic pressures, , are combined with the lateral static earth pressures, . It is important to know whether these pressures have been determined at service level or strength level so that the appropriate load combinations are used. Diaphragms must be designed to resist the combined lateral static and dynamic pressures and must transfer these forces to the basement walls parallel to the direction of analysis.

A

Reinforced concrete shear wall below

Plan

4.2.5 Flood and Tsunami Forces Where required, flood and tsunami forces determined in accordance with ASCE/SEI Chapters 5 and 6, respectively, produce in-plane forces on diaphragms. Hydrodynamic forces are caused by water moving at a moderate to high velocity above the ground level. Similar to wind forces, moving water impacts the upstream face of a building. If the upstream walls are not designed to fail during a design event, the walls

4-12

Section A-A

Figure 4.9 Out-of-plane Offset in the LFRS

Reinforced concrete shear wall above A

Design Guide for Reinforced Concrete Diaphragms transfer the hydrodynamic forces to the diaphragms below the water level, which in turn, transfer them to the walls in the direction parallel to the flow of water. Negative forces and drag forces are produced on the downstream side of the building and on the sides of the building, respectively. Impact forces must also be considered.

Diaphragm force Reinforced concrete shear �a�� �t�p�� Transfer force

Reinforced concrete �asement �a�� �t�p��

4.3 Transfer Forces In addition to the in-plane forces discussed in Section 4.2 of this publication, diaphragms may also be subjected to transfer forces. Illustrated in Figure 4.9 is a floor level in a building where an offset occurs in one of the reinforced concrete shear walls that is part of the LFRS. The forces from the reinforced concrete shear wall above must be transferred through the segment of diaphragm at the offset to the reinforced concrete shear wall below. The diaphragm must be designed to resist this transfer force in addition to all other applicable forces.

Figure 4.10 Transfer Forces to a Podium Slab

Reinforced concrete s�ear �a�� �t����

A similar vertical discontinuity is illustrated in Figure 4.10 where reinforced concrete shear walls that are part of the LFRS of a building above transfers forces through the reinforced concrete podium slab (diaphragm) to the reinforced concrete basement walls below. The diaphragm force shown in the figure includes the static soil pressure and dynamic seismic pressure (if applicable) that are transferred to the basement walls parallel to the direction of analysis (see Section 4.2.4 of this publication). The diaphragm must be designed to resist the effects from the transfer forces in combination with the diaphragm forces.

Transfer force

Transfer forces can also occur where the relative lateral stiffnesses of the vertical elements of the LFRS vary from story to story or where there is a vertical discontinuity, such as the setback in the building illustrated in Figure 4.11. In structures with a Type 4 horizontal structural irregularity in ASCE/SEI Table 12.3-1 (that is, an out-of-plane, offset ir-   regularity, which is illustrated in Figure 4.9), seismic transfer forces from the vertical seismic force-resisting elements above the diaphragm to the vertical seismic force-resisting elements below the diaphragm must be increased by the overstrength factor, , which is given in ASCE/SEI Table 12.2-1 based on the SFRS (ASCE/SEI 12.10.1.1). This force, , is in addition to the diaphragm inertial forces, , that originate on the diaphragm (see Figure 4.12), and the diaphragm must be designed for the effects caused by both forces. The redundancy factor, , for the building is to be applied to transfer forces, except where the transfer forces must be increased by in accordance with ASCE/SEI 12.10.1.1, as noted above, in which case . Note that amplification of transfer forces by

in dia-

Figure 4.11 Transfer Forces at a Building Setback

𝐹𝐹 

𝐹𝐹��  

Ω� 𝐹𝐹 

Figure 4.12 Seismic Transfer Force Due to Out-of-plane Offset in the SFRS

4-13

Design Guide for Reinforced Concrete Diaphragms Table 4.6 Summary of Wind Load Provisions in Chapter 30 of ASCE/SEI 7-16 for C&C of Building Structures Part

Applicability Building Type

Height Limit

Enclosed, low-rise

1

Partially enclosed, low-rise

and least horizontal dimension

Enclosed Partially enclosed Enclosed, low-rise 2

and least horizontal dimension

Enclosed Enclosed Partially enclosed Enclosed

5

Open

• Regular-shaped building • Building does not have response characteristics making it subject to across-wind loading, vortex shedding, or instability caused by galloping or flutter • Building is not located at a site where channeling effects or buffeting in the wake of upwind obstructions warrant special consideration • Building has a flat roof, gable roof, multispan gable roof, hip roof, monoslope roof, stepped roof, or sawtooth roof • Same first three conditions as in Part 1 • Building has a flat roof, gable roof with a roof slope less than or equal to 45 degrees, or hip roof with a roof slope less than or equal to 27 degrees • Same first three conditions as in Part 1 • Building has a flat roof, pitched roof, gable roof, hip roof, mansard roof, arched roof, or domed roof

3

4

Conditions

• Same first three conditions as in Part 1 • Building has a flat roof, gable roof, hip roof, monoslope roof, or mansard roof None

• Same first three conditions as in Part 1 • Building has a pitched free roof, monosloped free roof, or troughed free roof

phragms are not addressed for buildings with in-plane offsets, which is illustrated in Figure 4.11 for the case of a setback in a building. Reference 7 recommends using the out-of-plane offset amplification procedure discussed above for buildings with in-plane offsets. The above discussion on transfer forces focuses on buildings with reinforced concrete shear walls as the LFRS (or SFRS). The discussion is the same if reinforced concrete moment frames (or any other type of reinforced concrete LFRS or SFRS) are used instead of, or in combination with, the shear walls.

4.4 Anchorage and Connection Forces 4.4.1 Overview Nonstructural components, such as cladding, is commonly attached to reinforced concrete slabs over the height of a building. These components transfer wind pressures, inertial forces from earthquake shaking, or both, to the diaphragm through the connections between the two. 4.4.2 Wind Forces Illustrated in Figure 4.13 is a generic tie-back connection between a precast concrete wall panel and a reinforced concrete slab, which is representative of the many types of connections between cladding and diaphragms. Wind pressure normal to the surface of the panel can be calculated by one of the applicable methods in ASCE/SEI Chapter 30 for components and cladding (C&C) of building structures (see Table 4.6). In Part 3 of ASCE/SEI Chapter 30, wind pressure on C&C is determined by ASCE/SEI Equation (30.5-1): (4.18)

4-14

Design Guide for Reinforced Concrete Diaphragms Velocity pressure, , is calculated by Equation (4.2) and is equal to on windward walls and on leeward and side walls. External pressure coefficients are given in ASCE/SEI Figure 30.5-1 for walls based on the effective wind area, which is defined in ASCE/SEI 26.2 as the larger of the following: (1) total area tributary to the element and (2) span length of the element times one-third the span length. For cladding fasteners, such C&C wind pressure as the connector assembly illustrated in Figure 4.13, the effective wind area is equal to the area tributary to the fastener. Internal pressure coefficients are given in ASCE/ SEI Table 26.13-1 and velocity pressure can be taken as for calculation of positive and negative internal pressures on all walls for enclosed and partially enclosed buildings. Wind pressures on C&C using the provisions in Part 1 of ASCE/SEI Chapter 30 are calculated in a similar fashion [see ASCE/SEI Equation (30.3-1)].

Precast concrete wall panel

Connector assembly Reinforced concrete slab Slab reinforcement not shown for clarity

Wind connector force

Figure 4.13 Wind Connection Forces on a Diaphragm

The wind connector force shown in Figure 4.13 is determined based on the C&C wind pressure; this force is transferred from the precast concrete wall panel to the diaphragm via the connector assembly. The diaphragm must be designed for this force in combination with all other applicable forces.

4.4.3 Seismic Forces Nonstructural Walls Exterior nonstructural wall elements and connections to the supporting structural elements, including diaphragms, must be designed for a horizontal seismic design force, , equal to that determined by ASCE/SEI Equations (13.3-1) through (13.3-3) where is applied at the CM of the panel (ASCE/SEI 13.4.1 and 13.5.3): (4.19)

where



component importance factor, which is determined in accordance with ASCE/SEI 13.1.3 component operating weight, which in the case of nonstructural wall elements, is equal to the weight of the wall



component amplification factor, which is obtained from ASCE/SEI Table 13.5-1 for architectural components







component response modification factor, which is obtained from ASCE/SEI Table 13.5-1 for architectural compo nents height in structure of point of attachment of component with respect to the base of the building average roof height of the structure with respect to the base of the building

ASCE/SEI Equation (13.3-1) represents a trapezoidal distribution of floor accelerations within a structure. Horizontal seismic force distribution in accordance with ASCE/SEI Equations (13.3-1) through (13.3-3) is illustrated in ASCE/SEI Figure C13.3-2. For elements that are attached at more than one height, like the precast concrete wall panel in Figure 4.13 that is connected to the reinforced concrete slabs on adjoining floor levels, it is recommended that design be based on the average values of determined individually at each point of attachment using the entire component weight . Alternatively, may be determined at each point of attachment using that is tributary to the connector assembly (that is, the point of attachment).

4-15

Design Guide for Reinforced Concrete Diaphragms The reinforcing bars in the slab perpendicular to the edge under consideration must be designed to resist the force sion and must be fully developed for tension into the slab.

in ten-

The overstrength factor, , in ASCE/SEI Table 13.5-1 is applicable to anchorage of components to concrete where required by ASCE/SEI 13.4.2 or ACI 318. Seismic load effects including overstrength given in ASCE/SEI 12.4.3 are to be used, and the redundancy factor, , is permitted to be taken as 1.0 (ASCE/SEI 13.3.1.1). The force determined by Equation (4.19) is to be applied independently in at least two orthogonal horizontal directions in combination with service loads on the component. Also, the component must be designed for a concurrent vertical force equal to (ASCE/SEI 13.3.1.2). In lieu of determined by Equation (4.19), ASCE/SEI 13.3.1.4 permits design seismic forces for nonstructural components to be determined by one of the dynamic analysis methods given in that section. Requirements for nonstructural component attachments and anchorage are given in ASCE/SEI 13.4. Anchors in concrete are to be designed in accordance with Chapter 17 of ACI 318 or by one of the two exceptions given in ASCE/SEI 13.4.2. In addition to the seismic forces defined in ASCE/SEI 13.3.1 discussed above, nonstructural wall elements and their connections must satisfy the requirements of ASCE/SEI 13.5.3 pertaining to design for relative seismic displacements defined in ASCE/ SEI 13.3.2 and any movements caused by temperature changes. The five requirements in ASCE/SEI 13.5.3 must be satisfied for such elements. Regardless of the method used, seismic forces on nonstructural wall elements must be transferred through properly designed connector assemblies to the supporting diaphragm, and the diaphragm must be designed for these forces in combination with all other applicable forces.

Structural Walls Provisions for anchorage of structural walls into diaphragms are given in ASCE/SEI 12.11.2. The minimum anchorage force, determined by ASCE/SEI Equation (12.11-1):

, is

(4.20) where amplification factor for diaphragm flexibility when the connection is not at a flexible diaphragm

, which is permitted to be taken not larger than 1.0

span, in feet, of a flexible diaphragm that provides the lateral support for the wall, which is measured between vertical elements that provide lateral support to the diaphragm in the direction of analysis ( for rigid diaphragms)

weight of the wall tributary to the anchor

For anchorage that is not located at the roof diaphragm, the value of determined by Equation (4.20) is permitted to be multiplied by the factor where is the height of the anchor above the base of the structure and is the height of the roof above the base. Note that the reduced anchorage force must not be taken less than . �iaphragm �typ�� The reinforcing bars in the slab perpendicular to the structural wall under consideration must be designed to resist the force in tension and must be fully developed for tension into the slab and into the wall. Additional requirements for anchorage of reinforced concrete structural walls into diaphragms are given in ASCE/SEI 12.11.2.2.

4.5 Column Bracing Forces Diaphragms must be designed to resist forces from any sloped structural members in a building. An inclined column, such as the one illustrated in ACI Figure R12.1.1, transfers the horizontal components of the supported gravity and lateral forces, where applicable, into the reinforced concrete slab (diaphragm) in cases

4-16

Isolated frame

Isolated shear wall

Frame and shear wall connected by diaphragms

Figure 4.14 Diaphragm Forces Due to Displacement Compatibility

Design Guide for Reinforced Concrete Diaphragms where there are no other local structural elements to counteract these forces. The diaphragm must be designed to resist the thrust from the column in combination with all other applicable forces and must be able to transfer the thrust to elements of the LFRS below. The vertical components of the supported gravity and lateral forces from the inclined column and the other out-ofplane forces must also be considered in the design of the diaphragm. Diaphragms must also be designed for forces resulting from bracing vertical structural elements in a building. For example, in the case of a building frame system, it is assumed that all the lateral forces are resisted by the shear walls. When analyzed separately, shear walls and frames have different displacement profiles over the height of a building when subjected to lateral forces (see Figure 4.14). In an actual building, the diaphragms connect the columns and shear walls together, which means that these elements must displace the same amount at each level. Thus, the resulting transfer forces caused by displacement compatibility between the columns that are not part of the LFRS and the shear walls must be resisted by the diaphragms.

4.6 Out-Of-Plane Forces One of the main roles of a diaphragm is to support and transfer gravity loads to the supporting structural members in a building. The weight of the structure, superimposed dead loads, and live loads are typical out-of-plane gravity loads applied to the surface of a diaphragm. Dead and live loads are determined in accordance with ASCE/SEI Chapters 3 and 4, respectively. Additional loads that can occur on roof diaphragms are from snow and rain (ASCE/SEI Chapters 7 and 8), to name a few. Wind uplift loads are also applicable, as are vertical accelerations due to seismic effects. The structural analysis methods in ACI Chapter 6 can be used to analyze diaphragms for the effects of out-of-plane forces; included is a simplified method for nonprestressed, continuous one-way slabs. In addition to the methods in ACI Chapter 6, the Direct Design Method in ACI 8.10 is permitted to be used to analyze nonprestressed, two-way slab systems that satisfy the limits of ACI 8.10.2.

4.7 Collector Design Forces 4.7.1 Overview Where vertical elements of the LFRS do not extend the full depth of a diaphragm, collectors are needed to transfer forces between the diaphragm and the vertical elements of the LFRS (see ASCE/SEI Figure 12.10-1). Depending on several factors, a collector can be part of the slab or can be a beam. In general, collectors must be designed for the combined effects due to gravity loads (flexure and shear) and lateral loads (axial tension and compression forces caused by in-plane shear transfer in the diaphragm due to wind forces, seismic forces, or both). The in-plane diaphragm forces are determined using the methods presented in Section 4.2 of this publication. Once the forces are determined, the diaphragm is modeled and analyzed using the methods in ACI 12.4.2, and the axial forces in a collector can be determined based on the selected method of analysis. Chapter 6 of this publication covers the analysis procedures in ACI 12.4.2. These procedures can be used to determine collector axial forces in diaphragms subjected to wind and seismic forces in buildings assigned to SDC A or B. Collector design forces for buildings assigned to SDC C through F are given in the following section.

4.7.2 Collector Design Forces for Buildings Assigned to SDC C through F According to ASCE/SEI 12.10.2.1, collectors and their connections to the vertical elements of the SFRS in buildings assigned to SDC C, D, E, or F must be designed to resist the effects from the maximum of the following forces: 1. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with seismic forces determined by the ELF Procedure of ASCE/SEI 12.8 or the modal response spectrum analysis procedure of ASCE/SEI 12.9.1. In ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7, use using .

where

is determined

2. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with seismic forces determined by ASCE/SEI Equation (12.10-1) for diaphragms. In ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7, use using .

where

is determined

3. F  orces calculated using the load combinations of ASCE/SEI 2.3.6 with seismic forces determined by ASCE/SEI Equation (12.10-2), which is the lower-limit diaphragm force. In ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7, use using .

where

is determined

4-17

Design Guide for Reinforced Concrete Diaphragms The purpose of the overstrength requirements is to help ensure that inelastic behavior occurs in the ductile elements of the SFRS and not in the collectors or their connections. It is essential that the collectors and their connections perform as intended during a seismic event. In the first of these three force requirements, collectors are subjected to the seismic load effects caused by where is the seismic force at level , which is a fraction of the base shear, ; is applied simultaneously to each level of the overall analysis model of the building and is determined by ASCE/SEI Equations (12.8-11) and (12.8-12) (see Section 4.2.3 of this publication). Combined gravity and seismic load effects are determined using the seismic load combinations that include overstrength in ASCE/SEI 2.3.6. In this case, the redundancy factor, , can be taken as 1.0 (condition 5 in ASCE/SEI 12.3.4.1). Similarly, in the second force requirement, collectors are subjected to the seismic load effects caused by where is the seismic force applied to the diaphragm at level in the building and is determined by ASCE/SEI Equation (12.10-1). Combined gravity and seismic load effects are determined using the seismic load combinations that include overstrength in ASCE/SEI 2.3.6. Like in the first force requirement, can be taken as 1.0. In the third force requirement, is applied to the diaphragm, and the combined gravity and seismic load effects in the collector are determined using the seismic load combinations with seismic load effects in ASCE/SEI 2.3.6. The redundancy factor, , determined for the SFRS is applicable in the design of the collectors in this case. The collector is to be designed using whichever of these three requirements produces the critical combined effects. Where applicable, transfer forces must also be considered when determining forces in a collector (ASCE/SEI 12.10.2.1). As discussed in Section 4.3 of this publication, seismic transfer forces must be increased by the overstrength factor, , in structures with a Type 4 horizontal structural irregularity in ASCE/SEI Table 12.3-1 (that is, an out-of-plane, offset irregularity, which is illustrated in Figure 4.9). Amplification of transfer forces by in collectors is not addressed for buildings with in-plane offsets (Type 4 vertical structural irregularities in ASCE/SEI Table 12.3-2). Reference 7 recommends increasing the transfer forces by in such cases. Note that the redundancy factor, , applies to transfer forces in all other cases where is the same as that determined for the SFRS in accordance with ASCE/SEI 12.3.4.

4-18

Design Guide for Reinforced Concrete Diaphragms

Chapter 5 Load Combinations 5.1 Overview Once the applicable design forces have been determined using the information in Chapter 4 of this publication, the required strength of diaphragms, collectors, and their connections must be calculated using the strength design load combinations in ACI 5.3 and ASCE/SEI 2.3. A summary of these load combinations is given in the next section. Seismic load combinations for buildings assigned to Seismic Design Category (SDC) B, C, and D through F, are given in Sections 5.3, 5.4, and 5.5, respectively.

5.2 Strength Design Load Combinations The required strength, , is obtained by multiplying service-level (nominal) load effects determined in accordance with the governing building code by the load factors in ACI 5.3 or ASCE/SEI 2.3.1 and ASCE/SEI 2.3.6 for seismic load effects. Strength design load combinations are given in Table 5.1. Table 5.1 ACI 318 and ASCE/SEI 7 Strength Design Load Combinations ACI Equation Number

ASCE/SEI 7 Load Combination

5.3.1a

1

5.3.1b

2

5.3.1c

3

5.3.1d

4

5.3.1e

6

5.3.1f

5

5.3.1g

7

Load Combination

Exceptions and additions to these load combinations for nonprestressed members are given in ACI 5.3.3 through 5.3.10. For example, the load factor on the live load, , in ACI Equations (5.3.1c), (5.3.1d), and (5.3.1e) is permitted to be reduced to 0.5 except for parking garages, places of public assembly, and areas where is greater than 100 psf. Where the governing building code permits live load reduction, the reduced live load can be used in combination with the 0.5 load factor. Load factors are typically greater than or less than 1.0. Wind and seismic load effects are an exception to this: A load factor of 1.0 is used because wind and seismic effects are defined as effects resulting from strength-level loads. If the governing building code has provisions where the seismic effects, , are based on service-level seismic loads, then a higher load factor on is required (ACI R5.3.1). Similarly, if the wind effects, , are based on service-level loads, is to be used in place of in ACI Equations (5.3.1d) and (5.3.1f), and is to be used instead of in ACI Equation (5.3.1c) [ACI 5.3.5]. Wind and seismic loads act in more than one direction on a building, and this must be accounted for in the load combinations. ACI Equations (5.3.1d) and (5.3.1e) are to be used where gravity load effects and lateral load effects are additive, whereas ACI Equations (5.3.1f) and (5.3.1g) are applicable where dead load effects counteract the lateral load effects. In the absence of a general building code that prescribes seismic loads, the provisions in References 2 or 3 are to be used (ACI R5.2.2). Seismic load effect, , is defined in ASCE/SEI 12.4.2 as follows: • For use in ACI Equation (5.3.1e) or load combination 6 in ASCE/SEI 2.3.6 [ASCE/SEI Equation (12.4-1)]:

(5.1)

• For use in ACI Equation (5.3.1g) or load combination 7 in ASCE/SEI 2.3.6 [ASCE/SEI Equation (12.4-2)]:

(5.2)

5-1

Design Guide for Reinforced Concrete Diaphragms The horizontal seismic load effect,

, is determined by ASCE/SEI Equation (12.4-3):

(5.3) where is the redundancy factor determined in accordance with ASCE/SEI 12.3.4 and seismic forces on the structural members. The vertical seismic load effect,

are the effects of the horizontal

, is determined by ASCE/SEI Equation (12.4-4a):

(5.4) where

is the design spectral acceleration parameter at short periods (see Equation (4.6) of this publication).

The following seismic load combinations are to be used in lieu of those in Table 5.1 where seismic load effects with overstrength, , are required: • ACI Equation (5.3.1e) or load combination 6 in ASCE/SEI 2.3.6: (5.5)

• ACI Equation (5.3.1g) or load combination 7 in ASCE/SEI 2.3.6:

(5.6)

where •

for use in ACI Equation (5.3.1e) or load combination 6 in ASCE/SEI 2.3.6 [ASCE/SEI Equation (12.4-5)]

•

for use in ACI Equation (5.3.1g) or load combination 7 in ASCE/SEI 2.3.6 [ASCE/SEI Equation (12.4-6)]

•

[Equation (5.4)]

The effect of horizontal seismic forces including overstrength,

, is determined by ASCE/SEI Equation (12.4-7): (5.7)

where is the overstrength factor given in ASCE/SEI Table 12.2-1 based on the seismic force-resisting system (SFRS) and are the effects of horizontal seismic forces.

Seismic load combinations that are applicable to structures assigned to SDC B through F are given in the following sections for diaphragms and collectors subjected to in-plane seismic forces and to in-plane seismic transfer forces. Similar load combinations can be derived where seismic connection and anchorage forces and seismic column bracing forces must be included. 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, • Effects of service lateral loads due to fluids,

, in accordance with ACI 5.3.6

, in accordance with ACI 5.3.7

• Effects of service load due to lateral earth pressure,

, in accordance with ACI 5.3.8

• Effects due to flood loads in accordance with ACI 5.3.9 • Effects due to atmospheric ice loads in accordance with ACI 5.3.10 • For buildings subjected to tsunamis, the applicable load combinations are given in ASCE/SEI 6.8.3.3. It is possible that critical effects can occur when one or more of the variable loads are set equal to zero. ACI 5.3.2 requires that this type of investigation be performed in all cases. The strength design notional load combinations in ASCE/SEI 2.6.1 are applicable in the design of structural members in buildings assigned to any SDC. The notional loads, , include the following: (1) general structural integrity lateral forces, , determined by ASCE/SEI Equation (1.4-1), (2) connection forces determined by ASCE/SEI 1.4.3, and (3) structural wall anchorage forces determined by ASCE/SEI 1.4.4. Buildings assigned to SDC A need only comply with the general structural integrity requirements of ASCE/SEI 1.4; satisfying seismic requirements is not mandatory. For buildings where extraordinary loads and events must be considered, the capacity and residual capacity load combinations in ASCE/SEI Equations (2.5-1) and (2.5-2), respectively, must be used.

5-2

Design Guide for Reinforced Concrete Diaphragms 5.3 Seismic Load Combinations for SDC B . The redundancy factor,

Redundancy factor,

, is permitted to be taken as 1.0 in the following cases:

• Structures assigned to SDC B (condition 1 in ASCE/SEI 12.3.4.1). • Diaphragm forces,

, determined by ASCE/SEI Equations (12.10-1) through (12.10-3) [condition 7 in ASCE/SEI 12.3.4.1]

Vertical seismic load effect, . According to the second exception is ASCE/SEI 12.4.2.2, in ASCE/SEI Equations (12.4-1) and (12.4-2) for structures assigned to SDC B.

is permitted to be taken as zero

Seismic load combinations. Seismic load combinations with seismic load effect, , in ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7 are the following for structures assigned to SDC B: • ACI Equation (5.3.1e) or ASCE/SEI load combination 6: (5.8)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.9)

In these equations (as well as for all the equations in SDC C through F), the effects are to be determined for sidesway to the right (SSR) and sidesway to the left (SSL), that is, for . Where seismic load effects including overstrength are required, the following equations are applicable: • ACI Equation (5.3.1e) or ASCE/SEI load combination 6:

(5.10)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.11)



In-plane inertial and transfer forces. In buildings that have a Type 4 horizontal structural irregularity (out-of-plane offset irregularity) in ASCE/SEI Table 12.3.1 (see Figure 4.9), in-plane seismic transfer forces, , from the vertical seismic force-resisting elements above the diaphragm to the vertical seismic force-resisting elements below the diaphragm must be increased by (ASCE/SEI 12.10.1.1). This force, , is in addition to the diaphragm inertial force, , that originates on the diaphragm below (see Figure 4.12), and the diaphragm must be designed for the effects caused by both forces. Therefore, the in-plane seismic effect, , that is to be used in Equations (5.8) and (5.9) must include seismic effects from in-plane forces and . In buildings that do not have a Type 4 horizontal structural irregularity, the redundancy factor, , that is determined for the SFRS of the building must be applied to the transfer force, which in the case of buildings assigned to SDC B, is equal to 1.0. Thus, the force must be applied to the diaphragm in addition to , and the total in-plane seismic effect, , that is to be used in Equations (5.8) and (5.9) must include seismic effects from in-plane forces and . Direction of loading. For structures assigned to SDC B, design seismic forces are permitted to be applied independently in each of two orthogonal directions, and orthogonal interaction effects are permitted to be neglected (ASCE/SEI 12.5.2). Thus, the seismic effects in the diaphragm and collectors can be determined in each orthogonal direction independently.

5.4 Seismic Load Combinations for SDC C Redundancy factor,

. The redundancy factor,

, is permitted to be taken as 1.0 in the following cases:

• Structures assigned to SDC C (condition 1 in ASCE/SEI 12.3.4.1) • D  esign of collector elements, splices, and their connections for which the seismic load effects, including overstrength of ASCE/SEI 12.4.3, are required for design (condition 5 in ASCE/SEI 12.3.4.1) • Diaphragm forces,

, determined by ASCE/SEI Equations (12.10-1) through (12.10-3) [condition 7 in ASCE/SEI 12.3.4.1]

Vertical seismic load effect, . Vertical seismic load effects, Equations (12.4-1) and (12.4-2) for structures assigned to SDC C.

, determined by Equation (5.4) must be included in ASCE/SEI

Seismic load combinations. Seismic load combinations with seismic load effect, , in ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7 are the following for structures assigned to SDC C:

5-3

Design Guide for Reinforced Concrete Diaphragms • ACI Equation (5.3.1e) or ASCE/SEI load combination 6: (5.12)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.13)

Where seismic load effects including overstrength are required, which includes collectors and their connections (ASCE/SEI 12.10.2.1), the following equations are applicable: • ACI Equation (5.3.1e) or ASCE/SEI load combination 6:

(5.14)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.15)



In-plane inertial and transfer forces. In buildings that have a Type 4 horizontal structural irregularity, the in-plane seismic effect, , that is to be used in Equations (5.12) and (5.13) must include seismic effects from in-plane forces and . Similarly, in buildings that do not have a Type 4 horizontal structural irregularity, that is to be used in Equations (5.12) and (5.13) must include seismic effects from in-plane forces and . Direction of loading. Design seismic forces are permitted to be applied independently in each of two orthogonal directions for structures assigned to SDC C except where a Type 5 horizontal structural irregularity in ASCE/SEI Table 12.3-1 (nonparallel system irregularity) is present. In such cases, orthogonal load effects must be accounted for in the design of diaphragms and collectors using one of the two procedures in ASCE/SEI 12.5.3.1a or 12.5.3.1b. The orthogonal combination procedure in ASCE/ SEI 12.5.3.1a is discussed here. In the orthogonal combination procedure, a structure is analyzed independently in two orthogonal directions using a permitted analysis method in ASCE/SEI Table 12.6-1 (such as the Equivalent Lateral Force Procedure in ASCE/SEI 12.8, where applicable). Total diaphragm forces in each direction are then determined using inertial diaphragm forces, , determined by ASCE/ SEI Equations (12.10-1) through (12.10-3) plus any other applicable in-plane forces. Orthogonal load effects on the diaphragm are determined using 100 percent of the forces in one direction plus 30 percent of the forces in the perpendicular direction (that is, these forces are applied simultaneously to the diaphragm). This requirement is illustrated in Figure 5.1 for the case of inertial forces . Both cases in the figure must be investigated when designing the diaphragm, including the effects due to forces in the opposite directions shown. The load combinations that must be considered are given in ASCE/SEI C12.5.3.

0.3𝐹𝐹��  

𝐹𝐹�� 

𝐹𝐹�� 

0.3𝐹𝐹��  

Figure 5.1 Direction of Loading Requirements for Structures Assigned to SDC C with a Type 5 Horizontal Structural Irregularity

5.5 Seismic Load Combinations for SDC D, E, and F Redundancy factor, . For structures assigned to SDC D, E, or F, the redundancy factor, , must be determined in accordance with ASCE/SEI 12.3.4.2. In the design of collector elements and for the effects due to diaphragm forces , can be taken as 1.0 (see conditions 5 and 7, respectively, in ASCE/SEI 12.3.4.1). For structures assigned to SDC D with a Type 1b horizontal structural irregularity in ASCE/SEI Table 12.3-1 (extreme torsional irregularity; see Figure 4.7 of this publication), (ASCE/SEI 12.3.4.2). For all other structures assigned to SDC D and for structures assigned to SDC E or F (where Type 1b horizontal structural irregularities are not permitted; see ASCE/SEI 12.3.3.1), unless one of the two conditions in ASCE/SEI 12.3.4.2 is met, in which case .

5-4

Design Guide for Reinforced Concrete Diaphragms Vertical seismic load effect, . Vertical seismic load effects, , determined by Equation (5.4) must be included in ASCE/SEI Equations (12.4-1) and (12.4-2) for structures assigned to SDC D, E, or F. Seismic load combinations. Seismic load combinations with seismic load effect, , in ACI Equations (5.3.1e) and (5.3.1g) or ASCE/SEI load combinations 6 and 7 are the following for structures assigned to SDC D, E, or F: • ACI Equation (5.3.1e) or ASCE/SEI load combination 6: (5.16)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.17)



As in the case for SDC C, the following equations are applicable for structures assigned to SDC D, E, or F where seismic load effects including overstrength are required, which includes collectors and their connections (ASCE/SEI 12.10.2.1): • ACI Equation (5.3.1e) or ASCE/SEI load combination 6:

(5.18)

• ACI Equation (5.3.1g) or ASCE/SEI load combination 7:

(5.19)

Inertia forces determined in accordance with ASCE/SEI 12.10.1.1 must be increased by 25 percent when designing (1) connections of diaphragms to vertical elements of the SFRS and to collectors and (2) collectors and their connections, including connections to the vertical elements of the SFRS in structures with the following irregularities (ASCE/SEI 12.3.3.4): • Horizontal structural irregularity Type 1a, 1b, 2, 3, or 4 in ASCE/SEI Table 12.3-1 • Vertical structural irregularity Type 4 in ASCE/SEI Table 12.3-2 The 25 percent increase need not be applied in cases where seismic load effects including overstrength (ASCE/SEI 12.4.3) have been used in the design. In-plane inertial and transfer forces. In buildings that have a Type 4 horizontal structural irregularity, the in-plane seismic effect, , that is to be used in Equations (5.16) and (5.17) must include seismic effects from in-plane forces and . Similarly, in buildings that do not have a Type 4 horizontal structural irregularity, that is to be used in Equations (5.16) and (5.17) must include seismic effects from in-plane forces and where is the redundancy factor determined by ASCE/SEI 12.3.4.2 for the structure. Direction of loading. The direction of load requirements for diaphragms in buildings assigned to SDC D, E, or F are the same as those for SDC C (ASCE/SEI 12.5.4; see Section 5.4).

5-5

Design Guide for Reinforced Concrete Diaphragms

5-6

Design Guide for Reinforced Concrete Diaphragms

Chapter 6 Diaphragm Modeling and Analysis 6.1 Overview General modeling and analysis requirements for diaphragms are given in ACI 12.4.2. The provisions in ACI 12.4.2.2 through 12.4.2.4 are to be used where the requirements of the general building code, such as the IBC or ASCE/SEI 7, are not applicable. Some of the general analysis requirements in ACI Chapter 6 are applicable to diaphragms. The provisions for elastic analysis in ACI 6.6.1 through 6.6.3 can be applied because diaphragms are designed to remain essentially elastic when subjected to inplane wind and seismic forces. In-plane stiffness modeling for diaphragms is covered in Sections 6.2. Analysis methods are given in Section 6.3 and various diaphragm models are included in Sections 6.4 through 6.7.

6.2 In-plane Stiffness Modeling ACI 12.4.2.3 permits the use of any set of reasonable and consistent assumptions for in-plane stiffness (or, rigidity) of diaphragms. Distribution of forces in the diaphragm and displacements and forces in the vertical elements of the lateral forceresisting system (LFRS) are contingent on the in-plane stiffness of a diaphragm. In general, a diaphragm is considered to be rigid where the in-plane deflection due to lateral forces is relatively small compared to that of the vertical elements of the LFRS. For purposes of analysis, a rigid diaphragm is assumed to have an infinite in-plane stiffness, which means in-plane deflections are equal to zero; as such, rigid diaphragms displace and rotate as a rigid body when subjected to lateral forces, and the vertical elements of the LFRS move together accordingly (displacement compatibility). Also, lateral forces are distributed to the vertical elements of the LFRS in proportion to their relative rigidities (stiffnesses) and their location with respect to the center of rigidity (CR), which is the stiffness centroid within a diaphragm (see Section 6.3.3 below). Illustrated in Figure 6.1 is the deflected shape of a rigid diaphragm supported by three shear walls that all have the same rigidity. In this case, the resultant of the lateral forces acts through the CR at that level and the diaphragm undergoes only a rigid body displacement. If the resultant of the lateral forces did not act through the CR, the diaphragm would also rotate (see Section 6.3.3). In contrast, the in-plane deflection of a flexible diaphragm is relatively large compared to that of the vertical elements of the LFRS (see Figure 6.1), and the distribution of lateral forces to the vertical elements of the LFRS is independent of their relative rigidities. In such cases, lateral forces are distributed based on the tributary mass of the diaphragm to the vertical elements of the LFRS. For diaphragms of uniform material and weight, lateral forces can be distributed by tributary areas. Flexible diaphragms do not undergo rigid body rotation like rigid diaphragms. A diaphragm is classified as semirigid where the in-plane deflection of the diaphragm and the deflection of the vertical elements of the LFRS are of the same order of magnitude. According to ASCE/SEI 26.2, a concrete diaphragm with a span-to-depth ratio of 2 or less is permitted to be idealized as rigid when subjected to lateral wind forces. In the case of lateral seismic forces, it is permitted to assume that a concrete diaphragm is rigid when the following two conditions are satisfied (ASCE/SEI 12.3.1.2): (1) span-to-depth ratio is 3 or less and (2) structure has none of the horizontal irregularities in ASCE/SEI Table 12.3-1. When determining the spanto-depth ratio, the span is equal to the distance between lines of lateral resistance (such as walls and frames) in the direction of analysis. The depth is equal to the overall depth of the diaphragm in the direction of analysis. The reinforced concrete diaphragm illustrated in ACI Figure R12.4.2.3a may

Rigid Diaphragm

Flexible Diaphragm

Figure 6.1 Rigid and Flexible Diaphragms

6-1

Design Guide for Reinforced Concrete Diaphragms not be considered rigid because of its relatively large span-to-depth ratio, , where is the distance between the walls at each end, which have been designated to be part of the LFRS, and is the depth of the diaphragm in the direction of analysis. Similarly, ramps in parking structures may not be classified as rigid because of relatively long span-to-depth ratios that are common in such structures. A more precise method of determining diaphragm flexibility is given in ASCE/SEI 12.3.1.3, which is based on the maximum in-plane deflection of the diaphragm, , and the average story drift, ΔADVE (see ASCE/SEI Figure 12.3-1): A diaphragm is permitted to be idealized as flexible where δMDD / ΔADVE > 2 [ASCE/SEI Equation (12.3-1)]. Note that a rigid diaphragm is defined in IBC 1604.4 based on deflections and drifts as well: A diaphragm is rigid for the purpose of distribution of story shear and torsional moment when the lateral deformation of the diaphragm, , is less than or equal to two times the average story drift, ΔADVE. For diaphragms that cannot be idealized as either rigid or flexible, the in-plane stiffness of the diaphragm must be explicitly accounted for in the analysis (ASCE/SEI 12.3.1). A three-dimensional analysis that considers the stiffnesses of the diaphragms and the elements of the LFRS usually provides the most accurate distribution of forces in such cases. As discussed in Section 4.3 of this publication, vertical elements of the LFRS may be subjected to relatively large transfer forces in buildings with offsets or other types of discontinuities, such as the one illustrated in Figure 4.11. In buildings where the diaphragms are assumed to be rigid, the transfer forces that are obtained from analysis of the structure may be unnecessarily larger than they need to be at the levels of the discontinuities. It is recommended in such cases to model the diaphragm stiffness to obtain more realistic estimates of the forces in the diaphragms and the vertical elements of the LFRS. Other examples of Type 4 vertical irregularities where this type of analysis would be appropriate are illustrated in ASCE/SEI Figure C12.3-3. To account for cracking in reinforced concrete diaphragms, a stiffness modifier should be applied to the gross in-plane stiffness of the diaphragm. In the case of wind forces, the structure is typically assumed to respond in the elastic range, so using 50 percent of the gross in-plane moment of inertia of the diaphragm would be appropriate (see ACI 6.6.3.1.2). For seismic forces, it has been shown that stiffness modifiers typically fall in the range of 0.15 to 0.50 (see Reference 8).

6.3 Analysis Methods 6.3.1 Overview

6.3.2 Load Paths for Lateral Forces It is important to have a clear understanding of the complete gravity and lateral load paths in any building, from the application points on the structure to the termination points in the ground and every point in between. Illustrated in Figure 6.2 is the load path

6-2

Diaphragm Shear wall

Windward wall 𝑏𝑏

𝑏

Direction of wind

𝐹𝐹�����

Force to frame, 𝐹𝐹�����

Forces due to wind transferred to foundations

Force to wall, 𝐹𝐹����

Total wind force, 𝐹𝐹� � 𝑝𝑝� ℎ𝑏𝑏⁄𝑏 𝑝𝑝� ℎ⁄2

ℎ ⁄2

A general discussion on the load path of lateral forces in a reinforced concrete building is given in Section 6.3.2. Section 6.3.3 contains a procedure on how to distribute lateral forces to the members of the LFRS in a building with rigid diaphragms. ACI 318 analysis methods to determine internal forces in rigid diaphragms and collectors due to lateral forces are covered in Section 6.3.4. Information on beam, strutand-tie, finite element, and alternative models for diaphragms is given in Sections 6.4, 6.5, 6.6, and 6.7, respectively.

Frame

ℎ

Once the diaphragm and collector forces have been determined (see Chapter 4 of this publication) and the type of diaphragm has been established based on in-plane stiffness using the information in Section 6.2 (rigid, semirigid, or flexible), the diaphragm and any collectors must be analyzed for the effects due to inplane forces in combination with out-of-plane forces (see Chapter 5).

Leeward wall

Total wind pressure, 𝑝𝑝�

Spread footing

𝐹𝐹���� 𝐹𝐹����

𝑀𝑀����

Soil pressure �includes gra�it� forces on wall�

Figure 6.2 Path of Wind Forces in a Reinforced Concrete Building

Design Guide for Reinforced Concrete Diaphragms for wind in a one-story reinforced concrete building where it has been determined that the roof slab is a rigid diaphragm. The LFRS system consists of a moment frame and a shear wall in the direction of analysis (the lateral stiffness of a frame or wall about its weak axis is considered to be negligible compared to the lateral stiffness about its strong axis, so only those elements in the direction of analysis are assumed to be part of the LFRS in that direction). The load paths would be essentially the same in regularly-shaped, multistory buildings where dynamic effects caused by wind would be negligible. As discussed in Section 4.2.1 of this publication, wind forces are directly proportional to the areas of exposed surfaces that receive wind. The wind load path through this building is as follows (Note: Only the wind pressures acting on the LFRS in the direction of analysis are shown in Figure 6.2; the negative wind pressures acting over the surface of the roof slab and the side walls are not shown): 1. T  he windward and leeward walls, which are identical in this example and which are supported laterally by the roof diaphragm, are subjected to the tributary windward and leeward wind pressures, respectively, determined at the roof level in accordance with ASCE/SEI Chapter 27 or 28 (see Section 4.2.1 of this publication). The wind pressures acting on the thickness of the roof slab on both sides of the windward and leeward walls are assumed to be negligible in this example and are not shown in the figure. The summation of the windward and leeward pressures is the total wind pressure, , in the direction of analysis, and that pressure is applied on the windward wall, as shown in Figure 6.2. The total wind pressure is transferred to the edge of the roof diaphragm as a line load, which is equal to where the mean roof height of the building. 2. B  ecause the diaphragm is rigid, the total wind force, , is transferred from the diaphragm to the frame and wall based on their relative rigidities where is the width of the windward and leeward walls. In the typical case where the point of application of the total wind force and the CR do not coincide, the diaphragm undergoes rigid body displacement and rotation (see Section 6.3.3). If the diaphragm were flexible, the frame and shear wall would each resist and there would be no torsion. 3. T  he frame and wall transfer their respective wind forces to their foundations, which subsequently transmit them into the ground. In the case of the shear wall, which is supported on a spread footing in this example, the soil pressure distribution beneath the spread footing shown in Figure 6.2 includes the effects from gravity forces tributary to the wall. The reactions at the bases of the columns in the moment frame due to wind effects only are shown in the free-body diagram; the effects due to gravity forces are not shown. In all cases, the combined effects due to gravity and wind forces (and any other applicable forces) must be used to determine the size and required reinforcement for all the structural members in this load path. Because wind can occur from any direction, wind pressures must be determined in the opposite direction to that shown in Figure 6.2 and in the two perpendicular directions. Similar load paths must be established in those directions. The path of seismic forces through this building is very similar to the one for wind forces. The main difference is that the resultant of the in-plane inertial force, which is determined in accordance with ASCE/SEI 12.10.1 (see Section 4.2.3 of this publication), is applied at the center of mass (CM) at the roof level. Additional information on the horizontal distribution of lateral forces to the vertical elements of the LFRS is given in Section 6.3.3.

6.3.3 Horizontal Distribution of Lateral Forces Overview As noted previously, the horizontal distribution of lateral forces to the elements of the LFRS in a building with rigid diaphragms depends on the relative stiffness of these elements and their location with respect to the CR. The location of the CR must be determined on a floor/roof level prior to lateral force allocation. Center of Rigidity (CR) By definition, the CR is the point on a floor/roof level where the equivalent story stiffness is assumed to be located. It is often referred to as the stiffness centroid. For buildings with rigid diaphragms, application of a lateral force through that point produces only rigid body displacement of the story. Displacement and rotation occur where the lateral force is applied at any point other than the CR. Depending on the structural layout, the CR can be at different locations on different levels in a building. For the reinforced concrete building in Figure 6.3, it is assumed that only the shear walls labeled 1 to 4 are part of the LFRS. The following equations, which are applicable to any type of LFRS and not just to shear walls, can be used to locate the CR in the x-direction ( ) and in the y-direction ( ):

6-3

Design Guide for Reinforced Concrete Diaphragms

(6.1) (6.2) where



in-plane lateral stiffness of lateral-force-resisting element i in the y-direction in-plane lateral stiffness of lateral-force-resisting element i in the x-direction



distance in the x-direction from the origin to the centroid of lateral-force-resisting element i

distance in the y-direction from the origin to the centroid of lateral-force-resisting element i

The in-plane stiffness, , can be obtained by any rational method, including the approximate methods presented later in this section. In Figure 6.3, the origin from which all distances are measured is at the centroid of the lower-left column; in general, it can be positioned anywhere when determining the location of the CR.

③ 𝑥𝑥��

Figure 6.3 Location of Center of Rigidity (CR) 𝛿𝛿�

𝑉𝑉�

ℓ�

𝑉𝑉�

Figure 6.4 Lateral Displacement of a Story in a Rigid Frame 𝑉𝑉�

The following approximate methods can be used to determine in-plane stiffnesses of moment frames and shear walls. These stiffnesses can be utilized in equivalent beam models, where applicable, to determine in-plane internal forces in rigid diaphragms (see Section 6.4 of this publication).

6-4

𝑦𝑦�

𝑥𝑥�

Approximate In-plane Stiffness of Elements of the LFRS

Rigid Frames. For the two-dimensional rigid frame in Figure 6.4 subjected to the lateral force, , which produces the deflection, , the following equation can be used to determine an approximate equivalent story stiffness, :

𝑦𝑦��

𝑦𝑦�

①

②

𝛿𝛿�

(a)

𝛿𝛿�

ℎ�

If wall 3 (without an effective flange) and wall 4 have the same thickness and length, are constructed from the same concrete mix, and are located symmetrically in the plan of the building in the y-direction, the CR in the y-direction can be determined by inspection: it is located midway between the two walls. In the x-direction, the CR will most likely be closer to wall 2 than to wall 1 because wall 2 appears to be stiffer than wall 1 based on its length and thickness.

CR

ℎ�

All the elements of the LFRS that are parallel to the direction of analysis are included in Equations (6.1) and (6.2); it is commonly assumed that out-of-plane resistance of a 𝑥𝑥� lateral-force-resisting element is negligible compared to its in-plane resistance. Thus, when determining the location of the CR in the x-direction by Equation (6.1), only the stiffnesses of walls 1 and 2 are considered. Note that a portion of wall 3 (that is, an effective flange width) may be included when determining the stiffness of wall 1. Similarly, when determining the location of the CR in the y-direction by Equation (6.2), only the stiffness of wall 3 (including an effective flange width of wall 1, if desired) and wall 4 are considered.

④

𝑉𝑉�

Figure 6.5 Lateral Displacement of a Wall or Pier

(b)

Design Guide for Reinforced Concrete Diaphragms

(6.3) where



modulus of elasticity of concrete column length



column stiffness

moment of inertia of column in the direction of analysis

beam stiffness

moment of inertia of beam

beam length

The summations in Equation (6.3) pertain to all the columns and beams in a story. Solid Walls. The lateral stiffness of a solid wall or pier about its major axis is a function of both flexural and shear displacements and the support conditions (fixed or pinned) at the top and bottom of the member. Displacement components for the walls in Figure 6.5, which are subjected to the lateral force, , that produces the total displacements, , are given in Table 6.1. The wall on the left is assumed to be fixed at both ends and the one on the right is fixed at one end and pinned at the other. In the table, is the moment of inertia of the wall or pier in the direction of analysis, is the shear modulus of concrete (which is usually taken as ), and is the cross-sectional area of the wall or pier. The total stiffness of a wall or pier, , can be determined by the following equation: (6.4)

Table 6.1 Flexure and Shear Components of Lateral Deflection of a Solid Wall or Pier Flexural Deflection, δFi

Support Condition

Shear Deflection, δVi

Fixed at top Fixed at bottom Fixed at top Pinned at bottom

where is the flexural stiffness of the wall and is the shear stiffness of the wall. The method outlined above is valid for solid walls without any openings. An approximate method that yields accurate results for walls with one or more openings of any size and location can be found in Reference 9.

In the usual case where the resultant wind force does not act through the CR (either due to geometry or minimum eccentricity requirements of ASCE/SEI 7), a rigid diaphragm translates and rotates (see Figure 6.6 for the building in Figure 6.3). The wind force, , induces horizontal forces in the elements of the main wind force-resisting

④

�Term 2��

①

�Term 2��

CR �Term 1��

③ 𝐵𝐵⁄𝐵

𝑥𝑥̅�

�Term 1��

�Term 2��

𝑥𝑥̅�

�Term 2�� 𝑉𝑉�

②

���

It is generally assumed that total wind pressure (windward plus leeward pressures) at a roof or floor level in a building acts over the tributary story height and is uniformly distributed over the width of the building that is perpendicular to the pressure (see Figure 4.1). The resultant wind force acts through the centroid of the building face at that level, as shown in Figure 4.2 for the building in Figure 4.1.

