Design Guide on the ACI 318
Building Code Requirements
for Structural Concrete
A comprehensive guide to assist
design professionals on the design and
detailing of reinforced concrete buildings.
Based on ACI 318-19
2020
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Publication No:
DG-ACI318-19
ISBN: 978-1-943961-52-8
Copyright © 2020
By Concrete Reinforcing Steel Institute
First Edition 2020, First Printing
Contains errata material, December 2020
All rights reserved. This guide or any part thereof may not be reproduced in any
form without the written permission of the Concrete Reinforcing Steel
Institute.
Printed in the U.S.A
This publication is intended for the use of professionals competent to evaluate the significance and limitations of its contents and
who will accept responsibility for the application of the material it contains. The Concrete Reinforcing Steel Institute reports
the foregoing material as a matter of information and, therefore, disclaims any and all responsibility for application of the stated
principles or for the accuracy of the sources other than material developed by the Institute.
Founded in 1924, the Concrete Reinforcing Steel Institute (CRSI) is a technical institute and an ANSI-accredited Standards Developing Organization (SDO) that stands as the authoritative resource for information related to steel reinforced concrete construction.
Serving the needs of engineers, architects and construction professionals, CRSI offers many industry-trusted technical publications,
standards documents, design aids, reference materials and educational opportunities. CRSI Industry members include manufacturers, fabricators, material suppliers and placers of steel reinforcing bars and related products. Our Professional members are involved
in the research, design, and construction of steel reinforced concrete. CRSI also has a broad Region Manager network that supports
both members and industry professionals and creates awareness among the design/construction community through outreach
activities. Together, they form a complete network of industry information and support.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Foreword
The purpose of this Design Guide is to assist in the proper application of the design and detailing requirements
in Building Code Requirements for Structural Concrete (ACI 318-19) for cast-in-place concrete buildings with
nonprestressed steel reinforcement. Many design aids and worked-out examples are provided to make designing
and detailing reinforced concrete members simpler and faster. The goal is to acquire an understanding of the code
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In addition to structural engineers, this Design Guide is a valuable resource for educators, students, individuals
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6LQFHWKH¿UVW&56,'HVLJQ+DQGERRNLQXVHUVRI&56,SXEOLFDWLRQVKDYHEHHQFRRSHUDWLYHLQVXJJHVWLQJWR
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This professional assistance is very helpful, and is appreciated. Comments on this design guide are welcome so
that future editions can be further improved. Please direct all comments to Amy Trygestad, P.E., F.ACI, CRSI Vice
President of Engineering.
Acknowledgements
Special recognition is due to the following individuals who provided a review of the publication and made insightful
comments and suggestions for improvement:
David A. Fanella, S.E., P.E., F.ACI, F.ASCE, F.SEI, Senior Director of Engineering,
is the author of this Design Guide.
Amy Trygestad, P.E., F.ACI, Vice President of Engineering
Nathan Westin, P.E., Technical Director
Special recognition is also due to Dave Mounce, Director of Communications,
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and expertise is greatly appreciated.
Photo Credits
Front cover image courtesy of Skidmore, Owings & Merrill, SEOR
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Client: Kilroy Realty
General Contractor: Webcor Builders
Back cover image courtesy of DeSimone Consulting Engineers
Project Name: Panorama Tower
Client: Florida East Coast Realty (FECR)
General Contractor: Florida East Coast Realty (FECR)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Contents
Foreword
Chapter 3
Acknowledgements
Design Loads and Load Combinations
3.1 Overview
3-1
3.2 Design Loads
3-1
Chapter 1
3.3 Seismic Design Category
3-1
Introduction
3.4 Live Load Reduction
3-3
Photo Credits
1.1 Overview
1-1
3.5 Load Factors and Combinations
3-4
1.2 Example Buildings
1-2
3.6 Determination of Wind Forces
3-8
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3.7 Determination of Seismic Forces
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and Collectors
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1.3 Organization of This Design Guide
3.8 Examples
1-10
Chapter 2
Material Requirements and Strength
Reduction Factors
2.1 Overview
2-1
2.2 Material Requirements
2-1
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Seismic Forces: Diaphragms of
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4.2 Minimum Slab Thickness
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4.3 Required Strength
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Two-way Slabs
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5.1 Overview
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5.2 Minimum Slab Thickness
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4.6 Reinforcement Detailing
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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5.5 Determination of Required Reinforcement
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5.7 Design Procedure
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5.8 Examples
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 6
Beams
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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8-1
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Analysis Methods
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9.7 Reinforcement Detailing
9-28
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Chapter 12
Earthquake-Resistant Structures – Overview
12.1 Overview
12-1
12.2 Seismic Design Category
12-1
12.3 Design and Detailing Requirements
12-1
12.4 Structural Systems
12-2
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Earthquake-Resistant Structures – SDC B
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13.3.3 Columns
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Shear Strength Requirements
Columns Supporting Reactions from
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Chapter 14
Earthquake-Resistant Structures – SDC D, E
and F
14.1 Overview
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Determining the Required Transverse
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Appendix A
References
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A-1
Appendix B
Reinforcing Bar Data
Table B.1 ASTM Standard Reinforcing Bars
B-1
Table B.2 Overall Reinforcing Bar Diameters
B-2
Appendix C
Section Index
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 1
INTRODUCTION
1.1 Overview
7KHSXUSRVHRIWKLVGHVLJQJXLGHLVWRDVVLVWLQWKHSURSHUDSSOLFDWLRQRIWKHSURYLVLRQVLQWKHHGLWLRQRIBuilding Code Requirements for Structural Concrete (ACI 318-19) and Commentary (ACI 318R-19) for cast-in-place
concrete buildings with nonprestressed steel reinforcing bars (Reference 1; see Appendix A for a list of references in
this design guide).
The main goals of this publication are to provide:
•
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step-by-step design procedures and design aids that make designing and detailing reinforced
concrete buildings simpler and faster.
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Seismic Design Categories A through F.
Throughout this publication, emphasis is placed on how to correctly apply the ACI 318 requirements, and explanatory material is provided to supplement the requirements. Readers who are interested in the origins of the provisions
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examples are given throughout the chapters of this design guide that systematically show the design and detailing of
WKHVWUXFWXUDOPHPEHUVLQWKHVHV\VWHPVXVLQJWKHH[SODQDWRU\PDWHULDODQGGHVLJQDLGVLQWKH¿UVWSDUWRIWKHFKDSter. A number of the examples address certain requirements in ACI 318 not covered in any other reinforced concrete
resources. The titles for the examples are purposely very descriptive so users can quickly locate what they are looking for in the table of contents of this publication.
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318 provision: ACI 318 section numbers are given in the left column of the index and the corresponding chapters
and page numbers in this design guide are given in the right column of the index. This method of cross-referencing
between ACI 318 and this design guide is a very useful feature for users who are interested in a particular requirement.
Throughout this publication, section numbers from ACI 318-19 are referenced as illustrated by the following: SecWLRQRI$&,LVGHQRWHGDV$&,6LPLODUO\VHFWLRQQXPEHUVIURPWKH,QWHUQDWLRQDO%XLOGLQJ&RGH
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regardless of the edition of the IBC).
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In addition to practicing engineers, this design guide is a valuable resource for educators, students, individuals
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1.2 Example Buildings
1.2.1 Building #1 – 5-Story Office Building
1A
2A
3A
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A
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Figure 1.1 Building #1 – 5-story office building.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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3A
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3A
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5A
6A
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A
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A
N
A
C
A
B
A
۴‫ ܖܗܑܜܘ۽ ܏ܖܑܕ܉ܚ‬۱
Figure 1.1 Building #1 – 5-story office building (cont.).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1A
2A
3A
4A
5A
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Figure 1.1 Building #1 – 5-story office building (cont.).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 1.1 Design Information for Building #1
Design Data
Location
Latitude
Occupancy
, Longitude
Business Group B
Topography
Not situated on a hill, ridge, or escarpment
Exposure category
C
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Site Classes C and D (default)
Load Data
Superimposed dead load
Live load
Roof
Floor
Live load
Superimposed dead load
(includes
for partitions)
Cladding
Structural Systems
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Lateral
Moment-resisting frames in both directions
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1.2.2 Building #2 – 12-Story Emergency Operations Center
1A
2A
3A
4A
35ᇱ -6ᇳ
42ᇱ -6ᇳ
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42ᇱ -6ᇳ
A
G
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F
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system ሺtyp.ሻ
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N
A
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A
C
A
B
Pan depth
Slab
Thickness
A
Rib width
12
1
Pan width
Section through wide-module joist system
Figure 1.2 Building #2 – 12-story emergency operations center.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ͳA
ʹA
͵A
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Figure 1.2 Building #2 – 12-story emergency operations center (cont.).
Table 1.2 Design Information for Building #2
Design Data
Location
Latitude
, Longitude
Occupancy
Essential facility
Topography
Not situated on a hill, ridge, or escarpment
Exposure category
B
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Site Class D (stiff soil; determined)
Load Data
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Superimposed dead load
Live load
Floor
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Live load
Cladding
Structural Systems
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Wide-module joist system
Lateral
Building frame system in both directions
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1.2.3 Building #3 – 16-Story Residential Building
1A
2A
3A
4A
5A
A
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N
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Figure 1.3 Building #3 – 16-story residential building.
Table 1.3 Design Information for Building #3
Design Data
Location
Latitude
, Longitude
Occupancy
Residential Group R
Topography
Not situated on a hill, ridge, or escarpment
Exposure category
B
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1.2.4 Building #4 – 30-Story Office Building
1
2
40ᇱ -0ԣ
3
30ᇱ -0ԣ
4
40ᇱ -0ԣ
G
F
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N
9ᇱ -0ᇳ 8ᇱ -0ᇳ D
C
B
A
Story heights ൌ 11ᇱ -0ᇳ Figure 1.4 Building #4 – 30-story office building.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 1.4 Design Information for Building #4
Design Data
Location
Latitude
, Longitude
Occupancy
Business Group B
Topography
Not situated on a hill, ridge, or escarpment
Exposure category
B
6RLOFODVVL¿FDWLRQ
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Wide-module joist system
Lateral
Dual system in both directions
1.3 Organization of This Design Guide
Properties of concrete and nonprestressed steel reinforcing bars permitted to be used in the design of reinforced conFUHWHPHPEHUVDUHFRYHUHGLQ&KDSWHU$&,VWUHQJWKUHGXFWLRQIDFWRUVDUHDOVRFRYHUHGLQWKLVFKDSWHU
Presented in Chapter 3 are typical loads and required load combinations applicable in the design of reinforced
concrete buildings. Methods are given to determine the Seismic Design Category (SDC) of a building and live load
reduction on structural members. Step-by-step procedures on how to determine wind forces on the main wind force
resisting system (MWFRS), seismic forces on the seismic-force-resisting system (SFRS), and seismic design forces
on diaphragms (including collectors) are also presented along with examples illustrating the load calculations for
each of the example buildings.
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examples are provided for a one-way slab that is part of a wide-module joist system.
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Design aids are provided on how to determine the slab thickness based on serviceability and two-way shear strength
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method to determine the section properties of any critical section for two-way shear is also given, including properties for polygon-shaped sections, which are required where shear reinforcement is provided. Examples include
analysis and design for two-way slabs systems subjected to both gravity and lateral load effects, two-way shear
design for slabs with headed shear studs and stirrups, and analysis for two-way shear with a slab opening adjacent to
a column.
The design and detailing requirements for beams are covered in Chapter 6 and are applicable to buildings assigned
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economical detailing and design examples for wide-module joist systems and beams subjected to combined gravity
and lateral load effects.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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and spirals) for both nonslender and slender columns. Examples include the determination of nominal and design
strength interaction diagrams for rectangular and circular columns, columns subjected to combined gravity and latHUDOORDGHIIHFWVDQGWKHGHVLJQDQGGHWDLOLQJRIDFRUQHUFROXPQVXEMHFWHGWRELD[LDOÀH[XUHDQGVKHDU
Structural walls in buildings assigned to SDC A, B, or C are covered in Chapter 8. Information is given on the moPHQWPDJQL¿FDWLRQPHWKRGWKHDOWHUQDWLYHPHWKRGIRURXWRISODQHVOHQGHUZDOODQDO\VLVDQGWKHVLPSOL¿HGGHVLJQ
method. Typical reinforcement details are provided, including details for walls with openings, and examples are
given for nonslender and slender walls subjected to axial forces and in-plane and out-of-plane lateral forces.
Design and detailing requirements for diaphragms are given in Chapter 9 for buildings assigned to SDC A, B, or C.
Diaphragm modeling and analysis, in-plane stiffness modeling, and analysis methods for diaphragms with and without openings are the primary topics covered. Methods are presented on how to determine the center of rigidity, how
to distribute the in-plane forces to the vertical elements of the lateral force-resisting system, and how to determine
the required diaphragm and collector reinforcement.
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spread footings, and drilled piers (caissons) supporting buildings assigned to SDC A and B. Procedures are presentHGRQKRZWRVL]HWKHIRXQGDWLRQPHPEHUDQGKRZWRGHWHUPLQHDQGGHWDLOWKHUHTXLUHGUHLQIRUFHPHQWIRUPHPEHUV
subjected to axial forces and combined bending and axial forces.
Design and detailing requirements for beam-column and slab-column joints in buildings assigned to SDC A and B
are covered in Chapter 11. Step-by-step procedures are given on how to determine joint forces in moment frames
subjected to gravity loads only and to gravity and lateral loads. Examples are provided illustrating application of the
requirements.
An overview of the design and detailing requirements for structures assigned to SDC B through F is given in ChapWHU7KHUHTXLUHPHQWVEDVHGRQ6'&DUHSURYLGHGDORQJZLWKWKHDSSOLFDEOH$&,VHFWLRQQXPEHUVWKDWPXVW
EHVDWLV¿HG,QFOXGHGLVDQRYHUYLHZRIVHLVPLFIRUFHUHVLVWLQJV\VWHPVIRUUHLQIRUFHGFRQFUHWHEXLOGLQJVWKHFRUUHVSRQGLQJGHVLJQFRHI¿FLHQWVDQGIDFWRUVDQGWKHVWUXFWXUDOV\VWHPOLPLWDWLRQVIRUWKHVHV\VWHPV
The design and detailing requirements for frame members and foundations in buildings assigned to SDC B and C
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columns, and beam-column joints in intermediate moment frames and for drilled piers (caissons). Included are examples for the design of the structural members in an intermediate moment frame and a foundation seismic tie.
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SDC D, E, or F:
• beams, columns, and beam-column joints in special moment frames;
• special structural walls;
• coupling beams;
• wall piers;
• diaphragms;
• shallow and deep foundations (including foundation seismic ties);
• and members not designated as part of the SFRS.
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along with examples illustrating the proper application of the provisions.
References in this design guide are given in Appendix A. Appendix B contains properties of currently available steel
reinforcing bars. The section index described in Section 1.1 above is in Appendix C.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 2
MATERIAL REQUIREMENTS AND STRENGTH REDUCTION FACTORS
2.1 Overview
Properties of concrete and nonprestressed reinforcement permitted to be used in the design of reinforced concrete
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accordance with the requirements in ACI Chapter 19. Similarly, the design properties of reinforcement must be
VHOHFWHGLQDFFRUGDQFHZLWKWKHUHTXLUHPHQWVLQ$&,&KDSWHU $&, play a key role in the
Strength reduction factors, which are commonly referred to as resistance factors or
determination of the design strength of a reinforced concrete member. ACI strength reduction factors are given in
$&,&KDSWHUDQGDUHDOVRFRYHUHGLQWKLVFKDSWHU
2.2 Material Requirements
2.2.1 Concrete Design Properties
Specified Compressive Strength
Stress-strain curves for compression tests on concrete mixtures of varying compressive strengths are given in FigXUH
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Figure 2.1 Stress-strain curves for compression tests on
concrete mixtures of varying compressive strengths.
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QRUPDOZHLJKWDQGOLJKWZHLJKWFRQFUHWH>$&, D @7KHOLPLWVRQ for cast-in-place structural members
based on Seismic Design Category (SDC), including foundations supporting concrete structures with other than
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 2.1 Limits for Concrete Compressive Strength
Minimum f 'c (psi)
Application
General
Foundations for structures assigned to SDC A, B, or C
Foundations for structures assigned to SDC D, E, or F
Special moment frames
6SHFLDOVWUXFWXUDOZDOOVZLWK*UDGHRUUHLQIRUFHPHQW
6SHFLDOVWUXFWXUDOZDOOVZLWK*UDGHUHLQIRUFHPHQW
A minimum RISVLIRUVSHFLDOVWUXFWXUDOZDOOVZLWK*UDGHUHLQIRUFHPHQWLVVSHFL¿HGEHFDXVHWHVWGDWD
are not available for lower concrete strengths. In general, minimum concrete strengths are increased for special
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to enhance reinforcing bar anchorage and to improve structural performance by reducing the neutral axis depth.
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WKDWPXVWEHVSHFL¿HG>$&, E @([SRVXUHFDWHJRULHVDQGFODVVHVDUHGH¿QHGLQ$&,7DEOHDQG
structural members must be assigned an exposure class for the applicable exposure category. Based on the severity
based on durability requirements may be higher than that required for strucof the anticipated exposure class,
tural purposes.
Structural strength requirements may also have an effect on the minimum >$&, F @7KHRQO\ZD\WR
satisfy certain strength requirements for a structural member is to increase the compressive strength of the concrete.
No upper limits on DUHVSHFL¿HGH[FHSWIRUOLJKWZHLJKWFRQFUHWHLQVSHFLDOPRPHQWIUDPHVVSHFLDOVWUXFWXUDO
walls, and their foundations; in such cases, PXVWEHOHVVWKDQRUHTXDOWRSVL>$&, G @7KLVOLPLW
LVLPSRVHGEHFDXVHRIWKHYHU\OLPLWHGDPRXQWRIH[SHULPHQWDODQG¿HOGGDWDDYDLODEOHRQWKHEHKDYLRUDQGSHUIRUPDQFHRIPHPEHUVPDGHZLWKOLJKWZHLJKWFRQFUHWH+RZHYHUYDOXHVRI JUHDWHUWKDQSVLDUHSHUPLWWHG
if it can be shown that members made with lightweight concrete provide strength and toughness levels equal to or
exceeding those of comparable members made with normalweight concrete of the same compressive strength.
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$&,DQGIRUWHVWLQJDQGDFFHSWDQFHRIFRQFUHWHLQ$&,
Concrete compressive strength is typically determined for a concrete mixture on the basis of compression tests of
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FRQVWUXFWLRQGRFXPHQWV $&, Modulus of Elasticity
The modulus of elasticity of concrete,
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For concrete with
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For normalweight concrete
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2-2
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,Q(TXDWLRQ is the density (unit weight) of normalweight concrete or the equilibrium density of lightweight
has the units of pounds per square inch (psi).
concrete. In both equations,
It is permitted to specify
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IRUH[DPSOHZKHQFDOFXODWLQJWKHGHÀHFWLRQRIDEHDPVXSSRUWdesign conditions are sensitive to the value of
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EXLOGLQJVHH$&,5 Modulus of Rupture
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The modulus of rupture for concrete,
(2.3)
In this equation,
has the units of pounds per square inch (psi) and LVDPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, VHHWKH
following section).
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Lightweight Concrete Modification Factor
The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODtive to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium density, RIWKHFRQFUHWHPL[WXUHRU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[WXUH VHH$&,
9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOH D @DQGYDOXHVRI based on composiWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOH E @1RWHWKDW LVSHUPLWWHGWREHWDNHQDVIRU
OLJKWZHLJKWFRQFUHWH $&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH $&, Table 2.2 Values of Ȝ Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
Table 2.3 Values of Ȝ Based on Composition of Aggregates
Concrete
All-lightweight
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Sand-lightweight
Sand-lightweight, coarse blend
Composition of Aggregates
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WR(1)
WR (1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate.
(2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
2.2.2 Nonprestressed Steel Reinforcement
Material Properties
Yield Strength
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for a steel reinforcing bar is given in Figure
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• A rounded strain-hardening segment
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Figure 2.2 Stress-strain curve of a steel reinforcing bar with a well-defined yield strength.
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force hesitates corresponds to the yield strength of the reinforcing bar.
Test values of
The slope of the initial linear segment of the stress-strain curve is the modulus of elasticity,
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$&, is the slope tangent to the initial portion of the strain-hardening segment of the
The strain-hardening modulus,
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The stress-strain curves for some types of higher strength reinforcing bars can be more rounded in shape than the
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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method, which must be used to determine ZKHUHDVWUHVVVWUDLQFXUYHGRHVQRWKDYHDZHOOGH¿QHG\LHOGSRLQW
is reached.
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Figure 2.3 Rounded stress-strain curve for Grade 100 steel reinforcing bars.
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Tensile Property and Uniform Elongation Requirements
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 2.4 ASTM Specifications and Additional Requirements for Deformed Reinforcing Bars
Steel Type
ASTM Specification/Additional Requirements
Carbon steel
ASTM A615/ACI Table 20.2.1.3(a)
Low-alloy steel
ASTM A706/ACI 20.2.1.3(b)(i)-(iii)
Axle steel, rail steel, and bars from rail steel
ASTM A996/Bars from rail steel must be Type R
Stainless steel
ASTM A955
Low-carbon chromium steel
ASTM A1035
The additional requirements in ACI Table 20.2.1.3(a) for ASTM A615 reinforcement are given in Table 2.5.
Table 2.5 Modified Tensile Strength and Additional Tensile Strength Property Requirements for ASTM A615
Reinforcement
Grade 40
Grade 60
Grade 80
Grade 100
Minimum tensile strength (psi)
60,000
80,000
100,000
115,000
Minimum ratio of actual tensile strength
to actual yield strength
1.10
1.10
1.10
1.10
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Table 20.2.1.3(b) [see Table 2.6]. Bend test requirements for ASTM A706 Grade 100 reinforcement must
be the same as those for ASTM A706 Grade 80 reinforcement.
ii. 8QLIRUPHORQJDWLRQUHTXLUHPHQWVIRUDOOJUDGHVRI$670$UHLQIRUFHPHQWPXVWEHDVVSHFL¿HGLQ
ACI Table 20.2.1.3(c) [see Table 2.7] where the uniform elongation is determined as the elongation at the
maximum force sustained by the reinforcing bar test piece.
iii. For all grades of ASTM A706 reinforcement, the radius at the base of each bar deformation must be at
least 1.5 times the height of the deformation. This requirement applies to all deformations, including
transverse lugs, longitudinal ribs, grade ribs, grade marks, and intersections between deformations. Conformance must be assessed by measurements taken on newly-machined rolls used to manufacture reinforcing bars instead of measurements taken on reinforcing bar samples.
i.
Table 2.6 Tensile Property Requirements for ASTM A706 Grade 100 Reinforcement
Minimum tensile strength, psi
117,000
Minimum ratio of actual tensile strength to actual yield strength
1.17
Minimum yield strength, psi
100,000
Maximum yield strength, psi
118,000
Minimum fracture elongation in 8 in., percent
10
Table 2.7 Uniform Elongation Requirements for ASTM A706 Reinforcement
Bar Size
Minimum Uniform Elongation, Percent
Grade 60
Grade 80
Grade 100
#3 through #10
9
7
6
#11, #14, #18
6
6
6
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Properties
Stress-Strain Relationships
For deformed reinforcing bars, it is permitted to assume the stress in the reinforcing bars,
is proportional to
that is,
$&, )RUVWUDLQVJUHDWHUWKDQWKDWFRUUHstrain in cases where the strain is below
the increase in strength due to strain hardening is neglected for nominal strength calculasponding to
tions.
Modulus of Elasticity
As noted above, the modulus of elasticity of the reinforcing steel,
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LVJLYHQLQ$&,7DEOH E IRUSODLQUHLQIRUFHPHQW VHH7DEOH Table 2.8 Nonprestressed Deformed Reinforcing Bars
Usage
• Flexure
• Axial force
• Shrinkage and
temperature
• Lateral support
of longitudinal
bars
• Concrete
FRQ¿QHPHQW
Special
seismic
systems
Torsion
Anchor
reinforcement
Special moment frames
Special structural walls(1)
Other
Special seismic systems
Spirals
Other
Special
seismic
systems Shear
Maximum Value of f y or
f yt Permitted for Design
Calculations (psi)
Application
Special moment frames(6)
Special structural walls ASTM Specification
$ A$$$
$
A$$$
$
$$$$
$
$$$$
$$$$
Spirals
$$$$
Shear friction
Stirrups, ties, hoops
Longitudinal and transverse
$$$$
$$$$
$
$$$$
Special seismic systems
$ Other
$$$$
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 2.8 Nonprestressed Deformed Reinforcing Bars (cont.)
Usage
Application
Maximum Value of f y or
f yt Permitted for Design
Calculations (psi)
Regions designed
using the strut-andtie method
Longitudinal ties
Other
(1)
(2)
(3)
(4)
(5)
(6)
(7)
ASTM Specification
$$$$
All components of special structural walls, including coupling beams and wall piers.
ASTM A615 Grade 60 bars are permitted provided the requirements of ACI 20.2.2.5(b) are satisfied.
In slabs and beams that are not part of a special seismic system, reinforcing bars passing through or extending from special structural walls must satisfy ACI 20.2.2.5.
Longitudinal reinforcement with fy>80,000 psi is not permitted for intermediate moment frames and ordinary moment frames resisting earthquake demands,
The maximum design value of fy or fyt is 80,000 psi for use in design calculations when used for shear reinforcement in diaphragms and foundations for load combinations including earthquake forces where the diaphragms or foundations are part of a building with a special seismic system.
Shear reinforcement in this application includes stirrups, ties, hoops, and spirals in special moment frames.
Shear reinforcement in this application includes all transverse reinforcement in special structural walls, coupling beams, and wall piers. Diagonal bars in coupling
beams must comply with ASTM A706 or Footnote (2).
Table 2.9 Nonprestressed Plain Spiral Reinforcing Bars
Application
Maximum Value of f y or f yt
Permitted for Design
Calculations (psi)
ASTM Specification
Spirals in special
seismic systems
$$$
$
Spirals
$$$
$
Shear
Spirals
$$$
$
Torsion in nonprestressed
beams
Spirals
$$$
$
Usage
• Lateral support of
longitudinal bars
• &RQFUHWHFRQ¿QHPHQW
Special Seismic Systems and Anchor Reinforcement
$FFRUGLQJWR$&,GHIRUPHGQRQSUHVWUHVVHGORQJLWXGLQDOUHLQIRUFHPHQWUHVLVWLQJHDUWKTXDNHLQGXFHGEHQGing moments, axial forces, or both, in special seismic systems and anchor reinforcement in SDCs C, D, E, and F
must be in accordance with (a) or (b):
a. $
670$*UDGHRUIRUVSHFLDOVWUXFWXUDOZDOOVDQG*UDGHDQGIRUVSHFLDOPRPHQW
frames.
b. $670$*UDGHLI L WKURXJK LY DUHVDWLV¿HG $670$*UDGHDQG*UDGHDUHQRWSHUPLWWHG
in special seismic systems because of concern associated with low-cycle fatigue behavior).
i. Actual yield strength based on mill tests does not exceed E\PRUHWKDQSVL
ii. 5DWLRRIWKHDFWXDOWHQVLOHVWUHQJWKWRWKHDFWXDO\LHOGVWUHQJWKLVDWOHDVW
iii. 0LQLPXPIUDFWXUHHORQJDWLRQLQLQPXVWEHDWOHDVWSHUFHQWIRUEDUVL]HVWKURXJKDWOHDVW
SHUFHQWIRUEDUVL]HVWKURXJKDQGDWOHDVWSHUFHQWIRUEDUVL]HVDQG
iv. 0LQLPXPXQLIRUPHORQJDWLRQPXVWEHDWOHDVWSHUFHQWIRUEDUVL]HVWKURXJKDQGDWOHDVW
SHUFHQWIRUEDUVL]HVDQG
occurs at the tensile strength is commonly referred to as uniform elongation, which is the largest
The strain
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generally occurs right before the onset of necking. This is the maximum strain relied upon at a location where yielding of the reinforcing bar may occur (that is, in an anticipated plastic hinge region in the member).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The strain at the fracture point
is the total elongation over a prescribed gauge length extending across the fracture
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marks is re-measured. The total elongation is calculated as the percent increase in length relative to the original
gauge length:
(2.4)
2.2.3 Headed Shear Stud Reinforcement
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2.2.4 Durability of Steel Reinforcement
Specified Concrete Cover
Overview
Concrete cover, which is provided to reinforcing bars to protect them from weather and other effects, is measured
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JLYHQLQ$&,WKURXJK $&, $JUHDWHUFRYHUWKDQWKDWSUHVFULEHGLQWKHVHVHFWLRQVPD\
EHUHTXLUHGE\WKHJHQHUDOEXLOGLQJFRGHIRU¿UHSURWHFWLRQ&RQFUHWHÀRRU¿QLVKHVDUHSHUPLWWHGWREHSDUWRIWKH
UHTXLUHGFRYHUIRUQRQVWUXFWXUDOSXUSRVHV $&, Nonprestressed Cast-in-place Members
0LQLPXPFRQFUHWHFRYHUIRUQRQSUHVWUHVVHGFDVWLQSODFHFRQFUHWHPHPEHUVLVJLYHQLQ$&,7DEOH $&,
VHH7DEOH Table 2.10 Minimum Concrete Cover for Cast-in-place Nonprestressed Concrete Members
Concrete Exposure
Member
Reinforcement
Specified Cover (in.)
Cast against and permanently
in contact with the ground
All
All
Exposed to weather or in
contact with the ground
WKURXJKEDUV
All
EDUVDQGVPDOOHU
DQGEDUV
EDUVDQGVPDOOHU
Slabs, joists, and walls
Not exposed to weather or
in contact with the ground
Beams, columns, pedestals, Primary reinforcement, stirrups,
and tension ties
ties, spirals, and hoops
Deep Foundation Members
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2-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 2.11 Minimum Concrete Cover for Cast-in-place Deep Foundation Members
Concrete Exposure
Reinforcement
Specified Cover (in.)
Cast against and permanently in contact with the ground and not
enclosed by steel pipe, tube, permanent casing, or stable rock socket
All
Enclosed by steel pipe, tube, permanent casing, or stable rock socket
All
%XQGOHG5HLQIRUFLQJ%DUV
)RUEXQGOHGUHLQIRUFLQJEDUVWKHVSHFL¿HGFRQFUHWHFRYHULVHTXDOWRWKHVPDOOHURIWKHHTXLYDOHQWGLDPHWHURIWKH
EXQGOHRULQ $&, For bundled bars in members where the concrete is cast against and is permanently in contact with the ground, the
VSHFL¿HGFRYHULVLQ
Headed Shear Stud Reinforcement
7KHVSHFL¿HGFRQFUHWHFRYHUIRUKHDGVDQGEDVHUDLOVLQKHDGHGVKHDUVWXGDVVHPEOLHVPXVWEHDWOHDVWHTXDOWRWKDW
UHTXLUHGIRUWKHUHLQIRUFHPHQWLQWKHPHPEHU $&,VHH$&,)LJXUH5 )RUH[DPSOHWKHPLQLPXPFRQFUHWHFRYHUWRWKHKHDGVDQGUDLOVLVLQIRUDKHDGHGVKHDUVWXGDVVHPEO\XVHGDVVKHDUUHLQIRUFHPHQW
in a two-way reinforced concrete slab not exposed to the elements and reinforced with longitudinal reinforcement
VPDOOHUWKDQEDUV VHH7DEOH Corrosive Environments
For members in corrosive environments where it is expected the concrete will be exposed to external sources of
chlorides (such as deicing salts, brackish water, seawater, or spray from these sources) or other severe exposure
FRQGLWLRQVVSHFL¿HGFRQFUHWHFRYHUVKRXOGEHLQFUHDVHGDERYHWKHPLQLPXPYDOXHVVSHFL¿HGLQ$&,DV
GHHPHGQHFHVVDU\ $&, ,QVXFKFDVHVWKHFRQFUHWHVKRXOGEHSURSRUWLRQHGWRVDWLVI\WKHUHTXLUHPHQWV
IRUWKHDSSOLFDEOHH[SRVXUHFODVVVSHFL¿HGLQ$&,&KDSWHU
$VSHFL¿HGFRQFUHWHFRYHURIQRWOHVVWKDQLQLVUHFRPPHQGHGLQ$&,5IRUUHLQIRUFHPHQWLQZDOOV
DQGVODEVLQFRUURVLYHHQYLURQPHQWV)RURWKHUPHPEHUVDLQFRQFUHWHFRYHULVUHFRPPHQGHG
Nonprestressed Coated Reinforcement
Coated reinforcing bars are used in applications where corrosion resistance is required (such as parking structures or
other buildings and structures where the members are exposed to the elements). Nonprestressed coated reinforcing
EDUVPXVWFRQIRUPWR$&,7DEOH $&, =LQFFRDWHG KRWGLSSHGJDOYDQL]HG $670$
(SR[\FRDWHG$670$RU$
=LQFDQGHSR[\GXDOFRDWHG$670$
7KHDIRUHPHQWLRQHGFRDWHGEDUW\SHVPXVWFRQIRUPWRWKHUHTXLUHPHQWVLQ$&, D E RU F 2.3 Strength Reduction Factors
2.3.1 Overview
Strength reduction factors,
used in the determination of the design strength of a reinforced concrete member are
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2-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1. To account for the probability of under-strength members due to variations in material strengths and dimensions.
The strength of concrete can vary because concrete is a composite material made of constituent materials
whose properties vary. The strength of reinforcing steel can also vary but usually to a much lesser degree than
concrete.
To account for inaccuracies in design equations.
$QXPEHURIDVVXPSWLRQVDQGVLPSOL¿FDWLRQVDUHPDGHLQWKHHTXDWLRQVWRGHWHUPLQHQRPLQDOVWUHQJWKV$V
such, inaccuracies are introduced that must be accounted for when determining the design strength of a reinforced concrete member.
3. 7RUHÀHFWWKHDYDLODEOHGXFWLOLW\DQGUHTXLUHGUHOLDELOLW\RIWKHUHLQIRUFHGFRQFUHWHPHPEHUXQGHUWKHORDGHIfects being considered.
Reinforced concrete members that are more ductile, such as beams or slabs, are less sensitive to variations
in concrete strength compared to members that are less ductile, such as columns. Spiral reinforcement con¿QHVWKHFRQFUHWHLQDFROXPQEHWWHUWKDQWLHGUHLQIRUFHPHQW7KXVVSLUDOFROXPQVDUHPRUHGXFWLOHDQGKDYH
greater toughness than tied columns.
7RUHÀHFWWKHLPSRUWDQFHRIWKHUHLQIRUFHGFRQFUHWHPHPEHULQWKHVWUXFWXUH
The failure of a vertical member, such as a column or wall, in a structure is usually considered more detrimenWDOWKDQIDLOXUHRIDKRUL]RQWDOPHPEHUVXFKDVDEHDP
2.3.2 Strength Reduction Factors Based On Action or Structural Element
Strength reduction factors based on different types of actions and nonprestressed structural elements are given in
$&,7DEOH VHH$&,DQG7DEOHIRUWKHVWUHQJWKUHGXFWLRQIDFWRUVDSSOLFDEOHLQWKLVSXEOLFDWLRQ The strength reduction factor for shear in this table is not applicable to members in special moment frames (including beam-column joints), special structural walls (including diagonally reinforced coupling beams), and the other
PHPEHUVQRWHGLQ$&, VHH6HFWLRQRIWKLVSXEOLFDWLRQ Table 2.12 Strength Reduction Factors, Ҁ
Action
Strength Reduction Factor, Ҁ
Moment, axial force, or combined moment
and axial force
Shear
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VHH6HFWLRQRIWKLVSXEOLFDWLRQ
Torsion
Bearing
2.3.3 Strength Reduction Factors For Moment, Axial Force, Or Combined Moment and Axial Force
Strength reduction factors for reinforced concrete members subjected to bending moments, axial forces, or a combiQDWLRQRIERWKDUHJLYHQLQ$&,7DEOH VHH$&,7DEOHDQG)LJXUH Table 2.13 Strength Reduction Factors, Ҁ, For Moment, Axal Force, Or Combined Moment and Axial Force
Strength Reduction Factor, Ҁ
Net Tensile Strain, İt
Classification
Type of Transverse Reinforcement
Spirals Conforming to ACI 25.7.3
Other
0.75
0.65
Compression-controlled
Transition*
Tension-controlled
*For sections classified as transition, it is permitted to use
corresponding to compression-controlled sections.
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2-11
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 2.4 Variation of
with net tensile strain
The nominal strength of a member subjected to bending moments or combined bending moments and axial forces is
GHWHUPLQHGDVVXPLQJWKHVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHURIWKHVHFWLRQLVHTXDOWR VHH$&,
is the tensile strain calculated in the extreme tension reinforcement at nominal
7KHQHWWHQVLOHVWUDLQ
VWUHQJWK VHH)LJXUHIRUWKHFDVHRIDUHLQIRUFHGFRQFUHWHEHDPZLWKWZROD\HUVRIWHQVLRQUHLQIRUFHPHQWDWWKH
section under consideration).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
&RPSUHVVLRQFRQWUROOHGVHFWLRQVDUHGH¿QHGDVVHFWLRQVZKHUH
The term
is the value of the net tensile
VWUDLQLQWKHH[WUHPHOD\HURIORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQ
$&, (2.5)
)RU*UDGHUHLQIRUFHPHQW LVSHUPLWWHGWREHWDNHQHTXDOWR $&, 0HPEHUVVXEMHFWHGWRRQO\
axial compression, like columns, are compression-controlled. In such cases, a brittle compression failure is expected
with little or no warning prior to failure. As such, lower strength reduction factors are assigned to compressioncontrolled sections. Additionally, columns typically support areas much greater than beams, which are tensionFRQWUROOHGDQGDVQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHFRQVHTXHQFHVRIFROXPQIDLOXUHDUHJHQHUDOO\PRUH
severe than those attributed to beam failure.
7KLVOLPLWSURYLGHVVXI¿FLHQWGXFWLOLW\IRUPRVWDS$VHFWLRQLVGH¿QHGDVWHQVLRQFRQWUROOHGZKHUH
SOLFDWLRQVDQGHVVHQWLDOO\SURYLGHVDZDUQLQJWRIDLOXUH WKDWLVH[FHVVLYHFUDFNLQJDQGGHÀHFWLRQDUHH[SHFWHGSULRU
to failure). Beams and slabs must be tension-controlled.
A linear transition in the strength reduction factor is permitted between the limits for compression-controlled and
WHQVLRQFRQWUROOHGVHFWLRQV VHH)LJXUH ,WLVSHUPLWWHGWRXVHVWUHQJWKUHGXFWLRQIDFWRUVFRUUHVSRQGLQJWR
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$FFRUGLQJWR$&,GHYHORSPHQWOHQJWKVGRQRWUHTXLUHDVWUHQJWKUHGXFWLRQIDFWRU%HFDXVHODSVSOLFHOHQJWKV
are based on development lengths, a strength reduction factor is not required for lap splices as well. An allowance for
VWUHQJWKUHGXFWLRQLVLQFOXGHGLQWKHH[SUHVVLRQVLQ$&,&KDSWHUIRUGHWHUPLQLQJGHYHORSPHQWDQGVSOLFHOHQJWKV
2.3.4 Strength Reduction Factors For Shear In Structures Relying On Special Moment Frames and Special
Structural Walls
)RUVWUXFWXUHVWKDWXWLOL]HVSHFLDOPRPHQWIUDPHVDQGVSHFLDOVWUXFWXUDOZDOOVWRUHVLVWHDUWKTXDNHHIIHFWV the value of the strength reduction factor, IRUVKHDULVWREHPRGL¿HGLQDFFRUGDQFHZLWK$&,WKURXJK
$&, For any member designed to resist where the nominal shear strength of the member is less than the shear corresponding to the development of the nominal moment strength of the member, IRUVKHDUPXVWEHWDNHQDV
$&, 7KHQRPLQDOPRPHQWVWUHQJWKLQVXFKFDVHVLVWREHFDOFXODWHGXVLQJWKHIDFWRUHGD[LDOIRUFHVIURP
ORDGFRPELQDWLRQVLQ$&,7DEOHWKDWLQFOXGH This provision is applicable to shear-controlled members,
such as low-rise structural walls, portions of walls between openings, or diaphragms where nominal shear strength
is less than the shear corresponding to the development of nominal moment strength. In most cases, it would be
impractical to design the member otherwise.
For diaphragms, for shear must not exceed the value of for shear required for the vertical components of the
SULPDU\VHLVPLFIRUFHUHVLVWLQJV\VWHP 6)56 >$&,@7KLVSURYLVLRQLVLQWHQGHGWRLQFUHDVHWKHVWUHQJWK
for the walls assigned to resist the earthquake
of the diaphragm and its connections in buildings where
forces. This requirement is also applicable to foundation elements supporting the primary vertical elements of the
6)56 $&, For beam-column joints of special moment frames and diagonally reinforced coupling beams,
WR $&, for shear is equal
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 3
DESIGN LOADS AND LOAD COMBINATIONS
3.1 Overview
$FFRUGLQJWR$&,WKHORDGVDQGORDGFRPELQDWLRQVFRQVLGHUHGLQGHVLJQPXVWEHLQDFFRUGDQFHZLWK$&,&KDSWHU3UHVHQWHGLQWKLVFKDSWHUDUHWKHW\SLFDOORDGVDQGUHTXLUHGORDGFRPELQDWLRQVDSSOLFDEOHLQWKHGHVLJQRIUHinforced concrete buildings. Methods are given to determine the Seismic Design Category (SDC) of a building and
live load reduction on structural members. Step-by-step procedures on how to determine wind forces on the main
wind force resisting system (MWFRS), seismic forces on the seismic-force-resisting system (SFRS), and seismic
design forces on diaphragms (including collectors) are also presented.
3.2 Design Loads
In the design of reinforced concrete members, the loads that must be included are self-weight, applied loads, and the
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Table 3.1 Summary of Load Effects
Section/Chapter No.
Notation
Load Effect
IBC
ASCE/SEI 7
Dead loads
3.1
Weight of ice
Seismic load effects
1613
Seismic load effects including overstrength
1613
/RDGVGXHWRÀXLGVZLWKZHOOGH¿QHGSUHVVXUHVDQGPD[LPXPKHLJKWV
—
—
Flood loads
Loads due to lateral earth pressures, ground water pressure, or pressure
of bulk materials
5RRIOLYHORDGVJUHDWHUWKDQOEIWDQGÀRRUOLYHORDG
5RRIOLYHORDGVRIOEIW or less
Rain loads
1611
8
Snow loads
Cumulative effects of self-straining forces and effects
—
Loads due to wind pressure
±
Wind-on-ice loads
3.3 Seismic Design Category
The SDC of a building is determined in accordance with the general building code or determined by the building
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$6&(6(, $&, )LJXUHFDQEHXVHGWRGHWHUPLQHWKH6'&EDVHGRQWKHUHTXLUHPHQWVLQWKHVH
and
are the
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UHVSHFWLYHO\ZKLFKDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$6&(6(,
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3-1
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine ܵܵ and ܵ1 from Figures 22-1 through 22-8 ሺ11.4.2ሻ
EŽ
Determine the Risk Category
from Table 1.5-1
EŽ
EŽ
zĞƐ
Is ܵܵ ൑ 0.15 and ܵ1 ൑ 0.04? Structure is permitted to be assigned to
SDC A and must only comply with 11.7
Is the Risk Category I, II, or III?
Is ܵ1 ൒ 0.75? zĞƐ
zĞƐ
Is ܵ1 ൒ 0.75? EŽ
zĞƐ
Structure is assigned to SDC E
Structure is assigned to SDC F
Determineܵ‫ ܵܦ‬and ܵ‫ܦ‬1 E\
EŽ
Are all 4 conditions in 11.6 satisfied?
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UHVSHFWLYHO\
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Figure 3.1 Determination of Seismic Design Category (SDC).
Structural members in buildings assigned to SDC A must satisfy all the applicable requirements in ACI 318 except
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WKURXJK)WKHSURYLVLRQVRI&KDSWHUPXVWEHVDWLV¿HGLQDGGLWLRQWRWKHDSSOLFDEOHUHTXLUHPHQWVRIRWKHUFKDSWHUV
$&, 7KHUHTXLUHPHQWVRI$&,SHUWDLQWRVWUXFWXUDOPHPEHUVGHVLJQDWHGQRWWREHSDUWRIWKH6)56
in buildings assigned to SDC B through F.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3.4 Live Load Reduction
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and uniform roof live loads, respectively, may be used.
Determine ‫ ݋ܮ‬from Table 4.3-1
zĞƐ
Is ‫ ݋ܮ‬൐ 100 psf? EŽ
Live load reduction is not permitted*
EŽ
Is the member a one-way slab?
zĞƐ
‫ ܶ ܣ‬൑ 1.5ሺslab spanሻ2 EŽ
Is the structure a passenger vehicle garage?
zĞƐ
Live load reduction is not permitted*
Is the area classified as an assembly
occupancy?
EŽ
zĞƐ
Live load reduction is not permitted
EŽ
Is‫ ܶ ܣ ܮܮܭ‬൏ 400 sq ft?
zĞƐ
Live load reduction is not permitted
‫ ܮ‬ൌ ‫ ݋ܮ‬ቆ0.25 ൅
15
ඥ‫ܶܣ ܮܮܭ‬
ቇ ሾASCE/SEI Eq. ሺ4.7-1ሻሿ
൒ 0.50‫ ݋ܮ‬for members supporting one floor
* Live loads for members supporting two or more
floors are permitted to be reduced by a maximum of
20 percent, but the reduced live load must not be less
than L determined in accordance with ASCE/SEI 4.7.2.
൒ 0.40‫ ݋ܮ‬for members supporting two or more floors
Figure 3.2 Determination of live load reduction in accordance with ASCE/SEI 4.7.
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3-3
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Reduced uniform live load, LQDFFRUGDQFHZLWK$6&(6(,FDQEHGHWHUPLQHGXVLQJWKHÀRZFKDUWLQ)LJXUH
is the live load element
,QWKLV¿JXUH LVWKHXQLIRUPO\GLVWULEXWHGOLYHORDGLQ$6&(6(,7DEOH
is the tributary area in square
IDFWRULQ$6&(6(,7DEOH VHH$6&(6(,)LJXUH&DQG7DEOH DQG
FDQEHGHWHUPLQHGE\WKHÀRZFKDUWLQ)LJXUHZKHUH and
are
feet. Similarly, reduced roof live load,
and the factor
respectively. For pitched roofs, is equal to
reduction factors based on the tributary area,
WKHQXPEHURILQFKHVRIULVHSHUIRRWDQGIRUDUFKHVRUGRPHVLWLVWKHULVHWRVSDQUDWLRPXOWLSOLHGE\
Table 3.2 Live Load Element Factor, KLL
KLL
Element
Interior columns
Exterior columns without cantilever slabs
Edge columns with cantilever slabs
3
Corner columns with cantilever slabs
Edge beams without cantilever slabs
Interior beams
$OORWKHUPHPEHUVQRWLGHQWL¿HGLQFOXGLQJ
• Edge beams with cantilever slabs
• Cantilever beams
• One-way slabs
• Two-way slabs
• Members without provisions for continuous shear transfer normal to their span
1
3.5 Load Factors and Combinations
The load combinations needed to determine the required strength,
of a structural member are given in ACI Table
VHH7DEOH 7KHORDGFRPELQDWLRQVLQ,%&DUHWKHVDPHDVWKRVHLQ7DEOHZLWKWKHIROORZLQJ
exceptions:
1. The variable LQ,%&(TXDWLRQV DQG DUHQRWLQ$&,(TXDWLRQV F G DQG
H ,QVWHDGWKHORDGIDFWRURQWKHOLYHORDG LQWKH$&,ORDGFRPELQDWLRQVLVHTXDOWRZLWKWKHH[ception that the load factor on LVSHUPLWWHGWRHTXDOIRUDOORFFXSDQFLHVH[FHSWIRUSDUNLQJJDUDJHVDUHDV
occupied as places of public assembly, and areas where LVJUHDWHUWKDQOEIW $&, 7KLVH[FHStion makes these load combinations the same in the IBC and ACI 318.
The variable LQ,%&(TXDWLRQ LVQRWLQ$&,(TXDWLRQ H ,QVWHDGDORDGIDFWRURILVDSSOLHG
to $FFRUGLQJWRWKHVHFRQGH[FHSWLRQLQ$6&(6(,$6&(6(,ORDGFRPELQDWLRQ>ZKLFKLVWKH
or the sloped roof snow
VDPHDV$&,(TXDWLRQ H @ PXVWEHWDNHQDVHLWKHUWKHÀDWURRIVQRZORDG
7KLVPHDQVWKHEDODQFHGVQRZORDGGH¿QHGLQ$6&(6(,IRUÀDWURRIVDQGLQ$6&(6(,IRU
load,
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LVGH¿QHGLQWKHVDPHZD\ VHHH[FHSWLRQLQ$6&(6(, 'ULIWORDGVDQGXQEDODQFHGVQRZORDGVDUH
FRYHUHGE\$6&(6(,ORDGFRPELQDWLRQ
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3-4
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
EŽ
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EŽ
zĞƐ
EŽ
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Figure 3.3 Determination of roof live load reduction in accordance with ASCE/SEI 4.8.
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3-5
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.3 ACI and ASCE/SEI Strength Design Load Combinations
ACI Equation Number
ASCE/SEI 7 Load Combination
D
1
E
F
3
G
H
6
I
J
$FFRUGLQJWR$&,
Load Combination
must include the applicable effects from the following:
• &RQFHQWUDWHGOLYHORDGV $6&(6(,7DEOH
• 9HKLFXODUORDGV $6&(6(,
• &UDQHORDGV $6&(6(,
• /RDGVRQKDQGUDLOVJXDUGUDLOVDQGYHKLFXODUEDUULHUV\VWHPV $6&(6(,
• ,PSDFWHIIHFWV $6&(6(,
• Vibration effects
is provided at service-level loads,
Where wind load,
must be used in place of
G DQG I DQG
Seismic load effect,
must be used in place of
in ACI Equations
LQ$&,(TXDWLRQ F >$&,@
LVGH¿QHGLQ$6&(6(,DVIROORZV
• )RUXVHLQ$&,(TXDWLRQ H RUORDGFRPELQDWLRQLQ$6&(6(,
(3.1)
• )RUXVHLQ$&,(TXDWLRQ J RUORDGFRPELQDWLRQLQ$6&(6(,
(3.2)
where
LVSHUPLWWHGWREHWDNHQDVIRUEXLOGLQJV
assigned to SDC B or C)
6HLVPLFORDGFRPELQDWLRQVEDVHGRQ6'&XVLQJWKHSURYLVLRQVRI$6&(6(,DQGDUHJLYHQLQ7DEOH
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3-6
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.4 Seismic Load Combinations
SDC
ACI Equation
Number
ASCE/SEI 7 Load Combination
H
6
J
H
6
J
H
6
J
B
C
D, E, and F
Seismic load effect,
Load Combination
LVGH¿QHGLQ$6&(6(,DVIROORZV
• )RUXVHLQ$&,(TXDWLRQ H RUORDGFRPELQDWLRQLQ$6&(6(,
(3.3)
• )RUXVHLQ$&,(TXDWLRQ J RUORDGFRPELQDWLRQLQ$6&(6(,
(3.4)
where
Seismic load combinations based on SDC where seismic load effects including overstrength are required are given
LQ7DEOH
Table 3.5 Seismic Load Combinations Where Seismic Load Effects Including Overstrength are Required
SDC
B
C, D, E, and F
ACI Equation
Number
ASCE/SEI 7 Load Combination
H
6
J
H
6
J
Load Combination
The following load effects must be considered, where applicable, in combination with other load effects:
• Cumulative service effects of temperature, creep, shrinkage, and differential settlement,
ZLWK$&,
• (IIHFWVRIVHUYLFHODWHUDOORDGVGXHWRÀXLGV LQDFFRUGDQFHZLWK$&,
LQDFFRUGDQFHZLWK$&,
• Effects of service load due to lateral earth pressure,
• (IIHFWVGXHWRÀRRGORDGVLQDFFRUGDQFHZLWK$&,
• (IIHFWVGXHWRDWPRVSKHULFLFHORDGVLQDFFRUGDQFHZLWK$&,
/RDGFRPELQDWLRQVZKHUHÀRRGORDGV
in accordance
and atmospheric ice loads must be considered are given in Table 3.6.
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3-7
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.6 Strength Design Load Combination for Flood Loads and Atmospheric Ice Loads
Load Effect
Flood load,
ASCE/SEI load combination
V Zones or
Coastal A Zones
Noncoastal A
Zones
*
Load Combination
Atmospheric ice load
* Definitions of V Zones and Coastal A Zones are given in ASCE/SEI 5.2.
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WR]HUR,WLVSRVVLEOHWKDWWKHPRVWFULWLFDOORDGHIIHFWVRQDPHPEHURFFXUZKHQRQHRUPRUHYDULDEOHORDGVDUHQRW
present.
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3.6 Determination of Wind Forces
7KHÀRZFKDUWLQ)LJXUHFDQEHXVHGWRGHWHUPLQHWKHWRWDOGHVLJQZLQGSUHVVXUHVDWDKHLJKW above ground
OHYHORQWKH0:)56RIDQHQFORVHGRUSDUWLDOO\HQFORVHGULJLGRUÀH[LEOHUHLQIRUFHGFRQFUHWHEXLOGLQJZLWKRXWD
SDUDSHWDQGZLWKDQHVVHQWLDOO\ÀDWURRIXVLQJ3DUWLQ&KDSWHURI$6&(6(,7KHVHFWLRQ¿JXUHWDEOHDQG
HTXDWLRQQXPEHUVLQ)LJXUHDUHIURP$6&(6(,+RUL]RQWDOLQWHUQDOSUHVVXUHVFDQFHORXWZKHQGHWHUPLQLQJ
total wind pressures on the MWFRS of a building.
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level is determined by multiplying the total wind pressure at that level (windward plus leeward pressure) by the
tributary area, which is equal to the tributary story height times the width of the windward face of the building (see
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3.7 Determination of Seismic Forces
3.7.1 Seismic Forces on the SFRS
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3-8
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine the Risk Category of the
building in accordance with Table 1.5-1
Determine the basic wind speed, V, for
the applicable Risk Category from
Figs. 26.5-1A through 26.5-1D or from
Figs. 26.5-2A through 26.5-2D
Determine the wind directionality
factor, Kd, from Table 26.6-1 ሺ26.6ሻ
Determine the exposure category ሺ26.7ሻ
EŽ
Are all 5 conditions of 26.8.1 met?
zĞƐ
Determine ground elevation factor, Ke,
from Table 26.9-1 ሺ26.9ሻ
Determine velocity pressure exposure
coefficients, Kz and Kh, from
Table 26.10-1 ሺ26.10.1ሻ
Determine pressure coefficients, Cp, for
the windward and leeward walls from
Fig. 27.3-1
A
Figure 3.4 Determination of total wind pressure on the MWFRS of a building
in accordance with Part 1 in Chapter 27 of ASCE/SEI 7.
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3-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
Determine whether the building is rigid
or flexible in accordance with 26.11.2
EŽ
Is the building rigid?
Determine Gf from
26.11.5
zĞƐ
G ൌ 0.85
Determine G from
26.11.4
Figure 3.4 (cont.) Determination of total wind pressure on the MWFRS of a building
in accordance with Part 1 in Chapter 27 of ASCE/SEI 7.
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3-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ܮ‬
‫݌‬௛ ݄
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൰ ‫ܤ‬
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Figure 3.5 Determination of the total wind force at a level of the MWFRS.
3.7.2 Seismic Forces on Diaphragms, Chords, and Collectors
Floor and roof diaphragms must be designed to resist the effects due to out-of-plane gravity loads and in-plane latHUDOIRUFHVXVLQJWKHORDGFRPELQDWLRQVLQ$&, $&, )RUEXLOGLQJVDVVLJQHGWR6'&%WKURXJK)GLDSKUDJPVDUHWREHGHVLJQHGIRUWKHODUJHURIWKHIRUFHVLQ DQG EHORZ $6&(6(, determined from the structural analysis (see Figure 3.6).
1. Design seismic force,
GHWHUPLQHGE\$6&(6(,(TXDWLRQ Diaphragm design force,
where
weight tributary to the diaphragm at level
portion of the seismic base shear,
induced at level
(see Figure 3.6)
weight tributary to level
Minimum and maximum values of
DUHGHWHUPLQHGE\$6&(6(,(TXDWLRQV DQG UHVSHFtively, which are independent of the design base shear,
Minimum
Maximum
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3-11
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
zĞƐ
EŽ
Determine the Site Class in accordance
with 11.4.3 and Chapter 20
Structure is permitted to be assigned to
SDC A and must only comply with 11.7
Determine the SDC from Figure 3.1 of
this publication
Determine the response modification
coefficient, R, from Table 12.2-1 for the
appropriate structural system based on
the SDC
Determine the importance factor, Ie,
from Table 1.5-2 based on the Risk
Category
A
Figure 3.6 Determination of seismic forces on a SFRS in accordance
with the ELF Procedure in ASCE/SEI 12.8.
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3-12
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
Determine approximate fundamental
period of the building, Ta, in accordance
12.8.2.1 and set T ൌ Ta
Determine fundamental period of the
building, T, in accordance 12.8.2
Determine TL from Figs. 22-14 through
22-17
zĞƐ
EŽ
B
EŽ
EŽ
zĞƐ
zĞƐ
EŽ
zĞƐ
C
* Value of CS is permitted to be calculated using a value of 1.0 for SDS,
but not less than 0.7SDS, provided all criteria of 12.8.1.3 are met
Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance
with the ELF Procedure in ASCE/SEI 12.8.
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3-13
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
B
EŽ
EŽ
zĞƐ
zĞƐ
EŽ
C
Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance
with the ELF Procedure in ASCE/SEI 12.8.
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3-14
zĞƐ
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
C
Determine the effective seismic weight,
W, in accordance with 12.7.2
Determine base shear, V, by Eq. ሺ12.8-1ሻ:
V ൌ CsW
EŽ
EŽ
zĞƐ
zĞƐ
Exponent related to structure
period k ൌ 2 ሺ12.8.3ሻ
Exponent related to structure
period k ൌ 2 ሺ12.8.3ሻ
Exponent related to structure
period k ൌ 1 ሺ12.8.3ሻ
Exponent related to structure
period k ൌ 0.75 ൅ 0.5T
ሺ12.8.3ሻ
Figure 3.6 (cont.) Determination of seismic forces on a SFRS in accordance
with the ELF Procedure in ASCE/SEI 12.8.
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3-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In addition to
GLDSKUDJPVPXVWEHGHVLJQHGIRUDSSOLFDEOHWUDQVIHUIRUFHV VHH$6&(6(, Collectors must be provided where required to transmit forces between diaphragms and the vertical elements of the
/)56 $&, )RUEXLOGLQJVDVVLJQHGWR6'&&WKURXJK)FROOHFWRUVDQGWKHLUFRQQHFWLRQVWRWKHYHUWLFDO
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1. )RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKVHLVPLF
IRUFHVGHWHUPLQHGE\WKH(/)3URFHGXUHRI$6&(6(,RUWKHPRGDOUHVSRQVHVSHFWUXPDQDO\VLVSURFHGXUHRI$6&(6(,
where
,Q$&,(TXDWLRQV H DQG J RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH
is determined using
)RUFHVFDOFXODWHGXVLQJWKHVHLVPLFORDGHIIHFWVLQFOXGLQJRYHUVWUHQJWKRI$6&(6(,ZLWKVHLVPLF
IRUFHVGHWHUPLQHGE\$6&(6(,(TXDWLRQ IRUGLDSKUDJPV
where
,Q$&,(TXDWLRQV H DQG J RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH
is determined using
3. )RUFHVFDOFXODWHGXVLQJWKHORDGFRPELQDWLRQVRI$6&(6(,ZLWKVHLVPLFIRUFHVGHWHUPLQHGE\$6&(
6(,(TXDWLRQ ZKLFKLVWKHORZHUOLPLWGLDSKUDJPIRUFH
where
is
,Q$&,(TXDWLRQV H DQG J RU$6&(6(,ORDGFRPELQDWLRQVDQGXVH
determined using
Diaphragms and collectors in buildings assigned to SDC D, E, and F are designed in accordance with ACI Chapter
$&, 3.8 Examples
3.8.1 Example 3.1 – Determination of Wind Forces: Building #1
Determine the wind forces on the MWFRS in the north-south and east-west directions (see Figure 1.1). Assume the
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Step 1 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 2 – Determine the basic wind speed
ASCE/SEI Figure 26.5-1B
For the given location of this Risk Category II building,
Step 3 – Determine the wind directionality factor
ZKLFKFDQEHREWDLQHGIURP5HIRU ASCE/SEI Table 26.6-1
For the MWFRS of a building,
Step 4 – Determine the exposure category
ASCE/SEI 26.7
The exposure category is given as C in the design data.
Step 5 – Determine the topographic factor
ASCE/SEI 26.8
The building is not located on a hill or escarpment, so
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3-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 6 – Determine the ground elevation factor
The ground elevation factor,
ASCE/SEI Table 26.9-1
LVSHUPLWWHGWREHLQDOOFDVHV VHH1RWHLQ$6&(6(,7DEOH Step 7 – Determine the velocity pressure exposure coefficients
ASCE/SEI Table 26.10-1
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LQ$6&(6(,7DEOH
For Exposure C,
Values of
and
and
IURP$6&(6(,7DEOH
DUHJLYHQLQ7DEOH
Table 3.7 Velocity Pressure Exposure Coefficients, Kz, Building #1
Height Above Ground Level, z (ft)
Kz
Step 8 – Determine the velocity pressures
Values of
ASCE/SEI Eq. (26.10-1)
are given in Table 3.8.
Table 3.8 Velocity Pressures, qz, Building #1
Height Above Ground Level, z (ft)
Kz
qz (lb/ft2)
Step 9 – Determine the external pressure coefficients
ASCE/SEI Figure 27.3-1
For wind in the north-south direction:
• Windward wall:
• Leeward wall:
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3-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For wind in the east-west direction:
• Windward wall:
• Leeward wall:
Step 10 – Determine whether the building is rigid or not
ASCE/SEI 26.11.2
&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(,
1. Building height
Building height
(building length in each direction is constant
over the height, which means
is equal to the building length parallel to the wind direction)
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Therefore, the building is rigid in both directions.
Step 11 – Determine the gust-effect factor
For rigid buildings,
ASCE/SEI 26.11.1
LVSHUPLWWHGWREHWDNHQDV
Step 12 – Determine the design wind pressures
ASCE/SEI Eq. (27.3-1)
For wind in the north-south direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
For wind in the east-west direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
Total design wind pressures in the north-south and east-west directions are given in Table 3.9.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.9 Total Design Wind Pressures, p (lb/ft2), Building #1
Height Above Ground Level, (ft)
North-South
East-West
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LVHTXDOWROEIW for walls.
Step 13 – Determine the total design wind forces
Total design wind forces are obtained at each level by multiplying the total design wind pressures in Table 3.9 by
WKHWULEXWDU\DUHDIRUHDFKOHYHO VHH7DEOH Table 3.10 Total Design Wind Forces, Building 1
North-South
East-West
Height Above
Ground Level,
z (ft)
Tributary Area (sq ft)
Wind Force (kips)
Tributary Area (sq ft)
Wind Force (kips)
1,933.8
Ȉ
3.8.2 Example 3.2 – Determination of Wind Forces: Building #2
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Step 1 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For an essential facility, the Risk Category is IV.