���

Horizontal Distribution of Wind Forces

𝑒𝑒� 𝐵𝐵

Figure 6.6 Resultant Wind Force Not Acting Through the CR

6-5

Design Guide for Reinforced Concrete Diaphragms system (MWFRS) in the direction of the wind in proportion to the stiffness of those elements. Additionally, the torsional moment, , which is equal to in this case, induces horizontal forces in the elements of the MWFRS in the direction of the wind and in the direction perpendicular to the wind based on the stiffness of the elements, the location of the elements with respect to the CR, and the overall torsional resistance of all the elements of the MWFRS in both directions. The following equation can be used to determine the portion of the total story shear, by element i of the MWFRS considering both direct and torsional shear forces:

, in the y-direction that is resisted

(6.5)

where



perpendicular distance from the centroid of element i to the CR parallel to the x-axis perpendicular distance from the centroid of element i to the CR parallel to the y-axis



Term 1 in Equation (6.5) is the portion of the total story shear that is to be resisted by element i based on relative stiffness. Only the elements of the MWFRS parallel to the y-direction share this force (see Figure 6.6, which indicates Term 1 forces for walls 1 and 2 in the direction resisting the applied wind force, ). Term 2 is the shear force that is to be resisted by element i that is generated by the torsional moment . The elements of the MWFRS in both directions share the shear forces generated by this moment (see Figure 6.6, which indicates Term 2 forces for all four walls in the directions resisting the applied clockwise torsional moment). It is important to note that the forces in walls 3 and 4 are calculated using Term 2 with the numerator equal to instead of . The denominator of this term is the torsional stiffness of all the walls and is analogous to the polar moment of inertia of a section. Term 2 would be equal to zero in this equation if acts through the CR; in such cases, the floor would translate as a rigid body and the elements of the MWFRS would all displace an equal amount horizontally. Similarly, the portion of the total story shear, determined by the following equation:

, in the x-direction that is resisted by element i of the MWFRS can be

(6.6)

The forces in walls 1 and 2 in this case are calculated using Term 2 in this equation with the numerator equal to of .

instead

Term 2 forces in Equations (6.5) and (6.6) may be either positive or negative depending on the location of the element with respect to the CR. For example, for wind forces in the y-direction, the Term 2 force for wall 2 acts in the opposite direction of the Term 1 force (see Figure 6.6). To properly capture the maximum force effects on an element, wind acting in the opposite direction must also be considered. Horizontal Distribution of Seismic Forces The center of mass (CM) is the location on a level of a building where the mass of the entire story is assumed to be concentrated. Because the mass of the floor/roof system is usually much larger than the mass of the vertical elements (columns and walls), a good approximation for the location of the CM is at the centroid of the floor area where the same floor/roof system is used throughout the entire level. This is the location where the seismic force for the story is assumed to act (see Figure 4.6). In the case of rigid diaphragms, the seismic force induces a torsional moment where the CM and the CR do not coincide. In general, this inherent torsional moment, , is equal to the seismic force times the eccentricity between the CM and the CR that is perpendicular to the direction of analysis (see Figure 4.6). According to ASCE/SEI 12.8.4.2, an accidental torsional moment, , must also be included for buildings with diaphragms that are not flexible that possess one or more of the horizontal irregularities given in ASCE/SEI Table 12.3-1. This accidental torsional

6-6

Design Guide for Reinforced Concrete Diaphragms moment is determined based on the assumption that the CM is displaced each way from its actual location by a distance equal to 5 percent of the dimension of the structure perpendicular to the direction of analysis (see Figure 4.6). In such cases, the total torsional moment is equal to plus . Equations (6.5) and (6.6) can be used for horizontal seismic force distribution to the elements of the LFRS where the applicable total torsional moment is used in Term 2 of those equations.

6.3.4 ACI 318 Analysis Methods Overview According to ACI 12.4.2.4, in-plane design moments, shear forces, and axial forces in diaphragms must be determined using any method that satisfies equilibrium and the design boundary conditions. Methods of analysis based on the following models are permitted to be used: • A rigid diaphragm model where diaphragms can be idealized as rigid • A flexible diaphragm model where diaphragms can be idealized as flexible • A  bounding analysis in which the design values are the envelope of values obtained by assuming upper bound and lower bound in-plane stiffnesses for the diaphragm in two or more separate analyses • A finite element model considering diaphragm flexibility • A strut-and-tie model in accordance with ACI 23.3 A general overview of these models is given in Table 6.2. Table 6.2 Diaphragm Models Model

Applicability

Rigid diaphragm

For lateral wind forces, reinforced concrete slabs with a span-to-depth ratio of 2 or less. For lateral seismic forces, when the following two conditions are satisfied: (1) reinforced concrete slabs with a span-to-depth ratio of 3 or less and (2) structure possesses none of the horizontal irregularities in ASCE/SEI Table 12.3-1.

Beam models are commonly used to determine diaphragm internal forces in buildings without major irregularities and/or transfer forces

Flexible diaphragm

A diaphragm is permitted to be idealized as flexible where the ratio of the maximum in-plane deflection of the diaphragm, δMDD, and the average story drift, ΔADVE, is greater than 2

Beam models are commonly used to determine diaphragm internal forces in buildings without major irregularities and/or transfer forces

Suitable for semirigid diaphragms

Diaphragm internal forces are taken as the envelope of values from the following two analyses: (1) rigid diaphragm on flexible supports and (2) flexible diaphragm on rigid supports

Finite element

Suitable for any diaphragm

This model is especially useful for irregularlyshaped diaphragms and diaphragms with large transfer forces and/or openings

Strut-and-tie

Suitable for any diaphragm

Model should include consideration of force reversals that may occur under design load combinations

Bounding analysis

Notes

Four acceptable approaches for diaphragm modeling and analysis are given in ACI 12.5.1.3: • Beam model • Strut-and-tie model • Finite element model • Alternative models

6-7

Design Guide for Reinforced Concrete Diaphragms Information on these models, including methods to determine internal forces in rigid diaphragms and collectors, is given in Sections 6.4 through 6.7.

6.4 Beam Models 6.4.1 Overview Beam models are widely used to determine internal forces in rigid and flexible diaphragms in buildings without major irregularities and/or transfer forces. Diaphragms are treated as beams that span between the vertical elements of the LFRS, which are idealized as either rigid or flexible (spring) supports. The three beam models that are frequently used in the analysis of diaphragms are (1) the equivalent beam model with rigid supports, (2) the equivalent beam model with spring supports, and (3) the corrected equivalent beam model with spring supports. The first and last models are discussed in the following sections. The equivalent beam model with spring supports is similar to the equivalent beam model with rigid supports, except that the supports are modeled as springs that have stiffnesses equal to that of the vertical elements of the LFRS in the direction of analysis.

6.4.2 Equivalent Beam Model with Rigid Supports This model is usually appropriate for low-rise buildings with regular geometries where two or more lines of lateral force-resisting systems are provided in a given direction that have approximately the same relative rigidity.

①

In this model, roof and floor systems act as the web of the equivalent beam, which resists the design shear forces. For diaphragms that are analyzed by this method, the shear flow, , must

6-8

𝑻𝑻𝒖𝒖

𝑉𝑉� � 𝑉𝑉�,���

Chord force �t����

Uniform shear flow, 𝑣𝑣� � 𝑉𝑉�,��� ⁄𝐿𝐿

Collector element �t����

𝑪𝑪𝒖𝒖

�tructural wall �t����

𝑉𝑉� � 𝑉𝑉�,���

②

④

𝑤𝑤�

Diaphragm Forces The equivalent in-plane load, , in Figure 6.7 is uniformly distributed over the width of the diaphragm, which means the reactions in the supports (that is, walls 1 and 2) are equal. These reactions are the diaphragm forces that are transferred to the vertical elements of the LFRS, which in this case are walls. It is permitted to assume that the diaphragm resists in-plane moment and axial force in accordance with ACI 22.3 and 22.4, which includes the assumption that strains vary linearly over the depth of the diaphragm. Shear and moment diagrams for the diaphragm are also given in Figure 6.7.

𝐵𝐵

③

𝐿𝐿

Illustrated in Figure 6.7 is a reinforced concrete diaphragm and a LFRS consisting of structural (shear) walls. In the direction of analysis, the diaphragm is modeled as a beam whose depth is equal to the full diaphragm depth, . The equivalent beam in this case is simply-supported because the walls, which are assumed to be rigid supports in the direction of analysis, are at the perimeter of the diaphragm. This method can also be used where walls are not at the perimeter (which means the equivalent beam has one or two cantilevers) or where more than two walls are in the direction of analysis (which means the equivalent beam is continuous). Because the walls in Figure 6.7 do not extend the full depth of the diaphragm, collector elements are needed to collect the shear from the diaphragm and to transfer it to the walls.

𝑤𝑤�

Equivalent beam

𝑉𝑉�,���

𝑉𝑉� 𝑉𝑉�,��� 𝑀𝑀� 𝑀𝑀�,���

Figure 6.7 Equivalent Beam Model with Rigid Supports

Design Guide for Reinforced Concrete Diaphragms be uniform over the depth , which means in this case that . This maximum shear force must be less than or equal to the design shear strength of the diaphragm. Shear flow is transferred from the diaphragm to the vertical elements of the LFRS by shear friction. The diaphragm boundaries that are perpendicular to the lateral force act as the flanges of the equivalent beam (commonly referred to as chords) and resist the tension and compression forces that are induced in the diaphragm due to bending. These forces can be determined by dividing the maximum bending moment in the diaphragm by the distance between the forces (moment arm): (6.7)



In this equation, is the perpendicular distance between the chord forces, which is commonly taken as 95 percent of the total diaphragm depth in the direction of analysis (in this case, ). The maximum bending moment in this diaphragm is equal to . Reinforcement must be provided in the chords to resist the tension forces generated by this bending moment. Because wind and seismic forces can act in any direction, chord reinforcement is required at both boundaries of the diaphragm perpendicular to the direction of analysis. For the diaphragm in Figure 6.7, the same chord reinforcement can be used at both boundaries. Collector Forces As noted above, collectors (sometimes referred to as drag struts) must be provided where elements of the LFRS do not extend the full depth of a diaphragm. The purpose of collectors is to collect shear forces from the portions of the diaphragm that do not contain elements of the LFRS and to transfer those forces to the elements of the LFRS along that line by axial compression and tension. Shear flow is transferred from the diaphragm to the collectors by shear friction. A reinforced concrete beam or a portion of a reinforced concrete slab that is in line with the LFRS can be used as a collector. In cases where the beam or slab has the same width as the vertical element of the LFRS it frames into (for example, the width of the collector is equal to the thickness of the structural wall), all the collected forces are transferred directly into the member of the LFRS at its ends. This is illustrated in Figure 6.8 for shear wall 1 in Figure 6.7. The uniform factored shear force in the diaphragm, , is equal to the reaction in the wall, , divided by the depth of the diaphragm in the direction of analysis, . Similarly, the uniform factored shear force in the wall along its length is equal to . The net factored shear force at any point is equal to the difference between the unit factored shear forces in the wall and diaphragm at that point. The axial forces in the collectors are determined by summing the areas in the net shear force diagram. At the edges of the diaphragm, the axial force in the collectors is equal to zero (Note: In cases where wind and/or seismic forces from perimeter elements, such as cladding, must be collected, the forces at the ends of a collector may not be zero; see Section 4.4 of this publication for more information). The axial force increases linearly as shear is transferred from the diaphragm to the collectors. At the end of the wall located a distance from the edge of the diaphragm, the axial tension force in the collector is equal to . At the other end of the wall, the axial compression force is equal to at a distance from the edge of the diaphragm. Because wind and seismic forces can act in any direction, the axial forces in the collectors change from compression to tension and from tension to compression, so a collector must be designed for the most critical effects.

ℓ�

𝐿𝐿

ℓ�

ℓ�

Where a collector is assumed to be wider than the member of the LFRS element it frames into, a part of the collector force is transferred directly into the member of the LFRS at its ends and a part is transferred through shear-friction along the ① length of the vertical element of the LFRS (see ACI Figure R12.5.4.1 for the case of a wall located at the edge of a diaphragm). Wider collectors are usually required in buildings where it is not practical or possible (from a design or constructability perspective) to provide collector elements that are concentric with the vertical elements

𝑉𝑉� /𝐿𝐿 𝑉𝑉� /ℓ�

𝑉𝑉� ℓ�

𝑉𝑉� 𝐿𝐿

𝑉𝑉� 𝐿𝐿 𝑉𝑉� 𝑉𝑉� � ℓ� 𝐿𝐿

𝑉𝑉� � � ℓ� 𝐿𝐿

Collector element �t����

Unit Shear Forces Net Shear Forces

�

𝑉𝑉� 𝑉𝑉� 𝑉𝑉� � � ℓ � � � ℓ� ℓ� 𝐿𝐿 � 𝐿𝐿

𝑉𝑉� � � � ℓ� 𝐿𝐿

Collector Axial Forces

Figure 6.8 Unit Shear Forces, Net Shear Forces, and Collector Axial Forces in a Diaphragm

6-9

Design Guide for Reinforced Concrete Diaphragms

𝑡𝑡 𝐿𝐿

ℓ�

①

Collector element

𝑡𝑡⁄𝑡

𝑇𝑇� � 𝑇𝑇� � 𝑇𝑇�

𝑒𝑒

𝑇𝑇� 𝑇𝑇�

𝑉𝑉� /ℓ�

�𝑇𝑇� � 𝐶𝐶� �⁄ℓ� 𝑉𝑉� /𝐿𝐿

𝑇𝑇�

𝐶𝐶𝑑𝑑

𝐶𝐶� 𝐶𝐶�

Force Transfer to Wall �a�

𝐶𝐶� � 𝐶𝐶� � 𝐶𝐶�

Collector Axial Forces

Collector element 𝑡𝑡

𝐿𝐿� ⁄4

𝐿𝐿�

An effective width of slab, , adjacent to the shear wall or frame that is used to resist the collector forces must be determined where collectors are wider than the vertical elements of the LFRS that they frame into. Requirements for are not covered in ACI 318 or ASCE/SEI 7. Reference 10 recommends using equal to the width of the vertical element of the LFRS plus a width on either side of the vertical element equal to one-half the contact length between the diaphragm and the vertical element (see Figure 6.9(a) for the effective collector width for shear wall 1 in Figure 6.7 where ). ACI R12.5.4 recommends an effective slab width equal to basically the same as that in Reference 10. Illustrated in Figure 6.9(b) is the effective slab width based on the above recommendation for an interior shear wall with an opening in the diaphragm on one side of the wall

𝑏𝑏���

𝐿𝐿𝑤𝑤 ⁄2

ℓ�

𝑏𝑏��� � 𝑡𝑡 � �ℓ� ⁄2�

ℓ�

of the LFRS. An example of where wider collectors may be necessary is in a building with relatively thin walls where only a narrow strip of slab is available to resist relatively large collector forces. Another example is in a building assigned to SDC C through F where the collectors must be designed for axial forces calculated using the seismic load effects including overstrength (ASCE/SEI 12.10.2.1; see Section 4.7.2 of this publication for more information).

𝐿𝐿𝑤𝑤 ⁄2

𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒

𝐿𝐿𝑤𝑤 ⁄4

Also illustrated in Figure 6.9(a) is the force transfer to �b� wall 1. The tension and compression forces that are transferred directly into the ends of wall 1 are designatFigure 6.9 Effective Slab Width for Collectors Wider than the Vertical Elements of the LFRS. (a) Edge Wall (b) Interior Wall with Diaphragm ed and , respectively. The tension force, , Opening is resisted by reinforcement in the wall parallel to the direction of analysis and the compression force, , is resisted by the concrete. The remaining tension and compression forces and , respectively, are transferred through shear along , which is the length of wall 1. Shear-friction reinforcement is required to resist the total shear force ; this reinforcement is placed perpendicular to the wall (see ACI Figure R12.5.4.1). It is evident from Figure 6.9(a) that the forces and act at an eccentricity with respect to wall 1. If it is assumed that the reinforcement in the collector is uniformly distributed in the direction of analysis within (like that shown in ACI Figure R12.5.4.1), then the eccentricity, , is equal to , and the in-plane bending moment resulting from this eccentricity and the portion of the collector force that is not transferred directly into the end of the wall must be considered in design (see Section 8.6.3 of this publication for an approximate method on how to determine this in-plane bending moment). In general, the section of the collector that is concentric with the vertical element of the LFRS should be designed for a reasonable portion of the total collector force considering overall design and construction limitations. Reinforcing bar congestion at the ends of a shear wall, for example, is a practical limitation to consider when selecting the portion of the total tension force that will be transferred directly into the end of a wall. In some cases, the total collector force will need to be resisted entirely by the shear in the slab adjacent to the vertical element of the LFRS. Partial-depth collectors may be used instead of the full-depth collectors described above (ACI 12.5.4.1). A design procedure for this option is given in Reference 7.

6-10

Design Guide for Reinforced Concrete Diaphragms Transfer Forces In cases where an element of the LFRS transfers forces from above Diaphragm to a diaphragm below, like the reforce inforced concrete shear walls supported by the podium slab in Figure 4.10, collectors are required to transfer the shear forces from the walls into the diaphragm (see Figure 6.10). These transfer forces and the in-plane diaphragm forces at that level are then transferred to the basement walls in the direction of analysis.

Reinforced concrete shear �all �t�p�� Collector �t�p��

𝑉𝑉�

Reinforced concrete �asement �all �t�p��

The equivalent beam model may not be the most suitable method to use where relatively large transfer forces must be considered. Direction of Load Requirements Direction of load analyses must be performed for lateral forces where required (see Chapter 5 of this publication). In the case of wind forces, the four load cases in ASCE/SEI Figure 27.3-8 should be considered when determining the internal forces in a diaphragm. Generally, applying full wind pressures independently in each perpendicular direction results in the critical cases. Direction of load requirements for diaphragms in buildings subjected to seismic forces are given in Sections 5.3 through 5.5.

Collector Axial Forces

Figure 6.10 Collectors for Transferring Shear Forces from Structural Walls Above to Basement Walls Below

𝐹𝐹�  

6.4.3 Corrected Equivalent Beam Model with Spring Supports This model is best suited for buildings with rigid diaphragms and lateral force-resisting systems that have different stiffnesses. Effects of torsion are automatically accounted for in this method. Diaphragm Forces Just like in the equivalent beam model with rigid supports, the first step in determining the internal in-plane forces in a diaphragm based on the corrected equivalent beam model is to obtain the diaphragm forces that are transferred to the vertical elements of the LFRS. In multistory buildings, a threedimensional model of the building is usually constructed with the appropriate lateral forces applied at each level over the height of the building. Horizontal wind forces are applied using the information in Section 4.2.1 of this publication. Seismic forces are applied over the height in accordance with ASCE/SEI Equation (12.8-11) with the exception that at the diaphragm level under consideration is replaced with the diaphragm force determined by ASCE/SEI 12.10.1.1 (see Reference 11). This is illustrated in Figure 6.11 where is the diaphragm force applied to the structure at level (which is the diaphragm level under consideration) instead of . Where rigid diaphragms are assigned at the roof and floor levels, diaphragm forces are determined by making section cuts in the vertical elements of the LFRS immediately

𝐹𝐹�   𝐹𝐹��  

𝑥𝑥  𝑖𝑖  𝑉𝑉 

Figure 6.11 Vertical Seismic Force Distribution in a Multistory Building for Use in the Determination of Diaphragm Forces

𝑉𝑉���

𝑉𝑉�

𝑉𝑉� � 𝑉𝑉� � 𝑉𝑉���

Figure 6.12 Force Transferred Between a Diaphragm and a Vertical Element of the LFRS

6-11

Design Guide for Reinforced Concrete Diaphragms

Unlike rigid diaphragms, diaphragm forces in semirigid diaphragms can be obtained by making section cuts through the diaphragm adjacent to the vertical elements of the LFRS. To obtain more accurate results, a relatively fine finite element mesh should be used in the diaphragm at the locations of the vertical elements. In low-rise buildings with rigid diaphragms, the stiffnesses of the elements of the LFRS can be obtained using approximate analyses; such analyses do not result in a significant loss of accuracy. For example, reactions can be calculated by Equations (6.5) and (6.6) using approximate stiffnesses (such as those given in Section 6.3.3 of this publication). Otherwise, stiffnesses can be obtained from a more refined analysis. Once the diaphragm forces that are transferred to the vertical elements of the LFRS have been obtained using the appropriate method outlined above, the second step is to determine an equivalent in-plane distributed load on the diaphragm that is in equilibrium with these forces (reactions). The distributed load is usually trapezoidal, which accounts for any torsional moments (see Figure 6.13). The loads and at each end of the equivalent beam can be obtained by using the equations for force equilibrium and moment equilibrium and then solving these two equations for the two unknowns and . For the rigid diaphragm in Figure 6.13, the force and moment equations of equilibrium in the direction of analysis are the following where moments are summed about the left edge of the diaphragm:

④ 𝐿𝐿𝐿

above and below the diaphragm. This is illustrated in Figure 6.12 for shear wall 1 in Figure 6.7. The shear forces and in the wall above and below the diaphragm, respectively, are obtained from the lateral analysis of the entire building. The force , which in this case is equal to , is the diaphragm force that is transferred to the wall at this location.

𝑏𝑏�

𝑤𝑤�

⑤

① ⑥ 𝑅𝑅�

③

②

𝑏𝑏� 𝑅𝑅�

𝑉𝑉

𝑏𝑏�

⑦ 𝑅𝑅� 𝑤𝑤�

𝑉𝑉�,���

Shear

Moment

𝑀𝑀�,���

Figure 6.13 Equivalent Distributed Load, Shear Diagram, and Moment Diagram for a Rigid Diaphragm

(6.8)



(6.9) The resultant force of the trapezoidal distributed load is equal to the applied lateral force at that level obtained from analysis, which is reflected in Equation (6.8). Note that the moment caused by the forces in the elements of the LFRS in the direction perpendicular to the in-plane force is often ignored in overall horizontal force distribution (for example, the moment caused by the Term 2 forces in walls 3 and 4 in Figure 6.6 are not considered). This moment can be incorporated into the trapezoidal load, if desired. Once and have been determined, the final step is to construct shear and moment diagrams for the diaphragm (see Figure 6.13). The shear diagram is used in (1) checking the design shear strength of the diaphragm, (2) designing the connections

6-12

Design Guide for Reinforced Concrete Diaphragms

𝑻𝑻𝒖𝒖

𝑉𝑉�

The diaphragm depicted in Figure 6.15 is the same as the one in Figure 6.13 but with a relatively large opening in it. The forces at each edge of the opening are designated and , which can be obtained from the overall trapezoidal force distribution.

𝑉𝑉�

Collec�or elemen� ������

𝑻𝑻𝒖𝒖𝒖𝒖𝒖 𝑪𝑪𝒖𝒖𝒖𝒖𝒖

𝑪𝑪𝒖𝒖

𝑤𝑤�

②

��r�c��ral wall ������

Uniform shear flow, 𝑣𝑣� � 𝑉𝑉�⁄𝐿𝐿

④

𝐿

①

②

𝑏𝑏�

𝑤𝑤�

𝑏𝑏� 𝑅𝑅�

𝐴𝐴�

③

ℓ����

ℓ�

Figure 6.14 Force Distribution in a Diaphragm with a Relatively Large Opening

ℓ�

This analysis method can also be used for diaphragms with large openings (Reference 12). Illustrated in Figure 6.14 is a representation of the force distribution for the diaphragm in Figure 6.7 with a relatively large opening. For purposes of analysis, the diaphragm segments (commonly referred to as subdiaphragms) above and below the opening can be idealized as beams that are fixed at each end. It is assumed that the collector element on one side of the opening collects the uniform diaphragm shear on that side and transfers it to the subdiaphragms above and below the opening in proportion to their relative stiffness or mass. The collector on the other side of the opening then collects the shear from the subdiaphragms and transfers it to the portion of the diaphragm on that side of the opening. Thus, the loading on a subdiaphragm is based on the total applied force at that level and the relative stiffness or mass of the subdiaphragm. Secondary chord forces occur in each subdiaphragm due to local bending caused by this loading.

𝑪𝑪𝒖𝒖𝒖𝒖𝒖

ℓ����

①

𝑻𝑻𝒖𝒖𝒖𝒖𝒖

���dia�hra�m ������

𝐿𝐿

Diaphragms with Openings

Chord force ������

𝐵𝐵

③

𝐿𝐿

of the diaphragm to the vertical elements of the LFRS, and (3) determining the axial compression and tension forces in the collectors, if any. As noted previously, the moment diagram is used in determining the tension and compression forces in the chords, the former of which is needed to calculate the required chord reinforcement near the edges of the diaphragm.

𝐴𝐴�

𝑏𝑏�

𝑅𝑅� 𝑤𝑤�

𝑏𝑏����

𝑅𝑅� 𝑤𝑤�

𝑤𝑤�

Figure 6.15 Diaphragm in Figure 6.13 with a Relatively Large Opening

In the case of wind forces, the forces on the subdiaphragms above and below the opening can be approximately determined using the in-plane stiffness ratios of these subdiaphragms, and , which can be calculated by the following equations where it is assumed that the entire slab is of uniform thickness and material:





(6.10)

(6.11)

In the case of seismic forces, the forces on the subdiaphragms above and below the opening can be approximately determined based on mass. Assuming the diaphragm has the same thickness and material properties everywhere, the mass ratios of the subdiaphragms and can be calculated by the following equations:

(6.12)

(6.13)

6-13

Design Guide for Reinforced Concrete Diaphragms where and are the areas of the subdiaphragms above and below the opening, respectively (see Figure 6.15): (6.14)



(6.15)



Free-body diagrams of the top and bottom subdiaphragms are given in Figure 6.16; as noted previously, the ends of these elements are assumed to be fixed. The moment diagrams in the figure can be obtained from statics where the appropriate and determined by Equations (6.10) through (6.13) are used to obtain the forces at each end of the subdiaphragm. The secondary tension and compression chord forces are determined using Equation (6.7) where is the maximum positive moment in each subdiaphragm. The total tension chord force along the edge of the diaphragm at the center of the opening is obtained by adding the primary tension chord force, , to the secondary tension chord force, (see Figure 6.17). The primary tension chord force can be calculated by the following equation: (6.16)



𝑅𝑅��� � 𝑀𝑀�����

Tension chord reinforcement, , is determined using the larger of the following tension chord forces:

� 𝑀𝑀�����

𝑤𝑤��� � 𝑓𝑓��� 𝑤𝑤�

𝑤𝑤��� � 𝑓𝑓��� 𝑤𝑤� � 𝑀𝑀���

Moment � 𝑀𝑀�����

� 𝑀𝑀�����

�a� Top segment

𝑅𝑅���

𝑅𝑅���

𝑏𝑏����

� 𝑀𝑀�����

� 𝑀𝑀�����

𝑤𝑤��� � 𝑓𝑓��� 𝑤𝑤�

� 𝑀𝑀�����

𝑓𝑓��� �

� 𝑀𝑀�����

�b� Bottom segment

� ⎧ ℓ� ⎪ℓ� � ℓ� for wind �

𝑤𝑤��� � 𝑓𝑓��� 𝑤𝑤�

� 𝑀𝑀���

Moment

where is the moment at the center of the opening, which is not necessarily the maximum diaphragm moment, (see Figure 6.17). The secondary tension chord force is equal to the following (see Figure 6.16): (6.17)

𝑅𝑅���

𝑏𝑏����

�

⎨ 𝐴𝐴� for seismic ⎪ ⎩ 𝐴𝐴� � 𝐴𝐴�

𝑓𝑓��� �

� ⎧ ℓ� ⎪ℓ� � ℓ� for wind �

�

⎨ 𝐴𝐴� for seismic ⎪ ⎩ 𝐴𝐴� � 𝐴𝐴�

Figure 6.16 Free-body Diagrams and Moment Diagrams for Subdiaphragms in Figure 6.15

• The primary tension chord force based on the maximum diaphragm bending moment, : (6.18)



• T  he summation of the primary tension chord force at the location of the opening plus the secondary tension chord force at the opening: (6.19) Thus, the required area of chord reinforcement is equal to the following: (6.20)

6-14

Design Guide for Reinforced Concrete Diaphragms

𝐿

②

ℓ�

①

ℓ����

ℓ�

𝑻𝑻𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖

𝐿𝐿

Secondary chord forces also develop at the corners of the opening due to the negative moments that occur at these locations. For the diaphragm in Figure 6.15, the tension chord force that occurs along the bottom edge of the opening (that is, at the top of the bottom subdiaphragm) can be determined by the following equation (see Figure 6.16): (6.21) where it has been assumed that . The required chord reinforcement along this edge is equal to . Reinforcing bars must be provided along the entire face of the opening and must be developed into the slab.

𝑏𝑏�

𝑏𝑏�

𝑪𝑪𝒖𝒖𝒖𝒖𝒖

𝑻𝑻𝒖𝒖𝒖𝒖𝒖 𝑪𝑪𝒖𝒖𝒖𝒖𝒖

𝑏𝑏�

③

𝑪𝑪𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖𝒖

Moment

Because wind and seismic forces can act in any direction, the tension and compression edges of the diaphragm and opening also are compression and tension edges. The larger area of reinforcement determined from both analyses (or from analyses where interaction of orthogonal load effects must be considered) is provided along the edges of the diaphragm and opening for simpler detailing. The requirements in ACI 8.5.4 must be satisfied in any slab system with openings, regardless of the size of the opening.

𝑻𝑻𝒖𝒖𝒖𝒖𝒖

𝑀𝑀�𝒖��� 𝑀𝑀�

Figure 6.17 Determination of Chord Forces in a Diaphragm with an Opening

Collector Forces, Transfer Forces, and Direction of Loading Requirements The methods to determine collector forces and transfer forces given in Section 6.4.2 for the equivalent beam model with rigid supports can be used in the corrected equivalent beam model with spring supports. Similarly, the direction of loading requirements in Section 6.4.2 must also be satisfied.

6.5 Strut-and-Tie Models ACI 12.4.2.4(e) permits the use of strut-and-tie models that satisfy the general requirements of ACI 23.2 to determine in-plane design forces in diaphragms. ACI R12.4.2.4 points out the importance of considering force reversals in the model, which could occur under design load combinations. Reference 7 provides a general discussion on how strut-and-tie models can be used to understand reinforcement layouts in diaphragms with significant openings. Numerous resources are available that cover the basics of strut-and-tie modeling. As such, a discussion on this model is not provided here.

6.6 Finite Element Models As noted previously, finite element modeling is usually the most accurate way to determine in-plane design forces in diaphragms and collectors. Such models are especially useful when the diaphragm has an irregular shape, is subjected to large transfer forces, or has large openings or other types of irregularities. The acceptability requirements in ACI 6.9 must be satisfied when constructing a finite element model for diaphragms. Reference 7 recommends using a finite element mesh no larger than one-fifth to one-third of the bay length or wall length. As noted above, a finer mesh may be beneficial in certain cases, especially where section cuts are made through the diaphragm to determine shear distribution. To account for cracking, a stiffness modifier should be applied to the gross in-plane stiffness of a diaphragm. Recommendations for such modifiers are given in Section 6.2 of this publication.

6-15

Design Guide for Reinforced Concrete Diaphragms 6.7 Alternative Models ACI 12.5.1.3(e) permits reinforced concrete diaphragms to be analyzed by any method other than those described above provided that equilibrium is satisfied and design strengths are obtained that are at least equal to the required strengths for all structural elements in the load path. Several alternative methods can be found in Reference 7.

6-16

Design Guide for Reinforced Concrete Diaphragms

Chapter 7 Design Strength 7.1 Overview Basic performance requirements that must be satisfied for all reinforced concrete structural members and systems, including diaphragms and collectors, are given in ACI Chapter 4. The general requirement for strength design is the following (ACI 4.6.2): Design strength ≥ Required Strength

(7.1)

or (7.2) The design strength of a member is equal to the strength reduction factor, , times the nominal strength of the member, (ACI 4.6.1). Strength reduction factors and nominal strengths that are applicable to the design of diaphragms and collectors are included in this chapter. The information needed to determine required strength, , is given in Chapters 3 through 6 of this publication.

7.2 Strength Reduction Factors Strength reduction factors, which are commonly referred to as resistance factors or tion of the design strength of a reinforced concrete member.

-factors, play a key role in the determina-

According to ACI 12.5.1.2, strength reduction factors to be used in the design of diaphragms and collectors must be determined in accordance with ACI 21.2. A summary of pertinent strength reduction factors is given in Table 7.1. The quantities and in Table 7.1 are the net tensile strain in the extreme layer of longitudinal tension reinforcement at nominal strength and the net tensile strain in the longitudinal tension reinforcement used to define a compression-controlled section, respectively. For buildings where special moment frames, special structural walls, or both are used as the seismic force-resisting system (SFRS), the strength reduction factor for shear in diaphragms is equal to 0.60 where the nominal shear strength of the diaphragm is less than the shear corresponding to the development of the nominal moment strength of the diaphragm (ACI 21.2.4.1). Also, the strength reduction factor for diaphragm shear must not exceed the least value of used in the shear design of the vertical elements of the SFRS (ACI 21.2.4.2). Table 7.1 ACI 318 Strength Reduction Factors for Diaphragms and Collectors Action, Structural Element, or Net Tensile Strain, εt

Tension-controlled sections

Strength Reduction Factor, φ

0.90

Members with spiral reinforcement conforming to ACI 25.7.3 Transition(1) Other

Compression-controlled sections

Shear(2) (1) (2)

Members with spiral reinforcement conforming to ACI 25.7.3

0.75

Other

0.65 0.75

For sections classified as transition, it is permitted to use a strength reduction factor corresponding to compression-controlled sections. A strength reduction for shear equal to 0.60 must be used when the provisions of ACI 21.2.4.1 are applicable. See also the requirements of ACI 21.2.4.2.

7-1

Design Guide for Reinforced Concrete Diaphragms 7.3 Nominal Strength Table 7.2 Design Strength Requirements for Diaphragms

7.3.1 Overview The design strength of a diaphragm depends on the type of model that is used to determine the internal force distribution (see Sections 6.4 through 6.7 of this publication for permitted analysis models). Design strength requirements based on the model type are given in ACI 12.5.1.3 (see Table 7.2). The requirements in ACI 12.5.2 through 12.5.4 for diaphragms modeled as beams are given in the following sections.

Model Beam

ACI Reference Sections 12.5.2 through 12.5.4

Strut-and-tie

Section 23.3

Finite element

Chapter 22

Alternative

Section 12.5.1.3(d)

7.3.2 Moment and Axial Force ACI 12.5.2 permits a diaphragm that has been modeled as a beam to be designed for in-plane moment and axial force using the assumptions in ACI 22.3 (flexural strength) and 22.4 (axial strength or combined flexural and axial strength), which are the same assumptions used in the design of reinforced concrete beams, columns, and walls. This includes the assumption that strains vary linearly over the depth of the diaphragm when it is subjected to in-plane forces. Diaphragms in buildings assigned to SDC D, E, or F must also be designed for flexure using the requirements in ACI Chapter 12 (ACI 18.12.8.1).

7.3.3 Shear The design strength provisions for in-plane shear given in ACI 12.5.3 are based on the assumption that shear flow is approximately uniform over the depth of the diaphragm (see Section 6.4 and Figure 6.7). Nominal shear strength, , for cast-in-place reinforced concrete diaphragms in buildings assigned to SDC A through C is determined by ACI Equation (12.5.3.3): (7.3) where





gross area of the diaphragm (diaphragm thickness times width in the direction of analysis) modification factor that accounts for the reduced mechanical properties of lightweight concrete ratio of distributed slab transverse reinforcement to gross concrete area

The transverse reinforcement ratio, , is determined using the slab reinforcement that is parallel to the in-plane shear force. When calculating by Equation (7.3), it is conservative to check the design shear strength requirements assuming that is equal to zero. Also, in this equation is limited to 100 psi (ACI 12.5.3.4). The nominal shear strength, (12.5.3.4):

, cannot exceed the maximum nominal shear strength,

, determined by ACI Equation

(7.4) Nominal and maximum shear strengths per unit length for various diaphragm thicknesses assuming normalweight concrete with equal to 4,000 psi and 5,000 psi and with are given in Table 7.3. Nominal and maximum shear strength requirements for diaphragms in buildings assigned to SDC D through F are given in ACI 18.12.9.1 and 18.12.9.2, respectively, and are the same as those discussed above for buildings assigned to SDC A through C. Shear strength requirements must also be checked around any openings in a diaphragm. Even though the shear force at the edge of an opening may be smaller than that at other locations within the diaphragm, the length of the opening must be deducted from the overall depth of the diaphragm, which means and, thus, are smaller at that location. Consider the diaphragm in Figure 6.13, which contains a relatively large opening. The factored shear force along the interior edge of the opening is much smaller than the maximum factored shear force at wall 2, but the depth of the diaphragm that is available to resist this force is equal to minus the length of the opening in the direction of analysis, . Thus, along this edge, the nominal shear strength of the diaphragm is equal to the following: (7.5) where

7-2

is the thickness of the diaphragm.

Design Guide for Reinforced Concrete Diaphragms Table 7.3 Nominal and Maximum Shear Strengths of Diaphragms Slab Thickness, h (in.)

Vn (kips/ft)

Vn,max (kips/ft)

ƒc´ = 4,000 psi

ƒc´ = 4,000 psi

ƒc´ = 5,000 psi

4.5

6.8

ƒc´ = 5,000 psi

5.0

7.6

8.5

30.4

33.9

5.5

8.3

9.3

33.4

37.3

6.0

9.1

10.2

36.4

40.7

6.5

9.9

11.0

39.5

44.1

7.0

10.6

11.9

42.5

47.5

7.5

11.4

12.7

45.5

50.9

8.0

12.1

13.6

48.6

54.3

8.5

12.9

14.4

51.6

57.7

9.0

13.7

15.3

54.6

61.1

7.6

27.3

30.5

7.3.4 Shear Transfer According to ACI 12.3.7, shear transfer between diaphragms, collectors, and vertical elements of the LFRS must be satisfied using the shear-friction provisions of ACI 22.9 or by mechanical connectors or dowels. In cast-in-place construction, shear transfer requirements must typically be satisfied at construction joints between the elements. Where shear-friction provisions are used, the nominal shear strength across the assumed shear plane is determined by ACI Equation (22.9.4.2): (7.6) In this equation, is the coefficient of friction obtained from ACI Table 22.9.4.2, which depends on the contact surface condition of the concrete and is the area of shear-friction reinforcement that crosses the shear plane with the reinforcing bars oriented perpendicular to that plane. The value of across the assumed shear plane must not exceed the limits in ACI Table 22.9.4.4. The equations in this table are a function of the contact surface condition and the area of the concrete section that resists shear transfer, .

7.3.5 Collectors According to ACI 12.5.4.2, collectors are to be designed as tension members, compression members, or both in accordance with the design provisions for axial strength or combined flexural and axial strength in ACI 22.4. In cases where collectors are subjected to relatively significant effects from gravity forces in addition to those from tension and compression axial forces due to in-plane lateral forces, design strength interaction diagrams are often constructed to check the adequacy of a collector for all applicable load combinations, just like for columns. Unlike typical columns, both the axial compression and tension parts of the interaction diagram are generally needed for collectors. In addition to the strength requirements noted above, the shear strength requirements in ACI 7.5.3 for one-way slabs and in ACI 8.5.3 for two-way slabs must be satisfied where a portion of a slab is used as a collector. Where beams are utilized as collectors, the strength requirements for shear or for combined shear and torsion in ACI 9.5.3 and 9.5.4 must also be satisfied.

7-3

Design Guide for Reinforced Concrete Diaphragms

7-4

Design Guide for Reinforced Concrete Diaphragms

Chapter 8 Determining and Detailing the Required Reinforcement 8.1 Overview This chapter presents information on how to determine and detail the reinforcement for diaphragms and collectors in accordance with ACI 318 and ASCE/SEI 7 provisions for buildings assigned to Seismic Design Category (SDC) A through F. The following types of reinforcement are covered: chord reinforcement; diaphragm shear reinforcement; shear transfer reinforcement between the diaphragm and the vertical elements of the lateral force-resisting system (LFRS) and between the diaphragm and collectors; anchorage reinforcement; and collector reinforcement. A summary of the requirements is given in Section 8.10.

8.2 Diaphragm Reinforcement Limits According to ACI 12.6.1 and 18.12.7.1, diaphragms assigned to any SDC must contain at least the shrinkage and temperature reinforcement ratio in ACI Table 24.4.3.2. Additionally, the minimum area of flexural reinforcement, , in ACI 7.6 for one-way slabs and in ACI 8.6 for two-way slabs must be provided in diaphragms that are part of floor or roof construction (ACI 12.6.2). For deformed reinforcing bars with , is determined by the following:

(8.1)

where

is the gross area of the slab and

is the specified yield strength of the reinforcement.

Reinforcement that is required to resist in-plane forces in diaphragms must be in addition to the reinforcement that is required to resist other load effects, such as those from gravity (ACI 12.6.3). Shrinkage and temperature reinforcement, however, is permitted to also resist diaphragm in-plane forces.

8.3 Reinforcement Detailing Requirements Requirements for concrete cover, development lengths, splices, bar spacing, and reinforcement detailing for diaphragms and collectors in buildings assigned to SDC A through C are given in ACI 12.7. Similar requirements are provided in ACI 18.12.7 for buildings assigned to SDC D through F. A summary of these requirements is given in Table 8.1. At critical sections in diaphragms and collectors, tension and compression forces in the reinforcement must be adequately developed on each side of those sections. Tension reinforcement must extend beyond the point at which it is no longer required a distance equal to at least the tension development length, , determined in accordance with ACI 25.4.2 (ACI 12.7.3.3). Table 8.1 Reinforcement Detailing Requirements for Diaphragms and Collectors Seismic Design Category (SDC)

Requirement

A, B, and C

Concrete cover

ACI 20.6.1

Development lengths

ACI 25.4

Splices

ACI 25.5

D, E, and F

Minimum

ACI 25.2

Maximum

ACI 12.7.2.2

ACI 18.12.7.1 ACI 18.12.7.6(a)

One-way slabs

ACI 7.7

ACI 7.7

Two-way slabs

ACI 8.7

ACI 8.7 ACI 18.14.5

Spacing

Reinforcement detailing

ACI 20.6.1 ACI 18.12.7.6(a) ACI 25.4 ACI 18.12.7.3 ACI 25.5 ACI 18.12.7.3 ACI 18.12.7.4 ACI 25.2 ACI 18.12.7.6(a)

8-1

Design Guide for Reinforced Concrete Diaphragms The detailing requirements in ACI 18.12.7.5 are applicable to reinforced concrete collector elements in buildings assigned to SDC D, E, or F. Because of the relatively large forces that must be resisted, beams are usually provided as collector elements in such cases instead of only a portion of the slab. Longitudinal reinforcement in collectors must extend into the vertical elements of the LFRS a length equal to at least the greater of the two lengths given in ACI 12.5.4.3. This requirement is illustrated in ACI Figure R12.5.4.3. Additional information on detailing requirements for collectors is given in Section 8.9 of this publication.

8.4 Chord Reinforcement 8.4.1 Required Area of Chord Reinforcement As discussed in Chapter 6 of this publication, tension and compression chord forces occur along the edges of diaphragms and openings in diaphragms due to in-plane bending moments. Where beam models are used to approximate the internal force distribution within a diaphragm, Equation (6.7) in Section 6.4.2 can be used to determine these forces: (8.2) In this equation, is the maximum bending moment in the diaphragm due to the in-plane forces and dicular distance between the chord forces.

is the perpen-

The maximum tension chord force, , must be resisted by reinforcement that is perpendicular to the direction of the applied in-plane force. At any location within the diaphragm, must be less than or equal to the design tension strength of the chord reinforcement: (8.3) In this equation, for reinforcing bars in tension and , Equation (8.3) can be used to determine :

is the required area of chord reinforcement. For a given

(8.4) For diaphragms with openings, subdiaphragm tension chord forces are determined by Equation (8.2) where is the maximum bending moment in the subdiaphragm and is equal to 95 percent of the depth of the subdiaphragm in the direction of analysis. An approximate method to determine subdiaphragm bending moments is given in Section 6.4.4 of this publication. Once the subdiaphragm tension chord force is calculated, the required area of chord reinforcement along the edges of the opening perpendicular to the direction of analysis can be determined by Equation (8.4). Because wind and seismic forces can act in any direction, chord reinforcement is required along all edges of diaphragms and openings.

8-2

ℓ� ⁄4

Uniformly distributed chord reinforcement developed outside of reinforcement zone

ℓ� ⁄4

ℓ� ⁄4

ℓ�

ℓ�

Primary chord reinforcement is often concentrated near the edges of a diaphragm. In such cases, the moment arm that is used to determine in Equation (8.2) is usually approximated as 95 percent of the diaphragm depth in the direction of analysis. Alternatively, it is permitted to locate chord reinforcement within zones that extend from the tension edge of the diaphragm a distance equal to 25 percent the depth of the diaphragm in the direction of analysis (ACI 12.5.2.3); placing the chord reinforcement within these zones results in an essentially uniform shear flow over the depth of the diaphragm just like the

ℓ� ⁄4

8.4.2 Location of Chord Reinforcement

𝑏𝑏�

𝑏𝑏� Lateral force

Figure 8.1 Location of Chord Reinforcement

Zones for placement of chord reinforcement

Structural framing and other reinforcement not shown for clarity

Design Guide for Reinforced Concrete Diaphragms case where the chord bars are concentrated along the edges of the diaphragm. Where the depth of the diaphragm changes along its span, it is permitted to develop chord reinforcement into adjacent sections even where the reinforcement falls outside the 25 percent depth limit of the adjacent section (see Figure 8.1). According to ACI 12.6.3, chord reinforcement must be provided in addition to other required reinforcement, such as flexural reinforcement. However, as noted in Section 8.2, reinforcement designed to resist shrinkage and temperature is permitted to also resist diaphragm in-plane forces.

Chord reinforcement

Slab top reinforcement

Beam flexural reinforcement

Slab bottom reinforcement

Beam transverse reinforcement

Beam flexural reinforcement

Figure 8.2 Location of Chord Reinforcement in Systems with Perimeter Beams

In roof and floor systems without perimeter beams, the chord reinforcing bars are typically tied to the flexural reinforcement in the slab. The chord bars should be placed below the top flexural bars, above the bottom flexural bars, or both to minimize interference with other reinforcement and to reduce their contribution to the flexural strength of the slab. Where perimeter beams are present, one option is to locate the chord reinforcement within the slab outside of the beam cross-section; this is depicted in Figure 8.2 for the case of a two-way slab system. Alternatively, the top flexural reinforcement in the beam can be used to resist the tension chord force provided there is a sufficient amount to do so; that is, reinforcement in excess of that required for flexure can be used as chord reinforcement. In some cases, the beam reinforcement may not be sufficient and additional reinforcing bars must be provided. Longitudinal reinforcement that is added to a beam that is part of a special moment frame to resist tension chord forces must be included when determining the nominal flexural strength, , of the beam; is needed to (1) check the minimum flexural strength requirements in ACI 18.7.3 for columns in special moment frames and (2) calculate the design shear force in beams that are part of an intermediate moment frame (ACI 18.4.2.3). The additional reinforcement must also be included when determining the probable flexural strength, , of a beam in a special moment frame, which is needed to (1) calculate the design shear force on the beam in accordance with ACI 18.6.4 and (2) check the beam-column joint strength in accordance with ACI 18.8. Similarly, if reinforcement must be added to a slab to resist tension chord forces and that reinforcement is within the effective slab width defined in ACI 6.3.2 of a beam that is part of an intermediate or special moment frame, that reinforcement must be included in the calculation of for the beam to check the requirements of ACI 18.4.2.3 and 18.7.3, respectively. Situations may occur where elements that resist chord forces due to lateral forces in one direction act as collectors for lateral forces in the perpendicular direction. These elements must be designed and detailed for the critical effects due to lateral forces applied in each direction separately or in both orthogonal directions concurrently (see Chapter 5 of this publication for more information on when orthogonal load effects must be considered). For diaphragms with openings, chord reinforcement around openings that resists the subdiaphragm chord forces must extend a sufficient distance into the adjacent diaphragm so that the axial tension force in the reinforcement is developed through shear transfer. Thus, the required development length of these chord reinforcing bars, which is measured from the face of the opening, is determined by dividing the axial tension force in the bars by the shear capacity of the diaphragm, the latter of which being expressed in force per unit length.

8.4.3 Compression Chords For diaphragms in buildings assigned to SDC A, B, or C, there are no requirements that diaphragm chords subjected to compression forces be designed and detailed as columns (ACI R12.5.4.2). However, in cases where diaphragm boundaries (edges of a diaphragm or edges of an opening in a diaphragm) are subjected to relatively large compression forces compared to the axial compression strength of the boundary element, transverse reinforcement, such as hoops, should be used to confine the concrete. Such transverse reinforcement is usually feasible where there are beams along the diaphragm boundaries; without beams, the hoops must be developed in the slab, which is not always possible, especially for relatively thin slabs. In such cases, a thicker slab may be required if beams are not an option. In buildings assigned to SDC D, E, or F, strut-like elements around openings, diaphragm edges, or other discontinuities subjected primarily to axial forces must satisfy the transverse and longitudinal reinforcement requirements for collectors in ACI 18.12.7.5 and 18.12.7.6 (ACI 18.12.3.2).

8-3

Design Guide for Reinforced Concrete Diaphragms Illustrated in Figure 8.3 is a beam adjacent to a diaphragm opening in a building assigned to SDC D, E, or F. The amount of transverse reinforcement in the beam (strut) over the length of the opening must be determined by ACI 18.12.7.5 where the axial compressive stress exceeds (or where the design forces have been amplified by the overstrength factor, ). Additionally, the longitudinal reinforcement in the beam must satisfy the detailing requirements in ACI 18.12.7.6(a) or 18.12.7.6(b). The required transverse reinforcement should be extended past the edges of the opening a length equal to at least the tension development length, , of the longitudinal reinforcement in the beam or 12 in., whichever is greater. Beyond that length, the requirements of ACI 18.12.7.5 and 18.12.7.6 need not be satisfied. Information on how to determine the transverse reinforcement in accordance with these requirements is given in Section 8.9.3 of this publication for beams designated as collector elements.

A A

Diaphragm opening

Plan

Transverse reinforcement in accordance with ACI 18.12.7.5

8.5 Diaphragm Shear Reinforcement As discussed in Section 7.3.3 of this publication, the following requirements must be satisfied for design shear strength of diaphragms in buildings assigned to any SDC (see ACI 12.5.3 and 18.12.9):

Section A-A

Figure 8.3 Reinforcement Details for a Diaphragm Element in Accordance with ACI 18.12.3.2

(8.5)

Required shear strength, , due to in-plane forces is determined using one of the models in Chapter 6 of this publication. For diaphragms modeled as beams with chord reinforcement concentrated near it edges, is uniformly distributed over the depth of the diaphragm (see Figure 6.7). The uniform factored shear flow, , is equal to where is the overall depth of the diaphragm in the direction of analysis (Note: Where openings are present, the length of the opening in the direction of analysis must be subtracted from ). Strength reduction factor, , is equal to 0.60 or 0.75, and information on the correct value to use is given in Section 7.2. The term is the gross area of the diaphragm in the direction of analysis. Where there are openings in a diaphragm, must be based on the overall depth of the diaphragm minus the length of the openings in the direction of analysis. The shear reinforcement ratio, , is equal to the area of the uniformly distributed slab reinforcement that is oriented parallel to the shear force divided by the gross area of the slab perpendicular to that reinforcement. Equation (8.5) can be solved for the required : (8.6) It is evident from Equation (8.6) that shear reinforcement is not required where . Providing a slab thickness that is adequate for serviceability or, where applicable, for two-way shear is usually adequate for in-plane shear strength as well. Where in-plane shear reinforcement is required, it is usually combined with the flexural reinforcement in the slab in the direction of analysis (where two layers of flexural reinforcement are provided in a slab, the combination usually occurs with the bottom reinforcement). Thus, the total area of reinforcement that must be provided is equal to the area of shear reinforcement determined by Equation (8.6) plus the area of flexural reinforcement. Combining the reinforcement in this fashion generally results in simpler detailing and more efficient placement of the reinforcing bars in the field. Shear strength requirements in a diaphragm must be satisfied for in-plane forces applied in two orthogonal directions separately or in two orthogonal directions concurrently, where required (see Chapter 5 of this publication for more information on when orthogonal load effects must be considered). In any case, it is recommended to use the same amount of shear reinforcement in both directions, especially in relatively deep diaphragms where shear strength is more likely to be attained from distributed reinforcement in two directions instead of one (similar to in-plane shear resistance of squat reinforced concrete walls).

8-4

Design Guide for Reinforced Concrete Diaphragms 8.6 Shear Transfer Reinforcement 8.6.1 Overview Shear transfer reinforcement must be provided between diaphragms, collectors, and vertical elements of the LFRS. In addition to the in-plane shear strength requirements in Section 8.5 of this publication, shear transfer requirements must also be satisfied using the shear-friction provisions of ACI 22.9 or by mechanical connectors or dowels (ACI 12.5.3.7). In cast-in-place construction, shear transfer reinforcement is provided mainly at construction joints between the elements. The required amount of reinforcement depends on the overall layout of the elements, the magnitude of the factored shear force at the location of interest, the width of the collector with respect to the width of the vertical element of the LFRS it frames into, and the construction sequence. Methods on how to determine the required shear transfer reinforcement are given in the following sections.

8.6.2 Shear Transfer in a Diaphragm ③ 𝑉𝑉�

ℓ�

𝐿𝐿

Consider the reinforced concrete diaphragm in Figure 8.4 subjected to the in-plane force . The uniform factored shear flow, , in the diaphragm adjacent to Wall 1 is equal to where is the factored shear force in Wall 1 due to and is the overall depth of the diaphragm in the direction of analysis. If the width of the collector element is equal to the thickness of Wall 1, the uniform factored shear flow in this wall, , is equal to where is the length of Wall 1. At Wall 2, collector elements are not required because the wall extends the full depth of the diaphragm. The uniform factored shear flow in the diaphragm, , and in the wall, , are both equal to where is the factored shear force in Wall 2 due to . Note that and can be determined using any of the methods given in Chapter 6 of this publication.

①

𝐵𝐵

𝑣𝑣� � 𝑉𝑉� ⁄𝐿𝐿

𝑣𝑣� � 𝑉𝑉� ⁄𝐿𝐿 Collector element �t����

𝑉𝑉�

②

Structural �all �t����

④ 𝑉𝑉�

Figure 8.4 Shear Transfer in a Diaphragm

Shear transfer reinforcement must be provided between the diaphragm and the walls and the diaphragm and the collector elements. Where reinforcing bars are used, the design strength requirements for shear transfer are determined using the shearfriction provisions in ACI 22.9 (see Section 7.4 of this publication): (8.7) In this equation, the strength reduction factor, , is equal to 0.60 or 0.75 (information on the correct value to use is given in Section 7.2); is the coefficient of friction obtained from ACI Table 22.9.4.2, which depends on the contact surface condition of the concrete; and, is the area of shear-friction reinforcement that crosses the shear plane with the reinforcing bars oriented perpendicular to that plane. Equations to determine the maximum values of across the assumed shear plane are given in ACI Table 22.9.4.4. Equation (8.7) can be used to determine for a factored shear force, : (8.8) Equation (8.8) can be rewritten in terms of the uniform factored shear flow, , at the location of interest where units of force per unit length and has the units of square inches per unit length:

has the

(8.9) For the system depicted in Figure 8.4, the required for shear transfer between the diaphragm and Wall 1 is determined by Equation (8.9) using . Similarly, the required for shear transfer between the diaphragm and the collector elements at this location is determined by Equation (8.9) using . The method in which a building is constructed has a direct impact on the types and amount of shear-friction reinforcement, that needs to be provided between the diaphragm and vertical elements of the LFRS. Information on how to determine this reinforcement for two common construction methods is given on the next page.

,

8-5

Design Guide for Reinforced Concrete Diaphragms 8.6.3 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS – Construction Method A Overview Construction Method A is illustrated in Figure 8.5 for the case where the floor/roof system is a reinforced concrete two-way slab without beams and the LFRS consists of walls. The general construction sequence for this method is as follows:

�all reinforcement �t����

Cold joint

1. C  onstruct the reinforced concrete wall to the underside of the reinforced concrete slab; 2. Construct the reinforced concrete slab;

Slab reinforcement

3. C  onstruct the reinforced concrete wall above the reinforced concrete slab. Cold joint

A similar construction sequence would be employed where beams are part of the floor/roof system, moment frames are used as the LFRS instead of walls, or both. The following discussion is based on the LFRS consisting of shear walls but is equally applicable to the LFRS consisting of moment frames.

Shear Transfer Reinforcement at Face of Wall

Figure 8.5 Construction Method A

Support bar

Dowel bar

� �� �t�p� �

In this construction method, cold joints occur on the top and bottom surfaces of the slab along the length of the wall, and shear transfer reinforcement must be determined at the following locations: (1) the face of the wall, (2) the cold joint at the bottom surface of the slab, and (3) the cold joint at the top surface of the slab. The shear transfer reinforcement perpendicular to the face of the wall and perpendicular to the bottom surface of the slab must transfer the shear forces along the length of the wall from the diaphragm to the wall below. Similarly, the vertical reinforcement in the wall must transfer the shear force in the wall above through the top surface of the slab to the wall below. Shear transfer reinforcement must also be provided along the length of the collectors.

At the face of the wall, shear transfer occurs by shear-friction reinforcement through monolithic concrete. Dowel bars are commonly used to transfer the shear forces in such cases Figure 8.6 Shear Transfer Utilizing Dowel Bars – Construction Method A (see Figure 8.6). In this detail, the dowel bars are not spliced or tied to the top or bottom flexural reinforcement in the slab or to any other reinforcement; as such, a continuous support bar is required at the bend of the dowel bar mainly so that the horizontal leg is not displaced from its intended position during concrete placement. The dowel bar legs must be developed at least a tension development length, , into the wall and slab. The required area of dowel reinforcement, , at the face of the wall can be determined by Equation (8.9) where for concrete placed monolithically (ACI Table 22.9.4.2): (8.10) In this equation, is the modification factor that accounts for reduced mechanical properties of lightweight concrete (ACI Table 19.2.4.2) and is the length of the wall. The total factored shear force, , depends on the width of the collector relative to the thickness of the wall that it frames into. If the width of the collector is equal to the thickness of the wall, then where is the factored shear force in the wall determined from analysis. Where collector elements are wider than the thickness of the wall, a part of the collector axial force is transferred directly into the vertical element at its ends and a part is transferred by shear-friction along the length of the LFRS element [see Figure 6.9(a)]. A reasonable part of the total

8-6

Design Guide for Reinforced Concrete Diaphragms collector tension force that is assumed to be concentric with the vertical element of the LFRS is selected considering design and construction limitations and the remaining part is transferred by shear (see Section 8.9). It is evident from Figure 6.9(a) that the total factored shear force, , along the length of the wall is equal to the factored shear force, , in the wall plus the factored collector axial tension and compression forces transferred by shear and , respectively. Thus, the required area of dowel reinforcement, , at the face of the wall where the collector is wider than the vertical element of the LFRS can be determined by Equation (8.9): (8.11) In lieu of providing distinct dowel bars as described above, it may be possible to satisfy shear-friction requirements at the face of the wall using the reinforcement in the slab that is perpendicular to the face of the vertical elements of the LFRS, which for purposes of discussion here, is designated . Depending on the orientation of a one-way slab or one-way joist system with respect to the face of the wall, could be either the main flexural reinforcement in the slab (which may be one layer or two layers of reinforcing bars depending on the thickness of the slab) or temperature and shrinkage reinforcement. In the case of a two-way slab system, would be the top or bottom layer of flexural reinforcement. For a two-way joist system with a typical slab thickness of 4.5 in., would most likely be temperature and shrinkage reinforcement.

� ��

As noted previously, only temperature and shrinkage reinforcement is permitted to also resist diaphragm in-plane forces; reinforcement designed to resist diaphragm in-plane forces must be in addition to reinforcement designed to resist other load effects, like those from gravity (ACI 12.6.3). However, any excess flexural reinforcement in a slab at the location of interest that is not required for gravity loads under the wind and seismic load combinations can be used to resist inplane forces. For example, consider the bottom flexural reinforcement in a two-way slab that resists the effects from gravity loads only. Because positive bending moments due to gravity load effects are generally nominal at the edges of the slab adjacent to the wall, the tension force in the bottom slab reinforcement is essentially zero and, thus, the provided area of reinforcement at 90-deg this location is significantly more than that required hook A A for flexural strength. Therefore, the bottom flexural reinforcement in the slab can be used as shear-friction reinforcement. This is consistent with ACI 22.9.4.6, which indicates that shear-friction reinforcement can Shear ke� �optional� include all reinforcement that crosses the shear plane provided it is not used to resist direct tension. In this opDowel bar tion, the bottom bars are typically spliced to dowel bars of the same size and spacing that emanate from the wall below so that the shear force is transferred directly from the diaphragm to the wall (see Figure 8.7). Equation (8.9) can be used to determine the area of slab reinforcement, , that is needed to resist at the face of the wall:

Slab bottom reinforcement

(8.12) In cases where the width of the collector is equal to the width of the vertical element of the LFRS, in this equation is equal the factored shear force in the wall, . Where the collector is wider than the width of the vertical element, is equal to . If the area of reinforcement that is originally provided in the slab is less than that required by Equation (8.12), then one option is to provide reinforcement in the slab over the length of the wall that has an area greater than or

Dowel bar �t�p�� Class B lap splice

Section A-A

Figure 8.7 Shear Transfer Utilizing the Bottom Flexural Reinforcement in a Two-way Slab – Construction Method A

8-7

Design Guide for Reinforced Concrete Diaphragms equal to determined by Equation (8.12). This option is viable where temperature and shrinkage reinforcement was originally provided in the slab or where the tensile force in the flexural reinforcing bars is nominal. Another option, which is applicable to any flexural reinforcement irrespective of the tension force in the bars (including, for example, top reinforcement in a two-way slab), is to provide reinforcing bars in the slab over the length of the wall that have an area equal to at least determined by Equation (8.12) plus the area required for flexural strength at the critical section (see ACI 22.9.4.6). Of course, providing distinct dowel bars as described above is always an option. Shear Transfer Reinforcement at Bottom Surface of Slab At the cold joint between the wall and the bottom surface of the slab, shear-friction requirements along the length of the vertical element of the LFRS can be satisfied, in general, using all the reinforcement that crosses perpendicular to this shear plane, which in the case of Construction Method A, consists of dowel bars (either spliced to the reinforcement in the slab or not; see Figures 8.6 and 8.7) and the vertical reinforcement in the wall. Typically, a wall that is part of the LFRS is subjected to axial forces in combination with flexure, which must be resisted by the vertical reinforcement in the wall. These vertical bars must also resist shear forces from lateral load effects, so the tension forces in these bars are usually not nominal. As such, vertical wall reinforcement is usually not used as shear-friction reinforcement at this location. Where distinct dowel bars like those illustrated in Figure 8.6 are provided, the required area of dowel reinforcement, determined by Equation (8.9):

, is

(8.13) where that is to be used in this equation is based on the width of the collector with respect to the width of the vertical element of the LFRS, as noted above. For buildings assigned to SDC A, B, or C, in Equation (8.13) can be taken as or depending on whether condition (b) or condition (c) in ACI Table 22.9.4.2 is satisfied, respectively. According to ACI 18.12.10.1, construction joints in diaphragms in buildings assigned to SDC D, E, or F must satisfy condition (b) in ACI Table 22.9.4.2, that is, the hardened concrete must be clean, free of laitance, and intentionally roughened to a full amplitude of approximately ¼ in., which means in this case. In lieu of roughening the concrete, shear keys can be used (see the shear key option identified in Figure 8.7); in such cases, . Equation (8.9) can also be used to determine where the reinforcement in the slab is used as shear-friction reinforcement (see Figure 8.7 for the case of bottom flexural reinforcement in a two-way slab): (8.14) When considering the shear transfer requirements at the face of the wall and at the bottom surface of the slab, it is evident that determined by Equation (8.13) is greater than that determined by Equation (8.10) or (8.11) because at the bottom surface of the slab is less than that at the face of the wall. Similarly, determined by Equation (8.14) is greater than that determined by Equation (8.12). Thus, shear transfer requirements at both locations are satisfied by providing distinct dowel bars or reinforcement in the slab that has an area equal to at least that determined by Equation (8.13) or (8.14), respectively. Shear Transfer Reinforcement at Top Surface of Slab At the cold joint between the wall and the top surface of the slab, shear-friction requirements along the length of the wall must be satisfied using the reinforcement that crosses this shear plane, which is the vertical reinforcement in the wall, . Assuming the factored shear force in the wall above the slab is equal to , Equation (8.9) can be used to determine the required area of shear-friction reinforcement, : (8.15) Where the tension forces in the vertical reinforcement of the wall are not nominal, the total area of reinforcement crossing this joint, , must be equal to the area of the reinforcement determined by Equation (8.15) plus (see ACI 22.9.4.6): (8.16)

8-8

Design Guide for Reinforced Concrete Diaphragms 8.6.4 Required Shear Transfer Reinforcement Between the Diaphragm and the Vertical Elements of the LFRS – Construction Method B �all reinforcement �ty���

Overview In cases where the wall is constructed ahead of the slabs (for example, where slip forms are used to construct the wall), the cold joint is located at the face of the wall and shear-friction requirements must be satisfied at that location. Construction Method B is illustrated in Figure 8.8 for the case of a two-way slab system.

Form saver assembly – deformed bar with a 90-deg hook

Slab reinforcement

It is common for proprietary form saver systems to be used with this type of construction method mainly Cold joint Form saver assembly – for economic reasons. Form savers, which usually headed deformed bar consist of a coupler that is connected to an anchor, are typically nailed to the inside face of the wall formwork. Anchorage into the wall is achieved by straight deformed reinforcing bars, deformed reinforcing bars with standard hooks at the ends, or headed deformed Figure 8.8 Construction Method B reinforcing bars or shear studs, to name a few. Illustrated in Figure 8.8 are systems with deformed reinforcing bars with 90-degree hooks and headed deformed bars (Note: Usually one type of form saver system is used at any given location; the two systems shown in the figure is for illustration purposes only). When the wall form is stripped, the end of the coupler is exposed on the wall face and slab reinforcing bars are subsequently connected to the coupler. In addition to preventing costly penetrations in the wall forms, these types of systems eliminate dowel bars that protrude from the wall, which could be bent or damaged during construction prior to placement of the slab concrete. Mechanical connectors of this type must comply with the requirements of ACI 12.5.3.7(b), 18.2.7, 18.12.7.4, and 22.9.5. It is important to understand all aspects of the form saver system that is being specified. For example, threaded reinforcing bars may be used in these systems to make the connections at the couplers and it is imperative to know if the full area of the reinforcing bar is available or not at the critical section. It is strongly recommended to contact the manufacturer to obtain pertinent information on a form saver system. Shear Transfer Reinforcement at Face of Wall Shear-friction reinforcement must be provided at the face of the wall. Like in Method A, distinct dowel bars can be used in such cases, as illustrated in Figure 8.9. Note that form savers are required for each dowel bar that is anchored into the wall; the horizontal dowel bar leg is attached later along with the flexural reinforcement in the slab. The required area of dowel reinforcement,

, can be determined by Equation (8.9):

(8.17) In this equation, is the total factored shear force based on the width of the collector with respect to the width of the vertical element of the LFRS, as noted above for Method A, and is determined in the same manner as in Method A based on the SDC. Like in Method A, it may be possible to utilize the reinforcing bars in the slab as shear-friction reinforcement (see Figure 8.10 for the case of bottom flexural reinforcement being utilized in a two-way slab system). Equation (8.9) can be used to determine the area of reinforcement in the slab, , that is needed to resist at the face of the wall:

Form saver assembly – deformed bar with a 90-deg hook

Dowel bar

(8.18) Figure 8.9 Shear Transfer Utilizing Dowel Bars – Construction Method B

8-9

Design Guide for Reinforced Concrete Diaphragms If the area of reinforcement in the slab is not sufficient, it can be increased to that required by Equation (8.18) over the length of the wall. Like in Method A, another option is to provide reinforcing bars in the slab that have an area equal to at least determined by Equation (8.18) plus the area required for flexural strength at the critical section. Providing distinct dowel bars as noted above is always an option.

8.6.5 Required Shear Transfer Reinforcement Between the Diaphragm and the Collector Elements In general, it is likely that reinforcement in the slab alone is sufficient to satisfy shear-friction requirements between the diaphragm and collectors. Regardless of the construction method and the width of the collector relative to the width of the wall, Equation (8.19) can be used to determine the area of reinforcement in the slab, , that is needed to resist along the length of a collector:

Slab bottom reinforcement

Form saver assembly – deformed bar with a 90-deg hook

Figure 8.10 Shear Transfer Utilizing the Bottom Flexural Reinforcement in a Two-way Slab – Construction Method B

(8.19) In this equation, is the overall depth of the diaphragm in the direction of analysis. A value of is used because the collector element is either part of the slab or, in the case where beams are used as collectors, the beams and slab are cast at the same time (that is, there is no cold joint between these two elements). If the area of reinforcement originally provided in the slab is less than that required by Equation (8.19), then the area of slab reinforcement can be increased so that this equation is satisfied. A summary of the equations for shear transfer reinforcement is given in Table 8.2 for both construction methods where the LFRS consists of walls. The equations are the same for moment frames; the only difference is that the longitudinal reinforcement in a column is to be used as the shear transfer reinforcement at the top surface of the slab instead of the vertical reinforcement in a wall.

Table 8.2 Required Shear Transfer Reinforcement Shear Transfer

Diaphragm and Vertical Elements of the LFRS

Diaphragm and Collector Elements

Location

Face of wall and bottom surface of slab(1)

Type of shear transfer reinforcement

Avf Method A

Method B

Distinct dowel bars

Reinforcement in slab(2)

Top surface of slab

Wall vertical reinforcement

Interface between diaphragm and collector

Reinforcement in slab(2)

––––––—

(1) Bottom surface of slab is not applicable in Method B. (2) Where flexural reinforcement is utilized in a slab for shear transfer reinforcement, the tabulated equations for can be used assuming that the tension forces in the flexural reinforcement are nominal. Otherwise, the total area of reinforcement that must be provided is equal to the area of reinforcement determined by the applicable tabulated equation plus the area of flexural reinforcement at the critical section.

8-10

Design Guide for Reinforced Concrete Diaphragms 8.7 Required Reinforcement Due to Eccentricity of Collector Forces

Illustrated in Figure 8.11 is a portion of a diaphragm with a wall that is part of the LFRS located at its edge and a collector that is wider than the thickness of the wall. The internal forces acting on the free-body diagram of the diaphragm adjacent to the wall are indicated in the figure based on some simplifying, conservative assumptions. The eccentric bending moment, , due to these internal forces can be approximated by the following equation: (8.20)

ℓ�

𝑡𝑡

𝑇𝑇� 𝑇𝑇� 𝑉𝑉�

𝑉𝑉�

𝑉𝑉�

𝑀𝑀�

𝐶𝐶� 𝐶𝐶�

𝑏𝑏���

Figure 8.11 Internal Forces in a Diaphragm with a Collector that is Wider than the Vertical Element of the LFRS

In this equation, the eccentricity, , is the distance between the forces and and the centerline of the wall (which is equal to in the case of uniformly distributed reinforcing bars in the slab over the effective slab width, ) and is the shear strength of the diaphragm based on only reinforcing bars (that is, because tension forces are present):

𝐴𝐴��ecc� �ty���

ℓ�

(8.21) The area of the diaphragm, , is equal to the thickness of the slab times the slab length , is the thickness of the wall, and is the reinforcement ratio of the slab reinforcement parallel to .

𝑒𝑒

𝑡𝑡⁄𝑡

For collectors that are wider than the width of the vertical elements of the LFRS that they frame into, an in-plane bending moment is generated in a diaphragm adjacent to the vertical elements due in part to the forces and that act at an eccentricity from the centerline of the vertical elements [see Figure 6.9(a)]. A method to determine the reinforcement that is required to resist this bending moment is given below, which is based on the method given in Reference 10.

Other reinforcement not shown for clarity

The area of tension reinforcement, , that is required to resist can be determined by the following equation: (8.22) where for reinforcing bars in tension. This reinforcement is placed perpendicular to the face of the wall at both ends and must be developed into the slab and into the wall (see Figure 8.12).

Figure 8.12 Required Tension Reinforcement Where the Collector is Wider Than the Vertical Elements of the LFRS

8.8 Anchorage Reinforcement 8.8.1 Overview In addition to the in-plane effects described above in Section 8.6, connections between diaphragms and vertical elements of the LFRS must be designed to resist the effects due to general structural integrity forces and anchorage/connector forces, where applicable. Anchorage reinforcement must be provided to resist these forces, and methods to determine the required area are given in the following sections.

8.8.2 General Structural Integrity Forces Anchorage connections for structural walls must be capable of resisting a strength level horizontal force, , that is equal to the greater of or 5 psf where is the weight of the wall tributary to the reinforcement (ASCE/SEI 1.4.4). Because tension

8-11

Design Guide for Reinforced Concrete Diaphragms forces govern, the required area of anchorage reinforcement for general structural integrity, the following equation:

, can be determined by

(8.23)

8.8.3 Wind Forces In the case of wind, out-of-plane component and cladding (C&C) wind pressures, , on the vertical elements of the LFRS that are located at the perimeter of a building are determined using one of the methods in ASCE/SEI Chapter 30 (see Section 4.4.2 of this publication). Wind pressures acting away from the surface of the vertical element typically govern because such pressures create tension forces that must be resisted by anchorage reinforcement. The required area of anchorage reinforcement for wind forces, , can be determined by the following equation: (8.24) where

is the area that is tributary to the reinforcement and

for reinforcing bars in tension.

For the vertical elements of the LFRS, there are no requirements for simultaneous application of wind pressures on the main wind force-resisting system (MWFRS) and the C&C, so the required area of reinforcement between the diaphragm and the vertical elements of the LFRS is equal to the larger of the applicable shear transfer area of reinforcement determined in Sections 8.6.3 and 8.6.4 and the area of the wind anchorage reinforcement determined by Equation (8.24). Typically, the area of shear transfer reinforcement is determined first based on the in-plane shear due to wind forces and that area of reinforcement is used to subsequently check if the anchorage reinforcement requirements in Equation (8.24) are satisfied.

8.8.4 Seismic Forces Structural Walls The required area of seismic anchorage reinforcement for walls connected to rigid diaphragms in buildings assigned to SDC B and to SDC C, D, E, or F where the building does not have a Type 5 horizontal structural irregularity in ASCE/SEI Table 12.3-1 is determined by the following (ASCE/SEI 12.11.2.1):

(8.25)

In this equation, is the design spectral response acceleration parameter at short periods (ASCE/SEI 11.4.5), is the amplification factor for diaphragm flexibility determined by ASCE/SEI Equation (12.11-2), is the seismic importance factor (ASCE/ SEI Table 1.5-2), and is the weight of the wall tributary to the anchor. Where anchorage is not located at the roof, the first of these equations may be reduced by where is the height of the anchor above the base of the building and is the height of the roof above the base. In such cases, the reduced area of anchorage reinforcement must not be less than the area determined by the second of these equations. Seismic forces in this case are permitted to be applied independently in two orthogonal directions (ASCE/SEI 12.5.2 and 12.5.3). Like in the case of wind, the area of shear transfer reinforcement is determined first using the applicable equations in Sections 8.6.3 and 8.6.4 based on the in-plane shear due to seismic forces. That area of reinforcement is used to subsequently check if the anchorage reinforcement requirements in Equation (8.25) are satisfied. For buildings assigned to SDC C, D, E, or F where the building has a Type 5 horizontal structural irregularity in ASCE/SEI Table 12.3-1, the orthogonal combination procedure in ASCE/SEI 12.5.3.1(a) can be used to determine the in-plane and out-of-plane effects that must be considered to act concurrently on the vertical elements of the LFRS in cases where the Equivalent Lateral Force Procedure of ASCE/SEI 12.8 or Modal Response Spectrum Analysis of ASCE/SEI 12.9.1 are used to determine seismic load effects. In the case of structural walls where the required area of shear-friction reinforcement for in-plane shear due to seismic forces is determined in accordance with ACI 22.9.3, the total area of reinforcement that is required for the connection

8-12

Design Guide for Reinforced Concrete Diaphragms between the diaphragm and the wall is equal to 100 percent of the area of reinforcement required for in-plane shear determined by Sections 8.6.3 or 8.6.4 plus 30% of the area of reinforcement required for out-of-plane forces determined by Equation (8.25). The area of shear-friction reinforcement and the area of tension reinforcement are added together to obtain the total required area of reinforcement in accordance with ACI 22.9.4.6. Nonstructural Components The requirements of ASCE/SEI 13.4.1 must be satisfied for anchorage of nonstructural components, such as cladding, to a diaphragm. The seismic design force in the attachment is determined by ASCE/SEI Equation (13.3-1):



(8.26)

In this equation, is the component amplification factor, obtained from ASCE/SEI Tables 13.5-1 and 13.6-1 for architectural components and mechanical/electrical components, respectively; is the component operating weight; is the component response modification factor obtained from ASCE/SEI Tables 13.5-1 and 13.6-1; is the component importance factor determined in accordance with ASCE/SEI 13.1.3; is the height above the base where the component is attached to the structure; and, average roof height of the structure with respect to the base. Anchorage forces that subject the anchorage reinforcement to tension govern. Therefore, for the case of nonstructural components where is determined by Equation (8.26). As noted previously, the area of shear transfer reinforcement is usually determined first using the applicable equations in Sections 8.6.3 and 8.6.4 based on the in-plane shear due to seismic forces. That area of reinforcement is used to subsequently check if the anchorage reinforcement requirements are satisfied.

8.9 Collector Reinforcement 8.9.1 Overview In general, collectors must be designed as tension members, compression members, or both in accordance with the provisions of ACI 22.4 for members subjected to axial strength or combined flexural and axial strength (ACI 12.5.4.2). The types and amount of reinforcement and the detailing requirements that must be satisfied depend mostly on (1) the type of collector that is utilized (that is, a portion of the slab or a beam), (2) the width of the collector with respect to the width of the vertical element of the LFRS that it frames into, and (3) the SDC that the building is assigned to. Methods to determine the required reinforcement are given in the following sections.

8.9.2 Slabs Slab Width Equal to the Width of the Vertical Elements of the LFRS Where slabs are utilized as collector elements where the width of the collector is equal to the width of the vertical elements of the LFRS that they frame into, the entire factored axial tension and compression forces due to in-plane diaphragm forces are transferred directly into the ends of the vertical elements. The area of longitudinal reinforcement, , that is required to resist the factored axial tension force, , can be determined by the following equation: (8.27) where for reinforcing bars in tension. This reinforcement is provided in addition to the flexural reinforcement in the slab that is required for gravity loads. The size and number of the longitudinal bars in a collector should be selected considering the slab cross-sectional dimensions and the size and spacing of the reinforcing bars in the LFRS. Specifying relatively large longitudinal bars or a relatively large number of smaller bars may cause placement or congestion issues. The collector bars should be placed in the inner portion of

8-13

Design Guide for Reinforced Concrete Diaphragms the slab (below the top layers and/or above the bottom layers of the flexural reinforcement, where applicable) to minimize any possible out-of-plane bending in the slab due to the axial forces in the bars (see Figure 8.13 for the case of a two-way slab system). The applicable detailing requirements in Table 8.1 of this publication must be satisfied based on the SDC of the building. Collector longitudinal reinforcement must extend along the length of the vertical elements of the LFRS in accordance with the provisions in ACI 12.5.4.3. ACI Figure R12.5.4.3 illustrates force transfer from a collector into the columns of a moment frame. Some of the collector bars extend the full length of the moment frame and others are terminated where they are no longer needed based on the magnitude of , which is obtained from the collector force diagram (see, for example, Figure 6.8 of this publication) The bars that are cutoff must be developed at least a tension development length, , past the point they are no longer required. Note that full force transfer from the collector to the vertical elements of the LFRS is ensured by extending all the collector longitudinal bars over the entire length of the vertical elements.

Collector reinforcement, 𝐴𝐴��collector�

Figure 8.13 Longitudinal Reinforcement in a Collector that has the Same Width as the Vertical Element of the LFRS 𝑏𝑏���

For the case of axial compression forces, the factored axial compression force, , must be less than or equal to the design axial compression strength at zero eccentricity, :

Total collector reinforcement, 𝐴𝐴��collector� � 𝐴𝐴���� � 𝐴𝐴����

(8.28) In this equation, for compression-controlled sections (ACI Table 21.2.2), is equal to the thickness of the slab times the width of the vertical member of the LFRS that it frames into, and is the area of longitudinal reinforcement in . The design strength requirements for axial compression rarely govern.

𝐴𝐴����

𝐴𝐴����

For building assigned to SDC D, E, or F, the transverse reinforcement requirements of ACI 18.12.7.5 must be satisfied Figure 8.14 Longitudinal Reinforcement in a Collector that is Wider where the compressive stress in a collector exceeds Than the Vertical Element of the LFRS (in cases where the design forces have been amplified by the overstrength factor, , the limit of is increased to ). It is unlikely that slabs that have the same width as the vertical elements of the LFRS can be used as collectors because of the relatively large seismic forces that must be resisted; slabs that are wider than the vertical elements or beams must be used in such cases instead. Additional information on the determination of this transverse reinforcement for beams that are designated as collectors is given in Section 8.9.3. At splices and anchorage zones, the detailing requirements of either ACI 18.12.7.6(a) or 18.12.7.6(b) need to be satisfied for building assigned to SDC D, E, or F. The purpose of these requirements is to reduce the possibility of longitudinal bar buckling and to provide adequate bar development in these regions. Because ACI 18.12.7.6(b) requires the use of transverse reinforcement, which is difficult to develop and place in a slab, it is likely that the requirements of ACI 18.12.7.6(a) will need to be satisfied instead. Slab Width Greater Than the Width of the Vertical Elements of the LFRS Collectors that are wider than the width of the vertical elements of the LFRS are usually required in buildings where it is not practical or possible (from a design or constructability perspective) to provide collector elements that are concentric with the vertical elements of the LFRS. As noted previously, one option is to assume that a part of the total factored collector axial tension force, , is transferred directly into the vertical element at its ends ( ) and a part is transferred by shear-friction along the length of the LFRS element ( ), as illustrated in Figure 6.9(a) of this publication.

8-14

Design Guide for Reinforced Concrete Diaphragms The total area of longitudinal tension reinforcement, , that is required to resist can be determined by Equation (8.27). The axial force and the corresponding required area of tension reinforcement, , in the width of the vertical elements of the LFRS is selected considering design and construction limitations. Once the size and number of reinforcing bars are chosen based on , the area of reinforcement, , that is required in the effective slab width outside the width of the vertical elements is equal to minus the area of longitudinal reinforcement that is provided in the width of the vertical elements of the LFRS. The reinforcing bars corresponding to are usually uniformly distributed over the effective width of the collector (see Figure 8.14).

Seismic hook: 6𝑑𝑑� � 3" extension

Detail 𝐵𝐵

6𝑑𝑑� extension

��osstie ���� ��3�

Detail 𝐴𝐴

Detail 𝐶𝐶

The design axial compressive strength defined in Equation (8.27) must be checked for and using the appropriate and for each segment.

8.9.3 Beams

Figure 8.15 Examples of Hoops

𝐴𝐴����

𝑏𝑏���

As noted above, the transverse reinforcement requirements of ACI 18.12.7.5 must be satisfied where the compressive stress in a collector exceeds (in cases where the design forces have been amplified by the overstrength factor, , the limit of is increased to ) for buildings assigned to SDC D, E, or F. Also, the detailing requirements of either ACI 18.12.7.6(a) or 18.12.7.6(b) need to be satisfied at splices and anchorage zones. Providing transverse reinforcement is usually not practical or feasible in slabs, so the requirements of ACI 18.12.7.6(a) need to be implemented.

𝐴𝐴����

Beams that are designated as collector elements must be designed for the combined effects from flexure, shear, torsion, axial compression forces, and axial tension forces due 𝑏𝑏��� to gravity and lateral loads. Applicable design and detailing provisions in ACI Chapter 9 must be satisfied along with the detailing requirements in Table 8.1 of this publication. Longi𝐴𝐴�� � 𝑏𝑏��� � 𝑏𝑏��� tudinal reinforcement must be determined for the combined effects due to flexure, torsion, and axial compression and Figure 8.16 Confined Core Dimensions for Beams Where the Requiretension forces, and transverse reinforcement must be determents of ACI 18.12.7.5 Govern mined for combined effects due to shear and torsion, where applicable. Generally, a design strength interaction diagram is constructed, which includes both the axial compression and tension portions, to determine whether the collector is adequate for all the combined factored flexure and axial load effects. The information provided previously for slabs that are wider than the vertical elements of the LFRS is also applicable to beams. As noted in Section 8.9.2, the requirements of ACI 18.12.7.5 must be satisfied for collectors in buildings assigned to SDC D, E, or F where the compressive stress in the collector exceeds (in cases where the design forces have been amplified by the overstrength factor, , the limit of is increased to ). The combined compressive stress on a collector is calculated using the factored combined compression forces and a linearly elastic model based on gross section properties of the collector. The transverse reinforcement required by ACI 18.12.7.5 must be hoops where the compressive stress is larger than the applicable limiting value given above. Hoops are closed ties or continuously wound ties that are made up of one or several reinforcement elements having seismic hooks at both ends. It is permitted for hoops to be made up of two pieces of reinforcement: (1) a stirrup having seismic hooks at both ends and (2) a crosstie as defined in ACI 2.3. Illustrated in Figure 8.15 are examples of hoops that satisfy these requirements. The hoops formed by Details B and C are preferred over those formed by Detail A because they allow the longitudinal bars in the beam to be placed more easily and efficiently.

8-15

Design Guide for Reinforced Concrete Diaphragms

Class B lap splice per ACI 25.5.2 �typ. � Collector element �typ.�

Minimum reinforcement per ACI 24.4

A

𝑠𝑠 � 1�″

A

𝑠𝑠 � 1�″

3𝑑𝑑� 1.5″

Spacing � �

𝑥𝑥�

ℎ Other reinforcement not shown for clarity

Compressive stress*

2.5𝑑𝑑� Clear cover � � 2″

𝑥𝑥�

� 2″

Spacing � �

𝑥𝑥�

3𝑑𝑑� 1.5″

Transverse reinforcement, 𝐴𝐴��

𝑏𝑏�

Section A-A

Transverse reinforcement, 𝑨𝑨𝒔𝒔𝒔𝒔

� 0.2𝑓𝑓��

𝐴𝐴�� � 0.09𝑠𝑠𝑏𝑏� 𝑓𝑓�� ⁄𝑓𝑓��

� 0.15𝑓𝑓��

0.75�𝑓𝑓�� �𝑏𝑏� 𝑠𝑠⁄𝑠𝑠 �� � 𝐴𝐴�� � � 50 𝑏𝑏� 𝑠𝑠⁄𝑠𝑠 ��

Spacing, 𝒔𝒔

�smaller of ℎ and 𝑏𝑏� �⁄3 𝑠𝑠 � �6𝑑𝑑� 𝑠𝑠�

4″ � 𝑠𝑠� � 4 � ��14 � ℎ� �⁄3� � 6″ ℎ� � maximum of 𝑥𝑥� � 14″

𝑠𝑠 determined in accordance with ACI 22.5 � Maximum spacing in ACI Table 9.7.6.2.2

* Where design forces have been amplified by Ω� , limits of 0.2𝑓𝑓�� and 0.15𝑓𝑓�� are increased to 0.5𝑓𝑓�� and 0.4𝑓𝑓�� , respectively.

Figure 8.17 Requirements for Diaphragms and Collectors in Buildings Assigned to SDC D, E, or F

In the case of rectilinear hoops, the required area of transverse reinforcement, ACI Table 18.12.7.5:

, is determined by the following equation in

(8.29) In this equation, is the spacing of the transverse reinforcement and is the cross-sectional dimension of the collector core measured to the outside edges of the transverse reinforcement. The appropriate must be used when calculating ; these are identified by the same numbered subscripts in Figure 8.16. Transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided at sections where the combined factored axial compressive stress is less than (or where the design forces have been amplified by ).

8-16

Design Guide for Reinforced Concrete Diaphragms To avoid triggering the transverse reinforcement requirements outlined above, the collector can be sized, if possible, so that the factored compressive stress is less than : (8.30) Where the design forces have been amplified by .

,

is to be used in the denominator of Equation (8.30) instead of

The spacing and cover requirements in ACI 18.12.7.6(a) or the transverse reinforcement requirements in ACI 18.12.7.6(b) must be satisfied at splices and anchorage zones for collectors in buildings assigned to SDC D, E, or F. The minimum area of transverse reinforcement in ACI 18.12.7.6(b) corresponds to that in ACI 9.6.3.3 at sections in beams where the provisions of ACI 18.12.7.5 do not govern. Requirements for collectors and diaphragms in buildings assigned to SDC D, E, or F are given in Figure 8.17.

8.9.4 Subdiaphragms As discussed in Section 6.4.3 of this publication, collector elements are provided on each side of an opening in the direction of analysis to collect and transfer the uniform shear from the diaphragm on one side of the opening to the diaphragm on the other side via the subdiaphragms above and below the opening. The required area of collector reinforcement parallel to the direction of analysis is determined using Equation (8.27) where is equal to the shear force in the diaphragm at this location. This reinforcement typically extends the full depth of the diaphragm in the direction of analysis. The required chord reinforcement for the subdiaphragms due to lateral forces in one direction (see Section 8.4.1) act as the collector reinforcement along the edges of the opening for lateral forces in the perpendicular direction. The reinforcement along the edges of the opening must be designed and detailed for the critical effects due to lateral forces applied in each direction separately or in both orthogonal directions concurrently (see Chapter 5 of this publication for more information on when orthogonal load effects must be considered).

8.10 Summary of Design and Detailing Requirements A summary of the design and detailing requirements for diaphragms that utilize slabs as collectors where the width of the collector is equal to the width of the vertical elements of the LFRS is given in Figure 8.18 for the case where bottom flexural reinforcement in a two-way slab system is used as shear transfer reinforcement. The types of reinforcement are identified for Methods A and B along with the corresponding equations numbers from this chapter. Figure 8.19 provides similar information for diaphragms that utilize slabs as collectors that are wider than the width of the vertical elements of the LFRS. Design and detailing requirements for diaphragms that utilize beams as collectors are like those in Figures 8.18 and 8.19 for slabs. Unless beams are much wider than the vertical elements of the LFRS that they frame into, the in-plane moment due to the eccentricity of collector forces is usually relatively small; thus, the required tension reinforcement determined by Equation (8.22) is comparatively small, and separate reinforcement (identified by mark number 6 in Figure 8.18) is typically not required. As noted above, the design and detailing requirements in Figure 8.17 must be satisfied for collectors in buildings assigned to SDC D, E, or F.

8-17

Design Guide for Reinforced Concrete Diaphragms ①

②

Slab reinforcement

④ ⑤

Mark 1 2

③

A

A

3 4 5

④

Chord Shear

Type

Shear transfer between diaphragm and vertical elements of LFRS

Reinforcement

Equation No.

Method A a

b c

Face of wall

Bottom surface Top surface

Shear transfer between diaphragm and collectors Collector

8.12

8.4 8.6

Method B

8.14 8.16

8.18 ─

8.19

─

8.27

① Direction of analysis

3c

⑤

Slab reinforcement �typ.�

3a

⑤

Slab reinforcement �typ.�

3a

3b Dowel bar

Method B

Method A Section A-A

Figure 8.18 Design and Detailing Requirements – Slabs That Have the Same Width as the Vertical Elements of the LFRS

8-18

Design Guide for Reinforced Concrete Diaphragms ⑤

②

① Slab reinforcement

④ ⑥

Mark 1 2

③

A

A

3 4 5

④

6

Chord Shear

Type

Shear transfer between diaphragm and vertical elements of LFRS

Reinforcement

Equation No.

Method A a

b c

Face of wall

Bottom surface Top surface

Shear transfer between diaphragm and collectors Collector Tension as a result of in-plane moment due to eccentricity of collector forces

8.12

8.4 8.6

Method B

8.14 8.16

8.18 ─

8.19

─

8.27

8.22

① Direction of analysis

3c

⑤

3a

3b

⑤

Slab reinforcement �typ.�

Dowel bar

3a

Slab reinforcement �typ.�

Method B

Method A

Section A-A

Figure 8.19 Design and Detailing Requirements – Slabs That are Wider Than the Width of the Vertical Elements of the LFRS

8-19

Design Guide for Reinforced Concrete Diaphragms

8-20

Design Guide for Reinforced Concrete Diaphragms

Chapter 9 Design Procedure 9.1 Overview This chapter presents a step-by-step design procedure that can be used to design and detail reinforced concrete diaphragms in buildings assigned to any Seismic Design Category (SDC) based on the information presented in Chapters 2 through 8 of this publication. References are made in the steps to the equations, tables, and figures in these chapters, and, where applicable, to the flowcharts that are given at the end of this chapter.

9.2 Step 1 – Select the Materials The permissible specified compressive strength of the concrete can be selected from Table 2.1 based on the SDC of the building. Similarly, the permissible reinforcing bar type and specified yield strength can be selected from ACI Table 20.2.2.4a based on the SDC.

9.3 Step 2 – Determine the Diaphragm Thickness Determine the diaphragm thickness based on each of the following requirements: 1. Serviceability (Section 3.2) a. One-way slabs (Table 3.1) b. Two-way slabs (Tables 3.2 and 3.3 and Figure 3.1) 3. Out-of-plane shear (Section 3.4) a. One-way shear i. One-way slabs [Equation (3.9)] ii. Two-way slabs [Equation (3.10)] b. Two-way shear (Equations (3.11) and (3.12), and for flat plates, Figure 3.7) 3. Minimum thickness for buildings assigned to SDC D, E, or F (Section 3.6) 4. Fire resistance (Sections 3.5 and 3.6) The flowchart in Figure 9.1 can be used to determine the required thickness based on these requirements. After Step 5 is complete (see Section 9.6), the minimum thickness based on in-plane shear requirements needs to be determined using Equation (3.1) or (3.2). If it is found that in-plane shear requirements govern, the corresponding required slab thickness must be used in the reanalysis of the system and in all subsequent calculations. In the case of one-way slab systems, a practical initial estimate for the slab thickness is that required for serviceability. This is also appropriate for two-way slab systems with beams. In-plane and out-of-plane strength requirements are often satisfied based on that slab thickness. For two-way slab systems without beams, an initial slab thickness that satisfies two-way shear requirements at an edge column bending perpendicular to the edge is usually sufficient for in-plane strength requirements as well.

9.4 Step 3 – Determine the Diaphragm Design Forces The following design forces must be determined, where applicable: 1. In-plane a. b. c. d. e. f. g.

Wind (Section 4.2.1) General structural integrity (Section 4.2.2) Seismic (Section 4.2.3) Soil (Section 4.2.4) Flood and tsunami (Section 4.2.5) Transfer (Section 4.3) Anchorage and connection (Section 4.4)

h. Column bracing (Section 4.5) 2. Out-of-plane (Section 4.6) 3. Collector (Section 4.7) The flowchart in Figure 9.2 can be used to determine the required forces on a diaphragm.

9-1

Design Guide for Reinforced Concrete Diaphragms 9.5 S  tep 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces The in-plane stiffness of a diaphragm can be determined using an analysis model based on any set of reasonable and consistent assumptions. Diaphragms can be classified as rigid, semirigid, or flexible using the information obtained from the analysis. Section 6.2 contains information on how to classify diaphragms. An analysis model in ACI 12.4.2.4 must be selected to determine internal forces in the diaphragm. Information on beam models is given in Section 6.4. The flowchart in Figure 9.3 can be used to determine in-plane reactions in a diaphragm subjected to wind and inertial forces using the corrected equivalent beam model with spring supports in Section 6.4.3.

9.6 Step 5 – Determine the Combined Load Effects Combined load effects are determined using the applicable strength design load combinations in ACI 5.3 and ASCE/SEI 2.3. A summary of the strength design load combinations is given in Table 5.1. The seismic load combinations that are to be used in lieu of those in Table 5.1 where seismic load effects with overstrength are required are given in Section 5.2. Governing load combinations based on the SDC that the building is assigned to and corresponding direction of loading requirements are given in the following sections: 1. SDC A and B – Sections 5.2 and 5.3 2. SDC C – Sections 5.2 and 5.4 3. SDC D, E, and F – Sections 5.2 and 5.5 The flowchart in Figure 9.4 can be used to determine load combinations based on the SDC and whether orthogonal load interaction effects on a structure must be considered or not.

9.7 Step 6 – Determine the Chord Reinforcement The required area of chord reinforcement for in-plane bending moments in a rigid diaphragm based on a beam model can be determined using the information in Section 8.4. The flowchart in Figure 9.5 can be used to determine the required area and location of chord reinforcement.

9.8 Step 7 – Determine the Diaphragm Shear Reinforcement Information on how to calculate the required shear reinforcement in a diaphragm is given in Section 8.5. The flowchart in Figure 9.6 can be used to determine the required area of shear reinforcement.

9.9 Step 8 – Determine the Shear Transfer Reinforcement Section 8.6 contains information on how to determine the following types of shear transfer reinforcement: 1. B  etween the diaphragm and the vertical elements of the LFRS for two construction methods with collector elements that have the same width as and that are wider than the vertical elements of the LFRS that they frame into 2. Between the diaphragm and collector elements The flowchart in Figure 9.7 can be used to determine the required areas of shear transfer reinforcement.

9.10 Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces A method to determine the required area of reinforcement in a diaphragm due eccentricity of collector forces is given in Section 8.7. This is applicable where collector elements are wider than the vertical elements of the LFRS that they frame into. The flowchart in Figure 9.8 can be used to determine the area of reinforcement due to eccentricity of collector forces.

9.11 Step 10 – Determine the Anchorage and Connection Reinforcement Information on how to calculate the required area of anchorage and connection reinforcement due to wind and seismic forces is given in Section 8.8. The area of shear transfer reinforcement determined in Step 8 is usually determined first. It is subsequently compared to the required area of anchorage or connection reinforcement and the larger of the two areas must be used. The flowchart in Figure 9.9 can be used to determine the area of anchorage and connection reinforcement.

9-2

Design Guide for Reinforced Concrete Diaphragms 9.12 Step 11 – Determine the Collector Reinforcement Section 8.9 contains information on how to calculate the required area of reinforcement in slabs and beams that are designated as collector elements. Methods are given for collector elements that have the same width as and that are wider than the vertical elements of the LFRS that they frame into. The flowchart in Figure 9.10 can be used to determine the area of collector reinforcement.

Flowcharts

Given: 𝑓𝑓𝑐𝑐′ , 𝑓𝑓𝑦𝑦 , ℓ, and ℓ𝑛𝑛

No

Is the floor/roof system a one-way slab?

Determine minimum ℎ for two-way slabs using Table 3.2, Table 3.3, and Figure 3.1

Yes

Determine minimum ℎ using Table 3.1 �ℎ � ℎ1 �

�ℎ � ℎ1 �

Determine minimum ℎ based on one-way and two-way out-of-plane shear requirements in Section 3.4 �ℎ � ℎ2 � Determine minimum ℎ based on fire-resistance requirements �ℎ � ℎ3 �

No

Is the building assigned to SDC A, B, or C?

Minimum required ℎ � ma��ℎ1 , ℎ2 , ℎ3 , 2��∗

Yes

Minimum required ℎ � ma��ℎ1 , ℎ2 , ℎ3 �∗

*Minimum slab thickness based on the in-plane shear requirements in Section 3.3 must also be checked

Figure 9.1 Procedure to Determine Minimum Required Diaphragm Thickness

9-3

Design Guide for Reinforced Concrete Diaphragms Determine wind pressures at floor/roof levels using procedures in ASCE/SEI Chapter 27 or 28 [Section 4.2.1] and apply the total wind resultant force at the centroid of the windward face [Fig. 4.2]

No

Is the building assigned to SDC A?

Determine seismic forces 𝐹𝐹𝑥𝑥 over the height of the building using the ELF Procedure in ASCE/SEI 12.� �Sect. 4.2.3��1�

Yes

Determine general structural integrity lateral forces at each level �Sect. 4.2.2��2�

Determine diaphragm seismic forces 𝐹𝐹𝑝𝑝𝑝𝑝 at the roof/floor levels using ASCE/SEI 12.10 �E�. �4.16�� Seismic force is the larger of 𝐹𝐹𝑝𝑝𝑝𝑝 and 𝐹𝐹𝑥𝑥 applied at the CM at each level�3�

Determine static and dynamic soil lateral forces, where applicable �Sect. 4.2.4�

Determine flood and/or tsunami forces, where applicable �Sect. 4.2.5�

A �1� The

other analytical procedures in ASCE/SEI Table 12.6-1 may be used to determine seismic design forces. �2� General structural integrity forces are applicable to buildings assigned to any SDC but generally do not govern for buildings assigned to SDC B through F. �3� Seismic forces determined by ASCE/SEI 12.10.1.1 must be increased by 1.25 for the design of connections of diaphragms to vertical elements of the LFRS and to collectors in buildings assigned to SDC D through F for structures that have any of the irregularities in ASCE/SEI 12.3.3.4. Forces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 need not be increased by 1.25.

Figure 9.2 Procedure to Determine Required Forces on a Diaphragm

9-4

Design Guide for Reinforced Concrete Diaphragms A Determine wind transfer forces, where applicable �Sect. 4.3�

No

Does the structure have horizontal irregularity Type 4 in ASCE/SEI Table 12.3-1?

Determine seismic transfer forces ρF , where applicable �Sect. 4.3�

Yes

Determine seismic transfer forces Ω𝑜𝑜 𝐹𝐹𝐹𝐹 where applicable �Sect. 4.3�

Determine wind and seismic anchorage and connection forces, where applicable �Sect. 4.4�

Determine column bracing forces, where applicable �Sect. 4.5�

Determine out-of-plane forces �Sect. 4.��

Determine collector wind forces, and for buildings assigned to SDC C through F, determine collector seismic forces in accordance with ASCE/SEI 12.10.2.1 �Sect. 4.���4� �4� Seismic

forces determined by ASCE/SEI 12.10.1.1 must be increased by 1.25 for the design of collectors and their connections, including connections to vertical elements of the LFRS, in buildings assigned to SDC D through F for structures that have any of the irregularities in ASCE/SEI 12.3.3.4. Forces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 need not be increased by 1.25.