Step 2 – Determine the basic wind speed
ASCE/SEI Figure 26.5-1D
For the given location of this Risk Category IV building,
Step 3 – Determine the wind directionality factor
ZKLFKFDQEHREWDLQHGIURP5HIRU ASCE/SEI Table 26.6-1
For the MWFRS of a building,
Step 4 – Determine the exposure category
ASCE/SEI 26.7
The exposure category is given as B in the design data.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 5 – Determine the topographic factor
ASCE/SEI 26.8
The building is not located on a hill or escarpment, so
ASCE/SEI Table 26.9-1
Step 6 – Determine the ground elevation factor
The ground elevation factor,
LVSHUPLWWHGWREHLQDOOFDVHV VHH1RWHLQ$6&(6(,7DEOH Step 7 – Determine the velocity pressure exposure coefficients
ASCE/SEI Table 26.10-1
7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH
LQ$6&(6(,7DEOH
For Exposure B,
Values of
and
and
IURP$6&(6(,7DEOH
are given in Table 3.11.
Table 3.11 Velocity Pressure Exposure Coefficients, Kz, Building #2
Height Above Ground Level, z (ft)
Kz
Step 8 – Determine the velocity pressures
Values of
ASCE/SEI Eq. (26.10-1)
DUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.12 Velocity Pressures, qz, Building #2
Height Above Ground Level, z (ft)
Kz
qz (lb/ft2)
36.6
31.1
Step 9 – Determine the external pressure coefficients
ASCE/SEI Figure 27.3-1
For wind in the north-south direction:
Windward wall:
Leeward wall:
For wind in the east-west direction:
Windward wall:
Leeward wall:
Step 10 – Determine whether the building is rigid or not
ASCE/SEI 26.11.2
&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(,
1. Building height
Building height
over the height, which means
(building length in each direction is constant
is equal to the building length parallel to the wind direction)
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)RUDFRQFUHWHEXLOGLQJZLWKD0:)56RWKHUWKDQPRPHQWUHVLVWLQJIUDPHV>$6&(6(,(T @
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shear wall buildings.
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3-21
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUFRPSDULVRQSXUSRVHVWKHIXQGDPHQWDOIUHTXHQFLHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHHTXDOWR
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is used in subsenorth-south direction and rigid in the east-west direction. The approximate natural frequency,
quent calculations.
Step 11 – Determine the gust-effect factor
)RUÀH[LEOHEXLOGLQJV
Calculations for
ASCE/SEI 26.11.5
LVGHWHUPLQHGE\$6&(6(,(T in both the north-south and east-west directions are given in Table 3.13.
Table 3.13 Determination of Gust-Effect Factor, Gf , Building #2
Item
ASCE/SEI Reference
(T 7DEOH
7DEOH
7DEOH
7DEOH
(T (T (T Damping ratio
for concrete buildings
&
7DEOH
7DEOH
(T (T (T (table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.13 Determination of Gust-Effect Factor, Gf , Building #2 (cont.)
Item
ASCE/SEI Reference
(T D
(T D
(T D
(T (T Step 12 – Determine the design wind pressures
ASCE/SEI Eq. (27.3-1)
For wind in the north-south direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
For wind in the east-west direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
7RWDOGHVLJQZLQGSUHVVXUHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.14 Total Design Wind Pressures, p (lb/ft2), Building #2
Height Above Ground Level, z (ft)
North-South
East-West
7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK
LVHTXDOWROEIW for walls.
Step 13 – Determine the total design wind forces
7RWDOGHVLJQZLQGIRUFHVDUHREWDLQHGDWHDFKOHYHOE\PXOWLSO\LQJWKHWRWDOGHVLJQZLQGSUHVVXUHVLQ7DEOHE\
WKHWULEXWDU\DUHDIRUHDFKOHYHO VHH7DEOH Table 3.15 Total Design Wind Forces, Building 2
North-South
East-West
Height Above
Ground Level,
z (ft)
Tributary Area (sq ft)
Wind Force (kips)
Tributary Area (sq ft)
Wind Force (kips)
91.1
88.3
86.3
88.9
Ȉ
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3-24
963.8
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3.8.3 Example 3.3 – Determination of Wind Forces: Building #3
Determine the wind forces on the MWFRS in the north-south and east-west directions (see Figure 1.3). Assume the
RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Residential Group R occupancy, the Risk Category is II.
Step 2 – Determine the basic wind speed
ASCE/SEI Figure 26.5-1B
For the given location of this Risk Category II building,
ZKLFKFDQEHREWDLQHGIURP5HIRU Step 3 – Determine the wind directionality factor
ASCE/SEI Table 26.6-1
For the MWFRS of a building,
Step 4 – Determine the exposure category
ASCE/SEI 26.7
The exposure category is given as B in the design data.
Step 5 – Determine the topographic factor
ASCE/SEI 26.8
The building is not located on a hill or escarpment, so
Step 6 – Determine the ground elevation factor
The ground elevation factor,
ASCE/SEI Table 26.9-1
LVSHUPLWWHGWREHLQDOOFDVHV VHH1RWHLQ$6&(6(,7DEOH Step 7 – Determine the velocity pressure exposure coefficients
ASCE/SEI Table 26.10-1
7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH
LQ$6&(6(,7DEOH
For Exposure B,
Values of
and
and
IURP$6&(6(,7DEOH
are given in Table 3.16.
Table 3.16 Velocity Pressure Exposure Coefficients, Kz, Building #3
Height Above Ground Level, z (ft)
Kz
1.11
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.16 Velocity Pressure Exposure Coefficients, Kz, Building #3 (cont.)
Height Above Ground Level, z (ft)
Kz
Step 8 – Determine the velocity pressures
Values of
ASCE/SEI Eq. (26.10-1)
DUHJLYHQLQ7DEOH
Table 3.17 Velocity Pressures, qz, Building #3
Height Above Ground Level, z (ft)
Kz
qz (lb/ft2)
1.11
19.9
11.9
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 9 – Determine the external pressure coefficients
ASCE/SEI Figure 27.3-1
For wind in the north-south direction:
Windward wall:
Leeward wall:
For wind in the east-west direction:
Windward wall:
Leeward wall:
Step 10 – Determine whether the building is rigid or not
ASCE/SEI 26.11.2
&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(,
1. Building height
Building height
(building length in each direction is constant
over the height, which means
is equal to the building length parallel to the wind direction)
7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\EHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\
)RUDFRQFUHWHEXLOGLQJZLWKD0:)56RWKHUWKDQPRPHQWUHVLVWLQJIUDPHV>$6&(6(,(T @
7KHUHIRUHWKHEXLOGLQJLVÀH[LEOHLQERWKGLUHFWLRQV
,WFDQEHGHWHUPLQHGWKHEXLOGLQJLVDOVRÀH[LEOHXVLQJ$6&(6(,(T ZKLFKLVDSSOLFDEOHIRUFRQFUHWH
shear wall buildings.
)RUFRPSDULVRQSXUSRVHVWKHIXQGDPHQWDOIUHTXHQFLHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHHTXDOWR
+]DQG+]UHVSHFWLYHO\IURPDG\QDPLFDQDO\VLVRIWKHVWUXFWXUHZKLFKDOVRVLJQL¿HVWKHEXLOGLQJLVÀH[LEOH
is used in subsequent calculations.
in both directions. The approximate natural frequency,
Step 11 – Determine the gust-effect factor
)RUÀH[LEOHEXLOGLQJV
Calculations for
ASCE/SEI 26.11.5
LVGHWHUPLQHGE\$6&(6(,(T in both the north-south and east-west directions are given in Table 3.18.
Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3
Item
ASCE/SEI Reference
(T 7DEOH
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3 (cont.)
Item
ASCE/SEI Reference
7DEOH
7DEOH
7DEOH
(T (T (T Damping ratio
for concrete buildings
&
7DEOH
7DEOH
(T (T (T (T D
(T D
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.18 Determination of Gust-Effect Factor, Gf , Building #3 (cont.)
Item
ASCE/SEI Reference
(T D
(T (T Step 12 – Determine the design wind pressures, pz
ASCE/SEI Eq. (27.3-1)
For wind in the north-south direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
For wind in the east-west direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
Total design wind pressures in the north-south and east-west directions are given in Table 3.19.
Table 3.19 Total Design Wind Pressures, p (lb/ft2), Building #3
Height Above Ground Level, z (ft)
North-South
East-West
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.19 Total Design Wind Pressures, p (lb/ft2), Building #3 (cont.)
Height Above Ground Level, z (ft)
North-South
East-West
19.6
7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK
LVHTXDOWROEIW for walls.
Step 13 – Determine the total design wind forces
Total design wind forces are obtained at each level by multiplying the total design wind pressures in Table 3.19 by
WKHWULEXWDU\DUHDIRUHDFKOHYHO VHH7DEOH Table 3.20 Total Design Wind Forces, Building #3
North-South
East-West
Height Above
Ground Level,
z (ft)
Tributary Area (sq ft)
Wind Force (kips)
Tributary Area (sq ft)
Wind Force (kips)
13.9
18.8
Ȉ
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3.8.4 Example 3.4 – Determination of Wind Forces: Building #4
'HWHUPLQHWKHZLQGIRUFHVRQWKH0:)56LQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQV VHH)LJXUH $VVXPHWKH
RYHUDOOEXLOGLQJGLPHQVLRQVDUHIWLQWKHQRUWKVRXWKGLUHFWLRQDQGIWLQWKHHDVWZHVWGLUHFWLRQ
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 2 – Determine the basic wind speed
ASCE/SEI Figure 26.5-1B
For the given location of this Risk Category II building,
ZKLFKFDQEHREWDLQHGIURP5HIRU Step 3 – Determine the wind directionality factor
ASCE/SEI Table 26.6-1
For the MWFRS of a building,
Step 4 – Determine the exposure category
ASCE/SEI 26.7
The exposure category is given as B in the design data.
Step 5 – Determine the topographic factor
ASCE/SEI 26.8
The building is not located on a hill or escarpment, so
Step 6 – Determine the ground elevation factor
The ground elevation factor,
ASCE/SEI Table 26.9-1
LVSHUPLWWHGWREHLQDOOFDVHV VHH1RWHLQ$6&(6(,7DEOH Step 7 – Determine the velocity pressure exposure coefficients
ASCE/SEI Table 26.10-1
7KHYHORFLW\SUHVVXUHH[SRVXUHFRHI¿FLHQWVDUHGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQVZKLFKDUHJLYHQLQ1RWH
LQ$6&(6(,7DEOH
For Exposure B,
Values of
and
and
IURP$6&(6(,7DEOH
DUHJLYHQLQ7DEOH
Table 3.21 Velocity Pressure Exposure Coefficients, Kz, Building #4
Height A Height Above Ground Level, z (ft)und
Kz
1.39
1.38
1.36
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.21 Velocity Pressure Exposure Coefficients, Kz, Building #4 (cont.)
Height A Height Above Ground Level, z (ft)und
Kz
1.33
1.18
1.16
Step 8 – Determine the velocity pressures
Values of
ASCE/SEI Eq. (26.10-1)
DUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.22 Velocity Pressures, qz, Building #4
Height Above Ground Level, z (ft)
Kz
qz (lb/ft2)
1.39
1.38
1.36
1.33
1.18
1.16
19.9
18.3
16.3
Step 9 – Determine the external pressure coefficients
ASCE/SEI Figure 27.3-1
For wind in the north-south direction:
Windward wall:
Leeward wall:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For wind in the east-west direction:
Windward wall:
Leeward wall:
Step 10 – Determine whether the building is rigid or not
ASCE/SEI 26.11.2
&KHFNLIWKHQDWXUDOIUHTXHQF\FDQEHGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$6&(6(,
Building height
7KHUHIRUHWKHSURYLVLRQVRI$6&(6(,PD\QRWEHXVHGWRGHWHUPLQHWKHDSSUR[LPDWHQDWXUDOIUHTXHQF\
)URPDG\QDPLFDQDO\VLVRIWKHVWUXFWXUHWKHEXLOGLQJLVIRXQGWREHÀH[LEOHLQERWKGLUHFWLRQV
in the east-west direction).
north-south direction and
Step 11 – Determine the gust-effect factor
)RUÀH[LEOHEXLOGLQJV
Calculations for
in the
ASCE/SEI 26.11.5
LVGHWHUPLQHGE\$6&(6(,(T LQERWKWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH
Table 3.23 Determination of Gust-Effect Factor, Gf , Building #4
Item
ASCE/SEI Reference
26.11.5
Eq. (26.11-11)
Table 26.11-1
26.11.4
Table 26.11-1
Table 26.11-1
Table 26.11-1
Eq. (26.11-7)
Eq. (26.11-9)
Eq. (26.11-8)
Damping ratio
C26.11
for concrete buildings
Table 26.11-1
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.23 Determination of Gust-Effect Factor, Gf , Building #4 (cont.)
Item
ASCE/SEI Reference
Table 26.11-1
Eq. (26.11-16)
Eq. (26.11-14)
Eq. (26.11-13)
26.11.5
Eq. (26.11-15a)
26.11.5
Eq. (26.11-15a)
26.11.5
Eq. (26.11-15a)
Eq. (26.11-12)
Eq. (26.11-10)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 12 – Determine the design wind pressures, pz
ASCE/SEI Eq. (27.3-1)
For wind in the north-south direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
For wind in the east-west direction:
• Windward wall:
• Leeward wall:
• Total pressure on MWFRS
7RWDOGHVLJQZLQGSUHVVXUHVLQWKHQRUWKVRXWKDQGHDVWZHVWGLUHFWLRQVDUHJLYHQLQ7DEOH
Table 3.24 Total Design Wind Pressures, p (lb/ft2), Building #4
Height Above Ground Level, z (ft)
North-South
East-West
330.0
33.6
33.2
319.0
33.4
33.0
308.0
33.2
32.8
297.0
33.0
32.5
286.0
32.7
32.3
275.0
32.5
32.1
264.0
32.3
31.9
253.0
32.0
31.6
242.0
31.8
31.4
231.0
31.5
31.1
220.0
31.2
30.9
209.0
30.9
30.7
198.0
30.6
30.4
187.0
30.3
30.1
176.0
30.0
29.8
165.0
29.7
29.5
154.0
29.4
29.2
143.0
29.0
28.8
132.0
28.6
28.5
121.0
28.2
28.1
110.0
27.7
27.6
99.0
27.3
27.2
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.24 Total Design Wind Pressures, p (lb/ft2), Building #4 (cont.)
Height Above Ground Level, z (ft)
North-South
East-West
88.0
26.7
26.7
77.0
26.2
26.2
66.0
25.5
25.6
55.0
24.9
25.0
44.0
24.0
24.2
33.0
23.1
23.4
22.0
21.9
22.2
11.0
20.8
21.2
7KHWRWDOGHVLJQZLQGSUHVVXUHVDUHJUHDWHUWKDQWKHPLQLPXPZLQGSUHVVXUHSUHVFULEHGLQ$6&(6(,ZKLFK
LVHTXDOWROEIW for walls.
Step 13 – Determine the total design wind forces
7RWDOGHVLJQZLQGIRUFHVDUHREWDLQHGDWHDFKOHYHOE\PXOWLSO\LQJWKHWRWDOGHVLJQZLQGSUHVVXUHVLQ7DEOHE\
WKHWULEXWDU\DUHDIRUHDFKOHYHO VHH7DEOH Table 3.25 Total Design Wind Forces, Building #4
North-South
East-West
Height Above
Ground Level,
z (ft)
Tributary Area (sq ft)
Wind Force (kips)
Tributary Area (sq ft)
Wind Force (kips)
39.8
38.8
36.9
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.25 Total Design Wind Forces, Building #4 (cont.)
North-South
East-West
Height Above
Ground Level,
z (ft)
Tributary Area (sq ft)
Wind Force (kips)
Tributary Area (sq ft)
Wind Force (kips)
36.1
33.9
Ȉ
3.8.5 Example 3.5 – Determination of the Seismic Design Category: Building #1
Determine the SDC for (a) Site Class C, (b) Site Class D (default), and (c) Site Class E (see Figure 1.1).
'HVLJQGDWDDUHJLYHQLQ6HFW
Part (a): Site Class C
Step 1 – Determine the mapped acceleration parameters
ASCE/SEI 11.4.2
In lieu of determining the mapped acceleration parameters
and IURP$6&(6(,)LJXUHVDQGUHand
for the latitude and longitude of the site.
VSHFWLYHO\XVH5HIRU
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
SDC A.
and
ASCE/SEI 11.4.2
, the building is not permitted to be automatically assigned to
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 4 – Determine if the SDC is E or F
Because the Risk Category is II and
ASCE/SEI 11.6
the SDC is not E or F.
Step 5 – Determine the design acceleration parameters
For Site Class C and
IURP$6&(6(,7DEOH
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ASCE/SEI 11.4.4 and 11.4.5
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
IURP$6&(6(,7DEOH
For Site Class C and
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
Determine the approximate fundamental period,
$6&(6(, For concrete moment-resisting frames in both directions:
ASCE/SEI Eq. (12.8-7)
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is A.
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is A.
Therefore, the SDC is A for this building.
Part (b): Site Class D (default)
Step 1 – Determine the mapped acceleration parameters
From Step 1 in Part (a):
ASCE/SEI 11.4.2
and
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
SDC A.
and
ASCE/SEI 11.4.2
the building is not permitted to be automatically assigned to
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 4 – Determine if the SDC is E or F
Because the Risk Category is II and
ASCE/SEI 11.6
the SDC is not E or F.
Step 5 – Determine the design acceleration parameters
For Site Class D (default) and
ASCE/SEI 11.4.4 and 11.4.5
IURP$6&(6(,7DEOH
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3-39
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
IURP$6&(6(,7DEOH
For Site Class D (default) and
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
Determine the approximate fundamental period,
$6&(6(, From Step 6 in Part (a),
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is A.
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is B.
Therefore, the SDC is B for this building.
Part (c): Site Class E
Step 1 – Determine the mapped acceleration parameters
From Step 1 in Part (a):
ASCE/SEI 11.4.2
and
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
SDC A.
and
ASCE/SEI 11.4.2
the building is not permitted to be automatically assigned to
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 4 – Determine if the SDC is E or F
Because the Risk Category is II and
ASCE/SEI 11.6
the SDC is not E or F.
Step 5 – Determine the design acceleration parameters
For Site Class E and
IURP$6&(6(,7DEOH
For Site Class E and
IURP$6&(6(,7DEOH
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3-40
ASCE/SEI 11.4.4 and 11.4.5
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
$6&(6(, Determine the approximate fundamental period,
From Step 6 in Part (a),
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is B.
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is C.
Therefore, the SDC is C for this building.
3.8.6 Example 3.6 – Determination of the Seismic Design Category: Building #2
'HWHUPLQHWKH6'&IRU6LWH&ODVV' VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
ASCE/SEI 11.4.2
In lieu of determining the mapped acceleration parameters
and IURP$6&(6(,)LJXUHVDQGUHand
for the latitude and longitude of the site.
VSHFWLYHO\XVH5HIRU
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
SDC A.
and
ASCE/SEI 11.4.2
the building is not permitted to be automatically assigned to
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For this essential facility, the Risk Category is IV.
Step 4 – Determine if the SDC is E or F
Because the Risk Category is IV and
ASCE/SEI 11.6
the SDC is not E or F.
Step 5 – Determine the design acceleration parameters
ASCE/SEI 11.4.4 and 11.4.5
IURPOLQHDULQWHUSRODWLRQLQ$6&(6(,7DEOH
For Site Class D (stiff soil) and
For Site Class D (stiff soil) and
IURP$6&(6(,7DEOH
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3-41
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
Determine the approximate fundamental period,
$6&(6(, For all other structural systems in both directions:
ASCE/SEI Eq. (12.8-7)
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category IV, the SDC is C.
)URP$6&(6(,7DEOHZLWK
and Risk Category IV, the SDC is C.
Therefore, the SDC is C for this building.
3.8.7 Example 3.7 – Determination of the Seismic Design Category: Building #3
'HWHUPLQHWKH6'&IRU6LWH&ODVV' VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
ASCE/SEI 11.4.2
In lieu of determining the mapped acceleration parameters
and IURP$6&(6(,)LJXUHVDQGUHand
for the latitude and longitude of the site.
VSHFWLYHO\XVH5HIRU
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
to SDC A.
and
ASCE/SEI 11.4.2
the building is not permitted to be automatically be assigned
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Residential Group R occupancy, the Risk Category is II.
Step 4 – Determine if the SDC is E or F
Because the Risk Category is II and
ASCE/SEI 11.6
the SDC is not E or F.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 5 – Determine the design acceleration parameters
ASCE/SEI 11.4.4, 11.4.5, and 11.4.8
Because the Site Class is D and
DJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(,
PD\EHUHTXLUHG$VVXPHWKHVHFRQGH[FHSWLRQLQ$6&(6(,LVDSSOLFDEOHWKXVVXFKDQDQDO\VLVQHHG
not be performed.
For Site Class D (stiff soil) and
IURP$6&(6(,7DEOH
For Site Class D (stiff soil) and
IURPOLQHDULQWHUSRODWLRQLQ$6&(6(,7DEOH
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
$6&(6(, Determine the approximate fundamental period,
For all other structural systems in both directions:
ASCE/SEI Eq. (12.8-7)
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is D.
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is D.
Therefore, the SDC is D for this building.
3.8.8 Example 3.8 – Determination of the Seismic Design Category: Building #4
'HWHUPLQHWKH6'&IRU6LWH&ODVV' GHIDXOW >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
ASCE/SEI 11.4.2
In lieu of determining the mapped acceleration parameters
and IURP$6&(6(,)LJXUHVDQGUHand
for the latitude and longitude of the site.
VSHFWLYHO\XVH5HIRU
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
Because
to SDC A.
and
ASCE/SEI 11.4.2
the building is not permitted to be automatically be assigned
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Determine the Risk Category
ASCE/SEI Table 1.5-1
For the Business Group B occupancy, the Risk Category is II.
Step 4 – Determine if the SDC is E or F
ASCE/SEI 11.6
Because the Risk Category is II and
the SDC is not E or F.
Step 5 – Determine the design acceleration parameters
ASCE/SEI 11.4.4, 11.4.5, and 11.4.8
Because the Site Class is D and
QHHGQRWEHSHUIRUPHG
DJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(,
For Site Class D (default) and
IURP7DEOH$6&(6(,E\OLQHDULQWHUSRODWLRQ
For Site Class D (default) and
IURP$6&(6(,7DEOHE\OLQHDULQWHUSRODWLRQ
Step 6 – Check if the SDC can be determined by ASCE/SEI Table 11.6-1 alone
ASCE/SEI 11.6
&KHFNLIDOOIRXUFRQGLWLRQVLQ$6&(6(,DUHVDWLV¿HG
Determine the approximate fundamental period,
$6&(6(, For all other structural systems in both directions:
ASCE/SEI Eq. (12.8-7)
Because
VDWLV¿HG WKH6'&FDQQRWEHGHWHUPLQHGE\$6&(6(,7DEOHDORQH WKDWLVFRQGLWLRQLVQRW
Step 7 – Determine the SDC
ASCE/SEI Tables 11.6-1 and 11.6-2
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is D.
)URP$6&(6(,7DEOHZLWK
and Risk Category II, the SDC is D.
Therefore, the SDC is D for this building.
3.8.9 Example 3.9 – Determination of Seismic Forces: SFRS of Building #1 (Framing Option A)
'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV' GHIDXOW >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
)URP6WHSLQ3DUW E RI([DPSOH
and
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3-44
ASCE/SEI 11.4.2
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
ASCE/SEI 11.4.2
)URP6WHSLQ3DUW E RI([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$
Step 3 – Determine the Site Class
ASCE/SEI 11.4.3
From the design data, the Site Class is given as D (default).
Step 4 – Determine the acceleration parameters adjusted for site class effects
)URP6WHSRI([DPSOH
and
Step 5 – Determine the design acceleration parameters
)URP6WHSRI([DPSOH
ASCE/SEI 11.4.4
ASCE/SEI 11.4.5
and
Step 6 – Determine the SDC
ASCE/SEI 11.6
)URP6WHSRI([DPSOHWKH6'&LV%
Step 7 – Determine the response modification coefficient
ASCE/SEI Table 12.2-1
Because the building is assigned to SDC B, ordinary reinforced concrete moment frames may be used in both direcWLRQVZLWKQROLPLWDWLRQV 6)56& For this SFRS,
Step 8 – Determine the importance factor
ASCE/SEI Table 1.5-2
For Risk Category II buildings,
Step 9 – Determine the approximate period
ASCE/SEI 12.8.2.1
)URP6WHSRI([DPSOH
Step 10 – Determine the long-period transition period
)URP$6&(6(,)LJXUH
WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU Step 11 – Determine the seismic response coefficient
Because
ASCE/SEI Figures 22-14 through 22-17
ASCE/SEI 12.8.1.1
is determined as follows:
ASCE/SEI Eq. (12.8-3)
need not exceed the following:
ASCE/SEI Eq. (12.8-2)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Minimum
ASCE/SEI Eq. (12.8-5)
Therefore,
Step 12 – Determine the effective seismic weight
ASCE/SEI 12.7.2
7KHHIIHFWLYHVHLVPLFZHLJKWIRUWKLVEXLOGLQJLQFOXGHVWKHGHDGORDGRIWKHVWUXFWXUH DVVXPLQJDLQWKLFNVODE
DQGLQE\LQFROXPQV WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHOEIW partition weight
RQWKHÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ
7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKW
are given in Table 3.
Table 3.26 Seismic Forces and Story Shears, Building #1
Level
Story Weight,
wx (kips)
Height,
hx (ft)
wx hxk
Lateral Force,
Fx (kips)
Story Shear,
Vx (kips)
R
3
Ȉ
W Step 13 – Determine the seismic base shear
ASCE/SEI Eq. (12.8-1)
Step 14 – Determine the exponent related to the structure period
Step 15 – Determine the seismic forces at each level
Values of
DUHJLYHQLQ7DEOH
)RUH[DPSOHDWOHYHO
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ASCE/SEI 12.8.3
ASCE/SEI Eqs. (12.8-11) and (12.8-12)
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3.8.10 Example 3.10 – Determination of Seismic Forces: SFRS of Building #2
'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV' VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
From Step 1 in Example 3.6,
ASCE/SEI 11.4.2
and
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
ASCE/SEI 11.4.2
)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$
Step 3 – Determine the Site Class
ASCE/SEI 11.4.3
From the design data, the Site Class is given as D (stiff soil).
Step 4 – Determine the acceleration parameters adjusted for site class effects
)URP6WHSLQ([DPSOH
ASCE/SEI 11.4.4
and
Step 5 – Determine the design acceleration parameters
)URP6WHSLQ([DPSOH
ASCE/SEI 11.4.5
and
Step 6 – Determine the SDC
ASCE/SEI 11.6
)URP6WHSLQ([DPSOHWKH6'&LV&
Step 7 – Determine the response modification coefficient
ASCE/SEI Table 12.2-1
Because the building is assigned to SDC C, a building frame system with ordinary reinforced concrete shear (strucWXUDO ZDOOVPD\EHXVHGLQERWKGLUHFWLRQVZLWKQROLPLWDWLRQV 6)56% For this SFRS,
Step 8 – Determine the importance factor
ASCE/SEI Table 1.5-2
For Risk Category IV buildings,
Step 9 – Determine the approximate period
ASCE/SEI 12.8.2.1
From Step 6 in Example 3.6,
Step 10 – Determine the long-period transition period
)URP$6&(6(,)LJXUH
ASCE/SEI Figures 22-14 through 22-17
WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 11 – Determine the seismic response coefficient
Because
ASCE/SEI 12.8.1.1
is determined as follows:
ASCE/SEI Eq. (12.8-3)
need not exceed the following:
ASCE/SEI Eq. (12.8-2)
Minimum
ASCE/SEI Eq. (12.8-5)
Therefore,
Step 12 – Determine the effective seismic weight
ASCE/SEI 12.7.2
The effective seismic weight for this building includes the dead load of the structure (assuming a
ZLGHPRGXOHMRLVWV\VWHPLQE\LQLQWHULRUEHDPVLQE\LQHGJHEHDPVLQE\LQDQG
LQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQDQGLQE\LQFROXPQVLQVWRULHV
WKURXJKLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHVWKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHOEIWSDUWLWLRQZHLJKWRQWKHÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ
7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKW
are given in Table 3.
Table 3.27 Seismic Forces and Story Shears, Building #2
Level
Story Weight,
wx (kips)
Height,
hx (ft)
wx hxk
Lateral Force,
Fx (kips)
Story Shear,
Vx (kips)
R
11
9
8
6
99.8
1,369.8
3
Ȉ
W @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 13 – Determine the seismic base shear
ASCE/SEI Eq. (12.8-1)
Step 14 – Determine the exponent related to the structure period
Step 15 – Determine the seismic forces at each level
Values of
ASCE/SEI 12.8.3
ASCE/SEI Eqs. (12.8-11) and (12.8-12)
are given in Table 3.
)RUH[DPSOHDWOHYHO
3.8.11 Example 3.11 – Determination of Seismic Forces: SFRS of Building #3
'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV' VWLIIVRLOGHWHUPLQHG >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
)URP6WHSLQ([DPSOH
ASCE/SEI 11.4.2
and
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
ASCE/SEI 11.4.2
)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$
Step 3 – Determine the Site Class
ASCE/SEI 11.4.3
From the design data, the Site Class is given as D (stiff soil).
Step 4 – Determine the acceleration parameters adjusted for site class effects
)URP6WHSLQ([DPSOH
and
Step 5 – Determine the design acceleration parameters
)URP6WHSLQ([DPSOH
ASCE/SEI 11.4.4
ASCE/SEI 11.4.5
and
Step 6 – Determine the SDC
ASCE/SEI 11.6
)URP6WHSLQ([DPSOHWKH6'&LV'
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 7 – Determine the response modification coefficient
ASCE/SEI Table 12.2-1
7KHKHLJKWRIWKLVEXLOGLQJLVHTXDOWRIW%HFDXVHWKHEXLOGLQJLVDVVLJQHGWR6'&'DEXLOGLQJIUDPHV\VWHP
ZLWKVSHFLDOUHLQIRUFHGFRQFUHWHVKHDU VWUXFWXUDO ZDOOVPD\EHXVHGLQERWKGLUHFWLRQVZLWKDKHLJKWOLPLWRI
IW 6)56% For this SFRS,
Step 8 – Determine the importance factor
ASCE/SEI Table 1.5-2
For Risk Category II buildings,
Step 9 – Determine the approximate period
ASCE/SEI 12.8.2.1
)URP6WHSLQ([DPSOH
From a dynamic analysis of the structure, the periods in the north-south and east-west directions are determined to
EHVDQGVUHVSHFWLYHO\$FFRUGLQJWR$6&(6(,WKHIXQGDPHQWDOSHULRG must not exceed the
IURP$6&(6(,7DEOHDQGWKHDSSUR[LSURGXFWRIWKHFRHI¿FLHQWIRUXSSHUOLPLWRQFDOFXODWHGSHULRG
mate fundamental period,
In the north-south direction:
In the east-west direction:
where
Therefore,
IURP$6&(6(,7DEOHIRU
is permitted to be used to determine the base shear,
Step 10 – Determine the long-period transition period
)URP$6&(6(,)LJXUH
ASCE/SEI Figures 22-14 through 22-17
WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU Step 11 – Determine the seismic response coefficient
Because
in both directions.
ASCE/SEI 12.8.1.1
is determined as follows:
ASCE/SEI Eq. (12.8-3)
need not exceed the following:
ASCE/SEI Eq. (12.8-2)
Minimum
ASCE/SEI Eq. (12.8-5)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,WZDVDVVXPHGLQ6WHSRI([DPSOHWKDWDJURXQGPRWLRQKD]DUGDQDO\VLVLQDFFRUGDQFHZLWK$6&(6(,
ZDVQRWUHTXLUHGEHFDXVHWKHVHFRQGH[FHSWLRQLQ$6&(6(,LVDSSOLFDEOH7KHUHIRUHGHWHUPLQH based on
that exception:
Thus,
PXVWEHWDNHQDVWLPHVWKHYDOXHFRPSXWHGLQDFFRUGDQFHZLWK$6&(6(,(T Step 12 – Determine the effective seismic weight
ASCE/SEI 12.7.2
7KHHIIHFWLYHVHLVPLFZHLJKWIRUWKLVEXLOGLQJLQFOXGHVWKHGHDGORDGRIWKHVWUXFWXUH DVVXPLQJDQLQWKLFNVODE
LQE\LQDQGLQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQDQGLQE\
LQFROXPQVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFNZDOOVLQVWRULHV
WKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVDQGWKHZHLJKWRIWKHFODGGLQJ
7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKWDUHJLYHQLQ7DEOH
Table 3.28 Seismic Forces and Story Shears, Building #3
Level
Story Weight,
wx (kips)
Height,
hx (ft)
wx hxk
Lateral Force,
Fx (kips)
Story Shear,
Vx (kips)
R
16
398.3
168.3
13
11
118.1
9
8
6
3
Ȉ
W = 13,319
Step 13 – Determine the seismic base shear
ASCE/SEI Eq. (12.8-1)
Step 14 – Determine the exponent related to the structure period
ASCE/SEI 12.8.3
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 15 – Determine the seismic forces at each level
Values of
ASCE/SEI Eqs. (12.8-11) and (12.8-12)
DUHJLYHQLQ7DEOH
)RUH[DPSOHDWOHYHO
3.8.12 Example 3.12 – Determination of Seismic Forces: SFRS of Building #4
'HWHUPLQHWKHVHLVPLFIRUFHVRQWKH6)56IRU6LWH&ODVV' GHIDXOW >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the mapped acceleration parameters
From Step 1 in Example 3.8,
ASCE/SEI 11.4.2
and
Step 2 – Determine if the building is permitted to be automatically assigned to SDC A
ASCE/SEI 11.4.2
)URP6WHSLQ([DPSOHWKHEXLOGLQJLVQRWSHUPLWWHGWREHDXWRPDWLFDOO\DVVLJQHGWR6'&$
Step 3 – Determine the Site Class
ASCE/SEI 11.4.3
From the design data, the Site Class is given as D (default).
Step 4 – Determine the acceleration parameters adjusted for site class effects
)URP6WHSLQ([DPSOH
and
Step 5 – Determine the design acceleration parameters
)URP6WHSLQ([DPSOH
ASCE/SEI 11.4.4
ASCE/SEI 11.4.5
and
Step 6 – Determine the SDC
ASCE/SEI 11.6
)URP6WHSLQ([DPSOHWKH6'&LV'
Step 7 – Determine the response modification coefficient
ASCE/SEI Table 12.2-1
Because the building is assigned to SDC D, a dual system with special reinforced concrete shear (structural) walls
DQGVSHFLDOPRPHQWIUDPHVRIUHLQIRUFHGFRQFUHWHFDSDEOHRIUHVLVWLQJDWOHDVWSHUFHQWRIWKHSUHVFULEHGVHLVPLF
forces may be used with no limits in both directions (SFRS D3).
For this SFRS,
A building frame system with special reinforced concrete shear (structural) walls is not permitted because the buildLQJKHLJKWZKLFKLVHTXDOWRIWH[FHHGVWKHKHLJKWOLPLWRIIWIRUWKLVV\VWHPLQ6'&'
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 8 – Determine the importance factor
ASCE/SEI Table 1.5-2
For Risk Category II buildings,
Step 9 – Determine the approximate period
ASCE/SEI 12.8.2.1
From Step 6 in Example 3.8,
From a dynamic analysis of the structure, the periods in the north-south and east-west directions are determined to
EHVDQGVUHVSHFWLYHO\$FFRUGLQJWR$6&(6(,WKHIXQGDPHQWDOSHULRG must not exceed the
IURP$6&(6(,7DEOHDQGWKHDSSUR[LSURGXFWRIWKHFRHI¿FLHQWIRUXSSHUOLPLWRQFDOFXODWHGSHULRG
mate fundamental period,
In the north-south direction:
In the east-west direction:
where
Therefore,
directions.
IURP$6&(6(,7DEOHE\OLQHDULQWHUSRODWLRQIRU
is permitted to be used to determine the base shear,
in both the north-south and east-west
,QRUGHUWRXVHWKH(/)3URFHGXUHVWUXFWXUHVH[FHHGLQJIWLQKHLJKWZLWKQRVWUXFWXUDOLUUHJXODULWLHVPXVWVDWLVI\WKHIROORZLQJ $6&(6(,7DEOH Therefore, use
in both directions.
Step 10 – Determine the long-period transition period
)URP$6&(6(,)LJXUH
WKLVSHULRGFDQDOVREHGHWHUPLQHGIURP5HIRU Step 11 – Determine the seismic response coefficient
Because
ASCE/SEI Figures 22-14 through 22-17
ASCE/SEI 12.8.1.1
is determined as follows:
ASCE/SEI Eq. (12.8-3)
need not exceed the following:
ASCE/SEI Eq. (12.8-2)
Minimum
ASCE/SEI Eq. (12.8-5)
Thus,
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3-53
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 12 – Determine the effective seismic weight
ASCE/SEI 12.7.2
The effective seismic weight for this building includes the dead load of the structure (assuming wide-module roof
LQE\LQFROXPQVLQVWRULHVWKURXJKLQE\LQ
DQGÀRRUV\VWHPVZLWK
FROXPQVLQVWRULHVWKURXJKLQE\LQFROXPQVLQVWRULHVWKURXJKEHDPV LQFOXGLQJFROOHFWRU
EHDPV DUHDVZLGHDVWKHFROXPQVZLWKDGHSWKRILQFRXSOLQJEHDPVDUHZLGHDVWKHZDOOVZLWKDGHSWKRI
LQLQWKLFNZDOOVLQVWRULHVWKURXJKLQWKLFNZDOOVLQVWRULHVWKURXJKDQGLQWKLFN
ZDOOVLQVWRULHVWKURXJK WKHVXSHULPSRVHGGHDGORDGVRQWKHURRIDQGÀRRUVWKHZHLJKWRIWKHSDUWLWLRQVDQG
the weight of the cladding.
7KHHIIHFWLYHVHLVPLFZHLJKWVSHUÀRRUDQGWKHWRWDOHIIHFWLYHVHLVPLFZHLJKWDUHJLYHQLQ7DEOH
Table 3.29 Seismic Forces and Story Shears, Building #4
Level
Story Weight,
wx (kips)
Height,
hx (ft)
wx hxk
Lateral Force,
Fx (kips)
Story Shear,
Vx (kips)
R
196.1
163.1
19
98.3
18
16
13
11
9
8
16.9
6
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.29 Seismic Forces and Story Shears, Building #4 (cont.)
Level
Story Weight,
wx (kips)
Height,
hx (ft)
wx hxk
Lateral Force,
Fx (kips)
Story Shear,
Vx (kips)
3
Ȉ
W Step 13 – Determine the seismic base shear
ASCE/SEI Eq. (12.8-1)
Step 14 – Determine the exponent related to the structure period
Step 15 – Determine the seismic forces at each level
Values of
ASCE/SEI 12.8.3
ASCE/SEI Eqs. (12.8-11) and (12.8-12)
DUHJLYHQLQ7DEOH
For example, at the roof level:
3.8.13 Example 3.13 – Determination of Seismic Forces: Diaphragms of Building #1 (Framing Option A)
'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(,
IRU6LWH&ODVV' GHIDXOW >VHH)LJXUH@
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the seismic forces on the SFRS
ASCE/SEI 12.8.3
Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOH VHH7DEOH DQGDUH
JLYHQLQ7DEOH
Table 3.30 Seismic Diaphragm Design Forces, Building #1
(kips)
wpx
(kips)
ȈFi Ȉwi
Fpx
(kips)
Design
Force
(kips)**
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
R
Ȉwi
ȈFi
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 3.30 Seismic Diaphragm Design Forces, Building #1 (cont.)
(kips)
wpx
(kips)
ȈFi Ȉwi
Fpx
(kips)
Design
Force
(kips)**
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
3
Ȉwi
ȈFi
* Max.
** Max. of
Step 2 – Determine the seismic diaphragm forces
ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)
Diaphragm forces
RYHUWKHKHLJKWRIWKHEXLOGLQJDUHJLYHQLQ7DEOH9DOXHVRIWKHVWRU\ZHLJKW
and the
seismic lateral force, DUHJLYHQLQ7DEOHRI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO
is set equal to
,WLVHYLGHQWIURP7DEOHWKDWWKHPD[LPXP
the three top levels of the building. For example, at the roof level:
governs at
Also, minimum
Step 3 – Determine the seismic diaphragm design forces
Seismic diaphragm design forces are the larger of
JLYHQLQ7DEOH
and
ASCE/SEI 12.10.1.1
The design forces over the height of the building are
3.8.14 Example 3.14 – Determination of Seismic Forces: Diaphragms of Building #2
'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(,
VHH)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the seismic forces on the SFRS
ASCE/SEI 12.8.3
Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOH VHH7DEOH DQG
are given in Table 3.31.
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Table 3.31 Seismic Diaphragm Design Forces, Building #2
wpx
(kips)
Fpx
(kips)
ȈFi Ȉwi
(kips)
Design
Force
(kips)**
9
8
6
99.8
1,369.8
331.8
331.8
331.8
331.8
331.8
331.8
3
331.8
331.8
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
R
11
Ȉwi
ȈFi
* Min.
** Max. of
Step 2 – Determine the seismic diaphragm forces
ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)
Diaphragm forces
over the height of the building are given in Table 3.31. Values of the story weight,
and the
are given in Table 3.RI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO
seismic lateral force,
is set equal to
It is evident from Table 3.31 that the minimum
DOOOHYHOVRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO
governs at
Also, maximum
Step 3 – Determine the seismic diaphragm design forces
Seismic diaphragm design forces are the larger of
given in Table 3.31.
and
ASCE/SEI 12.10.1.1
The design forces over the height of the building are
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3.8.15 Example 3.15 – Determination of Seismic Forces: Diaphragms of Building #3
'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(,
VHH)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the seismic forces on the SFRS
ASCE/SEI 12.8.3
Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOH VHH7DEOH DQG
DUHJLYHQLQ7DEOH
Table 3.32 Seismic Diaphragm Design Forces, Building #3
(kips)
wpx
(kips)
ȈFi Ȉwi
Fpx
(kips)
Design
Force
(kips)**
398.3
168.3
13
3,936
11
118.1
168.3
168.3
9
8
6
3
13,319
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
R
16
Ȉwi
ȈFi
* Min.
** Max. of
Step 2 – Determine the seismic diaphragm forces
ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Diaphragm forces
RYHUWKHKHLJKWRIWKHEXLOGLQJDUHJLYHQLQ7DEOH9DOXHVRIWKHVWRU\ZHLJKW
and the
are given in Table 3.RI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO
seismic lateral force,
is set equal to
,WLVHYLGHQWIURP7DEOHWKDWWKHPLQLPXP
OHYHOVWKURXJKRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO
governs at
Also, maximum
Step 3 – Determine the seismic diaphragm design forces
Seismic diaphragm design forces are the larger of
JLYHQLQ7DEOH
and
ASCE/SEI 12.10.1.1
The design forces over the height of the building are
3.8.16 Example 3.16 – Determination of Seismic Forces: Diaphragms of Building #4
'HWHUPLQHWKHVHLVPLFGLDSKUDJPGHVLJQIRUFHVRYHUWKHKHLJKWRIWKHEXLOGLQJLQDFFRUGDQFHZLWK$6&(6(,
VHH)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the seismic forces on the SFRS
ASCE/SEI 12.8.3
Seismic forces RYHUWKHKHLJKWRIWKHEXLOGLQJDUHGHWHUPLQHGLQ6WHSRI([DPSOHLQWKHQRUWKVRXWKDQG
HDVWZHVWGLUHFWLRQV VHH7DEOH DQGDUHJLYHQLQ7DEOH
Table 3.33 Seismic Diaphragm Design Forces, Building #4
(kips)
wpx
(kips)
ȈFi Ȉwi
Fpx
(kips)
Design
Force
(kips)**
196.1
163.1
368.6
368.6
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
R
Ȉwi
ȈFi
(table continued on next page)
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Table 3.33 Seismic Diaphragm Design Forces, Building #4 (cont.)
wpx
(kips)
Fpx
(kips)
ȈFi Ȉwi
(kips)
Design
Force
(kips)**
98.3
16
13
11
9
8
16.9
6
3
Level
Story
Weight,
wi (kips)
Lateral
Force,
Fi (kips)
(kips)
19
18
Ȉwi
ȈFi
* Min.
** Max. of
Step 2 – Determine the seismic diaphragm forces
ASCE/SEI Eqs. (12.10-1), (12.10-2), and (12.10-3)
Diaphragm forces
over the height of the building are given in Table 3.33. Values of the story weight,
and the
seismic lateral force, DUHJLYHQLQ7DEOHRI([DPSOH7KHZHLJKWWULEXWDU\WRWKHGLDSKUDJPDWOHYHO
is set equal to
It is evident from Table 3.33 that the minimum
DOOOHYHOVRIWKHEXLOGLQJ)RUH[DPSOHDWWKHVHFRQGÀRRUOHYHO
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governs at
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Also, maximum
Step 3 – Determine the seismic diaphragm design forces
Seismic diaphragm design forces are the larger of
given in Table 3.33.
and
ASCE/SEI 12.10.1.1
The design forces over the height of the building are
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 4
ONE-WAY SLABS
4.1 Overview
One-way slabs are elements in one-way construction where the members are designed to support loads through
EHQGLQJLQDVLQJOHGLUHFWLRQ:KHUHWKHUDWLRRIWKHORQJWRWKHVKRUWVLGHRIDVODESDQHOLVJUHDWHUWKDQORDG
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2
3
κଵ E
Beam
ሺtyp. ሻ
κଶ One-way slab
κଶ Τκଵ ൐ 2
A
A
D
Section A-A
Figure 4.1 A one-way slab system.
The design and detailing of solid, cast-in-place one-way slabs with nonprestressed reinforcement are covered in this
FKDSWHU3URYLVLRQVIRURQHZD\VODEVDUHJLYHQLQ$&,&KDSWHU
4.2 Minimum Slab Thickness
2QHZD\VODEVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRWKDWDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG,QOLHXRIFDOFXODWLQJGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,DQGVXEVHTXHQWO\FKHFNLQJWKDWWKHGHÀHFWLRQV
GRQRWH[FHHGWKHOLPLWVLQ$&, $&, WKHPLQLPXPRYHUDOOVODEWKLFNQHVV for solid, nonprestressed
VODEVQRWVXSSRUWLQJRUDWWDFKHGWRSDUWLWLRQVRURWKHUW\SHVRIFRQVWUXFWLRQOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV
FDQEHGHWHUPLQHGXVLQJWKHOLPLWVLQ$&,7DEOHWKHVHOLPLWVDUHDSSOLFDEOHWRRQHZD\VODEVZLWKUHLQIRUFHPHQWWKDWKDVDVSHFL¿HG\LHOGVWUHQJWK HTXDOWRSVLDQGQRUPDOZHLJKWFRQFUHWH $&, 0RGL¿FDWLRQIDFWRUVIRUVODEVZLWKUHLQIRUFHPHQWRWKHUWKDQSVLDQGOLJKWZHLJKWFRQFUHWHDUHJLYHQLQ$&,
DQGUHVSHFWLYHO\
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
0LQLPXPVODEWKLFNQHVVHVEDVHGRQYDULRXVVXSSRUWFRQGLWLRQVDUHJLYHQLQ7DEOH,QWKHH[SUHVVLRQVIRUPLQLis the span length of the one-way slab in inches; for cantilevers, is the clear projection of the cantilever
mum
in inches. Also, and DUHWKHPRGL¿FDWLRQIDFWRUVIRUUHLQIRUFHPHQWJUDGHDQGOLJKWZHLJKWFRQFUHWHUHVSHFtively.
Table 4.1 Minimum Thickness of Solid, Nonprestressed One-way Slabs
Lightweight Concrete(1)
Normalweight Concrete
Support Condition
ƒy = 60 ksi
ƒy other than 60 ksi(2)
ƒy = 60 ksi(3)
ƒy other than 60 ksi(2),(3)
Simply supported
One end continuous
Both ends continuous
Cantilever
(1) Applicable where equilibrium density,
(2)
(3)
is in the range of 90 to 115 lb/ft3
For the usual case of continuous construction, the thickness of the slab, should be the same for all spans and
it should be determined on the basis of the span yielding the largest minimum depth; this results in economical
IRUPZRUN VHH5HIHUHQFH )RUWKHRQHZD\VODEV\VWHPLQ)LJXUHZLWKQRUPDOZHLJKWFRQFUHWHDQG*UDGH
UHLQIRUFHPHQWWKHPLQLPXPVODEWKLFNQHVVWREHXVHGIRUDOOWKHVSDQVLVHTXDOWRLQZKLFKLVURXQGHGXSWR
LQ6ODEWKLFNQHVVLVW\SLFDOO\VSHFL¿HGLQZKROHLQFKLQFUHPHQWVDOWKRXJKKDOILQFKLQFUHPHQWVDUHFRPPRQO\XVHG
in wide-module joist construction.
ͷᇱ ǦͲᇳ κΤͳͲ ൌ ͸ǤͲᇳ ͳͷᇱ ǦͲᇳ ͳͺᇱ ǦͲᇳ κΤʹͶ ൌ ͹Ǥͷᇳ κΤʹͺ ൌ ͹Ǥ͹ᇳ ͳͶᇱ Ǧ͸ᇳ κΤʹͶ ൌ ͹Ǥ͵ᇳ Figure 4.2 Calculation of minimum slab thickness for a one-way slab system.
Fire-resistance requirements of the general building code must also be considered when specifying a slab thickQHVV $&, 7KLVLVHVSHFLDOO\LPSRUWDQWIRURQHZD\VODEVWKDWDUHSDUWRIDMRLVWV\VWHPZKHUHWKHMRLVWVDUH
spaced relatively closely together, which means the required slab thickness for serviceability is relatively small. In
FHUWDLQVLWXDWLRQVWKHUHTXLUHGVODEWKLFNQHVVEDVHGRQ¿UHUHVLVWDQFHUHTXLUHPHQWVIRXQGLQ,%&7DEOH LV
JUHDWHUWKDQWKDWUHTXLUHGE\$&,IRUVHUYLFHDELOLW\ $&, 4.3 Required Strength
4.3.1 Analysis Methods
7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR
EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWK VHH$&,DQGUHVSHFWLYHO\ )RUWKHFDVHRIJUDYLW\ORDGVWKH
VLPSOL¿HGPHWKRGIRUDQDO\VLVRIFRQWLQXRXVEHDPVDQGRQHZD\VODEVLQ$&,LVSHUPLWWHGWREHXVHGSURYLGHG
WKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG>$&, D @
1. Members are prismatic
Gravity loads are uniformly distributed
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
3. The service live load,
is less than or equal to 3 times the service dead load,
There are at least two spans
7KHORQJHURIWZRDGMDFHQWVSDQVGRHVQRWH[FHHGWKHVKRUWHUVSDQE\PRUHWKDQSHUFHQW
and factored shear forces,
LQ$&,DQGUHVSHFWLYHThe approximate factored bending moments,
is the factored uniformly distributed gravity load on the one-way slab.
O\DUHJLYHQLQ)LJXUHZKHUH
Uniformly distributed gravity load, ‫ݓ‬௨ , with ‫ ܮ‬൑ 3‫ܦ‬
Two or more spans
Integral with
support
Prismatic members
Simple
support
Positive
moment
Spandrel
support
Column
support
Negative
moment
**
Shear
* For two-span condition, first interior negative moment ൌ ‫ݓ‬௨ κଶ௡ /9
κ௡,௔௩௚ ൌ ሺκ௡,ଵ ൅ κ௡,ଶ ሻ/2
** Applicable to slabs with spans equal to or less than 10 ft and beams where
the ratio of the sum of column stiffness to beam stiffness is greater than 8 at
each end of the span.
A Interior face of exterior support
B Exterior face of first interior support
C Other faces of interior supports
Figure 4.3 Simplified analysis method for continuous beams and one-way slabs.
4.3.2 Critical Sections for Flexure and Shear
,QDFFRUGDQFHZLWK$&,DQG
and
are permitted to be calculated at the faces of the supports
IRURQHZD\VODEVEXLOWLQWHJUDOO\ZLWKWKHVXSSRUWV VHH)LJXUH )RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVRFcur at the faces of the supports where negative moments are maximum and in the span where positive moments are
maximum. One-way slabs are permitted to be designed for the shear force at a critical section located a distance
IURPWKHIDFHRIWKHVXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG VHH)LJXUH • Support reaction, in direction of applied shear, introduces compression into the end region of the slab
• Loads are applied at or near the top surface of the slab
• No concentrated load occurs between the face of the support and the critical section
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ݓ‬௨ ƒ’’Ž‹‡†ƒ–‘”‡ƒ”–‘’
•—”ˆƒ ‡‘ˆ•Žƒ„
ܸ௨ ̷ˆƒ ‡
‘ ‡–”ƒ–‡†
Ž‘ƒ†ǡܲ௨ ܸ௨ ̷݀
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݀
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Figure 4.4 Critical section for shear in a one-way slab satisfying the conditions of ACI 7.4.3.2.
The critical section for shear must be taken at the face of the support where one or more of the three conditions in
$&,DUHQRWPHW)RUH[DPSOHWKHFULWLFDOVHFWLRQPXVWEHDWWKHIDFHRIWKHVXSSRUWIRUWKHIUDPLQJFRQ¿JXUDWLRQLQ)LJXUHEHFDXVHWKHVXSSRUWLQJPHPEHULVLQWHQVLRQ
‡•‹‘ˆ‘” ‡ǡܶ௨ ‫ݓ‬௨ ܸ௨ ̷ˆƒ ‡
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Figure 4.5 Critical section for shear in a one-way slab supported by a member in tension.
0RPHQWUHGLVWULEXWLRQLVQRWSHUPLWWHGZKHQEHQGLQJPRPHQWVDUHFDOFXODWHGXVLQJWKHVLPSOL¿HGPHWKRG $&,
:KHUHRQHRUPRUHRIWKHFRQGLWLRQVLQ$&,LVQRWVDWLV¿HGRQHRIWKHRWKHUIRXUPHWKRGVRIDQDO\VLV
and
at the critical sections.
JLYHQLQ$&,PXVWEHXVHGWRGHWHUPLQH
4.4 Design Strength
4.4.1 General
7KHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\VHFWLRQLQDRQHZD\VODEIRUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOH $&, @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(4.1)
(4.2)
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGRQHZD\VODEVPXVW
EHGHVLJQHGDVWHQVLRQFRQWUROOHGLQDFFRUGDQFHZLWK$&,7DEOH $&, WKXVWKHQHWWHQVLOHVWUDLQLQ
must be greater than or equal
the extreme layer of the longitudinal tension reinforcement at nominal strength,
and
$&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPHOD\HURIORQJLWXGLQDOWHQVLRQ
to
is equal to
$&, 7KHPRGXreinforcement corresponding to compression-controlled sections,
LVSHUPLWWHGWREHWDNHQDVSVLUHJDUGOHVVRIWKHJUDGHRIWKH
lus of elasticity of the reinforcement,
$&,7DEOH UHLQIRUFHPHQW $&, )RUVKHDU
Methods to determine the nominal strengths
and
DUHJLYHQLQ6HFWLRQVDQGUHVSHFWLYHO\
4.4.2 Nominal Flexural Strength
7KHQRPLQDOÀH[XUDOVWUHQJWK
of a rectangular section with one layer of tension reinforcement is determined in
DFFRUGDQFHZLWK$&,DQGLVEDVHGRQPRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQ VHH$&,DQG)LJXUH
(4.3)
ߝ௧ ‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ
݀ െ ሺܽȀʹሻ
ܿ
݀ ൌ ݀௧ ݄
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ܽ
ͲǤͺͷ݂௖ᇱ ͲǤͲͲ͵
ܶ ൌ ‫ܣ‬௦ ݂௬ ܾ
Figure 4.6 Strain and stress distributions at a positive moment section in a one-way slab.
In this equation, LVWKHWRWDODUHDRIWKHÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUH
are at a positive moment section and are equally applicable at negative moment sections.
)RURQHZD\VODEVWKHDYHUDJHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQUsing an approximate SURYLGHVVXI¿FLHQWDFFXUDF\
sion reinforcement, can be approximated as
ZKHQGHWHUPLQLQJWKHUHTXLUHGDPRXQWRIÀH[XUDOUHLQIRUFHPHQWLQW\SLFDORQHZD\VODEV\VWHPV
The depth of the equivalent stress block, is determined from force equilibrium, that is, it is determined by setequal to the tension force in the reinforcement,
ting the resultant compressive force in the concrete,
and solving for VHH)LJXUH (4.4)
,WLVDVVXPHGWKDWDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK is distributed over
where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV $&, the depth
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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SUHVVLRQIDFHRIWKHPHPEHUZKLFKLVW\SLFDOO\WDNHQDVLQIRUDRQHZD\VODE
is the width of the com-
is the factor relating the depth of the equivalent rectangular compression block,
The term
LVGH¿QHGLQ$&,7DEOHDVIROORZV $&, neutral axis,
to the depth of the
• For
• For
• For
4.4.3 Nominal Shear Strength
The nominal one-way shear strength,
DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, (4.5)
In this equation,
is the nominal shear strength provided by concrete and
is the nominal shear strength provided by shear reinforcement. Shear reinforcement is typically not used in one-way slabs because of the issues associated with cost, fabrication, and placement of such reinforcement in a relatively thin slab; where additional shear
strength is required, the slab thickness is often increased.
For nonprestressed members, LVGHWHUPLQHGE\$&,7DEOH $&, (TXDWLRQ F LQ$&,7DEOH
LVDSSOLFDEOHZKHUHQRVKHDUUHLQIRUFHPHQWLVXVHGLQDVHFWLRQ7KLVHTXDWLRQUHGXFHVWRWKHIROORZLQJIRU
LVHTXDOWR]HUR
the case where the axial force,
(4.6)
In this equation,
is the width of the web of the section; for one-way slabs,
must not be taken greater than
$FFRUGLQJWR$&,
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU
accounts for the phenomenon indicated in test results that the shear strength
attributed to concrete in members without shear reinforcement does not increase in direct proportion with member
GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(4.7)
,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK
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means that
For one-way slabs, this
The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH
to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium
RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[ VHH$&,DQG
density,
$&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOH D @DQGYDOXHVRI based on
FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOH E @1RWHWKDW is permitted to be taken as
IRUOLJKWZHLJKWFRQFUHWH $&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH $&, Table 4.2 Values of Ȝ Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 4.3 Values of Ȝ Based on Composition of Aggregates
Concrete
Composition of Aggregates
)LQH$670&
&RDUVH$670&
All-lightweight
)LQH&RPELQDWLRQRI$670&
and ASTM C33
&RDUVH$670&
/LJKWZHLJKW¿QHEOHQG
Fine: ASTM C33
&RDUVH$670&
Sand-lightweight
Sand-lightweight, coarse blend
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&
and ASTM C33
Ȝ
WR(1)
WR 1. Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate.
2. Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.
The term
LQ(TXDWLRQ LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
at the section divided by
$FFRUGLQJWR$&,5 PD\EHWDNHQDVWKHVXPRIWKHDUHDVRIWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW
ORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUV
with one layer of tension reinforcement, like one-way slabs, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDWVHFtion.
used to calculate DUHOLPLWHGWRSVLH[FHSWDVDOORZHGLQ$&,ZKLFKLVQRWDSSOLFDEOH
Values of
WRRQHZD\VODEV $&, 7KLVOLPLWDWLRQRQ is primarily due to the fact that there is a lack of test data and
SUDFWLFDOH[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\ for oneZD\VODEVXVXDOO\GRHVQRWH[FHHGSVL 5HIHUHQFH In situations where
increased, although increasing
WKHWKLFNQHVVRIWKHVODEDQGRUWKHFRQFUHWHFRPSUHVVLYHVWUHQJWKVKRXOGEH
is less effective than increasing (or, equivalently, ).
7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH
FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@
(4.8)
4.5 Determination of Required Reinforcement
4.5.1 Required Flexural Reinforcement
7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW , at a critical section of a one-way slab is determined based on the reand
, for tension-controlled sections. Required
can be determined
TXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV
E\WKHIROORZLQJHTXDWLRQZKLFKLVEDVHGRQ(TXDWLRQV DQG (4.9)
7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH
LVGH¿QHGDVIROORZV
(4.10)
where
for tension-controlled sections.
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@Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
It is important to check that FDOFXODWHGE\(TXDWLRQ LVDWOHDVWHTXDOWRWKHUHTXLUHGPLQLPXPDUHDRI
, which is equal to
where the gross area of the slab,
is equal to (ACI
ÀH[XUDOUHLQIRUFHPHQW
It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear
for tension-controlled sections (see
strain distribution where is the depth of the neutral axis when
$&,DQG)LJXUH (4.11)
Substituting
LQWR(TXDWLRQ ZLWK HTXDOWRWKDWIURP(TXDWLRQ WKHIROORZLQJHTXDWLRQFDQEH
corresponding to tension-controlled sections for one-way
XVHGWRFDOFXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
slabs with
(4.12)
)RU*UDGHUHLQIRUFHPHQW
duces to the following for these material properties:
and for
(TXDWLRQ UH(4.13)
If FDOFXODWHGE\(TXDWLRQ LVJUHDWHUWKDQ
the latter of which is essentially the maximum amount of
and must be inÀH[XUDOUHLQIRUFHPHQWSHUPLWWHGDWDQ\VHFWLRQWKHVHFWLRQLVQRWWHQVLRQFRQWUROOHG
creased accordingly to attain a tension-controlled section.
4.5.2 Minimum Shrinkage and Temperature Reinforcement
A minimum amount of shrinkage and temperature reinforcement must be provided in a one-way slab perpendicular
WRWKHÀH[XUDOUHLQIRUFHPHQWLQDFFRUGDQFHZLWK$&, $&, 7KHPLQLPXPDUHDRIVXFKUHLQIRUFHPHQW
$&, is equal to
The maximum center-to-center spacing of the shrinkage and temperature reinforcement is equal to the lesser of
DQGLQ $&, DQGWKHVHUHLQIRUFLQJEDUVPXVWEHGHVLJQHGWRGHYHORS in tension at all sections where
UHTXLUHG $&, 4.6 Reinforcement Detailing
4.6.1 Concrete Cover
5HLQIRUFLQJEDUVDUHSODFHGLQRQHZD\VODEVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UH
DQGRWKHUHIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&, $&, :KHQJHQHUDOEXLOGLQJFRGH
PLQLPXPFRYHUUHTXLUHPHQWVIRU¿UHSURWHFWLRQDUHJUHDWHUWKDQ$&,$&,UHTXLUHVWKHJUHDWHUFRQFUHWHFRYHUWKLFNQHVVWRJRYHUQ,%&7DEOH SURYLGHVPLQLPXPFRQFUHWHFRYHUUHTXLUHPHQWVIRU¿UHUDWLQJVUDQJLQJIURPKUWRKU)RUUHLQIRUFLQJEDUVLQRQHZD\VODEVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKH
JURXQGDQGDPD[LPXPKU¿UHUDWLQJWKHPLQLPXPFRYHULVHTXDOWRLQIRUDQGVPDOOHUEDUV $&,7DEOH
)RURQHZD\VODEVZLWKRXWWUDQVYHUVHUHLQIRUFHPHQWFRQFUHWHFRYHULVPHDVXUHGIURPWKHVXUIDFHRIWKH
FRQFUHWHWRWKHRXWPRVWOD\HURIUHLQIRUFHPHQW VHH)LJXUH 4.6.2 Minimum Spacing of Flexural Reinforcing Bars
0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&, $&, 7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRWKDWFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJ
bars.
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4-8
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‘˜‡”
Figure 4.7 Concrete cover for one-way slabs.
Ž‡š—”ƒŽ”‡‹ˆ‘” ‡‡–ሺ–›’Ǥሻ
݀௕ Š”‹ƒ‰‡ƒ†–‡’‡”ƒ–—”‡
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‫ۓ‬
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Figure 4.8 Minimum clear spacing requirements for reinforcing bars.
7KHVSDFLQJUHTXLUHPHQWVDUHJLYHQLQ)LJXUHIRUDRQHZD\VODEZKHUH
VL]HLQWKHFRQFUHWHPL[
is the nominal maximum aggregate
4.6.3 Maximum Spacing of Flexural Reinforcing Bars
0D[LPXPFHQWHUWRFHQWHUVSDFLQJRIUHLQIRUFLQJEDUVLVJLYHQLQ$&, $&, 7KHLQWHQWRIWKHVH
UHTXLUHPHQWVLVWRFRQWUROÀH[XUDOFUDFNLQJ,QJHQHUDODODUJHUQXPEHURI¿QHUFUDFNVLVSUHIHUDEOHWRDIHZZLGH
cracks mainly for reasons of durability and appearance.
The maximum center-to-center bar spacing, LVGHWHUPLQHGE\WKHHTXDWLRQVLQ$&,7DEOH $&, The following equation is applicable to deformed reinforcing bars:
(4.14)
In this equation, LVWKHOHDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKHVHFtion and LVWKHFDOFXODWHGVWUHVVLQWKHÀH[XUDOUHLQIRUFHPHQWFORVHWWRWKHWHQVLRQIDFHDWVHUYLFHORDGV LQSVL In lieu of calculating using the unfactored bending moment at that section, is permitted to be taken as
and
maximum center-to-center
$&, )RU*UDGHUHLQIRUFHPHQW
IURP(TXDWLRQ bar spacing
$FFRUGLQJWR$&,WKHPD[LPXPVSDFLQJRIGHIRUPHGÀH[XUDOUHLQIRUFHPHQWLQDRQHZD\VODELVHTXDO
and 18 in. For typical one-way slab thicknesses, the maximum spacing based on crack control
to the lesser of
requirements usually governs.
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4-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
4.6.4 Selection of Flexural Reinforcement
7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ DQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV
JLYHQLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\
In one-way slabs, LVW\SLFDOO\FDOFXODWHGSHUIRRWZLGWKRIVODE$EDUVL]HDQGVSDFLQJVKRXOGEHVHOHFWHGVRWKDW
is equal to or slightly greater than the required
considering minimum reinforcement requirethe provided
PHQWVDQGPLQLPXPDQGPD[LPXPEDUVSDFLQJUHTXLUHPHQWV7KHLQIRUPDWLRQLQ7DEOHFDQEHXVHGWRIDFLOLWDWH
VHOHFWLRQRIEDUVL]HDQGVSDFLQJDWDFULWLFDOVHFWLRQIRUÀH[XUHLQDRQHZD\VODE
Table 4.4 Area of Reinforcement (in.2) Based on Center-to-Center Spacing of the Bars
Spacing (in.)
Bar Size
4
5
6
7
8
9
10
11
12
4.6.5 Development of Flexural Reinforcement
Development of Deformed Bars in Tension
'HYHORSPHQWOHQJWKVRIGHIRUPHGUHLQIRUFHPHQWDUHJLYHQLQ$&, $&, 3URYLVLRQVIRUWKHGHYHOis
RSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHORSPHQWOHQJWK
GHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RUDORQJZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KH
UHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHUUHTXLUHPHQWVDUHFRYHUHG¿UVW
Method 1 – ACI 25.4.2.4
The development length in tension of a deformed reinforcing bar,
LVGHWHUPLQHGE\$&,(TXDWLRQ D (4.15)
7KHWHUPVLQ(TXDWLRQ DUHDVIROORZV $&,7DEOH 0RGL¿FDWLRQIDFWRUIRUOLJKWZHLJKWFRQFUHWH
crete:
7KLVIDFWRUUHÀHFWVWKHORZHUWHQVLOHVWUHQJWKRIOLJKWZHLJKWFRQ-
(4.16)
5HLQIRUFHPHQWJUDGHIDFWRU
development length:
This factor accounts for the effect of reinforcement yield strength on the required
(4.17)
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4-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
*UDGHÀH[XUDOUHLQIRUFHPHQWLVW\SLFDOO\XVHGLQRQHZD\VODEVVR
.
5HLQIRUFHPHQWFRDWLQJIDFWRU
This factor accounts for the reduced bond strength between the concrete and
HSR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFLQJEDUV
(4.18)
5HLQIRUFHPHQWVL]HIDFWRU
7KLVIDFWRUUHÀHFWVWKHPRUHIDYRUDEOHSHUIRUPDQFHRIVPDOOHUGLDPHWHUUHLQIRUFLQJEDUV
(4.19)
&DVWLQJSRVLWLRQIDFWRU
7KLVIDFWRUUHÀHFWVWKHDGYHUVHHIIHFWVWKDWFDQRFFXUWRWKHWRSUHLQIRUFHPHQWLQDPHPber due to vertical migration of water and mortar, which collect on the underside of the bars during placement of the
concrete:
(4.20)
7KLVIDFWRULVXVXDOO\IRURQHZD\VODEV$OVRDFFRUGLQJWRWKHIRRWQRWHLQ$&,7DEOHWKHSURGXFW
QHHGQRWH[FHHG
Spacing or cover dimension,
7KLVWHUPLVGH¿QHGDVIROORZV VHH)LJXUH (4.21)
ܿଵ For one-way slabs, the distance from the center of the bar to the nearest concrete surface typically governs.
ܿଶ ‫ݏ‬
‫ۓ‬
ۖ
ܿଵ ܿ௕ ൌ Ž‡••‡”‘ˆ ܿଶ ‫۔‬
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‫ݏە‬Τʹ
Figure 4.9 Spacing or cover dimension, cb.
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4-11
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7UDQVYHUVHUHLQIRUFHPHQWLQGH[
7KHWUDQVYHUVHUHLQIRUFHPHQWLQGH[UHSUHVHQWVWKHHIIHFWRIFRQ¿QLQJUHLQIRUFHPHQWDFURVVWKHSRWHQWLDOVSOLWWLQJSODQHVODUJHUDPRXQWVRIFRQ¿QLQJUHLQIRUFHPHQWUHGXFHWKHSRWHQWLDOIRU
splitting failure, thereby reducing the overall required development length. This index is determined by ACI EquaWLRQ E (4.22)
where
total cross-sectional area of all transverse reinforcement within a spacing
potential plane of splitting through the reinforcement being developed
that crosses the
center-to-center spacing of the transverse reinforcement
number of bars being developed across the plane of splitting
It is permitted to conservatively use
LIWUDQVYHUVHUHLQIRUFHPHQWLVSUHVHQWRUUHTXLUHG $&, )RU
. Also, for reinforcing bars
one-way slabs, where it is typical that transverse reinforcement is not used,
VSDFHGFORVHUWKDQLQRQFHQWHUWUDQVYHUVHUHLQIRUFHPHQWPXVWEHSURYLGHGVXFKWKDW
with
$&, 7KHFRQ¿QLQJWHUP
PXVWEHWDNHQOHVVWKDQRUHTXDOWRLQ(TXDWLRQ >$&,@7HVWV
KDYHVKRZQWKDWZKHQWKLVWHUPLVOHVVWKDQVSOLWWLQJIDLOXUHVDUHOLNHO\WRRFFXU$SXOORXWIDLOXUHRIWKHUHLQIRUFHPHQWLVPRUHOLNHO\ZKHQWKLVWHUPLVJUHDWHUWKDQVRDQLQFUHDVHLQWKHDQFKRUDJHFDSDFLW\GXHWRDQ
LQFUHDVHLQFRYHURUDPRXQWRIFRQ¿QLQJUHLQIRUFHPHQWLVQRWOLNHO\
used in calculating PXVWEHOHVVWKDQRUHTXDOWRSVL $&, 7KLVUHTXLUHPHQWLVDSValues of
SOLFDEOHWRDOOWKHUHLQIRUFHPHQWGHYHORSPHQWSURYLVLRQVLQ$&,
The tension development length LVSHUPLWWHGWREHUHGXFHGLQFDVHVZKHUHWKHÀH[XUDOUHLQIRUFHPHQWLVJUHDWHU
WKDQWKDWUHTXLUHGIURPDQDO\VLVH[FHSWIRUWKHVL[FDVHVLQ$&, $&, :KHUHSHUPLWWHGWKH
reduction factor applied to LVHTXDOWRWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWGLYLGHGE\WKHSURYLGHGDUHDRI
reinforcement.
for deformed bars in one-way slabs based on normalweight concrete with
, uncoated
Values of
,
, and
DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI
*UDGHUHLQIRUFHPHQW
are rounded up to the next whole number. Values of
for one-way slabs with coated bars can be obtained by
multiplying the tabulated values by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI for oneZD\VODEVPDGHRIOLJKWZHLJKWFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFHZLWK
(TXDWLRQ Table 4.5 Tension Development Length, ℓd, for Deformed Bars in One-way Slabs in Accordance with
ACI 25.4.2.4*
Bar Size
ℓd (in.)
* Normalweight concrete with
Uncoated Grade 60 reinforcement
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4-12
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Method 2 – ACI 25.4.2.3
7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUH. Two sets of spacing and cover cases are given in ACI Table
VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP
and
)RURQHZD\VODEVLWLVFRPPRQIRUWKHFOHDUVSDFLQJRIWKHEDUVEHLQJGHYHORSHGWREHDWOHDVW
7KHUHIRUHWKHIROORZLQJHTXDWLRQIURP$&,7DEOHIRUWKHFDVHRIDQG
the clear cover to be at least
for reinforcing bars in one-way slabs:
smaller longitudinal bars can be used to determine
(4.23)
Values of
for deformed bars in one-way slabs based on normalweight concrete with
, uncoated
DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI are rounded up to the next
*UDGHUHLQIRUFHPHQWDQG
for one-way slabs with coated bars can be obtained by multiplying the tabulated values
whole number. Values of
by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI for one-way slabs made of lightweight
DJJUHJDWHFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFHZLWK(TXDWLRQ Table 4.6 Tension Development Length, ℓd, for Deformed Bars in One-way Slabs in Accordance with
ACI 25.4.2.3*
Bar Size
ℓd (in.)
19
* Normalweight concrete with
Uncoated Grade 60 reinforcement
Development of Standard Hooks in Tension
+RRNVDUHSURYLGHGDWWKHHQGVRIUHLQIRUFLQJEDUVWRSURYLGHDGGLWLRQDODQFKRUDJHZKHUHGHYHORSPHQWOHQJWKFDQƒ”‘Ǥ
ͻͲǦ†‡‰”‡‡
‘‘
QRWEHDWWDLQHGZLWKVWUDLJKWEDUV6WDQGDUGKRRNVDUHGH¿QHGLQ$&,
VHH)LJXUH
͓ͻǡ͓ͳͲǡ
κௗ௛ ݀௕ κ௘௫௧ ൌ ͳʹ݀௕ ‫ܦ‬
ͻͲǦ†‡‰”‡‡ ‘‘
ͻͲǦ†‡‰”‡‡ ‘‘
͓ͳͶǡ͓ͳ
κௗ௛ ݀௕ ”‹–‹ ƒŽ•‡ –‹‘
‫ܦ‬
ƒ”‘Ǥ
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͸݀௕ ͓ͻǡ͓ͳͲǡ͓ͳͳ
ͺ݀௕ ƒƒ”‘Ǥ ͳͲ݀ ͓ͳͶǡ͓ͳͺ
௕
͓͵
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κ௘௫௧ ൌ Ͷ݀௕ ൒ ʹΦ̶
ͳͺͲǦ†‡‰”‡‡ ‘‘
Figure 4.10 Standard hooks in accordance with ACI 25.3.1.
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4-13
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The development length of a deformed reinforcing bar in tension with a standard hook,
LVJLYHQLQ$&,
(4.24)
7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRN VHH)LJXUH 7KH
PRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOH VHH$&,7DEOH Table 4.7 Modification Factors for Development of Hooked Bars in Tension
Modification Factor
Lightweight,
Epoxy,
&RQ¿QLQJ
reinforcement,
Location,
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG
reinforcement
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
or
Other
1.6
)RUDQGVPDOOHUGLDPHWHUKRRNHGEDUV WHUPLnating inside a column core with side cover normal
RU ZLWKVLGH
to the plane of the hook
cover normal to the plane of the hook
Other
Concrete strength,
The term
LVWKHWRWDOFURVVVHFWLRQDODUHDRIWLHVRUVWLUUXSVFRQ¿QLQJKRRNHGEDUVDQG
is the total cross-sectional area of hooked bars being developed at the same critical section. For one-way slabs, it is typical for
(that is, ties or stirrups are not present). The term is the center-to-center spacing of the hooked bars, which have a
nominal diameter
7KHUHTXLUHPHQWVLQ$&,DSSO\WRKRRNVDWGLVFRQWLQXRXVHQGVRIPHPEHUV WKDWLVDWHQGVRIVLPSO\
supported members, at the free ends of cantilevers, and at exterior joints where members do not extend beyond the
MRLQW ZKHUHERWKVLGHFRYHUDQGWRS RUERWWRP FRYHUWRWKHKRRNLVOHVVWKDQLQ7KHVHSURYLVLRQVDUHW\SLFDOO\
applicable at beam-column joints, and thus, do not apply to one-way slabs in continuous construction where con¿QHPHQWLVSURYLGHGE\WKHVODERQERWKVLGHVSHUSHQGLFXODUWRWKHSODQHRIWKHKRRN
for hooked deformed bars in one-way slabs based on normalweight concrete with
Values of
and side cover normal to the plane
XQFRDWHG*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKRRNHGEDUVSDFLQJ
DUHJLYHQLQ7DEOH&DOFXODWHGYDOXHVRI
are rounded up to the next whole number.
of the hook
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4-14
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 4.8 Tension Development Length, ℓdh, for Hooked Deformed Bars in One-way Slabs*
Bar Size
ℓdh (in.)
6
6
8
* Normalweight concrete with
Uncoated Grade 60 reinforcement
Side cover normal to the plane of the hook
Development of Headed Deformed Bars in Tension
3URYLVLRQVIRUWKHGHYHORSPHQWRIKHDGHGGHIRUPHGEDUVLQWHQVLRQDUHJLYHQLQ$&,([DPSOHVRIKHDGHG
UHLQIRUFLQJEDUVDUHJLYHQLQ)LJXUH
Figure 4.11 Headed reinforcing bars.
+HDGHGGHIRUPHGEDUVDUHSHUPLWWHGWREHXVHGRQO\ZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG
(a) %DUPXVWFRQIRUPWR$&,
(b) %DUVL]HPXVWEHRUVPDOOHU
must be at least
where
is the area of the bar
(c) Net bearing area of head,
(d) Concrete must be normalweight
where is the nominal diameter of the bar
(e) Clear cover to the bar must be greater than or equal to
(f) Center-to-center spacing between bars must be greater than or equal to
The development length of a headed deformed reinforcing bar in tension,
LVJLYHQLQ$&,
(4.25)
This development length is measured from the critical section to the bearing face of the head (see ACI Figure
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@Seismicisolation
4-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 4.9 Modification Factors for Development of Headed Bars in Tension
Modification Factor
Condition
Epoxy,
Parallel tie reinforcement,
Value of Factor
(SR[\FRDWHGRU]LQFDQG
epoxy dual-coated reinforcement
8QFRDWHGRU]LQFFRDWHG
JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
or
Other
1.6
For headed bars (1) terminating inside a
column core with side cover to bar
RU ZLWKVLGHFRYHUWREDU
Location,
Other
Concrete strength,
The term
is the total cross-sectional area of ties or stirrups acting as parallel tie reinforcement for headed bars
is the area of nonprestressed reinforcement in a tie or stirrup. For anchorages other than in beam-column
and
PXVWQRWEHFRQVLGHUHG $&, 7KHWHUP is the centerjoints (which is applicable to one-way slabs),
to-center spacing of the headed bars, which have a nominal diameter
7KHUHTXLUHPHQWVRI$&,DQGDUHQRWDSSOLFDEOHWRRQHZD\VODEV
Values of
for headed deformed bars in one-way slabs based on normalweight concrete with
, and side cover to the bar
XQFRDWHG*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKHDGHGEDUVSDFLQJ
are rounded up to the next whole number.
JLYHQLQ7DEOH&DOFXODWHGYDOXHVRI
are
Table 4.10 Tension Development Length, ℓdt, for Headed Deformed Bars in One-way Slabs*
Bar Size
ℓdt (in.)
#3
6
#4
6
#5
6
#6
8
* Normalweight concrete with
Uncoated Grade 60 reinforcement
Side cover normal to the bar
Development of Mechanically Anchored Deformed Bars in Tension
It is permitted to use any mechanical attachment or device capable of developing of the deformed bars provided
LWLVDSSURYHGE\WKHEXLOGLQJRI¿FLDOLQDFFRUGDQFHZLWK$&, $&, 7KHXVHRIPHFKDQLFDOGHYLFHV
WKDWGRQRWPHHWWKHUHTXLUHPHQWVLQ$&,RUDUHQRWGHYHORSHGLQDFFRUGDQFHZLWK$&,PD\EHXVHG
provided test results are available that demonstrate the ability of the head and bar system to develop or anchor the
required force in the bar.
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4-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Development of Positive and Negative Flexural Reinforcement
Flexural reinforcement must be properly developed or anchored in a reinforced concrete one-way slab for the slab
WRSHUIRUPDVLQWHQGHGLQDFFRUGDQFHZLWKWKHVWUHQJWKGHVLJQPHWKRG&ULWLFDOVHFWLRQVIRUGHYHORSPHQWRIÀH[XUDO
UHLQIRUFHPHQWRFFXUDWWKHIROORZLQJ $&, 1. Points of maximum stress (that is, sections of maximum bending moment)
Points along the span where adjacent reinforcement is terminated because it is no longer required to
UHVLVWÀH[XUH
In continuous one-way slabs subjected to uniform gravity loads, the maximum positive and negative bending moPHQWVW\SLFDOO\RFFXUQHDUPLGVSDQDQGDWWKHIDFHVRIWKHVXSSRUWVUHVSHFWLYHO\3RVLWLYHDQGQHJDWLYHÀH[XUDO
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this publication.
)RUWKHRQHZD\VODEV\VWHPLQ)LJXUHWKHWRWDOUHTXLUHGDUHDRIQHJDWLYHUHLQIRUFHPHQWIRUWKHPD[LPXP
at critical section A is equal to
and the total number of reinforcing
negative factored bending moment,
bars at this location is equal to ZKHUHDOOWKHEDUVDUHWKHVDPHVL]H6LPLODUO\WKHWRWDOUHTXLUHGDUHDRISRVLWLYH
at critical section C is equal to
and
reinforcement for the maximum positive factored bending moment,
the total number of reinforcing bars at this location is equal to Requirements for the development of these two
sets of reinforcing bars are discussed next.
݊ ൌ ݊ଵ ൅ ݊ଶ ݊ଵ ൒ ݊Τ3
‫ିܣ‬
௦ ሺ݊ barsሻ
‫ିܣ‬
௦ଵ ሺ݊ଵ barsሻ
‫ܣ‬ା
௦ ሺ‫ ݌‬barsሻ
‫ܣ‬ା
௦ଶ ሺ‫݌‬ଶ barsሻ
‫ିܣ‬
௦ଶ ሺ݊ଶ barsሻ
‫ ݌‬ൌ ‫݌‬ଵ ൅ ‫݌‬ଶ ‫݌‬ଵ ൒ ‫݌‬Τ4
൒ 6"
A
B
C
D
‫ܣ‬ା
௦ଵ ሺ‫݌‬ଵ barsሻ
൒ ݀ or 12݀௕ ൒ ݀ or 12݀௕ ݊ଶ bars
൒ κௗ ൒ κௗ ‫ݔ‬஺஻ ݊ଵ bars
‫ݔ‬஼஽ ൒ κௗ ሺ‫ܯ‬௨ା ሻ஽ ൒ ݀, 12݀௕ , or κ௡ /16
‫݌‬ଶ bars
ሺ‫ܯ‬௨ା ሻ஼ ‫ݔ‬஻ா E
Point of inflection
ሺ‫ܯ‬௨ି ሻ஻ ሺ‫ܯ‬௨ି ሻ஺ Figure 4.12 Development of flexural reinforcement in a one-way slab.
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4-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Negative Flexural Reinforcement
Assume a portion of
LVFXWRIIDWVHFWLRQ%ZKHUHLWLVQRORQJHUUHTXLUHGIRUÀH[XUDOVWUHQJWK7KLVPDNHV
and the number of
section B a critical section. At this location, the cutoff reinforcement has an area equal to
The area of the remaining portion of reinforcing bars is equal to
and
reinforcing bars is equal to
This reinforcement must be able to resist the
the corresponding number of reinforcing bars is equal to
at section B.
negative factored bending moment
bars must be adequately developed to the right of this section. This is
Because section B is a critical section,
achieved by extending the bars a minimum distance of SDVWVHFWLRQ%DVVKRZQLQ)LJXUHZKHUH is the
GHYHORSPHQWOHQJWKLQWHQVLRQRIDGHIRUPHGEDUZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, $FFRUGLQJWR$&,DWOHDVWRQHWKLUGRIWKHWRWDOQHJDWLYHUHLQIRUFHPHQWSURYLGHGDWDVXSSRUWPXVWKDYH
, and
SDVWWKHSRLQWRILQÀHFWLRQZKHUH is the clear
an embedment length equal to the larger of ,
span length measured face-to-face of supports (which in this case are beams). Therefore, to satisfy this requirement,
ZKLFKDFFRXQWVIRUSRVVLEOHVKLIWLQJRIWKHEHQGLQJPRPHQWGLDJUDPDWWKHSRLQWRILQÀHFWLRQGXHWRWKHDSSUR[L. The minimum length of
bars to the right
mate bending moment diagram customarily used in design,
RIFULWLFDOVHFWLRQ$LVHTXDOWRWKHIROORZLQJ $&,DQG (4.26)
In these equations,
is the distance from section A to the theoretical cutoff point at section B and
GLVWDQFHIURPVHFWLRQ%WRWKHSRLQWRILQÀHFWLRQDWVHFWLRQ(
is the
Because section A is a critical section, the bars cutoff at section B must be developed a distance equal to at least
EH\RQGWKDWVHFWLRQ$GGLWLRQDOO\$&,VWLSXODWHVWKDWWKHVHEDUVPXVWH[WHQGEH\RQGWKHSRLQWZKHUHWKH\DUH
The minimum length of
bars to the right
no longer required a distance equal to at least the greater of and
of critical section A is equal to the following:
(4.27)
5HLQIRUFLQJEDUVDUHQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRIWKHIROORZLQJFRQGLWLRQVDUH
VDWLV¿HG $&, where
is the design shear strength of the section at that point (see Sec1. At the cutoff point,
WLRQRIWKLVSXEOLFDWLRQ
)RUDQGVPDOOHUEDUVFRQWLQXLQJUHLQIRUFHPHQWSURYLGHVDWOHDVWGRXEOHWKHDUHDUHTXLUHGIRUÀH[XUHDWWKH
cutoff point and
7KHWKLUGFRQGLWLRQLQ$&,ZKLFKLVUHODWHGWRVWLUUXSDUHDLQWKHVHFWLRQLVW\SLFDOO\QRWDSSOLFDEOHWRRQH
way slabs.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The development of the negative reinforcing bars to the left of section A depends on its location within the framing
V\VWHP$WLQWHULRUMRLQWVOLNHWKHRQHGHSLFWHGLQ)LJXUHGHYHORSPHQWLVDFKLHYHGE\FRQWLQXLQJWKHQHJDtive reinforcing bars into the span to the left of the beam. In an end span where the slab terminates at edge beams, a
standard hook is provided at the ends of the negative reinforcing bars.
Positive Flexural Reinforcement
is cut off at section D. This makes section D a critical section. At this location, the reinAssume a portion of
and the number of reinforcing bars is equal to
The area of the remaining
forcement has an area equal to
and the corresponding number of reinforcing bars is equal to
portion of reinforcing bars is equal to
This reinforcement must be able to resist the negative factored bending moment
at section D.
The bars cut off at section D must be developed to a distance greater than or equal to
beyond the critical section C. Like in the case of the negative reinforcement, these bars must extend beyond the point they are no longer
$&, 7KHPLQLPXPOHQJWKRI bars to the left
required by a distance equal to the greater of and
of critical section C is equal to the following:
(4.28)
In this equation,
is the distance between critical section C and the theoretical cutoff point located at section D.
7KHUHTXLUHPHQWVRI$&,SHUWDLQLQJWRUHLQIRUFHPHQWWHUPLQDWHGLQDWHQVLRQ]RQHDUHDOVRDSSOLFDEOHLQWKLV
case.
$FFRUGLQJWR$&,DWOHDVWRQHIRXUWKRIWKHSRVLWLYHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQLQWRWKHVXSAlthough not common in typical reinforced concrete construction, at least one-third of the
port. Thus,
maximum positive reinforcement must extend along the slab bottom into simple supports.
7KHGLDPHWHURIWKHSRVLWLYHUHLQIRUFHPHQWLVOLPLWHGDWSRLQWVRILQÀHFWLRQLQDFFRUGDQFHZLWK$&,WKLV
requirement helps to ensure the bars are developed in a length short enough such that the moment capacity is greater
than the applied moment over the entire length of the slab:
(4.29)
In this equation,
LVWKHQRPLQDOÀH[XUDOVWUHQJWKRIWKHVODEVHFWLRQDVVXPLQJDOOWKHUHLQIRUFHPHQWDWWKDWVHFis the factored shear force at the section, and
is the embedment length of the reinforcetion is stressed to
Additional information on this topic
PHQWEH\RQGWKHLQÀHFWLRQSRLQWZKLFKLVOLPLWHGWRWKHJUHDWHURI and
FDQEHIRXQGLQ$&,5IRUEHDPV
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this publication).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
4.6.6 Splices of Reinforcement
Overview
6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQRQHZD\VODEVPXVWEHLQDFFRUGDQFHZLWK$&, $&, 7KHSULmary reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to
WKHFRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG
/DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQWDQGHDFK
type of splice is examined next.
Lap Splices
/DSVSOLFHVDUHIUHTXHQWO\VSHFL¿HGDQGDUHXVXDOO\WKHPRVWHFRQRPLFDOW\SHRIVSOLFH7KHUHDUHEDVLFDOO\WZR
types of lap splices: contact and noncontact. In a contact lap splice, the bars are generally in contact over a speci¿HGOHQJWKDQGDUHWLHGWRJHWKHU>VHH)LJXUH D @,QDQRQFRQWDFWODSVSOLFHWKHEDUVDUHQRWLQFRQWDFWDQGWKH
FHQWHUWRFHQWHUVSDFLQJRIWKHEDUVEHLQJVSOLFHGPXVWQRWH[FHHGWKHOLPLWVLQGLFDWHGLQ)LJXUH E VHH$&,
&RQWDFWODSVSOLFHVDUHXVXDOO\SUHIHUUHGEHFDXVHWKHEDUVDUHWLHGWRJHWKHUDQGDUHOHVVOLNHO\WRGLVSODFH
during construction. The minimum clear spacing between a contact lap splice and adjacent splices or bars is the
VDPHDVWKDWIRULQGLYLGXDOEDUV $&,VHH)LJXUH (a)
” lap length/5, 6"
(b)
Figure 4.13 Lap splices (a) Contact lap splice (b) Noncontact lap splice.
Lap splices should be provided at locations away from maximum stress (that is, away from sections of maximum
bending moment) and should be staggered wherever possible. Reinforcing bars provided for negative bending moments (top bars) should be spliced near the midspan of a member, and reinforcing bars provided for positive bending moments (bottom bars) should be spliced over the supports.
depends on the tension development length of the bars,
the area of
The required tension lap splice length,
reinforcement provided over the length of the splice, and the percentage of reinforcement spliced at any one locaWLRQ/DSVSOLFHVDUHFODVVL¿HGDV&ODVV$RU&ODVV%%HFDXVHH[SHULPHQWDOGDWDRQODSVSOLFHVZLWKDQG
EDUVDUHVSDUVHWKHXVHRIWHQVLRQODSVSOLFHVIRUWKHVHEDUVDUHSURKLELWHGH[FHSWDVSHUPLWWHGLQ$&,IRU
FRPSUHVVLRQODSVSOLFHV $&, The length of a tension lap splice is given as a multiple of
$&, (4.30)
(4.31)
When determining
to be used in determining
WKHLQPLQLPXPOHQJWKVSHFL¿HGLQ$&, E DQGWKH
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spaced closer than 6 in. on center, transverse reinforcement must be provided such that
bars with
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHGHIDXOWVSOLFHFODVVL¿FDWLRQIRUDWHQVLRQODSVSOLFHLV&ODVV%$&ODVV$VSOLFHLVSHUPLWWHGZKHUHERWKRIWKH
IROORZLQJFRQGLWLRQVDUHVDWLV¿HG $&,7DEOH over the entire splice length where
and
are the area of reinforcement provided at the splice location and the area of reinforcement required by analysis at the splice location,
respectively
• /HVVWKDQRUHTXDOWRSHUFHQWRIWKHUHLQIRUFHPHQWLVVSOLFHGZLWKLQWKHUHTXLUHGODSVSOLFHOHQJWK
•
:KHUHEDUVRIGLIIHUHQWVL]HDUHODSVSOLFHGLQWHQVLRQ
WKHVPDOOHUEDU $&, is equal to the greater of
of the larger bar and
of
Mechanical Splices
Mechanical splices are, in general, a complete assembly of a coupler, a coupling sleeve, or an end-bearing sleeve,
including any additional material or components, required to accomplish the splicing of the reinforcing bars. Such
RIWKHEDU $&, 7KLVLVWRHQVXUHWKDW\LHOGsplices must develop in tension or compression at least
ing occurs in the reinforcing bar adjacent to the mechanical splice prior to the failure of the splice.
,OOXVWUDWHGLQ)LJXUHDUHWKUHHRIWKHPRUHSRSXODUW\SHVRIPHFKDQLFDOVSOLFHV7KHFROGVZDJHGFRXSOLQJ
VOHHYHLQ)LJXUH D XVHVDK\GUDXOLFVZDJLQJSUHVVZLWKVSHFLDOGLHVWRGHIRUPWKHVOHHYHDURXQGWKHHQGVRI
the spliced bars to produce positive mechanical interlock with the reinforcing bars. The shear screw coupling sleeve
LQ)LJXUH E FRQVLVWVRIDFRXSOLQJVOHHYHZLWKVKHDUKHDGVFUHZVZKLFKDUHGHVLJQHGWRVKHDURIIDWDVSHFL¿HGWRUTXH7KHUHLQIRUFLQJEDUVDUHLQVHUWHGWRPHHWDWWKHFHQWHURIWKHFRXSOLQJVOHHYHDQGWKHQWKHVFUHZVDUH
tightened. The tightening process embeds the pointed screws into the bars. The non-upset straight thread coupler
LQ)LJXUH F FRQVLVWVRIDFRXSOHUZLWKLQWHUQDOVWUDLJKWWKUHDGVDWHDFKHQGWKDWMRLQVWKHWZRUHLQIRUFLQJEDUV
with matching external threads. Additional information on these and other mechanical splices can be found in Reference 9.
(a)
(b)
(c)
Figure 4.14 Mechanical splices (a) Cold-swaged coupling sleeve
(b) Shear screw coupling sleeve (c) Non-upset straight thread coupler.
Mechanical splices can be advantageous in a number of situations, including the following:
• Where long lap splices are needed
• Where lap splices cause reinforcement congestion
• :KHUHVSDFLQJRIWKHÀH[XUDOUHLQIRUFHPHQWLVLQVXI¿FLHQWWRSHUPLWODSVSOLFHV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Welded Splices
$&,SHUPLWVWKHXVHRIZHOGHGVSOLFHVZKLFKPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&,/LNHPHFKDQLFDOVSOLFHVDIXOOZHOGHGVSOLFHZKLFKLVJHQHUDOO\LQWHQGHGIRUDQGODUJHUEDUVPXVWEHDEOHWRGHYHORSDWOHDVW
of the bar.
4.6.7 Structural Integrity Reinforcement
5HTXLUHPHQWVIRUVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWLQFDVWLQSODFHRQHZD\VODEVDUHJLYHQLQ$&,7KHSXUpose of this reinforcement is to improve the redundancy and ductility in the structure so that in the event of damage
WRDPDMRUVXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFture will have a higher probability of maintaining overall stability.
7KUHHUHTXLUHPHQWVDUHJLYHQLQ$&,
1. Longitudinal structural integrity reinforcement must consist of at least one-quarter of the maximum positive
moment reinforcement and it must be continuous.
Longitudinal structural integrity reinforcement at noncontinuous supports must be anchored to develop
at the face of the support.
3. Where splices are needed in the continuous structural integrity reinforcement, the reinforcement must be
VSOLFHGQHDUWKHVXSSRUWV0HFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQ
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4.6.8 Recommended Flexural Reinforcement Details
5HFRPPHQGHGÀH[XUDOUHLQIRUFHPHQWGHWDLOVIRURQHZD\VODEVEDVHGRQWKHUHTXLUHPHQWVFRYHUHGDERYHLQFOXGLQJWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&,DUHJLYHQLQ)LJXUH7KHEDUOHQJWKVJLYHQLQWKH¿JXUH
are based on a one-way slab subjected to uniformly distributed gravity loads; these lengths can be used for one-way
VODEVWKDWKDYHEHHQGHVLJQHGXVLQJWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&, VHH6HFWLRQRIWKLVSXEOLFDtion). For one-way slabs subjected to the effects from other types of loads, required bar lengths must be determined
by calculations.
‘–‡ͳ
‘–‡ʹ
‘–‡Ͷ
‘–‡͵
‘–‡ʹ
‘–‡͵
–ƒ†ƒ”†
Š‘‘ሺ–›’Ǥሻ
൒ ‫ܣ‬ା௦ ȀͶ
൒ ͸̶
ሺ–›’Ǥሻ
‫ܣ‬ା௦ ‫ܣ‬ା௦ ൑ ͲǤͳʹͷκ௡ଵ κ௡ଵ ൑ ͲǤͳʹͷκ௡ଶ ‫ܣ‬ା௦ ‘–‡ͷ
κ௡ଶ ൑ ͲǤͳʹͷκ௡ଷ κ௡ଷ ‘–‡•ǣ
ሺͳሻ൒ κ௡ଵ ΤͶ
ሺʹሻ൒ ‰”‡ƒ–‡”‘ˆሺκ௡ଵ Τ͵ ǡ κ௡ଶ Τ͵ሻ
ሺ͵ሻ൒ ‰”‡ƒ–‡”‘ˆሺκ௡ଶ Τ͵ ǡ κ௡ଷ Τ͵ሻ
ሺͶሻŠ”‹ƒ‰‡ƒ†–‡’‡”ƒ–—”‡”‡‹ˆ‘” ‡‡–ሺ–›’Ǥሻ
ሺͷሻŽƒ••Žƒ’•’Ž‹ ‡‘”‡“—‹˜ƒŽ‡–
Figure 4.15 Recommended flexural reinforcement details for one-way slabs.
4.7 Design Procedure
7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIRQHZD\VODEV,QFOXGHGLQWKH¿JXUH
DUHWKHVHFWLRQQXPEHUVWDEOHQXPEHUVDQG¿JXUHQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFLQWKLVFKDSWHU
can be found.
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4-22
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1
EŽ
zĞƐ
EŽ
zĞƐ
1
A
Figure 4.16 Design procedure for one-way slabs.
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4-23
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
EŽ
zĞƐ
1
zĞƐ
EŽ
1
Detail the flexural reinforcement in
accordance with Sects. 4.6.5 – 4.6.8
Figure 4.16 (cont.) Design procedure for one-way slabs.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
4.8 Examples
4.8.1
Example 4.1 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2,
Normalweight Concrete
'HWHUPLQHWKHWKLFNQHVVRIWKHRQHZD\VODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDO
ÀRRUDVVXPLQJWKHMRLVWVDUHVSDFHGIWRQFHQWHUDQGWKHULEZLGWKLVHTXDOWRLQ VHH)LJXUH $OVRDVVXPH
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum slab thickness based on serviceability requirements
ACI 7.3.1.1
Because all span lengths are equal, the governing minimum slab thickness is based on a support condition of one
end continuous:
Table 4.1
7U\DLQWKLFNVODE
Step 2 – Check shear strength requirements
ACI 7.5.1.1
• Step 2a – Determine the maximum factored distributed load
For a one-foot design strip:
Maximum
LVGHWHUPLQHGE\$&,(T E Table 3.3
• 6WHSE±'HWHUPLQHLIWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFH
(1) Members are prismatic
Gravity loads are uniformly distributed
(3)
All spans are equal in length
ACI 6.5.1
7KHUHIRUHWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHG
• Step 2c – Determine the maximum factored shear force
Maximum RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUEHDP
ACI 6.5.4
Figure 4.3
where
is conservatively taken as the clear distance between the bottom instead of the top of the tapered ribs
VHH)LJXUH Although the critical section for shear is permitted to be taken a distance
FDVHLWLVFRQVHUYDWLYHO\WDNHQDWWKHIDFHRIWKHVXSSRUW $&, from the face of the support in this
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4-25
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• Step 2d – Check the shear strength requirements
ACI 7.5.1.1
Eq. (4.6)
Assuming
Eq. (4.7)
for normalweight concrete
ACI 19.2.4.3
$VVXPLQJPLQLPXPÀH[XUDOUHLQIRUFHPHQW
For shear,
ACI Table 21.2.1
Also, it is evident that
Eq. (4.8)
7KHUHIRUHDLQWKLFNVODEVDWLV¿HVVHUYLFHDELOLW\DQGVKHDUVWUHQJWKUHTXLUHPHQWV
1RWH7KHLQWKLFNVODEPXVWDOVREHFKHFNHGIRU¿UHUHVLVWDQFHUHTXLUHPHQWVEDVHGRQWKHUHTXLUHG¿UHUHVLVtance rating prescribed in the governing building code.
4.8.2
Example 4.2 – Determination of Minimum Slab Thickness: One-way Slab System, Building #2,
Lightweight Concrete
'HWHUPLQHLIDLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJLVDGHTXDWHDWWKHURRI
HTXDOWRSFI
DQG*UDGH
level assuming lightweight concrete with an equilibrium density,
UHLQIRUFHPHQW VHH)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the minimum slab thickness based on serviceability requirements
ACI 7.3.1.1
Because all span lengths are equal, minimum thickness based on a support condition of one end continuous governs:
Table 4.1
7KHUHIRUHDLQWKLFNVODEVDWLV¿HVVHUYLFHDELOLW\UHTXLUHPHQWV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Check shear strength requirements
ACI 7.5.1.1
• Step 2a – Determine the maximum factored distributed load
For a one-foot design strip:
Maximum
LVGHWHUPLQHGE\$&,(T F Table 3.3
• 6WHSE±'HWHUPLQHLIWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFH ACI 6.5.1
(1) Members are prismatic
Gravity loads are uniformly distributed
(3)
All spans are equal in length
7KHUHIRUHWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHG
• Step 2c – Determine the maximum factored shear force
Maximum RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUEHDP
ACI 6.5.4
Figure 4.3
where
is conservatively taken as the clear distance between the bottom instead of the top of the tapered ribs
VHH)LJXUH Although the critical section for shear is permitted to be taken a distance
FDVHLWLVFRQVHUYDWLYHO\WDNHQDWWKHIDFHRIWKHVXSSRUW $&, • Step 2d – Check the shear strength requirements
from the face of the support in this
ACI 7.5.1.1
Eq. (4.6)
Assuming
Eq. (4.7)
Table 4.2
$VVXPLQJPLQLPXPÀH[XUDOUHLQIRUFHPHQW
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For shear,
ACI Table 21.2.1
Also, it is evident that
Eq. (4.8)
7KHUHIRUHWKHLQWKLFNVODEVDWLV¿HVVKHDUVWUHQJWKUHTXLUHPHQWV
1RWH7KHLQWKLFNVODEPXVWDOVREHFKHFNHGIRU¿UHUHVLVWDQFHUHTXLUHPHQWVEDVHGRQWKHUHTXLUHG¿UHUHVLVtance rating prescribed in the governing building code.
4.8.3
Example 4.3 – Determination of Required Reinforcement: One-way Slab System, Building #2
'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWIRUWKHLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGDQG*UDGHUHLQIRUFHPHQW VHH
LQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWHZLWK
)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the factored bending moments along the span
ACI 6.5
,WZDVGHWHUPLQHGLQ6WHSERI([DPSOHWKDWWKHVLPSOL¿HGDQDO\VLVPHWKRGFDQEHXVHGWRGHWHUPLQHIDFWRUHG
bending moments and shear forces.
%HFDXVHWKHVODEVKDYHVSDQVOHVVWKDQIWWKHQHJDWLYHEHQGLQJPRPHQWVDWWKHFULWLFDOVHFWLRQV IDFHRIVXSSRUWV LVFDOFXODWHGLQ6WHSDRI([DPSOH
are equal to the following where
Figure 4.3
Similarly, the maximum positive bending moment occurs near midspan in an end span; this moment is used for all
spans:
Step 2 – Determine the required flexural reinforcement
ACI 7.4.1.2
Eq. (4.10)
Eq. (4.9)
For tension-controlled sections,
A summary of the required
ACI Table 21.2.2
LVJLYHQLQ7DEOHZKHUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 4.11 Summary of Required Flexural Reinforcement for the One-way Slab in Example 4.3
Location
Mu (ft-kips/ft)
Rn (psi)
As (in.2)
Face of support
Midspan
0LQLPXPÀH[XUDOUHLQIRUFHPHQW
Because
ACI 7.6.1.1
at all critical sections, use
Check if the sections are tension-controlled.
)RU*UDGHUHLQIRUFHPHQWDQG
Eq. (4.13)
Because
the sections are tension-controlled.
Step 3 – Select the reinforcing bars
Sect. 4.6.4
7U\EDUVVSDFHGDWLQRQFHQWHUIRUWKHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQW
Provided
required
Table 4.4
Check spacing requirements.
The minimum center-to-center spacing of the bars,
PD[LPXPDJJUHJDWHVL]H
LVHTXDOWRWKHJUHDWHURIWKHIROORZLQJDVVXPLQJDLQ
Figure 4.8
The maximum center-to-center spacing of the bars,
is equal to the minimum of
LQDQGWKHPLQLPXPRIWKHIROORZLQJDVVXPLQJDLQFRQFUHWHFRYHUDQG
Eq. (4.14)
7KHLQVSDFLQJRIWKHEDUVLVJUHDWHUWKDQWKHPLQLPXPVSDFLQJRILQDQGLVHTXDOWRWKHPD[LPXP
VSDFLQJRILQ
8VHEDUVVSDFHGDWLQRQFHQWHUIRUWKHQHJDWLYHDQGSRVLWLYHÀH[XUDOUHLQIRUFHPHQW
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Determine the shrinkage and temperature reinforcement
ACI 7.6.4.1
Minimum shrinkage and temperature reinforcement is equal to the following:
ACI 24.4.3.2
The maximum center-to-center spacing of the bars,
is equal to the following:
ACI 24.4.3.3
8VHEDUVVSDFHGDWLQRQFHQWHUIRUWKHVKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQW SURYLGHG
7KLVUHLQIRUFHPHQWLVSODFHGSHUSHQGLFXODUWRWKHÀH[XUDOUHLQIRUFHPHQW VHH)LJXUH
required
4.8.4
Example 4.4 – Determination of Lap Splice Lengths: One-way Slab System, Building #2
'HWHUPLQHWKHUHTXLUHGODSVSOLFHOHQJWKIRUWKHSRVLWLYHUHLQIRUFHPHQWLQWKHLQWKLFNVODEWKDWLVSDUWRIWKH
and
ZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWHZLWK
*UDGHUHLQIRUFHPHQW VHH)LJXUH 'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Determine the tension development length
ACI 25.4.2.4
Eq. (4.15)
For normalweight concrete,
Eq. (4.16)
)RU*UDGHUHLQIRUFHPHQW
Eq. (4.17)
For uncoated reinforcing bars,
Eq. (4.18)
)RUUHLQIRUFLQJEDUV
Eq. (4.19)
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Eq. (4.20)
Eq. (4.21)
There is no transverse reinforcement in this one-way slab, so
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4-30
Eq. (4.22)
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
PDWFKHVWKHYDOXHLQ7DEOH
The calculated value of
Step 2 – Determine the required lap splice length
ACI 25.5.2.1
$FFRUGLQJWRWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,DWOHDVWRQHTXDUWHURIWKHPD[LPXPSRVLWLYH
moment reinforcement must be used as structural integrity reinforcement and it must be continuous. Also, where
splices are needed, Class B tension lap splices must be provided for this reinforcement near the supports (joists).
Class B lap splice,
Eq. (4.31)
8VHDIWLQODSVSOLFHOHQJWKIRUWKHSRVLWLYHUHLQIRUFHPHQWLQWKLVRQHZD\VODE
4.8.5
Example 4.5 – Determination of Reinforcement Details: One-way Slab System, Building #2
'HWHUPLQHWKHUHLQIRUFHPHQWGHWDLOVIRUWKHLQWKLFNVODEWKDWLVSDUWRIWKHZLGHPRGXOHMRLVWV\VWHPLQ%XLOGLQJDWDW\SLFDOÀRRUDVVXPLQJQRUPDOZHLJKWFRQFUHWH
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
ͶΦᇳ 5HLQIRUFHPHQWGHWDLOVIRUWKLVRQHZD\VODEV\VWHPDUHJLYHQLQ)LJXUHZKLFKDUHEDVHGRQWKHLQIRUPDWLRQ
JLYHQLQ)LJXUHWKHGHWDLOVLQ)LJXUHFDQEHXVHGEHFDXVHWKHRQHZD\VODELQWKLVH[DPSOHLVVXEMHFWHGWR
uniformly distributed gravity loads.
͓͵̷ͳʹᇳ ሺ–›’Ǥ ሻ
Ͷᇱ ǦͲᇳ ͳᇱ Ǧʹᇳ ͳᇱ ǦͲᇳ ʹᇱ ǦͲᇳ –ƒ†ƒ”†
Š‘‘ሺ–›’Ǥሻ
͹ᇳ Ͷᇱ Ǧͷᇳ ͹ᇳ Ͷᇱ Ǧͷᇳ ͹ᇳ –Š‡””‡‹ˆ‘” ‡‡–‘–•Š‘™ˆ‘” Žƒ”‹–›
Figure 4.17 Reinforcement details for the one-way slab in Example 4.5.
Continuous positive reinforcement is provided because the span lengths are relatively small; this automatically satis¿HVRQHRIWKHVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVRI$&, VHH)LJXUH $OVRGHJUHHKRRNV
are provided at the ends of the positive reinforcement in the end span at the noncontinuous supports (joists), which
DOVRVDWLV¿HVRQHRIWKHVWUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWV$OORWKHUUHLQIRUFHPHQWLQWKHV\VWHPLVQRW
shown for clarity.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 5
TWO-WAY SLABS
5.1 Overview
Two-way slabs are elements in two-way construction where the slabs are designed to support loads through bending
LQWZRGLUHFWLRQV:KHUHWKHUDWLRRIWKHORQJWRWKHVKRUWVLGHRIDVODESDQHOLVRUOHVVORDGWUDQVIHULVE\EHQGLQJ
SUHGRPLQDWHO\LQWZRGLUHFWLRQVDQGWKHSDQHOLVGH¿QHGDVDWZRZD\VODE7KHPDLQÀH[XUDOUHLQIRUFHPHQWLQD
two-way slab system is parallel to the two orthogonal directions of load transfer.
'HVFULSWLRQVRIFRPPRQWZRZD\VODEV\VWHPVDUHJLYHQLQ7DEOH VHHDOVR)LJXUH Table 5.1 Two-way Slab Systems
System
Description
Flat plate
A two-way concrete slab supported directly on columns.
Flat slab
A two-way concrete slab supported directly on columns where the slab is
thickened around the columns; the thickened portions of the slab are called
drop panels.
Two-way beam-supported slab
A two-way concrete slab with column-line beams on all four sides of a
panel.
7ZRZD\MRLVW ZDIÀHVODE
Evenly spaced concrete joists spanning in both directions and a reinforced
concrete slab cast integrally with the joists; solid slab sections are provided
around the columns mainly to provide two-way shear resistance.
Flat plate voided concrete slab
A two-way concrete slab of uniform thickness containing regularly-spaced,
KROORZSODVWLFEDOOVPDGHRIKLJKGHQVLW\UHF\FOHGSRO\HWK\OHQH +'3( inside the concrete.
(a)
(b)
(c)
(d)
(e)
Figure 5.1 Two-way slab systems. (a) Flat plate. (b) Flat slab. (c) Two-way beam-supported slab.
(d) Two-way joist. (e) Flat plate voided concrete slab.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The design and detailing of solid, cast-in-place two-way slabs with nonprestressed reinforcement in buildings assigned to Seismic Design Category (SDC) A and B are covered in this chapter. Two-way slabs that are part of the
lateral force-resisting system (LFRS) in buildings assigned to SDC C are covered in Chapter 13 of this publication.
Provisions for two-way slabs are given in ACI Chapter 8.
5.2 Minimum Slab Thickness
5.2.1 Overview
7ZRZD\VODEVPXVWKDYHVXI¿FLHQWWKLFNQHVVVRDOODSSOLFDEOHVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG'HÀHFWLRQVDUHSHUPLWWHGWREHFDOFXODWHGLQDFFRUGDQFHZLWK$&,DQGVXEVHTXHQWO\FKHFNHGDJDLQVWWKH
OLPLWVLQ$&, $&, )RUVODEVZLWKRXWLQWHULRUEHDPVVSDQQLQJEHWZHHQVXSSRUWVRQDOOVLGHVZKHUH
in
WKHÀH[XUDOUHLQIRUFHPHQWKDVDVSHFL¿HG\LHOGVWUHQJWK JUHDWHUWKDQSVLWKHPRGXOXVRIUXSWXUH
instead of
VHH$&,DQG )RUHFRQRP\*UDGHUHLQIRUFHFDOFXODWLQJGHÀHFWLRQVLV
PHQWLVFRPPRQO\XVHGLQWZRZD\VODEV 5HIHUHQFH ,QOLHXRISHUIRUPLQJGHÀHFWLRQFRPSXWDWLRQVPLQLPXPVODEWKLFNQHVV can be obtained using the provisions in
$&,IRUQRQSUHVWUHVVHGVODEVZLWKRXWLQWHULRUEHDPVRULQ$&,IRUQRQSUHVWUHVVHGVODEVZLWKEHDPV
spanning between supports on all sides. Methods to determine minimum slab thicknesses for the two-way systems
LOOXVWUDWHGLQ)LJXUHDUHJLYHQEHORZ
7KHWKLFNQHVVRIDFRQFUHWHÀRRU¿QLVKLVSHUPLWWHGWREHLQFOXGHGLQ ZKHUHLWLVSODFHGPRQROLWKLFDOO\ZLWKWKHÀRRU
VODERUZKHUHWKHÀRRU¿QLVKLVGHVLJQHGWREHFRPSRVLWHZLWKWKHÀRRUVODELQDFFRUGDQFHZLWK$&, $&, Where single- or multiple-leg stirrups are used as part of the two-way shear resistance at critical sections in a twoZD\VODEWKHPLQLPXPVODEWKLFNQHVVUHTXLUHPHQWVLQ$&,DQGPXVWEHVDWLV¿HG,QSDUWLFXODUWKH
GLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQW must be at
least equal to the greater of the following:
(5.1)
In this equation, is the nominal diameter of the stirrups. For two-way slabs, an average can be approximated
which means the minimum WRVDWLVI\WKHUHTXLUHPHQWVRI$&,PXVWEHHTXDOWRWKH
as
This requirement essentially ensures the slab is thick enough so the stirrups
JUHDWHURILQDQG
considering the minimum inside bend diameters at the corners of the
FDQGHYHORSWKHVSHFL¿HG\LHOGVWUHQJWK
VWLUUXSV VHH$&, 5.2.2 Flat Plates
0LQLPXPVODEWKLFNQHVVIRUÀDWSODWHVLVSHUPLWWHGWREHGHWHUPLQHGXVLQJWKHSURYLVLRQVLQ$&,IRUH[WHULRU
DQGLQWHULRUSDQHOVZLWKRXWGURSSDQHOV VHH$&,7DEOHDQG7DEOH $VQRWHGSUHYLRXVO\*UDGHUHLQIRUFHPHQWLVRIWHQXVHGLQÀDWSODWHV\VWHPV
Table 5.2 Minimum Thickness of Flat Plates
Exterior Panels
fy (psi)
Interior Panels
Without Edge Beams
With Edge Beams*
* Slabs with beams between columns along exterior edges. Exterior panels are considered to be without edge beams where
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5-2
.
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In the expressions for minimum
is the clear span length in the long direction of the panel measured face-toIDFHRIVXSSRUWVLQLQFKHV0LQLPXPVODEWKLFNQHVVHVGHWHUPLQHGE\7DEOHPXVWEHJUHDWHUWKDQRUHTXDOWR
LQ>$&, D @
LQWKHIRRWQRWHRI7DEOHLVGH¿QHGDVWKHUDWLRRIWKHÀH[XUDOVWLIIQHVVRIWKHEHDPVHFWLRQWRWKH
The term
ÀH[XUDOVWLIIQHVVRIDZLGWKRIVODEERXQGHGODWHUDOO\E\FHQWHUOLQHVRIDGMDFHQWSDQHOVLIDQ\RQHDFKVLGHRIWKH
beam:
(5.2)
In this equation,
and
are the moduli of elasticity of the beam and slab concrete, respectively, which can be
FDOFXODWHGE\$&,(TXDWLRQ D ZKHUHWKHGHQVLW\ XQLWZHLJKW RIQRUPDOZHLJKWFRQFUHWHRUWKHHTXLOLELVEHWZHHQDQGOEIW3LQFOXVLYH>$&, D @
rium density of lightweight concrete,
(5.3)
FDQEHGHWHUPLQHGE\$&,(TXDWLRQ E >$&,
For normalweight concrete
E @
(5.4)
,Q(TXDWLRQV DQG construction,
and
have the units of pounds per square inch. For the typical case of monolithic
,QOLHXRIXVLQJ(TXDWLRQV RU LWLVSHUPLWWHGWRVSHFLI\
FRUGDQFHZLWK$&,
based upon testing of concrete mixtures in ac-
The terms and LQ(TXDWLRQ DUHWKHPRPHQWVRILQHUWLDRIWKHJURVVVHFWLRQVRIWKHEHDPDQGVODEDERXW
the centroidal axes, respectively.
݄
κଶ ‡–”‘‹†
Žƒ„ǡ‫ܫ‬௦ ‡ƒǡ‫ܫ‬௕ ݄௕ ‫ݕ‬௕ ܾ௪ ܾ௘ ൌ ܾ௪ ൅ ݄௕ ൑ ܾ௪ ൅ Ͷ݄
Figure 5.2 Effective beam and slab sections for computation of stiffness ratio, Įf , for edge beams.
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5-3
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A portion of the slab on the sides of a beam is permitted to be included when determining the section properties of
WKHVHFWLRQ $&, 7KHHIIHFWLYHVODEZLGWK permitted to be included in the determination of is the
OHVVHURIWKHIROORZLQJIRUDQHGJHEHDP VHH)LJXUH (5.5)
7KHWHUPVLQ(TXDWLRQ DUHGH¿QHGLQ)LJXUHIRUWKHFDVHZKHUHWKHEHDPSURMHFWVEHORZWKHVODE,QJHQeral, LVWKHJUHDWHURIWKHEHDPSURMHFWLRQDERYHRUEHORZWKHVODE VHH$&,)LJXUH5 7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQH and for an edge beam [Note: These equations are also
valid for interior beams where LVGHWHUPLQHGE\(TXDWLRQ @7KHWHUP is the distance from the bottom of
equal to that deterthe combined beam section [which is the beam plus the slab section with an effective width,
PLQHGE\(TXDWLRQ RU @WRWKHFHQWURLGRIWKHVHFWLRQ
TX OEIW Figure 5.3 Preliminary slab thickness for flat plate systems.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.3 Equations for Ib and Is for Edge and Interior Beams
yb
Ib
Is
)RUÀDWSODWHV\VWHPVZLWKHGJHEHDPVDSUHOLPLQDU\VODEWKLFNQHVVLVGHWHUPLQHGXVLQJWKHH[SUHVVLRQVLQ7DEOH
because the slab thickness, is not known at this stage; this assumption is checked after
assuming
has been calculated based on that slab thickness.
7KHWKLFNQHVVRIDÀDWSODWHLVXVXDOO\FRQWUROOHGE\WKHVHUYLFHDELOLW\UHTXLUHPHQWVLQ7DEOHIRUVSDQOHQJWKVLQ
WKHUDQJHRIWRIWDQGOLYHORDGVOHVVWKDQRUHTXDOWROEIW. Otherwise, two-way shear requirements typically govern the thickness. Shear stresses at edge and corner columns are particularly critical because of relatively
large bending moments transferred from the slab to the columns at these locations. Because two-way shear requirePHQWVDUHUHODWHGWRÀH[XUDOUHTXLUHPHQWVDFORVHGIRUPVROXWLRQIRUVODEWKLFNQHVVWKDWVDWLV¿HVERWKVHWVRIUHTXLUHPHQWVFDQQRWEHREWDLQHGXQOHVVVRPHVLPSOLI\LQJDVVXPSWLRQVDUHPDGH)LJXUHFDQEHXVHGWRGHWHUPLQH
DSUHOLPLQDU\VODEWKLFNQHVVIRUÀDWSODWHVFRQVLGHULQJWZRZD\VKHDUVWUHVVHV7KHLQIRUPDWLRQLQWKH¿JXUHLVEDVHG
on the following assumptions:
• 6TXDUHHGJHFROXPQVRIVL]H bending perpendicular to the edge with a three-sided critical section
• Column supports a tributary area
• Concrete is normalweight with
,QWKH¿JXUH is the total factored uniformly distributed load on the slab, which must include the factored selfZHLJKWRIWKHVODEWKLVZHLJKWFDQEHHVWLPDWHGXVLQJDVODEWKLFNQHVVEDVHGRQWKHH[SUHVVLRQVLQ7DEOH$
preliminary slab thickness, LVREWDLQHGE\DGGLQJLQWRWKHYDOXHRI DFTXLUHGIURP)LJXUHIRUDJLYHQ
and area ratio
For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUÀDWSODWHVZLWKHGJHEHDPVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRIWKLV
publication.
Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness
$&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHDELOLW\DQGRUVKHDUVWUHQJWKUHTXLUHPHQWV
:KHUHLWLVQRWYLDEOHWRLQFUHDVHWKHVODEWKLFNQHVVDQGRUWKHFROXPQVL]HWRVDWLVI\WZRZD\VKHDUUHTXLUHPHQWV
shear reinforcement or shear caps may be used. Information on common types of shear reinforcement is given in
6HFWLRQRIWKLVSXEOLFDWLRQ
5.2.3 Flat Slabs
0LQLPXPVODEWKLFNQHVVIRUÀDWVODEVLVSHUPLWWHGWREHGHWHUPLQHGXVLQJ$&,IRUH[WHULRUDQGLQWHULRU
SDQHOVZLWKGURSSDQHOV VHH$&,7DEOHDQG7DEOH 7KHH[SUHVVLRQVLQ7DEOHFDQQRWEHXVHGXQOHVV
WKHGLPHQVLRQVRIWKHGURSSDQHOVFRQIRUPWRWKHUHTXLUHPHQWVLQ$&,ZKLFKDUHLOOXVWUDWHGLQ)LJXUH
and
are the adjoining center-to-center span lengths (Note: The dimensional limitations of the
,QWKH¿JXUH
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UHLQIRUFHPHQWLVRIWHQXVHGLQÀDWVODEV\VWHPV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.4 Minimum Thickness of Flat Slabs
Exterior Panels
fy (psi)
Interior Panels
Without Edge Beams
With Edge Beams*
* Slabs with beams between columns along exterior edges. Exterior panels are considered to be without edge beams where
݄
൒ ͳǤʹͷ݄
.
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൒ κ஺ Ȁ͸
൒ κ஻ Ȁ͸
Figure 5.4 Dimensional requirements for drop panels.
/LNHLQWKHFDVHIRUÀDWSODWHV is the clear span length in the long direction of the panel measured face-to-face
RIVXSSRUWVLQLQFKHV0LQLPXPVODEWKLFNQHVVHVGHWHUPLQHGE\WKHH[SUHVVLRQVLQ7DEOHPXVWEHJUHDWHUWKDQRU
HTXDOWRLQ>$&, E @
7RDFKLHYHHFRQRPLFDOIRUPZRUNVWDQGDUGOXPEHUGLPHQVLRQVVKRXOGEHXVHGWRIRUPGURSSDQHOV 5HIHUHQFHV
DQG 'URSSDQHOKHLJKWV WKDWLVWKHGHSWKVRIWKHSURMHFWLRQVEHORZWKHVODE EDVHGRQDFWXDOOXPEHUGLPHQVLRQV
DQGLQWKLFNIRUPZRUNVKHDWKLQJDUHJLYHQLQ7DEOH)RUPZRUNFRVWVXQQHFHVVDULO\LQFUHDVHLIGURSSDQHO
KHLJKWVRWKHUWKDQWKRVHLQ7DEOHDUHVSHFL¿HG
Nominal
[
[
6x
8x
݄
Table 5.5 Drop Panel Height for Formwork Economy
Drop panel height
Actual lumber dimension
¾ԣ formwork sheathing ሺtyp.ሻ
Lumber Size
Lumber Size
Actual
Nominal
Actual
[
òƎ
[
òƎ
[
òƎ
[
óƎ
Drop Panel
Height
óƎ
óƎ
óƎ
Ǝ
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5-6
Drop Panel Height
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUÀDWVODEVZLWKHGJHEHDPVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRIWKLV
publication.
Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness
$&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHDELOLW\DQGRUVKHDUUHTXLUHPHQWV
5.2.4 Two-way Beam-Supported Slabs
0LQLPXPWKLFNQHVVUHTXLUHPHQWVIRUVODEVLQWZRZD\EHDPVXSSRUWHGVODEV\VWHPVDUHJLYHQLQ$&,DQG
is the average value of
for all beams on the
DUHVXPPDUL]HGLQ7DEOH VHH$&,7DEOH 7KHWHUP
HGJHVRIDSDQHO7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHVHFWLRQSURSHUWLHVRIWKHFRPELQHGEHDP
of the slab is determined
section and the slab for edge and interior beams. For edge beams, the effective width,
E\(TXDWLRQ DQGIRULQWHULRUEHDPV LVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ VHH$&,DQG)LJXUH
for the case of an interior beam projecting below the slab):
(5.6)
In general,
LVWKHJUHDWHURIWKHEHDPSURMHFWLRQDERYHRUEHORZWKHVODE VHH$&,)LJXUH5 Table 5.6 Minimum Thickness of Two-way Slabs with Beams Spanning Between Supports on All Sides
Įfm
Minimum Thickness, h
κଶ ‡–”‘‹†
݄
Žƒ„ǡ‫ܫ‬௦ ‡ƒǡ‫ܫ‬௕ ݄௕ ‫ݕ‬௕ ACI 8.3.1.1 applies
VHH7DEOHVDQG
ܾ௪ ܾ௘ ൌ ܾ௪ ൅ ʹ݄௕ ൑ ܾ௪ ൅ ͺ݄
Figure 5.5 Effective beam and slab sections for computation of stiffness ratio, Įf , for interior beams.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,QWKHH[SUHVVLRQVLQ7DEOH is the clear span in the long direction measured face-to-face of beams in inches
and is the ratio of the long-to-short dimensions of the clear spans in a panel. Also, has the units of pounds per
square inch.
the minimum
In edge panels with edge beams where the stiffness ratio
7DEOHPXVWEHLQFUHDVHGE\DWOHDVWSHUFHQW $&, determined by the equations in
For the usual case of continuous construction, it is typical for to be the same for all spans and for it to be determined on the basis of the span yielding the largest minimum depth; this results in economical formwork (see ReferHQFH )RUWZRZD\EHDPVXSSRUWHGVODEVPLQLPXPGHSWKUHTXLUHPHQWVIRUWKHEHDPVDUHJLYHQLQ6HFWLRQRI
this publication.
Fire-resistance requirements of the general building code must also be considered when specifying a slab thickness
$&, )LUHUHVLVWDQFHUHTXLUHPHQWVDUHXVXDOO\VDWLV¿HGE\SURYLGLQJDVODEWKLFNQHVVWKDWVDWLV¿HVVHUYLFHability requirements.
5.2.5 Two-way Joists
‘‡†‡’–Šǡ ݄௥ Žƒ„–Š‹ ‡••ǡ ݄
General requirements for nonprestressed two-way joist systems are given in ACI 8.8. This type of system is formed
E\DQGLQZLGHGRPHVUHVXOWLQJLQDQGIWPRGXOHVUHVSHFWLYHO\ VHH)LJXUH 6ODEWKLFNness, LVW\SLFDOO\FRQWUROOHGE\¿UHUHVLVWDQFHUHTXLUHPHQWVDQGWKHMRLVW ULE ZLGWK is set by the dome width,
WKDWLVVSHFL¿HG7KHRQO\GHSWKWREHGHWHUPLQHGWRVDWLVI\VHUYLFHDELOLW\UHTXLUHPHQWVLVWKHGRPHGHSWK
6WDQGDUGIRUPGLPHQVLRQVIRUWZRZD\MRLVWFRQVWUXFWLRQDUHJLYHQLQ7DEOH
‘‹•–™‹†–Šǡ ܾ௥ ͳʹ
ͳ
‘‡™‹†–Šǡ ܾௗ Figure 5.6 Cross-section of a two-way joist system.
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5-8
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.7 Standard Form Dimensions for Two-way Joist Construction
Dome Width, bd (in.)
Dome Depth, hr (in.)
Joist Width, br (in.)
6
8
)RUGHVLJQSXUSRVHVDWZRZD\MRLVWV\VWHPLVFRQVLGHUHGWREHDÀDWVODEZLWKWKHVROLGKHDGVDWWKHFROXPQVDFWLQJ
as drop panels. The cross-section of the two-way joist system is transformed into an equivalent section of uniform
thickness,
with the same width as the actual section, which is equal to
VHH)LJXUH 7KHHTXLYDOHQW
is determined by setting the moment of inertia of the equivalent section equal to the
uniform slab thickness,
gross moment of inertia of the actual two-way joist section,
(5.7)
Solving for
(5.8)
݄
Equations to determine IRUDWZRZD\MRLVWV\VWHPDUHJLYHQLQ7DEOHIRUWKHFDVHZKHUHWKHVLGHIDFHVRIWKH
GRPHVDUHVORSHG VHH)LJXUH ݄௥ ‡–”‘‹†
ͳʹ
ܾ௥ Τʹ
‫ݕ‬௕ ͳ
ܾௗ Figure 5.7 Determination of the gross moment of inertia, Ig for a two-way joist system.
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5-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.8 Equations to Determine Ig for Two-way Joist Systems
yb
Ig
(TXLYDOHQWVODEWKLFNQHVVHVIRUDWZRZD\MRLVWV\VWHPZLWKDIWPRGXOHDLQZLGHMRLVWDQGDLQWKLFNVODE
DUHJLYHQLQ7DEOH
Table 5.9 Equivalent Slab Thickness, te, for a Two-way Joist System with a 3-ft Module
Dome Depth, hd (in.)
Equivalent Slab Thickness, te (in.)
8
8.8
13.1
16
To satisfy serviceability requirements, an overall depth
must be provided such that the corresponding
equivalent thickness, ZKLFKLVGHWHUPLQHGE\(TXDWLRQ >RUZKHUHDSSOLFDEOHE\7DEOH@LVJUHDWHUWKDQ
RUHTXDOWRWKHPLQLPXPUHTXLUHGRYHUDOOGHSWKGHWHUPLQHGE\WKHDSSOLFDEOHH[SUHVVLRQLQ7DEOHXVLQJWKHORQFor economy in formwork, the largest required overall depth from all the spans should be
gest clear span length,
XVHGZKHUHYHUSRVVLEOH 5HIHUHQFH 7ZRZD\MRLVWV\VWHPVQRWVDWLVI\LQJWKHOLPLWDWLRQVRI$&,WKURXJKPXVWEHGHVLJQHGDVVODEVDQG
beams (ACI 8.8.1.8).
5.2.6 Flat Plate Voided Concrete Slabs
(PSLULFDOVHUYLFHDELOLW\UHTXLUHPHQWVIRUÀDWSODWHYRLGHGFRQFUHWHVODEVV\VWHPVDUHQRWVSHFL¿FDOO\SURYLGHGLQ$&,
%HFDXVHÀDWSODWHYRLGHGFRQFUHWHVODEV\VWHPVDUHYHU\VLPLODUWRWZRZD\VODEDQGWZRZD\MRLVWV\VWHPVWKH
IRUVODEVZLWK*UDGHUHLQIRUFHPHQW VHH7DEOH overall thickness can be initially estimated by
,QOLHXRIXVLQJDQHVWLPDWHGRYHUDOOVODEWKLFNQHVVLWLVUHFRPPHQGHGGHÀHFWLRQFDOFXODWLRQVEHSHUIRUPHGLQ
DFFRUGDQFHZLWK$&,,QDGGLWLRQWRIDFWRUVDFFRXQWLQJIRUFRQFUHWHFUDFNLQJWKHDQDO\VLVPXVWLQFOXGHWKH
stiffness reduction factor, which accounts for the presence of the voids in the slab; such factors can be found in the
PDQXIDFWXUHUV¶OLWHUDWXUH RQDYHUDJHWKLVIDFWRULVHTXDOWRDERXW &DOFXODWHGGHÀHFWLRQVPXVWEHOHVVWKDQRU
HTXDOWRWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQVLQ$&,7DEOH $&, The largest required slab thickness from all the panels should be used wherever possible for overall economy. AdGLWLRQDOO\WKHVDPHVL]HYRLGIRUPHUVVKRXOGEHXVHGDVRIWHQDVSUDFWLFDOWKURXJKRXWWKHSURMHFWWRIDFLOLWDWHSODFHPHQWLQWKH¿HOG
'HVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUÀDWSODWHYRLGHGFRQFUHWHVODEV\VWHPVFDQEHIRXQGLQ5HIHUHQFH
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5-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.3 Required Strength
5.3.1 Analysis Methods
7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUPXVW
EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWK VHH$&,DQGUHVSHFWLYHO\ 7KHIROORZLQJDQDO\VLVPHWKRGV
DUHUHOHYDQWWRWZRZD\VODEV $&,DQG (1) /LQHDUHODVWLF¿UVWRUGHUDQDO\VLV $&,
/LQHDUHODVWLFVHFRQGRUGHUDQDO\VLV $&,
(3) Inelastic analysis (ACI 6.8)
Finite element analysis (ACI 6.9)
'LUHFWGHVLJQPHWKRG ''0 >$&, D @
(6) (TXLYDOHQWIUDPHPHWKRG ()0 >$&, E @
For purposes of analysis, two-way slab panels, which are bounded by columns, beams, or wall centerlines on all
VLGHV $&, DUHSHUPLWWHGWREHGLYLGHGLQWRFROXPQVVWULSVDQGPLGGOHVWULSVWKHGH¿QLWLRQVRIZKLFKDUH
JLYHQLQ$&,DQGUHVSHFWLYHO\7KHZLGWKVRIWKHVHVWULSVGHSHQGRQWKHVSDQOHQJWKV and
is the length of the span in the direction moments are being determined, measured center-to-center of supwhere
is the length of the span in the direction perpendicular to
measured center-to-center of supports
ports, and
VHH)LJXUH $Q\FROXPQOLQHEHDPVSUHVHQWDUHWREHLQFOXGHGLQWKHFROXPQVWULS $&, ሺκଶ ሻ஺ κଵ ሺκଶ ሻ஻ ‫ݔ‬ଵ ‫݌݅ݎݐݏ݈݂݈݁݀݀݅݉ܽܪ‬
‫ݔ‬ଶ ‫ݔ‬ଶ ‫݊݉ݑ݈݋ܥ‬
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‫ܘܑܚܜܛܖ܏ܑܛ܍܌‬
‫ݔ‬ଵ ൌ Ž‡••‡”‘ˆ κଵ ΤͶ ‘”ሺκଶ ሻ஺ ȀͶ
‫ݔ‬ଶ ൌ Ž‡••‡”‘ˆ κଵ ΤͶ ‘”ሺκଶ ሻ஻ ȀͶ
Figure 5.8 Design strips, column strips, and middle strips in a two-way slab system.
For slabs that are part of the LFRS, the results from a gravity load analysis are permitted to be combined with those
IURPDODWHUDOORDGDQDO\VLVXVLQJWKHDSSOLFDEOHORDGFRPELQDWLRQVLQ$&,7DEOH $&, 7KH''0DQGWKH()0DUHQRWLQFOXGHGLQ$&,3URYLVLRQVIRUWKHVHPHWKRGVDUHJLYHQLQWKHWKURXJK
HGLWLRQVRI$&,7KH''0LVFRYHUHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHLQIRUPDWLRQLQWKDWVHFWLRQ
LVSULPDULO\IURP$&,LQ$&,
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5-11
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.3.2 Critical Sections for Flexure
Factored bending moments,
are permitted to be calculated at the faces of the supports for two-way slabs built
LQWHJUDOO\ZLWKWKHVXSSRUWV $&, )RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVLQWKHGHVLJQVWULSVRFFXUDWWKH
faces of the supports where negative moments are maximum and in the span where positive moments are maximum.
Studies of moment transfer between slabs and columns have shown that the factored slab moment resisted by the
GXHWRJUDYLW\ODWHUDODQGRURWKHUORDGHIIHFWVLVWUDQVIHUUHGE\DFRPELQDWLRQRIÀH[XUH
column at a joint,
DQGHFFHQWULFLW\RIVKHDU5HTXLUHPHQWVIRUWUDQVIHUE\ÀH[XUHDQGWUDQVIHUE\HFFHQWULFLW\RIVKHDUDUHJLYHQLQ$&,
DQGUHVSHFWLYHO\
7KHIUDFWLRQRIWKHIDFWRUHGVODEPRPHQWUHVLVWHGE\WKHFROXPQWUDQVIHUUHGE\ÀH[XUHLVHTXDOWR
factor LVFDOFXODWHGE\$&,(TXDWLRQ >$&,@
where the
(5.9)
In this equation, and are the dimensions of the critical section for two-way shear measured parallel and perpendicular to the direction of analysis, respectively. The critical section for two-way shear is located so its perimis a minimum where the perimeter need not approach closer than
to (1) edges or corners of columns,
eter,
FRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVRU FKDQJHVLQVODEWKLFNQHVVVXFKDVHGJHVRIFROXPQFDSLWDOVGURSSDQHOV
RUVKHDUFDSV $&, )RUVTXDUHRUUHFWDQJXODUFROXPQVFRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVWKHFULWLFDO
VHFWLRQLVSHUPLWWHGWREHGH¿QHGXVLQJVWUDLJKWVLGHV $&, )RUFLUFXODURUSRO\JRQVKDSHGFROXPQV
WKHFULWLFDOVHFWLRQLVSHUPLWWHGWREHGH¿QHGE\DVTXDUHFROXPQWKDWKDVWKHVDPHDUHDDVWKHDFWXDOFROXPQ $&,
3URSHUWLHVRIFULWLFDOVHFWLRQVDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
to be used to resist
$FFRUGLQJWR$&,WKHHIIHFWLYHVODEZLGWK
two-way slab systems without drop panels or shear caps:
is equal to the following for
(5.10)
ܿଶ ܿଵ ݀ Τʹሺ–›’Ǥሻ
ܾଵ ܾଶ ܾ௦௟௔௕ ൌ ܿଶ ൅ ͵݄
ߛ௙ ‫ܯ‬௦௖ ‘Ž—•–”‹’™‹†–Š
݄
ܾଵ ൌ ܿଵ ൅ ݀ Τʹ
ܾଶ ൌ ܿଶ ൅ ݀
ܾ௢ ൌ ʹܾଵ ൅ ܾଶ Figure 5.9 Effective slab width for transfer of moment at an edge column in a flat plate system.
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5-12
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For two-way systems with drop panels or shear caps,
is equal to the following:
(5.11)
In these equations, is the dimension of the column in the direction perpendicular to the direction of analysis,
is the thickness of the slab, and is the thickness of the drop panel or shear cap projection below the slab. For
columns with capitals, the width of the capital is to be used in these equations instead of
7KHHIIHFWLYHVODEZLGWKVIRUDQHGJHFROXPQLQDÀDWSODWHV\VWHPDQGIRUDQLQWHULRUFROXPQLQDÀDWSODWHV\VWHP
ZLWKDVKHDUFDSDUHLOOXVWUDWHGLQ)LJXUHVDQGUHVSHFWLYHO\
κ
ܿଵ κ
ܿଶ ܿଶ ൅ ͵ሺ݄ ൅ ݄ଵ ሻ
κ ൅ ͵݄
݄ଵ ݄
ܾ௦௟௔௕ ൌ Ž‡••‡”‘ˆ ൝
Š‡ƒ” ƒ’
ሺκ െ ܿଶ ͿȀʹ
Figure 5.10 Effective slab width for transfer of moment at an interior column
in a flat plate system with a shear cap.
The factor LVSHUPLWWHGWREHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHEDVHGRQWKHOLPLWDWLRQVJLYHQLQWKDW
in the table are the nomiWDEOHIRUGLIIHUHQWFROXPQORFDWLRQV $&,VHH7DEOH 7KHWHUPV and
QDOVKHDUVWUHQJWKSURYLGHGE\WKHFRQFUHWH $&, DQGWKHIDFWRUHGVKHDUVWUHVVRQWKHVODEFULWLFDOVHFWLRQIRU
two-way action, from the controlling load combination, without moment transfer, respectively. Information on how
WRFDOFXODWHWKHVHVWUHVVHVLVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
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5-13
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.10 Maximum Modified Values of Ȗf for Nonprestressed Two-way Slabs
vuv
İt within bslab
Maximum Modified Ȗf
Column Location
Span Direction
Corner column
Either direction
Perpendicular to the edge
Edge column
Parallel to the edge
Interior column
Either direction
,QRUGHUWREHDEOHWRXVHWKHPD[LPXPPRGL¿HGYDOXHVRI LQ7DEOHWKHQHWWHQVLOHVWUDLQLQWKHH[WUHPH
IRUWKHÀH[XUDOUHLQIRUFHPHQWLQ
must be
layer of longitudinal tension reinforcement at nominal strength,
is the minimum strain corresponding
greater than or equal to the strains indicated in the table. The strain
is the value of the net tensile strain in the extreme layer of longito a tension-controlled section where
WXGLQDOWHQVLRQUHLQIRUFHPHQWXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQ VHH$&,7DEOH results in a decrease in
which is the factor used to determine the fraction of the factored slab
Increasing
>VHH$&,(TXDWLRQ @,W
moment resisted by the column transferred by eccentricity of shear:
results in a decrease in the moment
and a corresponding decrease in factored
is evident that a decrease in
shear stresses, which may be advantageous in certain situations.
KDVEHHQGHWHUPLQHGWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWZLWKLQ
can be determined using
Once
(TXDWLRQV DQG LQ6HFWLRQRIWKLVSXEOLFDWLRQ)OH[XUDOUHLQIRUFHPHQWPD\QHHGWREHFRQFHQWUDWHG
to resist
$&, 7KLVFDQEHDFKLHYHGE\SURYLGLQJDSRUWLRQRIWKHQHJDWLYHFROXPQ
within
or by adding more reinforcement within
strip reinforcement at a closer spacing within
A reversal of moment can occur at slab-column joints that are part of the LFRS. In such cases, both top and bot$&,5UHFRPPHQGVDUDWLRRIWRSWRERWWRP
WRPÀH[XUDOUHLQIRUFHPHQWPXVWEHFRQFHQWUDWHGZLWKLQ
UHLQIRUFHPHQWRIDSSUR[LPDWHO\LQVXFKFDVHV
5.3.3 Critical Sections for Shear
One-way Shear
Factored shear forces,
are permitted to be calculated at the faces of the supports for two-way slabs built inteJUDOO\ZLWKWKHVXSSRUWV $&, Two-way slabs are permitted to be designed for the factored shear force at a critical section located a distance
IURPWKHIDFHRIWKHVXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG VHH)LJXUH (1) Support reaction, in direction of applied shear, introduces compression into the end region of the slab
Loads are applied at or near the top surface of the slab
(3) No concentrated load occurs between the face of the support and the critical section
The critical section for one-way shear must be taken at the face of the support where one or more of the three condiWLRQVLQ$&,DUHQRWPHW
Two-way Shear
The critical section for two-way shear is located so its perimeter,
is a minimum where the perimeter need not apWR HGJHVRUFRUQHUVRIFROXPQVFRQFHQWUDWHGORDGVRUUHDFWLRQDUHDVRU FKDQJHVLQ
proach closer than
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5-14
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
κଵ ”‹–‹ ƒŽ•‡ –‹‘
ܿଶ κଶ ܿଵ ݀
”‹„—–ƒ”›ƒ”‡ƒ
Figure 5.11 Critical section for one-way shear in a two-way slab that satisfies the conditions of ACI 8.4.3.2.
κଵ ܿଵ ൅ ݀
”‹–‹ ƒŽ•‡ –‹‘
ܿଶ ܿଶ ൅ ݀
κଶ ܿଵ ݀Ȁʹሺ–›’Ǥሻ
”‹„—–ƒ”›ƒ”‡ƒ
Figure 5.12 Critical section for two-way shear in a flat plate.
$FFRUGLQJWR$&,DFULWLFDOVHFWLRQZLWKVWUDLJKWVLGHVLVSHUPLWWHGIRUVTXDUHRUUHFWDQJXODUFROXPQV
concentrated loads, or reaction areas. The locations of the critical sections for two-way shear at an interior column
LQÀDWSODWHDQGÀDWVODEV\VWHPVDUHJLYHQLQ)LJXUHVDQGUHVSHFWLYHO\)RUÀDWVODEVVKHDUUHTXLUHPHQWV
from the face of the column and
from the face of the drop panel beneed to be checked at a distance
and are the average distances from the extreme comcause shear failure can occur at either location where
SUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWGH¿QHGLQ)LJXUH
A shear cap is a projection below the slab similar to a drop panel; unlike a drop panel, it is used exclusively to inFUHDVHWKHVODEVKHDUVWUHQJWKRIWKHVODE$FFRUGLQJWR$&,VKHDUFDSVPXVWH[WHQGKRUL]RQWDOO\IURPWKHIDFH
RIWKHFROXPQDGLVWDQFHHTXDOWRDWOHDVWWKHWKLFNQHVVRIWKHSURMHFWLRQEHORZWKHVODE VHH)LJXUHIRUWKHFDVH
of an interior column). Like slabs with drop panels, shear requirements must be checked at two locations.
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5-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ሺκȀ͵ሻ ൅ ݀ଵ κȀ͵
ܿଶ ܿଶ ൅ ݀ଶ κȀ͵
ሺκȀ͵ሻ ൅ ݀ଵ ܿଵ ൅ ݀ଶ ܿଵ ”‹–‹ ƒŽ
•‡ –‹‘•
݀ଶ ݀ଵ ”‘’’ƒ‡Ž
Figure 5.13 Critical sections for two-way shear in a flat slab.
ܿଵ ൅ ʹܾ௖௔௣ ൅ ݀ଵ ܿଵ ൅ ݀ଶ ܿଶ ܿଶ ൅ ݀ଶ ܿଶ ൅ ʹܾ௖௔௣ ൅ ݀ଵ ܿଵ ”‹–‹ ƒŽ•‡ –‹‘•
݀ଵ ݄ଵ ݄
݀ଶ Š‡ƒ” ƒ’
ܾ௖௔௣ ൒ ݄ଵ Figure 5.14 Critical sections for two-way shear at a shear cap.
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5-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ ܦ‬൅ ݀
‫ܦ‬
”‹–‹ ƒŽ•‡ –‹‘
݀
‘Ž— ƒ’‹–ƒŽ
Figure 5.15 Critical section for two-way shear at a column capital.
ሺκȀ͵ሻ ൅ ݀ଵ κȀ͵
‫ ܦ‬൅ ݀ଶ κȀ͵
ሺκȀ͵ሻ ൅ ݀ଵ ‫ܦ‬
”‹–‹ ƒŽ•‡ –‹‘•
݀ଶ ݀ଵ ”‘’’ƒ‡Ž
‘Ž— ƒ’‹–ƒŽ
Figure 5.16 Critical sections for two-way shear at a column capital and drop panel.
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5-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHFULWLFDOVHFWLRQIRUDQLQWHULRUFROXPQZLWKDFLUFXODUFDSLWDOLVLOOXVWUDWHGLQ)LJXUH%HFDXVHWKHFDSLWDOLV
part of the column and not part of the slab, shear requirements need only be checked at the critical section located
IURPWKHIDFHRIWKHFDSLWDO,QFDVHVZKHUHDFROXPQFDSLWDODQGGURSSDQHODUHERWKXWLOL]HGWKH
a distance
from the face of the capital and at the
shear strength must be checked at the critical section located a distance
IURPWKHIDFHRIWKHGURSSDQHO VHH)LJXUH critical section located a distance
&ULWLFDOVKHDUSHULPHWHUVDUHQRWDVFOHDUO\GH¿QHGLQFDVHVZKHUHWKHVODEHGJHVFDQWLOHYHUEH\RQGWKHIDFH V RIDQ
HGJHFROXPQ7KHIROORZLQJJHQHUDOJXLGHOLQHVFDQEHXVHGEDVHGRQWKHSURYLVLRQVRI$&,ZKLFKUHTXLUHV
be a minimum.
the perimeter of the critical section,
&RQVLGHUWKHHGJHFROXPQLQ)LJXUH7KHFULWLFDOVKHDUSHULPHWHUIRUWKLVFROXPQLVHLWKHUWKUHHVLGHGRUIRXUVLGed depending on the length of the cantilever (overhang), The following equation can be used to obtain minimum
(5.12)
Similar equations can be derived for corner columns.
ܿଵ ܿଶ ܿଶ ൅ ݀
ܿଵ ൅ ݀
ܿଵ ܿଶ ܿଵ ൅ ݀Ȁʹ
ܿଶ ൅ ݀
‫ ݔ‬൑ ܿଶ Ȁʹ ൅ ݀
‫ ݔ‬൐ ܿଶ Ȁʹ ൅ ݀
Žƒ„‡†‰‡
Žƒ„‡†‰‡
Š”‡‡Ǧ•‹†‡† ”‹–‹ ƒŽ’‡”‹‡–‡”
‘—”Ǧ•‹†‡† ”‹–‹ ƒŽ’‡”‹‡–‡”
Figure 5.17 Critical section perimeters for edge columns with slab cantilevers.
ܿଵ ൅ ݀
ܿଵ ൌ ξߨ ‫ܦ‬Τʹ
‫ܦ‬
݀Ȁʹ
݀Ȁʹ
ܿଵ ܿଵ ൅ ݀
”‹–‹ ƒŽ
•‡ –‹‘
Figure 5.18 Critical section for two-way shear in slabs with circular columns.
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5-18
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUFLUFXODURUUHJXODUSRO\JRQVKDSHGFROXPQVWKHFULWLFDOVHFWLRQVIRUWZRZD\VKHDUDUHSHUPLWWHGWREHGH¿QHG
DVVXPLQJDVTXDUHFROXPQWKDWKDVWKHVDPHDUHDDVWKHFLUFXODURUSRO\JRQVKDSHGFROXPQ $&, )RUD
WKHVL]HRIWKHVTXDUHFROXPQ is equal to the following:
circular column of diameter
(5.13)
The dimensions of the critical section in this case are
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FROXPQ 1RWHWKDWWKLVSURYLVLRQVSHFL¿FDOO\UHIHUVWRFLUFXODUFROXPQVDQGQRWFLUFXODUFROXPQFDSLWDOVWKHFULWLFDO
VHFWLRQIRUWKHODWWHULVFLUFXODUDVVKRZQLQ)LJXUHVDQG
Ž‘•‡†•–‹””—’ሺ–›’Ǥሻ
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݀ Τʹ
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ሺ–›’Ǥ ሻ
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•–‹””—’•ሺ–›’Ǥሻ
൑ ʹ݀
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”‹–‹ ƒŽ•‡ –‹‘
݀Ȁʹ
݀Ȁʹ
൑ ݀ Τʹ
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Figure 5.19 Critical sections for two-way shear in slabs reinforced with stirrups.
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5-19
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 5.20 Critical sections for two-way shear in slabs reinforced with headed shear stud reinforcement.
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RXWHUPRVWSHULSKHUDOOLQHRIVKHDUUHLQIRUFHPHQWPXVWDOVREHFRQVLGHUHG,QRUGHUWRPLQLPL]H the shape of this
critical section must be a polygon. The locations of the critical sections at interior, edge, and corner columns for
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5-20
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܾଵ ‫ݒ‬௨|஺஻
ܿଵ ‫ݒ‬௨|஼஽
Critical section
ܾଶ A
ܿଶ D
C ܿ ܿ B
஺஻
஼஽
ܿ஼஽ ܿ஺஻ Centroid
Interior Column
ܾଵ ܿଵ ܿ஼஽ ‫ݒ‬௨|஼஽
‫ݒ‬௨|஺஻
ܾଶ ܿଶ C
Critical section
A
D
B
ܿ஼஽ ܿ஺஻ ܿ஺஻ Centroid
Edge Column
Figure 5.21 Assumed distribution of shear stresses due to direct factored shear and eccentricity of shear.
The assumed distributions of shear stresses due to the combination of direct factored shear,
and eccentricity of
on the faces of the critical section for an interior column and an edge column bending perpendicular
shear,
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The total factored shear stresses on faces AB and CD of the critical section can be determined by the following
equations:
(5.14)
where
area of concrete section resisting shear transfer
perimeter of critical section for two-way shear
distance from centroid of critical section to face AB
distance from centroid of critical section to face CD
property of critical section analogous to the polar moment of inertia
Section Properties of Critical Sections
Section properties of the critical section for two-way shear for rectangular columns, which are applicable to critical
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sections located a distance
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5-21
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.11 Section Properties of the Critical Section for Rectangular Columns
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‫ܥ‬
‫ܣ‬
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†‰‡
Case
1
Section Property
2
Case 1: Interior rectangular column
Case 2: Edge rectangular column bending parallel to the edge
Table 5.11 Section Properties of the Critical Section for Rectangular Columns (cont.)
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Case
Section Property
3
4
Case 3: Edge rectangular column bending perpendicular to the edge
Case 4: Corner rectangular column bending perpendicular to the edge
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5-22
†‰‡
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Table 5.12 Section Properties of Polygon-Shaped Critical Sections
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‫ܤ‬
ܾଶ ܾଵ Case
Section Property
1
2
Case 1: Interior critical section with a rectangular column
Case 2: Edge critical section with a rectangular column bending parallel to the edge
(table continued on next page)
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sections with irregular geometries, such as polygon-shaped critical sections or critical sections affected by slab
RSHQLQJV 5HIHUHQFH (5.15)
(5.16)
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5-23
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.12 Section Properties of Polygon-Shaped Critical Sections (cont.)
ܾଵ ܿ஼஽ †‰‡
ܿ஺஻ ܿଵ ܾଶ †‰‡
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κଶ ሺܾଶ ሻ௣ ܿ஺஻ ܿ஺஻ ξʹሺܾଵ െ ܿଵ െ ܿ஺஻ ሻ
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ሺܾଵ ሻ௣ Case
Section
Property
3
4
Case 3: Edge critical section with a rectangular column bending perpendicular to the edge
Case 4: Corner critical section with a square column bending perpendicular to the edge
The summation occurs over the number of straight segments in the critical section where each segment has a length
are the coordinates of the ends of each straight segment along the and
equal to The terms
axes, respectively, with respect to the centroid of the critical section. The section properties LQ7DEOHDUH
FRQFHSWXDOO\GLIIHUHQWWKDQWKRVHLQ7DEOHGHWHUPLQLQJ for irregular critical sections using the methodology
to determine IRUUHJXODUO\VKDSHGFULWLFDOVHFWLRQVLVYHU\GLI¿FXOW7KHVHFWLRQSURSHUWLHVGHWHUPLQHGE\(TXDWLRQV DQG IRUDQ\VKDSHGFULWLFDOVHFWLRQDUHFRQVHUYDWLYH WKDWLVWKH\DUHVPDOOHUWKDQWKRVHGHWHUPLQHGE\WKHPHWKRGRORJ\XVHGIRUWKHUHJXODUO\VKDSHGFULWLFDOVHFWLRQVLQ7DEOH &DOFXODWLRQVIRUSRO\JRQ
shaped critical sections for two-way slabs with closed stirrups and headed shear stud reinforcement are given in
([DPSOHVDQGUHVSHFWLYHO\
Information on how to determine the critical section properties for rectangular corner columns is given in Reference
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5-24
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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corner circular columns, the critical section consists of a partial circular section plus two straight tangential segments perpendicular to and extending to the edge(s) of the slab. It is assumed the column is set back a distance equal
from the edge(s), which provides conservative section properties for columns with larger setbacks. Similar
to
section properties can be derived for circular columns at the edge(s) of the slab.
Table 5.13 Section Properties of the Critical Section for Circular Columns
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Section Property
1
2
Case 1: Interior circular column
Case 2: Edge circular column bending parallel to the edge
(table continued on next page)
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5-25
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.13 Section Properties of the Critical Section for Circular Columns (cont.)
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‫ ܦ‬൅ ݀
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Case
Section Property
3
4
Case 3: Edge circular column bending perpendicular to the edge
Case 4: Corner circular column bending perpendicular to the edge
5.3.4 Direct Design Method
Overview
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The DDM method gives reasonably conservative values of bending moments at the critical sections for two-way
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must be carried in each direction.
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5-26
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
κଵ ൒ ʹκଵ Ȁ͵
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Figure 5.22 Limitations of the Direct Design Method.
Determination of Factored Bending Moments in a Design Strip
The three fundamental steps in the DDM used to determine factored bending moments in a design strip are as follows:
in each span.
• Step 1: Determine the total factored static moment,
in a span is determined by the following equation:
The total factored static moment,
(5.17)
where
total factored uniformly distributed gravity load on the slab
clear span in the direction of analysis measured from face to face of columns, capitals, brackets,
VHH)LJXUHIRUWKHGH¿QLWLRQRIFOHDU
or walls, which must be taken greater than or equal to
nonrectangular
span for rectangular and nonrectangular columns; for the purpose of determining
columns are converted to square columns of the same cross-sectional area)
length of the span in the direction perpendicular to
)LJXUH
measured center-to-center of supports (see
is different on each side of the design strip, an average of these transverse span lengths must be used
Where
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equal to the distance from the edge of the slab to the centerline of the panel.
is determined for each clear span length; however,
For varying span lengths in the direction of analysis,
can be calculated based on the longest clear span and that value can conservatively be used for all spans in that
design strip.
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5-27
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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• 6WHS'LVWULEXWH
into negative and positive bending moments in each span.
into positive and negative bending moments at the critical sections within
&RHI¿FLHQWVDUHXVHGWRGLVWULEXWH
each span (that is, near or at midspan and the faces of the supports, respectively).
$VXPPDU\RIWKHEHQGLQJPRPHQWFRHI¿FLHQWVIRUDQHQGVSDQDUHJLYHQLQ7DEOHIRUWKHW\SLFDOFDVHRI
monolithic construction. A fully restrained exterior edge corresponds to a slab integrally constructed with a
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amount of rotation occurs at the slab-wall connection).
Table 5.14 Bending Moment Coefficients for an End Span
Flat Plate and Flat Slab
Two-way beam-supported slab
Without Edge Beams
With Edge Beams
Exterior Edge Fully
Restrained
Exterior negative
Positive
Interior negative
Location
)RULQWHULRUVSDQVWKHFRHI¿FLHQWVDUHHTXDOWRDQGDWWKHQHJDWLYHDQGSRVLWLYHORFDWLRQVLQWKHVSDQ
respectively.
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5-28
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The total factored bending moments in the design strips at the critical locations are obtained by multiplying
E\WKHFRHI¿FLHQWVLQ7DEOHIRUDQHQGVSDQDQGE\WKHDSSOLFDEOHFRHI¿FLHQWVQRWHGDERYHIRULQWHULRU
VSDQV)RUH[DPSOHWKHWRWDOIDFWRUHGGHVLJQVWULSEHQGLQJPRPHQWDWWKHH[WHULRUVXSSRUWRIDÀDWSODWHRUÀDW
slab without edge beams is equal to
• Step 3: Distribute the total negative and positive bending moments in the design strip to the column strips and
middle strips.
2QFHWKHWRWDOIDFWRUHGGHVLJQSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVKDYHEHHQGHWHUPLQHGLQ6WHSWKHVH
moments are distributed to the column strips and middle strips using percentages based on studies of linearly
elastic slabs with different beam stiffnesses. The percentages depend on the aspect ratio of the panel
and the presence of column-line beams in one or both directions.
Distribution to Column Strips
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Table 5.15 Percentages of Factored Bending Moments in Column Strips
Location
Percentage*
Exterior support
Positive
Interior support
*Where
where
In these equations,
LVWKHUDWLRRIWKHÀH[XUDOVWLIIQHVVRIWKHEHDPVHFWLRQWRWKHÀH[XUDOVWLIIQHVVRIDZLGWKRI
slab bounded laterally by centerlines of adjacent panels for the beams in the direction of analysis, and is calculated
E\(TXDWLRQ The term LVWKHUDWLRRIWKHWRUVLRQDOVWLIIQHVVRIDQHGJHEHDPVHFWLRQWRWKHÀH[XUDOVWLIIQHVVRIDVODEZLGWK
equal to the span length of the beam, measured center-to center of supports:
(5.18)
In this equation,
and
are the moduli of elasticity of the beam and slab concrete, respectively, which are
typically equal for monolithic construction; is the moment of inertia of the slab section using the full width of the
and is the cross-sectional constant determined by dividing the
slab tributary to the beam, that is,
beam section into its component rectangles, each having a smaller dimension and a larger dimension and by
summing the contribution of each rectangle:
(5.19)
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and
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5-29
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ݕ‬ଵ ‫ݔ‬ଶ ‫ݕ‬ଶ ‫ܥ‬஺ ൌ ൬1 െ 0.63
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൅ ൬1 െ 0.63 ൰
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‫ݕ‬ଶ 3
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‫ݔ‬ଵ ‫ݔ‬ଵଷ ‫ݕ‬ଵ
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‫ݔ‬ଵ ‫ݔ‬ଶ ‫ݕ‬ଶ ‫ݕ‬ଵ Case B
Figure 5.24 Calculation of cross-sectional constant
Values of
.
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In cases where FDOFXODWHGE\(TXDWLRQ LVJUHDWHUWKDQ PXVWEHWDNHQDVZKHQGHWHUPLQLQJWKH
SHUFHQWDJHRIWKHIDFWRUHGEHQGLQJPRPHQWDWWKHH[WHULRUVXSSRUWE\WKHHTXDWLRQLQ7DEOH
and without edge beams
the percentages to use for
For slabs without beams between supports
GLVWULEXWLRQRIWRWDOQHJDWLYHPRPHQWVWRFROXPQVWULSVDUHDQGIRUH[WHULRUDQGLQWHULRUVXSSRUWVUHVSHFWLYHly. It is evident that all the exterior negative factored moment is assigned to the column strip unless an edge beam
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which
Walls along column lines in the direction of analysis can be regarded as stiff beams with
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means
wall perpendicular to the direction of analysis and monolithic with the slab, it can be assumed the wall provides
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$QH[FHSWLRQWRWKHSHUFHQWDJHVLQ7DEOHRFFXUVZKHUHFROXPQVRUZDOOVKDYHDUHODWLYHO\ODUJHZLGWKSHUSHQdicular to the direction of analysis. In particular, where the transverse width of a column or wall extends a distance
the negative factored bending moments are to
JUHDWHUWKDQRUHTXDOWRSHUFHQWWKHZLGWKRIWKHGHVLJQVWULS
be uniformly distributed across
Distribution to Column-line Beams
For relatively stiff column-line beams
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percent of the factored column strip moments based on the applicable percentages determined by the equations in
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Where
WKHSHUFHQWDJHUHVLVWHGE\WKHEHDPFDQEHGHWHUPLQHGE\OLQHDULQWHUSRODWLRQEHWZHHQ
and
respectively.
DQG]HURSHUFHQWZKLFKFRUUHVSRQGWR
In addition to the factored bending moments from the column strip, column-line beams must also be designed to
resist the effects from any loads applied directly to the beam. Loads located on the slab outside of the beam width
must be distributed accordingly between the slab and beam.
Distribution to Middle Strips
The portions of the total factored design negative and positive bending moments that are not assigned to the column
strip are assigned to the half middle strips in the design strip. A middle strip adjacent and parallel to a panel edge
and supported by a wall must be designed for two times the factored bending moment assigned to the half middle
VWULSFRUUHVSRQGLQJWRWKH¿UVWURZRILQWHULRUVXSSRUWV
'HVLJQPRPHQWFRHI¿FLHQWVIRUWRWDOPRPHQWVFROXPQVWULSPRPHQWVDQGPLGGOHVWULSPRPHQWVDWWKHFULWLFDOVHFWLRQVLQ ÀDWSODWHVRUÀDWVODEVZLWKRXWHGJHEHDPV ÀDWSODWHVRUÀDWVODEVZLWKHGJHEHDPV ÀDWSODWHVRU
ÀDWVODEVZLWKDQHQGVSDQLQWHJUDOZLWKDUHLQIRUFHGFRQFUHWHZDOODQG WZRZD\EHDPVXSSRUWHGVODEVDUHJLYHQ
LQ7DEOHVDQGUHVSHFWLYHO\7KHVHFRHI¿FLHQWVDUHGHWHUPLQHGXVLQJWKHDSSOLFDEOHSHUFHQWDJHV
discussed previously.
Table 5.16 Design Moment Coefficients for Flat Plates or Flat Slabs Without Edge Beams
༃
༄
༅
༆
†’ƒ
ᬉ
–‡”‹‘”’ƒ
End Span
Slab Moments
Interior Span
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
Total Moment
Column Strip
Middle Strip
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.17 Design Moment Coefficients for Flat Plates or Flat Slabs With Edge Beams*
༃
༄
༅
†’ƒ
༆
ᬉ
–‡”‹‘”’ƒ
End Span
Slab Moments
Interior Span
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
Total Moment
Column Strip
M o
Middle Strip
*Applicable to edge beams with
For edge beams with
exterior negative column strip moment is equal to
Table 5.18 Design Moment Coefficients for Flat Plates or Flat Slabs With End Span Integral
with a Reinforced Concrete Wall
༃
༄
༅
†’ƒ
༆
ᬉ
–‡”‹‘”’ƒ
End Span
Slab Moments
Interior Span
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
Total Moment
Column Strip
Middle Strip
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.19 Design Moment Coefficients for Two-Way Beam-Supported Slabs
༃
༄
༅
†’ƒ
༆
ᬉ
–‡”‹‘”’ƒ
End Span
Span Ratio,
ℓ2 / ℓ1
Slab and Beam
Moments
Interior Span
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
Total Moment
Column
Strip
Beam*
Slab
Middle Strip
Column
Strip
Beam*
Slab
Middle Strip
Column
Strip
Beam*
Slab
Middle Strip
*Applicable to (1) edge beams with
and (2) column-line beams with
Determination of Factored Moments in Columns and Walls
Columns and walls supporting a two-way slab system must be designed to resist the appropriate negative factored
bending moments transferred from the slab. In lieu of a more exact analysis, the following equation can be used
transferred at an interior support due to factored gravity loads for the case of two
to determine the moment,
DGMRLQLQJVSDQVZLWKRQHVSDQORQJHUWKDQWKHRWKHU VHH)LJXUH (5.20)
In this equation,
and
are the uniformly distributed factored dead and live loads, respectively, on the longer
is the uniformly distributed factored dead load on the shorter span;
and
are the longer and shorter
span;
and
are the span lengths perpendicular to
and
respectively.
clear span lengths, respectively; and,
Where
and
(TXDWLRQ UHGXFHVWRWKHIROORZLQJ
(5.21)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄ଶ ሺܿଶ ሻ஺ ሺܿଵ ሻ஺ ሺ‫ܯ‬௨ ሻ஺ ‫ݍ‬஽௨ ൅ 0.5‫ݍ‬௅௨ ᇱ
‫ݍ‬஽௨
‫ܯ‬௦௖ ݄ଵ Joint A
ሺܿଶ ሻ஻ ሺ‫ܯ‬௨ ሻ஻ ሺܿଵ ሻ஻ κᇱ௡ ൏ κ௡ κ௡ κଵᇱ ൏ κଵ κଵ Joint A
ᇱ
‫ܯ‬௦௖ ൌ 0.07ሾሺ‫ݍ‬஽௨ ൅ 0.5‫ݍ‬௅௨ ሻκଶ κଶ௡ െ ‫ݍ‬஽௨
κᇱଶ ሺκᇱ௡ ሻଶ ሿ
ሺܿଶ ሻ஺ ሺܿଵ ሻଷ஺
ሺ݇ሻ஺ ൌ
12݄ଶ
ሺܿଶ ሻ஻ ሺܿଵ ሻଷ஻
ሺ݇ሻ஻ ൌ
12݄ଵ
ሺ݇ሻ஺
ሺ‫ܯ‬௨ ሻ஺ ൌ
‫ ܯ‬
ሺ݇ሻ஺ ൅ ሺ݇ሻ஻ ௦௖
ሺ݇ሻ஻
ሺ‫ܯ‬௨ ሻ஻ ൌ
‫ ܯ‬
ሺ݇ሻ஺ ൅ ሺ݇ሻ஻ ௦௖
Figure 5.25 Determination of factored moments at interior supporting members.
The moment
GHWHUPLQHGE\HLWKHU(TXDWLRQ RU LVGLVWULEXWHGWRWKHLQWHULRUVXSSRUWLQJPHPEHUV
above and below the slab in direct proportion to the members’ stiffnesses, which, in general, can be determined
by dividing the cross-sectional moment of inertia of the member in the direction of analysis by the length of the
PHPEHU WKLVLVEDVHGRQWKHDVVXPSWLRQWKHSUHGRPLQDQWFRQWULEXWLRQWRWKHWRWDOGLVSODFHPHQWLVGXHWRÀH[XUH
VHH)LJXUH :KHUHWKHFURVVVHFWLRQDOGLPHQVLRQVRIWKHVXSSRUWLQJPHPEHUVDERYHDQGEHORZWKHVODEDUH
is transferred on the basis of the inverse of the member lengths (that is, the longer member resists a
the same,
than the shorter member).
lesser amount of
At an exterior support, the total negative bending moment from the slab is transferred directly to the support. Like
interior supports, this moment is transferred to the members above and below the slab in direct proportion to their
stiffnesses. For two-way shear design at edge columns, the factored gravity bending moment to be transferred bemust be
when the DDM is used to
tween the slab and edge column bending perpendicular to the edge,
determine design strip bending moments.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determination of Factored Shear in Slabs with Beams
,QDGGLWLRQWRWKHDSSOLFDEOHEHQGLQJPRPHQWVLQ7DEOHEHDPVPXVWEHGHVLJQHGWRUHVLVWDSSOLFDEOHWULEXWDU\
factored shear forces due to the factored loads. The tributary areas for beams in a two-way beam-supported system
DUHJLYHQLQ)LJXUHWKHJUD\WULEXWDU\DUHDDSSOLHVWRWKHLQWHULRUEHDPWKDWVSDQVEHWZHHQFROXPQV$DQG%
EHDPVPXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHIDFWRUHGVKHDUIRUFHVFDXVHGE\WKH
Where
IDFWRUHGORDGVRQWKHWULEXWDU\DUHDWKLVPHDQVWKHVKHDUIRUFHVLQWKHVODEDURXQGWKHFROXPQDUHHTXDOWR]HUR
A
Tributary area
45 deg ሺtyp.ሻ
B
Figure 5.26 Tributary area for calculation of factored shear forces on an interior beam
in a two-way beam-supported slab.
Where
the shear forces resisted by the beam can be obtained by linear interpolation assuming the
(as noted above) and carries no load at
In
EHDPFDUULHVSHUFHQWRIWKHORDGZKHUH
FDVHVZKHUHWKHEHDPFDUULHVOHVVWKDQSHUFHQWRIWKHORDGWKHVKHDUIRUFHVQRWUHVLVWHGE\WKHEHDPVPXVWEH
resisted by the slab around the column. These shear forces and the applicable moments determined by the expresVLRQVLQ7DEOHFDXVHVKHDUVWUHVVHVLQWKHVODEZKLFKDUHFDOFXODWHGE\(TXDWLRQ In addition to the shear forces noted above, column-line beams must also be designed to resist the factored shear
forces due to any factored loads applied directly to the beam.
5.3.5 Lateral Loads
Effects from lateral loads on two-way systems can be determined using the analysis methods in ACI Chapter 6.
Regardless of the analysis method, the effects of axial loads, cracking, and effects of load duration must be taken
LQWRFRQVLGHUDWLRQVRGULIWFDXVHGE\ODWHUDOORDGVLVQRWXQGHUHVWLPDWHG:KHUHDOLQHDUHODVWLF¿UVWRUGHUDQDO\VLVLQ
DFFRUGDQFHZLWK$&,LVSHUIRUPHGXVLQJIDFWRUHGORDGVWKHVHFWLRQSURSHUWLHVLQ$&,DQGDUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
permitted to be used in lieu of performing a more rigorous analysis. In cases where service-level loads are applied to
WKHVWUXFWXUHWKHPRPHQWVRILQHUWLDJLYHQLQ$&,DUHDSSOLFDEOH6LPLODUO\ZKHUHDOLQHDUHODVWLFVHFRQG
RUGHUDQDO\VLVLVSHUIRUPHGLQDFFRUGDQFHZLWK$&,XVLQJIDFWRUHGORDGVLWLVSHUPLWWHGWRXVHWKHVHFWLRQSURSHUWLHVFDOFXODWHGLQDFFRUGDQFHZLWK$&, $&, ZKHUHVHUYLFHOHYHOORDGVDUHXVHGWKHSURYLVLRQV
RI$&,DUHDSSOLFDEOH
In two-way slab systems with column-line beams, the beams usually resist most or all of the lateral load effects
because they are typically much stiffer than the slab.
)RUÀDWSODWHV\VWHPVWKDWDUHSDUWRIWKHVHLVPLFIRUFHUHVLVWLQJV\VWHP 6)56 WKHPRPHQWRILQHUWLDIRUWKHVODE
members must be determined by a model in substantial agreement with results of comprehensive tests and analysis;
WKHPRPHQWVRILQHUWLDRIWKHRWKHUIUDPHPHPEHUVPXVWEHLQDFFRUGDQFHZLWK$&,DQG $&,
6.6.3.1.3). Several models satisfying this requirement are referenced in ACI R6.6.3.1.3. To account for cracking, the
slab moment of inertia is typically reduced between one-half and one-quarter of the uncracked moment of inertia.
When determining drifts or second-order effects in columns, a lower-bound slab stiffness should be used in the
analysis. In structures where slab-column frames interact with shear walls, a range of slab stiffnesses should be inYHVWLJDWHGLQRUGHUWRDVVHVVWKHLPSRUWDQFHRIWKHLQWHUDFWLRQ1RWHWKDWÀDWSODWHVDUHQRWSHUPLWWHGWREHSDUWRIWKH
SFRS in buildings assigned to Seismic Design Category (SDC) D, E, or F (see ACI Chapter 18).
5.4 Design Strength
5.4.1 General
7KHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGLQDWZRZD\VODEV\VWHPIRUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQ
LQ$&,7DEOH $&, (5.22)
(5.23)
(5.24)
(5.25)
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGWZRZD\VODEVPXVWEH
GHVLJQHGDVWHQVLRQFRQWUROOHGLQDFFRUGDQFHZLWK$&,7DEOH $&, WKXVWKHQHWWHQVLOHVWUDLQLQWKH
must be greater than or equal to
extreme layer of the longitudinal tension reinforcement at nominal strength,
and
$&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPHOD\HURIGHIRUPHGORQJLWXGLQDO
is equal to
$&, tension reinforcement corresponding to compression-controlled sections,
LVSHUPLWWHGWREHWDNHQDVSVLUHJDUGOHVVRI
where the modulus of elasticity of the reinforcement,
$&,7DEOH WKHJUDGHRIWKHUHLQIRUFHPHQW $&, )RUVKHDU
'HWHUPLQDWLRQRIQRPLQDOÀH[XUDODQGVKHDUVWUHQJWKVDUHJLYHQLQ6HFWLRQVDQGUHVSHFWLYHO\
5.4.2 Nominal Flexural Strength
7KHQRPLQDOÀH[XUDOVWUHQJWK
of a rectangular section with one layer of tension reinforcement is determined in
DFFRUGDQFHZLWK$&,DQGLVEDVHGRQPRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQ VHH$&,DQG)LJXUH
(5.26)
In this equation, LVWKHWRWDODUHDRIWKHÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUH
are at a positive moment section and are equally applicable at negative moment sections.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ
݀ െ ሺܽȀʹሻ
ܿ
݀ ൌ ݀௧ ݄
‫ܣ‬௦ ܽ ൌ ߚଵ ܿ
ܽ
ͲǤͺͷ݂௖ᇱ ͲǤͲͲ͵
ܶ ൌ ‫ܣ‬௦ ݂௬ ߝ௧ ܾ
Figure 5.27 Strain and stress distributions at a positive moment section in a two-way slab.
)RUWZRZD\VODEVWKHDYHUDJHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDO
)RUÀDWVODEVVDWLVI\LQJWKHGLPHQVLRQDOUHtension reinforcement, can be approximated as
TXLUHPHQWVLQ$&,IRUWKHGURSSDQHOVWKH XVHGLQGHWHUPLQLQJWKHQHJDWLYHÀH[XUDOVWUHQJWKDWWKHFULWLFDO
LVFDOFXODWHGEDVHGRQWKHOLPLWDWLRQLQ$&,WKHWKLFNQHVVRIWKHGURSSDQHOEHORZWKHVODE
section,
must be less than or equal to one-fourth the distance from the edge of the drop panel to the face of the column or
VHH)LJXUHIRUWKH
column capital. This requirement essentially limits the value of used in determining
case of an interior column with no capital).
݀
݄
൒ ͳǤʹͷ݄
”‘’’ƒ‡Ž
ܾௗ௥௢௣ ൒ κ஺ Ȁ͸
൒ κ஻ Ȁ͸
݀ ൌ ͳǤʹͷ݄ െ ݀௕ െ ‘˜‡”
൑ ݄ ൅ ൫ܾௗ௥௢௣ ΤͶ൯ െ ݀௕ െ ‘˜‡”
Figure 5.28 Determination of d for slabs with drop panels.
The depth of the equivalent stress block, is determined from force equilibrium, that is, it is determined by setequal to the tension force in the reinforcement,
ting the resultant compressive force in the concrete,
and solving for VHH)LJXUH (5.27)
,WLVDVVXPHGDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK is distributed over the
, where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV $&, 7KH
depth
PD[LPXPVWUDLQLQWKHFRQFUHWHLVDVVXPHGWREH $&, DQGWKHWHUP is the width of the compression face of the member, which is typically the width of the column strip or middle strip.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The term
is the factor relating the depth of the equivalent rectangular compression block,
LVGH¿QHGLQ$&,7DEOHDVIROORZV $&, neutral axis,
to the depth of the
• For
• For
• For
5.4.3 Nominal One-way Shear Strength
The nominal one-way shear strength,
DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, (5.28)
In this equation,
is the nominal shear strength provided by concrete and
is the nominal shear strength provided by shear reinforcement. For nonprestressed members, LVGHWHUPLQHGE\$&,7DEOH $&, $VVXPLQJVKHDUUHLQIRUFHPHQWLVQRWXVHGLQWZRZD\VODEVIRURQHZD\VKHDU(TXDWLRQ F LQ$&,7DEOHLV
LVHTXDOWR]HUR
applicable. This equation reduces to the following where the axial force,
(5.29)
In this equation, LVWKHZLGWKRIWKHSODQHH[WHQGLQJDFURVVWKHHQWLUHVODEZLGWK $&,VHH)LJXUH
$FFRUGLQJWR$&, must not be taken greater than
where
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU
accounts for the phenomenon indicated in test results that the shear strength
attributed to concrete in members without shear reinforcement does not increase in direct proportion with member
GHSWKWKLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(5.30)
,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK
ZD\VODEVZLWKDQRYHUDOOGHSWKOHVVWKDQDERXWLQ
This means
for two-
The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH
to normalweight concrete of the same compressive strength; it is determined based on either (1) the equilibrium
RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[ VHH$&,DQG
density,
$&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>VHH$&,7DEOH D @DQGYDOXHVRI based on
FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>VHH$&,7DEOH E @1RWHWKDW is permitted to be taken
DVIRUOLJKWZHLJKWFRQFUHWH $&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH $&, Table 5.20 Values of Ȝ Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.21 Values of Ȝ Based on Composition of Aggregates
Concrete
All-lightweight
/LJKWZHLJKW¿QHEOHQG
Sand-lightweight
Sand-lightweight, coarse blend
Composition of Aggregates
)LQH$670&
&RDUVH$670&
)LQH&RPELQDWLRQRI$670&DQG&
&RDUVH$670&
Fine: ASTM C33
&RDUVH$670&
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&DQG$670&
Ȝ
WR(1)
WR (1) /LQHDULQWHUSRODWLRQIURPWRLVSHUPLWWHGEDVHGRQWKHDEVROXWHYROXPHRIQRUPDOZHLJKW¿QHDJJUHJDWHDVDIUDFWLRQRIWKHWRWDODEVROXWHYROXPH
RI¿QHDJJUHJDWH
(2) /LQHDULQWHUSRODWLRQIURPWRLVSHUPLWWHGEDVHGRQWKHDEVROXWHYROXPHRIQRUPDOZHLJKWFRDUVHDJJUHJDWHDVDIUDFWLRQRIWKHWRWDODEVROXWH
volume of aggregate.
The term
LQ(TXDWLRQ LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
at the section divided by
$FFRUGLQJWR$&,5 PD\EHWDNHQDVWKHVXPRIWKHDUHDVRIWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW
ORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUV
with one layer of tension reinforcement, like two-way slabs, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDW
section, which has a width equal to
Values of
used to calculate DUHOLPLWHGWRSVLH[FHSWDVDOORZHGLQ$&, $&, 7KLV
is primarily due to a lack of test data and practical experience with concrete having compressive
limitation on
VWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\LWLVW\SLFDOIRU LQWZRZD\VODEVWREHOHVVWKDQRUHTXDOWR
SVL 5HIHUHQFH One-way shear rarely governs in a two-way slab system, so shear reinforcement for one-way shear is not required;
thus,
7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH
FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@
(5.31)
5.4.4 Nominal Two-way Shear Strength
Overview
For two-way shear, LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, 3URYLVLRQVWRGHWHUPLQHQRPLQDO
two-way shear strengths for slabs without and with shear reinforcement are given below.
Two-way Shear Strength Provided by Concrete in Slabs without Shear Reinforcement
)RUWZRZD\VODEV\VWHPVZLWKRXWVKHDUUHLQIRUFHPHQW(TXDWLRQ PXVWEHVDWLV¿HGDWWKHFULWLFDOVHFWLRQV
The stress corresponding to the nominal two-way shear
GH¿QHGLQ6HFWLRQRIWKLVSXEOLFDWLRQZKHUH
LVWKHOHDVWRIWKHYDOXHVFDOFXODWHGE\WKHHTXDWLRQVLQ$&,7DEOH
strength provided by the concrete,
$&, (5.32)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU LVGHWHUPLQHGE\(TXDWLRQ >$&,@DQG LVWKHPRGL¿FDWLRQ
IDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOZHLJKWFRQFUHWHRIWKH
VDPHFRPSUHVVLYHVWUHQJWK VHH7DEOHVDQG The term is the ratio of the long to short dimensions of a rectangular column, concentrated load, or reaction area.
The constant LVHTXDOWRIRULQWHULRUFROXPQVIRUHGJHFROXPQVDQGIRUFRUQHUFROXPQV5HIHUHQFHWR
references the
these types of columns does not suggest the actual location of a column in a building; instead,
number of critical sections available to resist shear around a column. For example, LVHTXDOWRIRUWKHLQWHULRUFROXPQLQ)LJXUHDGMDFHQWWRDUHODWLYHO\ODUJHRSHQLQJLQWKLVFDVHRQO\VLGHVRIWKHFULWLFDOVHFWLRQDUH
available to resist the shear stress. Additional information on the effects of openings in two-way slab systems is
JLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
’‡‹‰
”‹–‹ ƒŽ
•‡ –‹‘
Ƚ௦ ൌ ͵Ͳ
–‡”‹‘”
‘Ž—
Figure 5.29 Effect of an opening on the critical section of an interior column.
Two-way Shear Strength Provided by Concrete in Slabs with Shear Reinforcement
For two-way slab systems with shear reinforcement (stirrups or headed shear stud reinforcement), is determined
LQDFFRUGDQFHZLWK$&,(TXDWLRQVWRGHWHUPLQH at the various critical section locations are given in Table
VHH$&,7DEOH Table 5.22 Nominal Two-way Shear Strength Provided by Concrete in Slabs with Shear Reinforcement
Shear Reinforcement
Critical Sections
Stirrups
$&,DQG
)LJXUH
+HDGHGVKHDUVWXGUHLQIRUFHPHQW
$&,
)LJXUH
$&,
)LJXUH
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vc
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU GHWHUPLQHGE\(TXDWLRQ LVSHUPLWWHGWREHWDNHQDVZKHUHRQHRI
WKHIROORZLQJWZRFRQGLWLRQVLVVDWLV¿HG $&, 1. 6WLUUXSVGHVLJQHGDQGGHWDLOHGLQDFFRUGDQFHZLWK$&,DUHSURYLGHGZKHUH
6PRRWKKHDGHGVKHDUVWXGUHLQIRUFHPHQWZLWKDVWXGVKDIWOHQJWKOHVVWKDQRUHTXDOWRLQGHVLJQHGDQG
GHWLOHGLQDFFRUGDQFHZLWK$&,DUHSURYLGHGZKHUH
In these equations,
is the total area of stirrup legs or headed shear stud reinforcement on one peripheral line geometrically similar to the perimeter of the column section within the spacing Information on shear reinforcement
in two-way slabs is given below.
Two-way Shear Strength Provided by Single- or Multiple-leg Stirrups
Two-way shear strength of two-way slabs can be increased by shear reinforcement consisting of properly anchored
VLQJOHRUPXOWLSOHOHJVWLUUXSV VHH$&,DQG)LJXUH 6WLUUXSVDUHSHUPLWWHGWREHXVHGSURYLGHG of
where is the diameter of the stirrups (ACI
the two-way slab is greater than or equal to the larger of 6 in. or
VHH)LJXUH ηͳ2db
κext
ζ2 d
db (typ.)
κext
κext
κext
(a)
(b)
6d
For #3–#5 stirrups, κext = greater of ൜ b
3"
(c)
Figure 5.30 Stirrup shear reinforcement for use in two-way slabs. (a) Single-leg stirrup.
(b) Multiple-leg stirrup. (c) Closed stirrups.
ƒ ‡‘ˆ ‘Ž—
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ͳ͸݀௕
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Figure 5.31 Cover, spacing, and depth requirements for closed stirrups.
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5-41
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The nominal shear strength,
LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,
(5.33)
In this equation,
is the area of all stirrup legs on one peripheral line geometrically similar to the perimeter of the
FROXPQVHFWLRQ)RUH[DPSOHDWWKHLQWHULRUFROXPQGHSLFWHGLQ)LJXUHVWLUUXSVDUHSURYLGHGSDUDOOHOWRHDFK
column face. Along one peripheral line of stirrups, there are two vertical stirrup legs on each face, so for four faces,
where
is equal to the area of one stirrup leg.
The term is the spacing of the peripheral lines of stirrups in the direction perpendicular to the column face. The
ORFDWLRQDQGVSDFLQJRIWKHVWLUUXSVPXVWFRQIRUPZLWK$&,7DEOH VHH$&,DQG)LJXUH 0HWKods to determine DUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
Two-way Shear Strength Provided by Headed Shear Stud Reinforcement
+HDGHGVKHDUVWXGUHLQIRUFHPHQWFRQIRUPLQJWR$&,FDQEHXVHGDORQHRULQFRQMXQFWLRQZLWKVWLUUXSVGURS
SDQHOVVKHDUFDSVRUFROXPQFDSLWDOVWRLQFUHDVHWKHQRPLQDOWZRZD\VKHDUVWUHQJWKRIWZRZD\VODEV $&, $W\SLFDOKHDGHGVKHDUVWXGDUUDQJHPHQWLVVKRZQLQ)LJXUH/DUJHKHDGHGVWXGVDUHZHOGHGWRÀDWUDLOVDQG
WKHDVVHPEO\LVSRVLWLRQHGRQFKDLUVDURXQGWKHFROXPQVDQGVXEVHTXHQWO\QDLOHGWRWKHIRUPZRUN7KHVL]HDQG
spacing of the studs and the length of the base rail depend on the shear requirements.
‡ƒ†‡†•Š‡ƒ”•–—†
–‡‡Ž„ƒ•‡”ƒ‹Ž
Figure 5.32 Headed shear stud reinforcement.