Figure 9.2 (Continued) Procedure to Determine Required Forces on a Diaphragm

9-5

Design Guide for Reinforced Concrete Diaphragms Determine the reactions in the elements of the LFRS in the direction of analysis using computer software, or where applicable, an approximate method of horizontal force distribution like the one in Sect. 6.3.3 for all applicable in-plane forces on the diaphragm Determine the equivalent in-plane distributed load on the diaphragm based on the reactions in the elements of the LFRS �Sect. 6.4.3�

Construct in-plane shear and moment diagrams using the reactions and equivalent distributed load on the diaphragm

No

Does the diaphragm have one or more relatively large openings?

Determine the maximum tension and compression chord forces from �q. �6.��

Yes

Determine the maximum tension and compression chord forces using the procedure in Sect. 6.4.3

Determine the collector forces based on the width of the collector with respect to the width of the vertical element of the LFRS that it frames into using the information in Sect. 6.4.3 Figure 9.3 Procedure to Determine In-plane Reactions in a Rigid Diaphragm

9-6

Design Guide for Reinforced Concrete Diaphragms

No

No

Is the building assigned to SDC A?

Is the building assigned to SDC B?

Yes

Yes

Use the load combinations in Table 5.1 and ASCE/SEI 2.6.1 �Sect. 5.2��1� – �3�

Use the load combinations in Table 5.1 with seismic load combinations in Sect. 5.3�1� – �3�

No

Is the building assigned to SDC C?

Use the load combinations in Table 5.1 with seismic load combinations in Sect. 5.5�1�,�2�

No

Use the load combinations in Table 5.1 with seismic load combinations in Sect. 5.4�1�,�2�

Does the structure have a Type 5 horizontal structural irregularity?

Seismic forces may be applied independently in two orthogonal directions

Yes

Yes

Seismic forces must be applied in two orthogonal directions simultaneously using one of the procedures in ASCE/SEI 12.5.3.1

�1� Include

the applicable load effects in ACI 5.3.6 – 5.3.10. buildings where extraordinary loads and events must be considered, the capacity and residual capacity load combinations in ASCE/SEI Equations 2.5-1 and 2.5-2, respectively, must be used. �3� Lateral forces may be applied independently in two orthogonal directions. �2� For

Figure 9.4 Procedure to Determine Strength Design Load Combinations

9-7

Design Guide for Reinforced Concrete Diaphragms

Determine the applicable chord forces using the flowcharts in Figs. 9.2, 9.3, and 9.4 Determine the required chord reinforcement in the diaphragm and at the edges of any relatively large openings using �q. ��.4� �ocate the chord reinforcement at ��� the edges of the openings and �2� the edges of the diaphragm or within a distance of h/4 from the edges of the diaphragm��� Chord reinforcement must be provided in addtion to other required reinforcement�2�

No

Does the floor/roof system have perimeter beams?

Tie the chord reinforcing bars to either the top or bottom flexural reinforcement in the slab �similar to Fig. �.2�

Yes

Tie the chord reinforcing bars to either the top or bottom flexural reinforcement in the slab �Fig. �.2� Increase the amount of flexural reinforcement in the beam by the amount required for the chord reinforcement

(1)

The location of the chord reinforcement must correspond to the distance that was used in calculating the chord forces. (2) Shrinkage and temperature reinforcement may be used as chord reinforcement ��CI �2.�.3�.

Figure 9.5 Procedure to Determine Required Chord Reinforcement

9-8

Design Guide for Reinforced Concrete Diaphragms

Determine the maximum shear force Vu using the flowcharts in Figs. 9.2, 9.3, and 9.4

No

Are special moment frames and/or special structural walls used as the SFRS?

Yes

𝜙𝜙 � �.�� No

Is it anticipated that Vn of the diaphragm will be less than the shear corresponding to the development of Mn of the diaphragm?

𝜙𝜙 � 𝜙𝜙1 � �.��

No

Yes

𝜙𝜙 � 𝜙𝜙1 � �.��

Is 𝜙𝜙1 � 𝜙𝜙 � 𝜙𝜙2 where 𝜙𝜙2 � least value of 𝜙𝜙 for shear used for the vertical components of the SFRS?

Use 𝜙𝜙 � 𝜙𝜙1

Yes

Use 𝜙𝜙 � 𝜙𝜙2

𝜌𝜌𝑡𝑡 �

�𝑉𝑉𝑢𝑢 �𝜙𝜙𝐴𝐴𝑐𝑐𝑐𝑐 � � 2��𝑓𝑓𝑐𝑐′ 𝑓𝑓𝑦𝑦

Combine the required shear reinforcement, if any, with the flexural reinforcement in the slab in the direction of analysis Figure 9.6 Procedure to Determine Required Shear Reinforcement

9-9

Design Guide for Reinforced Concrete Diaphragms

Determine shear forces 𝑉𝑉𝑢𝑢 in the diaphragm and axial forces 𝑇𝑇𝑢𝑢 and 𝐶𝐶𝑢𝑢 in the collectors using the flowcharts in Figs. 9.2, 9.3, and 9.4

No

Is the width of the collector � the width of the vertical element of the LFRS that it frames into?

Yes

Select 𝑇𝑇𝑑𝑑 and 𝐶𝐶𝑑𝑑 considering design and construction limitations

𝑉𝑉𝑢𝑢�total� � 𝑉𝑉𝑢𝑢

𝑇𝑇𝑣𝑣 � 𝑇𝑇𝑢𝑢 � 𝑇𝑇𝑑𝑑

𝐶𝐶𝑣𝑣 � 𝐶𝐶𝑢𝑢 � 𝐶𝐶𝑑𝑑

𝑉𝑉𝑢𝑢 �total� � 𝑉𝑉𝑢𝑢 � 𝑇𝑇𝑣𝑣 � 𝐶𝐶𝑣𝑣

Shear transfer between diaphragm and vertical elements of the LFRS A Figure 9.7 Procedure to Determine Required Shear Transfer Reinforcement

9-10

Design Guide for Reinforced Concrete Diaphragms A Determine 𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙𝜙 Sect. �.� �see Fig. �.��

𝐴𝐴𝑠𝑠�dowel� �

𝑉𝑉𝑢𝑢�total� ⁄ℓ𝑤𝑤 𝜙𝜙𝜙𝜙𝑓𝑓𝑦𝑦

𝐴𝐴𝑠𝑠�slab� �

𝑉𝑉𝑢𝑢�total� ⁄ℓ𝑤𝑤 𝜙𝜙𝜙𝜙𝑓𝑓𝑦𝑦

For Construction Method A:

𝐴𝐴ℓ�total� �

𝑉𝑉𝑢𝑢�above� ⁄ℓ𝑤𝑤 � 𝐴𝐴ℓ 𝜙𝜙𝜙𝜙𝑓𝑓𝑦𝑦

Shear transfer between diaphragm and collector elements

𝐴𝐴𝑠𝑠�slab� �

𝑉𝑉𝑢𝑢 ⁄𝐿𝐿 1.4𝜙𝜙𝜙𝜙𝑓𝑓𝑦𝑦

Figure 9.7 (Continued) Procedure to Determine Required Shear Transfer Reinforcement

9-11

Design Guide for Reinforced Concrete Diaphragms

Determine shear forces 𝑉𝑉𝑢𝑢 in the diaphragm and axial forces 𝑇𝑇𝑢𝑢 and 𝐶𝐶𝑢𝑢 in the collectors using the flowcharts in Figs. 9.2, 9.3, and 9.4

Select 𝑇𝑇𝑑𝑑 and 𝐶𝐶𝑑𝑑 considering design and construction limitations 𝑇𝑇𝑣𝑣 � 𝑇𝑇𝑢𝑢 � 𝑇𝑇𝑑𝑑

𝐶𝐶𝑣𝑣 � 𝐶𝐶𝑢𝑢 � 𝐶𝐶𝑑𝑑

No

Is the number of sides that slab exists adjacent to the collector � 2�

𝑏𝑏𝑒𝑒𝑓𝑓𝑓𝑓 � 𝑡𝑡 � �ℓ𝑤𝑤 ⁄2�

Yes

𝑏𝑏𝑒𝑒𝑓𝑓𝑓𝑓 � 𝑡𝑡 � 2�ℓ𝑤𝑤 ⁄2�

𝑒𝑒 � 𝑏𝑏𝑒𝑒𝑓𝑓𝑓𝑓 ⁄2

𝑉𝑉𝑠𝑠 � 𝐴𝐴𝑐𝑐𝑐𝑐 𝜌𝜌𝑡𝑡 𝑓𝑓𝑦𝑦

𝑀𝑀𝑒𝑒 � �𝑇𝑇𝑣𝑣 � 𝐶𝐶𝑣𝑣 �𝑒𝑒 � 𝑉𝑉𝑠𝑠 ℓ𝑤𝑤

𝐴𝐴𝑠𝑠�ecc� �

𝑀𝑀𝑒𝑒 ⁄0.95ℓ𝑤𝑤 0.9𝑓𝑓𝑦𝑦

Figure 9.8 Procedure to Determine Required Reinforcement Due to Eccentricity of Collector Forces

9-12

Design Guide for Reinforced Concrete Diaphragms

Determine out-of-plane C&C wind pressure 𝑝𝑝𝑤𝑤 on the vertical elements of the LFRS located at the perimeter of the building �Sect. 4.4.2� Determine area tributary to the reinforcement 𝐴𝐴𝑡𝑡�wind�

𝐴𝐴𝑠𝑠�wind� �

Is the building assigned to SDC A?

No

No

𝑝𝑝𝑤𝑤 𝐴𝐴𝑡𝑡�wind� 0.9𝑓𝑓𝑦𝑦

Is the member a structural wall?

Yes

Yes

0.2𝑊𝑊𝑝𝑝 𝐹𝐹𝑝𝑝 � �reater of � 5 psf

0.4𝑆𝑆𝐷𝐷𝐷𝐷 𝑘𝑘𝑎𝑎 𝐼𝐼𝑒𝑒 𝑊𝑊𝑝𝑝 ⎧ ⎪ 0.9𝑓𝑓𝑦𝑦 �𝐴𝐴𝑠𝑠�seismic� � � �reater of ⎨ 0.2𝐼𝐼𝑒𝑒 𝑊𝑊𝑝𝑝 ⎪ ⎩ 0.9𝑓𝑓𝑦𝑦 ∗

𝐴𝐴𝑠𝑠�seismic� �

𝐴𝐴𝑠𝑠�integrity� �

𝐹𝐹𝑝𝑝 0.9𝑓𝑓𝑦𝑦

𝐹𝐹𝑝𝑝 0.9𝑓𝑓𝑦𝑦

where 𝐹𝐹𝑝𝑝 is determined by Eq. �8.2��

Provide the larger of the areas required for anchorage or connection forces and for shear transfer determined by Fig. 9.8 *For buildings assigned to SDC C, D, E, or F where the building has a Type 5 horizontal structural irregularity, orthogonal seismic loads must be applied concurrently on the diaphragm.

Figure 9.9 Procedure to Determine Required Anchorage and Connection Reinforcement

9-13

Design Guide for Reinforced Concrete Diaphragms

Determine shear forces 𝑉𝑉𝑢𝑢 in the diaphragm and axial forces 𝑇𝑇𝑢𝑢 and 𝐶𝐶𝑢𝑢 in the collectors using the flowcharts in Figs. 9.2, 9.3, and 9.4

No

Yes

Is the slab designated as the collector?

A Is the width of the slab � the width of the vertical element of the LFRS that it frames into?

No

𝐴𝐴𝑠𝑠�collector� �

𝑇𝑇𝑢𝑢 0.9𝑓𝑓𝑦𝑦

Yes

Select 𝑇𝑇𝑑𝑑 and 𝐶𝐶𝑑𝑑 considering design and construction limitations

𝑇𝑇𝑑𝑑 0.9𝑓𝑓𝑦𝑦

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�collector� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�collector� �

𝐴𝐴𝑠𝑠�𝑑𝑑� �

Is 𝜙𝜙𝜙𝜙𝑜𝑜 � 𝐶𝐶𝑢𝑢 ?

𝐴𝐴𝑠𝑠�collector� �

No

Yes

Increase material properties and/or collector size so that 𝜙𝜙𝜙𝜙𝑜𝑜 � 𝐶𝐶𝑢𝑢

Is the building assigned to SDC D, E, or F?

𝑇𝑇𝑢𝑢 0.9𝑓𝑓𝑦𝑦

𝐴𝐴𝑠𝑠�𝑣𝑣� � 𝐴𝐴𝑠𝑠�collector� � 𝐴𝐴𝑠𝑠�𝑑𝑑�

Yes

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�𝑑𝑑� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�𝑑𝑑 � �

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�𝑣𝑣� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�𝑣𝑣� �

No

Satisfy the detailing requirements of ACI 12.7

No

Is 𝑓𝑓𝑐𝑐 �

𝐶𝐶𝑢𝑢 or 𝐶𝐶𝑑𝑑 � �0.2𝑓𝑓𝑐𝑐′ �∗ ? 𝐴𝐴𝑔𝑔

Satisfy the detailing requirements of ACI 12.7 and either ACI 18.12.7.��a� or 18.12.7.��b� at splices and anchorage zones

Yes

Increase the thickness of the slab so that the transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided

*Replace 0.2𝑓𝑓𝑐𝑐′ with 0.5𝑓𝑓𝑐𝑐′ where design forces have been amplified by Ω𝑜𝑜 .

Figure 9.10 Procedure to Determine Required Collector Reinforcement

9-14

Design Guide for Reinforced Concrete Diaphragms A Determine 𝐴𝐴𝑠𝑠�collector� to satisfy combined flexure and torsion �where applicable� due to gravity loads and axial compression and tension forces due to lateral loads Is the width of the beam � the width of the vertical element of the LFRS that it frames into?

No

Select 𝑇𝑇𝑑𝑑 and 𝐶𝐶𝑑𝑑 considering design and construction limitations and determine 𝑇𝑇𝑑𝑑 𝐴𝐴𝑠𝑠�𝑑𝑑� � 0.9𝑓𝑓𝑦𝑦

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�collector� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�collector� �

No

Is 𝜙𝜙𝜙𝜙𝑜𝑜 � 𝐶𝐶𝑢𝑢 ?

Yes

𝐴𝐴𝑠𝑠�𝑣𝑣� � 𝐴𝐴𝑠𝑠�collector� � 𝐴𝐴𝑠𝑠�𝑑𝑑�

Increase material properties and/or collector size so that 𝜙𝜙𝜙𝜙𝑜𝑜 � 𝐶𝐶𝑢𝑢

Is the building assigned to SDC D, E, or F?

Yes

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�𝑑𝑑� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�𝑑𝑑� �

𝜙𝜙𝜙𝜙𝑜𝑜 � 0.�5�0.85𝑓𝑓𝑐𝑐′ �𝐴𝐴𝑔𝑔 � 𝐴𝐴𝑠𝑠�𝑣𝑣� � � 𝑓𝑓𝑦𝑦 𝐴𝐴𝑠𝑠�𝑣𝑣� �

Yes

No

Satisfy the detailing requirements of ACI 12.7

No

Is 𝑓𝑓𝑐𝑐 �

𝐶𝐶𝑢𝑢 or 𝐶𝐶𝑑𝑑 � �0.2𝑓𝑓𝑐𝑐′ �∗ ? 𝐴𝐴𝑔𝑔

Satisfy the detailing requirements of ACI 12.7 and either ACI 18.12.7.��a� or 18.12.7.��b� at splices and anchorage zones

Yes

Provide transverse reinforcement in accordance with ACI 18.12.7.5 �Fig. 8.17�**

*Replace 0.2𝑓𝑓𝑐𝑐′ with 0.5𝑓𝑓𝑐𝑐′ where design forces have been amplified by Ω𝑜𝑜 **Otherwise, increase the area of the beam so that the transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided

Figure 9.10 (Continued) Procedure to Determine Required Collector Reinforcement

9-15

Design Guide for Reinforced Concrete Diaphragms

9-16

Design Guide for Reinforced Concrete Diaphragms

Chapter 10 Examples 10.1 Overview This chapter contains five numerical examples that illustrate the design and detailing requirements presented in Chapters 2 through 8. The examples are organized in the following sections with respect to Seismic Design Category (SDC) for various types of lateral force-resisting systems (LFRS) and references are made to the flowcharts in Chapter 9 where applicable.

10.2 Example 10.1 – One-story Retail Building (SDC A) Design and detail the roof diaphragm for the one-story retail building depicted in Figure 10.1 given the design data below. The height of the building is 15 ft-0 in. and the roof is essentially flat except for ½ in. per foot slopes for drainage. Assume that all the walls contain similar distributions of window and door openings and that these openings have nominal impact on in-plane stiffness. 1A A

15� -0″

E

2A

17� -8″

3A

4A

17� -8″

14� -2″

⑤

A

②

③

A

C

13� -6″

④

A

14� -2″

N

16� � 16� column ������

①

B

56� -0″

13� -6″

D

8� ��ll ������

A

⑥

A

51� -0″

Figure 10.1 Plan of one-story retail building in Example 10.1.

Design Data • Site information Latitude = 42.05o, Longitude = –88.06o Exposure Category: Exposure B in all directions Soil classification: Site Class C • Loads Roof live load = 20 psf Roof superimposed dead load = 10 psf

10-1

Design Guide for Reinforced Concrete Diaphragms Snow loads in accordance with ASCE/SEI Chapter 7 Rain loads in accordance with ASCE/SEI Chapter 8 Wind loads in accordance with ASCE/SEI Chapters 26 and 27 Seismic loads in accordance with ASCE/SEI Chapters 11, 12, 20, and 22 • Lateral force-resisting system: Bearing wall system in both directions

Solution • Step 1 – Select the Materials Assume normalweight concrete with and Grade 60 reinforcement. The suitability of these materials will be checked once the SDC is determined (see Part 4 of Step 2 below). • Step 2 – Determine the Diaphragm Thickness The flowchart in Figure 9.1 is used to determine the minimum diaphragm thickness for this two-way slab system without edge beams. 1.

Determine minimum thickness

based on serviceability requirements.

Maximum clear span From Table 3.2 of this publication,

for an edge panel.

Therefore, 2. Determine minimum thickness

based on two-way shear requirements.

The interior columns are not part of the LFRS and unbalanced moments due to gravity loads are nominal in this case because the adjoining spans are either equal or are within 5 percent of each other. Thus, the minimum slab thickness is determined based on direct shear forces due to gravity loads only. (a) Determine gravity loads. i.

Dead load, Assuming a 7.0-in.-thick slab, the total dead load is as follows:

ii.

Roof live load, From the design data, roof live load

iii. Snow load, Snow load,

, is defined as either the flat roof snow load,

Because the roof is essentially flat,

, or the sloped roof snow load,

.

is applicable, which is determined by ASCE/SEI Equation (7.3-1):

Using surface roughness category B based on the exposure category given in the design data and assuming partial exposure for the roof, exposure factor from ASCE/SEI Table 7.3-1. Assuming the thermal condition is not specifically indicated in ASCE/SEI Table 7.3-2, thermal factor

.

For the retail occupancy, the building is assigned to Risk Category II in accordance with ASCE/SEI Table 1.5-1. According to ASCE/SEI Table 1.5-2, snow importance factor for Risk Category II buildings. Ground snow load tude of the site. Thus,

from ASCE/SEI Figure 7.2-1 or Reference 5 for the given latitude and longi.

Check minimum snow load, , for low-slope roofs (the roof in this example is low-slope because the slope is equal to 2.39 degrees, which is less than 15 degrees; see ASCE/SEI 7.3.4): Where

10-2

Design Guide for Reinforced Concrete Diaphragms Use

.

iv. Rain load, Design rain load,

, is determined by ASCE/SEI Equation (8.3-1):

Assume that the tributary area of each of the roof drains,

, is equal to 790 sq ft.

Also assume that closed scuppers are used as the secondary drainage system. The closed scuppers are 6.0 in. wide by 4.0 in. high . It has been determined that the vertical distance from the primary roof drain to the inlet of a scupper, equal to 6.0 in. (static head distance).

, is

The 15-minute precipitation intensity, , is equal to 6.37 in./h from Reference 5 or from Reference 12 for the given latitude and longitude of the site. Required flow rate,

, is determined by ASCE/SEI Equation (C8.3-1):

The hydraulic head, (Reference 13):

, can be determined by the following equation for closed scuppers where

Thus, for

and

: (initial assumption is correct)

Alternatively, can be obtained by interpolating the values in ASCE/SEI Table C8.3-3 for a 6.0-in. wide by 4.0in. high closed scupper with a flow rate ; linear interpolation results in Therefore, Note: The 2018 IBC requires the use of a 60-minute precipitation intensity for rain load calculations for both the primary and secondary drainage systems, which, for the given latitude and longitude of the site, is equal to 3.07 in./h from Reference 5 or Reference 12. Based on that intensity, , , and . The rain load determined in accordance with ASCE/SEI 7 is used throughout the remainder of this example. (b) Determine combined gravity load effects. The combined gravity load effects are determined using the load combinations in Table 5.1 of this publication. The applicable gravity load combinations for this example are given in Table 10.1. Maximum load effects are obtained using the design rain load, , because the factored rain load is greater than the factored snow load and the factored roof live load. Table 10.1 Strength Design Load Combinations for Gravity Loads in Example 10.1 ACI Equation Number

ASCE/SEI 7 Load Combination

5.3.1a 5.3.1c

1 3

Load Combination (psf)

(c) Check two-way shear requirements at an interior column assuming a 7.0-in. slab thickness. Factored shear force,

, at the critical section can be determined by the following equation (see Figure 3.4):

10-3

Design Guide for Reinforced Concrete Diaphragms The lengths of the critical section

and

are equal to the following for an interior column (see Table 3.4):

Columns B3 and D3 have the largest tributary areas, and the corresponding factored two-way shear force, equal to the following:

, is

Design two-way shear strength is the smallest value determined by the equations in ACI Table 22.6.5.2. Equation (a) in that table governs in the case of a square, interior column:

Two-way shear requirements are satisfied utilizing a 7.0-in.-thick slab Note: It can be determined that the maximum factored two-way shear stress due to unbalanced gravity moments at an interior column is approximately 4 psi; as expected, this stress has a minor impact on the overall shear strength requirements. 3. Determine minimum thickness

based on fire resistance requirements.

Assuming that the required fire-resistance rating of the roof slab is 2 hours for this occupancy, the minimum thickness from IBC Table 722.2.2.1 based on a concrete mix with siliceous aggregate. 4. Check if the minimum slab thickness requirements for SDC D, E, and F must be satisfied. The SDC is determined in accordance with IBC 1613.2.5 or ASCE/SEI 11.6 (see Step 3 in Section 4.2.3 of this publication). For the given latitude and longitude of the site, the earthquake spectral response acceleration parameters at short periods and at 1-second periods are and , respectively, from Reference 4 or Reference 5. For Site Class C, and from ASCE/SEI Tables 11.4-1 and 11.4-2, respectively. The earthquake spectral acceleration parameters at short periods and at 1-second periods, adjusted for site class effects, are determined by ASCE/ SEI Equations (11.4-1) and (11.4-2), respectively:

Design earthquake spectral response acceleration parameters at short periods and at 1-second periods are determined by ASCE/SEI Equations (11.4-3) and (11.4-4), respectively:

From ASCE/SEI Table 11.6-1, the SDC is A for 11.6-2, the SDC is A for

and Risk Category II. Similarly, from ASCE/SEI Table and Risk Category II.

Therefore, this building is assigned to SDC A, which means the minimum slab thickness requirements in ACI 18.12.6.1 for buildings assigned to SDC D, E, or F need not be satisfied. Also, the materials selected in Step 1 are permitted to be used for buildings assigned to SDC A where ordinary reinforced concrete shear walls are used. Use a slab thickness

10-4

Design Guide for Reinforced Concrete Diaphragms • Step 3 – Determine the Diaphragm Design Forces 1.

Determine the in-plane forces. The flowchart in Figure 9.2 is used to determine the in-plane diaphragm forces. Wind and general structural integrity lateral forces are applicable for this building that is assigned to SDC A. (a) Determine the wind forces. The wind load provisions in Part 1 of ASCE/SEI Chapter 27 are used to determine the wind forces for this building, which meets the conditions and limitations set forth in that part (see Table 4.1 of this publication). Design wind pressures, , on the windward and leeward walls are determined at the mean roof height using ASCE/SEI Equation (27.3-1). It is assumed that the following pressures are uniformly distributed over the tributary height of the roof level: Windward pressure: Leeward pressure: The steps in Section 4.2.1 of this publication are used to determine the design wind pressures in the north-south and east-west directions. Even though the building meets the conditions in Part 2 of ASCE/SEI Chapter 27 and in Parts 1 and 2 of ASCE/SEI Chapter 28, it is not recommended to use these provisions because L-, T-, and U-shaped buildings are considered to be outside the scope of those methods. i.

Determine the Risk Category of the building. It was determined in Part 2(a)iii of Step 2 above that the building is assigned to Risk Category II based on its occupancy.

ii.

Determine the basic wind speed,

.

For the given latitude and longitude of the site, iii. Determine total wind pressures,

from ASCE/SEI Figure 26.5-1B or Reference 5.

, in the north-south and east-west directions.

• For the main wind force resisting system (MWFRS) of a building structure, wind directionality factor from ASCE/SEI Table 26.6-1. • The exposure category is given as B in the design data. • A  ssuming the building is not sited on the upper half of an isolated hill, ridge, or escarpment, the topographic factor, , can be taken as 1.0 in accordance with ASCE/SEI 26.8. • A  ccording to Note 3 in ASCE/SEI Table 26.9-1, the ground elevation factor, elevations. • The gust-effect factor,

, can be taken as 1.0 for all

, is determined in accordance with ASCE/SEI 26.11.

The natural frequency of the building,

, is needed to determine

.

According to the definition in ASCE/SEI 26.2, this building is a low-rise building. Low-rise buildings are permitted to be considered rigid (that is, ; see ASCE/SEI 26.11.2). Therefore, can be taken as 0.85 in both directions (ASCE/SEI 26.11). • It is assumed that this building can be classified as enclosed according to the definition in ASCE/SEI 26.2 (see Table 4.5 of this publication). • For an enclosed building, the internal pressure coefficient,

(see ASCE/SEI Table 26.13.

• T  he velocity pressure exposure coefficient, , at the mean roof height of the building, which is 15 ft, is equal to 0.57 for Exposure B from ASCE/SEI Table 26.10-1. • T  he velocity pressure, (26.10-1):

, at the mean roof height of the building is determined by ASCE/SEI Equation

• External pressure coefficients,

(see ASCE/SEI Figure 27.3-1).

Because the building is not symmetrical in plan, all 4 wind directions normal to the walls must be consid-

10-5

Design Guide for Reinforced Concrete Diaphragms ered. However, the overall plan dimensions are used to determine external pressure coefficients on the walls, so one set of windward and leeward pressure coefficients is applicable in each direction. Windward wall:

(north-south and east-west wind)

Leeward wall: North-south wind: For

,

from linear interpolation

East-west wind: For

,

Therefore, the total wind pressures are equal to the following: North-south windward pressure:

North-south leeward pressure:

Total north-south wind pressure: < minimum wind pressure in ASCE/SEI 27.1.5 = 16.0 psf, use 16.0 psf East-west windward pressure:

East-west leeward pressure:

Total east-west wind pressure: < minimum wind pressure in ASCE/SEI 27.1.5 = 16.0 psf, use 16.0 psf iv. Determine the wind forces on the roof diaphragm in both directions. Wind forces are determined by multiplying the total wind pressure at the roof level by the corresponding tributary area (see Figure 4.1 of this publication): North-south wind: East-west wind: (b) Determine the general structural integrity lateral forces. Because the building is assigned to SDC A, the following general structural integrity lateral forces, 1.4.2 must be considered; seismic forces need not be calculated (ASCE/SEI 11.1):

, in ASCE/SEI

The portion of the dead load of the structure, , tributary to the roof is determined assuming that there are no openings in the 8.0-in. reinforced concrete walls: Weight of roof slab Weight of walls Weight of columns

10-6

Design Guide for Reinforced Concrete Diaphragms Therefore,

The in-plane wind forces are greater than the general structural integrity lateral forces. Because both types of forces have a load factor equal to 1.0 (see the applicable load combinations in ASCE/SEI 2.3.1 and 2.6.1), the diaphragm is designed for wind forces. 2.

Determine the out-of-plane forces. The out-of-plane forces on this roof diaphragm are determined in Part 2(a) of Step 2 above.

• S  tep 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces 1.

Determine the diaphragm classification. The information in Section 6.2 of this publication is used to determine the classification of this diaphragm. In the northsouth direction, the maximum span-to-depth ratio is equal to . In the east-west direction, the maximum span-to-depth ratio is equal to . Because the overall span-to-depth ratios of the diaphragm are less than 2, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to wind forces in both directions of analysis (ASCE/SEI 26.2).

2.

Select the diaphragm model. The corrected equivalent beam model with spring supports is selected for this diaphragm. As noted in Section 6.4.3 of this publication, this model is best suited for buildings with rigid diaphragms and lateral force-resisting systems with elements that have different stiffnesses, which is the case for the building in this example.

3.

Determine the diaphragm internal forces. The flowchart in Figure 9.3 is used to determine the internal forces in the diaphragm. (a) Determine the location of the CR. The location of the CR is determined using the information in Section 6.3.3 of this publication. In lieu of a more exact analysis, the in-plane stiffnesses of the walls are determined using the approximate method in Section 6.3.3 of this publication for this one-story building. As noted in the problem statement, it is assumed that typical door and window openings do not have a significant impact on the stiffness of a wall, so wall stiffnesses are calculated using the equations in Table 6.1 assuming that the walls do not have any openings. Wall stiffness calculations are given in Table 10.2 assuming that the walls are fixed at the top and pinned at the bottom and are subjected to unit horizontal forces. The modulus of elasticity of the concrete, , is carried as a constant through the calculations because it is the same for all walls.

Table 10.2 Stiffness Calculations for the Walls in Example 10.1 Wall 1 2 3 4 5 6

(in.4)

(in.2)

26,193,420 26,193,420 202,309,632 4,430,064 53,747,712 152,813,952

2,720 2,720 5,376 1,504 3,456 4,896

0.074 0.074 0.010 0.439 0.036 0.013

0.199 0.199 0.100 0.359 0.156 0.110

13.51 13.51 100.00 2.28 27.78 76.92

5.03 5.03 10.00 2.79 6.41 9.09

3.67 3.67 9.09 1.25 5.20 8.13

The location of the CR is determined by Equations (6.1) and (6.2) of this publication where the origin is taken at the intersection of column lines A and 1:

10-7

Design Guide for Reinforced Concrete Diaphragms

(b) Determine the wind forces in each of the walls. and

The north-south and east-west in-plane wind forces are applied at from column lines 1 and A, respectively.

The wind forces in each of the walls are determined using Equations (6.5) and (6.6) where and . Force calculations for the walls for wind in the south and west directions are given in Tables 10.3 and 10.4, respectively, where it is assumed that the applied in-plane wind forces of 6.10 kips and 6.70 kips are negative in those directions. The forces in these tables are in the directions that resist the applied in-plane forces and torsional moments where positive forces act in the north and east directions. Sample calculations for Wall 2 for wind in the south direction are given below. Table 10.3 Wind Forces in the Walls for Wind in the South Direction Term 1 Wall

(ft)

(ft)

0.0 15.0 50.3 — — —

— — — 27.7 55.3 0.0

3.67 3.67 9.09 — — —

— — — 1.25 5.20 8.13

S

16.43

14.58

(ft)

(ft)

31.2 16.2 19.1 — — —

— — — 5.6 33.2 22.1

(kips) (kips)

1 2 3 4 5 6 *

For walls, 4, 5, and 6, replace

with

Term 2

3,569 963 3,324 — — —

— — — 39 5,738 3,971

1.36 1.36 3.38 — — —

(kips) 0.24 0.12 –0.36 0.02 0.36 –0.38

1.60 1.48 3.02 0.02 0.36 –0.38

7,856**

9,748**

6.10

0.00

6.10

in this equation.

**

Table 10.4 Wind Forces in the Walls for Wind in the West Direction

Wall

1 2 3 4 5 6 *

(ft)

(ft)

(ft)

0.0 — 15.0 — 50.3 — — 27.7 — 55.3 — 0.0

3.67 3.67 9.09 — — —

— — — 1.25 5.20 8.13

S

16.43

14.58

For walls, 1, 2, and 3, replace

**

10-8

with

Term 1

Term 2

kips

kips

(ft)

31.2 16.2 19.1 — — —

— — — 5.6 33.2 22.1

(kips)

3,569 963 3,324 — — —

— — — 39 5,738 3,971

— — — 0.58 2.39 3.73

(0.24) 0.13 –0.37 0.02 0.36 –0.38

0.24 0.13 –0.37 0.60 2.75 3.35

7,856**

9,748**

6.70

0.00

6.70

in this equation.

Design Guide for Reinforced Concrete Diaphragms For Wall 2 with wind in the south direction:

Term 1 force: (the resisting force in this wall due to direct shear force acts in the north direction, which is positive) Term 2 force: (the resisting force in this wall due to torsional moment acts in the north direction, which is positive) Total force, Depicted in Figures 10.2 and 10.3 are the Term 1 and Term 2 forces in each of the walls for wind in the south and west directions, respectively. For wind forces in the north and east directions, the forces in the walls are in the opposite directions to those shown in the figure. (c) D  etermine the equivalent in-plane distributed loads on the diaphragm and construct the corresponding shear and moment diagrams. The equivalent in-plane distributed loads for wind in the south and west directions are determined using the information in Section 6.4.3 of this publication. The distributed loads are trapezoidal, which accounts for the eccentricities between the points of wind load application and the CR (see Figures 10.4 and 10.5 for wind in the south and west directions, respectively). 6.1 kips

25� -2″

⑤

1.36 kips 0.12 kips

0.36 kips

1.36 kips 0.24 kips

𝑦𝑦�� � 22.1�

④

0.02 kips

3.38 kips 0.36 kips

②

𝑒𝑒� � 6.0�

CR

③

N

Term 1 force

①

Term 2 force

0.38 kips 𝑥𝑥�� � 31.2�

⑥

Figure 10.2 Force allocation to the walls for wind in the south direction.

10-9

Design Guide for Reinforced Concrete Diaphragms

2.39 kips 0.36 kips

0.58 kips

0.02 kips

③

CR

6.7 kips

27� -8″

0.24 kips

𝑦𝑦�� � 22.1�

④

𝑒𝑒� � 5.6�

②

0.37 kips

0.13 kips

⑤

① 0.38 kips 3.73 kips

𝑥𝑥�� � 31.2�

N

Term 1 force Term 2 force

⑥

Figure 10.3 Force allocation to the walls for wind in the west direction. • Wind in the south direction The equations for force and moment equilibrium are solved simultaneously for the two unknowns where moments are summed about column line 1:

Therefore,

and

and

.

• Wind in the west direction The equations for force and moment equilibrium are solved simultaneously for the two unknowns where moments are summed about column line A:

Therefore,

and

and

.

(d) Determine the chord forces. • Wind in the south direction In the west segment of the diaphragm, the maximum moment is equal to 14.8 ft-kips (see Figure 10.4). Therefore, using Equation (6.7), the maximum chord forces are equal to the following:

10-10

Design Guide for Reinforced Concrete Diaphragms 1

4A

𝑤𝑤��� � 0.07 -kip⁄�t

𝑤𝑤��� � 0.17 -kip⁄�t N

15� -0″

𝑅𝑅� � 1.60 -kip 1.60

𝑅𝑅� � 1.48 -kip

35� -4″

𝑅𝑅� � 3.02 -kip

1.78 0.30

Shear �-kips

29.0

3.02

14.8

Moment �� -kips ft

30.3�

Figure 10.4 Equivalent distributed load, shear diagram, and moment diagram for wind in the south direction. In the east segment of the diaphragm:

• Wind in the west direction In the north segment of the diaphragm, the maximum moment is equal to 38.1 ft-kips (see Figure 10.5). Therefore, using Equation (6.7), the maximum chord forces are equal to the following:

In the south segment of the diaphragm:

(e) Determine the unit shear forces, net shear forces, and collector forces. • Wind in the south direction The maximum unit shear force in the diaphragm for wind in the south direction occurs at Wall 1:

10-11

Design Guide for Reinforced Concrete Diaphragms E

AA

𝑤𝑤��� � 0.15 -kip⁄�t

𝑤𝑤��� � 0.09 -kip⁄�t

E

𝑅𝑅� � 2.75 -kip

27� -8″

𝑅𝑅� � 0.60 -kip

2.75

𝑅𝑅� � 3.35 -kip

0.45

Shear �-kips

0.15

38.1

Moment �� -kips

27� -8″

38.0

38.9

3.35

ft

26.4�

31.3�

Figure 10.5 Equivalent distributed load, shear diagram, and moment diagram for wind in the west direction. The unit shear forces, net shear forces, and collector forces in the diaphragm at Wall 2 are given in Figure 10.6 assuming that the collector is the portion of the slab in line with Wall 2 and that the width of the collector is equal to the thickness of Wall 2. • Wind in the west direction The maximum unit shear force in the diaphragm for wind in the west direction occurs at Wall 5:

The unit shear forces, net shear forces, and collector forces in the diaphragm at Wall 4 are given in Figure 10.7 assuming that the collector is the portion of the slab in line with Wall 4 and that the width of the collector is equal to the thickness of Wall 4. • Step 5 – Determine Combined Load Effects Combined load effects are determined for this building that is assigned to SDC A using the applicable strength design load combinations in ACI 5.3 (see Table 5.1 of this publication) and ASCE/SEI 2.6.1 (see Figure 9.4). The governing combined out-of-plane load effects are determined in Part 2(b) of Step 2 above. The maximum factored uniform load on the roof slab due to dead and rain loads is equal to 184.2 psf (see Table 10.1).

10-12

Design Guide for Reinforced Concrete Diaphragms

27� -8″

0.03 kips⁄ft

0.05 kips⁄ft

0.02 kips⁄ft

1.48 � 0.05 kips⁄ft 27.7

②

④

0.6 kips

27� -8″

0.30 � 0.01 kips⁄ft 27.7

①

1.78 � 0.03 kips⁄ft 55.4

N

0.02 kips⁄ft

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

0.02 kips⁄ft

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

Figure 10.6 Unit shear forces, net shear forces, and collector forces in the diaphragm at Wall 2.

The governing in-plane load effects are due to wind in this example (see Parts 1(a) and 1(b) in Step 3 above). Thus, ACI Equation (5.3.1d) must be used where the load factor on is equal to 1.0. The shear and collector forces in the diaphragm are given in Figures 10.4 and 10.6 for wind in the south direction and in Figures 10.5 and 10.7 for wind in the west direction. • Step 6 – Determine the Chord Reinforcement The flowchart in Figure 9.5 is used to determine the required area of chord reinforcement in both directions. The chord forces in the north-south and east-west directions are determined in Part 3(d) of Step 4 above. 1.

Wind in the north-south direction In the north-south direction, the chord forces in the west and east segments of the diaphragm are essentially equal, and the required area of chord reinforcement is determined by Equation (8.4) of this publication:

2.

Wind in the east-west direction In the east-west direction, the required area of chord reinforcement in the north and south segments is equal to the following using the larger of the two calculated values of :

Because the required area of chord reinforcement is very small in both directions, it is safe to assume that the flexural reinforcement at the edges of the slab along column lines A, C, E, 1, 2, and 4 is adequate and no additional chord reinforcement is needed.

10-13

Design Guide for Reinforced Concrete Diaphragms 17� -8″

15� -0″

0.60 � 0.040 kips⁄ft 15.0

⑤

17� -8″

0.15 � 0.004 kips⁄ft 35.3

④

N

0.45 � 0.009 kips⁄ft 50.3

0.040 kips⁄ft

0.009 kips⁄ft

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

0.013 kips⁄ft

0.031 kips⁄ft

0.5 kips

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

0.013 kips⁄ft

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

Figure 10.7 Unit shear forces, net shear forces, and collector forces in the diaphragm at Wall 4. • Step 7 – Determine the Diaphragm Shear Reinforcement The flowchart in Figure 9.6 is used to determine the area of shear reinforcement, if required. The largest factored unit shear force in the diaphragm occurs along Wall 5 for wind in the west direction and is equal to 0.08 kips/ft (see Part 3(e) in Step 4 above). The design shear strength of the diaphragm is determined using Equation (7.3) of this publication where and assuming the shear reinforcement :

Alternatively, can be determined from Table 7.3 of this publication: , and 4,000-psi concrete. Thus,

for a 7.0-in. slab thickness, .

Therefore, the shear strength of the diaphragm is adequate without shear reinforcement. • Step 8 – Determine the Shear Transfer Reinforcement The flowchart in Figure 9.7 is used to determine the required areas of shear transfer reinforcement between the diaphragm and the walls and the diaphragm and the collector elements. 1.

Determine shear transfer reinforcement between the diaphragm and the walls. (a) Wind in the north-south direction The largest unit shear force occurs in Wall 1 and is equal to 0.06 kips/ft (see Part 3e of Step 4 above). The required area of shear-friction reinforcement between the diaphragm and Wall 1 can be determined by Equa-

10-14

Design Guide for Reinforced Concrete Diaphragms tion (8.9) of this publication assuming that there is a cold joint between the wall and the slab where the hardened concrete has not been intentionally roughened (that is, from ACI Table 22.9.4.2):

Because the required area of shear-friction reinforcement is very small, it is safe to assume that the bottom flexural reinforcement in the slab, which is doweled into the wall, can be used as the shear transfer reinforcement between the diaphragm and Wall 1. This is also applicable to Walls 2 and 3 because the unit shear forces in these walls are less than that in Wall 1. (b) Wind in the east-west direction The largest unit shear force occurs in Wall 5 and is equal to 0.08 kips/ft (see Part 3e of Step 4 above). Thus, the required area of shear-friction reinforcement is equal to the following:

The bottom flexural reinforcement in the slab, which is doweled into the wall, can be used as the shear transfer reinforcement between the diaphragm and Walls 4, 5, and 6. 2.

Determine shear transfer reinforcement between the diaphragm and the collectors. The largest unit shear force that occurs between the diaphragm and a collector occurs along column line 2 and is equal to 0.02 kips/ft (see Figures 10.6 and 10.7). As noted previously, this collector is part of the slab and has the same width as Wall 2, so from ACI Table 22.9.4.2 for concrete cast monolithically. Therefore, the required shear-friction reinforcement is determined by Equation (8.9):

Just like the shear-friction reinforcement between the diaphragm and the walls, the bottom flexural reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and both collectors in this example. • Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces Because the collectors are the same width as the walls that they frame into, there is no eccentricity of forces; thus, this type of reinforcement is not required. • Step 10 – Determine the Anchorage Reinforcement The flowchart in Figure 9.9 is used to determine the required area of anchorage reinforcement. For this building assigned to SDC A, both a wind anchorage force and a general structural integrity anchorage force for structural walls must be determined. 1.

Determine the wind connection force. , is determined on the walls using the information The out-of-plane components and cladding (C&C) wind pressure, provided in Section 4.4.2 of this publication. The provisions in Part 1 of Chapter 30 are selected to determine this wind pressure for this building that meets the criteria in ASCE/SEI 30.1 and 30.3 (see Table 4.6 of this publication). Design wind pressure

The velocity pressure, equal to 14.2 psf.

is determined by ASCE/SEI Equation (30.3-1):

, at the mean roof height of the building is determined in Part 1(a)iii of Step 3 above and is

The external pressure coefficients are determined from ASCE/SEI Figure 30.3-1 for walls. Assuming that the effective wind area for the anchorage reinforcing bars is less than 10 sq ft, the external pressure coefficients are the following for Zone 5, which is the zone with the highest wind pressure:

10-15

Design Guide for Reinforced Concrete Diaphragms

Note 5 in ASCE/SEI Figure 30.3-1 permits values of less than 10 degrees, which is applicable in this example:

to be reduced by 10 percent for buildings with roof slopes

Anchorage reinforcement is determined for wind acting away from the surface, which produces tension in the reinforcing bars. Thus, the governing is equal to the following for an enclosed building where the internal pressure coefficients are given in ASCE/SEI Table 26.13-1:

This pressure is greater than the minimum design pressure of 16.0 psf given in ASCE/SEI 30.2.2. 2.

Determine the general structural integrity anchorage force. According to ASCE/SEI 1.4.4, connections between walls and roof (or floors) must be capable of resisting a strength level horizontal force, , perpendicular to the plane of the wall equal to the greater of the following:

where 3.

is the weight of the wall, which is equal to 100 psf for the 8.0-in.-thick wall in this example.

Determine the required anchorage reinforcement. It is evident that the wind pressure governs, so the required anchorage reinforcement is determined by Equation (8.24) of this publication:

The required reinforcement for anchorage is equal to that required for shear transfer (see Step 8 above). Thus, the bottom flexural reinforcement in the slab can be used as the anchorage reinforcement along the lengths of all the walls in the building. • Step 11 – Determine the Collector Reinforcement The flowchart in Figure 9.10 is used to determine the required area of longitudinal reinforcement in the collectors along column lines C and 2. The collector along column line 2 has the largest axial force, which is equal to 0.6 kips (see Figures 10.6 and 10.7). The required area of longitudinal collector reinforcement is equal to the following:

Because the required area of collector reinforcement is very small, it is safe to assume that the flexural reinforcement in the slab can be used as the collector reinforcement along the lengths of both collectors. Comments. It is evident that the required areas of shear transfer, anchorage, and collector reinforcement are nominal in this example. As noted above, the excess flexural reinforcement in the slab can be used to satisfy the requirements for these types of reinforcement. It can be determined that all the bottom flexural reinforcement in the column and middle strips in both directions is the minimum required in accordance with ACI 8.6 for this two-way slab, which is more than adequate to satisfy the aforementioned requirements for the diaphragm

10.3 Example 10.2 – Seven-story Office Building (SDC B) Design and detail the roof diaphragm for the 7-story office building depicted in Figure 10.8 given the design data below. The height of the first story is 14 ft-0 in. and the typical story height is 12 ft-0 in. The roof is essentially flat except for ½ in. per foot slopes for drainage.

10-16

Design Guide for Reinforced Concrete Diaphragms 1A

20 -0″

2A

�

20 -0″

3A

�

20 -0″

4A

�

20 -0″

5A

�

20 -0″

6A

�

20 -0″

7A

24� -0″

G

�

24� -0″

F

24� -0″

E

24� -0″

D

24� � 24� column ������

N

28� � 20� beam ������

24� -0″

C

24� -0″

B

A

Figure 10.8 Roof plan of 7-story office building in Example 10.2.

Design Data • Site information Latitude = 37.54o, Longitude = −77.46o Exposure Category: Exposure C in all directions Soil classification: Site Class D (obtained from a geotechnical report based on measured soil properties) • Loads Roof live load = 20 psf Roof superimposed dead load = 10 psf Floor superimposed dead load = 20 psf Weight of partitions = 15 psf Weight of mechanical unit on roof that is supported directly on the building columns = 20 kips Cladding = 8 psf Snow loads in accordance with ASCE/SEI Chapter 7 Rain loads in accordance with ASCE/SEI Chapter 8

10-17

Design Guide for Reinforced Concrete Diaphragms Wind loads in accordance with ASCE/SEI Chapters 26 and 27 Seismic loads in accordance with ASCE/SEI Chapters 11, 12, 20, and 22 • Lateral force-resisting system: perimeter moment-resisting frames (all the beam-column frames are part of the LFRS)

Solution • Step 1 – Select the Materials and Grade 60 reinforcement. The suitability of these materials will be Assume normalweight concrete with checked once the SDC is determined (see Part 4 of Step 2 below). • Step 2 – Determine the Diaphragm Thickness The flowchart in Figure 9.1 is used to determine the minimum diaphragm thickness for this two-way slab system with edge beams. 1.

Determine minimum thickness

based on serviceability requirements.

Maximum clear span From Table 3.2 of this publication,

for an interior panel.

For an edge panel with beams at both edges, determine the ratio of the flexural stiffness of the edge beam to the flexural stiffness of the width of the slab bounded laterally by the centerline of the adjacent panel on the side of the edge beam, , in accordance with ACI Equation (8.10.2.7b):

Conservatively neglecting the effective flange width, the moment inertia of the beams,

Assuming an 8.0-in. slab thickness, the largest moment of inertia of the slab,

Therefore, assuming the same concrete mix is used everywhere (that is,

, is equal to the following:

, is equal to the following:

):

Thus, the edge panels are considered to have edge beams (see Footnote 4 in Table 3.2), and the minimum slab thickness for an exterior panel is equal to , which is the same as that for an interior panel (see Table 3.2). For the edge panels with an edge beam on one side only:

Therefore, 2.

Determine minimum thickness

based on two-way shear requirements.

Because there are perimeter beams and because adjoining bays have the same length in a given direction of analysis, the unbalanced moments at the columns due to gravity loads are nominal. Therefore, the minimum slab thickness is determined based on direct shear forces due to gravity loads only. (a) Determine gravity loads. i.

Dead load, Assuming an 8.0-in.-thick slab, the total dead load is as follows:

10-18

Design Guide for Reinforced Concrete Diaphragms ii.

Roof live load, From the design data, roof live load

.

iii. Snow load, Snow load,

, is defined as either the flat roof snow load,

Because the roof is essentially flat,

, or the sloped roof snow load,

.

is applicable, which is determined by ASCE/SEI Equation (7.3-1):

Using surface roughness category C based on the exposure category given in the design data and assuming partial exposure for the roof, exposure factor from ASCE/SEI Table 7.3-1. Assuming the thermal condition is not specifically indicated in ASCE/SEI Table 7.3-2, thermal factor

.

For the office occupancy, the building is assigned to Risk Category II in accordance with ASCE/SEI Table 1.5-1. According to ASCE/SEI Table 1.5-2, snow importance factor for Risk Category II buildings. Ground snow load tude of the site.

from Reference 5 or ASCE/SEI Figure 7.2-1 for the given latitude and longi-

Thus,

.

Check minimum snow load, , for low-slope roofs (the roof in this example is low-slope because the slope is equal to 2.39 degrees, which is less than 15 degrees; see ASCE/SEI 7.3.4): Where Use

.

iv. Rain load, Design rain load,

, is determined by ASCE/SEI Equation (8.3-1):

Assume that the tributary area of each of the roof drains,

, is equal to 2,400 sq ft.