/LNHUHLQIRUFLQJEDUVVXI¿FLHQWFRYHUPXVWEHSURYLGHGWRSURWHFWWKHDVVHPEO\IURPZHDWKHU¿UHDQGRWKHUHIIHFWV
VHH)LJXUH 0LQLPXPFRYHUUHTXLUHPHQWVIRUKHDGHGVKHDUVWXGUHLQIRUFHPHQWDUHJLYHQLQ$&,
WKHPLQLPXPFRYHUWRWKHKHDGVDQGEDVHUDLOVLVWKHVDPHDVWKDWUHTXLUHGIRUWKHÀH[XUDOUHLQIRUFHPHQWLQDWZR
ZD\VODEZKLFKLVJLYHQLQ$&,7DEOHIRUFDVWLQSODFHQRQSUHVWUHVVHGPHPEHUV)RUWZRZD\VODEVQRW
H[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQGWKHPLQLPXPFRYHULVLQIRUDQGVPDOOHUEDUV7KHRYHUDOOKHLJKWRIWKHKHDGHGVKHDUVWXGDVVHPEO\PXVWFRQIRUPWRWKHUHTXLUHPHQWVLQ$&,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ƒ ‡‘ˆ ‘Ž—
–—†ሺ•Šƒሻ
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‘˜‡”
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ƒ‹ŽŠ‘Ž‡
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൑ ݀ Τʹ
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‫ݏ‬൑ቐ
݀ Τʹ ™Š‡”‡‫ݒ‬௨ ൐ ߶͸ඥ݂௖ᇱ Figure 5.33 Cover and spacing requirements for headed shear stud reinforcement.
The nominal shear strength,
LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$&,(TXDWLRQ LVXVHG
WKLVHTXDWLRQLVWKHVDPHDV$&,(TXDWLRQ IRUVWLUUXSV>VHH(TXDWLRQ @7KHPDLQ
to determine
is the cross-sectional area of all headed shear studs on one
difference is that for the case of headed shear studs,
SHULSKHUDOOLQHJHRPHWULFDOO\VLPLODUWRWKHSHULPHWHURIWKHFROXPQVHFWLRQ)RUWKHLQWHULRUFROXPQLQ)LJXUH
where
is the area of one headed shear stud.
The term is the spacing of the peripheral lines of headed shear studs in the direction perpendicular to the column
IDFH7KHORFDWLRQDQGVSDFLQJRIWKHKHDGHGVKHDUVWXGVPXVWFRQIRUPZLWK$&,7DEOH VHH$&,
DQG)LJXUHVDQG 0HWKRGVWRGHWHUPLQH DUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
$&,(TXDWLRQ PXVWDOVREHVDWLV¿HGZKHUHKHDGHGVKHDUVWXGUHLQIRUFHPHQWLVSURYLGHG $&, (5.34)
Summary of Nominal Two-way Shear Strength Requirements
7KHLQIRUPDWLRQLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHQRPLQDOWZRZD\VKHDUVWUHQJWK
for two-way slab
for
systems without and with shear reinforcement. Included in this table are the maximum permitted values of
two-way slabs with shear reinforcement, which are equal to the values of LQ$&,7DEOHGLYLGHGE\WKH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.23 Nominal Two-way Shear Strength Requirements for Two-way Slabs
Shear Reinforcement
Critical Sections
None
$&,
)LJXUHV±
Nominal Strengths
$&,
)LJXUH
Stirrups
$&,
)LJXUH
+HDGHG6KHDU6WXG
Reinforcement
$&,
)LJXUH
$&,
)LJXUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.4.5 Openings in Two-way Slab Systems
Openings for mechanical ductwork, electrical and plumbing risers, stairs, elevators, and other elements are comPRQO\UHTXLUHGLQWZRZD\VODEV\VWHPV7KHHIIHFWVRIVXFKRSHQLQJVPXVWEHFRQVLGHUHGLQGHVLJQ $&, $FFRUGLQJWR$&,RSHQLQJVRIDQ\VL]HDUHSHUPLWWHGLQDWZRZD\VODEV\VWHPSURYLGHGDQDQDO\VLVRIWKH
V\VWHPZLWKWKHRSHQLQJVLVSHUIRUPHGWKDWVKRZVDOOVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWVDUHVDWLV¿HG)RUWZR
ZD\VODEV\VWHPVZLWKRXWEHDPVVXFKDQDQDO\VLVQHHGQRWEHSHUIRUPHGZKHQWKHSURYLVLRQVRI$&,DUH
VDWLV¿HGWKH¿UVWWKUHHRIWKHVHSURYLVLRQVDUHJLYHQLQ7DEOH VHH)LJXUH Table 5.24 Requirements for Openings in Two-way Slab Systems in Accordance with ACI 8.5.4.2
Location
Maximum Opening Size
Additional Requirements
2SHQLQJVRIDQ\VL]HDUHSHUPLWWHG
Total quantity of reinforcement
in the panel must be at least that
required for the panel without the
opening
Area common to intersecting
columns strips
One-eighth the width of the column
strip in either span
A quantity of reinforcement at
least equal to that interrupted by an
opening must be added to the sides
of the opening
Area common to one column
strip and one middle strip
Not more than one-fourth of the
reinforcement in either strip must
be interrupted by openings
A quantity of reinforcement at
least equal to that interrupted by an
opening must be added to the sides
of the opening
Area common to intersecting
middle strips
κଶ CS
κଵ MS
CS
κଵ MS
CS
κଶ CS
MS
CS
MS
CS
Any size opening permitted ሾACI 8.5.4.2ሺaሻሿ
Maximum opening width ൌ ሺCS widthሻ/8 ሾACI 8.5.4.2ሺbሻሿ
See ACI 8.5.4.2ሺcሻ
Figure 5.34 Permitted openings in two-way slab systems without beams in accordance with ACI 8.5.4.2.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The closer an opening is to a column, the greater effect it has on the critical shear perimeter available to resist shear
from the periphery of a column, concentrated load,
forces. In particular, where an opening is located closer than
RUUHDFWLRQDUHDWKHSURYLVLRQVRI$&,PXVWEHVDWLV¿HG>$&, G @7KHSHULPHWHURIWKHFULWLFDO
section must be reduced by a length equal to the projection of the opening formed by two straight lines that extend
from the centroid of the column, concentrated load, or reaction area and are tangent to the boundaries of the opening. The ineffective portions of DUHLOOXVWUDWHGLQ)LJXUHIRUDFROXPQLQDÀDWSODWHV\VWHPZLWKRSHQLQJVIRU
systems without and with shear reinforcement.
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Figure 5.35 Effect of openings on critical shear perimeter in two-way slabs.
(a) Without shear reinforcement. (b) With shear reinforcement.
The section properties of the remaining effective segments of the critical section can be determined using Equations
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RIWKHPRPHQWWUDQVIHUUHGWRWKHFROXPQE\HFFHQWULFLW\RIVKHDUDUHFDOFXODWHGE\(TXDWLRQ XVLQJWKHUHGXFHG
section properties of the critical section. The maximum shear stresses must be less than or equal to applicable design
LQ7DEOH
two-way shear strengths,
5.5 Determination of Required Reinforcement
5.5.1 Required Flexural Reinforcement
7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW
at a critical section in a column strip or middle strip in a two-way
and
for tension-controlled
VODEV\VWHPLVGHWHUPLQHGEDVHGRQWKHUHTXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV
VHFWLRQV8VLQJ(TXDWLRQV DQG WKHUHTXLUHG can be determined by the following equation:
(5.35)
7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH
LVGH¿QHGDVIROORZV
(5.36)
and
for tension-controlled sections. Factored bending moments,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHDUHDRIÀH[XUDOUHLQIRUFHPHQWDWDQ\FULWLFDOVHFWLRQLQDWZRZD\VODEPXVWQRWEHOHVVWKDQ
is equal to the thickness of the slab, times the width of the column strip or middle strip
where the gross area
(ACI 8.6.1.1).
FDQDOVREHGHWHUPLQHGE\(TXDWLRQ ZKHUH
7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWLQWKHVODEZLGWK
LQ(TXDWLRQ LVFDOFXODWHGXVLQJ
>VHH(TXDWLRQ @,QWKLVFDVH
is equal to the folORZLQJ VHH$&,DQG6HFWLRQRIWKLVSXEOLFDWLRQ (5.37)
In this equation,
for shear and
is the factored shear stress on the slab critical section for two-way action from the controlling load combination without moment transfer:
(5.38)
It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear
for tension-controlled sections (see
strain distribution where is the depth of the neutral axis when
)LJXUH (5.39)
Substituting
LQWR(TXDWLRQ ZLWK IURP(TXDWLRQ WKHIROORZLQJHTXDWLRQFDQEHXVHGWRFDOcorresponding to tension-controlled sections for two-way slabs with
FXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
(5.40)
)RU*UDGHUHLQIRUFHPHQW
SURSHUWLHV(TXDWLRQ UHGXFHVWRWKHIROORZLQJ
and for
For these material
(5.41)
In these equations, is the width of the column strip or middle strip. If FDOFXODWHGE\(TXDWLRQ LVJUHDWHU
the section is not tension-controlled
, and must be increased accordingly to attain a tensionthan
FRQWUROOHGVHFWLRQ$VLPLODUFKHFNPXVWEHSHUIRUPHGIRUWKHÀH[XUDOUHLQIRUFHPHQWZLWKLQ
5.5.2 Required Shear Reinforcement
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Stirrups
3ULRUWRGHWHUPLQLQJWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHVWLUUXSVLWVKRXOGEHYHUL¿HGWKDWWKHPD[LPXPVKHDU
in accordance with ACI Table
stress, DWWKHFULWLFDOVHFWLRQGH¿QHGE\$&,LVOHVVWKDQRUHTXDOWR
DQGRU must be increased accordingly to satisfy this requirement.
:KHUH
The total number of stirrup legs,
provided on one peripheral line geometrically similar to the perimeter of the
FROXPQGHSHQGRQWKHQXPEHURIVLGHVRIWKHFULWLFDOVHFWLRQ VHH)LJXUH $VVXPLQJDVWLUUXSEDUVL]HZLWKDQ
the following equation can be used to determine the required spacing, VHH7DEOH area equal to
(5.42)
Stirrups can be terminated where the design strength of the concrete can resist the factored shear stress without the
VHH)LJXUHDQG7DEOH contribution of the stirrups; this occurs where
Headed Shear Stud Reinforcement
3ULRUWRGHWHUPLQLQJWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHKHDGHGVKHDUVWXGUHLQIRUFHPHQWLWVKRXOGEHYHUL¿HG
in
that the maximum shear stress, DWWKHFULWLFDOVHFWLRQGH¿QHGE\$&,LVOHVVWKDQRUHTXDOWR
DQGRU must be increased accordingly to satisfy this
DFFRUGDQFHZLWK$&,7DEOH:KHUH
requirement.
provided on one peripheral line geometrically similar to the perimeter of
The number of headed shear studs,
WKHFROXPQGHSHQGRQ WKHQXPEHURIVLGHVRIWKHFULWLFDOVHFWLRQDQG WKHVSDFLQJUHTXLUHPHQWSDUDOOHOWRWKH
the
FROXPQIDFHLQ$&,7DEOH VHH)LJXUH $VVXPLQJDKHDGHGVKHDUVWXGZLWKDQDUHDHTXDOWR
following equation can be used to determine the required spacing, VHH7DEOH (5.43)
+HDGHGVKHDUVWXGVFDQEHWHUPLQDWHGZKHUHWKHGHVLJQVWUHQJWKRIWKHFRQFUHWHFDQUHVLVWWKHIDFWRUHGVKHDUVWUHVV
VHH)LJXUHDQG7DEOH
without the contribution of the headed shear studs; this occurs where
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VQRWHGSUHYLRXVO\$&,UHTXLUHVKHDGHGVKHDUVWXGUHLQIRUFHPHQWDQGVWXGDVVHPEOLHVFRQIRUPWR$670
is equal to
$ 5HIHUHQFH $FFRUGLQJWRWKDWGRFXPHQWWKHPLQLPXP\LHOGVWUHQJWKRIWKHVWXGPDWHULDO
SVL$OVRWKHDUHDRIDKHDGHGVKHDUVWXGVKRXOGEHREWDLQHGIURPPDQXIDFWXUHUV¶OLWHUDWXUH
5.6 Reinforcement Detailing
5.6.1 Concrete Cover
Reinforcing bars, stirrups, and headed shear stud assemblies are placed in two-way slabs with a minimum concrete
FRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHUHIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&,
$&, )RUUHLQIRUFLQJEDUVLQWZRZD\VODEVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQGWKHPLQLPXPFRYHULVHTXDOWRLQIRUDQGVPDOOHUEDUV $&,7DEOH &RQFUHWHFRYHULVPHDVXUHGIURP
the surface of the concrete to the following:
• 2XWHUPRVWOD\HURIÀH[XUDOUHLQIRUFHPHQWIRUWZRZD\VODEVZLWKRXWVKHDUUHLQIRUFHPHQW )LJXUH
• Outermost surface of the stirrup bars for two-way slabs with shear reinforcement consisting of stirrups (Figure
• Underside of the base rail and outermost surface of the head for two-way slabs with shear reinforcement conVLVWLQJRIKHDGHGVKHDUVWXGV )LJXUH
Ž‡š—”ƒŽ”‡‹ˆ‘” ‡‡–ሺ–›’Ǥሻ
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Figure 5.36 Minimum clear cover and spacing requirements for reinforcing bars in two-way slabs.
5.6.2 Minimum Spacing of Flexural Reinforcing Bars
0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&, $&, 7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJEDUV
7KHVSDFLQJUHTXLUHPHQWVDUHVXPPDUL]HGLQ)LJXUHIRUDWZRZD\VODEV\VWHPZKHUH
PD[LPXPDJJUHJDWHVL]HLQWKHFRQFUHWHPL[
is the nominal
5.6.3 Maximum Spacing of Flexural Reinforcing Bars
([FHSWIRUWZRZD\MRLVWV\VWHPVPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIÀH[XUDOUHLQIRUFLQJEDUVLVHTXDOWRWKH
and 18 in. at critical sections and
DQGLQDWRWKHUVHFWLRQV $&, ,QDGGLWLRQWRFRQWUROlesser of
ling crack widths, this requirement takes into consideration the effect that could be caused by loads concentrated on
small areas of the slab.
5.6.4 Selection of Flexural Reinforcement
7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ DQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV
JLYHQLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\
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5-49
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In two-way slabs, LVFDOFXODWHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSVDQGPLGGOHVWULSV$EDUVL]HDQG
is equal to or slightly greater than the required
considering
spacing should be selected so that the provided
PLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVDQGPLQLPXPDQGPD[LPXPEDUVSDFLQJUHTXLUHPHQWV7KHÀH[XUDOUHLQIRUFLQJEDUVDUHXQLIRUPO\GLVWULEXWHGLQWKHGHVLJQVWULSV(FRQRP\LVJHQHUDOO\DFKLHYHGE\XVLQJWKHODUJHVWVL]H
UHLQIRUFLQJEDUVVDWLVI\LQJVWUHQJWKDQGPD[LPXPVSDFLQJUHTXLUHPHQWVDQGWRUHSHDWEDUVL]HDQGVSDFLQJDVRIWHQ
as possible.
may need
,QRUGHUWRVDWLVI\(TXDWLRQ WKHSURYLGHGDUHDRIUHLQIRUFHPHQWLQWKHFROXPQVWULSZLWKLQ
WREHLQFUHDVHGWKLVLVW\SLFDOO\DFKLHYHGE\DGGLQJUHLQIRUFLQJEDUVRIWKHVDPHVL]HDVWKRVHLQWKHFROXPQVWULS
considering the requirements for bar spacing. Where additional reinforcing bars are required, the bars should be
and a special detail must be provided that clearly illustrates the extent and location of
uniformly spaced within
the reinforcing bars.
5.6.5 Corner Restraint in Slabs
At exterior corners of slabs supported by stiff elements such as walls and edge beams with
corner reinIRUFHPHQWPXVWEHSURYLGHGWKDWVDWLV¿HVWKHUHTXLUHPHQWVRI$&,7KLVUHLQIRUFHPHQWLVUHTXLUHGEHFDXVHWKH
stiff elements restrain the slab and cause additional bending moments at these locations.
Corner reinforcement must be provided in both the top and bottom of the slab, and the reinforcement in each layer
equal to the largest positive bending moment per
in each direction must be designed for a factored moment,
unit width in the slab panel. The top and bottom reinforcement must be placed parallel and perpendicular to the
GLDJRQDOUHVSHFWLYHO\IRUDGLVWDQFHRIDWOHDVWRQH¿IWKRIWKHORQJHURIWKHWZRVSDQOHQJWKVLQWKHFRUQHUSDQHO
>VHH)LJXUH D @,QOLHXRIWKHGLDJRQDOUHLQIRUFHPHQWOD\RXWUHLQIRUFHPHQWSODFHGSDUDOOHOWRWKHHGJHVLV
SHUPLWWHG VHH$&,DQG)LJXUH E @WKLVOD\RXWLVSUHIHUUHGFRPSDUHGWRWKHGLDJRQDOOD\RXWEHFDXVH
WKHUHLQIRUFLQJEDUVDUHVLPSOHUWRSODFHLQWKH¿HOG,QHLWKHURSWLRQWKHPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIWKH
corner reinforcement is equal to
κ஺ κ஺ κ஺ /5
κ஻ Top bars
Top and bottom bars in
both directions
κ஻ Bottom bars
κ஺ /5
κ஺ /5
κ஺ /5
ሺbሻ
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Notes
1.Applies to walls and edge beams where one or both edge beams has ߙ௙ ൐ 1.0
Ϯ͘κ஺ ൐ κ஻ 3.Maximum ‫ ݏ‬ൌ 2݄
Figure 5.37 Required reinforcement at corners of slabs supported by stiff edge members.
(a) Reinforcement parallel and perpendicular to the diagonal. (b) Reinforcement parallel to the slab edges.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.6.6 Termination of Flexural Reinforcement
5HTXLUHPHQWVIRUWHUPLQDWLRQORFDWLRQVRISRVLWLYHDQGQHJDWLYHÀH[XUDOUHLQIRUFHPHQWSHUSHQGLFXODUWRDGLVFRQWLQXRXVHGJHLQFROXPQDQGPLGGOHVWULSVDUHJLYHQLQ$&,)RUVODEVVXSSRUWHGGLUHFWO\RQHGJH VSDQGUHO EHDPVFROXPQVDQGZDOOVWKHSRVLWLYHÀH[XUDOUHLQIRUFHPHQWPXVWH[WHQGLQWRWKHVXSSRUWLQJPHPEHUDWOHDVW
LQIURPLWVIDFH VHH$&,DQG)LJXUH 6WUDLJKWKRRNHGRUKHDGHGEDUVPD\EHXVHGLQVXFKFDVHV
+RRNHGRUKHDGHGUHLQIRUFLQJEDUVPXVWEHSURYLGHGDWWKHHQGVRIQHJDWLYHUHLQIRUFHPHQWDQGWKHUHLQIRUFHPHQW
must be developed in tension at the face of the supporting member with development lengths determined in accorGDQFHZLWK$&, $&, ,QIRUPDWLRQRQKRZWRGHWHUPLQHWHQVLRQGHYHORSPHQWOHQJWKVIRUKRRNHGEDUV
DQGIRUKHDGHGEDUVFDQEHIRXQGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHLQIRUPDWLRQLQWKDWVHFWLRQIRURQHZD\
slabs is also applicable to two-way slabs.
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൒ ͸ԣ
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Figure 5.38 Termination of flexural reinforcement in a two-way slab supported
by an edge (spandrel) beam, column, or wall.
Anchorage of reinforcement is permitted within a slab where the slab is not supported by a spandrel beam or a wall
DWDGLVFRQWLQXRXVHGJHRUZKHUHDVODEFDQWLOHYHUVEH\RQGWKHVXSSRUW $&, )RUWZRZD\VODEV\VWHPVZLWKRXWEHDPVWKHÀH[XUDOUHLQIRUFHPHQWPXVWKDYHWKHPLQLPXPH[WHQVLRQVJLYHQLQ
$&,)LJXUH $&,VHH)LJXUH WKHVHH[WHQVLRQVDUHDSSOLFDEOHWRV\VWHPVVXSSRUWLQJRQO\
for the negative reinforcing bars, which governs
gravity loads. The purpose of the minimum extension length of
LVOHVVWKDQDERXWLVWRKHOSLQWHUFHSWSRWHQWLDOSXQFKLQJVKHDUFUDFNVWKDWFDQIRUPLQUHODWLYHO\
where
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PD\QRWEHVXI¿FLHQWWRLQWHUFHSWSRWHQWLDOSXQFKLQJVKHDU
the prescribed minimum extension length of
cracks in such cases.
Where a two-way slab resists the effects from combined gravity and lateral forces, the bar lengths must be based on
DQDO\VLVEXWPXVWQRWEHOHVVWKDQWKRVHJLYHQLQ$&,)LJXUH3UHFLVHORFDWLRQVRILQÀHFWLRQSRLQWVFDQQRW
be explicitly found using approximate methods, such as the DDM Method, because they depend on the ratio of the
panel dimensions, the ratio of live to dead load, and the continuity conditions at the edges of the panels. Also, other
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of the negative reinforcing bars in the column strip continuous; this is consistent with the requirement pertaining to
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
κ஺ଵ κ஺ଶ κ஻ଵ κ஻ଶ κ஺ଶ κ஺ଷ κ஺ଷ κ஻ଷ κ஻ଷ κ஻ଶ ൒ ‫ିܣ‬௦ଵ Τ2 ൒ ‫ିܣ‬௦ଶ Τ2 ൒ ‫ିܣ‬௦ଷ Τ2 ‫ିܣ‬௦ଷ ‫ିܣ‬௦ଶ ‫ିܣ‬௦ଵ ൒ 2 bars conforming
to ACI 8.7.4.2
Splices permitted in
this region ሺtyp.ሻ
൒ 6ԣ
κ௡ଵ κ௡ଶ κ௡ଷ Column Strip
κ஼ଵ κ஼ଶ ൒ max ሺ‫ܣ‬ା௦ଵ Τ2͕‫ܣ‬ା௦ଶ Τ2ሻ
‫ܣ‬ା௦ଵ ൒ 6ԣ
κ௡ଵ κ஼ଷ κ஼ଶ κ஼ଷ ‫ܣ‬ା௦ଶ κ஽ଵ κ஽ଶ ൒ 6ԣ
κ௡ଶ κ஽ଶ κ஽ଷ κ௡ଷ Middle Strip
Notes:
greater of 0.30κ௡ଵ and 5݀ for two-way slabs without drop panels
1. κ஺ଵ ൒ ቐ
greater of 0.33κ௡ଵ and 5݀ for two-way slabs with drop panels
greater of 0.30κ௡ଵ , 0.30κ௡ଶ , and 5݀ for two-way slabs without drop panels
2. κ஺ଶ ൒ ቐ
greater of 0.33κ௡ଵ , 0.33κ௡ଶ , and 5݀ for two-way slabs with drop panels
greater of 0.30κ௡ଶ , 0.30κ௡ଷ , and 5݀ for two-way slabs without drop panels
3. κ஺ଷ ൒ ቐ
greater of 0.33κ௡ଶ , 0.33κ௡ଷ , and 5݀ for two-way slabs with drop panels
4. κ஻ଵ ൒ 0.20κ௡ଵ
5. κ஻ଶ ൒ greater of 0.20κ௡ଵ and 0.20κ௡ଶ
6. κ஻ଷ ൒ greater of 0.20κ௡ଶ and 0.20κ௡ଷ
7. κ஼ଵ ൒ 0.22κ௡ଵ
8. κ஼ଶ ൒ greater of 0.22κ௡ଵ and 0.22κ௡ଶ
9. κ஼ଷ ൒ greater of 0.22κ௡ଶ and 0.22κ௡ଷ
10. κ஽ଵ ൑ 0.15κ௡ଵ
11. κ஽ଶ ൑ 0.15κ௡ଶ
12.κ஽ଷ ൑ 0.15κ௡ଷ
Figure 5.39 Minimum bar lengths for two-way slabs without beams.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.6.7 Splices of Reinforcement
6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQWZRZD\VODEVPXVWEHLQDFFRUGDQFHZLWK$&, $&, 7KHSULmary reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to
WKHFRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG
/DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQW7KH
LQIRUPDWLRQSURYLGHGLQ6HFWLRQRIWKLVSXEOLFDWLRQIRUVSOLFHVRIUHLQIRUFLQJEDUVLQRQHZD\VODEVLVDOVRDSplicable to two-way slabs.
5.6.8 Structural Integrity Reinforcement
6WUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVIRUFDVWLQSODFHWZRZD\VODEVDUHJLYHQLQ$&,7KHSXUpose of this reinforcement is to improve the redundancy and ductility in the structure so in the event of damage to
DPDMRUVXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFWXUH
will have a higher probability of maintaining overall stability.
7ZRUHTXLUHPHQWVDUHJLYHQLQ$&, VHH)LJXUH (1) $OOWKHERWWRPÀH[XUDOUHLQIRUFHPHQWLQWKHFROXPQVWULSVLQERWKGLUHFWLRQVPXVWEHFRQWLQXRXVRUVSOLFHGXVLQJPHFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQODSVSOLFHVLQDFFRUGDQFH
ZLWK$&,6SOLFHVPXVWEHORFDWHGLQDFFRUGDQFHZLWK$&,)LJXUH
At least two of the bottom reinforcing bars in the column strips in both directions must pass within the region
bounded by the longitudinal reinforcement of the column and must be anchored at exterior supports using
hooks or headed bars.
5.6.9 Shear Reinforcement Details
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RIWKLVSXEOLFDWLRQ 7KHVHGHWDLOVDUHJLYHQLQ)LJXUHVDQGIRUVWLUUXSVDQGLQ)LJXUHVDQG
IRUKHDGHGVKHDUVWXGV
5.7 Design Procedure
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DDM is used to determine the factored bending moments at the critical sections in the column and middle strips.
,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFFDQEHIRXQGLQ
this chapter.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1
EŽ
zĞƐ
EŽ
zĞƐ
1
A
Figure 5.40 Design procedure for two-way slabs.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
EŽ
zĞƐ
1
Detail the reinforcement in accordance
with Sect. 5.6
Figure 5.40 Design procedure for two-way slabs (cont.).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.8 Examples
5.8.1
Example 5.1 – Determination of Minimum Slab Thickness: Flat Plate System, Building #1
(Framing Option A)
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DQG*UDGHUHLQIRUFHPHQW
ZLWKLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum slab thickness based on serviceability requirements
The longest clear span length,
ACI 8.3.1.1
measured face-to-face of columns is equal to the following:
Figure 5.23
For exterior panels without edge beams:
Table 5.2
For interior panels:
7U\DLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHG
in ACI 8.3.1.1(a).
Step 2 – Determine a preliminary slab thickness based on two-way shear strength requirements
8VH)LJXUHWRGHWHUPLQHDSUHOLPLQDU\
based on gravity load effects.
Maximum
ACI Eq. (5.3.1b)
For the edge columns along column lines A or E, tributary area,
is equal to the following:
)RUWKHHGJHFROXPQVDORQJFROXPQOLQHVRUWULEXWDU\DUHD
is equal to the following:
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Sect. 5.2.2
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine the area ratio based on the largest tributary area:
Figure 5.3
Therefore,
3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDLQVODEWKLFNQHVVLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK
$¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQV VHH6HFW Because the columns and portions of the slab form the LFRS, the total two-way shear stress is due to the combination of gravity and lateral load effects.
Comments.7RLOOXVWUDWHWKHLPSDFWFROXPQVL]HKDVRQWKHUHTXLUHGVODEWKLFNQHVVGHWHUPLQHDSUHOLPLQDU\VODE
WKLFNQHVVDVVXPLQJLQVTXDUHHGJHFROXPQVLQVWHDGRIWKHLQVTXDUHFROXPQVXVHGLQWKLVH[DPSOH
For exterior panels without edge beams:
Figure 5.3
Therefore,
,QWKLVFDVHSUHOLPLQDU\FDOFXODWLRQVLQGLFDWHDLQVODEWKLFNQHVVLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDU
strength.
5.8.2
Example 5.2 – Determination of Minimum Slab Thickness: Flat Plate System with Edge Beams,
Building #1 (Framing Option B)
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ZLWKLQE\LQHGJHEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum slab thickness based on serviceability requirements
The longest clear span length,
ACI 8.3.1.1
measured face-to-face of columns is equal to the following:
Figure 5.23
For exterior panels with edge beams:
Table 5.2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$FFRUGLQJWRWKHIRRWQRWHLQ7DEOHWKLVPLQLPXPVODEWKLFNQHVVLVEDVHGRQWKHDVVXPSWLRQ
HGJHEHDPV&KHFNWKLVDVVXPSWLRQIRUWKHLQE\LQHGJHEHDPVDVVXPLQJDQLQWKLFNVODE
for the
Determine the effective slab width,
Eq. (5.5), Figure 5.2
Determine the section properties of the beam:
Table 5.3
Determine the moment inertia of the slab using the largest
the smallest
because this results in the maximum
and hence,
Therefore, assuming the same concrete mixture is used in the slab and beams
Eq. (5.2)
Thus, the assumption that
is correct.
For interior panels:
Table 5.2
7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHscribed in ACI 8.3.1.1(a).
Step 2 – Determine a preliminary slab thickness based on two-way
shear strength requirements
Sects. 5.3.3, 5.4.4
Because of the edge beams, two-way shear is not critical at the edge and corner columns.
Check two-way shear requirements at an interior column assuming the lateral load effects are resisted by the exterior frames, that is, the interior columns are subjected to gravity load effects only. The two-way shear stress due to
gravity load moment transfer at an interior column is not considered at this stage.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Maximum
ACI Eq. (5.3.1b)
Section properties of the critical section:
Table 5.11, Case 1
Factored shear force at the critical section:
Figure 5.12
Maximum shear stress,
Eq. (5.14)
Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following for
a square, interior column:
Eq. (5.32)
where
Eq. (5.30)
and
Table 5.20
3UHOLPLQDU\FDOFXODWLRQVLQGLFDWHDVODEWKLFNQHVVHTXDOWRLQLVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK
$¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVLQFOXGLQJWKH
VKHDUVWUHVVGXHWRWKHEHQGLQJPRPHQWVWUDQVIHUUHGIURPWKHVODE VHH6HFW 5.8.3
Example 5.3 – Determination of Minimum Slab Thickness: Two-way Beam-Supported Slab System,
Building #1 (Framing Option C)
Determine the minimum slab thickness for the two-way beam-supported slab system in Figure 1.1 (Framing Option
& DWDW\SLFDOÀRRUZLWKLQE\LQEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum slab thickness based on serviceability requirements
The longest clear span length,
ACI 8.3.1.2
measured face-to-face of beams is equal to the following:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The minimum slab thickness is determined on the basis of the stiffness ratio
Because the slab thickness is not
and
cannot be determined. In lieu of assuming a slab thickness, a minimum slab thickknown at this stage,
this assumption will be checked after the slab thickness has been calculated.
ness is determined assuming
Table 5.6
7U\DLQVODEWKLFNQHVVDQGFDOFXODWH
for the beams and
for the panels.
• (GJHEHDPVRQFROXPQOLQHVDQG
Determine the effective slab width,
Eq. (5.5), Figure 5.2
Determine the section properties of the beam:
Table 5.3
Determine the moment inertia of the slab,
Therefore,
Eq. (5.2)
• Edge beams on column lines A and E
%HFDXVHWKHEHDPVL]HRQFROXPQOLQHVDQGLVWKHVDPHDVWKDWRQFROXPQOLQHV$DQG(
Therefore,
Eq. (5.2)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• ,QWHULRUEHDPVRQFROXPQOLQHVWKURXJK
Determine the effective slab width,
Eq. (5.6), Figure 5.5
Determine the section properties of the beam:
Table 5.3
Determine the moment inertia of the slab,
Therefore,
Eq. (5.2)
• Interior beams along column lines B through D
%HFDXVHWKHEHDPVL]HRQFROXPQOLQHVWKURXJKLVWKHVDPHDVWKDWRQFROXPQOLQHV%WKURXJK'
Therefore,
Eq. (5.2)
Determine
for each type of panel:
• Corner panel:
• Edge panel on column line A or E:
• (GJHSDQHORQFROXPQOLQHRU
• Interior panel:
Because
for all panels, the initial assumption is correct.
7U\DLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHG
LQ$&,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine a preliminary slab thickness based on two-way shear
strength requirements
Sects. 5.3.3, 5.4.4
Because of the column-line beams, two-way shear in the slab is not critical at any of the columns.
:KHUHWKH''0LVXVHGFROXPQOLQHEHDPVPXVWEHGHVLJQHGWRUHVLVWSHUFHQWRIWKHIDFWRUHGVKHDUIRUFHGXH
7KLVFULWHULRQLVVDWLV¿HGIRUDOOWKHEHDPVLQWKLVH[DPSOHZKLFKPHDQVWKH
to gravity loads where
IDFWRUHGVKHDUIRUFHVLQWKHVODEDURXQGWKHFROXPQVDUHHTXDOWR]HUR7KHEHDPVPXVWEHGHVLJQHGIRUWKHFRPbined effects due to gravity and lateral loads.
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5.8.4
Example 5.4 – Determination of Minimum Slab Thickness: Flat Slab System With Edge Beams,
Building #1 (Framing Option D)
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ZLWKVWDQGDUGGURSSDQHOVLQE\LQHGJHEHDPVDQGLQE\LQFROXPQV$VVXPHQRUPDOZHLJKWFRQDQG*UDGHUHLQIRUFHPHQW
crete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum slab thickness based on serviceability requirements
ACI 8.3.1.1
Check the provided drop panel dimensions:
• At interior columns:
Figure 5.4
• At edge and corner columns:
7KXVWKHUHTXLUHPHQWVRI$&, E DUHVDWLV¿HGIRUWKHSODQGLPHQVLRQVRIWKHGURSSDQHOV2QFHDVODEWKLFNness is determined, the projection of the drop panel below the slab must be equal to at least one-fourth of the slab
WKLFNQHVVLQRUGHUWRVDWLVI\WKHUHTXLUHPHQWLQ$&, D The longest clear span length,
measured face-to-face of columns is equal to the following:
Figure 5.23
For exterior panels with edge beams:
Table 5.4
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HGJHEHDPV&KHFNWKLVDVVXPSWLRQIRUWKHLQE\LQHGJHEHDPVDVVXPLQJDQLQWKLFNVODE
for the
Determine the effective slab width,
Eq. (5.5), Figure 5.2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine the section properties of the beam:
Table 5.3
Determine the moment inertia of the slab using the largest
the smallest
because this results in the maximum
and hence,
Therefore, assuming the same concrete mixture is used in the slab and beams
Eq. (5.2)
Thus, the assumption that
is correct.
For interior panels:
Table 5.4
7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVJUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHscribed in ACI 8.3.1.1(b).
Step 2 – Determine a preliminary slab thickness based on two-way
shear strength requirements
Sects. 5.3.3, 5.4.4
Because of the edge beams, two-way shear is not critical at the edge and corner columns.
Check two-way shear requirements at an interior column assuming the lateral load effects are resisted by the exterior frames, that is, the interior columns are subjected to gravity load effects only. The two-way shear stress due to
gravity load moment transfer at an interior column is not considered at this stage.
• Check the shear stress at the critical section located a distance equal to
from the face of the column.
Figure 5.4
)RUIRUPZRUNHFRQRP\VHOHFWDLQGURSSDQHOKHLJKW
Table 5.5
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Maximum
ACI Eq. (5.3.1b)
Figure 5.13
Section properties of the critical section:
Table 5.11, Case 1
Factored shear force at the critical section:
Maximum shear stress,
Eq. (5.14)
Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following
for a square, interior column:
Eq. (5.32)
where
Eq. (5.30)
and
Table 5.20
• Check the shear stress at the critical section located a distance equal to
from the face of the drop panel.
Figure 5.13
Section properties of the critical section:
Table 5.11, Case 1
Factored shear force at the critical section:
Figure 5.13
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Maximum shear stress,
Eq. (5.14)
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$¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQVLQFOXGLQJWKH
VKHDUVWUHVVIURPWKHEHQGLQJPRPHQWVWUDQVIHUUHGIURPWKHVODE VHH6HFW 5.8.5
Example 5.5 – Determination of Minimum Thickness: Two-way Joist System,
Building #1 (Framing Option E)
Determine the minimum overall thickness for the two-way joist system in Figure 1.1 (Framing Option E) at a typiFDOÀRRUDVVXPLQJDIWPRGXOHDLQWKLFNVODEDQGLQE\LQFROXPQV$OVRDVVXPHQRUPDOZHLJKW
DQG*UDGHUHLQIRUFHPHQW
concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the minimum overall thickness based on serviceability requirements
ACI 8.3.1.1
7ZRZD\MRLVWV\VWHPVDUHFRQVLGHUHGDVÀDWVODEVIRUSXUSRVHVRIGHVLJQ
Sect. 5.2.5
The longest clear span length,
measured face-to-face of columns is equal to the following:
Figure 5.23
For exterior panels without edge beams:
Table 5.4
$LQGRPHGHSWKSOXVDLQVODESURYLGHVDQHTXLYDOHQWWKLFNQHVV
Table 5.9
The equivalent thickness is determined as follows:
Table 5.8, Figure 5.7
where
and
Eq. (5.8)
7U\DLQGRPHSOXVDLQVODEIRUDOOSDQHOV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine a preliminary overall thickness based on two-way shear
strength requirements
Check the shear stress at the critical section located a distance equal to
Maximum
Sects. 5.3.3, 5.4.4
from the face of an interior column.
ACI Eq. (5.3.1b)
Section properties of the critical section at an interior column:
Table 5.11, Case 1
Factored shear force at the critical section:
Maximum shear stress,
Eq. (5.14)
Assuming no shear reinforcement is used, the governing design two-way shear strength is equal to the following for
a square, interior column:
Eq. (5.32)
where
Eq. (5.30)
and
Table 5.20
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$¿QDORYHUDOOWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWDOOFROXPQV%His
cause the two-way joist system is also part of the LFRS and is subjected to lateral load effects, the maximum
larger than that calculated above for gravity loads only.
5.8.6
Example 5.6 – Determination of Minimum Slab Thickness: Flat Plate System, Building #3
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LQE\LQLQWHULRUDQGFRUQHUFROXPQVDQGLQE\LQHGJHFROXPQV$VVXPHQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the minimum slab thickness based on serviceability requirements
The longest clear span length,
ACI 8.3.1.1
is equal to the following for an exterior panel:
Figure 5.23
For exterior panels without edge beams:
Table 5.2
For interior panels:
7U\DQLQVODEWKLFNQHVVIRUDOOSDQHOV7KLVWKLFNQHVVLVVOLJKWO\OHVVWKDQWKDWUHTXLUHGE\$&,DQGLV
JUHDWHUWKDQWKHPLQLPXPWKLFNQHVVRILQSUHVFULEHGLQ$&, D Step 2 – Determine a preliminary slab thickness based on two-way
shear strength requirements
Sects. 5.3.3, 5.4.4
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loads only (the shear walls in the building resist lateral load effects).
Maximum
ACI Eq. (5.3.1b)
7KHPD[LPXPFRPELQHGVKHDUVWUHVVRFFXUVDWHGJHFROXPQV%&%DQG&EHQGLQJSHUSHQGLFXODUWRWKH
edge.
Section properties of the critical section:
Table 5.11, Case 3
Factored shear force at the critical section:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Assuming the DDM can be used, the required
at this location is equal to
Sect. 5.3.4
Eq. (5.17)
ACI Eq. (8.4.4.2.2)
Maximum shear stress,
Eq. (5.14)
Eq. (5.32)
where
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$¿QDOVODEWKLFNQHVVLVHVWDEOLVKHGRQFHWZRZD\VKHDUUHTXLUHPHQWVDUHFKHFNHGDWDOOFROXPQV VHH6HFW 5.8.7
Example 5.7 – Determination of Required Flexural Reinforcement: Flat Plate System,
Building #1 (Framing Option A), SDC A
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FROXPQV VHH)LJXUH $VVXPHWKH6LWH&ODVVLV&ZKLFKUHVXOWVLQWKHEXLOGLQJEHLQJDVVLJQHGWR6'&$>VHH
DQG*UDGHUHLQIRUFHPHQW
([DPSOH3DUW D @$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
ሺtyp.ሻ
A
E
ᇳ
D
ᇱ
23 -6
ሺtyp.ሻ
Design
strip
A
N
A
C
A
B
A
6ᇱ -7½ᇳ
½MS
11ᇱ -9ᇳ
CS
6ᇱ -7½ᇳ
½MS
Figure 5.41 Design strips for the flat plate system in Example 5.7.
Step 2 – Check if the Direct Design Method can be used to determine
gravity load bending moments
Sect. 5.3.4
Check the following limitations:
Figure 5.22
• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK
south and east-west directions, respectively.
• Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
• Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
VXSSRUWVQRWWRH[FHHG«
• &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ
centerlines of successive columns…None of the columns are offset.
• All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP
• 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG«
DVVXPLQJDQLQWKLFNVODE
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«7KHUHDUHQREHDPVLQWKLVÀRRUV\VWHPVRWKLV
in the two perpendicular directions:
limitation is not applicable.
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middle strips.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
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be combined with those due to the effects from lateral forces using the appropriate load combinations:
Dead load:
Eq. (5.17)
Live load:
These total factored static moments are the same for all spans of the interior design strip in the direction of analysis.
Step 4 – Distribute the total factored static moment to the column strips and middle strips
Table 5.16
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service dead and live load bending moments in the column strip and middle strip in an end span and interior span is
JLYHQLQ7DEOH
Table 5.25 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the
Flat Plate System in Example 5.7
End Span
Design Strip
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior
Negative
Positive
Interior
Negative
Column strip
Middle strip
Step 5 – Determine the bending moments due to wind forces
ACI 6.6
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7DEOH
$WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFHDQGDOLQHDU¿UVWRUGHUDQDO\VLVZDV
SHUIRUPHGXVLQJWKHQRUWKVRXWKZLQGORDGVLQ7DEOHDSSOLHGWRWKHFHQWURLGRIWKHEXLOGLQJIDFHDWWKHURRIDQG
ÀRRUOHYHOV VHH)LJXUH ,QWKHPRGHOWKHFROXPQVDUH¿[HGDWWKHEDVH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
AA
BA
CA
DA
EA
23ᇱ -6ᇳ
ሺtyp.ሻ
27.4 kips
4 @ 11ᇱ -6ᇳ ൌ 46ᇱ -0ᇳ
R
53.4 kips
5
51.6 kips
4
49.2 kips
3
14ᇱ -0ᇳ
51.3 kips
2
Figure 5.42 Total wind loads applied to Building #1.
The following reduced moments of inertia are used to account for cracked sections:
ACI Table 6.6.3.1.1(a)
• Columns:
• Slabs:
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wind loads assuming all the frames in the north-south direction are part of the LFRS. From the analysis, it was
determined the majority of the wind load effects at an interior design strip are resisted by slab strips centered on the
columns with widths approximately equal to the column strip width; thus, all the wind load effects are assigned to
the column strips in the direction of analysis in this example.
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WKHVHFRQGÀRRUOHYHO7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWK
the north direction and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when
looking at the east elevation of the building).
Table 5.26 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Flat Plate
System in Example 5.7
End Span
Design Strip
Column strip
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior
Negative
Positive
Interior
Negative
—
—
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUW D the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.
,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH
less than those due to wind forces, and, thus, need not be considered in this example.
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Step 7 – Determine the combined factored bending moments due to gravity and wind forces
ACI Table 5.3.1
7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKHEHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLV $&, ,WLVHYLdent at all locations that the maximum moments are equal to those from the combination of the gravity load effects.
Table 5.27 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in
Example 5.7
End Span
Load Combination
Location
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
SSR
83.8
SSL
83.8
SSR
SSL
Column strip
Middle strip
Column
strip
Middle strip
Column
strip
Interior Span
Middle strip
Step 8 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations where is the width of the column strip or middle strip and
Eq. (5.35)
Eq. (5.36)
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greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected
VRWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQWWKH
provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area of
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.28 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in
Example 5.7
b (in.)
As (in.2)*
Reinforcement*
Mu (ft-kips)
Location
Exterior
Negative
Column
strip
Positive
First Interior
Negative
Exterior
Negative
End Span
Middle
strip
Positive
First Interior
Negative
Column
strip
Positive
Negative
Interior Span
Middle
strip
Positive
Negative
* Min. As for the column strip
Min. As for the middle strip
Max. spacing
For
For
For the column strip:
For the middle strip:
Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements
ACI 8.4.2.2
• Edge columns
Determine the required reinforcement based on
Sect. 5.3.4
This moment is greater than 86.9 ft-kips, which is the moment due to the load combination
Table 5.27
Eq.(5.9)
where
Table 5.11, Case 3
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Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
The moment
must be transferred over the effective slab width
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
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FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQ VHH7DEOH )RUV\PPHWU\DGGEDUVDQG
check the bar spacing:
)RUZLWKLQWKHLQZLGWK
)RUZLWKLQWKH
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RIEDUVDUHUHTXLUHGDWWKHHGJHFROXPQVZLWKLQWKHFROXPQVWULSZLWKRIWKHEDUVFRQFHQWUDWHG
ZLWKLQDZLGWKRILQ
Check
within
ACI 8.6.1.2
Table 5.11, Case 3
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
and
Table 5.20
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Eq. (5.37)
Provided
within
Table 5.26
Therefore,
Eq. (5.38)
Provided
within
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ͳͳ̵Ǧͻԣ
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Figure 5.43 Reinforcement details for the top reinforcing bars at the edge columns
in the flat plate system in Example 5.7.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• Interior columns
Determine the required reinforcement based on the largest transfer moment at the interior columns.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to gravity
load effects and combined gravity and wind load effects can be determined from the following equations where
the spans in the direction of analysis and perpendicular to the direction of analysis are equal.
For gravity loads only:
Eq. (5.21)
)RUJUDYLW\DQGZLQGORDGVDWWKH¿UVWLQWHULRUFROXPQV
due to gravity load effects
due to wind load effects
Table 5.26
Total
For gravity and wind loads at the interior column:
due to gravity load effects
due to wind load effects
Total
Use
at all interior columns.
Eq. (5.9)
where
Table 5.11, Case 1
Determining whether
The moment
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
must be transferred over the effective slab width
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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satisfy moment transfer requirements at this location.
Check
within
ACI 8.6.1.2
Table 5.11, Case 1
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
and
Table 5.20
Eq. (5.37)
$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG
At the interior column, provided
within
within
Based on this load combination, additional reinforcement must be provided within
PXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,
to satisfy the mini-
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$WWKH¿UVWLQWHULRUFROXPQV
Table 5.26
At the interior column,
Therefore, maximum shear stress is equal to the following:
Eq. (5.38)
Eq. (5.37)
The required
from this load combination is less than the required
combination, so required
from the
load
To satisfy minimum reinforcement requirements within
EDUVPXVWEHSODFHGZLWKLQWKH
DWWKH¿UVWLQWHULRUFROXPQV
in. effective slab width
width of the column strip. For
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ͳͳԢǦͻԣ
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Figure 5.44 Reinforcement details for the top reinforcing bars at the first interior columns
in the flat plate system in Example 5.7.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS
Step 10 – Determine the reinforcement details
ACI 8.7
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EDUV VREDUVDUHPDGHFRQWLQXRXV
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A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
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)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͳͶǦ͓ͷ
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͵ᇱ ǦͲᇱᇱ ‹††Ž‡–”‹’
Figure 5.45 Reinforcement details for an interior design strip in the north-south direction.
for the flat plate system in Example 5.7.
5.8.8
Example 5.8 – Determination of Required Flexural Reinforcement: Flat Plate System With Edge
Beams, Building #1 (Framing Option B), SDC A
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V\VWHPLQ)LJXUH )UDPLQJ2SWLRQ% DWWKHVHFRQGÀRRUOHYHOZLWKDQLQWKLFNVODELQE\LQHGJH
EHDPVDQGLQE\LQFROXPQV VHH)LJXUH $VVXPHWKH6LWH&ODVVLV&$OVRDVVXPHQRUPDOZHLJKWFRQDQG*UDGHUHLQIRUFHPHQW
crete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
ሺtyp.ሻ
A
E
23ᇱ -6ᇳ
ሺtyp.ሻ
D
Design
strip
A
N
A
C
A
B
A
6ᇱ -7½ᇳ
½MS
11ᇱ -9ᇳ
CS
6ᇱ -7½ᇳ
½MS
Figure 5.46 Design strips for the flat plate system in Example 5.8.
Step 2 – Check if the Direct Design Method can be used to determine gravity load
bending moments
Sect. 5.3.4
Check the following limitations:
Figure 5.22
• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK
south and east-west directions, respectively.
• Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
• Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
VXSSRUWVQRWWRH[FHHG«
• &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ
centerlines of successive columns…None of the columns are offset.
• All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP
• 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG«
DVVXPLQJDQLQWKLFNVODE
• )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV
«7KHUHDUHRQO\SHULPHWHUEHDPVLQWKLVÀRRU
in the two perpendicular directions:
system, so this limitation is not applicable.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG
middle strips.
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
7KHÀDWSODWHV\VWHPLVQRWSDUWRIWKH/)56 RQO\WKHFROXPQVDQGHGJHEHDPVIRUPWKH/)56 WKHUHIRUHLWPXVW
resist the effects from gravity loads only.
Maximum
ACI Eq. (5.3.1b)
Total factored static moment
Eq. (5.17)
The total factored static moment is the same for all spans of the interior design strip in the direction of analysis.
Step 4 – Distribute the total factored static moment to the column strips and middle strips
Table 5.17
'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWSODWHV\VWHPZLWKHGJHEHDPVDUHJLYHQLQ7DEOH7KHWDEXODWHGH[WHULRU
where
is determined
QHJDWLYHFROXPQVWULSPRPHQWFRHI¿FLHQWVDUHDSSOLFDEOHLQFDVHVZKHUHWKHWHUP
by the following equation:
Eq. (5.18)
Eq. (5.5), Figure 5.2
Cross-sectional constant
is determined by the following equation:
Eq. (5.19)
Calculations for
DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW
Therefore, with the slab and beam cast monolithically with the same concrete mixture (that is,
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5-82
):
8.5ᇳ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
24.0ᇳ 15.5ᇳ 24.0
24.0ଷ ൈ 28.0
൰ൈ
቉
3
28.0
‫ܥ‬஺ ൌ ቈ൬1 െ 0.63
28.0ᇳ ൅ ቈ൬1 െ 0.63
8.5ᇳ Case A
8.5ଷ ൈ 15.5
8.5
൰ൈ
቉ ൌ 61,428 in.ସ 3
15.5
15.5ᇳ 15.5ᇳ ‫ܥ‬஻ ൌ ቈ൬1 െ 0.63
28.0ᇳ 15.5
15.5ଷ ൈ 28.0
൰ൈ
቉
28.0
3
൅ ቈ൬1 െ 0.63
Case B
8.5
8.5ଷ ൈ 43.5
൰ൈ
቉ ൌ 30,444 in.ସ 43.5
3
Figure 5.47 Calculations for cross-sectional constant C.
Because
, the exterior negative column strip bending moment is equal to the following:
Table 5.17
A summary of the design bending moments in the column strip and middle strip in an end span and in an interior
VSDQLVJLYHQLQ7DEOH
Table 5.29 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in
Example 5.8
End Span
Design Strip
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior
Negative
Positive
Interior Negative
Column strip
Middle strip
Step 5 – Determine the bending moments due to wind forces
'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ
7DEOH
Because it has been assumed all the wind load effects are resisted by the perimeter moment-resisting frames, the
VODEVUHVLVWRQO\JUDYLW\ORDGHIIHFWVZKLFKDUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUW D the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.
,WLVDOVRDVVXPHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH
resisted by the perimeter moment-resisting frames, and, thus, need not be applied to the slabs.
Step 7 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations where is the width of the column strip or middle strip and
Eq. (5.35)
Eq. (5.36)
$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH
greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected
VRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQW
the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area
FRUUHVSRQGLQJWRWHQVLRQFRQWUROOHGVHFWLRQVZKLFKLVGHWHUPLQHGE\(T RIÀH[XUDOUHLQIRUFHPHQW
WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,
Table 5.30 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System
in Example 5.8
Mu (ft-kips)
Location
Column
strip
End Span
Middle
strip
Interior Span
Column
strip
Middle
strip
Exterior
Negative
Positive
First Interior
Negative
Exterior
Negative
Positive
First Interior
Negative
Positive
Negative
Positive
Negative
b (in.)
As (in.2)*
Reinforcement*
3.33
* Min. As for the column strip
Min. As for the middle strip
Max. spacing
For
For
For the column strip:
For the middle strip:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements
ACI 8.4.2.2
• Edge columns
The factored slab moments resisted by the edge columns need not be checked in this example because of the
beams at the perimeter of the slab. These edge beams must be designed for shear forces and torsional moments
transferred from the slab.
• Interior columns
Determine the required reinforcement based on the largest transfer moment at the interior columns.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the
gravity load effects can be determined from the following equation where the spans in the direction of analysis
and perpendicular to the direction of analysis are equal:
Eq. (5.21)
Eq. (5.9)
where
Table 5.11, Case 1
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
must be transferred over the effective slab width
The moment
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKDXQLIRUPVSDFLQJRIEDUVLQWKHFROXPQVWULSRI
WKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQforcement is required to satisfy moment transfer requirements at these locations. Similar calculations show that
QRDGGLWLRQDOUHLQIRUFHPHQWLVUHTXLUHGDWWKHLQWHULRUFROXPQZKHUHEDUVDUHSURYLGHGLQWKHFROXPQVWULS
Check
within
ACI 8.6.1.2
Table 5.11, Case 1
Factored shear force at the critical section:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
Eq. (5.38)
where
Eq. (5.30)
and
Table 5.20
Eq. (5.37)
$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG
At the interior column, provided
within
within
Therefore, additional reinforcement must be provided within
PHQWVRI$&,
to satisfy the minimum reinforcement require-
EDUVPXVWEHSODFHGZLWKLQWKHLQHITo satisfy minimum reinforcement requirements within
DWWKH¿UVWLQWHULRUFROXPQVZLWKWKH
fective slab width
width of the column strip. For symmetry,
UHPDLQLQJEDUVWREHSODFHGZLWKLQWKH
DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH¿UVWLQWHULRU
FROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOIRUWKLV
FDVHLVVLPLODUWRWKDWVKRZQLQ)LJXUH
6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS)RUV\PPHWU\DGGEDUWRWKH
LQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKHLQWHULRUFROXPQZLWKEDUV
FRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK
Step 9 – Determine the reinforcement details
ACI 8.7
7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only.
A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
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5-86
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH)RUVLPSOHUGHWDLOLQJWKH
OHQJWKVRIWKHQHJDWLYHUHLQIRUFHPHQWDWDOOFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDUHVHWHTXDOWRSHUFHQWRIWKH
VHH)LJXUH 7KHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,
clear span length, which is greater than
DUHLQFOXGHG
ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹͳǦ͓ͷ
ͻǦ͓ͷ
ͳͻǦ͓ͷ
ʹᇱ Ǧʹᇱᇱ ʹǦ͓ͷ
ͻǦ͓ͷ
൒ ͸ԣ
ʹͳᇱ Ǧ͸ᇱᇱ ʹͳᇱ Ǧ͸ᇱᇱ ‘Ž—–”‹’
ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ͳͲǦ͓ͷ
ͳͲǦ͓ͷ
ͳͲǦ͓ͷ
ͷǦ͓ͷ
ͷǦ͓ͷ
൒ ͸ԣ
ʹͳᇱ Ǧ͸ᇱᇱ ͷǦ͓ͷ
൒ ͸ԣ
ͷǦ͓ͷ
͵ᇱ ǦͲᇱᇱ ͵ᇱ ǦͲᇱᇱ ʹͳᇱ Ǧ͸ᇱᇱ ͵ᇱ ǦͲᇱᇱ ͷǦ͓ͷ
͵ᇱ ǦͲᇱᇱ ‹††Ž‡–”‹’
Figure 5.48 Reinforcement details for an interior design strip in the north-south direction
for the flat plate system in Example 5.8.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.8.9
Example 5.9 – Determination of Required Flexural Reinforcement: Two-way Beam-Supported Slab
System, Building #1 (Framing Option C), SDC A
'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHWZRZD\
EHDPVXSSRUWHGVODEV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ& DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODE
LQE\LQEHDPVDQGLQE\LQFROXPQV VHH)LJXUH $VVXPHWKH6LWH&ODVVLV&$OVRDVVXPHQRUDQG*UDGHUHLQIRUFHPHQW
malweight concrete with
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
ሺtyp.ሻ
A
E
23ᇱ -6ᇳ
ሺtyp.ሻ
D
Design
strip
A
N
A
C
A
B
A
6ᇱ -7½ᇳ
½MS
11ᇱ -9ᇳ
CS
6ᇱ -7½ᇳ
½MS
Figure 5.49 Design strips for the two-way beam-supported slab system in Example 5.9.
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Check if the Direct Design Method can be used to determine
gravity load bending moments
Check the following limitations:
Sect. 5.3.4
Figure 5.22
• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK
south and east-west directions, respectively.
• Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
• Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
VXSSRUWVQRWWRH[FHHG«
• &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ
centerlines of successive columns…None of the columns are offset.
• All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP
• 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG«
DVVXPLQJDLQWKLFNVODE
• )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPVLQ
…Check the relative stiffnesses for interior, edge, and
the two perpendicular directions:
DUHGHWHUPLQHGLQ([DPSOH
corner panels where
Interior panel:
North-south interior beam:
East-west interior beam:
Edge panel (north or south face):
North-south interior beam:
East-west edge beam:
Edge panel (east or west face):
North-south edge beam:
East-west interior beam:
Corner panel:
North-south edge beam:
East-west edge beam:
%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG
middle strips.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
The slab in this system is not part of the LFRS (only the columns and beams form the LFRS); therefore, the slab
must resist the effects from gravity loads only.
Maximum
ACI Eq. (5.3.1b)
Total factored static moment
Eq. (5.17)
The total factored static moment is the same for all spans of an interior design strip in the direction of analysis.
Step 4 – Distribute the total factored static moment to the column strips and middle strips
7RWDOGHVLJQPRPHQWVDUHGHWHUPLQHGXVLQJ7DEOHDQGWKHLQIRUPDWLRQLQ6WHSRI6HFW$VXPPDU\RI
WKHWRWDOGHVLJQVWULSPRPHQWVLVJLYHQLQ7DEOH
Table 5.31 Total Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Beam-Supported Slab System in Example 5.9
End Span
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior Negative
Positive
Interior Negative
Determine the percentages for factored bending moments in the column strip:
Table 5.15
• Exterior support:
Interior beams in north-south direction:
Example 5.3
Eq. (5.18)
Eq. (5.5), Figure 5.2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Cross-sectional constant
is determined by the following equation:
Eq. (5.19)
DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW
7.0ᇳ Calculations for
24.0ᇳ 17.0ᇳ 24.0
24.0ଷ ൈ 28.0
൰ൈ
቉
28.0
3
‫ܥ‬஺ ൌ ቈ൬1 െ 0.63
28.0ᇳ ൅ ቈ൬1 െ 0.63
7.0ᇳ Case A
7.0
7.0ଷ ൈ 17.0
൰ൈ
቉ ൌ 60,791 in.ସ 17.0
3
17.0ᇳ 17.0ᇳ ‫ܥ‬஻ ൌ ቈ൬1 െ 0.63
28.0ᇳ ൅ ቈ൬1 െ 0.63
Case B
17.0
17.0ଷ ൈ 28.0
൰ൈ
቉
28.0
3
7.0
7.0ଷ ൈ 45.0
൰ൈ
቉ ൌ 32,956 in.ସ 45.0
3
Figure 5.50 Calculations for cross-sectional constant C.
With the slab and beam cast monolithically with the same concrete mixture (that is,
):
Therefore,
• Positive:
• Interior support:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine percentages for factored bending moments in the middle strip:
• Exterior support:
• Positive:
• Interior support:
Determine percentages for factored bending moments in the column-line beams:
WKHEHDPVPXVWUHVLVWRIWKHWRWDOFROXPQVWULSPRPHQWV
Because
• Exterior support:
• Positive:
• Interior support:
A summary of the design bending moments in the column strip and middle strip in an end span and interior span is
JLYHQLQ7DEOH
Table 5.32 Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Beam-Supported
Slab System in Example 5.9
End Span
Design Strip
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior
Negative
Positive
Interior Negative
Total
Total
Column
Beam
strip
Slab
Middle strip
The following illustrates the calculation of the design bending moments at the exterior negative support:
The bending moments at the other critical sections can be obtained in a similar fashion.
Step 5 – Determine the bending moments due to wind forces
ACI 6.6
'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ
7DEOH
Because it has been assumed all the wind load effects are resisted by the moment-resisting frames along all the colXPQOLQHVWKHVODEVUHVLVWRQO\JUDYLW\ORDGHIIHFWVZKLFKDUHJLYHQLQ7DEOH
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUW D the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.
@Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,WLVDOVRDVVXPHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH
resisted by the moment-resisting frames, and, thus, need not be applied to the slabs.
Step 7 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations:
Eq. (5.35)
Eq. (5.36)
where
is the width of the column strip or middle strip and
$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOHIRUWKHVODE7KHSURYLGHGDUHDVRIUHLQforcement are greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars
DUHVHOHFWHGVRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,
It is evident the provided areas of reinforcement at all critical sections in the column strip and middle strip are less
corresponding to tension-controlled sections, which is determined by
WKDQWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
(T WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,
Table 5.33 Required Flexural Reinforcement at the Second-Floor Level for the Two-way Beam-Supported
Slab System in Example 5.9
b (in.)
As (in.2)*
Reinforcement*
First Interior
Negative
Exterior
Negative
Mu (ft-kips)
Location
Exterior
Negative
Column
strip
End Span
Middle
strip
Positive
Positive
First Interior
Negative
Column
strip
Interior Span
Middle
strip
Positive
Negative
Positive
Negative
* Min. As for the column strip
Min. As for the middle strip
Max. spacing
For
For
For the column strip:
For the middle strip:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements
ACI 8.4.2.2
The factored slab moments resisted by the columns need not be checked in this example because of the column-line
beams. These beams must be designed for shear forces and torsional moments transferred from the slab.
Step 9 – Determine the reinforcement details
ACI 8.7
7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only.
A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
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Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
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span length, which is greater than
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͳͳǦ͓Ͷ
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Figure 5.51 Reinforcement details for an interior design strip in the north-south direction
for the two-way beam-supported slab system in Example 5.9.
5.8.10 Example 5.10 – Determination of Required Flexural Reinforcement: Flat Slab System With Edge
Beams, Building #1 (Framing Option D), SDC A
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Site Class is C. Also assume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
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ሺtyp.ሻ
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strip
A
N
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A
5ᇱ -3ᇳ ሺtyp.ሻ
A
B
A
6ᇱ -7½ᇳ
½MS
11ᇱ -9ᇳ
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6ᇱ -7½ᇳ
½MS
Figure 5.52 Design strips for the flat slab system in Example 5.10.
Step 2 – Check if the Direct Design Method can be used to determine gravity load
bending moments
Sect. 5.3.4
Check the following limitations:
Figure 5.22
• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK
south and east-west directions, respectively.
• Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
• Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
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centerlines of successive columns…None of the columns are offset.
• All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
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in the two perpendicular directions:
system, so this limitation is not applicable.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG
middle strips.
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
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slab must resist the effects from gravity loads only.
Maximum
ACI Eq. (5.3.1b)
Total factored static moment
Eq. (5.17)
The total factored static moment is the same for all spans of the interior design strip in the direction of analysis.