Also assume that 6-in.-diameter overflow standpipes are used as the secondary drainage system. It has been determined that the vertical distance from the primary roof drain to the inlet of the overflow standpipe, , is equal to 3.0 in. (static head distance). The 15-minute precipitation intensity, latitude and longitude of the site.

, is equal to 6.05 in./h from Reference 5 or Reference 12 for the given

Required flow rate,

, is determined by ASCE/SEI Equation (C8.3-1):

The hydraulic head,

, can be obtained from ASCE/SEI Table C8.3-1:

For a 6-in.-diameter overflow standpipe with a flow rate between 150 and 200 gal/min, Therefore, Note: The 2018 IBC requires the use of a 60-minute precipitation intensity for rain load calculations for both the primary and secondary drainage systems, which, for the given latitude and longitude of the site, is equal to 3.19 in./h from Reference 5 or Reference 12. Based on that intensity, , , and . The rain load determined in accordance with ASCE/SEI 7 is used throughout the remainder of this example. (b) Determine combined gravity load effects. The combined gravity load effects are determined using the load combinations in Table 5.1 of this publication. The applicable gravity load combinations for this example are given in Table 10.4. Maximum load effects are obtained

10-19

Design Guide for Reinforced Concrete Diaphragms using the design rain load, tored roof live load.

, because the factored rain load is greater than the factored snow load and the fac-

Table 10.4 Strength Design Load Combinations for Gravity Loads in Example 10.2 ACI Equation Number

ASCE/SEI 7 Load Combination

5.3.1a 5.3.1c

1 3

Load Combination (psf)

(c) Check two-way shear requirements at an interior column assuming an 8.0-in. slab thickness. Factored shear force, (see Figure 3.4):

, at the critical section of an interior column can be determined by the following equation

The lengths of the critical section

and

are equal to the following for an interior column (see Table 3.4):

Design two-way shear strength is the smallest value determined by the equations in ACI Table 22.6.5.2. Equation (a) in that table governs in the case of a square, interior column:

Two-way shear requirements are satisfied utilizing an 8.0-in.-thick slab 3. Determine minimum thickness

based on fire resistance requirements.

Assuming that the required fire-resistance rating of the roof slab is 2 hours for this occupancy, the minimum thickness from IBC Table 722.2.2.1 based on a concrete mix with siliceous aggregate. 4. Check if the minimum slab thickness requirements for SDC D, E, and F must be satisfied. The SDC is determined in accordance with IBC 1613.2.5 or ASCE/SEI 11.6 (see Step 3 in Section 4.2.3 of this publication). For the given latitude and longitude of the site, the earthquake spectral response acceleration parameters at short periods and at 1-second periods are and , respectively, from Reference 4 or Reference 5. For Site Class D, which as noted in the design data has been obtained based on measured soil properties at the site, and from ASCE/SEI Tables 11.4-1 and 11.4-2, respectively. The earthquake spectral acceleration parameters at short periods and at 1-second periods, adjusted for site class effects, are determined by ASCE/SEI Equations (11.4-1) and (11.4-2), respectively:

Design earthquake spectral response acceleration parameters at short periods and at 1-second periods are determined by ASCE/SEI Equations (11.4-3) and (11.4-4), respectively:

10-20

Design Guide for Reinforced Concrete Diaphragms From ASCE/SEI Table 11.6-1, the SDC is B for SEI Table 11.6-2, the SDC is B for

and Risk Category II. Similarly, from ASCE/ and Risk Category II.

Therefore, this building is assigned to SDC B, which means the minimum slab thickness requirements in ACI 18.12.6.1 for buildings assigned to SDC D, E, or F need not be satisfied. Also, the materials selected in Step 1 are permitted to be used for buildings assigned to SDC B where ordinary reinforced concrete moment frames are used. Use a slab thickness • Step 3 – Determine the Diaphragm Design Forces 1.

Determine the in-plane forces. The flowchart in Figure 9.2 is used to determine the in-plane diaphragm forces. Wind and seismic forces are applicable for this building that is assigned to SDC B. The general structural integrity forces in ASCE/SEI 1.4.2 must also be considered but are typically less than the required seismic forces for buildings assigned to SDC B. (a) Determine the wind forces. The wind load provisions in Part 1 of ASCE/SEI Chapter 27 are used to determine wind forces for this building, which meets the conditions and limitations set forth in that part (see Table 4.1 of this publication). The wind pressures on the windward and leeward faces of the building are determined at the mean roof height and at each of the floor levels using ASCE/SEI Equation (27.3-1). It is assumed that the following pressures are uniformly distributed over the tributary heights of the roof and floor levels: Windward pressure: Leeward pressure: The steps in Section 4.2.1 of this publication are used to determine the design wind pressures in the north-south and east-west directions. Even though the building meets the conditions in Part 2 of ASCE/SEI Chapter 27, it is not recommended to use these provisions because L-, T-, and U-shaped buildings are considered to be outside the scope of this method. i.

Determine the Risk Category of the building. It was determined in Part 2(a)iii of Step 2 above that the building is assigned to Risk Category II based in its occupancy.

ii.

Determine the basic wind speed,

.

For the given latitude and longitude of the site, iii. Determine total wind pressures,

from ASCE/SEI Figure 26.5-1B or Reference 5.

, in both the north-south and east-west directions.

• For the MWFRS of a building structure, wind directionality factor

from ASCE/SEI Table 26.6-1.

• The exposure category is given as C in the design data. • A  ssuming the building is not sited on the upper half of an isolated hill, ridge, or escarpment, the topographic factor, , can be taken as 1.0 in accordance with ASCE/SEI 26.8. • A  ccording to Note 3 in ASCE/SEI Table 26.9-1, the ground elevation factor, elevations. • The gust-effect factor,

, can be taken as 1.0 for all

, is determined in accordance with ASCE/SEI 26.11.

The natural frequency of the building, , is needed to determine . In lieu of a more exact analysis, an approximate natural frequency, , can be calculated using the provisions in ASCE/SEI 26.11.3 provided the building meets the limitations in ASCE/SEI 26.11.2.1. The first of these limitations is met because the height of the building is less than 300 ft. The second limitation requires that the height of the building be less than 4 times its effective length, , which is determined by ASCE/SEI Equation (26.11-1):

10-21

Design Guide for Reinforced Concrete Diaphragms In this equation, is the height above grade of level , is the length of the building at level in the direction of analysis, and is the number of stories in the building. In the north-south direction, the length of the building is the same at all levels, so . The same is true in the east-west direction where . Therefore, check the second limitation: In the north-south direction:

In the east-west direction:

Thus,

can be calculated using the provisions in ASCE/SEI 26.11.3.

ASCE/SEI Equation (26.11-3) can be used to determine frames:

for buildings with concrete moment-resisting

Therefore, the building is flexible and must be determined using the provisions of ASCE/SEI 26.11.5. In the north-south direction, and in the east-west direction, (calculations not shown here; see Step 3(e) in Section 4.2.1 of this publication). • It is assumed that this building can be classified as enclosed according to the definition in ASCE/SEI 26.2 (see Table 4.5 of this publication). • For an enclosed building, the internal pressure coefficient,

(see ASCE/SEI Table 26.13-1).

• T  he velocity pressure exposure coefficient, , over the height of the building for Exposure C is obtained from ASCE/SEI Table 26.10-1 (see Table 10.5). • The velocity pressure,

At

, over the height of the building is determined by ASCE/SEI Equation (26.10-1):

,

• External pressure coefficients, Windward wall:

(see ASCE/SEI Figure 27.3-1). (north-south and east-west wind)

Leeward wall: North-south wind: For

,

from linear interpolation

East-west wind: For

,

Therefore, the total wind pressures are equal to the following: North-south windward pressure:

North-south leeward pressure:

Total north-south wind pressure:

10-22

Design Guide for Reinforced Concrete Diaphragms East-west windward pressure:

East-west leeward pressure:

Total east-west wind pressure:

Wind pressures over the height of the building are given in Table 10.5. Table 10.5 Wind Pressures for the Building in Example 10.2 Level

Height above ground level, (ft)

R

86

7 6 5 4 3 2

74 62 50 36 26 14

Total North-South Wind Pressure (psf)

Total East-West Wind Pressure (psf)

1.23

39.7

40.5

1.19 1.14 1.09 1.03 0.95 0.85

38.9 37.9 36.9 35.6 34.0 31.9

39.6 38.6 37.6 36.4 34.8 32.8

iv. Determine the wind forces on the roof and floor diaphragms for wind in both directions. Wind forces over the height of the building are determined by multiplying the total wind pressures at the roof and floor levels by the corresponding tributary areas (see Figure 4.1 of this publication). For wind in the northsouth direction, the tributary area at a level is equal to 120.0 ft times the tributary story height. Similarly, for wind in the east-west direction, the tributary area is equal to 144.0 ft times the tributary story height. Wind forces over the height of the building are given in Table 10.6. Table 10.6 Wind Forces for the Building in Example 10.2 Level

Height above ground level, (ft)

R 7 6 5 4 3 2

86 74 62 50 36 26 14

Tributary Total North-South Height (ft) Wind Pressure (psf) 6.0 12.0 12.0 12.0 12.0 12.0 13.0

39.7 38.9 37.9 36.9 35.6 34.0 31.9

North-South Wind Force (kips) 28.6 56.0 54.6 53.1 51.3 49.0 49.8

Σ

342.4

Total East-West East-West Wind Wind Pressure Force (kips) (psf) 40.5 39.6 38.6 37.6 36.4 34.8 32.8

35.0 68.4 66.7 65.0 62.9 60.1 61.4 419.5

(b) Determine the seismic forces. The Equivalent Lateral Force (ELF) Procedure in ASCE/SEI 12.8 is permitted to be used to determine the seismic forces on the seismic force-resisting system (SFRS) over the height of this building that is assigned to SDC B (see Section 4.2.3 of this publication and ASCE/SEI Table 12.6-1). Diaphragm seismic forces are determined in accordance with ASCE/SEI 12.10.1.1.

10-23

Design Guide for Reinforced Concrete Diaphragms i.

Determine the seismic forces on the SFRS. The steps in Section 4.2.3 are used to determine the seismic forces on the SFRS over the height of the building. • The seismic ground motion values are determined in Part 4 of Step 2 above and are equal to the following: and . • It was determined in Part 2(a)iii of Step 2 above that the building is assigned to Risk Category II based in its occupancy. • The SDC is B for this building, as determined in Part 4 of Step 2 above. • Determine the seismic response coefficient, The seismic response coefficient,

.

, is determined by ASCE/SEI Equation (12.8-2):

Ordinary reinforced concrete moment frames are permitted to be used in buildings assigned to SDC B with no limitations (see ASCE/SEI Table 12.2-1). From ASCE/SEI Table 12.2-1, response modification coefficient, , is equal to 3 for this system. Also, the seismic importance factor, , is equal to 1.0 for Risk Category II buildings (see ASCE/SEI Table 1.5-2). Thus,

The value of need not exceed that determined by ASCE/SEI Equations (12.8-3) or (12.8-4), whichever is applicable. These equations include the long-period transition period, , and the period of the building, . The long-period transition period, , is determined using ASCE/SEI Figures 22-14 through 22-17. At this site, from ASCE/SEI Figure 22-14. This quantity can also be obtained from Reference 5. The fundamental period of the structure, , is determined in accordance with ASCE/SEI 12.8.2. According to that section, it is permitted to determine the approximate building period, , from ASCE/SEI Equation (12.8-7):

In this equation, is the vertical distance from the base of the building to the highest level of the SFRS. Because the moment frames are used over the entire height of the building, is equal to 86.0 ft. The approximate period parameters and are obtained from ASCE/SEI Table 12.8-2. For concrete momentresisting frames that resist 100 percent of the required seismic force and that are not enclosed or adjoined by components that are more rigid and will prevent the frames from deflecting when subjected to the seismic forces, and . Therefore,

This period is valid in both the north-south and east-west directions because reinforced concrete momentresisting frames are used in both directions. As such, the seismic forces in both directions are the same. Because

Also,

10-24

,

need not exceed that determined by ASCE/SEI Equation (12.8-3):

must not be less than that determined by ASCE/SEI Equation (12.8-5):

Design Guide for Reinforced Concrete Diaphragms Therefore, the seismic response coefficient, • Determine the effective seismic weight,

, is equal to 0.034. .

For this building, the weight of the slabs, beams, columns, partitions, cladding, and the mechanical unit on the roof plus the superimposed dead loads must all be included in . The snow load at the roof level need not be included because the flat roof snow load is less than 30 psf (ASCE/SEI 12.7.2(4); see Part 2(a)iii of Step 2 above). The story weights, , which are the portions of that are assigned to level in the building, are given in Table 10.7. Table 10.7 Seismic Forces and Story Shears on the SFRS for the Building in Example 10.2 Level

Story Weight, (kips)

Height, (ft)

1,985 2,284 2,284 2,284 2,284

86 74 62 50 38 26 14

R 7 6 5 4 3 2

2,284 2,314 15,719

Σ

• Determine the seismic base shear,

Seismic Force, (kips)

Story Shear, (kips)

397,936 382,897 310,200 240,143 173,236 110,285 53,488

127.5 122.7 99.4 77.0 55.5 35.3 17.1

127.5 250.2 349.6 426.6 482.1 517.4 534.5

1,668,185

534.5

.

The seismic base shear is determined by ASCE/SEI Equation (12.8-1):

• Distribute the seismic base shear, The seismic force, and (12.8-12):

In this equation,

, induced at any level of a building is determined by ASCE/SEI Equations (12.8-11)

is the exponent related to the period of the building,

–

for buildings where

–

for buildings where





–

, over the height of the building.

:

is to be determined by linear interpolation between 1 and 2 for buildings that have a period between 0.5 and 2.5 seconds or can be taken equal to 2



In this example, the building period is between 0.5 and 2.5 seconds and lation:

is determined by linear interpo-

The seismic forces over the height of the building are given in Table 10.7. Note that these seismic forces are greater than the general lateral structural integrity forces prescribed in ASCE/SEI 1.4.2, which are equal to at each level. ii.

Determine the seismic forces on the diaphragms. The information in Section 4.2.3 of this publication is used to determine the seismic forces on the diaphragms over the height of the building.

10-25

Design Guide for Reinforced Concrete Diaphragms According to ASCE/SEI 12.10, diaphragms are to be designed for the larger of the following forces: • Design seismic forces,

, acting on the SFRS at the levels of a building

• Diaphragm design forces,

, determined by ASCE/SEI Equations (12.10-1) through (12.10-3):

Minimum Maximum In these equation, is the portion of the effective seismic weight, , that is assigned to level (see the values of in Table 10.7) and is the weight that is tributary to the diaphragm at level , which can be taken as in buildings without structural walls (see the discussion in Section 4.2.3 of this publication). The forces, , are the portions of the seismic base shear, , induced at level (see the values of in Table 10.7). The diaphragm design forces, . At all other levels,

, over the height of the building are given in Table 10.8. At the roof level, .

The term that is related to the required minimum value of the design seismic force on the diaphragm is equal to the following:

The term that is related to the required maximum value of the design seismic force on the diaphragm is equal to the following:

It is evident from Table 10.8 that the minimum seismic diaphragm force is required at levels 1 through 4. Table 10.8 Design Seismic Forces on the Diaphragms for the Building in Example 10.2 Story Weight,

Seismic Force,

(kips)

(kips)

(kips)

(kips)

1,985 2,284 2,284 2,284 2,284 2,284 2,314

127.5 122.7 99.4 77.0 55.5 35.3 17.1

1,985 4,269 6,553 8,837 11,121 13,405 15,719

127.5 250.2 349.6 426.6 482.1 517.4 534.5

15,719

534.5

Level R 7 6 5 4 3 2 Σ * Minimum

(kips) 0.0642 0.0586 0.0533 0.0512* 0.0512* 0.0512* 0.0512*

governs.

The following are sample calculations for the design seismic diaphragm force at level 4:

10-26

127.5 133.8 121.9 116.9 116.9 116.9 118.5

Design Guide for Reinforced Concrete Diaphragms 2. Determine the out-of-plane forces. The out-of-plane forces on the roof diaphragm are determined in Part 2(a) of Step 2 above. • S  tep 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces 1.

Determine the diaphragm classification. The information in Section 6.1 of this publication is used to determine the classification of this diaphragm. In the northsouth direction, the span-to-depth ratio is equal to . In the east-west direction, the maximum span-to-depth ratio is equal to . Because the overall span-to-depth ratios of the diaphragm are less than 2, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to wind forces in both directions of analysis (ASCE/SEI 26.2). In the case of seismic forces, the diaphragm can be classified as rigid when the two conditions in ASCE/SEI 12.31.2 are satisfied. The first condition is satisfied because the span-to-depth ratio is less than 3 in both directions. For the second condition, check if the diaphragm has any of the horizontal irregularities in ASCE/SEI Table 12.3-1. A reentrant corner irregularity is defined to exist where both plan projections of the structure beyond a reentrant corner are greater than 15 percent of the plan dimension of the structure in the given direction. In the north-south direction, the length of the projection is equal to 24.0 ft which is greater than . In the east-west direction, the length of the projection is equal to 60.0 ft, which is greater than . Thus, the building has a reentrant corner irregularity. Therefore, ASCE/SEI 12.3.1.2 cannot be used to classify the diaphragm as rigid. A three-dimensional model of the building was constructed using Reference 14. In the model, the columns are fixed at the base (see ASCE/SEI 12.7.1) and the following reduced moments of inertia are used, which account for the effects of cracked sections (ASCE/SEI 12.7.3): • Columns that are part of the moment-resisting frames: • Beams: • Slabs (out-of-plane): • Slabs (in-plane): The columns that are not part of the LFRS are assigned stiffness modifiers equal to 0.001 to ensure that the effects from the lateral loads are resisted only by the moment-resisting frames. It was found that the in-plane forces based on rigid diaphragms assigned at each level are essentially identical to those based on semirigid diaphragms, so the diaphragms can be considered rigid.

2. Select the diaphragm model. The corrected equivalent beam model with spring supports is selected for this diaphragm. As noted in Section 6.4.3, this model is best suited for buildings with rigid diaphragms and lateral force-resisting systems with elements that have different stiffnesses, which is the case for the building in this example. 3. Determine the diaphragm internal forces. The flowchart in Figure 9.3 is used to determine the internal forces in the diaphragm. It is evident from the information Tables 10.6 and 10.8 that the seismic forces are greater than the wind forces at all levels of the building. Thus, internal forces in the roof diaphragm are determined using the design seismic forces in Table 10.8. Also, because the building is assigned to SDC B, the design seismic forces in the north-south and east-west directions are permitted to be applied independently in each of the two orthogonal directions and orthogonal interaction effects are permitted to be neglected (ASCE/SEI 12.5.2). (a) Determine the location of the CM. The location of the CM is determined based on the weight of the slab only (in typical cases, the weights of vertical elements and beams have a nominal effect on the location of the CM). The same slab thickness and concrete mix are used throughout the entire roof level, so the location of the CM is determined using the plan areas of the slab segments with the origin taken at the intersection of column lines 1 and A. In the north-south direction, due to symmetry. In the east-west direction, is determined as follows:

10-27

Design Guide for Reinforced Concrete Diaphragms From Reference 14, and weights of the slab, columns, and beams.

; these values are determined in the program based on the

(b) Determine the location of the CR. In the north-south direction, the CR is located 72.0 ft from column line A due to the symmetric distribution of the SFRS in the east-west direction. In the east-west direction, the CR is calculated using the approximate story stiffness for the frames on column lines 1 and 7, which are determined by Equation (6.3) of this publication:

For the frame along column line 1:

For the frame along column line 7:

Therefore, using Equation (6.1):

From Reference 14,

.

(c) Determine the seismic forces in each of the frames. Seismic forces are applied at the CM in both directions. According to ASCE/SEI 12.8.4.2, an accidental torsional moment must be included when evaluating whether a horizontal structural irregularity in ASCE/SEI Table 12.3-1 exists in a structure. It was determined in Part 1 of this step that a Type 2 reentrant corner irregularity is present. Check if a Type 1a torsional irregularity or a Type 1b extreme torsional irregularity exist by determining the lateral displacements and story drifts due to the seismic forces applied on the SFRS at the CM over the height of the building where the CM is displaced each way from its actual location a distance equal to 5 percent of the dimension of the structure perpendicular to the direction of analysis (see ASCE/SEI 12.8.4.2). The elastic displacements, , at each end of the building in both directions from the analysis are given in Table 10.9 assuming the effects of the seismic forces are resisted only by the moment frames along column lines A, B, F, G, 1, and 7. Also given in the table are and , which are the story drifts and the average of the story drifts at each end of the building, respectively. The term is the maximum of the story drifts at each end of the building. No torsional irregularities exist in either direction because the ratio (see ASCE/SEI Table 12.3-1).

at all levels of the building

Based on the descriptions in ASCE/SEI Table 12.3-1, it is evident that horizontal structural irregularity Types 3 through 5 do not exist for this structure.

10-28

Design Guide for Reinforced Concrete Diaphragms Table 10.9 Lateral Displacements and Story Drifts Due to Seismic Forces North-South Direction

East-West Direction

Story (in.)

(in.)

(in.)

(in.)

(in.)

7 6 5 4 3 2

2.51 2.35 2.10 1.74 1.31 0.84

0.16 0.25 0.36 0.43 0.47 0.48

3.04 2.84 2.53 2.10 1.59 1.10

0.20 0.31 0.43 0.51 0.58 0.58

0.18 0.28 0.40 0.47 0.53 0.53

1

0.36

0.36

0.43

0.43

0.40

(in.)

(in.)

(in.)

(in.)

(in.)

1.11 1.11 1.08 1.09 1.09 1.09

2.67 2.49 2.22 1.83 1.38 0.87

0.18 0.27 0.39 0.45 0.51 0.50

3.00 2.80 2.48 2.06 1.56 0.99

0.20 0.32 0.42 0.50 0.57 0.56

0.19 0.30 0.41 0.48 0.54 0.53

1.05 1.07 1.02 1.04 1.06 1.06

1.08

0.37

0.37

0.43

0.43

0.40

1.08

Accidental torsion in accordance with ASCE/SEI 12.8.4.2 need not be applied in the analysis for strength design or when checking the story drift limits prescribed in ASCE/SEI 12.12 because the structure, which is assigned to SDC B, does not have a Type 1b horizontal structural irregularity. Thus, in the east-west direction where there is no eccentricity between the CM and the CR , no torsional moment is generated. In the north-south direction, , which means a torsional moment is generated. The forces in each of the frames due to the 127.5-kip seismic diaphragm force at the roof level are determined using Equations (6.5) and (6.6) of this publication. The approximate stiffness of each of the frames along column lines A, B, F, and G is equal to , which is determined by Equation (6.3). Therefore, each of the frames resists because of the symmetrical distribution of the elements of the SFRS in the east-west direction and because there is no torsional moment. Forces in the frames for the 127.5-kip seismic force applied at the CM in the south direction are given in Table 10.10 where it is assumed that positive forces act in the north and east directions. The forces act in the directions that resist the applied in-plane force and torsional moment. Table 10.10 Forces in the Frames for the Seismic Force in the South Direction

Frame

(ft)

1

0.0

(ft) —

1.23

—

49.4

Term 2

(kips)

(kips)

—

75.0

-2.0

73.0

(ft)

(kips)

—

3,002

7

120.0

—

0.86

—

70.6

—

4,287

—

52.5

2.0

54.5

A

—

0.0

—

0.72

—

72.0

—

3,733

—

1.7

1.7

B

—

24.0

—

0.72

—

48.0

—

1,659

—

1.1

1.1

F

—

120.0

—

0.72

—

48.0

—

1,659

—

-1.1

-1.1

G

—

144.0

—

0.72

—

72.0

—

3,733

—

-1.7

-1.7

7,289**

10,784**

127.5

0.0

127.5

Σ *

(ft)

Term 1

2.09

2.88

For frames along column lines A, B, F, and G, replace

with

in this equation.

**

Depicted in Figure 10.9 are the Term 1 and Term 2 forces in each of the frames for the seismic force in the south direction. The largest difference between the forces in the moment frames obtained from the approximate method above and the forces from the three-dimensional analysis is about 5 percent. (d) D  etermine the equivalent in-plane distributed loads on the diaphragm and construct the corresponding shear and moment diagrams.

10-29

Design Guide for Reinforced Concrete Diaphragms 1A

20 -0″

2A

3A

�

20 -0″

�

20 -0″

4A

�

20 -0″

F

75.0 kips 2.0 kips

CR

𝑥𝑥�� � 4�.4�

𝑥𝑥�� � 54.0�

24� -0″

C

24� -0″

B

A

20 -0″

6A

�

20 -0″

7A

52.5 kips 2.0 kips

127.5 kips

N

CM 𝑒𝑒� � 4.6�

𝑦𝑦�� � 𝑦𝑦�� � 72.0�

24� -0″

D

�

1.1 kips

24� -0″ 24� -0″

E

5A

1.7 kips

24� -0″

G

�

1.1 kips

Term 1 force

1.7 kips

Term 2 force

Figure 10.9 Force allocation to the frames for the seismic force in the south direction.

The equivalent in-plane distributed loads for seismic forces in the south and west directions are determined using the information in Section 6.4.3 of this publication. • Seismic force in the south direction The equivalent in-plane load for the seismic force in the south direction is trapezoidal, which accounts for the eccentricity between the CM and the CR (see Figure 10.9). The equations for force and moment equilibrium are solved simultaneously for the two unknowns and where moments are summed about column line 1 (see Figure 10.10):

Therefore,

and

• Seismic force in the west direction In the west direction, the equivalent distributed load is uniformly distributed over the length of the diaphragm and is equal to (see Figure 10.11). The main reason for including the relatively large number of significant figures for the uniform load and the frame reactions in Figure 10.11 is to demonstrate that the relatively small net shear force over the length of the collector between column lines 1

10-30

Design Guide for Reinforced Concrete Diaphragms

𝑤𝑤�,� � 1.53 -kip⁄�t

7

1A

𝑤𝑤�,� � 0.�0 -kip⁄�t 120� -0″

𝑅𝑅� � 73.0 -kip

N

𝑅𝑅� � 54.5 -kip

73.0

Shear �-kips

1,923

Moment �� -kips

54.5

ft

55.7�

Figure 10.10 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the south direction. and 4 is equal to the net shear force over the length of the diaphragm adjacent to the moment frame between column lines 4 and 7 (see Figure 10.12). Using a smaller number of significant figures results in net shear forces that appear to be not equal; this difference is due to roundoff only. (e) Determine the chord forces. • Seismic force in the south direction The maximum moment is equal to 1,923 ft-kips and occurs at approximately 55.7 ft from column line 1 (see Figure 10.10). Therefore, using Equation (6.7), the maximum chord forces are equal to the following:

At 60.0 ft from column line 1 where the diaphragm depth is 96.0 ft in the direction of analysis, the moment is equal to 1,913 ft-kips and the chord forces are equal to the following: . • Seismic force in the west direction The maximum moment is equal to 1,530 ft-kips, which is located 72.0 ft from column line A (see Figure 10.11). Therefore, using Equation (6.7), the maximum chord forces are equal to the following:

10-31

Design Guide for Reinforced Concrete Diaphragms GA

24� -0″

𝑅𝑅� � 31.875 -kip 31.875

Shear �-kips

AA

𝑤𝑤� � 0.885 -kip⁄�t

𝑅𝑅� � 31.875 -kip

96� -0″

E 24� -0″

𝑅𝑅� � 31.875 -kip

𝑅𝑅� � 31.875 -kip

42.510

10.635

1,530

510 Moment �� -kips ft

Figure 10.11 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the west direction.

At 24.0 ft from column line A, the maximum chord forces are equal to the following:

(f) Determine the unit shear forces, net shear forces, and collector forces. • Seismic force in the south direction The maximum unit shear force in the diaphragm occurs along column line 7:

Collectors are not required in this direction of analysis because the moment frames extend the full depth of the diaphragm. • Seismic force in the west direction The maximum unit shear force in the diaphragm occurs along column lines A and G:

Collector elements are required along column lines B and F. The unit shear forces, net shear forces, and collector forces along column line F is given in Figure 10.12 assuming the collector is the portion of the slab in line with the frame along column lines F and that the width of the collector is equal to the width of the beam in the moment frame (the forces are the same for the collector along column line B).

10-32

Design Guide for Reinforced Concrete Diaphragms 1A

24� -0″

G

�

20 -0″

2A

3A

�

20 -0″

4A

�

20 -0″

10.635 � 0.177 kips⁄ft 60

F 42.510 � 0.354 kips⁄ft 120

20 -0″

5A

�

20 -0″

6A

�

20 -0″

7A

31.875 � 0.531 kips⁄ft 60

N

0.531 kips⁄ft

A

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

�

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

0.177 kips⁄ft

28� � 20� beam �t�p.�

0.354 kips⁄ft

0.177 kips⁄ft 0.177 kips⁄ft

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂 10.6 kips

Figure 10.12 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line F. • Step 5 – Determine Combined Load Effects Combined load effects are determined for this building that is assigned to SDC B using the applicable strength design load combinations in ACI 5.3 (see Table 5.1 of this publication and Figure 9.4). The governing combined out-of-plane load effects are determined in Part 2(a) of Step 2 above. The maximum factored uniform load on the roof slab due to dead and rain loads is equal to 177.8 psf (see Table 10.4). As noted previously, the governing in-plane load effects are due to design seismic forces. Thus, ACI Equation (5.3.1e) must be used, which reduces to the following in this example:

The effect of horizontal and vertical earthquake-induced forces, (5.3), and (5.4) of this publication):

, is determined by ASCE/SEI 12.4.2 (see Equations (5.1),

For buildings assigned to SDC B, the redundancy factor, , is permitted to be taken as 1.0 (ASCE/SEI 12.3.4.1). Also, according to the second exception in ASCE/SEI 12.4.2.2, the vertical seismic load effect, , is permitted to be taken as zero for buildings assigned to SDC B. Thus, in this example, where the effects of the horizontal seismic forces, , have been determined in Step 4 above. • Step 6 – Determine the Chord Reinforcement The flowchart in Figure 9.5 is used to determine the required area of chord reinforcement in both directions. The chord forces in the north-south and east-west directions are determined in Part 3(e) of Step 4 above. 1.

Seismic forces in the north-south direction

10-33

Design Guide for Reinforced Concrete Diaphragms For seismic forces in the north-south direction, the chord forces are equal to 14.1 kips and 21.0 kips, and the required areas of chord reinforcement are determined by Equation (8.4) of this publication: At 55.7 ft from column line 1:

At 60.0 ft from column line 1:

At column lines A and G, provide 1-#5 chord bar (provided ). These chord bars can be located just outside the cross-section of the beams in the moment frames (see Figure 8.2). Placing the bars at these locations makes the length of the moment arm equal to about 0.97 times the depth of the diaphragm, which is approximately equal to the assumed length of 0.95 times the depth of the diaphragm. Because the portions of the slab along column lines B and F act as collectors for seismic forces in the east-west direction, the larger of the area of chord reinforcement determined above for seismic forces in the north-south direction (which is equal to 0.39 in.2) and the area of collector reinforcement determined in Step 11 below will be provided along these column lines. 2. Seismic forces in the east-west direction For seismic forces in the east-west direction, the required areas of chord reinforcement are equal to the following: At 72.0 ft from column line A:

At 24.0 ft from column line A:

At column lines 1 and 7, provide 1-#5 bar (provided ) located just outside the cross-section of the beams in the moment frames. For simpler detailing, provide 1-#5 bar (provided ) adjacent to column line 4. These bars must be fully developed in tension into the slab. One method to accomplish this is to develop the axial tension force in the chord reinforcement through shear transfer. The required development length is determined by dividing the axial force in the reinforcing bar(s) by the shear capacity of the diaphragm. In this case, the axial tension force corresponding to 1-#5 chord bar is equal to . The design shear strength of the diaphragm is determined using Equation (7.3) of this publication where and the shear reinforcement, , is conservatively taken equal to zero:

Therefore, the development length . These reinforcing bars are extended one bay length to the north and south of column lines B and F, respectively (extending the reinforcement in this fashion corresponds more closely to the layout that could be obtained from a strut-and-tie model at that corner of the diaphragm). To ensure continuity at the corners, the chord reinforcement along column lines A and G is lap-spliced to the chord reinforcement along column line 4 using a Class B tension lap splice length. • Step 7 – Determine the Diaphragm Shear Reinforcement The flowchart in Figure 9.6 is used to determine the area of shear reinforcement, if required. The largest factored unit shear force in the diaphragm is equal to 0.57 kips/ft, which occurs along column line 7 for the seismic force in the south direction (see Part 3(f) in Step 4 above). The design shear strength of the diaphragm is determined using Equation (7.3) of this publication where based on ordinary reinforced concrete moment frames resisting

10-34

Design Guide for Reinforced Concrete Diaphragms the earthquake effects (ACI 21.2.4) and assuming the shear reinforcement above:

Alternatively, can be determined from Table 7.3 of this publication: , and 4,000 psi concrete. Thus,

. Using the information in Part 2 of Step 6

for an 8.0-in. slab thickness, .

Therefore, the shear strength of the diaphragm is adequate without shear reinforcement. • Step 8 – Determine the Shear Transfer Reinforcement The flowchart in Figure 9.7 is used to determine the required areas of shear transfer reinforcement between the diaphragm and the frames and between the diaphragm and the collector elements. 1.

Determine the shear transfer reinforcement between the diaphragm and the moment frames. (a) Seismic forces in the north-south direction The unit shear force in the frame along column line 7 is equal to 0.57 kips/ft (see Part 3(f) in Step 4 above). The required area of shear-friction reinforcement is determined by Equation (8.9) of this publication assuming that the slab and beams are cast monolithically (that is, from ACI Table 22.9.4.2):

Because the required area of shear-friction reinforcement is very small, it is safe to assume that the bottom flexural reinforcement in the slab (9-#4 bars in the column strips and in the middle strips, which is minimum reinforcement) can be used as the shear transfer reinforcement between the diaphragm and the frame along this column line. The bottom flexural reinforcement in the slab can also be used as the shear transfer reinforcement between the diaphragm and the frame along column line 1 where the unit shear force is equal to 0.51 kips/ft. (b) Seismic forces in the east-west direction A similar calculation is performed for the required shear-friction reinforcement between the diaphragm and the frames along column lines A and G where the unit shear force is equal to 0.53 kips/ft (see Part 3(f) in Step 4 above):

The bottom flexural reinforcement in the slab (10-#4 bars in the column strips and 9-#4 bars in the middle strips, the latter of which is minimum reinforcement) can be used as the shear transfer reinforcement between the diaphragm and the frames along these lines. It can also be used as the shear transfer reinforcement between the diaphragm and the frames along column lines B and F where the unit shear force is equal to 0.177 kips/ft (see Figure 10.12). 2.

Determine the shear transfer reinforcement between the diaphragm and the collectors. The largest unit shear force between the diaphragm and the collectors along column lines B and F is equal to (see Figure 10.12). As noted previously, the collector is part of the slab and has the same width as the beams in the moment frames, so from ACI Table 22.9.4.2 for concrete cast monolithically. Therefore, the required shear-friction reinforcement is determined by Equation (8.9):

Just like the shear-friction reinforcement between the diaphragm and the frames, the bottom flexural reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and both collectors. • Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces Because the collectors are the same width as the beams that they frame into, there is no eccentricity of forces; thus, this type of reinforcement is not required.

10-35

Design Guide for Reinforced Concrete Diaphragms • Step 10 – Determine the Anchorage Reinforcement The flowchart in Figure 9.9 is used to determine the required area of anchorage reinforcement. For this building assigned to SDC B, both a wind anchorage force and a seismic anchorage force must be determined. 1.

Determine the wind connection force. , is determined on the cladding using the informaThe out-of-plane components and cladding (C&C) wind pressure, tion provided in Section 4.4.2 of this publication. The provisions in Part 3 of Chapter 30 are selected to determine this wind pressure for this building that meets the criteria in ASCE/SEI 30.1 and 30.5 (see Table 4.6 of this publication). Design wind pressure

is determined by ASCE/SEI Equation (30.5-1):

Using and the information in Part 1(a) of Step 3 above, it can be determined that the velocity pressures at the second-floor level and at the mean roof height of the building are equal to 23.6 psf and 34.1 psf, respectively. The external pressure coefficients are determined from ASCE/SEI Figure 30.5-1. Assuming the effective wind area for the anchorage reinforcing bars is less than 10 sq ft, the external pressure coefficients are the following for Zone 5, which is the wall zone with the highest wind pressure:

Anchorage reinforcement is determined for wind acting away from the surface, which produces tension in the reinforcing bars. Thus, the governing is equal to the following for an enclosed building (internal pressure coefficients are given in ASCE/SEI Table 26.13-1):

This pressure is greater than the minimum design pressure of 16.0 psf given in ASCE/SEI 30.2.2. 2.

Determine the seismic connection force. The anchorage of the cladding (nonstructural component) to the diaphragm must satisfy the requirements of ASCE/ SEI 13.4. According to ASCE/SEI 13.4.1, the seismic design force in the attachment must be determined by ASCE/SEI Equation (13.3-1):

From ASCE/SEI Table 13.5-1, which contains coefficients for architectural components, component amplification factor, , is equal to 1.0 and component response modification factor, , is equal to 2.5 for exterior nonstructural wall elements and connections. The component importance factor, , is equal to 1.0 in this case because none of the conditions in ASCE/SEI 13.1.3 are met. Because the connection of the cladding occurs at the roof height, , , and is equal to the following:

This force is between the lower and upper limits prescribed in ASCE/SEI 13.3.3.1. 3.

Determine the required anchorage reinforcement. It is evident that the wind pressure governs, so the required anchorage reinforcement is determined by Equation (8.24) of this publication:

10-36

Design Guide for Reinforced Concrete Diaphragms 1A

20 -0″

24� -0″

G

2A

�

�

20 -0″

3A

20 -0″

F 24� -0″

Similar to Fig. 8.2 �typ.�

24� -0″ D

20 -0″

�

20 -0″

6A

�

20 -0″

7A

③

②

①

5A

�

①

②

E

4A

�

②

② 24� � 24� column �typ.�

①

N

24� -0″

28� � 20� beam �typ.�

24� -0″

C ②

24� -0″

B ②

2� -4″

A

1-#5

③

①

Detail A �typ.�

②

1. ① ─ 1-#5 chord reinforcement 2. ② ─ Shear transfer reinforcement – bottom flexural reinforcement in slab 3. ③ ─ 1-#6 chord/collector reinforcement 4. Provide standard 90-deg hooks at the ends of all bars. 5. Provide Class B lap splices where required. 6. Other reinforcement not shown for clarity.

Detail A

Figure 10.13 Reinforcement details for the diaphragm at the roof level in Example 10.2.

The required anchorage reinforcement is equal to that required for shear transfer at the perimeter of the building (see Step 8 above). • Step 11 – Determine the Collector Reinforcement The flowchart in Figure 9.10 is used to determine the required area of longitudinal reinforcement in the collectors along column lines B and F. The axial force is equal to 10.6 kips (see Figure 10.12). The required area of longitudinal collector reinforcement is equal to the following:

10-37

Design Guide for Reinforced Concrete Diaphragms

The required chord reinforcement along these column lines for seismic forces in the north-south direction is equal to 0.39 in.2, which is greater than the collector reinforcement for seismic forces in the perpendicular direction (see Step 6(a) above). Because orthogonal load effects need not be considered for this building, provide 1-#6 bar (provided ) along column lines B and F between column lines 1 and 7. This reinforcing bar can be located within the cross-section of the beam in the ordinary moment frame between column lines 4 and 7 and is in addition to the required longitudinal flexural reinforcing bars in that member. Reinforcement details for the roof diaphragm are given in Figure 10.13.

10.4 Example 10.3 – Eighteen-story Residential Building (SDC C) Design and detail the diaphragm at the second-floor level for the 18-story residential building depicted in Figure 10.14 given the design data below. The height of the first story is 12 ft-6 in. and the typical story height is 9 ft-8 in. The roof is essentially flat except for ½ in. per foot slopes for drainage.

A

1

22� -0″

3A

22� -0″

4A

22� -0″

5A

�

24 -0″

F

22� -0″

2A

A

16� � 60� columns �e�els 10 � 18 20� � 60� columns �e�els 1 � �

A

�

24 -0″

E

10� walls �e�els 10 � 18 12� walls �e�els 1 � � N

A

�

24 -0″

D

20� � 30� beams �e�els 10 � 18 24� � 36� beams �e�els 1 � �

24� � 24� columns �e�els 10 � 18 28� � 28� columns �e�els 1 � �

A

�

24 -0″

C

A

�

24 -0″

B

A

10-38

Figure 10.14 Typical floor plan of 18-story residential building in Example 10.3.

Design Guide for Reinforced Concrete Diaphragms Design Data • Site information Latitude = 36.11o, Longitude = −115.14o Exposure Category: Exposure B in all directions Soil classification: Site Class C • Loads Roof live load = 100 psf (roof areas used for public assembly and for mechanical equipment) Roof superimposed dead load = 20 psf Floor live load = 40 psf in private rooms and corridors serving them = 100 psf in public rooms and corridors serving them Floor superimposed dead load = 20 psf Cladding = 15 psf Snow loads in accordance with ASCE/SEI Chapter 7 Rain loads in accordance with ASCE/SEI Chapter 8 Wind loads in accordance with ASCE/SEI Chapters 26 and 27 Seismic loads in accordance with ASCE/SEI Chapters 11, 12, 20, and 22 • Lateral force-resisting systems: In the north-south direction: moment-resisting frames along column lines 1 and 5 In the east-west direction: building frame system with ordinary reinforced concrete shear walls along column lines C and D

Solution • Step 1 – Select the Materials Assume the following material properties: Floor members: normalweight concrete with Columns and walls between stories 1 and 9: normalweight concrete with Columns and walls between stories 10 and 18: normalweight concrete with Grade 60 reinforcement The suitability of these materials will be checked once the SDC is determined (see Part 4 of Step 2 below). • Step 2 – Determine the Diaphragm Thickness The flowchart in Figure 9.1 is used to determine the minimum diaphragm thickness for this two-way slab system. 1.

Determine minimum thickness Maximum clear span From Table 3.2 of this publication, beams).

based on serviceability requirements. , which occurs at levels 10 through 18 (see Figure 10.14). for edge panels (which do not have edge

It can be determined by calculating the immediate and time-dependent deflections in accordance with ACI 24.2.3 and 24.2.4, respectively, that an 8.5-in.-thick slab is adequate to satisfy the maximum permissible deflections in ACI Table 24.2.2 at all levels of the building. 2.

Determine minimum thickness

based on two-way shear requirements.

Two-way shear requirements are checked at a typical interior column and at an edge column that is not part of the LFRS. It can be determined that on a typical floor where the live load is equal to 40 psf, shear strength requirements are satisfied using an 8.5-in. slab. In areas where the live load is 100 psf, shear reinforcement is required. Headed shear

10-39

Design Guide for Reinforced Concrete Diaphragms stud reinforcement in accordance with the strength requirements of ACI 22.6.8 can be used to supplement the shear strength of the 8.5-in. slab where the live load is equal to 100 psf. Therefore, 3.

Determine minimum thickness

based on fire resistance requirements.

Assuming that the required fire-resistance rating of the roof and floor slabs is 2 hours for this occupancy, the minimum thickness from IBC Table 722.2.2.1 based on a concrete mix with siliceous aggregate. 4.

Check if the minimum slab thickness requirements for SDC D, E, and F must be satisfied. The SDC is determined in accordance with IBC 1613.2.5 or ASCE/SEI 11.6 (see Step 3 in Section 4.2.3 of this publication). For the given latitude and longitude of the site, the earthquake spectral response acceleration parameters at short periods and at 1-second periods are and , respectively, from Reference 4 or Reference 5. For Site Class C, and from ASCE/SEI Tables 11.4-1 and 11.4-2, respectively. The earthquake spectral acceleration parameters at short periods and at 1-second periods, adjusted for site class effects, are determined by ASCE/SEI Equations (11.4-1) and (11.4-2), respectively:

Design earthquake spectral response acceleration parameters at short periods and at 1-second periods are determined by ASCE/SEI Equations (11.4-3) and (11.4-4), respectively:

From ASCE/SEI Table 11.6-1, the SDC is C for SEI Table 11.6-2, the SDC is C for

and Risk Category II. Similarly, from ASCE/ and Risk Category II.

Therefore, this building is assigned to SDC C, which means the minimum slab thickness requirements in ACI 18.12.6.1 for buildings assigned to SDC D, E, or F need not be satisfied. Also, the materials selected in Step 1 are permitted to be used for buildings assigned to SDC C where intermediate reinforced concrete moment frames are used in the north-south direction and a building frame system with ordinary reinforced concrete shear walls are used in the east-west direction. Use a slab thickness h = 8.5 in. • Step 3 – Determine the Diaphragm Design Forces 1.

Determine the in-plane forces. The flowchart in Figure 9.2 is used to determine the in-plane diaphragm forces. Wind and seismic forces are applicable for this building that is assigned to SDC C. The general structural integrity forces in ASCE/SEI 1.4.2 must also be considered but are typically less than the required seismic forces for buildings assigned to SDC C. (a) Determine the wind forces. The wind load provisions in Part 1 of ASCE/SEI Chapter 27 are used to determine wind forces for this building, which meets the conditions and limitations set forth in that part (see Table 4.1 of this publication). The basic wind speed, , is equal to 98 mph from ASCE/SEI Figure 26.5-1B or Reference 5 for the given latitude and longitude of the site. Note that this Risk Category II, enclosed building is flexible in both directions based on the applicable equations for the approximate natural frequency, , in ASCE/SEI 26.11.3 (the building meets the limitations in ASCE/SEI 26.11.2.1, which permits to be determined by the equations in ASCE/SEI 26.11.3):

10-40

Design Guide for Reinforced Concrete Diaphragms For the concrete moment-resisting frames in the north-south direction [ASCE/SEI Equation (26.11-3)]:

For the concrete shear walls in the east-west direction [ASCE/SEI Equation (26.11-4)]:

ASCE/SEI Equation (26.11-5) can also be used to determine for the concrete shear walls in the east-west direction; this results in the building being flexible in that direction as well. The provisions of ASCE/SEI 26.11.5 are used to determine the gust-effect factor for flexible buildings, , which results in the following (calculations not shown here): in the north-south direction and in the east-west direction assuming a damping ratio, , equal to 0.015 for concrete structures (see ASCE/SEI C26.11). The wind forces in the north-south and east-west directions are given in Table 10.11. Table 10.11 Wind Forces for the Building in Example 10.3 Level

Height above ground level, (ft)

Total North-South Wind Force (kips)

Total East-West Wind Force (kips)

R 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

176.83 167.17 157.50 147.83 138.17 128.50 118.83 109.17 99.60 89.63 80.17 70.50 60.83 51.17 41.50 31.83 22.17 12.50

12.0 23.7 23.5 23.2 22.9 22.6 22.3 22.0 21.6 21.2 20.8 20.4 19.9 19.3 18.7 17.9 17.0 18.4

16.8 33.3 33.0 32.6 32.2 31.8 31.4 31.0 30.5 30.0 29.5 28.9 28.2 27.5 26.6 25.6 24.4 26.6

367.4

519.9

Σ

(b) Determine the seismic forces. The Equivalent Lateral Force (ELF) Procedure in ASCE/SEI 12.8 is permitted to be used to determine the seismic forces in the north-south and east-west directions on the SFRS over the height of the building that is assigned to SDC C (see ASCE/SEI Table 12.6-1 and Section 4.2.3 of this publication). Diaphragm seismic forces are determined in accordance with ASCE/SEI 12.10.1.1. i.

Determine the seismic forces on the SFRS. The steps in Section 4.2.3 are used to determine the seismic forces on the SFRS over the height of the building in both directions.

10-41

Design Guide for Reinforced Concrete Diaphragms The seismic ground motion values are determined in Part 4 of Step 2 above and are equal to the following: and . • Determine the seismic response coefficient, The seismic response coefficient,

, in the north-south direction.

, is determined by ASCE/SEI Equation (12.8-2):

Intermediate reinforced concrete moment frames must be used (as a minimum) in buildings assigned to SDC C with no limitations (see ASCE/SEI Table 12.2-1). From ASCE/SEI Table 12.2-1, the response modification coefficient, , is equal to 5 for this system. Also, the seismic importance factor, , is equal to 1.0 for Risk Category II buildings (see ASCE/SEI Table 1.5-2). Thus,

The value of need not exceed that determined by ASCE/SEI Equations (12.8-3) or (12.8-4), whichever is applicable. These equations include the long-period transition period, , and the period of the building, . The long-period transition period, , is determined using ASCE/SEI Figures 22-14 through 22-17. At this site, from ASCE/SEI Figure 22-14. This quantity can also be obtained from Reference 5. The fundamental period of the structure, , is determined in accordance with ASCE/SEI 12.8.2. According to that section, it is permitted to determine the approximate building period, , from ASCE/SEI Equation (12.8-7):

In this equation, is the vertical distance from the base of the building to the highest level of the SFRS. Because the moment frames are used over the entire height of the building, is equal to 176.83 ft. The approximate period parameters and are obtained from ASCE/SEI Table 12.8-2. For concrete momentresisting frames that resist 100 percent of the required seismic force and that are not enclosed or adjoined by components that are more rigid and will prevent the frames from deflecting when subjected to the seismic forces, and . Therefore,

Because

Also,

,

need not exceed that determined by ASCE/SEI Equation (12.8-3):

must not be less than that determined by ASCE/SEI Equation (12.8-6):

Therefore, the seismic response coefficient, • Determine the seismic response coefficient, The seismic response coefficient,

10-42

, is equal to 0.020 in the north-south direction. , in the east-west direction.

, is determined by ASCE/SEI Equation (12.8-2):

Design Guide for Reinforced Concrete Diaphragms A building frame system with ordinary reinforced concrete shear walls can be used (as a minimum) in buildings assigned to SDC C with no limitations (see ASCE/SEI Table 12.2-1). From ASCE/SEI Table 12.2-1, the response modification coefficient, , is equal to 5 for this system. Also, the seismic importance factor, , is equal to 1.0 for Risk Category II buildings (see ASCE/SEI Table 1.5-2). Thus,

The approximate building period,

where

and

Because

,

Also,

, is determined by ASCE/SEI Equation (12.8-7):

for “all other structural systems.” Therefore,

need not exceed that determined by ASCE/SEI Equation (12.8-3):

must not be less than that determined by ASCE/SEI Equation (12.8-6):

Therefore, the seismic response coefficient, • Determine the effective seismic weight,

, is equal to 0.036 in the east-west direction. .

For this building, the weight of the slabs, beams, columns, walls, cladding, and the superimposed dead loads must all be included in . It can be determined that snow load at the roof level need not be included because the flat roof snow load is less than 30 psf [the ground snow load at this site is equal to 5 psf from ASCE/SEI Figure 7.2-1 or Reference 5; see ASCE/SEI 12.7.2(4)]. The story weights, , which are the portions of that are assigned to level in the building, are given in Tables 10.12 and 10.13. • Determine the seismic base shear,

.

The seismic base shear is determined by ASCE/SEI Equation (12.8-1). In the north-south direction:

In the east-west direction:

• Distribute the seismic base shear, The seismic force, and (12.8-12):

, over the height of the building.

, induced at any level of a building is determined by ASCE/SEI Equations (12.8-11)

10-43

Design Guide for Reinforced Concrete Diaphragms Table 10.12 North-South Seismic Forces and Story Shears on the SFRS for the Building in Example 10.3 Level

Story Weight, (kips)

Height, (ft)

R 18 17 16 15 14 13 12

1,627 1,831 1,831 1,831 1,831 1,831 1,831 1,831

176.83 167.17 157.50 147.83 138.17 128.50 118.83 109.17 99.60 89.63 80.17 70.50 60.83 51.17 41.50 31.83 22.17 12.50

1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831

11 10 9 8 7 6 5 4 3 2

1,957 Σ

32,880

Seismic Force, (kips)

Story Shear, (kips)

6,419,143 6,603,006 6,002,570 5,423,860 4,867,998 4,334,444 3,824,460 3,339,286

78.6 80.9 73.5 66.4 59.6 53.1 46.9 40.9

78.6 159.5 233.0 299.4 359.0 412.1 459.0 499.9

2,883,389 2,435,644 2,037,546 1,658,788 1,310,017 993,375 710,502 464,758 260,563 111,339

35.3 29.8 25.0 20.3 16.1 12.2 8.7 5.7 3.2 1.4

535.2 565.0 590.0 610.3 626.4 638.6 647.3 653.0 656.2 657.6

53,680,688

657.6

Table 10.13 East-West Seismic Forces and Story Shears on the SFRS for the Building in Example 10.3 Level R 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

Height, (ft)

1,627 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831

176.83 167.17 157.50 147.83 138.17 128.50 118.83 109.17 99.60 89.63 80.17 70.50 60.83 51.17 41.50 31.83 22.17 12.50

1,957 Σ

10-44

Story Weight, (kips)

32,880

Seismic Force, (kips)

Story Shear, (kips)

996,224 1,045,694 971,217 897,830 825,660 754,620 684,852 616,505 550,212 482,759 420,400 358,462 298,534 240,917 185,810 133,723 85,396 44,850

122.9 129.0 119.8 110.8 101.9 93.1 84.5 76.1 67.9 59.6 51.9 44.2 36.8 29.7 22.9 16.6 10.5 5.5

122.9 251.9 371.7 482.5 584.4 677.5 762.0 838.1 906.0 965.6 1,017.5 1,061.7 1,098.5 1,128.2 1,151.1 1,167.7 1,178.2 1,183.7

9,593,665

1,183.7

Design Guide for Reinforced Concrete Diaphragms In this equation,

is the exponent related to the period of the building:

–

for buildings where

–

for buildings where





–

is to be determined by linear interpolation between 1 and 2 for buildings that have a period between 0.5 and 2.5 seconds or can be taken equal to 2



In the north-south direction: .

.

In the east-west direction: The seismic forces over the height of the building are given in Tables 10.12 and 10.13 for seismic forces in the north-south and east-west directions, respectively. Note that these seismic forces are greater than the general lateral structural integrity forces prescribed in ASCE/SEI 1.4.2, which are equal to at each level. ii.

Determine the seismic forces on the diaphragms. The information in Section 4.2.3 of this publication is used to determine the seismic forces on the diaphragms over the height of the building. According to ASCE/SEI 12.10, diaphragms are to be designed for the larger of the following forces: • Design seismic forces, • Diaphragm design forces,

, acting on the SFRS at the levels of a building , determined by ASCE/SEI Equations (12.10-1) through (12.10-3):

Minimum Maximum In these equation, is the portion of the effective seismic weight, , that is assigned to level (see the values of in Tables 10.12 and 10.13) and is the weight that is tributary to the diaphragm at level , which can be taken as (see the discussion in Section 4.2.3 of this publication). The forces, , are the portions of the seismic base shear, , induced at level (see the values of in Tables 10.12 and 10.13). The diaphragm design forces, , over the height of the building are given in Table 10.14 and 10.15 for seismic forces in the north-south and east-west directions, respectively. The term that is related to the required minimum value of the design seismic force on the diaphragm is equal to the following:

The term that is related to the required maximum value of the design seismic force on the diaphragm is equal to the following:

It is evident from Tables 10.14 and 10.15 that the minimum seismic diaphragm force is required at all levels for seismic forces in both the north-south and east-west directions. 2. Determine the out-of-plane forces. (a) Dead load,

(b) Live load, From the design data:

10-45

Design Guide for Reinforced Concrete Diaphragms Table 10.14 Design Seismic Forces in the North-South Direction on the Diaphragms for the Building in Example 10.3 Level

Story Weight, (kips)

Seismic Force, (kips)

(kips)

(kips)

R 18 17 16 15 14 13 12

1,627 1,831 1,831 1,831 1,831 1,831 1,831 1,831

78.6 80.9 73.5 66.4 59.6 53.1 46.9 40.9

1,627 3,458 5,289 7,120 8,951 10,782 12,613 14,444

78.6 159.5 233.0 299.4 359.0 412.1 459.0 499.9

0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866*

140.9 158.6 158.6 158.6 158.6 158.6 158.6 158.6

11 10 9 8 7 6 5 4 3 2

1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,957

35.3 29.8 25.0 20.3 16.1 12.2 8.7 5.7 3.2 1.4

16,275 18,106 19,937 21,768 23,599 25,430 27,261 29,092 30,923 32,880

535.2 565.0 590.0 610.3 626.4 638.6 647.3 653.0 656.2 657.6

0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866*

158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 169.5

32,880

657.6

Σ * Minimum

(kips)

governs.

Table 10.15 Design Seismic Forces in the East-West Direction on the Diaphragms for the Building in Example 10.3 Level

Story Weight, (kips)

Seismic Force, (kips)

(kips)

(kips)

R 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

1,627 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,831 1,957

122.9 129.0 119.8 110.8 101.9 93.1 84.5 76.1 67.9 59.6 51.9 44.2 36.8 29.7 22.9 16.6 10.5 5.5

1,627 3,458 5,289 7,120 8,951 10,782 12,613 14,444 16,275 18,106 19,937 21,768 23,599 25,430 27,261 29,092 30,923 32,880

122.9 251.9 371.7 482.5 584.4 677.5 762.0 838.1 906.0 965.6 1,017.5 1,061.7 1,098.5 1,128.2 1,151.1 1,167.7 1,178.2 1,183.7

32,880

1,183.7

Σ * Minimum

10-46

governs.

(kips) 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866* 0.0866*

140.9 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 158.6 169.5

Design Guide for Reinforced Concrete Diaphragms Floor live load

in private rooms and corridors serving them in public rooms and corridors serving them

(c) Determine combined gravity load effects. The combined gravity load effects are determined using the load combinations in Table 5.1 of this publication. Maximum load effects are obtained using ACI Equation (5.3.1b): In private rooms and corridors serving them: In public rooms and corridors serving them: • Step 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces 1.

Determine the diaphragm classification. The information in Section 6.1 of this publication is used to determine the classification of this diaphragm. In the northsouth direction, the span-to-depth ratio is equal to . In the east-west direction, the span-todepth ratio is equal to . Because the span-to-depth ratios of the diaphragm are less than 2, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to wind forces in both directions of analysis (ASCE/SEI 26.2). In the case of seismic forces, the diaphragm can be classified as rigid when the two conditions in ASCE/SEI 12.31.2 are satisfied. The first condition is satisfied because the span-to-depth ratio is less than 3 in both directions. For the second condition, check if the diaphragm has any of the horizontal irregularities in ASCE/SEI Table 12.3-1. Based on the descriptions given in ASCE/SEI Table 12.3-1, it is evident that no horizontal structural irregularities exist for this structure, including Type 3 diaphragm discontinuity irregularity because (1) the area of the opening is less than 50 percent of the gross enclosed diaphragm area and (2) the stiffness of the diaphragm is essentially the same at all levels of the building except for the roof where there are no major openings in the slab and the stiffness is not more than 50 percent greater than the stiffness of the diaphragm with the opening on level 18. Therefore, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to seismic forces in both directions.

2. Select the diaphragm model. The corrected equivalent beam model with spring supports is selected for this diaphragm. As noted in Section 6.4.3, this model is best suited for buildings with rigid diaphragms and lateral force-resisting systems with elements that have different stiffnesses, which is the case for the building in this example. 3. Determine the diaphragm internal forces. The flowchart in Figure 9.3 is used to determine the internal forces in the diaphragm. It is evident from the information in Table 10.11 for wind forces and Tables 10.14 and 10.15 for seismic forces that the seismic forces are greater than the wind forces at all the levels of the building. Thus, internal forces in the diaphragm at the second-floor level are determined using the design seismic forces in Tables 10.14 and 10.15. Also, because the building does not have a Type 5 horizontal structural irregularity as defined in ASCE/SEI Table 12.3-1, the design seismic forces in the north-south and east-west directions are permitted to be applied independently in each of the two orthogonal directions and orthogonal interaction effects are permitted to be neglected (ASCE/SEI 12.5.3). (a) Determine the location of the CM. The location of the CM is determined based on the weight of the slab only (in typical cases, the weights of vertical elements and beams have a nominal effect on the location of the CM). The same slab thickness and concrete mix are used throughout the entire second-floor level, so the location of the CM is determined using the plan areas of the slab segments with the origin taken at the intersection of column lines 1 and A. It is assumed for purposes of calculations that the entire bay bounded by column lines C, D, 2, and 3 that contains the elevator, stair, and mechanical chase openings is open.

From symmetry,

.

10-47

Design Guide for Reinforced Concrete Diaphragms (b) Determine the location of the CR. from column line A due to the symmetric distribution of the structural In the north-south direction, walls in the east-west direction. There is no eccentricity between the CM and the CR in this direction, which means no inherent torsional moment is generated. Similarly, in the east-west direction, from column line 1 due to the symmetric distribution of the intermediate moment frames in the north-south direction. The eccentricity between the CM and CR in this direction is , which generates a torsional moment. (c) Determine the seismic forces in the walls and intermediate moment frames. Seismic forces are applied at the CM in both directions. Accidental torsion in accordance with ASCE/SEI 12.8.4.2 need not be applied in the analysis for strength design or when checking the story drift limits prescribed in ASCE/ SEI 12.12 because the structure, which is assigned to SDC C, does not have a Type 1a or Type 1b horizontal structural irregularity. • Seismic force in the south direction Force calculations for the walls and frames are given in Table 10.16 where it is assumed that the applied in-plane seismic force of 169.5 kips in the south direction is negative. Wall and frame stiffnesses are calculated using the applicable equations in Section 6.3.3 of this publication. The forces in Table 10.16 are in the directions that resist the applied in-plane force and torsional moment where positive forces act in the north direction. Table 10.16 Forces in the Frames and Walls for the Seismic Force in the South Direction Term 1 Frame/ Wall

(ft)

(ft)

1 5 C D

0.0 88.0 — —

— — 48.0 72.0

8.54 8.54 — —

— — 9.57 9.57

Σ

17.08

19.14

(ft)

(ft)

44.0 44.0 — —

— — 12.0 12.0

(kips)

(kips)

*For walls along column lines C and D, replace **

with

Term 2

16,533 16,533 — —

— — 1,378 1,378

84.75 84.75 — —

(kips) -1.10 1.10 0.30 -0.30

83.65 85.85 0.30 -0.30

33,066**

2,756**

169.50

0.00

169.50

in this equation.

• Seismic force in the west direction In the west direction, there is no eccentricity between the CM and the CR, so each wall resists a force equal to at the second-floor level. For comparison purposes, a three-dimensional model of the building was constructed using Reference 14. In the model, the columns and walls are fixed at the base (ASCE/SEI 12.7.1), rigid diaphragms are assigned at all levels in the building, and the following reduced moments of inertia are used, which account for the effects of cracked sections (ASCE/SEI 12.7.3): • Columns that are part of the moment-resisting frames: • Beams: • Walls: • Slabs (out-of-plane): • Slabs (in-plane): The columns that are not part of the moment-resisting frames are assigned stiffness modifiers equal to 0.001 to ensure that the effects from the lateral loads are resisted only by the structural walls and by the moment-resisting frames. The results obtained from the three-dimensional model are essentially the same as those determined from the approximate method above (there is less than a 0.25% difference between the results).

10-48

Design Guide for Reinforced Concrete Diaphragms

A

1

3A

22� -0″

22� -0″

4A

22� -0″

5A

A

24� -0″

F

22� -0″

2A

0.30 kips

84.75 kips 1.10 kips

A

CR

169.5 kips

N

CM

𝑥𝑥�� � 44.0�

𝑥𝑥�� � 44.6�

0.30 kips

𝑒𝑒� � 0.6�

A

24� -0″

C

A

24� -0″

B

𝑦𝑦�� � 𝑦𝑦�� � 60.0�

A

24� -0″

D

84.75 kips 1.10 kips

24� -0″

E

A

Term 1 force Term 2 force

Figure 10.15 Force allocation to the frames and walls for the seismic force in the south direction. The diaphragm forces at the second-floor level are obtained from the three-dimensional model of the building, which is subjected to the forces, , from the ELF Procedure where at the second-floor level is replaced with at this level (see Tables 10.14 and 10.15 and Figure 6.11 of this publication). For purposes of analysis, an additional force equal to is applied at the CM of the second-floor level in the north-south direction and an additional force equal to is applied at the CM in the east-west direction. Because rigid diaphragms are specified in the model, the forces in the diaphragm are determined using the shear forces in the vertical members of the SFRS immediately above and below the second-floor level. (d) D  etermine the equivalent in-plane distributed loads on the diaphragm and construct the corresponding shear and moment diagrams.

10-49

Design Guide for Reinforced Concrete Diaphragms 1

𝑤𝑤� � 1.�3 -kip⁄�t

88� -0″

𝑅𝑅� � 84.8 -kip

5 N

𝑅𝑅� � 84.8 -kip

84.8

Shear �-kips

1,742

Moment �� -kips

1,866

84.8

ft

33� -0″

Figure 10.16 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the south direction. The equivalent in-plane distributed loads for seismic forces in the south and west directions are determined using the information in Section 6.4.3 of this publication. • Seismic force in the south direction The equivalent in-plane load for the seismic force in the south direction is trapezoidal, which accounts for the eccentricity between the CM and the CR (see Figure 10.15). Because this eccentricity is relatively small, it is sufficiently accurate to use an equivalent uniform load equal to instead of a trapezoidal load (there is a difference of less than 0.1% between the maximum diaphragm moments calculated using the trapezoidal and uniform loads). The equivalent uniform load and the corresponding shear and moment diagrams are depicted in Figure 10.16. • Seismic force in the west direction In the west direction, the equivalent distributed load is uniformly distributed over the length of the diaphragm and is equal to (see Figure 10.17). (e) Determine the chord forces. Chord forces must be determined in both directions considering the opening in the diaphragm. • Seismic force in the south direction The maximum moment in the diaphragm is equal to 1,866 ft-kips, which is located at midspan (see Figure 10.16). Therefore, using Equation (6.7), the maximum tension chord force at column line A is equal to the following (see Figure 10.18):

10-50

Design Guide for Reinforced Concrete Diaphragms F

A

𝑤𝑤� � 1.41 -kip⁄�t 48� -0″

24� -0″

𝑅𝑅� � 84.8 -kip 𝑅𝑅� � 84.8 -kip

E 48� -0″

67.8

Shear �-kips

Moment �� -kips ft

17.0 17.0

67.8

1,627

1,525

Figure 10.17 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the west direction.

The primary tension chord force at the center of the opening at column line A, which is located 33.0 ft from column line 1, can be determined by Equation (6.16) [see Figure 10.18]:

The secondary tension chord force at the center of the opening, tion (6.17):

, at column line A is determined by Equa-

In this equation, is the positive bending moment in the subdiaphragm bounded by column lines A, C, 2, and 3. It is assumed that the subdiaphragms are fixed at both ends and are subjected to a portion of the total uniform diaphragm load, , based on the mass of the segment (see Figure 6.16 of this publication). In this direction of analysis, the subdiaphragms to the north and south of the opening have the same mass (area), so each segment resists , which is uniformly distributed over the 22.0-ft wide opening. From statics, is equal to the following for a beam that is fixed at both ends with a uniformly distributed load over its entire length:

10-51

Design Guide for Reinforced Concrete Diaphragms 1

2

22� -0� �t�i. s

F

𝐶𝐶�,���� � 15.3 -kip

3

𝐶𝐶� � 16.4 -kip

4

5

𝐶𝐶�,� � 0.9 -kip

E

C

kip

84.8

CM

84.8

12� -0�

kip

s

169.5 kips

s

𝑇𝑇�,� � 0.9 -kip

D

N

24� -0� �t�i.s

𝐶𝐶�,� � 0.4 -kip

B

𝑇𝑇�,� � 0.4 -kip

A

𝑇𝑇�,���� � 15.3 -kip

𝑇𝑇� � 16.4 -kip

1,742

Moment �� -kips

1,866

ft

33� -0�

44� -0�

Figure 10.18 Chord forces for the seismic force in the south direction. Therefore, the secondary tension chord force at the center of the opening is equal to the following:

The total tension chord force at the center of the opening is equal to , which is less than the 16.4-kip tension chord force determined above for the overall diaphragm. The larger of the two tension chord forces is used to determine the required chord reinforcement (see Step 6 below). Secondary tension chord forces develop at the corners of the opening due to the negative bending moments in the subdiaphragm bounded by column lines D, F, 2 and 3. The tension chord force that occurs along column line D is determined by Equation (6.21):

10-52

Design Guide for Reinforced Concrete Diaphragms 2

1 F

5

4

3

22� -0� �t�i. s

N

24� -0� �t�i.s

84.8

kip

𝐶𝐶� � 19.5 -kip

𝐶𝐶�,� � 1.1 -kip

169.5 kips

12� -0�

𝐶𝐶�,� � 0.5 -kip

CM

1,627

s

𝐶𝐶�,���� � 18.2 -kip

C

84.8 kips 𝑇𝑇�,� � 0.5 -kip

𝑇𝑇�,���� � 18.2 kips

D

𝑇𝑇�,� � 1.1 -kip

𝑇𝑇� � 19.5 -kip

E

1,525

B

A

Moment �� t-kips f

Figure 10.19 Chord forces for the seismic force in the west direction.

• Seismic force in the west direction The maximum moment is equal to 1,627 ft-kips, which is located at column lines C and D (see Figure 10.17). Using Equation (6.7), the maximum tension chord force at column line 1 is equal to the following (see Figure 10.19):

The primary tension chord force at the center of the opening, which is located 12.0 ft from column line C or D (that is, at midspan), can be determined by Equation (6.16) [see Figure 10.19]:

The secondary tension chord force at the center of the opening, tion (6.17):

, at column line 1 is determined by Equa-

In this equation, is the positive bending moment in the subdiaphragm bounded by column lines 1, 2, C, and D. It is assumed that the subdiaphragms are fixed at both ends and are subjected to a portion of the total uniform diaphragm load, , based on the mass of the segment (see Figure 6.16 of this publication). In this

10-53

Design Guide for Reinforced Concrete Diaphragms direction of analysis, the subdiaphragm bounded by column lines 1, 2, C, and D has a mass (area) that is equal to one-half of the subdiaphragm bounded by column lines 3, 5, C, and D, so the subdiaphragm to the west of the opening resists over the 24.0-ft width. From statics, is equal to the following for a beam that is fixed at both ends with a uniformly distributed load over its entire length:

Therefore, the secondary tension chord force at the center of the opening is equal to the following:

The total tension chord force at the center of the opening is equal to , which is less than the 19.5-kip tension chord force determined above for the overall diaphragm. The larger of the two tension chord forces is used to determine the required chord reinforcement (see Step 6 below). Secondary tension chord forces develop at the corners of the opening due to the negative bending moments in the subdiaphragm bounded by column lines 3, 5, C, and D. The tension chord force that occurs along column line 3 is determined by Equation (6.21):

(f) Determine the unit shear forces, net shear forces, and collector forces. • Seismic force in the south direction Along column lines 1 and 5, the unit shear forces are equal to the following:

Along column line 2, (see Figure 10.16). The unit shear force is equal to the following where the 24.0-ft opening length is deducted from the overall depth of the diaphragm in that direction:

Collectors are not required along column lines 1 and 5 because the moment frames extend the entire depth of the diaphragm in this direction. Collectors are required along the west and east edges of the opening (column lines 2 and 3) to transfer shear forces from one side of the opening to the other via the subdiaphragms. Collector reinforcement is determined using the maximum shear force at the edge of the opening at the west edge (column line 2), which is equal to 42.3 kips (see Figure 10.16). • Seismic force in the west direction Along column lines C and D, the maximum shear force in the slab is equal to 67.8 kips (see Figure 10.17), which is distributed over a length of 88.0 ft. Therefore, the unit shear forces are equal to the following:

Just to the north of column line C and to the south of column line D, the shear force is equal to 17.0 kips, which is distributed over a length of ; thus, the unit shear forces at these locations are the following:

Collector elements are required along column lines C and D to transfer the forces from the diaphragm to the structural walls. The unit shear forces, net shear forces, and collector forces along column line D are given in Figure 10.20 assuming that the collectors are the portions of the slab in line with the wall along this column line and that the widths of the collectors are equal to the thickness of the wall (the force diagrams are the same along column line C). Note that in Figure 10.20, the uniform loads and axial forces are shown with more significant fig-

10-54

Design Guide for Reinforced Concrete Diaphragms

A

1

22� -0″

3A

22� -0″

4A

22� -0″

5A

24� -0″

F

22� -0″

2A

A

N

24� -0″

E

A

67.80 � 0.771 kips⁄ft 88.0

D

84.75 � 3.852 kips⁄ft 22.0

16.95 � 0.257 kips⁄ft 66.0

A

3.852 kips⁄ft

0.771 kips⁄ft

1.028 kips⁄ft

B 𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

1.028 kips⁄ft

3.081 kips⁄ft

A

C 𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

1.028 kips⁄ft

45.17 kips

1.028 kips⁄ft

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂 22.62 kips

Figure 10.20 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line D. ures after the decimal place than are used in the above calculations. The main reason for this is to demonstrate that the maximum axial collector force is equal to the same value−45.17 kips−regardless of which end of the net shear force diagram is used to calculate it. Using a smaller number of significant figures results in two slightly different values of the maximum collector force based on the net shear force areas to the west and to the east of column line 3; this difference is due to roundoff only. • Step 5 – Determine Combined Load Effects Combined load effects are determined for this building that is assigned to SDC C using the applicable strength design load combinations in ACI 5.3 (see Table 5.1 of this publication and Figure 9.4). The governing combined out-of-plane load effects are determined in Part 2 of Step 3 above. The maximum factored uniform load on the second-floor slab due to dead loads and a 100-psf live load is equal to 311.6 psf. As noted previously, the governing in-plane load effects are due to design seismic forces. Thus, ACI Equation (5.3.1e) must be used, which reduces to the following in this example for the second-floor diaphragm:

10-55

Design Guide for Reinforced Concrete Diaphragms The effect of horizontal and vertical earthquake-induced forces, (5.3), and (5.4) of this publication):

, is determined by ASCE/SEI 12.4.2 (see Equations (5.1),

For buildings assigned to SDC C, the redundancy factor, , is permitted to be taken as 1.0 (ASCE/SEI 12.3.4.1). Thus, where the effects of the horizontal seismic forces, , have been determined in Step 4 above. In buildings assigned to SDC C, collectors and their connections must be designed for the maximum of the three forces given in ASCE/SEI 12.10.2.1: 1.

 orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces F determined by the ELF Procedure of ASCE/SEI 12.8 or the modal response spectrum analysis procedure of ASCE/SEI 12.9.1. For the intermediate reinforced concrete moment frames in the north-south direction, the overstrength factor, , is equal to 3.0 (see ASCE/SEI Table 12.2-1). Therefore, the required in-plane diaphragm force at the second-floor level is equal to (see Table 10.12). Similarly, for the building frame system with ordinary reinforced concrete shear walls in the east-west direction, , and the required in-plane diaphragm force at the second-floor level based on this requirement is equal to (see Table 10.13).

2. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces determined by ASCE/SEI Equation (12.10-1). Using the information in Table 10.15, the diaphragm force, , at the second-floor level in both the north-south and east-west directions determined by ASCE/SEI Equation (12.10-1) is equal to the following:

The required diaphragm in-plane force including overstrength is equal to quirement.

based on this re-

3. F  orces calculated using the load combinations of ACE/SEI 2.3.6 with the seismic forces determined by ASCE/SEI Equation (12.10-2). The diaphragm force, Table 10.15).

, at the second-floor level based on ASCE/SEI Equation (12.10-2) is equal to 169.5 kips (see

Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the effects due to the 176.3-kip in-plane diaphragm force stipulated in the second requirement. It is determined in Part 3(f) of Step 4 above that the axial force in the collector due to the 169.5-kip diaphragm in-plane force is equal to 45.2 kips (see Figure 10.20). Thus, the required axial force in the collector according to the second requirement is equal to . • Step 6 – Determine the Chord Reinforcement The flowchart in Figure 9.5 is used to determine the required area of chord reinforcement in both directions. The chord forces for seismic forces in the north-south and east-west directions are determined in Part 3(e) of Step 4 above. 1.

Seismic forces in the north-south direction For seismic forces in the north-south direction, the maximum chord force is equal to 16.4 kips, and the required areas of chord reinforcement is determined by Equation (8.4) of this publication:

At the edges of the opening:

10-56

Design Guide for Reinforced Concrete Diaphragms At column lines A and F, provide 1-#5 chord bar (provided the flexural reinforcement at the edges of the slab.

). These chord bars can be tied to

It is evident that the required secondary chord reinforcement at the edges of the diaphragm at column lines C and D is nominal. The required collector reinforcement determined in Step 11 below for seismic forces in the east-west direction will be provided along these edges of the opening and will be sufficient for the secondary chord reinforcement. 2. Seismic forces in the east-west direction For seismic forces in the east-west direction, the required area of chord reinforcement is equal to the following:

At the edges of the opening:

At column lines 1 and 5, provide 2-#4 bars (provided ), which can be located just outside the cross-section of the edge beams in the intermediate moment frames (see Figure 8.2). Similar to seismic forces in the north-south direction, the required amount of secondary chord reinforcement along the slab edges adjacent to the opening at column lines 2 and 3 is nominal. The required collector reinforcement determined in Step 11 below for seismic force shear transfer in the north-south direction will be provided along these edges of the opening. • Step 7 – Determine the Diaphragm Shear Reinforcement The flowchart in Figure 9.6 is used to determine the area of shear reinforcement, if required. The largest factored unit shear force in the diaphragm is equal to 0.77 kips/ft, which occurs along column lines C and D for the seismic force in the south direction (see Part 3(f) in Step 4 above). The design shear strength of the diaphragm is determined using Equation (7.3) of this publication where based on ordinary reinforced concrete shear walls resisting the earthquake effects (ACI 21.2.4) and assuming the shear reinforcement :

Alternatively, can be determined from Table 7.3 of this publication: , and 4,000 psi concrete. Thus,

for an 8.5-in. slab thickness, .

Therefore, the shear strength of the diaphragm is adequate without shear reinforcement. • Step 8 – Determine the Shear Transfer Reinforcement The flowchart in Figure 9.7 is used to determine the required areas of shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS and between the diaphragm and the collectors. 1.

Determine the shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS. (a) Seismic forces in the north-south direction The unit shear force in the intermediate moment frames along column lines 1 and 5 is equal to 0.7 kips/ft (see Part 3(f) in Step 4 above). The required area of shear-friction reinforcement is determined by Equation (8.9) of this publication assuming that the slab and the beams of the intermediate moment frames are cast monolithically (that is, from ACI Table 22.9.4.2):

Because the required area of shear-friction reinforcement is very small, it is safe to assume that the bottom flexural reinforcement in the slab oriented in the east-west direction can be used as the shear transfer reinforcement between the diaphragm and the intermediate moment frames.

10-57

Design Guide for Reinforced Concrete Diaphragms (b) Seismic forces in the east-west direction A similar calculation is performed for the required shear-friction reinforcement between the diaphragm and the walls along column lines C and D where the unit shear force is equal to 3.9 kips/ft (see Figure 10.20). Assume that the slab and walls are not cast at the same time and that Construction Method A is used (see Section 8.6.3 of this publication for a description of this construction method). At the cold joint that occurs between the wall and the bottom of the slab, from ACI Table 22.9.4.2 (see Figure 8.5). The required area of shear-friction reinforcement is equal to the following for the case where the collectors are the same width as the thickness of the walls:

The bottom flexural reinforcement in the slab that is oriented in the north-south direction is 11-#5 bars in the column strips and 8-#5 bars in the middle strips for the 100 psf live load. The width of the column and middle strips is equal to 11.0 ft, so the minimum provided area of reinforcement is equal to . Thus, the bottom flexural reinforcement can be used as the shear transfer reinforcement between the diaphragm and the walls because the flexural demand on these reinforcing bars is nominal for gravity loads at the faces of the walls. If the live load on the floor areas adjacent to the wall is equal to 40 psf instead of 100 psf, 8-#5 bars are required in the column and middle strips; this reinforcement is also adequate for shear transfer for the same reason. In lieu of using the slab flexural reinforcement, #4 dowel bars spaced at 12 in. on center can be provided over the length of the wall (provided ). 2. Determine the shear transfer reinforcement between the diaphragm and the collectors. According to ASCE/SEI 12.10.2.1, the shear transfer reinforcement, which provides the connection between the diaphragm and the collectors, must be based on the largest of the three forces determined in accordance with that section. It is shown in Step 5 above that the seismic force determined by the second requirement governs (that is, the governing in-plane seismic force on the diaphragm is equal to the force determined by ASCE/SEI Equation (12.10-1) multiplied by the overstrength factor, ). Based on a diaphragm force equal to 169.5 kips, the unit shear force between the diaphragm and the collectors along column lines C and D is equal to (see Figure 10.20). In Step 5 above, it is determined that the seismic force in the diaphragm including overstrength is equal to 176.3 kips. Therefore, the required unit shear force is equal to . As noted previously, the collectors are part of the slab and have the same widths as the thickness of the walls, so from ACI Table 22.9.4.2 for concrete cast monolithically. Therefore, the required shear-friction reinforcement is determined by Equation (8.9):

Just like the shear-friction reinforcement between the diaphragm and the intermediate moment frames in the northsouth direction, the bottom flexural reinforcement in the slab that is oriented in the north-south direction can be used as the shear transfer reinforcement between the diaphragm and both collectors. • Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces Because the width of the collectors is equal to the thickness of the walls, there is no eccentricity of forces; thus, this type of reinforcement is not required. • Step 10 – Determine the Anchorage Reinforcement The flowchart in Figure 9.9 is used to determine the required area of anchorage reinforcement. For this building that is assigned to SDC C, both wind and seismic anchorage requirements must be considered. 1.

Determine the wind connection force. The out-of-plane components and cladding (C&C) wind pressure, , is determined on the cladding using the information provided in Section 4.4.2 of this publication. The provisions in Part 3 of Chapter 30 are selected to determine this wind pressure for this building that meets the criteria in ASCE/SEI 30.1 and 30.5 (see Table 4.6 of this publication).

10-58

Design Guide for Reinforced Concrete Diaphragms Design wind pressure

is determined by ASCE/SEI Equation (30.5-1):

Using and the information in Part 1(a) of Step 3 above, it can be determined that the velocity pressures at the second-floor level and at the mean roof height of the building are equal to 12.0 psf and 24.3 psf, respectively. The external pressure coefficients are determined from ASCE/SEI Figure 30.5-1. Assuming the effective wind area for the anchorage reinforcing bars is less than 10 sq ft, the external pressure coefficients are the following for Zone 5, which is the wall zone with the highest wind pressure:

Anchorage reinforcement is determined for wind acting away from the surface, which produces tension in the reinforcing bars. Thus, the governing is equal to the following for an enclosed building (internal pressure coefficients are given in ASCE/SEI Table 26.13-1):

This pressure is greater than the minimum design pressure of 16.0 psf given in ASCE/SEI 30.2.2. 2. Determine the seismic connection force. The anchorage of the cladding (nonstructural component) to the diaphragm must satisfy the requirements of ASCE/ SEI 13.4. According to ASCE/SEI 13.4.1, the seismic design force in the attachment must be determined by ASCE/SEI Equation (13.3-1):

From ASCE/SEI Table 13.5-1, which contains coefficients for architectural components, the component amplification factor, , is equal to 1.0 and the component response modification factor, , is equal to 2.5 for exterior nonstructural wall elements and connections. The component importance factor, , is equal to 1.0 in this case because none of the conditions in ASCE/SEI 13.1.3 are met. Because the connection of the cladding occurs at the second-floor level, , and is equal to the following:

This force is less than the lower limit prescribed in ASCE/SEI 13.3.3.1, which is equal to must be used.

, so

3. Determine the required anchorage reinforcement. It is evident that the wind pressure governs, so the required anchorage reinforcement is determined by Equation (8.24) of this publication:

Because the required area of anchorage reinforcement is very small, it is safe to assume that the excess flexural reinforcement in the slab may be used to satisfy this requirement.

10-59

Design Guide for Reinforced Concrete Diaphragms • Step 11 – Determine the Collector Reinforcement 1.

Collectors along column lines C and D. The flowchart in Figure 9.10 is used to determine the required area of longitudinal reinforcement in the collectors along column lines C and D. The maximum axial force in the collectors is equal to 47.0 kips (see Step 5 above). The required area of longitudinal collector reinforcement is equal to the following:

Provide 2-#6 bars (provided ) along column lines C and D. These reinforcing bars are in addition to any other reinforcement in the slab and must be placed within the 12.0-in. thickness of the walls. It is important to ensure that the collector reinforcement does not cause any congestion issues within the wall or within the 28.0-in. square columns at the ends of the wall. If congestion issues or any other problems occur, the collectors can be made wider than the thickness of the walls (this would require reanalysis of the force distribution in the diaphragm; see the comments section below). Collector reinforcement must extend along the length of the walls in accordance with ACI 12.5.4.3. Extending all the collector bars the full length of the walls ensures uniform shear flow across the wall segment below the slab, which must resist the shear force from the wall above plus the tension force transferred by the collector. It may be possible to terminate one of the collector reinforcing bars along the depth of the diaphragm based on the magnitude of the axial tension force in the collector (see Figure 10.20; note that the axial forces in the figure must be increased by the factor to account for overstrength, as shown in Part 2 of Step 8 above). One of the #6 bars can be theoretically cut off at the location where 1-#6 bar is adequate to resist the corresponding axial tension force, which is equal to . From Figure 10.20 with the axial forces equal to 1.04 times those shown in the figure, the theoretical cut off point is located approximately 21.7 ft to the east of column line 3. In accordance with ACI 12.5.4.3, the 1-#6 bar must extend past this cut off point at least a tension development length, , which is equal to 1 ft-7 in. (see ACI 25.4.2.3). Therefore, 1-#6 bar must extend at least to the east of column line 3. Similarly, the 1-#6 bar must extend at least to the west of column line 3. 2. Collectors along column lines 2 and 3. It is assumed that reinforcement in the slab transfers the shear force at the edge of the opening to the subdiaphragms above and below the opening, which subsequently transfer the total shear force to reinforcement at the other edge of the opening (see Section 8.9.4 of this publication). The largest shear force adjacent to the opening occurs along column line 2 for the seismic force in the south direction and is equal to (see Part 3(f) in Step 4 above). The required area of collector reinforcement is determined using Equation (8.27) assuming that the uniform shear flow in the diaphragm at this location is transferred to the collector reinforcing bars as a tension force:

Provide 2-#6 bars (provided ) along the edges of the opening at column lines 2 and 3. These bars must extend the full depth of the diaphragm in the north-south direction to ensure shear transfer from one side of the opening to the other. Collector reinforcing bars must be spliced using a Class B tension lap splice or a Type 2 mechanical connector. Reinforcement details for the second-floor diaphragm are given in Figure 10.21. For simpler detailing, the 2-#6 collector bars on column lines C and D are extended the full depth of the diaphragm; 1-#6 bar is not cutoff as determined in Part 1 of Step 11 above. Comments. It is assumed in this example that the building is constructed utilizing Construction Method A. If Construction Method B were used for the walls instead (that is, the walls are constructed ahead of the rest of the structure), the analysis and design is essentially the same as that outlined above for Construction Method A. The only difference is in the reinforcement details at the interface of the diaphragm and the walls: The cold joint occurs at this interface and a proprietary form-saver is likely to be used to transfer the shear from the diaphragm to the walls (see Figures 8.10 and 8.18).

10-60

Design Guide for Reinforced Concrete Diaphragms

A

1

22� -0″

3A

22� -0″

22� -0″

A

22� -0″

5A

③ Similar to Fig. �.2 �typ.�

④

24� -0″

E

4A

①

24� -0″

F

2A

④

A

④

24� -0″

D

③

②

②

N

A

③

C

A

24� -0″

④

24� -0″

B

A

①

A

1. ① ─ 1-#5 chord reinforcement 2. ② ─ 2-#4 chord reinforcement 3. ③ ─ 2-#6 collector reinforcement 4. ④ ─ Shear transfer reinforcement – bottom flexural reinforcement in the slab; at the walls at column lines C and D, #4 @ 12″ dowel bars can be used instead. 5. Provide standard 90-deg hooks at the ends of all bars. 6. Provide Class B lap splices or Type 2 mechanical connectors where required. 7. Other reinforcement not shown for clarity.

Figure 10.21 Reinforcement details for the diaphragm at the second-floor level in Example 10.3. It was determined in Part 1 of Step 11 above that 2-#6 bars are required as longitudinal reinforcement for the collectors along column lines C and D. These bars were designed assuming that the maximum axial force of 47.0 kips from the collectors was transferred directly into the 12.0-in. thickness of the walls. If it is determined that this reinforcement causes congestion or any other type of construction issue, the collectors can be made wider than the thickness of the walls. Assume 5-#4 collector bars are used (provided ) instead of the 2-#6 bars, and that 1-#4 bar is provided within the 12.0-in. wall thickness; the remaining 4-#4 bars are uniformly distributed within the effective slab width, , which in this case is equal to (see Figure 6.9). For 1-#4 bar, suming

. Given then

and and

, and as.

10-61

Design Guide for Reinforced Concrete Diaphragms

A

1

22� -0″

3A

22� -0″

22� -0″

4A

22� -0″

5A

�

24 -0″

F

2A

A

N

Bottom flexural reinforcement in slab

2-#4 �t��.�

A

D

A

4-#4

A

1-#4 1� -0″

12� -0″

4-#4

1-#4

8.5″

�

24 -0″

E

Slab reinforcement �t��.�

Section A-A

Figure 10.22 Reinforcement details for the collector along column line D assuming the collector is wider than the wall. The total shear force that must be transferred between the diaphragm and the walls is equal to 3.9 kips/ft (see Figure 10.20) plus , which is equal to 6.3 kips/ft. The required shear-friction reinforcement is equal to the following:

In areas where the live load is equal to 100 psf, 11-#5 bottom bars ( ) are provided in the column and middle strips, which are adequate for shear-friction requirements (see Part 1(b) of Step 8 above). Elsewhere, 8-#5 bottom bars are provided, which are also adequate ( ). The in-plane bending moment, , due to the eccentricity, , between the portion of the collector force that is not transferred directly in to the ends of the walls and the centerline of the walls can be determined by Equation (8.20) of this publication:

In this equation, , equal to zero,

10-62

. Conservatively taking the shear strength of the diaphragm due to the reinforcement, is equal to the following:

Design Guide for Reinforced Concrete Diaphragms The required reinforcement,

, is determined by Equation (8.22):

Provide 2-#4 bars ( ) placed perpendicular to the face of the walls at both ends; these bars must be developed for tension into the slab and into the wall. Reinforcement details for this case are given in Figure 10.22.

10.5 Example 10.4 – Thirty-story Office Building (SDC D) Design and detail the diaphragm at the second-floor level for the 30-story office building depicted in Figure 10.23 given the design data below. The height of the first story is 14 ft-0 in. and the typical story height is 11 ft-0 in. The roof is essentially flat except for ½ in. per foot slopes for drainage. 1 G

40� -0″

2

30� -0″

3

40� -0″

4

F

25� -0� �t����

E Wide-module joists �t���� N

D

C

B

A

36� � 36� columns �e�els 20 � 30, 𝑓𝑓�� � 4,000 �si 42� � 42� columns �e�els 11 � 1�, 𝑓𝑓�� � 6000 �si 48� � 48� columns �e�els 1 � 10, 𝑓𝑓�� � 8,000 �si 10� walls �e�els 20 � 30, 𝑓𝑓�� � 4,000 �si 14� walls �e�els 11 � 1�, 𝑓𝑓�� � 6000 �si 18� walls �e�els 1 � 10, 𝑓𝑓�� � 8,000 �si

Wide-module joists: 24 � 4�5 � � � 53, 𝑓𝑓�� � 4,000 �si

Beams: 56� � 28�5� , 𝑓𝑓�� � 4,000 �si

Figure 10.23 Typical floor plan of the 30-story office building in Example 10.4.

10-63

Design Guide for Reinforced Concrete Diaphragms Design Data • Site information Latitude = 36.17, Longitude = −115.16o Exposure Category: Exposure B in all directions Soil classification: Site Class D (default) • Loads Roof live load = 20 psf Roof superimposed dead load = 25 psf Floor live load = 50 psf + 15 psf for partitions Floor/roof framing: wide-module joists 24 + 4.5 × 7 + 53 (weight = 101.3 psf) Floor superimposed dead load = 15 psf Cladding = 10 psf Snow loads in accordance with ASCE/SEI Chapter 7 Rain loads in accordance with ASCE/SEI Chapter 8 Wind loads in accordance with ASCE/SEI Chapters 26 and 27 Seismic loads in accordance with ASCE/SEI Chapters 11, 12, 20, and 22 • Lateral force-resisting systems: Dual system with special reinforced concrete shear walls and special reinforced concrete moment frames capable of resisting at least 25 percent of the prescribed seismic forces in both directions. The special moment frames are on column lines 1, 4, A, C, E, and G.

Solution • Step 1 – Select the Materials Assume the following material properties: Floor members: normalweight concrete with Columns and walls between stories 1 and 10: normalweight concrete with Columns and walls between stories 11 and 20: normalweight concrete with Columns and walls between stories 21 and 30: normalweight concrete with Grade 60 reinforcement The suitability of these materials will be checked once the SDC is determined (see Part 4 of Step 2 below). • Step 2 – Determine the Diaphragm Thickness The flowchart in Figure 9.1 is used to determine the minimum diaphragm thickness for the one-way slab spanning between the ribs of the wide-module joist system. 1.

Determine minimum thickness Maximum clear span From Table 3.1 of this publication,

based on serviceability requirements. between the ribs. for one end continuous.

A 4.5-in. slab thickness is provided, which is greater than the minimum thickness that is required for serviceability. 2. Determine minimum thickness

based on one-way shear requirements.

It can be determined that one-way shear requirements are satisfied for the 4.5-in. slab at both the roof and floor levels. Therefore,

10-64

Design Guide for Reinforced Concrete Diaphragms 3. Determine minimum thickness

based on fire resistance requirements.

Assuming that the required fire-resistance rating of the roof and floor slabs is 2 hours for this occupancy, the minimum thickness from IBC Table 722.2.2.1 based on a concrete mix with carbonate aggregate. It is assumed that the 4.5-in. thickness is acceptable to the building official in this case; if not, the slab thickness must be increased to at least 5.0 in. 4. Check if the minimum slab thickness requirements for SDC D, E, and F must be satisfied. The SDC is determined in accordance with IBC 1613.2.5 or ASCE/SEI 11.6 (see Step 3 in Section 4.2.3 of this publication). For the given latitude and longitude of the site, the earthquake spectral response acceleration parameters at short periods and at 1-second periods are and , respectively, from Reference 4 or Reference 5. For Site Class D, and by linear interpolation from ASCE/SEI Tables 11.4-1 and 11.4-2, respectively. The earthquake spectral acceleration parameters at short periods and at 1-second periods, adjusted for site class effects, are determined by ASCE/SEI Equations (11.4-1) and (11.4-2), respectively:

Design earthquake spectral response acceleration parameters at short periods and at 1-second periods are determined by ASCE/SEI Equations (11.4-3) and (11.4-4), respectively:

From ASCE/SEI Table 11.6-1, the SDC is D for and Risk Category II. Similarly, from ASCE/SEI Table 11.6-2, the SDC is D for and Risk Category II. Therefore, this building is assigned to SDC D, which means the minimum slab thickness requirements in ACI 18.12.6.1 for buildings assigned to SDC D, E, or F must be satisfied. According to that section, concrete slabs serving as diaphragms must be at least 2 in. thick. Providing a 4.5-in.-thick slab satisfies this requirement. Also, the materials selected in Step 1 are permitted to be used for special moment frames and special structural walls in buildings assigned to SDC D (ACI 18.2.5). Use a slab thickness h = 4.5 in. • Step 3 – Determine the Diaphragm Design Forces The flowchart in Figure 9.2 is used to determine the in-plane diaphragm forces. Wind and seismic forces are applicable for this building that is assigned to SDC D. The general structural integrity forces in ASCE/SEI 1.4.2 must also be considered but are typically less than the required seismic forces for buildings assigned to SDC D. 1.

Determine the in-plane forces. (a) Determine the wind forces. The wind load provisions in Part 1 of ASCE/SEI Chapter 27 are used to determine wind forces for this building, which meets the conditions and limitations set forth in that part (see Table 4.1 of this publication). The basic wind speed, , is equal to 98 mph from ASCE/SEI Figure 26.5-1B or Reference 5 for the given latitude and longitude of the site. This building, which is enclosed and is assigned to Risk Category II, has a height that is greater than the height limit of 300 ft in ASCE/SEI 26.11.2.1; therefore, the equations in ASCE/SEI 26.11.3 are not permitted to be used to determine the approximate natural frequency of the building, . However, according to ASCE/SEI C26.11, ASCE/ SEI Equation (26.11-4) can be used to calculate for concrete buildings that are less than about 400 ft in height, which is applicable in this example:

10-65

Design Guide for Reinforced Concrete Diaphragms Therefore, the building is flexible in both directions (

is the same in both directions).

The provisions of ASCE/SEI 26.11.5 are used to determine the gust-effect factor for flexible buildings, , which results in the following (calculations not shown here): in the north-south direction and in the east-west direction assuming a damping ratio, , equal to 0.015 for concrete structures (see ASCE/SEI C26.11). The wind forces in the north-south and east-west directions are given in Table 10.17. Table 10.17 Wind Forces for the Building in Example 10.4 Level

Height above ground level, (ft)

Total North-South Wind Force (kips)

R

333

22.3

31.4

30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

322 311 300 289 278 267 256 245 234 223 212 201 190 179 168 157 146 135 124 113 102 91 80 69 58 47 36 25 14

44.3 44.0 43.7 43.4 43.1 42.8 42.5 42.1 41.8 41.4 41.0 40.7 40.3 39.8 39.4 38.9 38.5 37.9 37.4 36.8 36.2 35.6 34.8 34.0 33.1 32.1 30.9 29.4 31.2

62.4 62.1 61.7 61.3 60.9 60.4 60.0 59.6 59.1 58.6 58.1 57.6 57.1 56.5 55.9 55.3 54.7 54.0 53.3 52.5 51.7 50.8 49.9 48.8 47.6 46.2 44.6 42.6 45.6

1,139.4

1,620.3

Σ

Total East-West Wind Force (kips)

(b) Determine the seismic forces. The first step is to check if the Equivalent Lateral Force (ELF) Procedure in ASCE/SEI 12.8 can be used to determine the seismic forces on the SFRS over the height of this building in the north-south and east-west directions (see Section 4.2.3 of this publication). The height of this building is 333 ft and it does not possess any of the horizontal and vertical structural irregularities in ASCE/SEI Tables 12.3-1and 12.3-2, respectively. According to ASCE/SEI Table 12.6-1, the ELF Procedure is permitted to be used in this case if the fundamental period, , of the building is less than where .

10-66

Design Guide for Reinforced Concrete Diaphragms The fundamental period, , is determined in accordance with ASCE/SEI 12.8.2. It is permitted to determine the approximate building period, , by ASCE/SEI Equation (12.8-7):

In this equation, is the vertical distance from the base of the building to the highest level of the SFRS. Because the special moment frames and special structural walls are used over the entire height of the building, is equal to 333 ft. The approximate period parameters and are obtained from ASCE/SEI Table 12.8-2. For all other structural systems that are not explicitly listed in that table, and . Therefore,

Therefore, the ELF Procedure can be used. Diaphragm seismic forces are determined in accordance with ASCE/SEI 12.10.1.1. i.

Determine the seismic forces on the SFRS. The steps in Section 4.2.3 are used to determine the seismic forces on the SFRS over the height of the building in both directions. The seismic ground motion values are determined in Part 4 of Step 2 above and are equal to the following: and . The seismic base shears are the same in both directions because the same SFRS is used in both directions. • Determine the seismic response coefficient, The seismic response coefficient,

.

, is determined by ASCE/SEI Equation (12.8-2):

As noted in the design data, a dual system with special reinforced concrete shear walls and special reinforced concrete moment frames capable of resisting at least 25 percent of the prescribed seismic forces are used in both directions. This system is permitted for buildings assigned to SDC D with no limitations (see ASCE/SEI Table 12.2-1). From ASCE/SEI Table 12.2-1, response modification coefficient, , is equal to 7. Also, the seismic importance factor, , is equal to 1.0 for Risk Category II buildings (see ASCE/SEI Table 1.5-2). Thus,

The value of need not exceed that determined by ASCE/SEI Equations (12.8-3) or (12.8-4), whichever is applicable. These equations include the long-period transition period, , and the period of the building, . The long-period transition period, , is determined using ASCE/SEI Figures 22-14 through 22-17. At this site, from ASCE/SEI Figure 22-14. This quantity can also be obtained from Reference 5. Because

Also,

,

need not exceed that determined by ASCE/SEI Equation (12.8-3):

must not be less than that determined by ASCE/SEI Equation (12.8-6):

10-67

Design Guide for Reinforced Concrete Diaphragms Therefore, the seismic response coefficient, • Determine the effective seismic weight,

, is equal to 0.026.