Step 4 – Distribute the total factored static moment to the column strips and middle strips
Table 5.17
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where
is determined
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by the following equation:
Eq. (5.18)
Eq. (5.5), Figure 5.2
Cross-sectional constant
is determined by the following equation:
Eq. (5.19)
Calculations for
DUHJLYHQLQ)LJXUHZKHUHLWLVIRXQGWKDW
Therefore, with the slab and beam cast monolithically with the same concrete (that is,
):
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8.0ᇳ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
24.0ᇳ 16.0ᇳ 24.0
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28.0
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Case B
8.0
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3
Figure 5.53 Calculations for cross-sectional constant C.
Because
, the exterior negative column strip bending moment is equal to the following:
Table 5.17
A summary of the design bending moments in the column strip and middle strip in an end span and interior span is
given in Table 5.34.
Table 5.34 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Slab System
in Example 5.10
End Span
Design Strip
1
Exterior Negative
Interior Span
2
3
4
5
Positive
First Interior
Negative
Positive
Interior Negative
Column strip
Middle strip
Step 5 – Determine the bending moments due to wind forces
ACI 6.6
Design wind forces in the north-south direction on Building #1 are determined in Example 3.1 and are given in
Table 3.10.
Because it has been assumed all the wind load effects are resisted by the perimeter moment-resisting frames, the
slabs resist only gravity load effects, which are given in Table 5.34.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUW D the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.
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resisted by the perimeter moment-resisting frames, and, thus, need not be applied to the slabs.
Step 7 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations:
Eq. (5.35)
Eq. (5.36)
where
is the width of the column strip or middle.
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negative critical sections in the column strip:
Figure 5.28
where
At all other critical sections,
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greater than or equal to then the minimum required in accordance with ACI 8.6.1.1 and the number of bars are
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evident the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than
corresponding to tension-controlled sections, which is determined by Eq.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.35 Required Flexural Reinforcement at the Second-Floor Level for the Flat Slab System
in Example 5.10
Mu (ft-kips)
Location
Exterior
Negative
Column
strip
Positive
First Interior
Negative
Exterior
Negative
End
Span
Middle
strip
Positive
First Interior
Negative
Interior
Span
Column
strip
Middle
strip
Positive
Negative
Positive
Negative
b (in.)
d (in.)
As (in.2)*
Reinforcement*
* Min. As for the column strip at negative critical sections
Min. As for the column strip at positive critical sections
Min. As for the middle strip
Max. spacing
For
For
For the column strip at negative critical sections:
For the column strip at positve critical sections:
For the middle strip:
Step 8 – Check that the flexural reinforcement is adequate for moment transfer requirements
ACI 8.4.2.2
• Edge columns
The factored slab moments resisted by the edge columns need not be checked in this example because of the
beams at the perimeter of the slab. These edges beams must be designed for shear forces and torsional moments
transferred from the slab.
• Interior columns
Determine the required reinforcement based on the largest transfer moment at the interior columns.
transferred to an interior column due to the
In lieu of a more exact analysis, the factored slab moment,
gravity load effects can be determined from the following equation where the spans in the direction of analysis
and perpendicular to the direction of analysis are equal:
Eq. (5.21)
Eq. (5.9)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
where
Table 5.11, Case 1
Determining whether
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must be transferred over the effective slab width
The moment
Eq. (5.11)
Determine the required
Eq. (5.36)
Eq. (5.35)
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WKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVRQRDGGLWLRQDOUHLQIRUFHment is required to satisfy moment transfer requirements at these locations. Similar calculations show that no
additional reinforcement is required at the interior column.
Check
within
ACI 8.6.1.2
Determine the governing factored two-way shear stress,
at the interior columns.
Table 5.11, Case 1
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
and
Table 5.20
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Thus,
Eq. (5.37)
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At the interior column, provided
within
within
Additional reinforcement must be provided in
to satisfy the minimum reinforcement requirements of ACI
EDUVPXVWEHSODFHGZLWKLQWKHLQHITo satisfy minimum reinforcement requirements within
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fective slab width
width of the column strip. For symmetry,
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Step 9 – Determine the reinforcement details
ACI 8.7
7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHFDQEHXVHGEHFDXVHWKHVODELVVXEMHFWHGWRWKHHIIHFWVIURPXQLformly distributed gravity loads only.
A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
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ACI Table 25.4.2.5
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5-102
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
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clear span length, which is greater than
DUHLQFOXGHG
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Figure 5.54 Reinforcement details for an interior design strip in the north-south direction
for the flat slab system in Example 5.10.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.8.11 Example 5.11 – Determination of Required Flexural Reinforcement: Two-way Joist System,
Building #1 (Framing Option E), SDC A
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concrete with
1A
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ᇱ
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ᇳ
A
50 -0
ሺtyp.ሻ
A
18ᇱ -6ᇳ
N
A
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7ᇱ -10ᇳ
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15ᇱ -6ᇳ
ᇱ
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A
Design strip
ᇱ
31 -4
ᇳ
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D
A
Figure 5.55 Design strips for the two-way joist system in Example 5.11.
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
Step 2 – Check if the Direct Design Method can be used to determine
gravity load bending moments
Sect. 5.3.4
Check the following limitations:
Figure 5.22
•
There must be 3 or more continuous spans in each direction…There are 3 continuous spans in the north-south
and east-west directions.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
•
Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
•
Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
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•
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centerlines of successive columns…None of the columns are offset.
•
All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
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•
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•
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in the two perpendicular directions:
limitation is not applicable.
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middle strips.
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
Because the two-way joist system is part of the LFRS, service dead and live load bending moments are determined,
which will be combined with those due to the effects from lateral forces:
Dead load:
Eq. (5.17)
Live load:
These total factored static moments are the same for all spans of the interior design strip in the direction of analysis.
Step 4 – Distribute the total factored static moment to the column strips and middle strips
Table 5.16
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.36 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the
Two-way Joist System in Example 5.11
End Span
1
Design Strip
Exterior Negative
Interior Span
2
3
4
5
Positive
First Interior
Negative
Positive
Interior Negative
Column strip
Middle strip
Step 5 – Determine the bending moments due to wind forces
ACI 6.6
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͵
ʹͺǤͺ‹’•
ͳͶᇱ ǦͲᇳ
ʹ
Figure 5.56 Total wind loads applied to Building #1.
The following reduced moments of inertia are used to account for cracked sections:
ACI Table 6.6.3.1.1(a)
• Columns:
• Two-way joists:
$¿QLWHHOHPHQWDQDO\VLVZDVSHUIRUPHGWRGHWHUPLQHWKHH[WHQWRYHUZKLFKWKHVODEV\VWHPUHVLVWVWKHHIIHFWVIURP
the wind loads. From the analysis, it was determined the majority of the wind load effects at an interior design strip
are resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus,
all the wind load effects are assigned to the column strips in the direction of analysis in this example.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHEHQGLQJPRPHQWVLQWKHFROXPQVWULSGXHWRZLQGORDGVDUHJLYHQLQ7DEOHIRUWKHHQGDQGLQWHULRUVSDQV
7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHHDVWGLUHFWLRQDQG
the west direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the south
elevation of the building).
Table 5.37 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Two-way Joist
System in Example 5.11
End Span
1
Design Strip
Exterior Negative
Interior Span
2
3
4
5
Positive
First Interior
Negative
Positive
Interior Negative
ņ
Column strip
ņ
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV&IRUWKLVEXLOGLQJ)URP([DPSOH3DUW D the building is assigned to SDC A for this case. Therefore, effects due to seismic forces need not be considered.
,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH
less than those due to wind forces, and, thus, need not be considered in this example.
Step 7 – Determine the combined factored bending moments due to gravity and wind forces
ACI Table 5.3.1
7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKHEHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLV $&, ,WLVHYLdent that at all locations, the maximum moments are equal to those from the combination of the gravity load effects.
Table 5.38 Design Bending Moments (ft-kips) at the Second-Floor Level for the Two-way Joist System in
Example 5.11
End Span
Load Combination
Location
1
Exterior
Negative
First Interior
Negative
4
5
Positive
Interior
Negative
SSR
SSL
SSR
SSL
Middle strip
Middle strip
Column
strip
Positive
3
Column strip
Column
strip
2
Interior Span
Middle strip
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 8 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations:
Eq. (5.35)
Eq. (5.36)
,QWKLVH[DPSOHWKHUHDUHULEVLQWKHFROXPQVWULSDQGULEVLQWKHPLGGOHVWULS ULEVDUHLQHDFKRIWKHKDOI
middle strips).
• &ROXPQVWULSÀH[XUDOUHLQIRUFHPHQW
is resisted by the solid area of concrete
At the negative critical sections, the total negative factored moment
is determined using
and
(the width
around the column, and
RIWKHVROLGKHDGSHUSHQGLFXODUWRWKHGLUHFWLRQRIDQDO\VLVLVLQZKLFKLVLQOHVVWKDQWKHZLGWKRIWKH
FROXPQVVWULSWKHLQZLGWKLVXVHGWRFDOFXODWH ).
At the positive critical sections, the total positive factored moment
is determined for each rib using
the 6 ribs, and
is shared equally between
and
• 0LGGOHVWULSÀH[XUDOUHLQIRUFHPHQW
At the negative critical sections, the total negative factored moment
and
mined using
LVUHVLVWHGE\ULEVDQG
At the positive critical sections, the total positive factored moment
is determined per rib using
and
is deter-
LVVKDUHGHTXDOO\EHWZHHQWKHULEV
and
7KHPLQLPXPUHTXLUHGÀH[XUDOUHLQIRUFHPHQWDWQHJDWLYHFULWLFDOVHFWLRQVLQWKHFROXPQDQGPLGGOHVWULSVLV
equal to the following:
ACI 8.6.1.1
$WDOORWKHUFULWLFDOVHFWLRQVWKHPLQLPXPUHTXLUHGÀH[XUDOUHLQIRUFHPHQWFDQEHGHWHUPLQHGXVLQJWKHUHTXLUHments for beams:
ACI 9.6.1.2
$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOHZKLFKLVDSSOLFDEOHIRUERWKWKHHQG
and interior spans. It can be determined all sections are tension-controlled.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.39 Required Flexural Reinforcement at the Second-Floor Level for the Two-way Joist System in
Example 5.11
b (in.)
As (in.2)*
Reinforcement*
Exterior Negative
Positive
ULE
First Interior Negative
Positive
ULE
First Interior Negative
Mu (ft-kips)
Location
Column strip
Exterior Negative
Middle strip
* Min. As for negative moment in the column strip
Min. As for negative moment in the middle strip
minimum number of bars
Min. As for all other critical sections
;
Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements
• Edge columns
Determine the required reinforcement based on
ACI 8.4.2.2
Sect. 5.3.4
7KLVPRPHQWLVJUHDWHUWKDQIWNLSVZKLFKLVWKHPRPHQWGXHWRWKHORDGFRPELQDWLRQ
Table 5.38
Eq. (5.9)
where
Table 5.11, Case 3
Determining whether
The moment
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
must be transferred over the effective slab width
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH
FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQ VHH7DEOH )RUV\PPHWU\DGGEDUDQG
check bar spacing:
)RUZLWKLQWKHLQZLGWK
)RUZLWKLQWKH
Check
,
within
ACI 8.6.1.2
Table 5.11, Case 3
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
Table 5.20
Eq. (5.37)
Provided
within
quired based on this load combination.
, so no additional reinforcement is re-
Table 5.37
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
Eq. (5.38)
Provided
within
No additional reinforcement is required to satisfy moment transfer requirements at the edge columns.
• Interior columns
Determine the required reinforcement based on the largest transfer moment at the interior columns.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the
gravity load effects and gravity and wind load effects can be determined from the following equations where the
spans in the direction of analysis and perpendicular to the direction of analysis are equal.
For gravity loads only:
Eq. (5.21)
For gravity and wind loads:
due to gravity load effects
due to wind load effects
Table 5.37
Total
Therefore, use
at the interior columns.
Eq. (5.9)
where
Table 5.11, Case 1
Determining whether
The moment
can be increased to the values in A&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
must be transferred over the effective slab width
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKWKHEDUVXQLIRUPO\VSDFHGZLWKLQWKHFROXPQVWULS
RIWKHEDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHment is required to satisfy moment transfer requirements at this location.
Check
within
ACI 8.6.1.2
Table 5.11, Case 1
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
and
Table 5.20
Eq. (5.37)
Table 5.37
Therefore, the maximum shear stress is equal to the following:
Eq. (5.38)
Eq. (5.37)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Thus, the
Provided
load combination governs with respect to minimum reinforcement requirements.
within
Thus, no additional reinforcement is required to satisfy moment transfer requirements at the interior columns.
Step 10 – Determine the reinforcement details
ACI 8.7
7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQWKHLQWKLFNVODEVKRZQLQ)LJXUHFDQQRWEHXVHGLQWKHFROXPQ
VWULSEHFDXVHRIWKHHIIHFWVGXHWRZLQGORDGV7RDFFRXQWIRUWKHZLQGORDGHIIHFWVSHUFHQWRIWKHWRSEDUVLQWKH
column strips are made continuous over the spans. In this case, the maximum amount of top reinforcement occurs at
WKH¿UVWLQWHULRUVXSSRUW EDUV VREDUVDUHPDGHFRQWLQXRXV
The remaining bars in the column strip and the top bars in the middle strip can be terminated at the locations identi¿HGLQ)LJXUH
According to ACI 8.8.1.6, at least one bottom bar in each rib must be continuous and must be anchored to develop
at the faces of the supports for structural integrity. The other bottom bars can be cutoff in the column strips and
PLGGOHVWULSVLQDFFRUGDQFHZLWKWKHOHQJWKVVKRZQLQ)LJXUH
A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUH
ʹᇱ Ǧ͸ᇱᇱ ͳͷᇱ Ǧͻᇱᇱ ʹᇱ Ǧ͸ᇱᇱ ͳͷᇱ Ǧͻᇱᇱ ͸ᇱ Ǧʹᇱᇱ ͺǦ͓͹
ͳͺǦ͓͹
͹Ǧ͓͹
͹Ǧ͓͹
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ͻᇱ ǦͲᇱᇱ ͺᇱ ǦͲᇱᇱ ൒ ͸ԣ
Ͷ͹ᇱ Ǧ͸ᇱᇱ ‘Ž—–”‹’
ʹᇱ Ǧ͸ᇱᇱ ͳͲᇱ Ǧ͸ᇱᇱ ʹᇱ Ǧ͸ᇱᇱ ͳͲᇱ Ǧ͸ᇱᇱ ʹͳǦ͓ͷ
ʹͳǦ͓ͷ
൒ ͸ᇱᇱ ʹǦ͓ͺȀ”‹„
ͻᇱ ǦͲᇱᇱ ൒ ͸ԣ
ͺᇱ ǦͲᇱᇱ Ͷ͹ᇱ Ǧ͸ᇱᇱ ‹††Ž‡–”‹’
Figure 5.57 Reinforcement details for an interior design strip in the east-west direction
for the two-way joist system in Example 5.11.
5.8.12 Example 5.12 – Determination of Required Flexural Reinforcement: Flat Plate System, Building #1
(Framing Option A), SDC B
'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLQDQLQWHULRUGHVLJQVWULSLQWKHQRUWKVRXWKGLUHFWLRQIRUWKHÀDW
SODWHV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQGLQE\LQ
and
FROXPQV VHH)LJXUH $VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the widths of the column strip and middle strip
ACI 8.4.1.5, 8.4.1.6
Figure 5.8
Step 2 – Check if the Direct Design Method can be used to determine gravity load
bending moments
Sect. 5.3.4
Check the following limitations:
Figure 5.22
• 7KHUHPXVWEHRUPRUHFRQWLQXRXVVSDQVLQHDFKGLUHFWLRQ«7KHUHDUHDQGFRQWLQXRXVVSDQVLQWKHQRUWK
south and east-west directions, respectively.
• Successive panel lengths measured center-to-center of supports in each direction must not differ by more than
one-third the longer span…All the panel lengths are equal in each direction.
• Panels must be rectangular with the ratio of longer to shorter panel dimensions, measured center-to-center of
VXSSRUWVQRWWRH[FHHG«
• &ROXPQRIIVHWPXVWQRWH[FHHGSHUFHQWRIWKHVSDQLQWKHGLUHFWLRQRIWKHRIIVHWIURPHLWKHUD[LVEHWZHHQ
centerlines of successive columns…None of the columns are offset.
• All loads must be due to gravity only and uniformly distributed over an entire panel…This method will be used
RQO\IRUJUDYLW\ORDGVZKLFKDUHXQLIRUPO\GLVWULEXWHGRYHUWKHHQWLUHÀRRUV\VWHP
• 7KHXQIDFWRUHGOLYHORDGPXVWQRWH[FHHGWLPHVWKHXQIDFWRUHGGHDGORDG«
DVVXPLQJDQLQWKLFNVODE
• )RUDSDQHOZLWKEHDPVEHWZHHQVXSSRUWVRQDOOVLGHVWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGIRUWKHEHDPV
«7KHUHDUHQREHDPVLQWKLVÀRRUV\VWHPVRWKLV
in the two perpendicular directions:
limitation is not applicable.
%HFDXVHDOOWKHOLPLWDWLRQVDUHVDWLV¿HGWKH''0FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVLQWKHFROXPQDQG
middle strips.
Step 3 – Determine the total factored static moment in each span
Figure 5.8
Figure 5.23
%HFDXVHWKHÀDWSODWHLVSDUWRIWKH/)56VHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVDUHGHWHUPLQHGZKLFKZLOO
be combined with those due to the effects from lateral forces:
Dead load:
Eq. (5.17)
Live load:
These total factored static moments are the same for all spans of the interior design strip in the direction of analysis.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Distribute the total factored static moment to the column strips and middle strips
Table 5.16
'HVLJQPRPHQWFRHI¿FLHQWVIRUDÀDWSODWHV\VWHPZLWKRXWHGJHEHDPVDUHJLYHQLQ7DEOH$VXPPDU\RIWKH
service dead and live load bending moments in the column strip and middle strip in an end span and interior span is
JLYHQLQ7DEOH
Table 5.40 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the
Flat Plate System in Example 5.12
End Span
Design Strip
1
Exterior Negative
Interior Span
2
3
4
5
Positive
First Interior
Negative
Positive
Interior Negative
Column strip
Middle strip
Step 5 – Determine the bending moments due to wind forces
ACI 6.6
'HVLJQZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ
7DEOH
$WKUHHGLPHQVLRQDOPRGHORIWKHEXLOGLQJZDVFRQVWUXFWHGXVLQJ5HIHUHQFHDQGDOLQHDU¿UVWRUGHUDQDO\VLVZDV
SHUIRUPHGXVLQJWKHQRUWKVRXWKZLQGORDGVLQ7DEOHDSSOLHGWRWKHFHQWURLGRIWKHEXLOGLQJIDFHDWWKHURRIDQG
ÀRRUOHYHOV VHH)LJXUH ,QWKHPRGHOWKHFROXPQVDUH¿[HGDWWKHEDVH
The following reduced moments of inertia are used to account for cracked sections:
ACI Table 6.6.3.1.1(a)
• Columns:
• Slabs:
$¿QLWHHOHPHQWDQDO\VLVZDVSHUIRUPHGWRGHWHUPLQHWKHH[WHQWRYHUZKLFKWKHVODEUHVLVWVWKHHIIHFWVIURPWKH
wind loads. From the analysis, it was determined the majority of the wind load effects at an interior design strip are
resisted by slab strips centered on the columns with widths approximately equal to the column strip width; thus, all
the wind load effects are assigned to the column strips in the direction of analysis in this example.
The bending moments in the column strip due to wind loads in an end span and interior span are given in Table
7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHQRUWKGLUHFtion and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at the
east elevation of the building).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.41 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Flat Plate System in Example 5.12
End Span
Design Strip
1
Exterior Negative
Interior Span
2
3
4
5
Positive
First Interior
Negative
Positive
Interior Negative
ņ
Column strip
ņ
Step 6 – Determine the bending moments due to seismic forces
$VQRWHGLQWKHH[DPSOHVWDWHPHQWLWLVDVVXPHGWKH6LWH&ODVVLV'IRUWKLVEXLOGLQJ)URP([DPSOH3DUW E the building is assigned to SDC B for this case. Therefore, effects due to seismic forces must be considered.
'HVLJQVHLVPLFIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQRQ%XLOGLQJDUHGHWHUPLQHGLQ([DPSOHDQGDUHJLYHQLQ
7DEOH
7KHWKUHHGLPHQVLRQDOPRGHOGHVFULEHGLQ6WHSZDVXVHGWRSHUIRUPWKHOLQHDU¿UVWRUGHUDQDO\VLVXVLQJWKH
QRUWKVRXWKVHLVPLFORDGVLQ7DEOHZKLFKDUHDSSOLHGWRWKHFHQWHURIPDVVDWHDFKOHYHO ULJLGGLDSKUDJPVDUH
assigned at each level). Like in the case of wind loads, the majority of the seismic load effects at an interior design
strip are resisted by slab strips centered on the columns with widths approximately equal to the column strip width;
thus, all the seismic load effects are assigned to the column strips in the direction of analysis in this example.
The bending moments in the column strip due to seismic loads in an end span and interior span are given in Table
7KH³SOXVPLQXV´VLJQSUHFHGLQJWKHWDEXODWHGYDOXHVVLJQL¿HVWKHVHLVPLFORDGVFDQDFWLQERWKWKHQRUWK
direction and the south direction (that is, sidesway to the right (SSR) and sidesway to the left (SSL) when looking at
the east elevation of the building).
Table 5.42 Bending Moments (ft-kips) due to Seismic Loads at the Second-Floor Level for the Flat Plate
System in Example 5.12
End Span
Design Strip
Column strip
Interior Span
1
2
3
4
5
Exterior Negative
Positive
First Interior
Negative
Positive
Interior Negative
ņ
ņ
,WFDQEHGHWHUPLQHGWKHODWHUDOIRUFHVEDVHGRQWKHJHQHUDOVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$6&(6(,DUH
less than those due to wind and seismic forces, and, thus, need not be considered in this example.
&RPSDULQJ7DEOHVDQGLWLVHYLGHQWWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHODUJHUWKDQWKRVH
GXHWRZLQGORDGHIIHFWV%HFDXVHERWKW\SHVRIHIIHFWVDUHDVVLJQHGDORDGIDFWRUHTXDOWRLQWKHGHVLJQORDG
combinations, only the bending moments due to the seismic load effects are considered in the remainder of this
example.
Step 7 – Determine the combined factored bending moments due to gravity and
seismic forces
ACI Table 5.3.1
7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH7KHVHLVPLFORDG
FRPELQDWLRQVDUHJLYHQLQ7DEOHIRU6'&%7KHJUDYLW\ORDGPRPHQWVIURPWKH''0DUHFRPELQHGZLWKWKH
EHQGLQJPRPHQWIURPWKHODWHUDOORDGDQDO\VLV VHH$&, @Seismicisolation
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5-117
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.43 Design Bending Moments (ft-kips) at the Second-Floor Level for the Flat Plate System in
Example 5.12
End Span
Load Combination
Location
1
2
3
4
5
Exterior
Negative
Positive
First Interior
Negative
Positive
Interior
Negative
SSR
83.8
SSL
83.8
Column strip
Middle strip
Column
strip
Interior Span
Middle strip
SSR
Column
strip
SSL
Middle strip
Step 8 – Determine the required flexural reinforcement
7KHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQVLQWKHFROXPQVWULSDQGPLGGOHVWULSXVLQJ
the following equations where is the width of the column strip or middle strip and
Eq. (5.35)
Eq. (5.36)
$VXPPDU\RIWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWLVJLYHQLQ7DEOH7KHSURYLGHGDUHDVRIUHLQIRUFHPHQWDUH
greater than or equal to the minimum required in accordance with ACI 8.6.1.1 and the number of bars are selected
VRWKDWWKHSURYLGHGVSDFLQJRIWKHUHLQIRUFLQJEDUVLVOHVVWKDQWKHPD[LPXPUHTXLUHGLQ$&,,WLVHYLGHQW
the provided areas of reinforcement at all critical sections in the column strip and middle strip are less than the area
FRUUHVSRQGLQJWRWHQVLRQFRQWUROOHGVHFWLRQVZKLFKLVGHWHUPLQHGE\(T RIÀH[XUDOUHLQIRUFHPHQW
WKHUHIRUHDOOWKHFULWLFDOVHFWLRQVDUHWHQVLRQFRQWUROOHGZKLFKVDWLV¿HV$&,1RWHWKDWWKHERWWRPEDUV
LQWKHHQGVSDQUHTXLUHGIRUWKHIDFWRUHGSRVLWLYHPRPHQWDUHDOVRDGHTXDWHWRUHVLVWWKHIWNLSSRVLWLYHPRPHQW
DWWKHIDFHRIWKHH[WHULRUVXSSRUWGXHWRPRPHQWUHYHUVDO VHH7DEOH @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.44 Required Flexural Reinforcement at the Second-Floor Level for the Flat Plate System in
Example 5.12
b (in.)
As (in.2)*
Reinforcement*
3.11
Mu (ft-kips)
Location
Exterior
Negative
Column
strip
Positive
First Interior
Negative
Exterior
Negative
End Span
Middle
strip
Positive
First Interior
Negative
Column
strip
Positive
Negative
Interior Span
Middle
strip
Positive
Negative
* Min. As for the column strip
Min. As for the middle strip
Max. spacing
For
For
For the column strip:
For the middle strip:
Step 9 – Check that the flexural reinforcement is adequate for moment transfer requirements
ACI 8.4.2.2
• Edge columns
Determine the reinforcement based on the largest transfer moment at the edge columns:
Table 5.44
Eq. (5.9)
where
Table 5.11, Case 3
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The moment
must be transferred over the effective slab width
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV3URYLGHWKHEDUVE\FRQFHQWUDWLQJRIWKH
FROXPQVWULSEDUVZLWKLQWKHLQZLGWKRYHUWKHFROXPQ VHH7DEOH )RUV\PPHWU\DGGEDUDQG
check the bar spacing:
)RUZLWKLQWKHLQZLGWK
)RUZLWKLQWKH
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RIEDUVDUHUHTXLUHGDWWKHHGJHFROXPQVZLWKLQWKHFROXPQVWULSZLWKRIWKHEDUVFRQFHQWUDWHG
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Check
within
ACI 8.6.1.2
Table 5.11, Case 3
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
where
Eq. (5.30)
Table 5.20
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Eq. (5.37)
Provided
within
Based on this load combination, no additional reinforcement must be provided to satisfy the minimum reinIRUFHPHQWUHTXLUHPHQWVRI$&,
Table 5.42
Therefore,
Eq. (5.38)
Provided
within
5HLQIRUFHPHQWGHWDLOVIRUWKHWRSEDUVDWWKHHGJHFROXPQVDUHWKHVDPHDVWKRVHVKRZQLQ)LJXUH
• Interior columns
Determine the required reinforcement based on the largest transfer moment at the interior columns.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the
gravity load effects and gravity and seismic load effects can be determined from the following equations where
the spans in the direction of analysis and perpendicular to the direction of analysis are equal.
For gravity loads only:
Eq. (5.21)
)RUJUDYLW\DQGVHLVPLFORDGVDWWKH¿UVWLQWHULRUFROXPQV
due to gravity load effects
due to seismic load effects
Table 5.42
Total
For gravity and seismic loads at the interior column:
due to gravity load effects
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
due to seismic load effects
Total
Therefore, use
at all interior columns.
Eq. (5.9)
where
Table 5.11, Case 1
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
must be transferred over the effective slab width
The moment
Eq. (5.10)
Determine the required
Eq. (5.36)
Eq. (5.35)
7KLVUHTXLUHGDUHDRIVWHHOLVHTXLYDOHQWWREDUV:LWKXQLIRUPEDUVSDFLQJLQWKHFROXPQVWULSRIWKH
EDUVDUHZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKH¿UVWLQWHULRUFROXPQVVRQRDGGLWLRQDOUHLQIRUFHPHQW
is required to satisfy moment transfer requirements at these locations. Similarly, no additional reinforcement is
required to satisfy moment transfer requirements at the interior column.
Check
within
ACI 8.6.1.2
Table 5.11, Case 1
Factored shear force at the critical section:
Therefore,
Eq. (5.38)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
where
Eq. (5.30)
and
Table 5.20
Eq. (5.37)
$WWKH¿UVWLQWHULRUFROXPQVSURYLGHG
At the interior column, provided
within
within
Based on this load combination, additional reinforcement is required to satisfy the minimum reinforcement
UHTXLUHPHQWVRI$&,
$WWKH¿UVWLQWHULRUFROXPQV
Table 5.42
At the interior column,
Therefore, the maximum shear stress is equal to the following:
Eq. (5.38)
Eq. (5.37)
The required
from this load combination is less than the required
combination, so required
from the
load
To satisfy minimum reinforcement requirements within
EDUVPXVWEHSODFHGZLWKLQWKHLQ
DWWKH¿UVWLQWHULRUFROXPQVZLWK
effective slab width
width of the column strip. For symWKHUHPDLQLQJEDUVWREHSODFHGZLWKLQWKH
PHWU\DGGEDUWRWKHLQZLGWK7KXVEDUVPXVWEHSURYLGHGZLWKLQWKHFROXPQVWULSDWWKH
¿UVWLQWHULRUFROXPQVZLWKEDUVFRQFHQWUDWHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK7KHUHLQIRUFHPHQWGHWDLOIRUWKLVFDVHLVVLPLODUWRWKDWVKRZQLQ)LJXUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6LPLODUO\EDUVPXVWEHSODFHGZLWKLQWKHLQHIIHFWLYHVODEZLGWKDWWKHLQWHULRUFROXPQ7KHUHPDLQLQJEDUVDUHSURYLGHGZLWKLQWKHLQZLGWKRIWKHFROXPQVWULS
Step 10 – Determine the reinforcement details
ACI 8.7
7KHOHQJWKVRIWKHWRSUHLQIRUFLQJEDUVLQ)LJXUHFDQQRWEHXVHGLQWKHFROXPQVWULSEHFDXVHRIWKHHIIHFWVGXH
WRVHLVPLFORDGV7RDFFRXQWIRUWKHVHLVPLFORDGHIIHFWVSHUFHQWRIWKHWRSEDUVLQWKHFROXPQVWULSVDUHPDGH
FRQWLQXRXVRYHUWKHVSDQV,QWKLVFDVHWKHPD[LPXPDPRXQWRIWRSUHLQIRUFHPHQWRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWV EDUV VREDUVDUHPDGHFRQWLQXRXV
7KHUHPDLQLQJEDUVLQWKHFROXPQVWULSDQGWKHEDUVLQWKHPLGGOHVWULSFDQEHWHUPLQDWHGDWWKHORFDWLRQVLGHQWL¿HG
LQ)LJXUHWKLV¿JXUHDOVRLQFOXGHVWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&,
A Class B tension lap splice is provided over the supports for the bottom bars in the column strip. The required lap
splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
5HLQIRUFHPHQWGHWDLOVIRUWKHFROXPQVWULSDQGPLGGOHVWULSDUHJLYHQLQ)LJXUHZKLFKDUHWKHVDPHDVWKRVHLQ
)LJXUHIRU6'&$
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ͸ᇱ Ǧ͸ᇱᇱ ͳͶǦ͓ͷ
ͺǦ͓ͷ
ͳʹǦ͓ͷ
ͷǦ͓ͷ
ʹᇱ Ǧʹᇱᇱ ʹǦ͓ͷ
ͻǦ͓ͷ
൒ ͸ԣ
ʹͳᇱ Ǧ͸ᇱᇱ ʹͳᇱ Ǧ͸ᇱᇱ ‘Ž—–”‹’
ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ͷᇱ ǦͲᇱᇱ ͻǦ͓ͷ
ͻǦ͓ͷ
ͻǦ͓ͷ
ͷǦ͓ͷ
ͶǦ͓ͷ
൒ ͸ԣ
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൒ ͸ԣ
ͶǦ͓ͷ
͵ᇱ ǦͲᇱᇱ ͵ᇱ ǦͲᇱᇱ ʹͳᇱ Ǧ͸ᇱᇱ ͶǦ͓ͷ
͵ᇱ ǦͲᇱᇱ ͵ᇱ ǦͲᇱᇱ ‹††Ž‡–”‹’
Figure 5.58 Reinforcement details for an interior design strip in the north-south direction
for the flat plate system in Example 5.12.
Comments. Throughout this example, increasing WRWKHYDOXHVLQ$&,7DEOHZDVQRWFRQVLGHUHG$VDQ
exercise, determine if FDQEHLQFUHDVHGWRWKHPD[LPXPPRGL¿HGYDOXHRILQ7DEOHIRUWKHHGJHFROumns bending perpendicular to the edge for a typical interior design strip in the north-south direction.
From Step 9, maximum
for an edge column, which corresponds to the load combination
Assuming there is no shear reinforcement in the slab, LVGHWHUPLQHGE\(T )RUDVTXDUHHGJHFROXPQ
EHQGLQJSHUSHQGLFXODUWRWKHHGJHWKH¿UVWRIWKHWKUHHHTXDWLRQVJRYHUQV
Therefore,
Determine
within
EDUVDUHSURYLGHGZLWKLQ
):
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Thus,
FDQEHWDNHQDVDWWKHHGJHFROXPQVLQWKLVH[DPSOH
If
DQGEDUVDUHUHTXLUHGZLWKLQWKHLQHIIHFWLYHVODEZLGWK
3URYLGLQJWKLVDGGLWLRQDOÀH[XUDOUHLQIRUFHPHQWDWWKHHGJHFROXPQVUHVXOWV
FRPSDUHGWREDUVZLWK
LQWKHVKHDUVWUHVVFDXVHGE\HFFHQWULFLW\RIVKHDURQWKHVHFULWLFDOVHFWLRQVHTXDOWR]HUREHFDXVH
this shear stress is usually relatively large because
DWWKHHGJHFROXPQVLQÀDWSODWHV\Vand, thus,
WHPVZLWKRXWHGJHEHDPVLVUHODWLYHO\ODUJH VHH([DPSOH 5.8.13 Example 5.13 – Check of Shear Strength Requirements: Flat Plate System, Building #1
(Framing Option A), SDC B
Check the shear strength requirements at the columns in an interior design strip in the north-south direction for the
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DQG*UDGH
LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
reinforcement.
Edge Columns
Step 1 – Check one-way shear strength requirements
The critical section is located a distance
ACI 8.4.3
from the face of the column.
Because the panel is rectangular, one-way shear strength requirements must be checked at critical sections A and B
governs in this case.
VHH)LJXUH $OVRWKHJUDYLW\ORDGFRPELQDWLRQ
2ᇱ -0ᇱᇱ 2ᇱ -0ᇱᇱ 25ᇱ -0ᇱᇱ A
B
11ᇱ -9ᇱᇱ ݀ ൌ 8.25ᇱᇱ ݀ ൌ 8.25ᇱᇱ Figure 5.59 Critical sections for one-way shear at an edge column in Example 5.13.
Maximum
ACI Eq. (5.3.1b)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Maximum factored shear force on critical section A:
Maximum factored shear force on critical section B:
Design shear strength:
Eq. (5.29)
Eq. (5.30)
Table 5.20
)URP([DPSOHWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKHHGJHFROXPQFRQVLVWVRI
EDUV EDUVLQWKHFROXPQVWULSDQGEDUVLQWKHPLGGOHVWULS LQWKHQRUWKVRXWKGLUHFWLRQ VHH)LJXUH Thus,
2QHZD\VKHDUUHTXLUHPHQWVDUHDOVRVDWLV¿HGDWFULWLFDOVHFWLRQ% FDOFXODWLRQVQRWVKRZQKHUH Step 2 – Check two-way shear strength requirements
The critical section is located a distance
ACI 8.4.4
from the face of the column.
)DFWRUHGVKHDUIRUFHDWWKHFULWLFDOVHFWLRQ VHH)LJXUH For the gravity load combination where the DDM is used to determine design bending moments,
Dead load:
Eq. (5.17)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹᇱ ǦͲᇱᇱ ͳͳᇱ Ǧͻᇱᇱ ܾଶ ൌ ܿଶ ൅ ݀
ൌ ʹͶǤͲ ൅ ͺǤʹͷ ൌ ͵ʹǤʹͷᇱᇱ ʹᇱ ǦͲᇱᇱ ʹͷᇱ ǦͲᇱᇱ ܾଵ ൌ ܿଵ ൅ ሺ݀Ȁʹሻ
ൌ ʹͶǤͲ ൅ ሺͺǤʹͷȀʹሻ ൌ ʹͺǤͳ͵ᇱᇱ Figure 5.60 Critical section for two-way shear at an edge column in Example 5.13.
Live load:
Table 5.11, Case 3
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.42
Total factored shear force:
For other than gravity load combinations,
is equal to that from the actual load combination:
Table 5.44
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
6KHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWWKHH[WHULRUFROXPQV
First interior columns
Step 1 – Check one-way shear strength requirements
The critical section is located a distance
ACI 8.4.3
from the face of the column.
Because the panel is rectangular, one-way shear strength requirements must be checked at critical sections A and B
governs in this case.
VHH)LJXUH $OVRWKHJUDYLW\ORDGFRPELQDWLRQ
Maximum
ACI Eq. (5.3.1b)
Maximum factored shear force on critical section A:
Maximum factored shear force on critical section B:
Design shear strength:
Eq. (5.29)
Eq. (5.30)
Table 5.20
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
23ᇱ -6ᇱᇱ 2ᇱ -0ᇱᇱ 2ᇱ -0ᇱᇱ 25ᇱ -0ᇱᇱ A
B
݀ ൌ 8.25ᇱᇱ ݀ ൌ 8.25ᇱᇱ Figure 5.61 Critical sections for one-way shear at the first interior column in Example 5.13.
)URP([DPSOHWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKH¿UVWLQWHULRUFROXPQFRQVLVWV
RIEDUV EDUVLQWKHFROXPQVWULSDQGEDUVLQWKHPLGGOHVWULS LQWKHQRUWKVRXWKGLUHFWLRQ VHH
)LJXUH Thus,
$VVXPLQJWKHWHQVLRQUHLQIRUFHPHQWLQWKHGHVLJQVWULSDWWKHORFDWLRQRIWKH¿UVWLQWHULRUFROXPQFRQVLVWVRI
bars in the east-west direction:
Step 2 – Check two-way shear strength requirements
The critical section is located a distance
from the face of the column.
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ACI 8.4.4
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)DFWRUHGVKHDUIRUFHDWWKHFULWLFDOVHFWLRQ VHH)LJXUH ʹ͵ᇱ Ǧ͸ᇱᇱ ʹᇱ ǦͲᇱᇱ ʹᇱ ǦͲᇱᇱ ʹͷᇱ ǦͲᇱᇱ ܾଶ ൌ ܿଶ ൅ ݀
ൌ ʹͶǤͲ ൅ ͺǤʹͷ ൌ ͵ʹǤʹͷᇱᇱ ܾଵ ൌ ܿଵ ൅ ݀
ൌ ʹͶǤͲ ൅ ͺǤʹͷ
ൌ ͵ʹǤʹͷᇱᇱ Figure 5.62 Critical section for two-way shear at the first interior column in Example 5.13.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the
gravity load effects can be determined from the following equation where the spans in the direction of analysis
and perpendicular to the direction of analysis are equal:
Eq. (5.21)
Table 5.11, Case 1
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
Table 5.42
Total factored shear force:
From gravity load effects:
Eq. (5.21)
From seismic load effects:
Total
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
6KHDUVWUHQJWKUHTXLUHPHQWVDUHVDWLV¿HGDWWKH¿UVWLQWHULRUFROXPQV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5.8.14 Example 5.14 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (Framing
Option A), SDC B, Shear Cap
Check the two-way shear strength requirements at an edge column in an interior design strip in the north-south diUHFWLRQIRUWKHÀDWSODWHV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDQLQWKLFNVODE
DVKHDUFDSZLWKDLQSURMHFWLRQEHORZWKHVODEDQGLQE\LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVR
assume normalweight concrete with
Step 1 – Check two-way shear stress at the critical section around the column
Given
Assume
ACI 8.4.4
Figure 5.14
at the three faces of the column.
from the face of the column (see
Š‡ƒ” ƒ’
ʹͲᇱᇱ ʹͷᇱᇱ ʹ͸Ǥʹͷᇱᇱ ʹͻǤͳ͵ᇱᇱ The critical section is located a distance
)LJXUH ”‹–‹ ƒŽ
•‡ –‹‘•
ʹͲᇱᇱ ͵Ͳᇱᇱ ͵ʹǤͷᇱᇱ ͵ͺǤʹͷᇱᇱ Figure 5.63 Critical sections for two-way shear at the edge column in Example 5.14.
Factored shear force at the critical section:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For the gravity load combination where the DDM is used to determine design bending moments,
Figure 5.8
Figure 5.23
Dead load:
Eq. (5.17)
Live load:
Table 5.11, Case 3
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH
Total factored shear force:
For other than gravity load combinations,
is equal to that from the actual load combination.
For an edge column subjected to gravity loads effects:
Table 5.16
From seismic load effects,
Table 5.42
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
Step 2 – Check two-way shear stress at the critical section around the shear cap
ACI 8.4.4
Figure 5.63
Factored shear force at the critical section:
Table 5.11, Case 3
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
Table 5.42
Total factored shear force:
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
7ZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HGDWWKHHGJHFROXPQV
5.8.15 Example 5.15 – Check of Shear Strength Requirements: Flat Plate System, Building #1 (Framing
Option A), SDC B, Slab Opening
Check two-way shear strength requirements at an edge column in an interior design strip in the north-south direction
IRUWKHÀDWSODWHV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODEDQG
LQE\LQFROXPQVJLYHQWKHLQGLDPHWHUVODERSHQLQJLQ)LJXUH$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPH
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
Step 1 – Check if the size of the opening is permitted
ACI 8.5.4.2(b)
The 8-in.-diameter opening occurs within two intersecting column strips. Without performing an analysis of the
V\VWHPZLWKWKHRSHQLQJWKHPD[LPXPRSHQLQJVL]HSHUPLWWHGLQVXFKFDVHVLVRQHHLJKWKWKHZLGWKRIWKHJRYHUQing column strip:
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ͳʹᇱᇱ ͳʹᇱᇱ ͺᇱᇱ ͸ᇱᇱ ʹͶᇱᇱ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Figure 5.64 Slab opening in Example 5.15.
Figure 5.8
This opening is permitted at the indicated location because the 8-in. diameter is less than
Step 2 – Check if portions of the critical section are ineffective
A portion of
umn:
is considered ineffective where an opening is located closer than
ACI 22.6.4.3
from the periphery of the col-
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enclosed by straight lines projecting from the centroid of the column and tangent to the boundaries of the opening
must be considered ineffective.
VHH)LJXUH %DVHGRQ
The length of the critical section adjacent to the opening is
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹͺǤͳ͵ᇱᇱ ༃
ܿ஺஻ ൌ ͻǤ ͸ᇱᇱ ༃
ܽ
ܽ
Ͷᇱᇱ Ͷᇱᇱ ͸ᇱᇱ ʹͶᇱᇱ ͵ʹǤʹͷᇱᇱ ༄
༄
ͺǤʹͷΤʹ ൌ ͶǤͳ͵ԣ
”‹–‹ ƒŽ•‡ –‹‘
‡ˆˆ‡ –‹˜‡Ž‡‰–Š‘ˆ ”‹–‹ ƒŽ•‡ –‹‘ ൌ ͸ǤͲ‹Ǥ
ܽ ൌ ሺ͵ʹǤʹͷ െ ͸ǤͲሻΤʹ ൌ ͳ͵Ǥͳ͵‹Ǥ
Figure 5.65 Ineffective portion of bo for the edge column in Example 5.15.
Step 3 – Check two-way shear strength requirements
ACI 8.4.4
ACI Eq. (5.3.1b)
Factored shear force at the critical section:
For the gravity load combination where the DDM is used to determine design bending moments,
Dead load:
Eq. (5.17)
Live load:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Section properties of the critical section are determined based on the straight segments of the effective portions
RIWKHVHFWLRQ VHH)LJXUH Eq. (5.15)
Segment 1:
Segment 2:
Eq. (5.16)
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.42
Total factored shear force:
For other than gravity load combinations,
is equal to that from the actual load combination:
Table 5.44
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
6KHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGDWWKHH[WHULRUFROXPQVEHFDXVH
nation
for the gravity load combi-
Comments.&RPSDULQJWKHUHVXOWVIURPWKLVH[DPSOHWRWKRVHLQ([DPSOHZKHUHWKHUHLVQRRSHQLQJDGMDFHQW
WRWKHHGJHFROXPQLWLVHYLGHQWWKHLQGLDPHWHURSHQLQJKDVDVLJQL¿FDQWLPSDFWRQWZRZD\VKHDUVWUHQJWK
The section properties of the critical section are determined in this example using the conservative method outlined
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Figure 5.65
For the two faces
For the two faces
Therefore,
Total factored shear stress due to gravity loads:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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If the concrete compressive strength cannot be increased, it may be possible to satisfy two-way shear strength
requirements by investigating whether FDQEHLQFUHDVHGWR WKHUHE\GHFUHDVLQJ WR]HUR LQDFFRUGDQFH
ZLWK$&,,WLVGHWHUPLQHGLQWKH&RPPHQWVVHFWLRQDWWKHHQGRI([DPSOHWKDW FDQEHWDNHQDV
SURYLGHGWKHDPRXQWRIQHJDWLYHÀH[XUDOUHLQIRUFHPHQWDWWKHFULWLFDOVHFWLRQLVLQFUHDVHGDFFRUGLQJO\7KHUHIRUHDVthe maximum two-way shear stress
for the gravity load
suming
ZLWKWKHRSHQLQJZKLFKPHDQVWZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HG
combination
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detailing requirements, a quantity of reinforcement at least equal to that interrupted by the opening must be added
RQWKHVLGHVRIWKHRSHQLQJ>$&, E @
5.8.16 Example 5.16 – Check of Shear Strength Requirements: Flat Slab System With Edge Beams,
Building #1 (Framing Option D), SDC A, Circular Columns
Check two-way shear strength requirements at an interior column in an interior design strip in the north-south direcWLRQDWWKHVHFRQGÀRRUOHYHOIRUWKHÀDWVODEV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ' ZLWKDQLQWKLFNVODED
GURSSDQHOZLWKDLQSURMHFWLRQEHORZWKHVODEDQGLQFLUFXODUFROXPQV$VVXPHWKH6LWH&ODVVLV&$OVR
Interior design strips are not part of the LFRS.
assume normalweight concrete with
Step 1 – Check two-way shear stress at the critical section around the column
Maximum
ACI 8.4.4
ACI Eq. (5.3.1b)
Figure 5.13
Determine the section properties of the critical section assuming a square column of equivalent area (see Figure
ACI 22.6.4.1.2
Table 5.11, Case 1
Factored shear force at the critical section:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 5.66 Critical sections for the interior column in Example 5.15.
In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the gravity
load effects can be determined from the following equation where the spans in the direction of analysis and perpendicular to the direction of analysis are equal:
Eq. (5.21)
where
Determining whether
Figure 5.23
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Table 5.11, Case 1
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
Step 2 – Check two-way shear stress at the critical section around the drop panel
ACI 8.4.4
Figure 5.13
Section properties of the critical section:
Table 5.11, Case 1
Factored shear force at the critical section:
The transfer moment
stress.
is negligible at this location and is not included in the calculation of the maximum shear
Maximum shear stress,
Eq. (5.14)
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Comments. Instead of calculating the section properties of the critical section using an equivalent square column,
calculate the section properties based on the circular section and check two-way shear requirements.
Table 5.13, Case 1
Factored shear force at the critical section:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Total factored shear stress:
Eq. (5.14)
7ZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVFDQEHVDWLV¿HGE\LQFUHDVLQJWKHGURSSDQHOSURMHFWLRQEHORZWKHVODEWRLQ
5.8.17 Example 5.17 – Determination of Shear Reinforcement: Flat Plate System, Building #1 (Framing
Option A), SDC B, Stirrups
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DQG*UDGHUHLQIRUFHPHQW3URYLGHVKHDUUHLQIRUFHPHQWLQWKHIRUPRIFORVHGVWLUUXSVLI
with
needed.
Edge columns
Step 1 – Check two-way shear strength requirements
ACI 8.4.4
Factored shear force at the critical section:
For the gravity load combination where the DDM is used to determine design bending moments,
Figure 5.8
Figure 5.23
Dead load:
Eq. (5.17)
Live load:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.11, Case 3
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH
Total factored shear force:
For other than gravity load combinations,
is equal to that from the actual load combination:
Total factored shear stress:
Eq. (5.14)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Eq. (5.32)
7ZRZD\VKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGDWWKHHGJHFROXPQVEDVHGRQERWKORDGFRPELQDWLRQV
Step 2 – Determine the size and spacing of closed stirrups
ACI 8.7.6
• Step 2a – Check if stirrups can be utilized in the slab
ACI 22.6.7.1
$VVXPLQJVWLUUXSV
Thus, stirrups are permitted to be used to increase the design two-way shear strength.
• Step 2b – Check the maximum shear strength permitted with stirrups
ACI Table 22.6.6.3
From Step 1, maximum
• Step 2c – Determine the design shear strength of the concrete with stirrups
ACI Table 22.6.6.1
Table 5.23
where
Eq. (5.30)
and
Table 5.20
Note: LVSHUPLWWHGWREHWDNHQDVIRUWZRZD\VODEVZLWKVWLUUXSVWKDWVDWLVI\WKHSURYLVLRQVRI$&,
D • Step 2d – Determine the required area and spacing of the stirrups
Maximum stirrup spacing
ACI 22.6.7.2
Figure 5.19
Try
Total required area of stirrups,
for the three-sided, rectangular critical section located a distance
WKHIDFHRIWKHFROXPQ VHH)LJXUH from
Table 5.23
8VHVWLUUXSVVSDFHGDWLQRQFHQWHU
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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ͳ͸ᇱᇱ ʹͲᇱᇱ Figure 5.67 Closed stirrup shear reinforcement at the edge columns in Example 5.17.
• Step 2e – Determine the distance from the faces of the column where the stirrups can be terminated
Stirrups can be terminated where the design strength of the concrete can resist the factored shear stress without
stirrups.
At the critical section located a distance
from the outermost peripheral line of stirrups:
Table 5.23
Assume 6 stirrups are provided parallel to each face of the column and check if the maximum factored shear
from the centerline of the outermost peripheral line of
stress at the critical section located a distance of
.
stirrups is less than or equal to
6HFWLRQSURSHUWLHVRIWKHSRO\JRQDOFULWLFDOVHFWLRQ VHH)LJXUH Table 5.12, Case 3
Eq. (5.15)
Segment 1:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Segment 2:
Segment 3:
Eq. (5.16)
The transfer moment
tively taken as
LVHVVHQWLDOO\]HURDWWKLVFULWLFDOVHFWLRQEXWIRUFDOFXODWLRQSXUSRVHVLVFRQVHUYD-
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
7KHUHIRUHWKHFORVHGVWLUUXSVSURYLGHGRQHDFKIDFHRIWKHFROXPQDUHDGHTXDWH
First interior columns
Check two-way shear strength requirements
ACI 8.4.4
Factored shear force at the critical section:
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In lieu of a more exact analysis, the factored slab moment,
transferred to an interior column due to the
gravity load effects can be determined from the following equation where the spans in the direction of analysis
and perpendicular to the direction of analysis are equal:
Eq. (5.21)
Table 5.11, Case 1
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
Table 5.42
Total factored shear force:
From gravity load effects:
Eq. (5.21)
From seismic load effects:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Total
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
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5.8.18 Example 5.18 – Determination of Shear Reinforcement: Flat Plate System, Building #1
(Framing Option A), SDC B, Headed Shear Studs
Check two-way shear strength requirements at the edge columns in an interior design strip in the north-south direcWLRQIRUWKHÀDWSODWHV\VWHPLQ)LJXUH )UDPLQJ2SWLRQ$ DWWKHVHFRQGÀRRUOHYHOZLWKDLQWKLFNVODE
DQGLQE\LQFROXPQV$VVXPHWKH6LWH&ODVVLV'$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
Provide shear reinforcement in the form of headed shear studs if needed with
Step 1 – Check two-way shear strength requirements
ACI 8.4.4
Table 5.11, Case 3
Factored shear force at the critical section:
For the gravity load combination where the DDM is used to determine design bending moments,
:
Figure 5.8
Figure 5.23
Dead load:
Eq. (5.17)
Live load:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 5.11, Case 3
Determining whether
FDQEHLQFUHDVHGWRWKHYDOXHVLQ$&,7DEOHLVQRWFRQVLGHUHGLQWKLVH[DPSOH
ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
Eq. (5.32)
where
$VVXPHWKHEHQGLQJPRPHQWVGXHWRVHLVPLFORDGHIIHFWVDUHWKHVDPHDVWKRVHJLYHQLQ7DEOH
Total factored shear force:
For other than gravity load combinations,
is equal to that from the actual load combination:
Total factored shear stress:
Eq. (5.14)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Eq. (5.32)
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Step 2 – Determine the size and spacing of headed shear stud reinforcement
• Step 2a – Check the maximum shear strength permitted with headed shear studs
ACI 8.7.7
ACI Table 22.6.6.3
From Step 1, maximum
• Step 2b – Determine the design shear strength of the concrete with headed shear studs
ACI Table 22.6.6.1
where
Eq. (5.30)
Table 5.20
and
and
Note: LVSHUPLWWHGWREHWDNHQDVIRUWZRZD\VODEVZLWKVPRRWKKHDGHGVKHDUVWXGUHLQIRUFHPHQWWKDW
VDWLV¿HVWKHSURYLVLRQVRI$&, E WKLVSURYLVLRQLVVDWLV¿HGLQWKLVH[DPSOH VHH6WHSFEHORZ • Step 2c – Determine the required size and spacing of the headed shear studs
ACI 22.6.8
Figure 5.20
Provide three lines of headed shear studs on each face of the column:
Assuming ½-in.-diameter studs
, the required spacing is the following:
Table 5.23
In this equation,
is the cross-sectional area of all the headed studs on one peripheral line parallel to the peULPHWHURIWKHFROXPQVHFWLRQ VHH)LJXUH Because
Figure 5.20
$VVXPLQJDLQVSDFLQJFKHFNWKHUHTXLUHPHQWVRI$&,
ACI Eq. (22.6.8.3)
8VHòLQGLDPHWHUKHDGHGVKHDUVWXGVVSDFHGDWLQRQFHQWHU
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ʹʹǤͲᇱᇱ Figure 5.68 Headed shear stud reinforcement at the edge columns in Example 5.18.
• Step 2d – Determine the distance from the faces of the column where the headed shear studs can
be terminated
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stress without the headed shear studs.
At the critical section located a distance
from the outermost peripheral line of headed shear studs:
Table 5.23
$VVXPHKHDGHGVKHDUVWXGVDUHSURYLGHGLQHDFKOLQHSDUDOOHOWRHDFKIDFHRIWKHFROXPQDQGFKHFNLIWKHPD[from the centerline of the outermost
imum factored shear stress at the critical section located a distance of
.
peripheral line of headed studs is less than or equal to
6HFWLRQSURSHUWLHVRIWKHSRO\JRQDOFULWLFDOVHFWLRQ VHH)LJXUH Table 5.12, Case 3
Eq. (5.15)
Segment 1:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Segment 2:
Segment 3:
Eq. (5.16)
The transfer moment
taken as
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ACI Eq. (8.4.4.2.2)
Total factored shear stress:
Eq. (5.14)
7KHUHIRUHWKHòLQGLDPHWHUKHDGHGVKHDUVWXGVVSDFHGDWLQRQFHQWHUZLWKOLQHVRIKHDGHGVWXGVRQHDFK
face of the column are adequate.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 6
BEAMS
6.1 Overview
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direction of load transfer.
The design and detailing of cast-in-place beams with nonprestressed reinforcement are covered in this chapter for
where
is the gross cross-sectional area of
members subjected to a factored axial force, , less than
the beam (see ACI 9.3.3.1). Provisions for beams are given in ACI Chapter 9, which are applicable to members in
buildings assigned to Seismic Design Category (SDC) A and B.
Nonprestressed one-way joist systems consist of a monolithic combination of regularly spaced ribs with a clear
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once popular but is generally not used anymore. Wide-module joist systems, which are also referred to as skip joist
V\VWHPVDUHIRUPHGE\LQRULQZLGHSDQIRUPVDQGKDYHHVVHQWLDOO\WDNHQWKHSODFHRIVWDQGDUGRQHZD\
joist systems. The evenly spaced joists (or, ribs) support the loads from a one-way slab cast integrally with the joists
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ULEVLQ$&,IRUVWDQGDUGRQHZD\MRLVWV\VWHPVWKHQRPLQDOVKHDUVWUHQJWKRIWKHFRQFUHWH , is not permitWHGWREHLQFUHDVHGE\WLPHVWKHYDOXHSUHVFULEHGLQ$&, $&, WKXVWKHMRLVWVDQGEHDPVLQWKH
wide-module system must be designed in accordance with the provisions in ACI Chapter 9 excluding the requirements in ACI 9.8.
Figure 6.1 A wide-module joist system.
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6-1
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.2 Sizing the Cross-Section
6.2.1 Determining the Beam Depth
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Minimum beam depths based on various support conditions are given in Table 6.1. In the expressions for minimum
, is the span length of the beam in inches; for cantilevers, is the clear projection of the cantilever in inches.
Also, and DUHWKHPRGL¿FDWLRQIDFWRUVIRUUHLQIRUFHPHQWJUDGHDQGOLJKWZHLJKWFRQFUHWHUHVSHFWLYHO\
Table 6.1 Minimum Depth of Nonprestressed Beams
Normalweight Concrete
Support Condition
ƒy = 60 ksi
ƒy other than 60 ksi(2)
Lightweight Concrete(1)
ƒy = 60 ksi(3)
ƒy other than 60 ksi(2),(3)
Simply supported
One end continuous
Both ends continuous
Cantilever
(1)
(2)
(3)
Applicable where equilibrium density,
is in the range of 90 to 115 lb/ft3
For the usual case of continuous construction, the depth of the beams, , must be the same for all spans and it
should be determined on the basis of the span yielding the largest minimum depth; this results in economical formZRUN VHH5HIHUHQFH 7KHPLQLPXP for an end span condition governs in the case of equal end and interior
times the end span, the minimum for the
spans. In cases where the interior span is greater than
interior span condition governs.
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wide-module joist construction (see below).
As noted in Section 6.1, standard pan forms are used in constructing wide-module joist systems. The pan depths
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systems because structural requirements are minimal. Therefore, the overall depth of the system (slab thickness plus
pan depth) must be greater than or equal to the applicable minimum depth given in Table 6.1 to satisfy serviceability
requirements. To achieve economical formwork, the depths of the beams must be the same depth as the joists: The
formwork associated with beams deeper than the joists is more complex to build and additional shoring is required
to support the beam formwork, which results in unnecessary increases in construction time and cost. For economy,
the joists typically span in the long direction, which means their depth usually controls the overall depth of the
wide-module system.
Fire-resistance requirements of the general building code must also be considered when specifying a beam depth
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smaller than that required by ACI 9.3.1 for serviceability.
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6-2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 6.2 Standard form dimensions for wide-module joist systems.
7KHGHSWKRIDQ\EHDPFDQEHGHWHUPLQHGEDVHGRQWKHGHÀHFWLRQUHTXLUHPHQWVLQ$&,)RUEHDPVVXSSRUWLQJ
FRQVWUXFWLRQOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV VXFKDVJODVVSDUWLWLRQZDOOV GHÀHFWLRQFDOFXODWLRQVVKRXOG
EHSHUIRUPHGLQDFFRUGDQFHZLWKWKDWVHFWLRQ0D[LPXPSHUPLVVLEOHGHÀHFWLRQVDUHJLYHQLQ$&,7DEOHIRU
PHPEHUVVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVDQGIRUPHPEHUVWKDWDUHQRW0HWKRGVRQKRZWRFDOFXODWHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUEHDPVDUHJLYHQLQ
6HFWLRQRIWKLVSXEOLFDWLRQ7KLVLQIRUPDWLRQLVDOVRDSSOLFDEOHWRFDOFXODWLRQRIGHÀHFWLRQVIRURQHZD\VODEV
6.2.2 Determining the Beam Width
The width of a beam, FDQEHGHWHUPLQHGEDVHGRQÀH[XUDOVWUHQJWKUHTXLUHPHQWVWKDWLVE\VDWLVI\LQJWKHIRO$WWKLVVWDJHRIWKHGHVLJQWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW
lowing strength design equation:
LVW\SLFDOO\QRWNQRZQ+RZHYHUDUDQJHIRU can be determined based on required minimum and maximum
DUHDVRIUHLQIRUFHPHQW7KHPLQLPXPDUHDRIÀH[XUDOUHLQIRUFHPHQWLVVSHFL¿HGLQ$&,$OVREHDPVVXEMHFWHG
to relatively small or no axial loads must be designed as tension-controlled sections (ACI 9.3.3.1), which essentially
VHWVDQXSSHUOLPLWRQWKHDUHDRIÀH[XUDOUHLQIRUFHPHQWSURYLGHGLQDVHFWLRQ7KHSURYLGHGDUHDRIUHLQIRUFHPHQW
PXVWEHZLWKLQWKLVUDQJH)RU*UDGHUHLQIRUFHPHQWDQGDVSHFL¿HGFRQFUHWHFRPSUHVVLYHVWUHQJWK , equal to
SVLPLQLPXPDQGPD[LPXP are equal to the following:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Using an initial estimate of
IRUWKHDUHDRIWKHÀH[XUDOUHLQIRUFHPHQW ZKLFKLVDSSUR[LPDWHO\HTXDOWRWKH
can be obtained by setting
and solving for
average of the minimum and maximum values),
For beams with
and
In this equation,
has the units of in.-kips and
:
have the units of inches.
The above equation can be rewritten in the following form where
units of inches:
has the units of ft-kips and
have the
(6.1)
The largest factored bending moment
along the spans should be used in Equation (6.1) to determine
For
where is the depth of
beams with one layer of longitudinal tension reinforcement, can be taken as
WKHEHDPGHWHUPLQHGIRUVHUYLFHDELOLW\ VHH6HFWLRQRIWKLVSXEOLFDWLRQ DQGLQLVWKHVXPRIWKHFRQFUHWH
FRYHULQDFFRUGDQFHZLWK$&,7DEOHIRUEHDPVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG . To
LQ WKHGLDPHWHURIDVWLUUXS LQ DQGRQHKDOIWKHGLDPHWHURIDORQJLWXGLQDOEDU
achieve economical formwork, the same VKRXOGEHXVHGIRUDOOEHDPVDORQJWKHVSDQ$OVRWKHVDPHEHDPVL]H
should be used as often as possible throughout the structure.
Providing a beam width greater than or equal to that determined by Equation (6.1) results in cross-sectional dimensions satsifying both strength and serviceability requirements.