.

For this building, the weight of the slabs, beams, columns, walls, partitions, cladding, and the superimposed dead loads must all be included in . Snow load at the roof level need not be included because the flat roof snow load is less than 30 psf [the ground snow load at this site is equal to 5 psf from ASCE/SEI Figure 7.2-1 or Reference 5; see ASCE/SEI 12.7.2(4)]. The story weights, , which are the portions of that are assigned to level in the building, are given in Table 10.18. Table 10.18 Seismic Forces and Story Shears on the SFRS for the Building in Example 10.4 Level

Story Weight, (kips)

Height, (ft)

R 30 29 28 27

4,094 4,160 4,160 4,160 4,160

333 322 311 300 289

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

4,160 4,160 4,160 4,160 4,160 4,238 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,459 4,579 4,579 4,579 4,579 4,579 4,579 4,579 4,579 4,733

278 267 256 245 234 223 212 201 190 179 168 157 146 135 124 113 102 91 80 69 58 47 36 25 14

Σ

130,678

• Determine the seismic base shear,

Seismic Force, (kips)

Story Shear, (kips)

29,613,348 28,583,329 27,102,960 25,650,088 24,225,181

263.6 254.4 241.2 228.3 215.6

263.6 518.0 759.2 987.5 1,203.1

22,828,736 21,461,277 20,123,359 18,815,573 17,538,546 16,598,445 15,742,862 14,510,414 13,313,212 12,152,202 11,028,413 9,942,974 8,897,125 7,892,244 6,929,869 6,172,307 5,419,122 4,550,970 3,736,787 2,979,955 2,284,628 1,656,076 1,101,320 630,404 268,357

203.2 191.0 179.1 167.5 156.1 147.7 140.1 129.2 118.5 108.2 98.2 88.5 79.2 70.2 61.7 54.9 48.2 40.5 33.3 26.5 20.3 14.7 9.8 5.6 2.3

1,406.3 1,597.3 1,776.4 1,943.9 2,100.0 2,247.7 2,387.8 2,517.0 2,635.5 2,743.7 2,841.9 2,930.4 3,009.6 3,079.8 3,141.5 3,196.4 3,244.6 3,285.1 3,318.4 3,344.9 3,365.2 3,379.9 3,389.7 3,395.3 3,397.6

381,750,083

3,397.6

.

The seismic base shear is determined by ASCE/SEI Equation (12.8-1).

• Distribute the seismic base shear,

10-68

, over the height of the building.

Design Guide for Reinforced Concrete Diaphragms The seismic force, and (12.8-12):

In this equation,

, induced at any level of a building is determined by ASCE/SEI Equations (12.8-11)

is the exponent related to the period of the building:

–

for buildings where

–

for buildings where





– is to be determined by linear interpolation between 1 and 2 for buildings that have a period between 0.5 and 2.5 seconds or can be taken equal to 2

In this case: The seismic forces, , over the height of the building are given in Table 10.18. Note that these seismic forces are greater than the general lateral structural integrity forces prescribed in ASCE/SEI 1.4.2, which are equal to at each level. ii.

Determine the seismic forces on the diaphragms. The information in Section 4.2.3 of this publication is used to determine the seismic forces on the diaphragms over the height of the building. According to ASCE/SEI 12.10, diaphragms are to be designed for the larger of the following forces: • Design seismic forces, • Diaphragm design forces,

, acting on the SFRS at the levels of a building , determined by ASCE/SEI Equations (12.10-1) through (12.10-3):

Minimum Maximum In these equation, is the portion of the effective seismic weight, , that is assigned to level (see the values of in Table 10.18) and is the weight that is tributary to the diaphragm at level , which can be taken as (see the discussion in Section 4.2.3 of this publication). The forces, , are the portions of the seismic base shear, , induced at level (see the values of in Table 10.18). The diaphragm design forces,

, over the height of the building are given in Table 10.19.

The term that is related to the required minimum value of the design seismic force on the diaphragm is equal to the following:

The term that is related to the required maximum value of the design seismic force on the diaphragm is equal to the following:

It is evident from Table 10.19 that the minimum seismic diaphragm force is required at all levels in both the north-south and east-west directions. 2. Determine the out-of-plane forces. (a) Dead load,

10-69

Design Guide for Reinforced Concrete Diaphragms Table 10.19 Design Seismic Forces on the Diaphragms for the Building in Example 10.4 Level

Story Weight, (kips)

Seismic Force, (kips)

R 30 29 28 27 26 25 24

4,094 4,160 4,160 4,160 4,160 4,160 4,160 4,160

263.6 254.4 241.2 228.3 215.6 203.2 191.0 179.1

4,094 8,254 12,414 16,574 20,733 24,893 29,053 33,213

263.6 518.0 759.2 987.5 1,203.1 1,406.3 1,597.3 1,776.4

0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034*

423.3 430.1 430.1 430.1 430.1 430.1 430.1 430.1

23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2

4,160 4,160 4,238 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,343 4,459 4,579 4,579 4,579 4,579 4,579 4,579 4,579 4,579 4,733

167.5 156.1 147.7 140.1 129.2 118.5 108.2 98.2 88.5 79.2 70.2 61.7 54.9 48.2 40.5 33.3 26.5 20.3 14.7 9.8 5.6 2.3

37,373 41,533 45,771 50,114 54,456 58,799 63,141 67,484 71,827 76,169 80,512 84,854 89,313 93,892 98,471 103,050 107,629 112,208 116,787 121,366 125,945 130,678

1,943.9 2,100.0 2,247.7 2,387.8 2,517.0 2,635.5 2,743.7 2,841.9 2,930.4 3,009.6 3,079.8 3,141.5 3,196.4 3,244.6 3,285.1 3,318.4 3,344.9 3,365.2 3,379.9 3,389.7 3,395.3 3,397.6

0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034* 0.1034*

430.1 430.1 438.2 449.1 449.1 449.1 449.1 449.1 449.1 449.1 449.1 449.1 461.1 473.5 473.5 473.5 473.5 473.5 473.5 473.5 473.5 489.4

130,678

3,397.6

Σ * Minimum

(kips)

(kips)

(kips)

governs.

(b) Live load,

(c) Determine combined gravity load effects. The combined gravity load effects are determined using the load combinations in Table 5.1 of this publication. Maximum load effects are obtained using ACI Equation (5.3.1b):

• Step 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces 1.

Determine the diaphragm classification. The information in Section 6.1 of this publication is used to determine the classification of this diaphragm. In the northsouth direction, the maximum span-to-depth ratio is equal to . In the east-west direction, the

10-70

Design Guide for Reinforced Concrete Diaphragms maximum span-to-depth ratio is equal to . Because the span-to-depth ratios of the diaphragm are less than 2, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to wind forces in both directions of analysis (ASCE/SEI 26.2). In the case of seismic forces, the diaphragm can be classified as rigid when the two conditions in ASCE/SEI 12.31.2 are satisfied. The first condition is satisfied because the span-to-depth ratio is less than 3 in both directions. For the second condition, check if the diaphragm has any of the horizontal irregularities in ASCE/SEI Table 12.3-1. Based on the descriptions given in ASCE/SEI Table 12.3-1, it is evident that no horizontal structural irregularities exist for this structure, including Type 3 diaphragm discontinuity irregularity because (1) the total area of the openings is less than 50 percent of the gross enclosed diaphragm area and (2) the stiffness of the diaphragm is essentially the same at all levels of the building except for the roof where there are no major openings in the slab and the stiffness is not more than 50 percent greater than the stiffness of the diaphragm with the openings on level 30. Therefore, the reinforced concrete slab can be classified as a rigid diaphragm when subjected to seismic forces in both directions. 2. Select the diaphragm model. An equivalent beam model is selected for this diaphragm. The forces in the walls and frames are determined using a three-dimensional model of the building (see Part 3 of this step below, which describes how the diaphragm forces are obtained from the analysis). 3. Determine the diaphragm internal forces. The flowchart in Figure 9.3 is used to determine the internal forces in the diaphragm. It is evident from the information in Table 10.17 for wind forces and Table 10.19 for seismic forces that the seismic forces are greater than the wind forces at all levels of the building. Thus, internal forces in the diaphragm at the secondfloor level are determined using the design seismic forces in Table 10.19. Also, because the building does not have a Type 5 horizontal structural irregularity as defined in ASCE/SEI Table 12.3-1, the design seismic forces in the northsouth and east-west directions are permitted to be applied independently in each of the two orthogonal directions and orthogonal interaction effects are permitted to be neglected (ASCE/SEI 12.5.3). A three-dimensional model of the building was constructed using Reference 14. In the model, the columns and walls are fixed at the base (ASCE/SEI 12.7.1), rigid diaphragms are assigned at all levels in the building, and the following reduced moments of inertia are used, which account for the effects of cracked sections (ASCE/SEI 12.7.3): •

Columns:

•

Beams:

•

Walls:

•

Slabs (out-of-plane):

•

Slabs (in-plane):

(a) Determine the location of the CM. From symmetry,

and

.

(b) Determine the location of the CR. From symmetry,

and

.

There is no eccentricity between the CM and the CR in both directions, which means no inherent torsional moments are generated. (c) Determine the seismic forces in the special structural walls and the special moment frames. Seismic forces are applied at the CM in both directions. Accidental torsion in accordance with ASCE/SEI 12.8.4.2 need not be applied in the analysis for strength design or when checking the story drift limits prescribed in ASCE/ SEI 12.12 because the structure, which is assigned to SDC D, does not have a Type 1a or Type 1b horizontal structural irregularity. The diaphragm force, , at the second-floor level is equal to 489.4 kips in both directions (see Table 10.19). The diaphragm forces at the second-floor level are obtained from the three-dimensional model of the building, which is subjected to the forces, , from the ELF Procedure where at the second-floor level is replaced with at this level (see Tables 10.18 and 10.19 and Figure 6.11 of this publication). For purposes of analysis, an additional force equal to is applied at the CM of the second-floor level in the model.

10-71

Design Guide for Reinforced Concrete Diaphragms Because rigid diaphragms are specified in the model, the forces in the diaphragm are determined using the shear forces in the vertical members of the SFRS immediately above and below the second-floor level (see Figure 6.12). For example, consider the special moment frame along column line A, which resists a portion of the eastwest seismic force based on its relative rigidity. To determine the diaphragm force at this location, section cuts are made in the columns immediately above and below the second-floor level. The shear forces obtained from analysis are given in Figure 10.24. Immediately above the second-floor level, the total shear force is equal to . Similarly, the total shear force below this level is equal to 89.2 kips. Therefore, to maintain equilibrium, the force in the diaphragm is equal to . This force is shown as a reaction on the free-body diagram of the diaphragm in Figure 10.26 below. 1

40� -0″

2

30� -0″

3

40� -0″

6.6 kips

13.1 kips

13.1 kips

21.2 kips

23.4 kips

23.4 kips

4

6.6 kips

2nd Level 21.2 kips

����e �� di�p����� � 2 � �21.2 � 23.4� � 2 � �6.6 � 13.1� � 4�.� kips

Figure 10.24 Shear forces in the columns of the special moment frame along column line A due to the seismic force in the west direction. The seismic forces in the vertical elements of the SFRS from analysis are the reactions on the free-body diagrams of the diaphragms given in Figures 10.25 and 10.26 for the north-south and east-west directions of analysis, respectively (see below). (d) D  etermine the equivalent in-plane distributed loads on the diaphragm and construct the corresponding shear and moment diagrams. The equivalent in-plane distributed loads for seismic forces in the south and west directions are determined using the information in Section 6.4.3 of this publication. It is evident that the equivalent loads are uniformly distributed over the widths of the diaphragm in both directions. • Seismic force in the south direction The equivalent uniform load in this direction of analysis is equal to load and the corresponding shear and moment diagrams are depicted in Figure 10.25.

. This

• Seismic force in the west direction The equivalent uniform load in this direction of analysis is equal to load and the corresponding shear and moment diagrams are depicted in Figure 10.26.

. This

(e) Determine the chord forces. Chord forces must be determined in both directions considering the openings in the diaphragm. Based on the results from the analysis, these openings have a nominal effect on the overall behavior of the diaphragm. As such, chord forces are determined disregarding the openings, which means special chord and collector reinforcement at the edges of the openings need not be determined. Typically, the moment arm that is used to determine the tension chord force is equal to 95 percent of the depth of the diaphragm in the direction of analysis. If that depth is used in this example, the chord reinforcement will be located within the 4.5-in. slab outside of the cross-section of the beams in the special moment frames in both directions, similar to the detail in Figure 8.2. • Seismic force in the south direction

10-72

Design Guide for Reinforced Concrete Diaphragms 1

𝑅𝑅� � 13.7 -kip

40� -0″

𝑅𝑅� � 231.0 -kip

13.7 Shear �-kips

Moment �� -kips ft

4

𝑤𝑤� � 4.45 -kip⁄�t

21

30� -0″

𝑅𝑅� � 231.0 -kip

40� -0″

N

𝑅𝑅� � 13.7 -kip

66.7

164.3

2,511 3,011

Figure 10.25 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the south direction. The maximum moment in the diaphragm is equal to 3,011 ft-kips, which occurs at column lines 2 and 3 (see Figure 10.25). Therefore, using Equation (6.7), the maximum tension chord force at column line A is equal to the following:

• Seismic force in the west direction The maximum moment in the diaphragm is equal to 3,189 ft-kips, which occurs at the following two locations: 23.1 ft to the south of column line E and 23.1 ft to the north of column line C. Therefore, using Equation (6.7), the maximum tension chord force at column line 1 is equal to the following (see Figure 10.26):

(f) Determine the unit shear forces, net shear forces, and collector forces. • Seismic force in the south direction Along column lines 1 and 4, the unit shear forces are equal to the following:

Along column lines 2 and 3,

10-73

Design Guide for Reinforced Concrete Diaphragms G

A

𝑤𝑤� � 3.26 -kip⁄�t 25� -0″

25� -0″

25� -0″

𝑅𝑅� � 49.8 -kip 𝑅𝑅� � 156.5 -kip 124.7 49.8

𝑅𝑅� � 12.8 -kip

25� -0″

25� -0″

𝑅𝑅� � 156.5 -kip

𝑅𝑅� � 32.0 -kip

𝑅𝑅� � 32.0 -kip

𝑅𝑅� � 49.8 -kip

75.1 43.1

Shear �-kips

25� -0″

E

31.8

6.4

3,189

3,183

2,324

Moment �� -kips ft

380

225

Figure 10.26 Equivalent distributed load, shear diagram, and moment diagram for the seismic force in the west direction. The unit shear forces in the walls are based on the shear forces in each of the wall segments obtained from the analysis. For the 12.5-ft-long wall segments along column lines 2 and 3, the shear force in each segment is equal to 69.3 kips, so the unit shear forces are equal to the following:

For the 25.0-ft-long wall segments along column lines 2 and 3, the shear force in the wall is equal to , and the unit shear stress is equal to the following:

The unit shear forces, net shear forces, and collector forces along column line 2 are given in Figure 10.27 (the forces are the same along column line 3). Note that in Figure 10.27, the uniform loads and axial forces are shown with more significant figures after the decimal place than are used in the above calculations. The main reason for this is to demonstrate that the maximum axial collector force is equal to the same value−40.40 kips−regardless of which end of the net shear force diagram is used to calculate it. Using a smaller number of significant figures results in two slightly different values of the maximum collector force based on the net shear force areas; this difference is due to roundoff only. Collectors are not required along column lines 1 and 4 because the special moment frames extend the entire depth of the diaphragm in this direction.

10-74

Design Guide for Reinforced Concrete Diaphragms

66.7 � � 0.521 kips⁄ft 30� -0″ �2 � 11� 150

G

69.3 � 5.544 kips⁄ft 12.5

E

3.696 kips⁄ft

92.4 � 3.696 kips⁄ft 25.0

5.544 kips⁄ft

1.095 kips⁄ft

69.3 � 5.544 kips⁄ft 12.5

1.616 kips⁄ft

4.449 kips⁄ft

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

3.928 kips⁄ft

1.616 kips⁄ft

34.51 kips 40.40 kips

14.43 kips

1.616 kips⁄ft

25.97 kips

1.616 kips⁄ft

2.080 kips⁄ft

25.97 kips

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

F

25� -0� �t�p.�

11� -0� �t�p.�

5.544 kips⁄ft

1.616 kips⁄ft 1.095 kips⁄ft

4.449 kips⁄ft

3.928 kips⁄ft

2

1.616 kips⁄ft

14.43 kips

40.40 kips 34.51 kips

1.616 kips⁄ft

164.3 � 1.095 kips⁄ft 150

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

D

C

B

A

Figure 10.27 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line 2.

Collectors are required along column lines 2 and 3. The maximum collector axial force is equal to 40.4 kips, which occurs near column lines B and F (see Figure 10.27 and Step 5 below). These beams must be designed for combined flexure due to gravity loads and the 40.4-kip axial tension and compression force due to in-plane seismic forces. In lieu of transferring all or a portion of these axial forces into the walls, assume the entire force is to be resisted by shear in the slab adjacent to the walls. This shear force causes an unbalanced moment in the slab, which must be resisted by additional reinforcement (see Figure 8.12 of this publication and Step 9 below). • Seismic force in the west direction The unit shear forces in the diaphragm are equal to the following: – Along column lines A and G: ­– Along column lines B and F: ­– Along column lines C and E:

10-75

Design Guide for Reinforced Concrete Diaphragms 1 G

�

40 -0″

2

3

�

30 -0″

�

40 -0″

4

124.7 � 1.134 kips⁄ft 110.0

F

156.5 � 5.217 kips⁄ft 30.0

31.8 � 0.3�8 kips⁄ft 80.0

2

5.217 kips⁄ft

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

1.532 kips⁄ft

1.134 kips⁄ft

1.532 kips⁄ft

4.083 kips⁄ft

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

1.532 kips⁄ft

1.532 kips⁄ft 61.28 kips

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

36� � 36� columns �e�els 20 � 30, 𝑓𝑓�� � 4,000 psi 42� � 42� columns �e�els 11 � 1�, 𝑓𝑓�� � 6000 psi 48� � 48� columns �e�els 1 � 10, 𝑓𝑓�� � 8,000 psi 10� walls �e�els 20 � 30, 𝑓𝑓�� � 4,000 psi 61.28 14� walls �e�els 11 � 1�, 𝑓𝑓�� � 6000 psikips 18� walls �e�els 1 � 10, 𝑓𝑓�� � 8,000 psi

Wide-module joists: 24 � 4.5 � 7 � 53, 𝑓𝑓�� � 4,000 psi

Beams: 56� � 28.5� , 𝑓𝑓�� � 4,000 psi

Figure 10.28 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line F. The unit shear forces in the walls along column lines B and F and in the wall along column line D are equal to the following:

The unit shear forces, net shear forces, and collector forces along column line F are given in Figure 10.28 (the forces are the same along column line B). Note that in Figure 10.28, the uniform loads and axial forces are shown with more significant figures after the decimal place than are used in the above calculations. The main reason for this is to demonstrate that the maximum axial collector force is equal to the same value−61.28 kips−regardless of which end of the net shear force diagram is used to calculate it. Using a smaller number of significant figures results in two slightly different values of the maximum collector force based on the net shear force areas; this difference is due to roundoff only. Collectors are not required along column lines A, C, E, and G because the special moment frames extend the entire depth of the diaphragm in this direction.

10-76

Design Guide for Reinforced Concrete Diaphragms Collectors are required along column lines B, D, and F. For the collector beams along column line D, it is assumed that the portion of the collector that is concentric with the wall transfers its entire axial force into the web. This is a reasonable assumption to make because the required axial force in the collector is equal to , which is relatively small (see Step 5 below); it is very likely that the required reinforcement in the collector beam for combined flexure (due to gravity loads) and axial tension and compression forces (due to in-plane seismic forces) can be accommodated within the wall with no construction issues. For the collector beams along column lines B and F, the collector axial force is equal to 61.3 kips. These beams must be designed for combined flexure due to gravity loads and the 61.3-kip axial tension and compression force due to in-plane seismic forces. In lieu of transferring all or a portion of this axial force into the wall, assume the entire force is to be resisted by shear in the slab adjacent to the walls. This shear force causes an unbalanced moment in the slab, which must be resisted by additional reinforcement (see Figure 8.12 of this publication and Step 9 below). • Step 5 – Determine Combined Load Effects Combined load effects are determined for this building that is assigned to SDC D using the applicable strength design load combinations in ACI 5.3 (see Table 5.1 of this publication and Figure 9.4). The governing combined out-of-plane load effects are determined in Part 2 of Step 3 above. The maximum factored uniform load on the second-floor slab due to dead loads and a 65-psf live load is equal to 243.6 psf. As noted previously, the governing in-plane load effects are due to design seismic forces. Thus, ACI Equation (5.3.1e) must be used, which reduces to the following in this example for the second-floor diaphragm:

The effect of horizontal and vertical earthquake-induced forces, (5.3), and (5.4) of this publication):

, is determined by ASCE/SEI 12.4.2 (see Equations (5.1),

For buildings assigned to SDC D, the redundancy factor, , is determined in accordance with ASCE/SEI 12.3.4. Because this building meets the conditions in ASCE/SEI 12.3.4.2, . Thus, where the effects of the horizontal seismic forces, , have been determined in Step 4 above. In buildings assigned to SDC D, collectors and their connections must be designed for the maximum of the three forces given in ASCE/SEI 12.10.2.1: 1.

 orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces F determined by the ELF Procedure of ASCE/SEI 12.8 or the modal response spectrum analysis procedure of ASCE/SEI 12.9.1. For dual systems with special reinforced concrete shear walls and special reinforced concrete moment frames capable of resisting at least 25 percent of the prescribed seismic forces, the overstrength factor, , is equal to 2.5 from ASCE/ SEI Table 12.2-1. The required in-plane diaphragm force at the second-floor level based on this requirement is equal to (see Table 10.18).

2. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces determined by ASCE/SEI Equation (12.10-1). Using the information in Table 10.19, the diaphragm force, tion (12.10-1) is equal to the following:

, at the second-floor level determined by ASCE/SEI Equa-

The required diaphragm in-plane force including overstrength is equal to requirement.

based on this

3. F  orces calculated using the load combinations of ACE/SEI 2.3.6 with the seismic forces determined by ASCE/SEI Equation (12.10-2).

10-77

Design Guide for Reinforced Concrete Diaphragms The diaphragm force, (see Table 10.19).

, at the second-floor level determined by ASCE/SEI Equation (12.10-2) is equal to 489.4 kips

Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the effects due to the 489.4-kip in-plane diaphragm force stipulated in the third requirement. Axial tension and compression forces are determined in Part 3(f) of Step 4 above for the 489.4-kip diaphragm force. The collector beams must be designed for these axial forces in combination with the effects due to the tributary factored gravity loads along the length of the member. • Step 6 – Determine the Chord Reinforcement The flowchart in Figure 9.5 is used to determine the required area of chord reinforcement in both directions. The chord forces for seismic forces in the north-south and east-west directions are determined in Part 3(e) of Step 4 above. 1.

Seismic forces in the north-south direction For seismic forces in the north-south direction, the maximum chord force is equal to 21.1 kips, and the required areas of chord reinforcement is determined by Equation (8.4) of this publication:

At column lines A and G, provide 2-#4 chord bars (provided ), which can be located just outside the cross-section of the beams in the special moment frames (see Figure 8.2). 2. Seismic forces in the east-west direction For seismic forces in the east-west direction, the required area of chord reinforcement is equal to the following:

At column lines 1 and 4, provide 2-#5 bars (provided ), which can be located just outside the cross-section of the beams in the special moment frames (see Figure 8.2). • Step 7 – Determine the Diaphragm Shear Reinforcement The flowchart in Figure 9.6 is used to determine the area of shear reinforcement, if required. The largest factored unit shear force in the diaphragm is equal to 1.1 kips/ft, which occurs along column lines B and F for the seismic force in the west direction (see Part 3(f) in Step 4 above). The strength reduction factor, , for shear strength of a diaphragm must not exceed the least value of for shear used for the vertical elements of the SFRS (ACI 21.2.4.2). In this example, the least value of corresponds to the special structural walls, which is equal to 0.60 in accordance with ACI 21.2.4.1. Thus, the design shear strength of the diaphragm is determined by Equation (7.3) of this publication using and assuming the shear reinforcement :

Therefore, the shear strength of the diaphragm is adequate without shear reinforcement. • Step 8 – Determine the Shear Transfer Reinforcement The flowchart in Figure 9.7 is used to determine the required areas of shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS and between the diaphragm and the collectors. 1.

Determine the shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS. (a) Seismic forces in the north-south direction • Special moment frames along column lines 1 and 4 The unit shear force in the special moment frames along column lines 1 and 4 is equal to 0.1 kips/ft (see Part 3(f) in Step 4 above). The required area of shear-friction reinforcement is determined by Equation (8.9) of this publication assuming that the slab and the beams of the special moment frames are cast monolithically (that is, from ACI Table 22.9.4.2):

10-78

Design Guide for Reinforced Concrete Diaphragms

Because the required area of shear-friction reinforcement is very small, it is safe to assume that the temperature and shrinkage reinforcement in the 4.5-in. slab that is oriented in the east-west direction can be used as the shear transfer reinforcement between the diaphragm and the special moment frames. • 12.5-ft special structural walls along columns 2 and 3 As noted in Part 3(f) of Step 4 above, it is assumed that the entire 40.4-kip axial force in the collectors is resisted by shear in the slab adjacent to the 12.5-ft wall segments (that is, ). Therefore, and (see Figure 10.27). The total unit shear force in each of the 12.5-ft wall segments is equal to . Assuming the building is constructed using Construction Method A (see Figure 8.5 of this publication), the required shear-friction reinforcement is equal to the following where the coefficient of friction, , is equal to in accordance with the construction joint requirement of ACI 18.12.10.1:

According to ACI 7.6.4.1, the minimum area of temperature and shrinkage reinforcement in the 4.5-in. slab is equal to the following:

Although #5 bars at 12.0 in. are adequate, provide #5 bars spaced at 10 in. on center (provided ) oriented in the east-west direction in the 4.5-in. slab over the length of these wall segments for simpler detailing (that is, this reinforcement matches that required for the 30.0-ft wall segments along column lines B and F determined below). • 25-ft special structural walls along columns 2 and 3 The total unit shear force in the 25.0-ft wall segments is equal to Figure 10.27), and the required area of shear-friction reinforcement is equal to the following:

(see

Although #5 bars spaced at 18 in. on center are adequate (provided ), provide #5 bars spaced at 10 in. on center oriented in the east-west direction in the 4.5-in. slab over the length of these wall segments for simpler detailing (that is, this reinforcement matches that required for the 30.0-ft wall segments along column lines B and F determined below). (b) Seismic forces in the east-west direction • Special moment frames along column lines A and G The required shear-friction reinforcement between the diaphragm and the special moment frames along column lines A and G is equal to the following (see Part 3(f) in Step 4 above):

Because only one layer of flexural reinforcement is provided in the 4.5-in. slab in this direction, add the required area of shear-transfer reinforcement to the required area of flexural reinforcement, and provide at least that total area of reinforcement along the length of these special moment frames for combined flexure and shear transfer. For the given superimposed dead loads and the live loads, the required flexural reinforcement is equal to the minimum required flexural reinforcement, which is (see ACI Table 7.6.1.1). Therefore, the total area of reinforcement that must be provided is equal to . Provide #4 bars spaced at 12 in. on center oriented in the north-south direction in the 4.5-in. slab over the length of these special moment frames (provided ). Although #3 bars spaced at 12 in.

10-79

Design Guide for Reinforced Concrete Diaphragms are adequate, #4 bars are provided to match the required reinforcement for the special moment frames along column lines C and E (see below). • Special moment frames along column lines C and E For the special moment frames along column lines C and E:

The total area of reinforcement that must be provided for combined flexure and shear transfer reinforcement is equal to . Provide #4 bars spaced at 12 in. on center oriented in the north-south direction in the 4.5-in. slab over the length of these special moment frames (provided ). • Special structural wall along column line D For the wall along column line D:

The total area of reinforcement that must be provided for combined flexure and shear transfer reinforcement is equal to . Provide #4 bars spaced at 12 in. on center oriented in the north-south direction in the 4.5-in. slab over the length of this wall (provided ). This reinforcement matches the other required reinforcement in this direction. • Special structural walls along column line B and F The total unit shear force in each of the 30.0-ft wall segments is equal to (see Figure 10.28), and the required area of shear-friction reinforcement is equal to the following:

The total area of reinforcement that must be provided for combined flexure and shear transfer reinforcement is . Provide #5 bars spaced at 10 in. on center oriented in the north-south direction in the 4.5-in. slab over the length of this wall (provided ). 2. Determine the shear transfer reinforcement between the diaphragm and the collectors. According to ASCE/SEI 12.10.2.1, the shear transfer reinforcement, which provides the connection between the diaphragm and the collectors, must be based on the largest of the three forces determined in accordance with that section. It is shown in Step 5 above that the seismic force determined by the third requirement governs (that is, the governing in-plane seismic force on the diaphragm is equal to the force determined by ASCE/SEI Equation (12.10-2) using the load combinations of ASCE/SEI 2.3.6). Based on a diaphragm force equal to 489.4 kips, the maximum unit shear force between the diaphragm and the collector beams oriented in the north-south direction occurs along column lines 2 and 3 and is equal to 1.6 kips/ft (see Figure 10.27). Therefore, the required shear-friction reinforcement is determined by Equation (8.9):

The temperature and shrinkage reinforcement oriented in the east-west direction can be used as the shear transfer reinforcement between the diaphragm and the collector beams along column lines 2 and 3. The maximum unit shear force between the diaphragm and the collector beams oriented in the east-west direction occurs along column lines B and F and is equal to 1.5 kips/ft (see Figure 10.28). Therefore, the required shear-friction reinforcement is equal to the following:

10-80

Design Guide for Reinforced Concrete Diaphragms

The total area of reinforcement that must be provided for combined flexure and shear transfer reinforcement is . Provide #4 bars spaced at 12 in. on center oriented in the north-south direction in the 4.5-in. slab over the length of these collector beams (provided ). • Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces 1.

Collector beams along column lines 2 and 3 As noted in Part 3(f) in Step 4 above, the total collector axial force is resisted by shear in the slab adjacent to the walls along column lines 2 and 3. The reinforcement, , that is required to resist the eccentric bending moment, , generated by the maximum shear force is determined by Equation (8.22) of this publication where is calculated by Equation (8.20):

Assuming that the longitudinal reinforcement in the collector beams is uniformly distributed in the cross-section, the eccentricity, , between the centerlines of the beams and the walls is equal to 9.0 in. Conservatively assuming the shear strength of the diaphragm,

, is equal to zero,

is equal to the following:

The #5 bars spaced at 10 in. on center that is provided for shear transfer between the diaphragm and the walls along column lines 2 and 3 is adequate to resist this eccentric bending moment (see Part 1(a) of Step 8 above). 2. Collector beams along column lines B and F. As noted in Part 3(f) in Step 4 above, the total collector axial force is resisted by shear in the slab adjacent to the walls along column lines B and F. The reinforcement, , that is required to resist the eccentric bending moment, , generated by the maximum shear force is determined by Equation (8.22) of this publication where is calculated by Equation (8.20):

Assuming that the longitudinal reinforcement in the collector beams is uniformly distributed in the cross-section, the eccentricity, , between the centerlines of the beams and the walls is equal to 9.0 in. With

,

and

are calculated as follows:

The longitudinal reinforcement in the collector beams along column lines 2 and 3 is adequate to resist this eccentric bending moment. • Step 10 – Determine the Anchorage Reinforcement The flowchart in Figure 9.9 is used to determine the required area of anchorage reinforcement. For this building that is assigned to SDC D, both wind and seismic anchorage requirements must be considered. 1.

Determine the wind connection force. The out-of-plane components and cladding (C&C) wind pressure, , is determined on the cladding using the information provided in Section 4.4.2 of this publication. The provisions in Part 3 of Chapter 30 are selected to determine this wind pressure for this building that meets the criteria in ASCE/SEI 30.1 and 30.5 (see Table 4.6 of this publication).

10-81

Design Guide for Reinforced Concrete Diaphragms Design wind pressure

is determined by ASCE/SEI Equation (30.5-1):

Using and the information in Part 1(a) of Step 3 above, it can be determined that the velocity pressures at the second-floor level and at the mean roof height of the building are equal to 12.0 psf and 29.1 psf, respectively. The external pressure coefficients are determined from ASCE/SEI Figure 30.5-1. Assuming that the effective wind area for the anchorage reinforcing bars is less than 10 sq ft, the external pressure coefficients are the following for Zone 5, which is the wall zone with the highest wind pressure:

Anchorage reinforcement is determined for wind acting away from the surface, which produces tension in the reinforcing bars. Thus, the governing is equal to the following for an enclosed building (internal pressure coefficients are given in ASCE/SEI Table 26.13-1):

This pressure is greater than the minimum design pressure of 16.0 psf given in ASCE/SEI 30.2.2. 2. Determine the seismic connection force. The anchorage of the cladding (nonstructural component) to the diaphragm must satisfy the requirements of ASCE/ SEI 13.4. According to ASCE/SEI 13.4.1, the seismic design force in the attachment must be determined by ASCE/SEI Equation (13.3-1):

From ASCE/SEI Table 13.5-1, which contains coefficients for architectural components, the component amplification factor, , is equal to 1.0 and the component response modification factor, , is equal to 2.5 for exterior nonstructural wall elements and connections. The component importance factor, , is equal to 1.0 in this case because none of the conditions in ASCE/SEI 13.1.3 are met. Because the connection of the cladding occurs at the second-floor level, , and is equal to the following:

This force is less than the lower limit prescribed in ASCE/SEI 13.3.3.1, which is equal to must be used.

, so

3. Determine the required anchorage reinforcement. It is evident that the wind pressure governs, so the required anchorage reinforcement is determined by Equation (8.24) of this publication:

Because the required area of anchorage reinforcement is very small, it is safe to assume that the temperature and shrinkage reinforcement that is perpendicular to column lines 1 and 4 and the #4 bars spaced at 12 in. on center that are perpendicular to column lines A and G may be used to satisfy this requirement.

10-82

Design Guide for Reinforced Concrete Diaphragms • Step 11 – Determine the Collector Reinforcement Reinforcement is determined for the 56 in. by 28.5 in. collector beams along column line B and F. These beams must be designed for the combined effects from gravity loads (bending moments and shear forces) and seismic forces (axial compression and tension). A summary of the axial forces, bending moments, and shear forces is given in Table 10.20. Table 10.20 Summary of Design Axial Forces, Bending Moments, and Shear Forces for the Collector Beams Along Column Lines B and F Load Case

Axial Force (kips)

Bending Moment (ft-kips) Negative

Positive

Shear Force (kips)

Dead

0

194.8

131.4

34.5

Live

0

41.6

28.0

7.4

±61.3

0

0

0

1

0

272.7

184.0

48.3

2

0

300.3

202.5

53.2

3

±61.3

274.0

184.8

48.6

4

±61.3

155.8

105.1

27.6

Seismic Load Combination

The design strength interaction diagram for the collector beam reinforced with 7-#7 top bars, 7-#7 bottom bars, and 1-#5 bar on each side is given in Figure 10.29. It is evident from the figure that the longitudinal reinforcement is adequate for the load combinations in Table 10.20. 3,500 3,000 2,500

Axial Force �- ips

2,000

k

1,500 1,000 500 0

0

+ + ++ ++ +250 +

500

750

1,000

1,250

1,500

1,750

-500 -1,000

Bending Moment �� t- ips f

k

Figure 10.29 Design strength interaction diagram for the collector beams along column lines B and F.

10-83

Design Guide for Reinforced Concrete Diaphragms 1

2

40� -0″

G

3

30� -0″

①

F

4

40� -0″

⑥

25� -0� �typ.�

E

Similar to Fig. 8.2 �typ.�

C

B

②

⑤

N

③

④ A A ①

4.5″

A

⑥

1. ① ─ 2-#4 chord reinforcement 2. ② ─ 2-#5 chord reinforcement 3. ③ ─ #4 @ 12″ flexural reinforcement in slab 4. ④ ─ #3 @ 12″ temperature and shrinkage reinforcement 5. ⑤ ─ #5 @ 10″ shear-transfer reinforcement at all walls 6. ⑥ ─ #5 @ 10″ shear-transfer reinforcement 7. Provide standard 90-deg hooks at the ends of all bars. 8. Provide Class B lap splices or Type 2 mechanical connectors where required. 9. Other reinforcement not shown for clarity.

7-#7

2′-41/2″

D

②

2-#5

#3 @ 8″ 4′-8″

7-#7

Section A-A

Figure 10.30 Reinforcement details for the diaphragm at the second-floor level in Example 10.4. Check if confinement reinforcement in accordance with ACI 18.12.7.5. must be provided. The maximum compressive stress in the collector is equal to the following:

Thus, transverse reinforcement satisfying ACI 18.12.7.5 need not be provided. The limit of is used because the maximum axial force in the collector is based on ASCE/SEI Equation (12.10-2) with the load combinations determined in accordance with ASCE/SEI 2.3.6 (see Step 5 above).

10-84

Design Guide for Reinforced Concrete Diaphragms The maximum shear force in the collector is equal to 48.6 kips when the member is subjected to combined gravity and seismic effects (see Table 10.20). Because the collector is subjected to significant axial tension, the design shear strength of the concrete is set equal to zero. The required spacing of #3 ties with 4 legs is the following:

Provide #3 ties with 4 legs spaced at 8.0 in. on center over the entire length of the collector beams. Note that this transverse reinforcement satisfies the detailing requirement of ACI 18.12.7.6(b) at splices and anchorage zones: Provided

Reinforcement for the other collector beams can be obtained in a similar fashion. Reinforcement details for the second-floor diaphragm are given in Figure 10.30.

10.6 Example 10.5 – Five-story Residential Building (SDC D) Design and detail the diaphragm at the second-floor level in the 5-story residential building depicted in Figure 10.31 (next page) given the design data below. All stories have a height of 10 ft-0 in. The roof is essentially flat except for ½ in. per foot slopes for drainage.

Design Data • Site information Latitude = 34.09o, Longitude −118.27o Exposure Category: Exposure B in all directions Soil classification: Site Class D (determined by geotechnical investigation) • Loads Roof live load = 20 psf Roof superimposed dead load = 20 psf Floor live load = 40 psf in private rooms and corridors serving them = 100 psf in public rooms and corridors serving them Floor superimposed dead load = 20 psf Cladding = 10 psf Snow loads in accordance with ASCE/SEI Chapter 7 Rain loads in accordance with ASCE/SEI Chapter 8 Wind loads in accordance with ASCE/SEI Chapters 26 and 27 Seismic loads in accordance with ASCE/SEI Chapters 11, 12, 20, and 22 • Lateral force-resisting systems: Building frame system with special reinforced concrete shear walls in both directions

10-85

Design Guide for Reinforced Concrete Diaphragms 2

1

3

H

20� -0� ������

4

5

6

7

20� � 20� column ������

G

24� � 24� column ������

F 12� wall ������

N

22� -0� ������

E

D

C

B

A

Solution

Figure 10.31 Floor plans of the 5-storyRoof/Levels residential 3-5 building in Example 10.5 (Floors 3-R).

• Step 1 – Select the Materials Assume normalweight concrete with and Grade 60 reinforcement. The suitability of these materials will be checked once the SDC is determined (see Part 4 of Step 2 below). • Step 2 – Determine the Diaphragm Thickness The flowchart in Figure 9.1 is used to determine the minimum diaphragm thickness for this two-way slab system without edge beams. 1.

Determine minimum thickness

based on serviceability requirements.

Maximum clear span From Table 3.2 of this publication, Therefore,

10-86

for an edge panel.

Design Guide for Reinforced Concrete Diaphragms 1

2

3

H

20� -0� ������

5

4

6

7

20� � 20� column ������

G

24� � 24� column ������

F 12� wall ������

N

22� -0� ������

E

D

C 36� � 30� beam ������

B

A

12� wall below ������

Figure 10.31 (continued) Floor plans of the 5-story building in Example 10.5 (Floor 2). Levelresidential 2

2. Determine minimum thickness

based on two-way shear requirements.

The columns that are not part of the SFRS must resist the effects from gravity loads plus the deformation compatibility requirements in ACI 18.14 (ASCE/SEI 12.12.5). Two-way shear requirements are checked for gravity loads at a typical interior column and at an edge column. It can be determined that on a typical floor where the live load is equal to 40 psf, shear strength requirements are satisfied using an 8.0-in. slab. In areas where the live load is 100 psf, shear reinforcement is required. Headed shear stud reinforcement in accordance with the strength requirements of ACI 22.6.8 can be used to supplement the shear strength of the 8.0-in. slab where the live load is equal to 100 psf. Therefore, (Note: The requirements in ACI 18.14.5 for slab-column connections must also be satisfied. If additional shear reinforcement is required, provide headed shear stud reinforcement in addition to that required for gravity loads.). 3. Determine minimum thickness

based on fire resistance requirements.

Assuming that the required fire-resistance rating of the slab is 2 hours for this occupancy, the minimum thickness from IBC Table 722.2.2.1 based on a concrete mix with siliceous aggregate.

10-87

Design Guide for Reinforced Concrete Diaphragms 4. Check if the minimum slab thickness requirements for SDC D, E, and F must be satisfied. The SDC is determined in accordance with IBC 1613.2.5 or ASCE/SEI 11.6 (see Step 3 in Section 4.2.3 of this publication). For the given latitude and longitude of the site, the earthquake spectral response acceleration parameters at short periods and at 1-second periods are and , respectively, from Reference 4 or Reference 5. For Site Class D based on properties determined from a geotechnical report, and from ASCE/SEI Tables 11.4-1 and 11.4-2, respectively. Because is greater than or equal to 0.2, a ground motion hazard analysis in accordance with ASCE/SEI 21.2 must be performed (see item 3 in ASCE/SEI11.4.8). Assume that such an analysis is not required in this case. The earthquake spectral acceleration parameters at short periods and at 1-second periods, adjusted for site class effects, are determined by ASCE/SEI Equations (11.4-1) and (11.4-2), respectively:

Design earthquake spectral response acceleration parameters at short periods and at 1-second periods are determined by ASCE/SEI Equations (11.4-3) and (11.4-4), respectively:

From ASCE/SEI Table 11.6-1, the SDC is D for and Risk Category II. Similarly, from ASCE/SEI Table 11.6-2, the SDC is D for and Risk Category II. Therefore, this building is assigned to SDC D, which means the minimum slab thickness requirements in ACI 18.12.6.1 for buildings assigned to SDC D, E, or F must be satisfied. According to that section, concrete slabs serving as diaphragms must be at least 2 in. thick. Providing an 8.0-in.-thick slab satisfies this requirement. Also, the materials selected in Step 1 are permitted to be used for special structural walls in buildings assigned to SDC D (ACI 18.2.5). Use a slab thickness h = 8.0 in. • Step 3 – Determine the Diaphragm Design Forces The flowchart in Figure 9.2 is used to determine the in-plane diaphragm forces. Wind and seismic forces are applicable for this building that is assigned to SDC D. The general structural integrity forces in ASCE/SEI 1.4.2 must also be considered but are typically less than the required seismic forces for buildings assigned to SDC D. It is evident from Figure 10.31 that the structural walls on column lines B and G stop at the second-floor level. In the first story, these walls occur on column lines A and H, respectively. This constitutes a Type 4 horizontal structural irregularity and the transfer forces from the structural walls on column lines B and G to the diaphragm at the second-floor level must also be considered. 1.

Determine the in-plane forces. (a) Determine the wind forces. The wind load provisions in Part 1 of ASCE/SEI Chapter 27 are used to determine wind forces for this building, which meets the conditions and limitations set forth in that part (see Table 4.1 of this publication). The wind forces in the north-south and east-west directions are given in Table 10.21. The basic wind speed, , is equal to 95 mph from ASCE/SEI Figure 26.5-1B or Reference 5 for the given latitude and longitude of the site. This Risk Category II, enclosed building is rigid in both directions where the approximate natural frequency, , is equal to .

10-88

Design Guide for Reinforced Concrete Diaphragms Table 10.21 Wind Forces for the Building in Example 10.5 Level

Height above ground level, (ft)

Total North-South Wind Force (kips)

Total East-West Wind Force (kips)

R 5 4 3 2

50 40 30 20 10

10.2 19.7 18.7 17.4 16.6

13.7 26.3 25.1 23.5 22.5

82.6

111.1

Σ (b) Determine the seismic forces.

The first step is to check if the Equivalent Lateral Force (ELF) Procedure in ASCE/SEI 12.8 can be used to determine the seismic forces on the SFRS over the height of this building in the north-south and east-west directions (see Section 4.2.3 of this publication). The height of this building is 50 ft and it possesses a Type 4 horizontal structural irregularity in accordance with ASCE/SEI Table 12.3-1. According to ASCE/SEI Table 12.6-1, the ELF Procedure is permitted to be used in this case. The fundamental period, , is determined in accordance with ASCE/SEI 12.8.2. It is permitted to determine the approximate building period, , by ASCE/SEI Equation (12.8-7):

In this equation, is the vertical distance from the base of the building to the highest level of the SFRS. Because the special structural walls are used over the entire height of the building, is equal to 50 ft. The approximate period parameters and are obtained from ASCE/SEI Table 12.8-2. For all other structural systems that are not explicitly listed in that table, and . Therefore,

Diaphragm seismic forces are determined in accordance with ASCE/SEI 12.10.1.1. i.

Determine the seismic forces on the SFRS. The steps in Section 4.2.3 are used to determine the seismic forces on the SFRS over the height of the building in both directions. The seismic ground motion values are determined in Part 4 of Step 2 above and are equal to the following: and . The seismic base shears are the same in both directions because the same SFRS is used in both directions. • Determine the seismic response coefficient, The seismic response coefficient,

.

, is determined by ASCE/SEI Equation (12.8-2):

As noted in the design data, building frame systems with special reinforced concrete shear walls are used in both directions. This system is permitted for buildings assigned to SDC D with a height limit of 160 ft (see ASCE/SEI Table 12.2-1). From ASCE/SEI Table 12.2-1, response modification coefficient, , is equal to 6. Also, the seismic importance factor, , is equal to 1.0 for Risk Category II buildings (see ASCE/SEI Table 1.5-2). Thus,

10-89

Design Guide for Reinforced Concrete Diaphragms The value of need not exceed that determined by ASCE/SEI Equations (12.8-3) or (12.8-4), whichever is applicable. These equations include the long-period transition period, , and the period of the building, . The long-period transition period, , is determined using ASCE/SEI Figures 22-14 through 22-17. At this site, from ASCE/SEI Figure 22-14. This quantity can also be obtained from Reference 5. Because

Also,

,

need not exceed that determined by ASCE/SEI Equation (12.8-3):

must not be less than that determined by ASCE/SEI Equation (12.8-5):

Because

,

must not be less than that determined by ASCE/SEI Equation (12.8-6):

Therefore, the seismic response coefficient, • Determine the effective seismic weight,

, is equal to 0.230.

.

For this building, the weight of the slabs, beams, columns, walls, cladding, and the superimposed dead loads must all be included in . Snow load at the roof level need not be included because the flat roof snow load is less than 30 psf [the ground snow load at this site is equal to zero from ASCE/SEI Figure 7.2-1 or Reference 5; see ASCE/SEI 12.7.2(4)]. The story weights, , which are the portions of that are assigned to level in the building, are given in Table 10.22. Table 10.22 Seismic Forces and Story Shears on the SFRS for the Building in Example 10.5 Level

Story Weight, (kips)

Height, (ft)

R 5 4 3 2

2,548 2,824 2,824 2,824 2,932

50.0 40.0 30.0 20.0 10.0

Σ

13,952 • Determine the seismic base shear,

Seismic Force, (kips)

Story Shear, (kips)

127,400 112,960 84,720 56,480 29,320

995.0 882.2 661.7 441.1 229.0

995.0 1,877.2 2,538.9 2,980.0 3,209.0

410,880

3,209.0

.

The seismic base shear is determined by ASCE/SEI Equation (12.8-1).

• Distribute the seismic base shear, The seismic force, and (12.8-12):

10-90

, over the height of the building.

, induced at any level of a building is determined by ASCE/SEI Equations (12.8-11)

Design Guide for Reinforced Concrete Diaphragms In this equation,

is the exponent related to the period of the building:

–

for buildings where

–

for buildings where





is to be determined by linear interpolation between 1 and 2 for buildings that have a period between 0.5 and 2.5 seconds or can be taken equal to 2

In this case:

because

.

The seismic forces, , over the height of the building are given in Table 10.22. Note that these seismic forces are greater than the general lateral structural integrity forces prescribed in ASCE/SEI 1.4.2, which are equal to at each level. ii.

Determine the seismic forces on the diaphragms. The information in Section 4.2.3 of this publication is used to determine the seismic forces on the diaphragms over the height of the building. According to ASCE/SEI 12.10, diaphragms are to be designed for the larger of the following forces: Design seismic forces,

, acting on the SFRS at the levels of a building

Diaphragm design forces,

, determined by ASCE/SEI Equations (12.10-1) through (12.10-3):

Minimum Maximum In these equation, is the portion of the effective seismic weight, , that is assigned to level (see the values of in Table 10.22) and is the weight that is tributary to the diaphragm at level , which can be taken as (see the discussion in Section 4.2.3 of this publication). The forces, , are the portions of the seismic base shear, induced at level (see the values of in Table 10.22). The diaphragm design forces,

, over the height of the building are given in Table 10.23.

Table 10.23 Design Seismic Forces on the Diaphragms for the Building in Example 10.5 Level

Story Weight, (kips)

Seismic Force, (kips)

R 5 4 3 2

2,548 2,824 2,824 2,824 2,932

995.0 882.2 661.7 441.1 229.0

13,952

3,209.0

Σ * Minimum

(kips) 2,548 5,372 8,196 11,020 13,952

(kips) 995.0 1,877.2 2,538.9 2,980.0 3,209.0

(kips) 0.3905 0.3494 0.3098 0.2758* 0.2758*

995.0 986.8 874.8 778.9 808.7

governs.