The following equation can be used to determine
pressive strength, and reinforcement ratio
for a beam with any grade of reinforcing steel, concrete com:
(6.2)
where
has the units of ft-kips; and
have the units of psi; and and
have the units of inches. The
and
$&, DQGOHVV
reinforcement ratio must be greater than or equal to the greater of
(ACI 9.3.3.1) where the term LVGH¿QHGLQ$&,7DEOHDVIROORZV $&,
than or equal to
• For
• For
• For
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6-4
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(TXDWLRQV RU FDQDOVREHXVHGWRGHWHUPLQHWKHUHTXLUHGZLGWKRIWKHMRLVWVLQDZLGHPRGXOHMRLVWV\VWHP
-RLVWZLGWKFDQEHWDLORUHGWRVDWLVI\YLUWXDOO\DQ\UHTXLUHPHQW,QXVXDOVLWXDWLRQVWKHWKLQQHVWSUDFWLFDOZLGWKLV
W\SLFDOO\DGHTXDWHIRUVWUXFWXUDOUHTXLUHPHQWV*HQHUDOO\DLQZLGHMRLVWLVXVHGZLWKDLQSDQDQGDLQ
ZLGHMRLVWLVXVHGZLWKDLQSDQUHVXOWLQJLQIWDQGIWPRGXOHVUHVSHFWLYHO\&ROXPQOLQHMRLVWVFDQEHPDGH
part of the lateral force-resisting system (LFRS) and any practical width (not necessarily the same as the adjoining
joists) can be provided to resist combined gravity and lateral load effects.
6.2.3 General Guidelines for Sizing Beams for Economy
7KHIROORZLQJJHQHUDOJXLGHOLQHVIRUVL]LQJEHDPVVKRXOGEHIROORZHGIRUHFRQRP\ VHH5HIHUHQFH • Use whole-inch increments for beam dimensions, except for the depth of beams in wide-module joist systems
where half-inch beam depths are used for formwork economy where applicable.
• 8VHEHDPZLGWKVLQPXOWLSOHVRIRULQ
• 8VHFRQVWDQWEHDPVL]HIURPVSDQWRVSDQDQGYDU\WKHUHLQIRUFHPHQWDVUHTXLUHG
• Use wide, shallower beams rather than narrow, deeper beams.
• 8VHEHDPZLGWKVDWOHDVWLQZLGHUWKDQWKHVXSSRUWLQJFROXPQZLGWKVZKHUHYHUSRVVLEOH
• Use uniform width and depth of beams throughout the building, wherever possible.
6.3 Required Strength
6.3.1 Analysis Methods
Overview
7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR
EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWK VHH$&,DQGUHVSHFWLYHO\ Bending Moments and Shear Forces
)LYHPHWKRGVRIDQDO\VLVDUHJLYHQLQ$&,WRGHWHUPLQHWKHIDFWRUHGEHQGLQJPRPHQWVDQGVKHDUIRUFHVDWWKH
FULWLFDOVHFWLRQVLQDEHDP VHH6HFWLRQRIWKLVSXEOLFDWLRQIRUWKHORFDWLRQVRIWKHFULWLFDOVHFWLRQVIRUÀH[XUH
and shear).
)RUWKHFDVHRIJUDYLW\ORDGVWKHVLPSOL¿HGPHWKRGIRUDQDO\VLVRIFRQWLQXRXVEHDPVDQGRQHZD\VODEVLQ$&,
LVSHUPLWWHGWREHXVHGWRGHWHUPLQHEHQGLQJPRPHQWVDQGVKHDUIRUFHVSURYLGHGWKHFRQGLWLRQVLQ$&,DUH
VDWLV¿HG>$&, D @
(1) Members are prismatic
Gravity loads are uniformly distributed
(3) The service live load, , is less than or equal to 3 times the service dead load,
There are at least two spans
7KHORQJHURIWZRDGMDFHQWVSDQVGRHVQRWH[FHHGWKHVKRUWHUVSDQE\SHUFHQW
, and factored shear forces, LQ$&,DQGUHVSHFWLYHO\
The approximate factored bending moments,
is the factored uniformly distributed gravity load on the conDUHJLYHQLQ)LJXUHRIWKLVSXEOLFDWLRQZKHUH
tinuous beams.
0RPHQWUHGLVWULEXWLRQLVQRWSHUPLWWHGZKHQEHQGLQJPRPHQWVDUHFDOFXODWHGXVLQJWKHVLPSOL¿HGPHWKRG $&,
:KHUHRQHRUPRUHRIWKHFRQGLWLRQVLQ$&,LVQRWVDWLV¿HGRQHRIWKHRWKHUIRXUPHWKRGVRIDQDO\VLVJLYHQLQ
and
at the critical sections.
$&,PXVWEHXVHGWRGHWHUPLQH
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6-5
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Torsional Moments
Factored torsional moments, , can be obtained using the analysis methods in ACI Chapter 6. It is permitted to
WDNHWKHWRUVLRQDOORDGLQJIURPDVODEDVXQLIRUPO\GLVWULEXWHGDORQJWKHVSDQRIDEHDP $&, Once LVGHWHUPLQHGDWWKHFULWLFDOVHFWLRQ VHH6HFWLRQRIWKLVSXEOLFDWLRQIRUWKHORFDWLRQRIWKHFULWLFDOVHFtion for torsion), the next step is to determine if torsional effects on a beam need to be considered or not. Torsional
where
is the threshold torsional moeffects on a reinforced concrete beam need not be considered if
PHQW $&, )RUQRQSUHVWUHVVHGPHPEHUVZLWKVROLGFURVVVHFWLRQV LVJLYHQLQ$&,7DEOH D (6.3)
7KH¿UVWRIWKHVHHTXDWLRQVLVDSSOLFDEOHWRPHPEHUVQRWVXEMHFWHGWRDQD[LDOIRUFH7KHVHFRQGHTXDWLRQLVDSSOLFD(in pounds), which is taken as positive for compresble where the member is subjected to a factored axial force
, and negative for tension forces.
sion forces acting on the gross area of the beam,
The term LVWKHPRGL¿FDWLRQIDFWRUUHÀHFWLQJWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYH
to normalweight concrete of the same compressive strength, and is determined based on either (1) the equilibrium
density, RIWKHFRQFUHWHPL[RU WKHFRPSRVLWLRQRIWKHDJJUHJDWHLQWKHFRQFUHWHPL[ VHH$&,DQG
$&, 9DOXHVRI based on DUHJLYHQLQ7DEOH>$&,7DEOH D @DQGYDOXHVRI based on
FRPSRVLWLRQRIDJJUHJDWHVDUHJLYHQLQ7DEOH>$&,7DEOH E @1RWHWKDW is permitted to be taken as
IRUOLJKWZHLJKWFRQFUHWH $&, DQGLVHTXDOWRIRUQRUPDOZHLJKWFRQFUHWH $&, Table 6.2 Values of
Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
Table 6.3 Values of
Based on Composition of Aggregates
Concrete
Composition of Aggregates
Ȝ
All-lightweight
)LQH$670&
&RDUVH$670&
/LJKWZHLJKW¿QHEOHQG
)LQH&RPELQDWLRQRI$670&DQG&
&RDUVH$670&
WR(1)
Sand-lightweight
Fine: ASTM C33
&RDUVH$670&
Sand-lightweight, coarse blend
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&DQG$670&
WR(2)
(1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate.
(2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.
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6-6
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The term
is the area enclosed by the outside perimeter of the concrete cross-section, and
is the outside perimeter of the concrete cross-section. For beams cast monolithically with a slab, a portion of the slab may be able to
FRQWULEXWHWRWRUVLRQDOUHVLVWDQFH7KHRYHUKDQJLQJÀDQJHZLGWK permitted to be used in the calculation of
LVJLYHQLQ$&,
and
(6.4)
is the greater of the projection of the beam above or below the slab (see Figure 6.3).
ܾ௪ ൅ ʹܾ௘ ൌ ܾ௪ ൅ ʹ݄௕ ൑ ܾ௪ ൅ ͺ݄
݄௕ ൑ Ͷ݄
݄௕ଵ ൑ Ͷ݄
ܾ௪ ݄
݄௕ଵ ൐ ݄௕ଶ ݄௕ ݄௕ଵ ൅ ݄ ൅ ݄௕ଶ ݄
݄௕ ൑ Ͷ݄
ܾ௪ ൅ ܾ௘ ൌ ܾ௪ ൅ ݄௕ଵ ൑ ܾ௪ ൅ Ͷ݄
݄௕ଶ where
ܾ௪ ܾ௘ ൌ ݄௕ ൑ Ͷ݄
ܾ௘ ൌ ݄௕ଵ ൑ Ͷ݄
‫ܣ‬௖௣ ൌ ܾ௪ ݄௕ ൅ ሺܾ௪ ൅ ʹܾ௘ ሻ݄
‫݌‬௖௣ ൌ ʹሺ݄ ൅ ݄௕ ൅ ܾ௪ ൅ ʹܾ௘ ሻ
‫ܣ‬௖௣ ൌ ܾ௪ ሺ݄௕ଵ ൅ ݄ ൅ ݄௕ଶ ሻ ൅ ݄ܾ௘ ‫݌‬௖௣ ൌ ʹሺ݄௕ଵ ൅ ݄ ൅ ݄௕ଶ ൅ ܾ௪ ൅ ܾ௘ ሻ
۷‫ܚܗܑܚ܍ܜܖ‬۰‫ܕ܉܍‬
۳‫܍܏܌‬۰‫ܕ܉܍‬
Figure 6.3 Torsional section properties Acp and pcp for interior and edge beams
cast monolithically with the slab.
Torsional section properties
and
for an interior beam and an upturned edge beam cast monolithically with
the slab are also given in Figure 6.3. Note that the torsional section properties for the edge beam are applicable for
and
LQFOXGLQJDYDOXHRI]HUR
corresponds to the case where the bottom of the slab is
any value of
FRUUHVSRQGVWRWKHFDVHZKHUHWKHWRSRIWKHVODELVÀXVKZLWKWKHWRS
ÀXVKZLWKWKHERWWRPRIWKHEHDPDQG
of the beam).
calculated
7KHFRQWULEXWLRQRIWKHRYHUKDQJLQJÀDQJH V PXVWEHQHJOHFWHGLQFDVHVZKHUHWKHSDUDPHWHU
FDOFXODWHGIRUWKHVDPHVROLGEHDPLJQRULQJWKHÀDQJHV>$&,
IRUDVROLGEHDPZLWKÀDQJHVLVOHVVWKDQ
for the beam with
E @)RUKROORZVHFWLRQVWKHRYHUKDQJLQJÀDQJH V PXVWEHQHJOHFWHGZKHUH
is the area of the concrete only and does
ÀDQJHVLVOHVVWKDQWKDWIRUWKHVDPHEHDPLJQRULQJWKHÀDQJHVZKHUH
not include the area of the void.
For nonprestressed members with hollow cross-sections,
LVJLYHQLQ$&,7DEOH E (6.5)
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6-7
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Once it has been established that torsional effects must be considered (that is, where
), the next step is to
determine whether IURPDQDO\VLVFDQEHUHGXFHGLQDFFRUGDQFHZLWKWKHSURYLVLRQVLQ$&, $&, 7KHEHDPVXSSRUWLQJWKHFDQWLOHYHUVODELQ$&,)LJXUH5DLVXQDEOHWRUHGLVWULEXWHWKHHIIHFWVGXHWRWKHWRUsional moment (it is not part of an indeterminate structure where redistribution of the applied torsional moment can
from analysis. This type of torsion is referred to as
occur); thus, it must be designed for the torsional moment
equilibrium torsion because LVUHTXLUHGWRPDLQWDLQHTXLOLEULXP $&, 7KHVSDQGUHOEHDPLQ$&,)LJXUH5ELVSDUWRIDQLQGHWHUPLQDWHVWUXFWXUHZKHUHUHGLVWULEXWLRQRIWRUVLRQDO
moments can occur. This type of torsion is referred to as compatibility torsion. In this case, the beam can be deequal to the design cracking torsional moment,
, which is less than that obsigned for a torsional moment
is determined by the equations in ACI
WDLQHGIURPDQDO\VLV $&, )RUERWKVROLGDQGKROORZVHFWLRQV
7DEOH $&, (6.6)
7KH¿UVWRIWKHVHHTXDWLRQVLVDSSOLFDEOHWRPHPEHUVQRWVXEMHFWHGWRDQD[LDOIRUFH7KHVHFRQGHTXDWLRQLVDSSOL(in pounds), which is taken as positive for comcable where the member is subjected to a factored axial force
, and negative for tension forces. Note that
for
pression forces acting on the gross area of the beam,
members with solid cross-sections.
Adjoining members must be designed for the redistributed bending moments and shear forces due to application
RIWKHFRPSDWLELOLW\WRUVLRQDOPRPHQW $&, 7KHEHDPVIUDPLQJLQWRWKHVSDQGUHOEHDPLQ$&,)LJXUH
5EPXVWEHGHVLJQHGIRUDFRQFHQWUDWHGPRPHQWDWWKHLUHQGVHTXDOWRWKHFRPSDWLELOLW\WRUVLRQDOPRPHQW
; that moment is redistributed to the positive and interior negative critical sections of the beam and are algebraically combined with the bending moments corresponding to gravity and any other applicable loads. Typically,
the negative moment at the exterior support of the beam framing into the spandrel beam is reduced compared to that
prior to redistribution and the positive and negative moments at the other critical sections are increased.
from analysis is less than
In cases where torsional effects must be considered and
designed to resist the factored torsional moment from analysis.
, the section should be
6.3.2 Critical Sections for Flexure, Shear, and Torsion
,QDFFRUGDQFHZLWK$&,DQG
, , and
are permitted to be calculated at the faces of
WKHVXSSRUWVIRUEHDPVEXLOWLQWHJUDOO\ZLWKWKHVXSSRUWV)RUÀH[XUDOGHVLJQWKHFULWLFDOVHFWLRQVRFFXUDWWKHIDFHV
of the supports where negative moments are maximum and in the span where positive moments are maximum.
Beams are permitted to be designed for the shear force at a critical section located a distance
VXSSRUWZKHUHWKHFRQGLWLRQVLQ$&,DUHVDWLV¿HG VHH)LJXUH from the face of the
(1) Support reaction, in direction of applied shear, introduces compression into the end region of the beam
Loads are applied at or near the top surface of the beam
(3) No concentrated load occurs between the face of the support and the critical section
The critical section for shear must be taken at the face of the support where one or more of the three conditions in
$&,DUHQRWPHW)RUH[DPSOHWKHFULWLFDOVHFWLRQPXVWEHDWWKHIDFHRIWKHVXSSRUWIRUWKHIUDPLQJFRQ¿JXUDWLRQLQ)LJXUHEHFDXVHWKHVXSSRUWLQJPHPEHULVLQWHQVLRQ
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6-8
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‘ ‡–”ƒ–‡†
Ž‘ƒ†ǡܲ௨ ‫ݓ‬௨ ƒ’’Ž‹‡†ƒ–‘”‡ƒ”–‘’
•—”ˆƒ ‡‘ˆ„‡ƒ
ܸ௨ ̷݀
ܸ௨ ̷ˆƒ ‡
”‹–‹ ƒŽ•‡ –‹‘
݀
൐ ݀
Figure 6.4 Critical section for shear in a beam satisfying the conditions of ACI 9.4.3.2.
‡•‹‘ˆ‘” ‡ǡܶ௨ ‫ݓ‬௨ ܸ௨ ̷ˆƒ ‡
”‹–‹ ƒŽ•‡ –‹‘
Figure 6.5 Critical section for shear in a beam supported by a member in tension.
Beams are permitted to be designed for the torsional moment at a critical section located a distance from the face
RIWKHVXSSRUWSURYLGHGQRFRQFHQWUDWHGWRUVLRQDOPRPHQWVRFFXUZLWKLQWKDWGLVWDQFH $&, RWKHUZLVHWKH
critical section must be taken at the face of the support. The edge beam in Figure 6.6 is subjected to a uniformly
, from the beam framdistributed torsional moment, , from the slab and a concentrated torsional moment,
ing into it. The critical section for torsion can be taken a distance from the face of the column in this case because
occurs at a distance greater than from the face of the column. As such, the edge beam can be designed for
).
the torsional moment at the location of the critical section (that is, at
6.3.3 Redistribution of Moments in Continuous Flexural Members
%HQGLQJPRPHQWVFDOFXODWHGLQDFFRUGDQFHZLWK$&,DWVXSSRUWVRIFRQWLQXRXVÀH[XUDOPHPEHUVDUHSHUPLWWHG
WREHLQFUHDVHGRUGHFUHDVHGLQDFFRUGDQFHZLWK$&,H[FHSWZKHUHWKHPRPHQWVKDYHEHHQFRPSXWHGXVLQJWKHDSSUR[LPDWHFRHI¿FLHQWVLQ$&,,WLVFXVWRPDU\WRUHGXFHWKHQHJDWLYHPRPHQWVDWWKHVXSSRUWVZKLFK
results in an increase of the positive moments in the span.
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6-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
”‹–‹ ƒŽ•‡ –‹‘
݀
ܶ௨ ̷ˆƒ ‡
ܶ௨ ̷݀
ܶ௨ǡ௕௘௔௠ ‫ݐ‬௨ ൐ ݀
Figure 6.6 Critical section for torsion in a beam where a concentrated torsional moment
occurs at a distance greater than d from the face of the support.
7KHPD[LPXPSHUFHQWDJHRIUHGLVWULEXWLRQSHUPLWWHGLVWKHOHVVHURIWLPHVWKHQHWWHQVLOHVWUDLQLQWKHUHLQRUSHUFHQW
forcement,
Moment redistribution is dependent on adequate ductility in plastic hinge regions, which develop at points of maximum moment and which cause a shift in the elastic bending moment diagram. Thus, redistribution of negative moments is only permitted where LVJUHDWHUWKDQRUHTXDOWRDWWKHVHFWLRQLQZKLFKWKHPRPHQWLVUHGXFHG
$GMXVWPHQWVRIWKHQHJDWLYHPRPHQWVDWWKHVXSSRUWVDUHPDGHIRUHDFKORDGLQJFRQ¿JXUDWLRQFRQVLGHULQJSDWWHUQ
live loading. It is important to ensure that static equilibrium is maintained at all joints before and after moment
redistribution. Thus, a decrease in the negative moments at the supports warrants an increase in the positive moment
in the span under consideration.
are equal to
where
Adjusted negative bending moments at the faces of the supports,
is the lesser of
DQGDQG
is the factored negative moment at the face of the support from
analysis.
The adjusted positive moments are obtained based on equilibrium considering the adjusted negative bending moments at the faces of the supports.
6.4 Design Strength
6.4.1 General
)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\
VHFWLRQLQDEHDP $&, (6.7)
(6.8)
(6.9)
(6.10)
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,1RQSUHVWUHVVHGEHDPVZLWK
PXVWEHGHVLJQHGDVWHQVLRQFRQWUROOHGVHFWLRQVLQDFFRUGDQFHZLWK$&,7DEOH $&, thus, the net tensile strain in the extreme layer of the longitudinal tension reinforcement at nominal strength, ,
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6-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
must be greater than or equal to
and
$&,7DEOH 7KHQHWWHQVLOHVWUDLQLQWKHH[WUHPH
, is equal to
layer of longitudinal tension reinforcement corresponding to compression-controlled sections,
$&, 7KHPRGXOXVRIHODVWLFLW\RIWKHUHLQIRUFHPHQW LVSHUPLWWHGWREHWDNHQDV
$&,7DEOH SVLUHJDUGOHVVRIWKHJUDGHRIWKHUHLQIRUFHPHQW $&, )RUVKHDUDQGWRUVLRQ
Methods to determine the nominal strengths
tively.
,
, and
DUHJLYHQLQ6HFWLRQVDQGUHVSHF-
6.4.2 Nominal Flexural Strength
Rectangular Sections with Tension Reinforcement Only
Single Layer of Tension Reinforcement. 7KHQRPLQDOÀH[XUDOVWUHQJWK
, of a rectangular section with one
LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,DQGLVEDVHGRQ
layer of tension reinforcement and with
PRPHQWHTXLOLEULXPRIDUHFWDQJXODUVHFWLRQ VHH$&,DQG)LJXUH (6.11)
ܾ
‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ
݀ െ ሺܽȀʹሻ
ܿ
ܽ
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ܽ ൌ ߚଵ ܿ
ͲǤͺͷ݂௖ᇱ ͲǤͲͲ͵
‫ܣ‬௦ ߝ௧ ܶ ൌ ‫ܣ‬௦ ݂௬ Figure 6.7 Strain and stress distributions at a positive moment section in a beam.
In this equation, LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQW7KHVWUDLQDQGVWUHVVGLVWULEXWLRQVLQ)LJXUHDUHDW
a positive moment section and are equally applicable at negative moment sections.
The depth of the equivalent stress block, , is determined from force equilibrium, that is, it is determined by set, equal to the tension force in the reinforcement,
ting the resultant compressive force in the concrete,
, and solving for VHH)LJXUH (6.12)
,WLVDVVXPHGDXQLIRUPVWUHVVHTXDOWRSHUFHQWRIWKHFRQFUHWHFRPSUHVVLYHVWUHQJWK , is distributed over the
, where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LV $&, 7KH
depth
XOWLPDWHVWUDLQLQWKHFRQFUHWHLVDVVXPHGWREH $&, DQGWKHWHUP is the width of the compression
face of the member.
Multiple Layers of Tension Reinforcement. Under certain conditions, the required tension reinforcement cannot
¿WZLWKLQDVLQJOHOD\HU)RUH[DPSOHLQEHDPVZLWKDZLGWKUHVWULFWLRQGXHWRDUFKLWHFWXUDOFRQVWUDLQWVLWPD\QRWEH
SRVVLEOHWRVHOHFWDSUDFWLFDOEDUVL]HZKLOHVDWLVI\LQJWKHPLQLPXPFOHDUVSDFLQJUHTXLUHPHQWVEHWZHHQWKHEDUVLQ
$&,,QVXFKFDVHVWKHORQJLWXGLQDOEDUVDUHSURYLGHGLQPRUHWKDQRQHOD\HU
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHQRPLQDOÀH[XUDOVWUHQJWK
, for sections with multiple layers of reinforcement is determined in the same way
as that for sections with a single layer of reinforcement. The main difference is that for beams with two or more layers, the reinforcement in the inner layer(s) may not yield. Therefore, it is important to check whether all the reinis then determined accordingly based on this check.
forcement in the section yields or not;
&RQVLGHUWKHUHFWDQJXODUEHDPLQ)LJXUHZLWKWZROD\HUVRIUHLQIRUFHPHQW$VVXPHDOOWKHEDUVDUHWKHVDPHVL]H
occurs at a distance
from the
and are located below the neutral axis. Also assume the yield strain
and :
H[WUHPHFRPSUHVVLRQ¿EHU7KHIROORZLQJUHODWLRQVKLSLVHVWDEOLVKHGEHWZHHQ
(6.13)
ܾ
‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ
݀ െ ሺܽȀʹሻ
݀௧ ݀௧௬ ܿ
ܽ
݀
݄
ߝ௧௬ ‫ܣ‬௦ ܽ ൌ ߚଵ ܿ
ͲǤͺͷ݂௖ᇱ ͲǤͲͲ͵
ߝ௦ ܶ ൌ ‫ܣ‬௦ ݂௬ ߝ௧ Figure 6.8 A reinforced concrete beam with multiple layers of tension reinforcement.
Solving for
results in the following:
(6.14)
)RU*UDGHUHLQIRUFHPHQW(TXDWLRQ UHGXFHVWRWKHIROORZLQJ
(6.15)
Reinforcement located a distance equal to or greater than
IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVDQG
is
calculated by Equation (6.11) assuming all the reinforcement is concentrated at , which is the distance from the
H[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIDOOWKHUHLQIRUFHPHQW VHH)LJXUH :KHUHWKHUHLQIRUFHPHQWLQRQH
or more layers does not yield, it are separated from the reinforcement in the layer(s) that yield, and the actual stress
is used to calculate
.
in those bars
Rectangular Sections with Tension and Compression Reinforcement
5HLQIRUFHPHQWLVW\SLFDOO\DGGHGWRWKHFRPSUHVVLRQ]RQHRIDEHDPVHFWLRQIRUWKHIROORZLQJUHDVRQV WR
, where the dimensions of the beam are limited (due to architectural
LQFUHDVHWKHQRPLQDOÀH[XUDOVWUHQJWK
is greater than that correspondconstraints, for example) and the area of tensile reinforcement required to resist
LQJWRDWHQVLRQFRQWUROOHGVHFWLRQDQG WRKHOSUHGXFHORQJWHUPGHÀHFWLRQVRIWKHEHDP VHH6HFWLRQRIWKLV
SXEOLFDWLRQIRUDGLVFXVVLRQRQORQJWHUPGHÀHFWLRQV %HDPVZLWKERWKWHQVLRQDQGFRPSUHVVLRQUHLQIRUFHPHQWDUH
commonly referred to as doubly reinforced beams.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
5HLQIRUFHPHQWLQWKHFRPSUHVVLRQ]RQHFRQWULEXWHVWR
, though the contribution is usually relatively small. Additionally, is larger when compression reinforcement is used in a section, which results in more ductile behavior.
Transverse reinforcement in the form of stirrups are typically required in beams to resist factored shear forces. Stirrups must be anchored at both the top and bottom of the section to longitudinal reinforcement to properly develop
WKHPLQWHQVLRQ VHH6HFWLRQRIWKLVSXEOLFDWLRQ 7KXVUHLQIRUFLQJEDUVLQWKHFRPSUHVVLRQ]RQHPXVWEH
provided wherever stirrups are required. As noted above, the contribution of the compression reinforcement to
LVXVXDOO\UHODWLYHO\VPDOODQGWKXVLVRIWHQQRWFRQVLGHUHGLQGHVLJQ+RZHYHULIFRPSUHVVLRQUHLQIRUFHPHQWLV
.
needed to make the section tension-controlled, the following information can be used to determine
Nominal Flexural Strength When the Compression Reinforcement Yields. Strain and stress distributions in a
doubly reinforced section are given in Figure 6.9 where the total area of compression reinforcement, , is located
from the
a distance IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU$VVXPLQJWKH\LHOGVWUDLQ , occurs at a distance
H[WUHPHFRPSUHVVLRQ¿EHUWKHIROORZLQJUHODWLRQVKLSLVHVWDEOLVKHGEHWZHHQ and :
ͲǤͺͷ݂௖ᇱ ݀௬ᇱ ܿ
ߝ௦ᇱ ߝ௧௬ ‫ܥ‬௦ᇱ ൌ ‫ܣ‬ᇱ௦ ݂௦ᇱ ‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾܽ
݀ െ ሺܽȀʹሻ
݄
݀ ൌ ݀௧ ‫ܣ‬ᇱ௦ ܽ
ͲǤͲͲ͵
ܽ ൌ ߚଵ ܿ
ܾ
݀ᇱ (6.16)
‫ܣ‬௦ ߝ௧ ܶ ൌ ‫ܣ‬௦ ݂௬ Figure 6.9 Strain and stress distributions in a doubly reinforced concrete beam.
Solving for
results in the following:
(6.17)
)RU*UDGHUHLQIRUFHPHQW(TXDWLRQ UHGXFHVWRWKHIROORZLQJ
(6.18)
Compression reinforcement located a distance equal to or less than IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGV
, and the depth of the equivalent stress block, , can be obtained by satisfying force equilibrium (that is, by
and solving for ):
setting
(6.19)
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6-13
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHQRPLQDOÀH[XUDOVWUHQJWKLQWKLVFDVHFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ
(6.20)
Nominal Flexural Strength When the Compression Reinforcement Does Not Yield. When the compression rein, the depth of the stress block, , cannot be determined by Equation (6.19) because
forcement does not yield
is unknown. A relationship between
and can be obtained from strain compatibility:
the magnitude of
(6.21)
LQWR(TXDWLRQ DQGVROYLQJIRU
Substituting
results in the following:
(6.22)
From force equilibrium,
can be obtained from the following equation:
(6.23)
where
ing equations.
and
Note that
and
have the units of kips per square inch in the preced-
7KHQRPLQDOÀH[XUDOVWUHQJWKFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ
(6.24)
where
and the terms
and
DUHGHWHUPLQHGE\(TXDWLRQV DQG UHVSHFWLYHO\
T-Beams and Inverted L-Beams with Tension Reinforcement
Effective Flange Width. In typical reinforced concrete construction, the beams and slabs are cast together with
reinforcement extending between the members, which results in a monolithic structure. Therefore, the beams do not
act alone but rather work with a portion of the slab to resist the effects from the applied loads.
,QJHQHUDOWKHHIIHFWLYHÀDQJHZLGWK , of a beam (that is, the portion of the slab that works with the beam)
, slab thickness
, and clear span length of the
depends on the clear distance between adjoining beam webs
. Dimensional limits for for nonprestressed beams supporting monolithic slabs on both sides of the
beam
beam web (interior beams or T-beams) and on one side of the beam web (edge beams or inverted L-beams) are
JLYHQLQ$&, VHH)LJXUH Nominal Flexural Strength – Flange in Tension. 7KHÀDQJHRID7EHDPRUDQLQYHUWHG/EHDPLVLQWHQVLRQDW
locations of negative moment in a continuous system (that is, at the faces of the supports). The strain and stress distributions for a T-beam in this case are illustrated in Figure 6.11. An inverted L-beam has similar distributions.
7KHQRPLQDOÀH[XUDOVWUHQJWKLQWKLVFDVHLVGHWHUPLQHGE\(TXDWLRQ IRUDUHFWDQJXODUVHFWLRQZLWKDVLQJOH
If needed, the contribution of the positive reinforcement in the web (not shown
layer of reinforcement using
FDQEHGHWHUPLQHGXVLQJ(TXDWLRQ RU in Figure 6.11) can be included and
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܾ௪ଵ ݄
ܾ௙ଶ ܾ௙ଵ ܾ௪ଶ ‫ݏ‬௪ଵ ͸݄
ܾ௙ଵ ൌ ܾ௪ଵ ൅ Ž‡ƒ•–‘ˆ ‫ݏ‬௪ଵ Τʹ ‫ ۔‬
ۖ
‫ە‬κ௡ଵ Τͳʹ
‫ۓ‬
ۖ
ܾ௪ଷ ‫ݏ‬௪ଶ ͳ͸݄
ܾ௙ଶ ൌ ܾ௪ଶ ൅ Ž‡ƒ•–‘ˆ ሺ‫ݏ‬௪ଵ Τʹሻ ൅ ሺ‫ݏ‬௪ଶ Τʹሻ
‫۔‬
ۖ
‫ ە‬κ௡ଶ ΤͶ
‫ۓ‬
ۖ
Figure 6.10 Effective flange widths for T-beams and inverted L-beams.
ܾ௙ ݄
ߝ௧ ܶ ൌ ‫ܣ‬௦ ݂௬ ܽ
ܿ
ܾ ൌ ܾ௪ ݀ െ ሺܽȀʹሻ
ܽ ൌ ߚଵ ܿ
݀ ൌ ݀௧ ‫ܣ‬௦ ‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾ௪ ܽ
ͲǤͲͲ͵Ͳ
ܽ
ܾ ൌ ܾ௙ ܽ ൌ ߚଵ ܿ
Figure 6.11 Strain and stress distributions in a T-beam with the flange in tension.
ͲǤͲͲ͵Ͳ
݀ െ ሺܽȀʹሻ
݀ ൌ ݀௧ ݄
ܿ
‫ ܥ‬ൌ ͲǤͺͷ݂௖ᇱ ܾ௙ ܽ
‫ܣ‬௦ ߝ௧ ܾ௪ ܶ ൌ ‫ܣ‬௦ ݂௬ Figure 6.12 Strain and stress distributions in a T-beam with the flange in compression and D”K.
Nominal Flexural Strength – Flange in Compression. At locations of positive moment in a continuous system,
DSRUWLRQRIWKHÀDQJHRUWKHHQWLUHÀDQJHRID7EHDPRULQYHUWHG/EHDPLVLQFRPSUHVVLRQ7KHGHWHUPLQDWLRQRI
depends on whether the depth of the equivalent stress block, LVOHVVWKDQRUJUHDWHUWKDQWKHÀDQJHWKLFNQHVV
'HSWKRI6WUHVV%ORFN/HVV7KDQRU(TXDOWRWKH)ODQJH7KLFNQHVV
. The depth of the equivalent stress block,
LVLQLWLDOO\GHWHUPLQHGE\(TXDWLRQ IRUVHFWLRQVZLWKRQHOD\HURIUHLQIRUFHPHQWRUE\(TXDWLRQ IRU
WKHFRPSUHVVLYH]RQHLVUHFWDQJXODUZLWKDZLGWKHTXDOWR
doubly reinforced sections. If it is found that
for sections with one layer of reinforcement. SimiVHH)LJXUH (TXDWLRQ FDQEHXVHGWRGHWHUPLQH
ODUO\(TXDWLRQ RU FDQEHXVHGLIFRPSUHVVLRQUHLQIRUFHPHQWLVLQFOXGHG
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6-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
'HSWKRI6WUHVV%ORFN*UHDWHU7KDQWKH)ODQJH7KLFNQHVV
. Where GHWHUPLQHGE\(TXDWLRQ RU
IDOOVZLWKLQWKHZHEWKHFRPSUHVVLYH]RQHLV7RU/VKDSHGDVRSSRVHGWRUHFWDQJXODU VHH)LJXUHIRUD
7EHDP 7KHQRPLQDOÀH[XUDOVWUHQJWKRIWKHVHFWLRQLQWKLVFDVHFDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ
(6.25)
where
and
ܾ ൌ ܾ௙ ‫ܥ‬
݆݀
݀ ൌ ݀௧ ܿ
݄
ܽ ൌ ߚଵ ܿ
ͲǤͲͲ͵Ͳ
‫ܣ‬௦ ܶ ൌ ‫ܣ‬௦ ݂௬ ߝ௧ ܾ௪ Figure 6.13 Strain and stress distribution in a T-beam with the flange in compression and a>h.
6.4.3 Nominal Shear Strength
Overview
The nominal one-way shear strength,
DWDVHFWLRQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, (6.26)
where
is the nominal shear strength provided by concrete and
reinforcement.
is the nominal shear strength provided by shear
Nominal Shear Strength Provided by Concrete
For nonprestressed members, LVGHWHUPLQHGE\WKHDSSOLFDEOHHTXDWLRQLQ$&,7DEOH $&, 7KH
HTXDWLRQVLQWKDWWDEOHDUHJLYHQLQ7DEOHDORQJZLWKWKHPD[LPXPSHUPLWWHG VSHFL¿HGLQ$&,
Table 6.4 Nominal Shear Strength Provided by Concrete, Vc
Area of Shear Reinforcement, Av*
*Minimum area of shear reinforcement,
**
Vc**
, is determined by ACI Table 9.6.3.4
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6-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The minimum area of shear reinforcement in a beam,
LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, VHH6HFWLRQRIWKLVSXEOLFDWLRQ 7KHUHTXLUHGDUHDRIVKHDUUHLQIRUFHPHQW , is often greater than or equal to
at the critical section.
7KHPRGL¿FDWLRQIDFWRU WKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWR
QRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ
Table 6.3 based on composition of aggregates in the concrete mix.
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength
attributed to concrete in members without shear reinforcement does not increase in direct proportion with member
GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(6.27)
,WLVHYLGHQWIURP(TXDWLRQ WKDW LVOHVVWKDQIRUPHPEHUVZLWK
EHDPVZLWKDQRYHUDOOGHSWKOHVVWKDQDERXWLQ$VQRWHGDERYHEHFDXVH
, this factor is usually not applicable when determining
for beams.
This means
for
is often greater than or equal to
LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by
where
is the width
The term
RIWKHZHERIWKHEHDP$FFRUGLQJWR$&,5 may be taken as the sum of the areas of the longitudinal
ÀH[XUDOUHLQIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU)RUPHPEHUVZLWKRQHOD\HURIWHQVLRQUHLQIRUFHPHQW LVWKHWRWDODUHDRIÀH[XUDOUHLQIRUFHPHQWDWWKDW
section.
$FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken
, and is to be taken as negative for tenas positive for compression forces acting on the gross area of the beam,
sion forces.
used to calculate DUHOLPLWHGWRSVL $&, H[FHSWDVDOORZHGLQ$&, ZKLFK
Values of
WREHJUHDWHUWKDQSVLSURYLGHGWKHPLQLPXPVKHDUUHLQIRUFHPHQWLQ$&,RUWKH
permits values of
PLQLPXPWUDQVYHUVHWRUVLRQDOUHLQIRUFHPHQWLQ$&,LIDSSOLFDEOHLVSURYLGHGLQWKHVHFWLRQ 7KLVOLPLWDWLRQ
is primarily due to the fact that there is a lack of test data and practical experience with concrete having comon
SUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL)RUHFRQRP\ IRUEHDPVVKRXOGQRWH[FHHGSVL 5HIHUHQFH 7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH
FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@
(6.28)
In typical situations, beams are subjected to shear forces acting along one axis only, so the requirements in ACI
DQGIRULQWHUDFWLRQRIVKHDUIRUFHVDORQJRUWKRJRQDOD[HVDUHXVXDOO\QRWDSSOLFDEOH
Nominal Shear Strength Provided by Shear Reinforcement
The nominal shear strength provided by shear reinforcement, LVWREHGHWHUPLQHGE\$&, $&, The following types of shear reinforcement are permitted in nonprestressed members:
(1) 6WLUUXSVWLHVRUKRRSVSHUSHQGLFXODUWRWKHORQJLWXGLQDOD[LVRIWKHPHPEHU>$&, D @
Welded wire reinforcement with wires located perpendicular to the longitudinal axis of the member [ACI
E @
(3) 6SLUDOUHLQIRUFHPHQW>$&, F @
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6-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,QFOLQHGVWLUUXSVPDNLQJDQDQJOHRIDWOHDVWGHJUHHVZLWKWKHORQJLWXGLQDOD[LVRIWKHPHPEHUDQGFURVVLQJ
WKHSODQHRIWKHSRWHQWLDOVKHDUFUDFN $&,
/RQJLWXGLQDOUHLQIRUFHPHQWEHQWDQDQJOHRIGHJUHHVRUPRUHZLWKUHVSHFWWRWKHORQJLWXGLQDOD[LVRIWKH
PHPEHU $&,
6WLUUXSVRULHQWHGSHUSHQGLFXODUWRWKHD[LVRIWKHPHPEHUDQGDQFKRUHGWRWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQW
are the most commonly used type of shear reinforcement for beams in buildings assigned to SDC A or B. In buildings assigned to SDC C and above, spirals or hoops in accordance with the provisions in Chapter 18 of ACI 318
PXVWEHSURYLGHG VHH&KDSWHUVDQGRIWKLVSXEOLFDWLRQ ,QFOLQHGVWLUUXSVDQGEHQWORQJLWXGLQDOEDUVDUHHVsentially not used any longer and, thus, are not covered here.
7KHQRPLQDOVKHDUVWUHQJWKSURYLGHGE\VWLUUXSVLVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(6.29)
In this equation,
is the total area of shear reinforcement within the center-to-center spacing, , of the stirrups.
)RUWKHWZROHJJHGVWLUUXSVGHSLFWHGLQ)LJXUH ZKLFKDUHFRPPRQO\UHIHUUHGWR8VWLUUXSVEHFDXVHRIWKHLU
where
is the area of the stirrup bar. In general,
is equal to
times the number of stirrup
shape),
OHJVSURYLGHG $&, ‫ܣ‬௕ ‫ܣ‬௩ ൌ ʹ‫ܣ‬௕ –Š‡””‡‹ˆ‘” ‡‡–‘–•Š‘™ˆ‘” Žƒ”‹–›
‫ݏ‬
Figure 6.14 Two-legged U-stirrup configuration.
Locating stirrups solely around the perimeter of a wide beam is not fully effective. Thus, stirrup legs are required
in the interior of a wide beam. The maximum stirrup spacing across the width of a member is given in ACI Table
WKLVUHTXLUHPHQWHVVHQWLDOO\WUDQVODWHVWRDPLQLPXPQXPEHURIVWLUUXSOHJVQHHGHGDFURVVWKHZLGWKRI
DEHDP,QVXFKFDVHVVKHDUUHLQIRUFHPHQWXVXDOO\FRQVLVWVRIVHWVRI8VWLUUXSV)RUWKHFRQ¿JXUDWLRQLOOXVWUDWHG
LQ)LJXUHDVLQJOHRSHQVWLUUXSH[WHQGVWKHIXOOQHWZLGWKRIWKHEHDP WKHQHWZLGWKLVHTXDOWRWKHIXOOEHDP
ZLGWKPLQXVWKHWRWDOVLGHFRYHU $VWLUUXSFDSFRQVLVWLQJRIDKRUL]RQWDOEDUZLWKDGHJUHHKRRNDWRQHHQGDQG
DGHJUHHKRRNDWWKHRWKHUHQGLVSURYLGHGDWWKHWRSRIWKHFRQ¿JXUDWLRQZKLFKDOVRH[WHQGVWKHIXOOQHWZLGWKRI
the beam. Providing a full-width stirrup helps in maintaining the required concrete cover and facilitates installation
RIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDP7ZRVHWVRILGHQWLFDO8VWLUUXSVZLWKGHJUHHKRRNVDWHDFKHQG
.
DUHSODFHGV\PPHWULFDOO\ZLWKLQWKHLQWHULRURIWKHEHDP,QWKLVFRQ¿JXUDWLRQ
In relatively narrow beams, a continuous bent bar, which forms a series of single-leg stirrups along the length of
WKHPHPEHUFDQEHXVHGDVVKHDUUHLQIRUFHPHQWZKHUHUHTXLUHG $&,5VHH)LJXUH $&, F SHUPLWVDQGEDUVWREHXVHGDVVKHDUUHLQIRUFHPHQWLQVWDQGDUGMRLVWFRQVWUXFWLRQ ZKLFKLVGH¿QHGLQ$&,
9.8) where the bar is anchored by a standard hook, which is not necessarily engaging any of the longitudinal rein-
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Figure 6.15 Multiple-leg stirrup configuration for wide beams.
–ƒ†ƒ”†Š‘‘
‫ܣ‬௕ ‫ݏ‬
Figure 6.16 Single-leg stirrup configuration for joists and narrow beams.
IRUFHPHQW$OWKRXJKWKLVW\SHRIVKHDUUHLQIRUFHPHQWLVSHUPLWWHGVSHFL¿FDOO\IRUVWDQGDUGMRLVWFRQVWUXFWLRQLWLV
assumed that it can be used in joists that are part of a wide-module joist system or for any narrow beam where U.
VWLUUXSVFDQQRWEHXVHG,QWKLVFRQ¿JXUDWLRQ
6.4.4 Nominal Torsional Strength
Where torsional effects must be considered
, nominal torsional strength
is determined in accordance
ZLWK$&, $&, )RUQRQSUHVWUHVVHGPHPEHUV LVHTXDOWRWKHIROORZLQJ $&, (6.30)
The term LVWKHJURVVDUHDRIWKHEHDPFURVVVHFWLRQHQFORVHGE\WKHWRUVLRQDOVKHDUÀRZSDWKDIWHUFUDFNLQJ,Q
is permitted to be approximated as
where
is the area enclosed by
lieu of determining it by analysis,
WKHFHQWHUOLQHRIWKHRXWPRVWFORVHGWUDQVYHUVHWRUVLRQDOUHLQIRUFHPHQWLQWKHVHFWLRQ VHH$&,DQG$&,
where
and are the distances
)LJXUH5 )RUWKHLQWHULRUDQGHGJHEHDPVLQ)LJXUH
between the centerlines of the outmost closed stirrups in the section. Closed stirrups must be used because cracking
due to torsional moments can occur on all faces of a beam. The perimeter of the centerline of the outermost closed
transverse torsional reinforcement, LVDOVRJLYHQLQ)LJXUHIRUVROLGVHFWLRQV1RWHWKDWWKHGH¿QLWLRQVRI
and
LQ)LJXUHDUHWKHVDPHIRUKROORZVHFWLRQV
The term
is the area of one leg of the outermost closed stirrups, which have a center-to-center spacing of
along the length of the member. Regardless of the total number of stirrup legs, only the outermost legs resist torsional effects; this is consistent with the thin-walled tube methodology that forms the basis of the torsional provisions in
$&,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ݔ‬ଵ ‫ݕ‬ଵ ݄௕ ‫ݕ‬ଵ ݄௕ ݄
݄
‫ݔ‬ଵ ݀௦ ܿ
݀௦ ܿ
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ܿ ൌ Ž‡ƒ” ‘˜‡”–‘ Ž‘•‡†•–‹””—’
۷‫ܚܗܑܚ܍ܜܖ‬۰‫ܕ܉܍‬
۳‫܍܏܌‬۰‫ܕ܉܍‬
‫ݔ‬ଵ ൌ ܾ௪ െ ʹܿ െ ݀௦ ‫ݕ‬ଵ ൌ ݄௕ ൅ ݄ െ ʹܿ െ ݀௦ ‫ܣ‬௢௛ ൌ ‫ݔ‬ଵ ‫ݕ‬ଵ ‫݌‬௛ ൌ ʹሺ‫ݔ‬ଵ ൅ ‫ݕ‬ଵ ሻ
Figure 6.17 Torsional section properties Aoh and ph.
The total area of longitudinal reinforcement required to resist the effects of torsion, , must be uniformly distribXWHGDURXQGWKHSHULPHWHURIWKHEHDPDQGLVFRPELQHGZLWKWKHORQJLWXGLQDOUHLQIRUFHPHQWUHTXLUHGIRUÀH[XUH VHH
Section 6.5.4 of this publication).
The angle is related to the orientation of the compression diagonals that are part of the space truss assumed to
UHVLVWWKHHIIHFWVIURPWRUVLRQDIWHUFUDFNLQJ WKHWUDQVYHUVHDQGORQJLWXGLQDOUHLQIRUFHPHQWDUHWKHRWKHUSDUWVRIWKH
WUXVVVHH)LJXUH $&, D SHUPLWV to be taken as 45 degrees for nonprestressed members instead
of determining it from analysis.
‘‰‹–—†‹ƒŽ„ƒ”•
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‫ݕ‬ଵ ‘ ”‡–‡ ‘’”‡••‹‘
†‹ƒ‰‘ƒŽ•
‫ݔ‬ଵ Figure 6.18 Space truss analogy for a reinforced concrete beam subjected to torsional loading.
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6-20
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In order to help reduce unsightly cracking and to prevent crushing of the inclined concrete compression struts due
to shear and torsion, the cross-sectional dimensions of solid and hollow sections are limited by ACI Equations
D DQG E UHVSHFWLYHO\ $&, (6.31)
(6.32)
The terms on the left-hand side of these equations are the stresses due to shear and torsion, respectively.
In solid sections, shear stresses act over the full width of the section, while those due to torsion are resisted by a
WKLQZDOOHGWXEH>VHH$&,)LJXUH5 E @7KXVWKHVWUHVVHVFDQQRWEHGLUHFWO\DGGHGWRJHWKHU(TXDWLRQ represents an elliptical interaction between shear and torsional stresses.
For hollow sections, the shear and torsional stresses are directly additive on one side wall of the section [see ACI
)LJXUH5 D @7KLVOLQHDULQWHUDFWLRQRIVWUHVVHVLVHYLGHQWLQ(TXDWLRQ )RUKROORZVHFWLRQVZKHUHWKH
ZDOOWKLFNQHVVYDULHVDURXQGWKHSHULPHWHURIWKHVHFWLRQ(TXDWLRQ PXVWEHHYDOXDWHGZKHUHWKHVXPPDWLRQ
RIWKHVKHDUDQGWRUVLRQDOVWUHVVHVRQWKHOHIWKDQGVLGHRIWKHHTXDWLRQLVDPD[LPXP $&, $OVRZKHUH
, the torsional stress
LQ(TXDWLRQ PXVW
the wall thickness of a hollow section is less than
where is the thickness of the wall of the hollow section at the location where stresses are
be taken as
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the cross-sectional dimensions of a section typically has a greater impact than increasing .
Solid sections with an aspect ratio
(where is the width of that part of the cross-section containing the
closed stirrups provided to resist torsion) are permitted to be designed by a procedure other than that given in ACI
318 provided the procedure has demonstrated that the results are in substantial agreement with results from compreKHQVLYHWHVWV $&, ,QVXFKFDVHVWKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,DQGWKHGHWDLOLQJUHTXLUHPHQWVRI$&,DQGQHHGQRWEHVDWLV¿HG2QHVXFKGHVLJQSURFHGXUHLVUHIHUHQFHGLQ$&,
5
6.5 Determination of Required Reinforcement
6.5.1 Required Flexural Reinforcement
Rectangular Sections with Tension Reinforcement Only
Single Layer of Tension Reinforcement. 7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW , at a critical section of a
EHDPZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHPHQWLVGHWHUPLQHGEDVHGRQWKHUHTXLUHGDQGGHVLJQÀH[XUDOVWUHQJWKV
and
IRUWHQVLRQFRQWUROOHGVHFWLRQV8VLQJ(TXDWLRQV DQG WKHUHTXLUHG can be determined
by the following equation:
(6.33)
7KHQRPLQDOVWUHQJWKFRHI¿FLHQWRIUHVLVWDQFH
LVGH¿QHGDVIROORZV
(6.34)
where
for tension-controlled sections.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
It is important to check that
calculated by Equation (6.33) is at least equal to the following minimum area of
$&, ÀH[XUDOUHLQIRUFHPHQW
(6.35)
)RUSVLFRQFUHWHDQG*UDGHUHLQIRUFHPHQW
LWHGWRDPD[LPXPRISVL
governs. The value of
LQ(TXDWLRQ LVOLP-
0LQLPXPÀH[XUDOUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ QHHGQRWEHVDWLV¿HGDWDQ\VHFWLRQZKHUHDWOHDVW
is provided where LVFDOFXODWHGE\(TXDWLRQ >$&,@
an area of reinforcement equal to
It is also important to verify the section is tension-controlled. The following relationship is obtained from the linear
for tension-controlled sections (see
strain distribution where is the depth of the neutral axis when
$&,DQG)LJXUH (6.36)
Substituting
LQWR(TXDWLRQ ZLWK from Equation (6.36), the following equation can be used to
, corresponding to tension-controlled sections for beams with
FDOFXODWHWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW
:
(6.37)
)RU*UDGHUHLQIRUFHPHQW
SURSHUWLHV(TXDWLRQ UHGXFHVWRWKHIROORZLQJ
and for
For these material
(6.38)
If
calculated by Equation (6.33) is greater than
ZKLFKLVHVVHQWLDOO\WKHPD[LPXPDPRXQWRIÀH[XUDOUHLQ, and the dimensions of the beam
forcement permitted at any section, the section is not tension-controlled
PXVWEHPRGL¿HGDFFRUGLQJO\WRDWWDLQDWHQVLRQFRQWUROOHGVHFWLRQ
is determined in the same way as for sections with
Multiple Layers of Tension Reinforcement. The required
one layer of tension reinforcement using and (see Figure 6.8). For beams with two layers of tension reinforceusing
+RZHYHULWLVSHUPLWWHGWRXVH instead of the smaller
ment, it is conservative to determine
this may be advantageous in cases where the section is at or very close to the tension-controlled strain limit,
The reinforcement ratio
tion (
based on
; see Figure 6.8):
is equal to the following considering force equilibrium on the sec(6.39)
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6-22
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Similarly, the reinforcement ratio
to the following:
corresponding to a tension-controlled section based on
is equal
(6.40)
The following relationship between
and
LVREWDLQHGIURP(TXDWLRQV DQG (6.41)
The preceding discussion assumes the inner layer of reinforcement yields, which is advantageous for design purposIURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVZKHUH
es. Reinforcement located a distance equal to or greater than
LVFDOFXODWHGE\(TXDWLRQ RUE\(TXDWLRQ IRU*UDGHUHLQIRUFHPHQW VHH6HFWLRQRIWKLV
publication). Unless reinforcement is provided on the sides of a beam, it is common for all layers of reinforcement
to yield in normally proportioned beams.
Rectangular Sections with Tension and Compression Reinforcement
For rectangular sections where it has been determined that the section is not tension-controlled (that is,
), compression reinforcement may be added to the section so that it becomes tension-controlled (see
6HFWLRQRIWKLVSXEOLFDWLRQDQG)LJXUH which means the section is designed at the
A common design method in this situation is to set
, correWHQVLRQFRQWUROOHGQHWWHQVLOHVWUDLQOLPLW7KHQH[WVWHSLVWRGHWHUPLQHWKHQRPLQDOÀH[XUDOVWUHQJWK
using Equation (6.11):
sponding to
(6.42)
where
)RUDVHFWLRQZLWKDVLQJOHOD\HURIWHQVLRQUHLQIRUFHPHQWWKHUHLQIRUFHPHQWUDWLRWREHXVHGLQ(TXDWLRQ LV
the following:
(6.43)
where
LVGHWHUPLQHGE\(TXDWLRQ In situations where compression reinforcement is required, it is common for two layers of tension reinforcement to
EHQHHGHG,QWKLVFDVHWKHUHLQIRUFHPHQWUDWLRWREHXVHGLQ(TXDWLRQ LVWKHIROORZLQJ>VHH(TXDWLRQ @
(6.44)
7KHUHTXLUHGQRPLQDOÀH[XUDOVWUHQJWKUHVLVWHGE\WKHFRPSUHVVLRQUHLQIRUFHPHQW
and
:
UHTXLUHGQRPLQDOÀH[XUDOVWUHQJWK
, is the difference between the
(6.45)
where
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6-23
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In general, the stress in the compression reinforcement,
, can be determined by the following equation:
(6.46)
where
. Compression reinforcement located a distance equal to or less than
IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVZKHUH LVGHWHUPLQHGE\(TXDWLRQ RU >VHH6HFWLRQ
RIWKLVSXEOLFDWLRQ@
The required area of compression reinforcement,
, is determined from strength design:
(6.47)
Finally, the total required area of tension reinforcement, , is the summation of the area of reinforcement correassuming
:
sponding to the tension-controlled strain limit and
(6.48)
where LVGHWHUPLQHGE\(TXDWLRQ RU IRUDVLQJOHOD\HURUWZROD\HUVRIWHQVLRQUHLQIRUFHPHQWUHVSHFLVGHWHUPLQHGE\(TXDWLRQ tively, and
T-Beams and Inverted L-Beams with Tension Reinforcement
Nominal Flexural Strength – Flange in Tension. 7KHÀDQJHRID7EHDPRUDQLQYHUWHG/EHDPLVLQWHQVLRQDW
ORFDWLRQVRIQHJDWLYHPRPHQWLQDFRQWLQXRXVV\VWHP VHH6HFWLRQRIWKLVSXEOLFDWLRQ 2QFHWKHHIIHFWLYHÀDQJHZLGWK LVGHWHUPLQHG VHH)LJXUH WKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW
is calculated using Equation (6.33) for a rectangular section with a single layer of reinforcement (see Figure 6.11). If
and
are determined
needed, the contribution of compression reinforcement in the web can be included where
E\(TXDWLRQV DQG UHVSHFWLYHO\
7KHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVLQ$&,DUHDOVRDSSOLFDEOH>VHH(TXDWLRQ @
Nominal Flexural Strength – Flange in Compression. $WORFDWLRQVRISRVLWLYHPRPHQWDSRUWLRQRIWKHÀDQJHRU
WKHHQWLUHÀDQJHRID7EHDPRULQYHUWHG/EHDPLVLQFRPSUHVVLRQ$VVXPLQJWKHVHFWLRQLVWHQVLRQFRQWUROOHGZLWK
), FDQEHGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ VHH)LJXUH rectangular section behavior (that is,
(6.49)
where
and
If GHWHUPLQHGE\(TXDWLRQ LVOHVVWKDQRUHTXDOWR , the initial assumption that the section behaves as a
can be determined by Equation (6.33).
rectangular section is correct and
If it is found that
WKHLQLWLDODVVXPSWLRQRIUHFWDQJXODUEHKDYLRULVQRWFRUUHFWDQGWKHFRPSUHVVLYH]RQHLV
, and the
T- or L-shaped (see Figure 6.13). In such cases, the next step is to determine the area of reinforcement,
FRUUHVSRQGLQJWRWKHRYHUKDQJLQJÀDQJH V design strength,
(6.50)
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6-24
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(6.51)
The moment strength of the beam web,
section:
, is determined by subtracting the design strength
from
at the
(6.52)
The area of flexural reinforcement,
and
, required to resist
is determined by Equations (6.34) and (6.33) where
The total required area of flexural reinforcement is the sum of the areas corresponding to the contributions from the
overhanging flange(s) and the web:
(6.53)
Once
is determined by Equation (6.53), the assumption that the section is tension-controlled needs to be
where
is determined by Equation (6.37) or (6.38) with
checked, that is, it must be verified that
If the section is not tension-controlled, compression reinforcement may be added to the section to make it
tension-controlled or the dimensions of the beam must be modified.
The minimum reinforcement requirements in ACI 9.6.1.2 are also applicable [see Equation (6.35)].
6.5.2 Required Shear Reinforcement
The required area
and spacing of shear reinforcement perpendicular to the axis of a member is determined by
Equation (6.8), which can be rewritten as follows:
(6.54)
The nominal shear strength provided by the concrete, , is obtained from Table 6.4 and the nominal shear strength
provided by the shear reinforcement (stirrups), , is calculated by Equation (6.29).
𝑑𝑑
A summary of the shear requirements in ACI 9.6.3, 9.7.6, and 22.5 is given in Table 6.5 assuming the effects from
any torsional moments, if present, need not be considered. Note that these requirements are not applicable to the
beam types in ACI Table 9.6.3.1, which are cases where the minimum shear reinforcement requirements of ACI
Also, “Along the length” in Table 6.5 refers to the maximum spacing of
9.6.3.4 need not be satisfied if
legs of shear reinforcement along the length of the member and “Across the width” refers to the maximum spacing
of legs of shear reinforcement across the width of the member (see Figure 6.19).
𝑠𝑠��� �
𝑑𝑑
⎧lesser of � � for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑
24
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⎩
12″
𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝐭𝐭𝐭𝐭𝐭𝐭 𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰𝐰
𝑠𝑠��� �
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⎧lesser of � � for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑
24
⎪
⎨
⎪lesser of �𝑑𝑑/4 for 𝑉𝑉� � �𝑉𝑉� � �4�𝑓𝑓�� 𝑏𝑏� 𝑑𝑑
⎩
12″
𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀𝐀 𝐭𝐭𝐭𝐭𝐭𝐭 𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥𝐥
Figure 6.19 Maximum spacing requirements for legs of shear reinforcement.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.5 Shear Reinforcement Requirements for Beams*
Case
1
Required Shear
Strength, Vu
Stirrup spacing, s
Required Shear
Reinforcement, Av
Required
Maximum
None
ņ
ņ
• Along the length:
• Across the width:
• For
±$ORQJWKHOHQJWK
±$FURVVWKHZLGWK
3
• For
±$ORQJWKHOHQJWK
±$FURVVWKHZLGWK
*Minimum reinforcement requirements in this table are applicable in all cases except for those in ACI Table 9.6.3.1.
3ULRUWRFDOFXODWLQJWKHUHTXLUHGVKHDUUHLQIRUFHPHQWLWLVLPSRUWDQWWRFKHFNWKDW(TXDWLRQ LVVDWLV¿HG $&,
7KHVL]HRIWKHVHFWLRQPXVWEHLQFUHDVHGZKHUH
)RUHFRQRP\VWLUUXSVL]HDQGVSDFLQJ DORQJWKHOHQJWKDQGDFURVVWKHZLGWKRIWKHPHPEHU DUHHVWDEOLVKHGDWWKH
FULWLFDOVHFWLRQ¿UVW$WVHFWLRQVDZD\IURPWKHFULWLFDOVHFWLRQWKHVSDFLQJFDQEHLQFUHDVHGEXWWKHVWLUUXSEDUVL]H
and number of legs should be the same as those at the critical section. Stirrup spacing should be changed as few
times as possible over the required length; it is good practice to provide no more than 3 different stirrup spacings
ZKHUHUHTXLUHG DOWKRXJKGLIIHUHQWVSDFLQJVDUHSUHIHUUHGLISRVVLEOHRUSUDFWLFDO ZLWKWKH¿UVWVWLUUXSORFDWHGQR
PRUHWKDQLQIURPWKHIDFHRIVXSSRUW/DUJHUVWLUUXSEDUVDWZLGHUVSDFLQJDUHXVXDOO\PRUHFRVWHIIHFWLYHWKDQ
smaller bars at closer spacing; the latter require disproportionally higher costs for fabrication and placement.
7KHVHJPHQWVDORQJWKHVSDQZKHUHVKHDUUHLQIRUFHPHQWLVDQGLVQRWUHTXLUHGEDVHGRQWKHWKUHHFDVHVLQ7DEOH
DUHLGHQWL¿HGLQ)LJXUHIRUDUHLQIRUFHGFRQFUHWHEHDPVXEMHFWHGWRDIDFWRUHGXQLIRUPO\GLVWULEXWHGJUDYLW\
. In this case, two stirrup spacings are provided along the length of the member assuming the three requireload,
PHQWVLQ$&,SHUWDLQLQJWRWKHORFDWLRQRIWKHFULWLFDOVHFWLRQIRUVKHDUDUHVDWLV¿HG WKHVSDFLQJ corre,
sponds to the factored shear force at the critical section located a distance from the face of the support
corresponds to that based on minimum shear reinforcement
DQG WKHVSDFLQJ
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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ͳ
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‫ܣ‬௩ǡ௠௜௡ ൌ ‰”‡ƒ–‡”‘ˆ ቐ
ͷͲܾ௪ ‫ݏ‬Τ݂௬௧ ͵
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൒ ‫ܣ‬௩ǡ௠௜௡ ߶݂௬௧ ݀
Figure 6.20 Segments along the span of a reinforced concrete beam
where shear reinforcement is and is not required.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7RIDFLOLWDWHGHWHUPLQDWLRQRIWKHUHTXLUHGVL]HDQGVSDFLQJRIWKHVKHDUUHLQIRUFHPHQWYDOXHVRI
are given
where LVWDNHQDVDQG ZKLFK
LQ7DEOHXVLQJWKHHTXDWLRQVLQ7DEOHIRUVWLUUXSVSDFLQJV
provides a practical range for 7KHVHYDOXHVDUHEDVHGRQ*UDGHVWLUUXSVZLWKOHJVDQGDUHLQGHSHQGHQWRI
PHPEHUVL]HDQGFRQFUHWHFRPSUHVVLYHVWUHQJWK)RUVWLUUXSFRQ¿JXUDWLRQVZLWKPRUHWKDQOHJVWKHWDEXODWHG
YDOXHVFDQEHLQFUHDVHGDFFRUGLQJO\ IRUH[DPSOHIRUDVWLUUXSFRQ¿JXUDWLRQZLWKOHJVWKHFRUUHVSRQGLQJYDOXH
LVHTXDOWRWZRWLPHVWKHWDEXODWHGYDOXH $VWLUUXSVL]HDQGVSDFLQJLVVHOHFWHGIURP7DEOHWKDW
of
equal to or slightly greater than that required from analysis. For example, assume
provides a value of
)URP7DEOH8VWLUUXSV OHJV VSDFHGDW
provides
from analysis the required
where LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKH
longitudinal tension reinforcement in the beam.