The term that is related to the required minimum value of the design seismic force on the diaphragm is equal to the following:

10-91

Design Guide for Reinforced Concrete Diaphragms The term that is related to the required maximum value of the design seismic force on the diaphragm is equal to the following:

It is evident from Table 10.23 that the minimum seismic diaphragm force is required at the second and third levels in both the north-south and east-west directions. According to ASCE/SEI 12.3.3.4, the design forces from ASCE/SEI 12.10.1.1 for connections of diaphragms to vertical elements of the LFRS and to collectors and for collectors and their connections must be increased 25 percent for buildings with the horizontal and vertical irregularities listed in that section, including Type 4 horizontal irregularities. This increase, which may be applicable in this example for seismic forces in the eastwest direction, need not be applied in cases where the forces have been determined using the seismic load combinations including overstrength. It is shown in the next section and in Step 5 below that the overstrength factor, , must be applied in this example, so the 1.25 force increase is not applicable. iii. Determine the transfer forces on the second-floor diaphragm. For structures that have a Type 4 horizonal structural irregularity, which is applicable for the east-west direction of analysis in this example, the transfer forces from the vertical elements of the SFRS above the diaphragm to the vertical elements of the SFRS below the diaphragm must be increased by the overstrength factor, , which is equal to 2.5 for building frame systems with special reinforced concrete shear walls (see ASCE/SEI 12.10.1.1, ASCE/SEI Table 12.2-1, and Section 4.3 of this publication). To satisfy this requirement in the design of the second-floor diaphragm, the forces on levels 3, 4, 5, and the roof in Table 10.22 must be increased by 2.5; the loads effects from these transfer forces are combined with those from the inertial force at the second-floor level from Table 10.23. Details of the analysis and calculation of the transfer forces are given in Step 4 below. 2. Determine the out-of-plane forces. The combined gravity load effects are determined using the load combinations in Table 5.1 of this publication. Maximum load effects are obtained using ACI Equation (5.3.1b): In private rooms and corridors serving them: In public rooms and corridors serving them: • S  tep 4 – Determine the Classification of the Diaphragm, Select the Diaphragm Model, and Determine Diaphragm Internal Forces 1.

Determine the diaphragm classification. The information in Section 6.1 of this publication is used to determine the classification of this diaphragm. In the case of seismic forces, the diaphragm can be classified as rigid when the two conditions in ASCE/SEI 12.31.2 are satisfied. Because the structure possesses a Type 4 horizontal structural irregularity, the diaphragm cannot be classified as rigid. As such, semirigid diaphragms are assigned at each level of the building (ASCE/SEI 12.7.3). Using semirigid diaphragms automatically accounts for the stiffness characteristics of the diaphragms.

2. Select the diaphragm model. A finite element model is selected to determine the in-plane forces in the semirigid diaphragms. The forces in the walls are determined using a three-dimensional model of the building, which, according to ASCE/SEI 12.7.3, must be used in cases where a Type 4 horizontal irregularity is present (see Part 3 of this step below, which describes how the diaphragm forces are obtained from the analysis). 3. Determine the diaphragm internal forces. The flowchart in Figure 9.3 is used to determine the internal forces in the diaphragm. It is evident from Tables 10.21 and 10.23 that the seismic forces are significantly greater than the wind forces at all levels of the building. Thus, internal forces in the diaphragm at the second-floor level are determined using the design seismic forces. Also, because the building does not have a Type 5 horizontal structural irregularity as defined in ASCE/ SEI Table 12.3-1 and none of the special structural walls are part of two or more intersecting SFRSs, the design seismic forces in the north-south and east-west directions are permitted to be applied independently in each of the two orthogonal directions and orthogonal interaction effects are permitted to be neglected (ASCE/SEI 12.5.3).

10-92

Design Guide for Reinforced Concrete Diaphragms A three-dimensional model of the building was constructed using Reference 14. In the model, the columns and walls are pinned at the base of the building, semirigid diaphragms modeled as membrane elements are assigned at all levels, and the following reduced moments of inertia are used, which account for the effects of cracked sections (ASCE/SEI 12.7.3): •

Beams at second-floor level:

•

Walls:

•

Slabs (out-of-plane):

•

Slabs (in-plane): The columns are assigned stiffness modifiers equal to 0.001 to ensure that the effects from the lateral loads are resisted only by the special structural walls.

(a) Determine the location of the CM. From symmetry,

and

.

(b) Determine the location of the CR. The CR is only applicable to rigid diaphragms. In the case of semirigid diaphragms, the in-plane slab deformation is variable, so the determination of the location of the CR is not pertinent. (c) Determine the seismic forces in the special structural walls. Seismic forces are applied at the CM in both directions. Accidental torsion in accordance with ASCE/SEI 12.8.4.2 need not be applied in the analysis for strength design or when checking the story drift limits prescribed in ASCE/ SEI 12.12 because the structure, which is assigned to SDC D, does not have a Type 1a or Type 1b horizontal structural irregularity. The diaphragm force, , at the second-floor level is equal to 808.7 kips in both directions (see Table 10.23). In the north-south direction, the in-plane diaphragm forces at the second-floor level are obtained from the threedimensional model of the building, which is subjected to the forces, , from the ELF Procedure where at the second-floor level is replaced with at this level [see Tables 10.22 and 10.23 and Figure 10.32(a)]. For purposes of analysis, an additional force equal to is applied at the CM of the second-floor level in the model. The in-plane diaphragm forces are obtained in a similar manner in the east-west direction. However, as noted previously, the forces, , from the ELF Procedure on levels 3, 4, 5, and the roof in Table 10.22 must be increased by the overstrength factor because of the Type 4 horizontal irregularities in this direction of analysis [see Figure 10.32(b)]. Just like in the north-south direction, the force at the second-floor level is replaced with at this level, and an additional force equal to is applied at the CM of the second-floor level in the model. Figures 10.33 and 10.34 (next page) contain the shear forces in the special structural walls from the analysis for seismic forces applied in the south and west directions, respectively. (d) Determine the chord forces. The “section cut” feature in Reference 14 is used to determine the maximum in-plane bending moments in the diaphragm in both directions of analysis. To obtain the bending moments using this feature, a section cut is drawn on the diaphragm parallel to the direction of analysis. The “Section Cut Forces” box opens after the section cut is drawn, and the z-axis moments in that box represent the in-plane bending moments at that location. • Seismic force in the south direction The maximum moment in the diaphragm is equal to 14,960 ft-kips, which occurs at column line 4 (see Figure 10.35, which is obtained from Reference 14). Therefore, using Equation (6.7), the maximum tension chord force at column line A is equal to the following:

10-93

Design Guide for Reinforced Concrete Diaphragms N-S

995.0 kips 

2,487.5 kips 

R 

882.2 kips 

2,205.5 kips 

5 

661.7 kips 

808.7 kips 

4 

1,102.8 kips 

3 

3 

808.7 kips 

2 

R  5 

1,654.3 kips 

4 

441.1 kips 

E-W 

2 

�𝒃𝒃� 

�𝒂𝒂� 

Figure 10.32 Vertical distribution of seismic forces for the design of the second-floor diaphragm. (a) North-south direction. (b) East-west direction.

R 

5  4  3  2 

A

B

C

D

E

F

249.0 kips 

249.0 kips 

636.3 kips 

636.3 kips 

947.0 kips 

947.0 kips 

468.7 kips 

G

H

468.7 kips 

749.2 kips 

749.2 kips 

Elevations 1/7 

Figure 10.33 Shear forces in the special structural walls for the seismic force in the south direction. • Seismic force in the west direction The maximum moment in the diaphragm is equal to 27,205 ft-kips, which occurs at column line G. Therefore, using Equation (6.7), the maximum tension chord force at column line 1 is equal to the following:

(e) Determine the unit shear forces, net shear forces, and collector forces. • Seismic force in the south direction At column lines 1 and 7, the total shear force in the diaphragm is equal to 395.6 kips, which can be obtained by drawing a section cut adjacent to either of these column lines. Thus, the unit shear force in the diaphragm at each of these column lines is equal to the following:

Assuming that the collectors that are required along these column lines transfer their entire axial forces into the ends of the walls, the unit shear force that must be transferred between the diaphragm and each of the walls is equal to the following, which can be obtained by drawing a section cut adjacent to either of these column lines (the feasibility of this assumption is checked below):

The unit shear forces, net shear forces, and collector forces along column line 1 are given in Figure 10.36 (the forces are the same along column line 7).

10-94

Design Guide for Reinforced Concrete Diaphragms

R 

1

2

3

4

5

6

7

6

7

6

7

5  4  3 

2 

R 

680.0 kips 

1

680.0 kips 

Elevation H 

2

3

4

5

351.8 kips 

5  4 

351.8 kips 

628.2 kips 

628.2 kips 

853.7 kips 

853.7 kips 

806.7 kips 

3 

2 

806.7 kips 

Elevation G 

R  5  4  3  2 

1

2

3

4

5

270.2 kips 

270.2 kips 

779.3 kips 

779.3 kips 

544.8 kips 

544.8 kips 

1,008.8 kips 

1,008.8 kips 

1,386.9 kips 

Elevation E 

1,386.9 kips 

Figure 10.34 Shear forces in the special structural walls for the seismic force in the west direction. It is determined in Step 5 below that the collectors must be designed for the load effects based on the second requirement in ASCE/SEI 12.10.2.1, which includes application of the overstrength factor, . Thus, the required diaphragm force to be used in designing the collectors is equal to 1,685.9 kips (see Step 5). Reanalysis of the structure using this diaphragm force at the second-floor level results in the following axial forces in the collectors at the ends of the walls:

Check the transverse reinforcement requirements in ACI 18.12.7.5 assuming the axial compressive force is transferred directly into the end of the wall from the collector, which is the 12-in. wide by 8-in. deep slab segment in line with the wall:

10-95

Design Guide for Reinforced Concrete Diaphragms

Figure 10.35 Section cut at column line 4 for determination of the in-plane bending moment in the diaphragm for the seismic force in the south direction.

where the limit of

is used because the design forces are amplified by

.

Because , transverse reinforcement in accordance with ACI 18.12.7.5 must be provided. However, providing such reinforcement in an 8.0-in.-thick slab is not feasible. Consequently, a collector that is wider than the wall will be used, and the axial compression force, , that will be transferred directly into the ends of the walls will be selected so that transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided; the remaining portion of the axial force will be transferred from the diaphragm to the walls by shear. Assume that is transferred directly into the end of the wall. The corresponding axial compressive stress is equal to the following:

Thus, transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided. For and , the total unit shear force in the walls is determined based on the assumptions that and , which means and (see Figure 10.37). Thus, the total shear force that must be transferred between the diaphragms and the walls is equal to 9.0 kips/ft (see Figure 10.36) plus the portion of the collector axial force that is transferred by shear adjacent to the walls, which is :

10-96

Design Guide for Reinforced Concrete Diaphragms 1

8.99 kips⁄ft

197.8 � 22.0

8.99 kips⁄ft

113.04 kips

28.25 kips

113.04 kips

D

N

28.25 kips

22� -0� �t�p.�

E

2.57 kips⁄ft

6.42 kips⁄ft

8.99 kips⁄ft

395.6 � 154.0

6.42 kips⁄ft

197.8 � 22.0

2.57 kips⁄ft

F

2.57 kips⁄ft

G

8.99 kips⁄ft

2.57 kips⁄ft

H

2.57 kips⁄ft

C

B

A

Unit Shear Forces

Net Shear Forces

Collector Forces

Figure 10.36 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line 1. Note that the same total unit shear forces in the walls are obtained for assuming that .

and

• Seismic force in the west direction The diaphragm forces obtained from the analysis include the effects from the Type 4 horizontal irregularities that occur at column lines A/B and G/H. The following maximum unit shear forces in the diaphragm are obtained by drawing section cuts adjacent to these column lines: – Along column line H: – Along column line G: – Along column line E: Assuming that the collectors that are required along these column lines transfer their entire axial forces into the

10-97

Design Guide for Reinforced Concrete Diaphragms 𝐶𝐶� � 𝐶𝐶� ��������� � �������� 𝐶𝐶� 𝑒𝑒

𝐶𝐶� � 𝑇𝑇�

22� -0�

𝑉𝑉�

𝐶𝐶� � 2���2����� 𝐶𝐶� � ���2����� 𝐶𝐶� � ����0����� 𝑇𝑇� � ���2����� 𝑇𝑇� � ���2����� 𝑇𝑇� � 0 𝑉𝑉� � ����������

𝑇𝑇� � ��𝑇𝑇� �������� � �������� 𝑇𝑇�

Figure 10.37 Force transfer in the diaphragm and walls along column line 1 or 7.

2

1

3 34.47 kips⁄ft

H

34.47 kips⁄ft

𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔𝐔

20� -0� �t�p.�

4

11.49 kips⁄ft

7

6 34.47 kips⁄ft

N 34.47 kips⁄ft

11.49 kips⁄ft 22.98 kips⁄ft

22.98 kips⁄ft

𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍𝐍

5

11.49 kips⁄ft

11.49 kips⁄ft

11.49 kips⁄ft

229.8 kips

229.8 kips

𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂𝐂

229.8 kips

229.8 kips

Figure 10.38 Unit shear forces, net shear forces, and collector forces in the diaphragm along column line H.

10-98

Design Guide for Reinforced Concrete Diaphragms ends of the walls, the unit shear forces that must be transferred between the diaphragm and the walls are equal to the following:

The unit shear forces, net shear forces, and collector forces along column line H are given in Figure 10.38 (next page). Similar forces can be determined along column line E. It is shown in Step 5 below that the collectors must be designed for the load effects based on the second requirement in ASCE/SEI 12.10.2.1, which includes application of the overstrength factor, . To properly account for this requirement in the design of the collectors, the building must be reanalyzed for the vertical force distribution in Figure 10.32 with the exception that the seismic force at the second-floor level is set equal to 1,685.9 kips (see Step 5). From reanalysis of the structure, the axial forces in the collectors along column lines H, G, and E are equal to 240.0 kips, 290.0 kips, and 161.5 kips, respectively. Check the transverse reinforcement requirements in ACI 18.12.7.5: For the 12-in. wide by 8-in. deep slab segments in line with the walls along column line H:

For the 36-in. wide by 30-in. deep collector beams along column line G:

For the 12-in. wide by 8-in. deep slab segments in line with the walls along column line E:

In the calculation of the maximum compressive stresses, the limit of are amplified by .

is used because the design forces

It is evident that transverse reinforcement conforming to ACI 18.12.7.5 need not be provided for the collectors along column lines E and G, but must be provided for the collectors along column line H. However, providing such reinforcement in an 8.0-in.-thick slab is not feasible. For the collectors along column line H, a collector that is wider than the wall will be used, and the axial compression force, , that will be transferred directly into the ends of the walls will be selected so that transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided; the remaining portion of the axial force will be transferred from the diaphragm to the walls by shear. Assume that is transferred directly into the ends of the walls at column line H. The corresponding axial compressive stress is equal to the following:

Thus, transverse reinforcement in accordance with ACI 18.12.7.5 need not be provided. For

, the total unit shear force in these walls is determined based on the assumptions that and , which means . Thus, the total shear force that must be transferred between the diaphragms and the walls is equal to 34.5 kips/ft (see Figure 10.38) plus the portion of the collector axial force that is transferred by shear adjacent to the walls, which is :

10-99

Design Guide for Reinforced Concrete Diaphragms • Step 5 – Determine Combined Load Effects Combined load effects are determined for this building that is assigned to SDC D using the applicable strength design load combinations in ACI 5.3 (see Table 5.1 of this publication and Figure 9.4). The governing combined out-of-plane load effects are determined in Part 2 of Step 3 above. The maximum factored uniform load on the second-floor slab due to dead loads and a 100-psf live load is equal to 304.0 psf. As noted previously, the governing in-plane load effects are due to design seismic forces. Thus, ACI Equation (5.3.1e) must be used, which reduces to the following in this example for the second-floor diaphragm:

The effect of horizontal and vertical earthquake-induced forces, (5.3), and (5.4) of this publication):

, is determined by ASCE/SEI 12.4.2 (see Equations (5.1),

For buildings assigned to SDC D, the redundancy factor, , is determined in accordance with ASCE/SEI 12.3.4. Because this building meets the conditions in ASCE/SEI 12.3.4.2, . Thus, where the effects of the horizontal seismic forces, , have been determined in Step 4 above. In buildings assigned to SDC D, collectors and their connections must be designed for the maximum of the three forces given in ASCE/SEI 12.10.2.1: 1.

 orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces F determined by the ELF Procedure of ASCE/SEI 12.8 or the modal response spectrum analysis procedure of ASCE/SEI 12.9.1. For building frame systems with special reinforced concrete shear walls, the overstrength factor, , is equal to 2.5 from ASCE/SEI Table 12.2-1. The required in-plane diaphragm force at the second-floor level based on this requirement is equal to (see Table 10.22).

2. F  orces calculated using the seismic load effects including overstrength of ASCE/SEI 12.4.3 with the seismic forces determined by ASCE/SEI Equation (12.10-1). Using the information in Table 10.19, the diaphragm force, tion (12.10-1) is equal to the following:

, at the second-floor level determined by ASCE/SEI Equa-

The required diaphragm in-plane force including overstrength is equal to requirement.

based on this

3. F  orces calculated using the load combinations of ACE/SEI 2.3.6 with the seismic forces determined by ASCE/SEI Equation (12.10-2). The diaphragm force, (see Table 10.23).

, at the second-floor level determined by ASCE/SEI Equation (12.10-2) is equal to 808.7 kips

Therefore, the collectors and their connections to the vertical elements of the SFRS must be designed for the effects due to the 1,685.9-kip in-plane diaphragm force stipulated in the second requirement. In the east-west direction, the effects due to the transfer forces from the Type 4 horizontal structural irregularities, which are increased by in accordance with ASCE/SEI 12.10.1.1, must also be considered. • Step 6 – Determine the Chord Reinforcement The flowchart in Figure 9.5 is used to determine the required area of chord reinforcement in both directions. The chord forces for seismic forces in the north-south and east-west directions are determined in Part 3(d) of Step 4 above. 1.

Seismic forces in the north-south direction For seismic forces in the north-south direction, the maximum chord force is equal to 102.3 kips (see Part 3(d) in Step 4 above), and the required area of chord reinforcement at column lines A and H is determined by Equation (8.4) of this publication:

10-100

Design Guide for Reinforced Concrete Diaphragms

Collector reinforcement is also required at these locations for seismic forces in the east-west direction, which is determined in Step 11 below. The larger of the two required areas of reinforcement will be provided along these column lines. 2. Seismic forces in the east-west direction For seismic forces in the east-west direction, the required area of chord reinforcement at column lines 1 and 7 is equal to the following:

Collector reinforcement is also required at these locations for seismic forces in the north-south direction, which is determined in Step 11 below. The larger of the two required areas of reinforcement will be provided along these column lines. • Step 7 – Determine the Diaphragm Shear Reinforcement The flowchart in Figure 9.6 is used to determine the area of shear reinforcement, if required. The largest factored unit shear force in the diaphragm is equal to 11.5 kips/ft, which occurs along column lines H and G for the seismic force in the west direction (see Part 3(e) in Step 4 above). Note that the shear force based on amplification of the diaphragm force at the second-floor level by need not be considered when checking shear strength of the diaphragm. The strength reduction factor, , for shear strength of a diaphragm must not exceed the least value of for shear used for the vertical elements of the SFRS (ACI 21.2.4.2). In this example, the least value of corresponds to the special structural walls, which is equal to 0.60 in accordance with ACI 21.2.4.1. Thus, the design shear strength of the diaphragm is determined using . The required area of shear reinforcement,

, can be determined by Equation (7.3) of this publication:

Solving for

This reinforcement ratio corresponds to #3 bars spaced at 10 in. on center

.

Check the maximum shear strength requirements of ACI 18.12.9.2:

The #3 bars spaced at 10.0 in. on center can be placed in the slab in addition to the required flexural reinforcement in the east-west direction in the areas where such reinforcement is required, that is, in areas where . In lieu of providing additional reinforcement, any excess area of flexural reinforcement in the slab can be used to resist the required shear forces. In the north-south direction, the largest shear force in the slab is equal to 2.6 kips/ft (see Figure 10.36). Determine suming :

as-

10-101

Design Guide for Reinforced Concrete Diaphragms The design shear strength of the concrete alone is adequate to satisfy shear strength requirements in this direction. • Step 8 – Determine the Shear Transfer Reinforcement The flowchart in Figure 9.7 is used to determine the required areas of shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS and between the diaphragm and the collectors. 1.

Determine the shear transfer reinforcement between the diaphragm and the vertical elements of the SFRS. (a) Seismic forces in the north-south direction The unit shear force in the special structural walls along column lines 1 and 7 is equal to 17.1 kips/ft (see Part 3(e) in Step 4 above). The required area of shear-friction reinforcement is determined by Equation (8.9) of this publication assuming that Construction Method A is used (see Section 8.6.3 of this publication for a description of this construction method) and that the cold joint between these elements satisfies the construction joint requirements in ACI 18.12.10, that is, from ACI Table 22.9.4.2:

Provide #5 bottom bars spaced at 7.0 in. on center (provided length of the walls or provide separate dowel bars of the same bar size and spacing.

) in the slab over the

(b) Seismic forces in the east-west direction • Special structural walls along column line H The required shear-friction reinforcement between the diaphragm and the special structural walls along column line H is equal to the following (see Part 3(e) in Step 4 above):

Provide #6 bottom bars spaced at 4.0 in. on center (provided the length of the walls or provide separate dowel bars of the same bar size and spacing.

) in the slab over

• Special structural walls along column line E For the special structural walls along column line E:

Provide #5 bottom bars spaced at 6.0 in. on center (provided the length of the walls or provide separate dowel bars of the same bar size and spacing.

) in the slab over

2. Determine the shear transfer reinforcement between the diaphragm and the collectors. According to ASCE/SEI 12.10.2.1, the shear transfer reinforcement which provides the connection between the diaphragm and the collectors must be based on the largest of the three forces determined in accordance with that section. It is shown in Step 5 above that the seismic force determined by the second requirement governs. (a) Seismic forces in the north-south direction To account for the required amplification of by in accordance with ASCE/SEI 12.10.2.1, the structure was reanalyzed with a diaphragm force of 1,685.9 kips applied at the CM of the second-floor level (see Step 5 above). Therefore, the required shear-friction reinforcement between the collectors and the slab along column lines 1 and 7 is determined by Equation (8.9) where is obtained by drawing a section cut adjacent to column line 1 (see Figure 10.36):

In this equation, because the concrete is cast monolithically. The bottom flexural reinforcement in the slab (which has an area equal to ) can be used as the shear transfer reinforcement between the diaphragm and the collectors along column lines 1 and 7.

10-102

Design Guide for Reinforced Concrete Diaphragms (b) Seismic forces in the east-west direction The required shear-friction reinforcement between the collectors and the slab along column lines A/H and B/G is equal to the following where is obtained by drawing a section cut adjacent to these column lines:

The bottom flexural reinforcement in the slab (which is equal to ) can be used as the shear transfer reinforcement between the diaphragm and the collectors along these column lines. It can be determined that the bottom flexural reinforcement in the slab is also adequate along column line D. • Step 9 – Determine the Reinforcement Due to Eccentricity of Collector Forces 1.

Seismic forces in the north-south direction For the collectors along column lines 1 and 7, it was determined in Part 3(e) of Step 4 above that and . The reinforcement, , that is required to resist the eccentric bending moment, , generated by the maximum shear force is determined by Equation (8.22) of this publication where is calculated by Equation (8.20):

The effective width of the collector, , is equal to (see Figure 6.9). Assuming that the longitudinal reinforcement in the collector is uniformly distributed in the cross-section, the eccentricity, , between the centerlines of the collector and the wall is equal to . Conservatively assuming the shear strength of the diaphragm,

Provide 2-#7 bars (provided

, is equal to zero,

is equal to the following:

) at the ends of each wall (see Figure 8.12).

2. Seismic forces in the east-west direction For the collectors along column line H, it was determined in Part 3(e) of Step 4 above that that and . The effective width of the collector, , is equal to (see Figure 6.9). Assuming that the longitudinal reinforcement in the collector is uniformly distributed in the cross-section, the eccentricity, , between the centerlines of the collector and the wall is equal to . Conservatively assuming the shear strength of the diaphragm,

, is equal to zero,

is equal to the following:

.

Provide 2-#6 bars (provided

) at the ends of each wall (see Figure 8.12).

• Step 10 – Determine the Anchorage Reinforcement The flowchart in Figure 9.9 is used to determine the required area of anchorage reinforcement. For this building that is assigned to SDC D, both wind and seismic anchorage requirements must be considered. 1.

Determine the wind connection force. The out-of-plane components and cladding (C&C) wind pressure, , is determined on the walls using the information provided in Section 4.4.2 of this publication. The provisions in Part 1 of Chapter 30 are selected to determine this wind pressure for this building that meets the criteria in ASCE/SEI 30.1 and 30.3 (see Table 4.6 of this publication).

10-103

Design Guide for Reinforced Concrete Diaphragms Design wind pressure

is determined by ASCE/SEI Equation (30.3-1):

It can be determined that the velocity pressure at the mean roof height of the building,

, is equal to 15.9 psf.

The external pressure coefficients are determined from ASCE/SEI Figure 30.3-1 for walls. Assuming that the effective wind area for the anchorage reinforcing bars is less than 10 sq ft, the external pressure coefficients are the following for Zone 5, which is the zone with the highest wind pressure:

Note 5 in ASCE/SEI Figure 30.3-1 permits values of less than 10 degrees, which is applicable in this example:

to be reduced by 10 percent for buildings with roof slopes

Anchorage reinforcement is determined for wind acting away from the surface, which produces tension in the reinforcing bars. Thus, the governing is equal to the following for an enclosed building where the internal pressure coefficients are given in ASCE/SEI Table 26.13-1:

This pressure is greater than the minimum design pressure of 16.0 psf given in ASCE/SEI 30.2.2. 2. Determine the seismic anchorage force. The anchorage of structural walls and the transfer of design forces into the diaphragm must satisfy the requirements of ASCE/SEI 12.11.2. According to ASCE/SEI 12.11.2.1, the design seismic anchorage force, , is determined by ASCE/ SEI Equation (12.11-1):

The amplification factor for diaphragm flexibility, , which is determined by ASCE/SEI Equation (12.11-2), is permitted to be taken as 1.0 for diaphragms that are not flexible. The term is the weight of the wall that is tributary to the anchor (reinforcing bar). Therefore,

Where the diaphragms are not flexible at all levels of a building, it is permitted to multiply the value of by the factor where is the height of the anchor above the base of the structure and is the height of the roof above the base. At the second-floor level of this building,

3. Determine the required anchorage reinforcement.

The reinforcing bars in the slab that are provided for flexure or for shear transfer can be used as the anchorage reinforcement. The additional requirements in ASCE/SEI 12.11.2.2 for anchorage of concrete walls to diaphragms in build-

10-104

Design Guide for Reinforced Concrete Diaphragms ings that are assigned to SDC D must also be satisfied. Full development of the reinforcing bars in tension satisfies these requirements. • Step 11 – Determine the Collector Reinforcement 1.

Seismic forces in the north-south direction The flowchart in Figure 9.10 is used to determine the required area of longitudinal reinforcement in the collectors along column lines 1 and 7. It was determined in Part 3(e) of Step 4 above that the maximum axial tension force in the collectors, , is equal to 236.2 kips and that the axial tension force transferred directly into the ends of the walls, , is equal to 59.2 kips. The corresponding required areas of longitudinal collector reinforcement are equal to the following:

Thus, the required area of collector reinforcement in the effective slab width,

, is equal to

It was determined in Step 6 above that the required area of chord reinforcement along column lines 1 and 7 for seismic forces in the east-west direction is equal to , which is greater than that required for the maximum axial tension force in the collector for seismic forces in the north-south direction. Therefore, provide 6-#4 bars (provided ) in the width of the wall and 18-#4 bars (provided ) within the effective slab width of the collectors. Check the design axial compression strength of the collectors. For the axial compression forces that are transferred directly into the ends of the wall (see Equation (8.28) of this publication):

For the axial compression forces that are transferred into the effective slab width (width of slab resisting is equal to ):

2. Seismic forces in the east-west direction • Collectors along column line H It was determined in Part 3(e) of Step 4 above that the total axial tension force in the collectors, , is equal to 240.0 kips and that the axial tension force transferred directly into the ends of the walls, , is equal to 170.0 kips. The corresponding required areas of longitudinal collector reinforcement are equal to the following:

Thus, the required area of collector reinforcement in the effective slab width,

, is equal to

10-105

Design Guide for Reinforced Concrete Diaphragms It was determined in Step 6 above that the required area of chord reinforcement along column line H for seismic forces in the north-south direction is equal to , which is less than that required for the total axial tension force in the collector for seismic forces in the east-west direction. Therefore, provide 6-#7 bars (provided ) in the width of the wall and 7-#4 bars (provided ) within the effective slab width of the collectors. Check the design axial compression strength of the collectors. For the axial compression forces that are transferred directly into the ends of the wall (see Equation (8.28) of this publication):

For the axial compression forces that are transferred into the effective slab width (width of slab resisting is equal to ):

• Collectors along column line E The total axial tension force is transferred directly into the ends of the walls (see Part 3(e) of Step 4 above). The required area of longitudinal collector reinforcement is equal to the following:

Provide 6-#7 bars (provided

) in the width of the wall.

• Collectors along column lines B and G The 36 in. by 30 in. collector beams along column line G must be designed for (1) the combined effects from gravity loads (bending moments and shear forces due to the weight of the walls, the gravity loads that are tributary to these walls from levels 3 through the roof, and the tributary gravity loads at level 2) and (2) seismic forces (axial compression and tension). A summary of the axial forces, bending moments, and shear forces is given in Table 10.24 (next page). 3,000 2,500 2,000

Axial Force �- ips

1,500

k

1,000 500 0

-500

+ + 0

250

+

+ 500 +

750

+ ++ 1,000 +

1,250

1,500

1,750

2,000

2,250

-1,000 -1,500

Bending Moment �� t- ips f

k

Figure 10.39 Design strength interaction diagram for the collector beams along column lines B and H.

10-106

Design Guide for Reinforced Concrete Diaphragms Table 10.24 Summary of Design Axial Forces, Bending Moments, and Shear Forces for the Collector Beams Along Column Line G Load Case

Axial Force (kips)

Bending Moment (ft-kips) Negative

Positive

Shear Force (kips)

Dead

0

638.3

398.9

177.3

Live

0

128.3

80.2

35.6

±290.0

0

0

0

1

0

893.6

558.5

248.2

2

0

971.2

607.0

269.7

3

±290.0

1,008.8

630.5

280.2

4

±290.0

395.8

247.3

109.9

Seismic Load Combination

The design strength interaction diagram for the collector beam reinforced with 7-#11 top bars, 7-#11 bottom bars, and 3-#5 side bars each side is given in Figure 10.39. It is evident from the figure that the longitudinal reinforcement is adequate for the load combinations in Table 10.24. It was determined in Part 3(e) of Step 4 above that transverse reinforcement satisfying ACI 18.12.7.5 need not be provided. The maximum shear force in the collector is equal to 280.2 kips when the member is subjected to combined gravity and seismic effects (see Table 10.24). Because the collector is subjected to significant axial tension, the design shear strength of the concrete is set equal to zero. The required spacing of #5 ties with 7 legs is the following:

Check maximum shear strength requirements of ACI 22.5.1.2:

Provide #5 ties with 7 legs spaced at 7.0 in. on center over the entire length of the collector beams. Note that this transverse reinforcement satisfies the detailing requirement of ACI 18.12.7.6(b) at splices and anchorage zones:

10-107

Design Guide for Reinforced Concrete Diaphragms

Provided

Reinforcement details for the second-floor diaphragm are given in Figure 10.40. 2

1 H

⑥ �typ.�

B

B

3

④

20� -0� �typ.�

5

4

①

⑥ �typ.�

C

③

A

A

⑦

E 22� -0� �typ.� D

⑦

② ⑤

⑧ �typ.� ⑦

③

⑧ �typ.�

⑧ �typ.�

F

④

C

G

7

6

⑤ ②

N

⑧ �typ.�

⑤

⑤

⑦

C

B

A

⑥ �typ.�

④

①

1. ① ─ 6-#7 � 7-#4 chord�collector reinforcement 2. ② ─ 6-#7 collector reinforcement 3. ③ ─ 6-#4 � 18-#4 chord�collector reinforcement 4. ④ ─ #6 @ 4″ shear-transfer reinforcement 5. ⑤ ─ #5 @ 6″ shear-transfer reinforcement 6. ⑥ ─ 2-#6 reinforcement due to eccentricity of collector forces

⑥ �typ.�

④

7. ⑦ ─ #5 @ 7″ shear-transfer reinforcement 8. ⑧ ─ 2-#7 reinforcement due to eccentricity of collector forces 9. Provide standard 90-deg hooks at the ends of all bars. 10. Provide Class B lap splices or Type 2 mechanical connectors where required. 11. Other reinforcement not shown for clarity.

Figure 10.40 Reinforcement details for the diaphragm at the second-floor level in Example 10.5

10-108

Design Guide for Reinforced Concrete Diaphragms

1� -0″

12� -0″

9-#4

8.0″

3-#4 3-#4 #5 @ 7″

Slab reinforcement �t��.�

Section A-A

7-#4

8.0″

3-#7

1� -0″

11� -0″

3-#7 #6 @ 4″

Slab reinforcement �t��.�

Section B-B 1� -0″

2� -6″

8.0″

7-#11 3-#5 �t��.�

#5 @ 7″ �

3 -0″

7-#11

Slab reinforcement �t��.�

Section C-C

Figure 10.40 (continued) Reinforcement details for the diaphragm at the second-floor level in Example 10.5

10-109

Design Guide for Reinforced Concrete Diaphragms

10-110

Design Guide for Reinforced Concrete Diaphragms

Chapter 11 References 1.

American Concrete Institute (ACI). 2014. “Building Code Requirements for Structural Concrete and Commentary.” ACI 31814, Farmington Hills, MI.

2. International Code Council (ICC). 2018. “International Building Code.” Washington, D.C. 3. American Society of Civil Engineers (ASCE). 2017. “Minimum Design Loads and Associated Criteria for Buildings and Other Structures.” ASCE/SEI 7-16, Reston, VA. 4. United States Geological Survey (USGS). U.S. Seismic Design Maps. https://earthquake.usgs.gov/designmaps/beta/us/. 5. American Society of Civil Engineers (ASCE). ASCE 7 Hazard Tool. https://asce7hazardtool.online/. 6. Building Seismic Safety Council (BSSC). 2009. “NEHRP Recommended Seismic Provisions for New Buildings and Other Structures.” FEMA P-750, Part 3, Resource Paper 12, Washington, D.C. 7.

National Institute of Standards and Technology (NIST). 2016. “Seismic Design of Cast-in-place Concrete Diaphragms, Chords, and Collectors: A Guide for Practicing Engineers.” Second edition, GCR 16-917-42, NEHRP Seismic Technical Brief No. 3, produced by the Applied Technology Council for the National Institute of Standards and Technology, Gaithersburg, MD.

8. Nakaki, S.D. 2000. “Design Guidelines for Precast and Cast-in-place Concrete Diaphragms. EERI Professional Fellowship Report, Earthquake Engineering Research Institute, Oakland, CA. 9. Concrete Reinforcing Steel Institute (CRSI). 2017. “Design and Detailing of Low-Rise Reinforced Concrete Buildings.” Schaumburg, IL. 10. Structural Engineers Association of California (SEAOC). 2005. “Using a Concrete Slab as a Seismic Collector.” Structural Engineers Association of California, Sacramento, CA. 11. International Code Council (ICC). 2015. “Guide to the Design of Common Irregularities in Buildings – 2012/2015 IBC and ASCE/SEI 7-10.” National Council of Structural Engineers Associations, Chicago, IL. 12. National Oceanic and Atmospheric Administration (NOAA). National Weather Service Precipitation Frequency Data Server, Hydrometeorological Design Studies Center, https://hdsc.nws.noaa.gov/hdsc/pfds/index.html. 13. Factory Mutual Insurance Company. 2016. “Roof Loads for New Construction”. FM Global Property Loss Prevention Data Sheets 1-54. Johnston, RI. 14. Computers and Structures, Inc. (CSI). 2016. ETABS – Integrated Analysis, Design and Drafting of Building Systems, Version 16.2.1, Walnut Creek, CA.

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Design Guide for Reinforced Concrete Diaphragms

11-2

Design Guide for Reinforced Concrete Diaphragms

Notations = component amplification factor = tributary area supported by a column in a two-way slab system, in.2 = area of subdiaphragm, in.2 = area of an individual bar, in.2 = area of concrete section resisting shear transfer, in.2 = gross area of concrete diaphragm, not to exceed the thickness times the width of the diaphragm in the direction of analysis, in.2 = gross area of concrete section, in.2 = gross area of the wall in which

is identified, in.2 , in.2

= sum of the gross surface areas of the building envelope (walls and roof) not including = gross cross-sectional area of a wall or pier, in.2 = area of vertical reinforcement in a wall, in.2 = total area of vertical reinforcement crossing the joint between a wall and a slab, in.2 = total area of openings in a wall that receives positive external pressure, in.2 = sum of the areas of openings in the building envelope (walls and roof) not including

, in.2

= area of longitudinal reinforcement, in.2 = area of chord reinforcement, in.2 = area of collector reinforcement, in.2 = area of tension reinforcement required to resist

, in.2

= area of dowel reinforcement, in.2 = area of tension reinforcement required to resist in-plane bending moment

, in.2

= total cross-sectional area of transverse reinforcement, including crossties, within spacing dimension , in.2

and perpendicular to

= area of anchorage reinforcement for general structural integrity, in.2 = minimum area of flexural reinforcement, in.2 = area of anchorage reinforcement in a slab due to seismic forces, in.2

N-1

Design Guide for Reinforced Concrete Diaphragms

= area of flexural reinforcement in a slab, in.2 = area of tension reinforcement required to resist

, in.2

= area of anchorage reinforcement in a slab due to wind forces, in.2 = area tributary to anchorage reinforcement in a slab due to wind forces, in.2 = area of shear reinforcement within spacing

, in.2

= area of shear-friction reinforcement, in.2 = torsional amplification factor = cross-sectional dimension of member core measured to the outside edges of the transverse reinforcement composing area , in. = effective slab width for collectors that are wider than the vertical elements of the lateral force-resisting system that they frame in to, in. = perimeter of critical section for two-way shear in slabs and footings, in. = length of an opening in a diaphragm, in. = web width of beam section, in. = dimension of the critical section

measured in the direction of the span for which moments are determined, in.

= dimension of the critical section

measured in the direction perpendicular to

, in.

= horizontal dimension of building measured normal to wind direction, in. = diaphragm span length perpendicular to the direction of analysis, in. = distance from the centroid of the critical section to face

of the critical section, in.

= distance from the centroid of the critical section to face

of the critical section, in.

= dimension of rectangular or equivalent rectangular column measured in the direction of the span for which moments are being determined, in. = dimension of rectangular or equivalent rectangular column measured in the direction perpendicular to

, in.

= factored collector compression force transferred directly into the end of the vertical element of the lateral forceresisting system, lb = exposure factor = net pressure coefficient to be used in determination of wind loads for open buildings = external pressure coefficient to be used in determination of wind loads for buildings

N-2

Design Guide for Reinforced Concrete Diaphragms

= seismic response coefficient = thermal factor = factored compression chord force, lb = factored primary compression chord force at an opening, lb = factored secondary compression chord force at an opening, lb = factored collector compression force transferred to the vertical element of the lateral force-resisting system by shear, lb = vertical distribution factor for seismic loads = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in. = perpendicular distance between chord forces, in. = nominal diameter of bar, in. = additional depth of water on the undeflected roof above the inlet of the secondary drainage system at its design flow (hydraulic head), in. = depth of water on the undeflected roof up to the inlet of the secondary drainage system when the primary drainage system is blocked (static head), in. = effect of service dead load = eccentricity parallel to the x-axis, in. = eccentricity parallel to the y-axis, in. = effect of horizontal and vertical earthquake-induced forces = modulus of elasticity of concrete, psi = modulus of elasticity of beam concrete, psi = modulus of elasticity of slab concrete, psi = horizontal seismic load effect = seismic load effects including overstrength = effect of horizontal seismic forces including overstrength = modulus of elasticity of reinforcement, psi = vertical seismic load effect

N-3

Design Guide for Reinforced Concrete Diaphragms

= stiffness or mass ratio corresponding to the subdiaphragm below an opening = specified compressive strength of concrete, psi = stiffness or mass ratio corresponding to the subdiaphragm above an opening = specified yield strength of reinforcement, psi = specified yield strength of transverse reinforcement, psi = in-plane transfer force to a diaphragm, lb = short-period site coefficient = design force applied to level

, lb

= horizontal seismic design force on a nonstructural component, lb = diaphragm design force at level

, lb

= diaphragm design force at level

, lb

= total windward plus leeward design wind force, lb = long-period site coefficient = lateral seismic force induced on the seismic force-resisting system at level

, lb

= gust-effect factor = shear modulus of concrete, psi = gust-effect factor for the main wind force resisting systems of flexible buildings and other structures = product of external pressure coefficient and gust-effect factor to be used in determination of wind loads for buildings = product of internal pressure coefficient and gust-effect factor to be used in determination of wind loads for buildings = overall thickness, height, or depth of member, in. = mean roof height of a building, in. = maximum center-to-center spacing of longitudinal bars laterally supported by corners of crossties or hoop legs around the perimeter of a beam, in. = column length, in. = precipitation intensity, in./h

N-4

Design Guide for Reinforced Concrete Diaphragms

= moment of inertia of section about centroidal axis, in.4 = moment of inertia of gross section of beam about centroidal axis, in.4 = moment of inertia of gross section of column about centroidal axis, in.4 = seismic importance factor = moment of inertia of gross concrete section about centroidal axis, neglecting reinforcement, in.4 = seismic component importance factor = snow importance factor = moment of inertia of gross section of slab about centroidal axis, in.4 = property of the critical section analogous to the polar moment of inertia, in.4

k

= exponent related to the structure period = amplification factor for diaphragm flexibility = beam stiffness, lb/in. = column stiffness, lb/in. = in-plane flexural stiffness of shear wall

, lb/in.

= horizontal ground acceleration divided by the acceleration due to gravity = in-plane lateral stiffness of lateral-force-resisting element

, lb/in.

= in-plane lateral stiffness of lateral-force-resisting element

in the x-direction, lb/in.

= in-plane lateral stiffness of lateral-force-resisting element

in the y-direction, lb/in.

= in-plane shear stiffness of shear wall

, lb/in.

= wind directionality factor = ground elevation factor = velocity pressure exposure coefficient evaluated at height = velocity pressure exposure coefficient evaluated at height = topographic factor

N-5

Design Guide for Reinforced Concrete Diaphragms

= span length of beam or one-way slab, in. = development length in tension of deformed bar, in. = development length in compression of deformed bar, in. = development length in tension of deformed bar with a standard hook, measured from outside end of hook, point of tangency, toward critical section, in. = beam length, in. = length of clear span measured face-to-face of supports, in. = length of opening in a diaphragm, in. = compression lap splice length, in. = tension lap splice length, in. = length of shear wall in direction of analysis, in. = length of span in direction moments are being determined, measured center-to-center of supports, in. = length of span in direction perpendicular to

L

, measured center-to-center of supports, in.

= effect of service live load = diaphragm span length in direction of analysis, in. = horizontal dimension of building measured parallel to the wind direction, in. = span of a flexible diaphragm that provides lateral support for a structural wall, ft = effect of service roof live load = eccentric in-plane bending moment in a diaphragm, in.-lb = nominal flexural strength at section, in.-lb = probable flexural strength of a member, in.-lb = factored slab moment that is resisted by the column at a joint, in.-lb = inherent torsional moment, in.-lb = accidental torsional moment, in.-lb = factored moment at section, in.-lb

N-6

Design Guide for Reinforced Concrete Diaphragms

= maximum in-plane factored bending moment in a diaphragm, in.-lb = natural frequency of a building or structure, Hz = notional load for structural integrity = design pressure to be used in determination of wind loads for buildings, psf = flat roof snow load, psf = ground snow load, psf = design wind pressure evaluated at height

above ground, psf

= minimum snow load, psf = sloped roof snow load, psf = total windward plus leeward design wind pressure evaluated at height

, psf

= total windward plus leeward design wind pressure, psf = design wind pressure evaluated at height

, psf

= nominal axial strength of member, lb = nominal axial strength at zero eccentricity, lb = factored axial force; to be taken as positive for compression and negative for tension, lb = velocity pressure, psf = velocity pressure evaluated at height

, psf

= velocity pressure for internal pressure determination, psf = factored load per unit area, lb/ft2 = velocity pressure evaluated at height

above ground, psf

= effects due to horizontal seismic forces = cumulative effect of service rain load = response modification coefficient = reactions in the vertical elements of the lateral force resisting system, lb

N-7

Design Guide for Reinforced Concrete Diaphragms

= component response modification factor = diaphragm reduction factor = center-to-center spacing of items, such as longitudinal reinforcement and transverse reinforcement, in. = center-to-center spacing of transverse reinforcement, in. = effect of service snow load = nominal moment, shear, axial, torsional, or bearing strength = design, 5% damped, spectral response acceleration parameter at short periods = design, 5% damped, spectral response acceleration parameter at a period of 1 s = mapped MCER, 5% damped, spectral response acceleration parameter at short periods adjusted for site class effects = mapped MCER, 5% damped, spectral response acceleration parameter at a period of 1 s adjusted for site class effects = mapped MCER, 5% damped, spectral response acceleration parameter at short periods = mapped MCER, 5% damped, spectral response acceleration parameter at a period of 1 s = fundamental period of a building, s = approximate fundamental period of a building, s = factored collector tension force transferred directly into the end of the vertical element of the lateral force-resisting system, lb = long-period transition period, s = nominal tension strength of chord reinforcement, lb = factored tension chord force, lb = factored secondary tension chord force at an opening, lb = factored primary tension chord force at an opening, lb = factored collector tension force transferred to the vertical element of the lateral force-resisting system by shear, lb = strength of a member or cross section required to resist factored loads or related internal moments and forces in such combinations as stipulated in ACI 318 = stress corresponding to nominal two-way shear strength provided by concrete, psi = factored shear stress at section, psi

N-8

Design Guide for Reinforced Concrete Diaphragms

= factored shear flow in diaphragm, lb/ft = basic wind speed obtained from Reference 2 or 3, mph = seismic base shear, lb = nominal shear strength provided by concrete, lb = lateral force on lateral-force-resisting element

, lb

= portion of the total story shear

in the x-direction resisted by element

of the lateral force-resisting system, lb

= portion of the total story shear

in the y-direction resisted by element

of the lateral force-resisting system, lb

= nominal shear strength, lb = maximum nominal shear strength, lb = nominal shear strength provided by shear reinforcement, lb = factored shear force at section, lb = maximum factored shear force in diaphragm, lb = total factored shear force, lb = total story shear in the x-direction, lb = total story shear in the y-direction, lb = distributed loads at the ends of an equivalent beam, lb/ft = density, unit weight, of normalweight concrete or equilibrium density of lightweight concrete, lb/ft3 = weight that is located or assigned to level

in a building, lb

= weight that is tributary to the diaphragm at level

in a building, lb

= equivalent in-plane distributed load, lb/ft = weight that is located or assigned to level

in a building, lb

= distributed loads at the ends of a diaphragm opening, lb/ft = effect of wind loads = effective seismic weight, lb

N-9

Design Guide for Reinforced Concrete Diaphragms

= total resultant wind force evaluated at height

, lb

= component operating weight, lb = location of the center of rigidity in the x-direction, in. = distance in the x-direction from the origin to the centroid of lateral-force-resisting element = perpendicular distance from element

, in.

to the center of rigidity parallel to the x-axis, in.

= location of the center of rigidity in the y-direction, in. = distance in the y-direction from the origin to the centroid of lateral-force-resisting element = perpendicular distance from element

, in.

to the center of rigidity parallel to the y-axis, in

= height above ground level, in. = nominal height of the atmospheric boundary layer, in. = 3-s gust-speed power law exponent = ratio of flexural stiffness of beam section to flexural stiffness of a width of slab bounded laterally by centerlines of adjacent panels, if any, on each side of the beam = average value of

for all beams on edges of a panel

= constant used to calculate

in slabs and footings

= ratio of long to short clear span length for a two-way slab = ratio of long to short dimensions of the sides of a column, concentrated load, or reaction area = unit weight of soil, pcf = factor used to determine the fraction of

transferred by eccentricity of shear at slab-column connections

= average story drift, in. = flexural deflection of a shear wall, in. = lateral deflection of a rigid frame or shear wall, in. = maximum in-plane deflection of a diaphragm, in. = shear deflection of a shear wall, in. = dynamic seismic lateral earth pressure, psf

N-10

Design Guide for Reinforced Concrete Diaphragms

= net tensile strain in extreme layer of longitudinal tension reinforcement at nominal strength = value of net tensile strain in the extreme layer of longitudinal tension reinforcement used to define a compression-controlled section = angle of plane of roof from horizontal, deg = modification factor to reflect the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength = coefficient of friction = ratio of

to

= redundancy factor based on the extent of structural redundancy present in a building = ratio of area of distributed transverse reinforcement to gross concrete area perpendicular to that reinforcement = strength reduction factor = amplification factor to account for overstrength of the seismic force resisting system determined in accordance with the general building code

N-11

Design Guide for Reinforced Concrete Diaphragms

Notes

N-12

Description of Manual This guide is the definitive resource on the design and detailing of diaphragms in cast-in-place reinforced concrete buildings. The requirements in ACI 318-14 are clearly summarized in figures and tables for quick reference. Comprehensive methods are provided on how to (1) determine diaphragm thickness based on strength and serviceability requirements; (2) calculate in-plane and collector forces based on ASCE/SEI 7-16 requirements; (3) model and analyze diaphragms; (4) determine the required reinforcement based on two different types of common construction methods; and (5) economically detail the required reinforcement based on the latest ACI 318 requirements. A step-by-step design procedure is provided that can be used for buildings assigned to Seismic Design Categories A through F. Numerous design aids and worked-out examples illustrate the code requirements for low-, mid-, and high-rise buildings, including buildings with irregularities.

ISBN 9781943961467

❘

❘

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933 North Plum Grove Road Schaumburg, IL 60173 Tel. 847.517.1200 www.crsi.org

9 781943 961467

10-DG-RC-DIAPHRAGMS-2019