Table 6.6 Values of Vu –ҀVc for Grade 60 Stirrups*
Stirrup spacing, s
#3 U-stirrups
#4 U-stirrups
#5 U-stirrups
19.8
39.6
111.6
*Tabulated values are for 2 legs of vertical shear reinforcement. Multiply the tabulated values by
total number of stirrup legs provided across the width of the member.
for stirrup arrangements with other than 2 legs where
is the
The distance from the face of the support where minimum shear reinforcement can be used is determined by setting
. The following equation is apthe factored shear force, , equal to the design shear strength of the concrete,
SOLFDEOHIRUWKHEHDPLQ)LJXUH
(6.55)
where
is the distance from the face of the support to the section where minimum shear reinforcement can be
results in the following:
XVHG WKDWLVWKHGLVWDQFHWRWKHRQVHWRI&DVH 6ROYLQJIRU
(6.56)
Similarly, the distance,
from the face of the support to the section where shear reinforcement is no longer required (that is, the distance to the onset of Case 1) can be determined from the following equation:
(6.57)
In general, the following equation can be used to determine the distance, , from the face of the support to the sec,
WLRQZKHUHWKHVWLUUXSFRQ¿JXUDWLRQFDQEHVSDFHGDW along the span for a given area of shear reinforcement,
where is greater than that determined at the critical section:
(6.58)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
It is important to ensure the value of XVHGLQ(TXDWLRQ LVOHVVWKDQRUHTXDOWRWKHJRYHUQLQJ
and
can be used is
YDOXHVJLYHQLQ7DEOH)RUH[DPSOHWKHGLVWDQFHIURPWKHIDFHRIWKHVXSSRUWZKHUH
determined as follows:
(6.59)
6.5.3 Required Torsion Reinforcement
Transverse Reinforcement
The required area
and spacing of transverse torsional reinforcement perpendicular to the axis of a member are
calculated by Equation (6.9) where LVGHWHUPLQHGE\WKH¿UVWRIWKHHTXDWLRQVLQ(TXDWLRQ (6.60)
As noted previously,
is the area of one leg of the outermost closed stirrups, which have a center-to-center spacing of along the length of the member. Regardless of the total number of stirrup legs across the width of the beam,
only the outermost legs resist torsional effects.
The maximum spacing of transverse torsional reinforcement is equal to the lesser of
DQGLQ $&,
The minimum area of transverse reinforcement for the combined effects of shear and torsion is covered in Section
RIWKLVSXEOLFDWLRQ
Longitudinal Reinforcement
The longitudinal reinforcement required for torsion, , is also calculated by Equation (6.9) where
PLQHGXVLQJERWKRIWKHHTXDWLRQVLQ(TXDWLRQ is deter-
(6.61)
In this equation,
LVFDOFXODWHGE\(TXDWLRQ The minimum area of longitudinal reinforcement,
LVJLYHQLQ$&,
(6.62)
It is permitted to reduce LQWKHÀH[XUDOFRPSUHVVLRQ]RQHVRIDEHDPE\DQDUHDHTXDOWR
where
is the factored bending moment occurring at the section simultaneously with $&, 7KHUHGXFHGDUHDRI
GHWHUPLQHGE\(TXDWLRQ longitudinal reinforcement must not be taken less than
6.5.4 Reinforcement Requirements for Combined Flexure, Shear, and Torsion
The amounts of transverse and longitudinal reinforcement required to resist the combined effects from bending moPHQWVVKHDUIRUFHVDQGWRUVLRQDOPRPHQWVDUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The total required area of transverse reinforcement per stirrup leg is equal to the required area of transverse reLQIRUFHPHQWIRUVKHDU>(TXDWLRQ @SOXVWKHUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWIRUWRUVLRQ>(TXDWLRQ
@:KHUHWRUVLRQDOUHLQIRUFHPHQWLVUHTXLUHGWKHPLQLPXPDPRXQWRIWUDQVYHUVHUHLQIRUFHPHQWSHUOHJLV
GHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ $&, (6.63)
Longitudinal torsional reinforcement, LVDGGHGWRWKDWUHTXLUHGIRUÀH[XUH
If torsional reinforcement is remust be distributed around the perimeter of the beam, so an area of steel equal to
is added to each
quired,
IDFHRIWKHEHDP $&, 7KXVDWWKHWRSDQGERWWRPIDFHVWKHWRWDOUHTXLUHGDUHDRIUHLQIRUFHPHQWLVHTXDOWR
and
, respectively. Similarly, at least
must be provided on the side faces or must
be added to any other longitudinal reinforcement on the side faces.
$SSHQGL[%LQ5HIHUHQFHFRQWDLQVWDEOHVWRIDFLOLWDWHWRUVLRQGHVLJQIRUHGJH VSDQGUHO EHDPV7RUVLRQDOVHFWLRQ
SURSHUWLHVDUHJLYHQIRUFRPPRQVODEWKLFNQHVVHVDQGHGJHEHDPVL]HV$OVRLQFOXGHGDUHWKHFRUUHVSRQGLQJYDOXHV
for threshold torsion, cracking torsion, and required transverse and longitudinal reinforcement. No calculations are
UHTXLUHGWRREWDLQDQ\RIWKHVHLWHPVIRUDJLYHQEHDPVL]HDQGVODEWKLFNQHVV
6.6 Reinforcement Detailing
6.6.1 Concrete Cover
5HLQIRUFLQJEDUVDUHSODFHGLQEHDPVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU
HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&, $&, )RUEHDPVZLWKWUDQVYHUVHUHLQIRUFHment that enclose the longitudinal reinforcing bars, concrete cover is measured from the surface of the concrete to
WKHRXWHUHGJHRIWKHWUDQVYHUVHUHLQIRUFHPHQW VHH)LJXUH 7KHPLQLPXPFRYHUWRWKHWUDQVYHUVHUHLQIRUFHPHQWLVHTXDOWRLQIRUUHLQIRUFLQJEDUVLQEHDPVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG $&,
7DEOH ‘˜‡”
‘˜‡”
Figure 6.21 Concrete cover for beams.
Cover requirements for beams subjected to fatigue in structures designed to be watertight (such as tanks) or in structures subjected to very aggressive exposures (such as concrete exposed to moisture or external sources of chlorides)
DUHJLYHQLQ$&,
6.6.2 Flexural Reinforcement Spacing
Minimum Spacing of Flexural Reinforcing Bars
0LQLPXPFOHDUVSDFLQJRISDUDOOHOUHLQIRUFLQJEDUVLQDVLQJOHKRUL]RQWDOOD\HULVJLYHQLQ$&, $&, 7KHVHOLPLWVKDYHEHHQHVWDEOLVKHGSULPDULO\VRFRQFUHWHFDQÀRZUHDGLO\LQWRWKHVSDFHVEHWZHHQDGMRLQLQJUHLQforcing bars.
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6-30
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
0LQLPXPVSDFLQJUHTXLUHPHQWVDUHJLYHQLQ)LJXUHZKHUH
LVWKHQRPLQDOPD[LPXPDJJUHJDWHVL]HLQWKH
is the diameter of the stirrup; is the inside bend radius
concrete mixture; is the clear cover to the stirrup;
IRUDQGVWLUUXSVDQG IRUDQGVWLUUXSVVHH$&,7DEOH
of the stirrup (which is equal to
DQG is the overall diameter of the longitudinal reinforcing bars (the overall diameter of a reinforcing bar
LQFOXGHVWKHGHIRUPDWLRQVRQWKHEDUVHH7DEOH 7KHPLQLPXPFOHDUVSDFLQJEHWZHHQSDUDOOHOUHLQIRUFLQJEDUV
SODFHGLQRUPRUHKRUL]RQWDOOD\HUVLVLQ $&,VHH)LJXUH ,QVXFKFDVHVORQJLWXGLQDOUHLQIRUFHment in the upper layers must be placed directly above reinforcement in the bottom layer.
൒ ͳԣ
ܿ௦ ݀௦ ‫ݎ‬
݀௕ ݀௕ ͳԣ
‹‹— Ž‡ƒ”•’ƒ ‡ൌ ‰”‡ƒ–‡”‘ˆ ݀௕ ሺͶΤ͵ሻ݀௔௚௚ Figure 6.22 Minimum clear spacing requirements for reinforcing bars.
Nominal
Diameter
A
A
Overall
Diameter
Section A–A
Table 6.7 Overall Reinforcing Bar Diameter
Bar Size
Approximate Diameter to Deformations (in.)
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.7 Overall Reinforcing Bar Diameter (cont.)
Bar Size
Approximate Diameter to Deformations (in.)
1
WKDWFDQ¿WLQD
The following equation can be used to determine the maximum number of reinforcing bars,
VLQJOHOD\HUEDVHGRQWKHPLQLPXPVSDFLQJOLPLWVLQ$&,
(6.64)
,QWKLVHTXDWLRQ³FOHDUVSDFH´UHIHUVWRWKHJUHDWHURIWKHWKUHHPLQLPXPFOHDUVSDFHVJLYHQLQ)LJXUH7KH
GHWHUPLQHGE\(TXDWLRQ LVURXQGHGGRZQWRWKHQH[WZKROHQXPEHU
value of
7KHPD[LPXPQXPEHURIEDUVWKDWFDQ¿WLQDVLQJOHOD\HUEDVHGRQWKHPLQLPXPVSDFLQJUHTXLUHPHQWVLQ$&,
DUHJLYHQLQ7DEOHEDVHGRQWKHIROORZLQJ
• 2YHUDOOEDUGLDPHWHULVXVHGIRUWKHORQJLWXGLQDOUHLQIRUFHPHQW 7DEOH
• Clear cover to the stirrup
• 1RPLQDOPD[LPXPDJJUHJDWHVL]H
• VWLUUXSVXVHGIRUDQGORQJLWXGLQDOEDUVDQGVWLUUXSVXVHGIRUDQGODUJHUORQJLWXGLQDOEDUV
Table 6.8 Maximum Number of Reinforcing Bars Permitted in a Single Layer*
Beam Width (in.)
Bar
Size
12
14
16
18
20
22
24
26
28
30
36
42
48
6
8
11
16
8
9
11
13
19
6
8
9
11
3
6
8
9
11
18
3
6
8
9
11
16
19
3
6
8
8
9
3
6
8
9
11
13
3
3
6
8
11
13
*Overall bar diameter is used for the longitudinal reinforcement (Table 6.7)
Clear cover to the stirrup
Nominal maximum aggregate size
#3 stirrups are used for #4, #5, and #6 longitudinal bars and #4 stirrups are used for #7 and larger longitudinal bars
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Maximum Spacing of Flexural Reinforcing Bars for Crack Control
0D[LPXPFHQWHUWRFHQWHUVSDFLQJRIUHLQIRUFLQJEDUVLVJLYHQLQ$&, $&, 7KHLQWHQWRIWKHVHUHTXLUHPHQWVLVWRFRQWUROÀH[XUDOFUDFNLQJ,QJHQHUDODODUJHUQXPEHURI¿QHUFUDFNVDUHSUHIHUDEOHWRDIHZZLGH
cracks mainly for reasons of durability and appearance.
The maximum center-to-center bar spacing, LVGHWHUPLQHGE\WKHHTXDWLRQVLQ$&,7DEOH $&, The following equation is applicable to deformed reinforcing bars:
(6.65)
In this equation, LVWKHOHDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKH
The term
VHFWLRQ)RUH[DPSOHIRUDEHDPZLWKDLQFRQFUHWHFRYHUWRVWLUUXSV
LVWKHFDOFXODWHGVWUHVVLQWKHÀH[XUDOUHLQIRUFHPHQWFORVHWWRWKHWHQVLRQIDFHDWVHUYLFHORDGV WKDWLV is equal to
the unfactored bending moment at the section divided by the product of the area of reinforcement and the internal
$&, )RU*UDGHUHLQIRUFHmoment arm). In lieu of calculating , it is permitted to be taken as
Assuming
, the maximum center-to-center bar spacing
ment,
IURP(TXDWLRQ 1RWHWKDW LVLQGHSHQGHQWRIWKHVL]HRIWKHORQJLWXGLQDOUHLQIRUFLQJEDUVLQWKHVHFWLRQ
The following equation can be used to determine the minimum number of reinforcing bars,
VLQJOHOD\HUWRVDWLVI\WKHFUDFNFRQWUROUHTXLUHPHQWVRI$&,
, required in a
(6.66)
In this equation, LVGHWHUPLQHGE\(TXDWLRQ 7KHFDOFXODWHGYDOXHRI
number.
is rounded up to the next whole
7KHPLQLPXPQXPEHURIEDUVWKDWFDQ¿WLQDVLQJOHOD\HUEDVHGRQWKHFUDFNFRQWUROUHTXLUHPHQWVRI$&,DUH
given in Table 6.9 based on the following:
• *UDGHUHLQIRUFHPHQWZLWK
• Overall bar diameter is used for the longitudinal reinforcement 7DEOH
• /HDVWGLVWDQFHIURPWKHVXUIDFHRIWKHÀH[XUDOUHLQIRUFHPHQWWRWKHWHQVLRQIDFHRIWKHVHFWLRQ
Table 6.9 Minimum Number of Reinforcing Bars Required in a Single Layer*
Beam Width (in.)
12
14
16
18
20
22
24
26
28
30
36
42
48
3
3
3
3
3
6
*Grade 60 reinforcement with
Overall bar diameter is used for the longitudinal reinforcement (Table 6.7)
Least distance from the surface of the flexural reinforcement to the tension face of the section
Distribution of Tension Reinforcement in Flanges of T-Beams
5HTXLUHPHQWVIRUWKHFRQWURORIFUDFNLQJLQWKHÀDQJHVRI7EHDPVLQWHQVLRQ WKDWLVDWQHJDWLYHEHQGLQJUHJLRQV DUH
JLYHQLQ$&, VHH)LJXUH 7KHEHDPÀH[XUDOUHLQIRUFHPHQWLQWKHÀDQJH VODE , must be uniformly
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
GLVWULEXWHGRYHUDZLGWKHTXDOWRWKHOHVVHURIWKHHIIHFWLYHÀDQJHZLGWK GH¿QHGLQ$&, VHH)LJXUH where LVWKHOHQJWKRIWKHFOHDUVSDQRIWKHEHDPPHDVXUHGIDFHWRIDFHRIVXSSRUWV VHH)LJXUH and
ܾ௙ ݄
κ௡ /10
‫ܣ‬௦ Additional ‫ܣ‬௦
Additional ‫ܣ‬௦
Additional ‫ܣ‬௦ ሺtotalሻ ൒ 0.0018ൣܾ௙ െ ሺκ௡ Τ10ሻ൧݄
Other reinforcement not shown for clarity
Figure 6.23 Distribution of tension reinforcement in the flange of a T-beam.
$GGLWLRQDOUHLQIRUFHPHQWVDWLVI\LQJ$&, VKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQW PXVWEHSURYLGHGLQ
WKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH
Crack Control Reinforcement in Deep Flexural Members
For beams with an overall depth, , greater than 36 in., longitudinal skin reinforcement must be uniformly distribfrom the tension face to help control the width of
uted on both faces of the beam for a distance equal to at least
WKHFUDFNVWKDWFDQIRUPRQWKHVHIDFHV $&, 7KHPD[LPXPFHQWHUWRFHQWHUVSDFLQJRIWKHVNLQUHLQIRUFHPHQWLVHTXDOWRWKDWUHTXLUHGIRUFUDFNFRQWUROLQ$&,ZKHUH is the clear cover from the side face to the
edge of the skin reinforcement.
‹”‡‹ˆ‘” ‡‡–ሺ–›’Ǥ ሻ
‫ݏ‬
݄ ൐ ͵͸ԣ
7KHH[WHQWRIWKHVNLQUHLQIRUFHPHQWLVLOOXVWUDWHGLQ$&,)LJXUH5DWSRVLWLYHDQGQHJDWLYHEHQGLQJUHJLRQV
DORQJWKHVSDQRIWKHEHDP7KHDUHDRIVNLQUHLQIRUFHPHQWLVQRWVSHFL¿HGLQ$&,KRZHYHUDFFRUGLQJWR
$&,5WREDUVDUHW\SLFDOO\SURYLGHG$QDOWHUQDWHFRQ¿JXUDWLRQLVVKRZQLQ)LJXUHZKHUHLWLV
assumed skin reinforcement is provided over the entire span.
ܿ௖ ͶͲǡͲͲͲ
൰ െ ʹǤͷܿ௖
‫ͳۓ‬ͷ ൬
݂௦
ۖ
‫ ݏ‬൑ Ž‡••‡”‘ˆ
‫۔‬
ͶͲǡͲͲͲ
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Figure 6.24 Skin reinforcement for deep flexural members.
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6-34
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
It is permitted to include the skin reinforcement when calculating the nominal moment strength of the beam,
provided a strain compatibility analysis of the section is performed.
,
6.6.3 Selection of Flexural Reinforcement
7KHVL]HDQGVSDFLQJRIWKHUHLQIRUFLQJEDUVDWDFULWLFDOVHFWLRQIRUÀH[XUHPXVWEHGHWHUPLQHGEDVHGRQWKHUHTXLUHGDUHDRIUHLQIRUFHPHQWDQGWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVJLYHQLQ6HFWLRQRIWKLV
publication.
Selecting the number of reinforcing bars in a single layer within the limits of Table 6.8 and Table 6.9 provides autoPDWLFFRQIRUPLW\ZLWKWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\JLYHQ
the assumptions noted in those tables.
,QJHQHUDORYHUDOOFRVWVDYLQJVLVXVXDOO\DFKLHYHGE\VSHFLI\LQJWKHODUJHVWSUDFWLFDOORQJLWXGLQDOEDUVL]HVWKDWVDWLVI\ERWKVWUHQJWKDQGVHUYLFHDELOLW\UHTXLUHPHQWV8VLQJWKHVDPHEDUVL]HVDQGOHQJWKVZKHUHYHUSRVVLEOHDOVRKDV
a positive impact on cost savings.
6.6.4 Development of Flexural Reinforcement
Overview
'HYHORSPHQWRIGHIRUPHGUHLQIRUFHPHQWLQEHDPVIRUÀH[XUHPXVWEHLQDFFRUGDQFHZLWK$&, $&, 7KHFDOFXODWHGWHQVLOHRUFRPSUHVVLYHIRUFHLQWKHORQJLWXGLQDOÀH[XUDOUHLQIRUFHPHQWDWHDFKVHFWLRQLQDEHDPPXVW
be developed on each side of that section by embedment length, hooks, headed deformed bars, mechanical devices,
RUDQ\FRPELQDWLRQWKHUHRI $&,$&, ,WLVLPSRUWDQWWRQRWHWKHIROORZLQJSURYLVLRQVJLYHQLQ$&,DQGZKHQGHWHUPLQLQJ
development lengths:
• +RRNHGDQGKHDGHGSRUWLRQVRIDUHLQIRUFLQJEDUPXVWQRWEHXVHGWRGHYHORSWKHUHLQIRUFLQJEDULQFRPSUHVVLRQ
• Development lengths do not require a strength reduction factor,
• Values of
XVHGWRFDOFXODWHGHYHORSPHQWOHQJWKVDUHOLPLWHGWRSVL
,QIRUPDWLRQRQKRZWRGHWHUPLQHWKHUHTXLUHGGHYHORSPHQWOHQJWKVIRUÀH[XUDOUHLQIRUFHPHQWLQDEHDPLVJLYHQ
below.
Development of Deformed Bars in Tension
3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHvelopment length, LVGHWHUPLQHGXVLQJWKHHTXDWLRQVLQ$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU
UHTXLUHPHQWVDUHFRYHUHG¿UVW
Method 1 – ACI 25.4.2.4
The development length in tension of a deformed reinforcing bar,
LVGHWHUPLQHGE\$&,(TXDWLRQ D (6.67)
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6-35
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• 0
RGL¿FDWLRQIDFWRUIRUOLJKWZHLJKWFRQFUHWH
concrete:
7KLVIDFWRUUHÀHFWVWKHORZHUWHQVLOHVWUHQJWKRIOLJKWZHLJKW
(6.68)
• 5
HLQIRUFHPHQWJUDGHIDFWRU
. This factor accounts for the effect of reinforcement yield strength on the
required development length:
(6.69)
• 5
HLQIRUFHPHQWFRDWLQJIDFWRU . This factor accounts for the reduced bond strength between the concrete
DQGHSR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFLQJEDUV
(6.70)
• 5
HLQIRUFHPHQWVL]HIDFWRU
reinforcing bars:
7KLVIDFWRUUHÀHFWVWKHPRUHIDYRUDEOHSHUIRUPDQFHRIVPDOOHUGLDPHWHU
(6.71)
• &
DVWLQJSRVLWLRQIDFWRU 7KLVIDFWRUUHÀHFWVWKHDGYHUVHHIIHFWVWKDWFDQRFFXUWRWKHWRSKRUL]RQWDO
reinforcement in a member due to vertical migration of water and mortar, which collect on the underside
of the bars during placement of the concrete:
(6.72)
$FFRUGLQJWRWKHIRRWQRWHLQ$&,7DEOH
• Spacing or cover dimension,
QHHGQRWH[FHHG
7KLVWHUPLVGH¿QHGDVIROORZV VHH)LJXUH (6.73)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿଵ ݀௕ ܿଶ ‫ݏ‬
‫ۓ‬
ۖ
ܿଵ
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ۖ
‫ݏە‬Τʹ
Figure 6.25 Spacing or cover dimension, cb.
• 7UDQVYHUVHUHLQIRUFHPHQWLQGH[
7KHWUDQVYHUVHUHLQIRUFHPHQWLQGH[UHSUHVHQWVWKHUROHRIFRQ¿QLQJ
UHLQIRUFHPHQWDFURVVWKHSRWHQWLDOVSOLWWLQJSODQHVODUJHUDPRXQWVRIFRQ¿QLQJUHLQIRUFHPHQWUHGXFHVWKH
potential for splitting failure, thereby reducing the overall required development length. This index is
GHWHUPLQHGE\$&,(TXDWLRQ E (6.74)
where
total cross-sectional area of all transverse reinforcement within a spacing
tial plane of splitting through the reinforcement being developed
that crosses the poten-
center-to-center spacing of the transverse reinforcement
number of bars being developed along the plane of splitting
if transverse reinforcement is present or required. For reinforcing bars
It is permitted to conservatively use
VSDFHGFORVHUWKDQLQRQFHQWHUWUDQVYHUVHUHLQIRUFHPHQWPXVWEHSURYLGHGDORQJWKHGHwith
$&,DQG velopment or lap splice length such that
PXVWEHWDNHQOHVVWKDQRUHTXDOWRLQ(TXDWLRQ >$&,@,WKDV
7KHFRQ¿QLQJWHUP
EHHQVKRZQWKDWZKHQWKLVWHUPLVOHVVWKDQVSOLWWLQJIDLOXUHVDUHOLNHO\WRRFFXU$SXOORXWIDLOXUHRIWKHUHLQIRUFHPHQWLVPRUHOLNHO\ZKHQWKLVWHUPLVJUHDWHUWKDQVRDQLQFUHDVHLQWKHDQFKRUDJHFDSDFLW\GXHWRDQ
LQFUHDVHLQFRYHURUDPRXQWRIFRQ¿QLQJUHLQIRUFHPHQWLVQRWOLNHO\
The tension development length GHWHUPLQHGE\$&,(TXDWLRQ D LVSHUPLWWHGWREHUHGXFHGLQFDVHV
ZKHUHWKHÀH[XUDOUHLQIRUFHPHQWLVJUHDWHUWKDQWKDWUHTXLUHGIURPDQDO\VLVH[FHSWIRUWKHVL[FDVHVLQ$&,
$&, 7KHUHGXFWLRQIDFWRUDSSOLHGWR LVHTXDOWRWKHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQW
GLYLGHGE\WKHSURYLGHGDUHDRIUHLQIRUFHPHQW7KHUHGXFHGGHYHORSPHQWOHQJWKPXVWQRWEHWDNHQOHVVWKDQLQ
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6-37
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Values of
IRUGHIRUPHGEDUVLQEHDPVDUHJLYHQLQ7DEOHEDVHGRQWKHIROORZLQJDVVXPSWLRQV
• Normalweight concrete with
• 8QFRDWHG*UDGHUHLQIRUFHPHQW
•
• Concrete cover controls the determination of
•
• VWLUUXSVDUHXVHGIRUDQGORQJLWXGLQDOEDUVDQGVWLUUXSVDUHXVHGIRUDQd larger longitudinal
bars
Calculated values of
are rounded up to the next whole number. Values of
for beams with coated bars can be
obtained by multiplying the tabulated values by the appropriate value of LQ(TXDWLRQ 6LPLODUO\YDOXHVRI
IRUEHDPVPDGHRIOLJKWZHLJKWFRQFUHWHFDQEHREWDLQHGE\GLYLGLQJWKHWDEXODWHGYDOXHVE\LQDFFRUGDQFH
with Equation (6.68).
Table 6.10 Tension Development Length, ℓd , for Deformed Bars in Beams in Accordance with ACI 25.4.2.4*
ℓd (in.)
Bar Size
Top
Other
19
33
68
18
36
*Normalweight concrete with
Uncoated Grade 60 reinforcement
Concrete cover controls the determination of
#3 stirrups used for #4, #5, and #6 longitudinal bars and #4 stirrups used for #7 and larger longitudinal bars
Tabulated values for
Reference 9.
EDVHGRQWKHUHTXLUHPHQWVRI$&,IRUDYDULHW\RIRWKHUFRQGLWLRQVDUHJLYHQLQ
Method 2 – ACI 25.4.2.3
7KHPHWKRGLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHVHOHFWHG
7KHWZRVSDFLQJDQGFRYHUFDVHVLQ$&,7DEOHDUHJLYHQLQ
YDOXHVRIWKHFRQ¿QLQJWHUP
Table 6.11.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.11 Tension Development Length, ℓd , for Deformed Bars in Beams in Accordance with ACI 25.4.2.3
Spacing and Cover
Case 1
#6 and Smaller Bars
#7 and Larger Bars
Condition 1
1. Clear spacing of bars being developed
or lap spliced
Clear cover
3. Stirrups or ties throughout
not less
than the required minimum
&RQGLWLRQ
1. Clear spacing of bars being developed
or lap spliced
Clear cover
&DVH
Other conditions
In order to determine XVLQJWKHHTXDWLRQVLQ&DVHWKHVSDFLQJFRYHUDQGFRQ¿QHPHQWRIWKHEDUVEHLQJ
GHYHORSHGPXVWPHHWDOOWKHUHTXLUHPHQWVXQGHU&RQGLWLRQRU&RQGLWLRQ VHH)LJXUH ,WLVSUHVXPHGWKDW
WKHVHFRQGLWLRQVRFFXUIUHTXHQWO\LQSUDFWLFH:KHQWKHUHTXLUHPHQWVRIHLWKHU&RQGLWLRQRU&RQGLWLRQDUHPHWLW
WKDWYDOXHLVVXEVWLWXWHGLQWR(TXDWLRQ ZKLFKUHVXOWVLQWKHHTXDWLRQVLQWKH
is assumed
¿UVWURZRI7DEOHIRU&DVH
൒‹‹—‫ܣ‬௩ ݀௕ Ž‡ƒ”•’ƒ ‡൒ ݀௕ Ž‡ƒ” ‘˜‡”൒ ݀௕ ݀௕ Ž‡ƒ” ‘˜‡”൒ ݀௕ Ž‡ƒ”•’ƒ ‡൒ ʹ݀௕ ‘†‹–‹‘ͳ
‘†‹–‹‘ʹ
Figure 6.26 Spacing, cover, and confinement conditions of ACI 25.4.2.3.
&DVHLVDSSOLFDEOHZKHUHWKHUHTXLUHPHQWVLQHLWKHU&RQGLWLRQRU&RQGLWLRQLQ&DVHDUHQRWVDWLV¿HGLQWKLV
case, it is assumed
Tension development lengths, , must be the greater of the values deterPLQHGE\WKHHTXDWLRQVLQ7DEOHDQGLQ
Tabulated values for
tions.
EDVHGRQWKHUHTXLUHPHQWVRI$&,DUHJLYHQLQ5HIHUHQFHIRUDYDULHW\RIFRQGL-
Development of Standard Hooks in Tension
+RRNVDUHSURYLGHGDWWKHHQGVRIUHLQIRUFLQJEDUVWRSURYLGHDGGLWLRQDODQFKRUDJHZKHUHGHYHORSPHQWOHQJWKFDQnot be attained with straight bars. Standard hook geometry for development of deformed bars in tension is given in
$&,7DEOH $&, @Seismicisolation
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6-39
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The development length of a deformed reinforcing bar in tension with a standard hook,
, is given in ACI
(6.75)
7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRN VHH)LJXUH 7KH
PRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOH VHH$&,7DEOH Table 6.12 Modification Factors for Development of Hooked Bars in Tension
Modification Factor
Lightweight,
Epoxy,
&RQ¿QLQJUHLQIRUFHPHQW
Location,
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG
reinforcement
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
or
Other
1.6
)RUDQGVPDOOHUKRRNHGEDUV
terminating inside a column core with side cover
normal to the plane of the hook
or
with side cover normal to the plane of the hook
Other
Concrete strength,
The term
LVWKHWRWDOFURVVVHFWLRQDODUHDRIWLHVRUVWLUUXSVFRQ¿QLQJWKHKRRNHGEDUVDQG
is the total crosssectional area of the hooked bars being developed at the same critical section. The term is the center-to-center
are given in ACI
spacing of the hooked bars, which have a nominal diameter . Detailing requirements for
VHH$&,)LJXUHV5DDQG5EIRUFRQ¿QLQJUHLQIRUFHPHQWSODFHGSDUDOOHODQGSHUSHQGLFXODU
to hooked bars being developed, respectively).
7KHGHWDLOLQJUHTXLUHPHQWVLQ$&,DSSO\WRKRRNHGEDUVDWGLVFRQWLQXRXVHQGVRIPHPEHUV WKDWLVDWHQGV
of simply-supported members, at the free ends of cantilevers, and at exterior joints where members do not extend
EH\RQGWKHMRLQW ZKHUHERWKVLGHFRYHUDQGWRS RUERWWRP FRYHUWRWKHKRRNLVOHVVWKDQLQ
for hooked deformed bars in beams based on normalweight concrete with
, uncoated
Values of
, and side cover normal to the plane of the
*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKRRNHGEDUVSDFLQJ
are given in Table 6.13. Calculated values of
are rounded up to the next whole number.
hook
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.13 Tension Development Length, ℓdh , for Hooked Deformed Bars in Beams*
Bar Size
ℓdh (in.)
#4
6
#5
8
#6
10
#7
13
#8
15
#9
18
#10
22
#11
25
*Normalweight concrete with
Uncoated Grade 60 reinforcement
Center-to-center hooked bar spacing
Side cover normal to the plane of the hook
Development of Headed Deformed Bars in Tension
3URYLVLRQVIRUWKHGHYHORSPHQWRIKHDGHGGHIRUPHGEDUVLQWHQVLRQDUHJLYHQLQ$&,([DPSOHVRIKHDGHG
UHLQIRUFLQJEDUVDUHJLYHQLQ)LJXUH
+HDGHGGHIRUPHGEDUVDUHSHUPLWWHGWREHXVHGRQO\ZKHQWKHFRQGLWLRQVRI$&,DUHVDWLV¿HG
(a) %DUPXVWFRQIRUPWR$&,
(b) %DUVL]HPXVWEHRUVPDOOHU
, must be at least
where
is the area of the bar
(c) Net bearing area of head,
(d) Concrete must be normalweight
where is the nominal diameter of the bar
(e) Clear cover to the bar must be greater than or equal to
(f) Center-to-center spacing of the bars must be greater than or equal to
The development length of a headed deformed reinforcing bar in tension,
LVJLYHQLQ$&,
(6.76)
This development length is measured from the critical section to the bearing face of the head (see ACI Figures
5DDQG5E 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ7DEOH VHH$&,7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.14 Modification Factors for Development of Headed Bars in Tension
Modification Factor
Condition
Epoxy,
Parallel tie reinforcement,
Location,
Value of Factor
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG
reinforcement
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
or
Other
1.6
For headed bars (1) terminating inside a column core
RU ZLWKVLGHFRYHUWR
with side cover to bar
bar
Other
Concrete strength,
The term
DFFRXQWVIRUFRQ¿QLQJHIIHFWVSURYLGHGE\WUDQVYHUVHUHLQIRUFHPHQWRULHQWHGSDUDOOHOWRWKHGHYHORSPHQWOHQJWKRIKHDGHGEDUV&RQ¿QHPHQWUHTXLUHPHQWVIRUKHDGHGQHJDWLYHPRPHQWUHLQIRUFHPHQWLQDEHDP
WHUPLQDWHGLQDMRLQWDUHJLYHQLQ$&,$WEHDPFROXPQMRLQWVWKHWRWDOFURVVVHFWLRQDODUHDRIWUDQVYHUVH
must be located within
of the centerline of the headed bar toward the middle of the joint
reinforcement,
is greater than or
where LVWKHQRPLQDOGLDPHWHURIWKHKHDGHGEDU>VHH$&,)LJXUH5 D @:KHUH
or where the center-to center spacing of the headed bars is greater than or equal to
The
equal to
is the total cross-sectional area of the headed bars being developed.
term
For other than beam-column joints,
must not be considered and
$&, provided
Values of
for headed deformed bars in beams based on normalweight concrete with
, and side cover to the bar
*UDGHUHLQIRUFHPHQWFHQWHUWRFHQWHUKHDGHGEDUVSDFLQJ
are rounded up to the next whole number.
7DEOH&DOFXODWHGYDOXHVRI
Table 6.15 Tension Development Length, ℓd t , for Headed Deformed Bars in Beams*
Bar Size
ℓdt (in.)
#4
6
#5
6
#6
8
#7
9
#8
11
#9
14
#10
16
#11
19
*Normalweight concrete with
Uncoated Grade 60 reinforcement
Side cover normal to the bar
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6-42
, uncoated
are given in
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Development of Mechanically Anchored Deformed Bars in Tension
Any mechanical attachment or device capable of developing of the deformed bars is permitted to be used providHGLWLVDSSURYHGE\WKHEXLOGLQJRI¿FLDOLQDFFRUGDQFHZLWK$&, $&, 7KHXVHRIPHFKDQLFDOGHYLFHV
WKDWGRQRWPHHWWKHUHTXLUHPHQWVLQ$&,RUWKDWDUHQRWGHYHORSHGLQDFFRUGDQFHZLWK$&,PD\EH
used provided test results are available that demonstrate the ability of the head and bar system to develop or anchor
the required force in the bar.
Development of Positive and Negative Flexural Reinforcement
Flexural reinforcement must be properly developed or anchored in a reinforced concrete beam so that the beam
SHUIRUPVDVLQWHQGHGLQDFFRUGDQFHZLWKWKHVWUHQJWKGHVLJQPHWKRG&ULWLFDOVHFWLRQVIRUGHYHORSPHQWRIÀH[XUDO
UHLQIRUFHPHQWRFFXUDWWKHIROORZLQJ $&, (1) Points of maximum stress (that is, sections of maximum bending moment)
Points along the span where adjacent reinforcement is terminated because it is no longer required
WRUHVLVWÀH[XUH
In continuous beams subjected to uniform gravity loads, the maximum positive and negative bending moments
W\SLFDOO\RFFXUQHDUPLGVSDQDQGDWWKHIDFHVRIWKHVXSSRUWVUHVSHFWLYHO\3RVLWLYHDQGQHJDWLYHÀH[XUDOUHLQIRUFLQJ
EDUVPXVWEHGHYHORSHGRUDQFKRUHGRQERWKVLGHVRIWKHVHFULWLFDOVHFWLRQV $&, 7KHUHTXLUHGDUHDRIÀH[XUDOUHLQIRUFHPHQWDWDFULWLFDOVHFWLRQFDQEHGHWHUPLQHGXVLQJWKHPHWKRGVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
݊ ൌ ݊ଵ ൅ ݊ଶ ݊ଵ ൒ ݊Τ3
‫ିܣ‬
௦ ሺ݊ barsሻ
‫ିܣ‬
௦ଵ ሺ݊ଵ barsሻ
‫ܣ‬ା
௦ ሺ‫ ݌‬barsሻ
‫ܣ‬ା
௦ଶ ሺ‫݌‬ଶ barsሻ
‫ିܣ‬
௦ଶ ሺ݊ଶ barsሻ
‫ ݌‬ൌ ‫݌‬ଵ ൅ ‫݌‬ଶ ‫݌‬ଵ ൒ ‫݌‬Τ4
A
൒ 6"
B
D
C
‫ܣ‬ା
௦ଵ ሺ‫݌‬ଵ barsሻ
൒ ݀ or 12݀௕ ൒ ݀ or 12݀௕ ݊ଶ bars
‫ݔ‬஼஽ ൒ κௗ ൒ κௗ ‫ݔ‬஺஻ ݊ଵ bars
൒ κௗ ሺ‫ܯ‬௨ା ሻ஽ ൒ ݀, 12݀௕ , or κ௡ /16
‫݌‬ଶ bars
ሺ‫ܯ‬௨ା ሻ஼ ‫ݔ‬஻ா E
Point of inflection
ሺ‫ܯ‬௨ି ሻ஻ ሺ‫ܯ‬௨ି ሻ஺ Figure 6.27 Development of flexural reinforcement in a beam.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUWKHFRQWLQXRXVEHDPLQ)LJXUHWKHWRWDOUHTXLUHGDUHDRIQHJDWLYHUHLQIRUFHPHQWIRUWKHPD[LPXPQHJDWLYH
, at critical section A is equal to
, and the total number of reinforcing bars at
factored bending moment,
this location is equal to . Similarly, the total required area of positive reinforcement for the maximum positive fac, at critical section C is equal to
, and the total number of reinforcing bars at this
tored bending moment,
location is equal to . Requirements for the development of these two sets of reinforcing bars are discussed next.
Negative Flexural Reinforcement
Assume a portion of
LVFXWRIIDWVHFWLRQ%ZKHUHLWLVQRORQJHUUHTXLUHGIRUÀH[XUDOVWUHQJWK7KLVPDNHVVHFand the number of reinforcing
tion B a critical section. At this location, the reinforcement has an area equal to
and the correbars is equal to . The area of the remaining portion of reinforcing bars is equal to
. This reinforcement must be able to resist the negative
sponding number of reinforcing bars is equal to
at section B.
factored bending moment
Because section B is a critical section,
bars must be adequately developed to the right of this section. This is
achieved by extending the bars a minimum distance of SDVWVHFWLRQ%DVVKRZQLQ)LJXUHZKHUH is the
GHYHORSPHQWOHQJWKLQWHQVLRQRIDGHIRUPHGEDUZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&, $FFRUGLQJWR$&,DWOHDVWRQHWKLUGRIWKHWRWDOQHJDWLYHUHLQIRUFHPHQWSURYLGHGDWDVXSSRUWPXVWKDYHDQ
, and
SDVWWKHSRLQWRILQÀHFWLRQZKHUH is the clear span
embedment length equal to the greater of ,
length measured face-to-face of supports (which in this case are columns). Therefore, to satisfy this requirement,
ZKLFKDFFRXQWVIRUSRVVLEOHVKLIWLQJRIWKHEHQGLQJPRPHQWGLDJUDPDWWKHSRLQWRILQÀHFWLRQGXHWRWKHDSSUR[L. The minimum length of
bars to the right
mate bending moment diagram customarily used in design,
RIFULWLFDOVHFWLRQ$LVHTXDOWRWKHIROORZLQJ $&,DQG (6.77)
In these equations,
is the distance from section A to the theoretical cutoff point at section B and
GLVWDQFHIURPVHFWLRQ%WRWKHSRLQWRILQÀHFWLRQDWVHFWLRQ(
is the
Because section A is a critical section, the bars cutoff at section B must be developed a distance equal to at least
EH\RQGWKDWVHFWLRQ$GGLWLRQDOO\$&,VWLSXODWHVWKHVHEDUVPXVWH[WHQGEH\RQGWKHSRLQWZKHUHWKH\DUHQR
(except at supports of simply-supported spans
longer required a distance equal to at least the greater of and
bars to the right of critical section A is equal to the foland at free ends of cantilevers). The minimum length of
lowing:
(6.78)
)OH[XUDOWHQVLRQUHLQIRUFHPHQWLVQRWSHUPLWWHGWREHWHUPLQDWHGLQDWHQVLRQ]RQHXQOHVVRQHRIWKHIROORZLQJFRQGLWLRQVDUHVDWLV¿HG $&, (1) At the cutoff point,
WLRQRIWKLVSXEOLFDWLRQ where
is the design shear strength of the section at that point (see Sec-
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6-44
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUDQGVPDOOHUEDUVFRQWLQXLQJUHLQIRUFHPHQWSURYLGHVDWOHDVWGRXEOHWKHDUHDUHTXLUHGIRUÀH[XUHDWWKH
.
cutoff point and
(3) Stirrup or hoop area in excess of that required for shear and torsion is provided along each terminated bar
from the cutoff point. The excess area of the stirrups or hoops must be greater
over a distance equal to
where the center-to-center spacing must be less than or equal to
and
than or equal to
The development of the negative reinforcing bars to the left of section A depends on its location within the framing
V\VWHP$WLQWHULRUMRLQWVOLNHWKHRQHGHSLFWHGLQ)LJXUHGHYHORSPHQWLVDFKLHYHGE\FRQWLQXLQJWKHQHJDWLYH
reinforcing bars into the span to the left of the column. In an end span where the beam terminates at an edge column, a standard hook is provided at the ends of the negative reinforcing bars.
Positive Flexural Reinforcement
is cut off at section D. This makes section D a critical section. At this location, the reinAssume a portion of
and the number of reinforcing bars is equal to . The area of the remaining
forcement has an area equal to
and the corresponding number of reinforcing bars is equal to
portion of reinforcing bars is equal to
. This reinforcement must be able to resist the negative factored bending moment
at section D.
beyond the critical secThe bars cut off at section D must be developed to a distance greater than or equal to
tion C. Like in the case of the negative reinforcement, these bars must extend beyond the point they are no longer
$&, 7KHPLQLPXPOHQJWKRI bars to the left
required by a distance equal to the greater of and
of critical section C is equal to the following:
(6.79)
In this equation,
is the distance between critical section C and the theoretical cutoff point located at section D.
7KHUHTXLUHPHQWVRI$&,SHUWDLQLQJWRUHLQIRUFHPHQWWHUPLQDWHGLQDWHQVLRQ]RQHDUHDOVRDSSOLFDEOHLQWKLV
case.
$FFRUGLQJWR$&,DWOHDVWRQHIRXUWKRIWKHSRVLWLYHUHLQIRUFHPHQWPXVWH[WHQGDWOHDVWLQLQWRWKHVXS. Note that if a beam is part of the LFRS, this
SRUW H[FHSWDWVLPSOHVXSSRUWVVHH$&, 7KXV
reinforcement must be anchored to develop at the support.
7KHGLDPHWHURIWKHSRVLWLYHUHLQIRUFHPHQWLVOLPLWHGDWSRLQWVRILQÀHFWLRQ DQGDWVLPSOHVXSSRUWV LQDFFRUGDQFH
ZLWK$&,WKLVUHTXLUHPHQWKHOSVWRHQVXUHWKHEDUVDUHGHYHORSHGLQDOHQJWKVKRUWHQRXJKVXFKWKDWWKH
PRPHQWFDSDFLW\LVJUHDWHUWKDQWKHDSSOLHGPRPHQWRYHUWKHHQWLUHOHQJWKRIWKHEHDP VHH$&,)LJXUH5 (6.80)
In this equation,
LVWKHQRPLQDOÀH[XUDOVWUHQJWKRIWKHEHDPVHFWLRQDVVXPLQJDOOWKHUHLQIRUFHPHQWDWWKDWVHFis the factored shear force at the section, and
is the embedment length of the reinforcetion is stressed to ,
. Additional information on this topic can be
PHQWEH\RQGWKHLQÀHFWLRQSRLQWZKLFKLVWKHJUHDWHURI and
IRXQGLQ$&,5
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.6.5 Splices of Deformed Reinforcement
Overview
6SOLFHVRIGHIRUPHGUHLQIRUFHPHQWLQEHDPVPXVWEHLQDFFRUGDQFHZLWK$&, $&, 7KHSULPDU\
reasons for splicing reinforcement are based on (1) restrictions related to transporting the reinforcing bars to the
FRQVWUXFWLRQVLWHDQG OLPLWDWLRQVUHODWHGWRKDQGOLQJDQGSODFLQJWKHUHLQIRUFLQJEDUVLQWKH¿HOG
/DSVSOLFHVPHFKDQLFDOVSOLFHVDQGZHOGHGVSOLFHVDUHFRPPRQW\SHVRIVSOLFHVIRUÀH[XUDOUHLQIRUFHPHQW
Lap Splices
Lap splices are usually the most economical type of splice. There are basically two types of lap splices: contact and
QRQFRQWDFW,QDFRQWDFWODSVSOLFHWKHEDUVDUHJHQHUDOO\LQFRQWDFWRYHUDVSHFL¿HGOHQJWKDQGDUHWLHGWRJHWKHU
>VHH)LJXUH D @,QDQRQFRQWDFWODSVSOLFHWKHEDUVDUHQRWLQFRQWDFWDQGWKHFHQWHUWRFHQWHUVSDFLQJRIWKH
EDUVEHLQJVSOLFHGPXVWQRWH[FHHGWKHOLPLWVLQGLFDWHGLQ)LJXUH E VHH$&, &RQWDFWODSVSOLFHVDUH
usually preferred because the bars are tied together and are less likely to displace during construction. The minimum
clear spacing between a contact lap splice and adjacent splices or bars is the same as that for individual bars (ACI
VHH)LJXUH Lap splices must be provided at locations away from maximum stress (that is, maximum bending moment) and
should be staggered wherever possible. Reinforcing bars provided for negative bending moments (top bars) are
spliced near the midspan of a member, and reinforcing bars provided for positive bending moments (bottom bars)
are spliced over the supports.
, depends on the tension development length of the bars, , the area of
The required tension lap splice length,
reinforcement provided over the length of the splice, and the percentage of reinforcement spliced at any one locaWLRQ%HFDXVHH[SHULPHQWDOGDWDRQODSVSOLFHVZLWKDQGEDUVDUHVSDUVHWKHXVHRIWHQVLRQODSVSOLFHVIRU
WKHVHEDUVDUHSURKLELWHGH[FHSWDVSHUPLWWHGLQ$&, $&, /DSVSOLFHVDUHFODVVL¿HGDV&ODVV$RU&ODVV%7KHOHQJWKRIDODSVSOLFHLVJLYHQDVDPXOWLSOHRI
(ACI
(6.81)
(6.82)
When determining
to be used in determining
WKHLQPLQLPXPOHQJWKVSHFL¿HGLQ$&, E DQG
WKHH[FHVVUHLQIRUFHPHQWPRGL¿FDWLRQIDFWRURI$&,DUHQRWDSSOLFDEOH $&, $OVRIRUUHLQIRUFspaced closer than 6 in. on center, transverse reinforcement must be provided along
ing bars with
$&, the splice length such that
7KHGHIDXOWVSOLFHFODVVL¿FDWLRQIRUDODSVSOLFHLV&ODVV%,IERWKRIWKHIROORZLQJFRQGLWLRQVDUHVDWLV¿HGD&ODVV$
VSOLFHLVSHUPLWWHG VHH$&,7DEOH over the entire splice length where
and
are the area of reinforcement provided at the splice location and the area of reinforcement required by analysis at the splice location,
respectively
• /HVVWKDQRUHTXDOWRSHUFHQWRIWKHUHLQIRUFHPHQWLVVSOLFHGZLWKLQWKHUHTXLUHGODSVSOLFHOHQJWK
•
:KHUHEDUVRIGLIIHUHQWVL]HDUHODSVSOLFHGLQWHQVLRQ
RIWKHVPDOOHUEDU $&, is equal to the greater of (1)
RIWKHODUJHUEDUDQG Mechanical Splices
Mechanical splices are, in general, a complete assembly of a coupler, a coupling sleeve, or an end-bearing sleeve,
including any additional material or components, required to accomplish the splicing of the reinforcing bars. Acof the bar. This
FRUGLQJWR$&,PHFKDQLFDOVSOLFHVPXVWGHYHORSLQWHQVLRQRUFRPSUHVVLRQDWOHDVW
is to ensure yielding occurs in the reinforcing bar adjacent to the mechanical splice prior to the failure of the splice.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,OOXVWUDWHGLQ)LJXUHDUHWKUHHRIWKHPRUHSRSXODUW\SHVRIPHFKDQLFDOVSOLFHV7KHFROGVZDJHGFRXSOLQJVOHHYH
LQ)LJXUH D XVHVDK\GUDXOLFVZDJLQJSUHVVZLWKVSHFLDOGLHVWRGHIRUPWKHVOHHYHDURXQGWKHHQGVRIWKHVSOLFHG
bars to produce positive mechanical interlock with the reinforcing bars. The shear screw coupling sleeve in Figure
E FRQVLVWVRIDFRXSOLQJVOHHYHZLWKVKHDUKHDGVFUHZVZKLFKDUHGHVLJQHGWRVKHDURIIDWDVSHFL¿HGWRUTXH
The reinforcing bars are inserted to meet at the center of the coupling sleeve and then the screws are tightened. The
WLJKWHQLQJSURFHVVHPEHGVWKHSRLQWHGVFUHZVLQWRWKHEDUV7KHQRQXSVHWVWUDLJKWWKUHDGFRXSOHULQ)LJXUH F consists of a coupler with internal straight threads at each end that joins the two reinforcing bars with matching external threads. Additional information on these and other mechanical splices can be found in Reference 8.
Mechanical splices can be advantageous in a number of situations, including the following:
• When long lap splices are needed
• When lap splices cause reinforcement congestion
• :KHUHVSDFLQJRIWKHÀH[XUDOUHLQIRUFHPHQWLVLQVXI¿FLHQWWRSHUPLWODSVSOLFHV
Mechanical splices need not be staggered unless the splices are in tension tie members (that is, members having an
D[LDOWHQVLOHIRUFHVXI¿FLHQWWRFUHDWHWHQVLRQRYHUWKHFURVVVHFWLRQDOHYHORIVWUHVVLQWKHUHLQIRUFHPHQWVXFKWKDW
every bar should be fully effective, and limited concrete cover on all sides) where the minimum stagger between
VSOLFHVLQDGMDFHQWEDUVPXVWEHDWOHDVWLQLQDFFRUGDQFHZLWK$&, $&, Welded Splices
7KHXVHRIZHOGHGVSOLFHVDUHSHUPLWWHGWREHXVHGSURYLGHGWKH\FRQIRUPWRWKHUHTXLUHPHQWVRI$&, $&,
/LNHPHFKDQLFDOVSOLFHVDIXOOZHOGHGVSOLFHZKLFKLVJHQHUDOO\LQWHQGHGIRUDQGODUJHUEDUVPXVWEH
of the bar.
able to develop at least
Welded splices need not be staggered unless the splices are in tension tie members.
6.6.6 Longitudinal Torsional Reinforcement
Where it has been determined the effects of torsion must be considered (see Section 6.3.1 of this publication), the
GHWDLOLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HGIRUWKHORQJLWXGLQDOWRUVLRQDOUHLQIRUFHPHQW$VXPPDU\RI
WKHUHTXLUHPHQWVLQ$&,DQGLVJLYHQLQ)LJXUH,QWKH¿JXUH is the center-to-center spacing
of the required combined transverse reinforcement. The longitudinal torsional reinforcement is combined with that
UHTXLUHGIRUÀH[XUH VHH6HFWLRQRIWKLVSXEOLFDWLRQ ‘‰‹–—†‹ƒŽ”‡‹ˆ‘” ‡‡–†‹•–”‹„—–‡†
ƒ”‘—†–Š‡’‡”‹‡–‡”‘ˆ–Š‡ Ž‘•‡†•–‹””—’•
™‹–Šƒ–Ž‡ƒ•–‘‡„ƒ”’Žƒ ‡†‹‡ƒ Š ‘”‡”
൑ ͳʹԣ
ͲǤͲͶʹ‫ݏ‬
݀௕ ൒ ‰”‡ƒ–‡”‘ˆ ൝
͵Τͺ‹Ǥ
ܾ௧ Figure 6.28 Details for longitudinal torsional reinforcement.
beyond the point is reLongitudinal torsional reinforcement must be provided for a distance equal to at least
TXLUHG $&, 7KHWHUP is the width of that part of the cross-section that contains the closed stirrups resistLQJWRUVLRQ VHH)LJXUH 7KLVGLVWDQFHLVODUJHUWKDQWKDWXVHGIRUVKHDUUHLQIRUFHPHQWDQGÀH[XUHUHLQIRUFHPHQW
because torsional diagonal cracks develop in a helical form around a member. Also, the longitudinal reinforcement
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6-47
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
must be fully developed for tension at both ends of the member; proper anchorage is especially important at the
HQGVZKHUHWKHODUJHVWWRUVLRQDOPRPHQWVXVXDOO\RFFXU $&, 6.6.7 Transverse Reinforcement
Overview
7KHGHWDLOLQJUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGIRUWUDQVYHUVHUHLQIRUFHPHQWUHTXLUHGIRUVKHDURUIRU
shear and torsion. Where shear and torsion must be considered, the most restrictive detailing requirements of ACI
DSSO\ $&,DQG Shear Reinforcement
Shear reinforcement must be properly anchored and developed to be fully effective. Requirements for the developPHQWRIVWLUUXSVDUHJLYHQLQ$&,DQGDUHLOOXVWUDWHGLQ)LJXUH
݄ത
݄/2
Standard hook in accordance
with ACI 25.3.2 ሺtyp. ሻ
݄/2
ܾ௪ ݄ത ൒
0.014݀௕ ݂௬௧
ߣඥ݂௖ᇱ
for #6, #7, and #8 stirrups with ݂௬௧ ൐ 40,000 psi
Figure 6.29 Anchorage details for U-stirrups.
In general, the stirrups must extend as close to the compression and tension surfaces of the member as cover and
other constraints permit and must extend at least a distance IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU $&, Each bend in the continuous portion of a single- or multiple-leg U-stirrup and each bend in a closed stirrup must
HQFORVHDORQJLWXGLQDOEDU $&, 0LQLPXPLQVLGHEHQGGLDPHWHUVDQGVWDQGDUGKRRNJHRPHWU\IRUVWLUUXSV
WLHVDQGKRRSVDUHJLYHQLQ$&,7DEOH
7KHEHDPGHSWKWKDWPXVWEHSURYLGHGWRVDWLVI\WKHPLQLPXPHPEHGPHQWOHQJWKIRUDQGVWLUUXSEDUV
LVJLYHQLQ7DEOH>$&, E @7KHEHDPGHSWKVLQWKHWDEOHDUHEDVHGRQQRUPDOwith
*UDGHVWLUUXSEDUVDQGDLQFRYHUWRWKHVWLUUXSKRRN%HDPVZLWK
weight concrete with
No
RYHUDOOGHSWKVOHVVWKDQWKRVHLQ7DEOHDUHQRWSHUPLWWHGWRXVHDQGVWLUUXSVZLWK
PLQLPXPGHSWKUHTXLUHPHQWVDUHJLYHQIRUEHDPVZLWK VWLUUXSEDUVDQGVPDOOHURIDQ\JUDGHDQG DQGVWLUUXSVZLWK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.16 Minimum Beam Depth to Anchor #6, #7, and #8 Stirrups*
Stirrup Bar
Minimum Beam Depth, h (in.)
#6
23
#7
27
#8
30
*Normalweight concrete with
Grade 60 stirrup bars
1.5-in. cover to the stirrup hooks
%HDPVPXVWKDYHVXI¿FLHQWZLGWK , to allow for bend radii at corners of stirrups. Minimum beam widths correVSRQGLQJWRVWLUUXSEDUVL]HDUHJLYHQLQ7DEOH
Table 6.17 Minimum Beam Width for Stirrup Development
Stirrup Bar
Minimum Beam Width, bw (in.)
#3
10
#4
12
#5
14
#6
18
#7
20
#8
22
Closed stirrups consisting of pairs of U-stirrups are permitted to be used except for torsion or integrity reinforcebut not
PHQW $&, 7KHOHJVRIWKHVWLUUXSVPXVWEHODSVSOLFHGZLWKDVSOLFHOHQJWKHTXDOWRDWOHDVW
OHVVWKDQLQZKHUHWKHWHQVLRQGHYHORSPHQWOHQJWK LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, VHH)LJXUH
,QFDVHVZKHUHWKHUHTXLUHGODSOHQJWKFDQQRW¿WZLWKLQDPHPEHUWKDWKDVDGHSWKRIDWOHDVWLQSDLUVRI
8VWLUUXSVFDQVWLOOEHXVHGSURYLGHGWKHIRUFHLQHDFKOHJLVHTXDOWRRUOHVVWKDQOE)RU*UDGHUHLQIRUFHPHQWRQO\DVWLUUXSVDWLV¿HVWKLVUHTXLUHPHQW
Figure 6.30 A closed stirrup formed by pairs of U-stirrups.
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6-49
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Torsion Reinforcement
5HTXLUHPHQWVIRUVWLUUXSVVXEMHFWHGWRWRUVLRQDOPRPHQWVDUHJLYHQLQ$&,$VQRWHGSUHYLRXVO\FORVHG
stirrups must be used in such cases because cracking can occur on all faces of a beam. The corners of a beam are
susceptible to spalling due to the inclined compressive stresses occurring due to torsion. Where spalling is not
UHVWUDLQHGE\DQDGMDFHQWVODEERWKHQGVRIDVWLUUXSPXVWWHUPLQDWHZLWKGHJUHHVWDQGDUGKRRNVDURXQGORQJLWXGLQDOEDUVLQWKHVHFWLRQ>$&, D @6SDOOLQJLVHVVHQWLDOO\SUHYHQWHGZKHUHDVODELVSUHVHQWRQRQHRUERWK
VLGHVRIDEHDPZHEVRGHJUHHVWDQGDUGKRRNVLQDFFRUGDQFHZLWK$&,DUHSHUPLWWHGLQVXFKFDVHV>$&,
E @'HWDLOVIRUWUDQVYHUVHWRUVLRQUHLQIRUFHPHQWZKHUHVSDOOLQJLVXQUHVWUDLQHGDQGUHVWUDLQHGDUHJLYHQLQ
Figure 6.31.
”‘••–‹‡
ͳ͵ͷǦ†‡‰Š‘‘ሺ–›’Ǥሻ
”‘••–‹‡
ͻͲǦ†‡‰Š‘‘
Ǧ•–‹””—’
Ǧ•–‹””—’
ሺ„ሻ
ሺƒሻ
Figure 6.31 Details for transverse torsion reinforcement.
(a) Spalling is unrestrained. (b) Spalling is restrained.
7ZRSLHFHFORVHGVWLUUXSVDUHSHUPLWWHGWREHXVHG $&, 7KHFORVHGVWLUUXSLQ)LJXUH D FRQVLVWVRI
D8VWLUUXSZLWKGHJUHHKRRNVDWERWKHQGVDQGDFURVVWLHZLWKGHJUHHKRRNVDWERWKHQGV7KHGHJUHH
hooks are required on both ends of both pieces of reinforcement because spalling is unrestrained. Similarly, Figure
E FRQVLVWVRID8VWLUUXSZLWKGHJUHHKRRNVDWERWKHQGVDQGDFURVVWLHZLWKDGHJUHHKRRNRQRQHHQG
DQGDGHJUHHKRRNRQWKHRWKHUHQG7KHGHJUHHKRRNRIWKHFURVVWLHLVORFDWHGRQWKHVLGHDGMDFHQWWRWKHVODE
which provides restraint against spalling. Using two-piece closed stirrups are preferred because closed, one-piece
VWLUUXSVPDNHLWGLI¿FXOWWRSODFHWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHEHDPZKHUHWKHUHLQIRUFHPHQWLVEXLOWLQ
place (as opposed to preassembled cages).
Like longitudinal reinforcement required for torsion, transverse torsional reinforcement must extend a distance of at
EH\RQGWKHSRLQWLWLVUHTXLUHGE\DQDO\VLV $&, least
6.6.8 Structural Integrity Reinforcement
6WUXFWXUDOLQWHJULW\UHLQIRUFHPHQWUHTXLUHPHQWVIRUFDVWLQSODFHEHDPVDUHJLYHQLQ$&,7KHSXUSRVHRIWKLV
reinforcement is to improve the redundancy and ductility in the structure so that in the event of damage to a major
VXSSRUWLQJHOHPHQWRUDQDEQRUPDOORDGLQJHYHQWWKHUHVXOWLQJGDPDJHPD\EHORFDOL]HGDQGWKHVWUXFWXUHZLOOKDYH
a higher probability of maintaining overall stability.
A summary of the structural integrity reinforcement requirements is given in Table 6.18.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.18 Structural Integrity Reinforcement Requirements for Beams
Requirement
ACI Section No.
Perimeter
Beams
1. At least one-quarter of the maximum positive moment reinforcement, but not
OHVVWKDQEDUVPXVWEHFRQWLQXRXV
At least one-sixth of the negative moment reinforcement at a support, but not
OHVVWKDQEDUVPXVWEHFRQWLQXRXVDQG
3. Longitudinal structural integrity reinforcement must be enclosed by closed
VWLUUXSVLQDFFRUGDQFHZLWK$&,RUE\KRRSVDORQJWKHFOHDUVSDQRI
the beam.
Other Than
Perimeter
Beams
1. At least one-quarter of the maximum positive moment reinforcement, but not
OHVVWKDQEDUVPXVWEHFRQWLQXRXVRU
Longitudinal reinforcement must be enclosed by closed stirrups in accordance
ZLWK$&,RUE\KRRSVDORQJWKHFOHDUVSDQRIWKHEHDP
Longitudinal structural integrity reinforcement must pass through the region bounded by the
longitudinal reinforcement in the column.
Longitudinal structural integrity reinforcement at noncontinuous supports must be anchored to
develop at the face of the support.
Splices for longitudinal structural integrity reinforcement must occur at or near a support for
positive moment reinforcement and at or near midspan for negative moment reinforcement.
Splices for longitudinal structural integrity reinforcement must be mechanical or welded splices
LQDFFRUGDQFHZLWK$&,RU&ODVV%WHQVLRQODSVSOLFHVLQDFFRUGDQFHZLWK$&,
6.6.9 Flexural Reinforcement Requirements for SDC B
For beams in buildings assigned to SDC B that are part of an ordinary moment frame resisting seismic load effects,
WKHGHWDLOLQJUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG7ZRFRQWLQXRXVÀH[XUDOEDUVPXVWEHSURYLGHGDWERWK
WKHWRSDQGERWWRPIDFHVRIWKHEHDP7KHFRQWLQXRXVERWWRPEDUVPXVWKDYHDQDUHDHTXDOWRDWOHDVWSHUFHQWWKH
maximum area of the bottom bars along the span and must be anchored to develop in tension at the face of the
support.
These provisions are intended to improve continuity and improve lateral force resistance and structural integrity.
6.6.10 Recommended Flexural Reinforcement Details
5HFRPPHQGHGÀH[XUDOUHLQIRUFHPHQWGHWDLOVIRUEHDPVEDVHGRQWKHUHTXLUHPHQWVFRYHUHGLQWKHSUHYLRXVVHFWLRQVRIWKLVFKDSWHULQFOXGLQJWKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVRI$&,DUHJLYHQLQ)LJXUH7KHEDU
OHQJWKVJLYHQLQWKH¿JXUHDUHEDVHGRQDEHDPVXEMHFWHGWRXQLIRUPO\GLVWULEXWHGJUDYLW\ORDGVZKHUHWKHVKDSHRI
WKHPRPHQWGLDJUDPLVNQRZQWKHVHOHQJWKVFDQEHXVHGIRUEHDPVGHVLJQHGXVLQJWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&, VHH6HFWLRQRIWKLVSXEOLFDWLRQ )RUEHDPVVXEMHFWHGWRWKHHIIHFWVIURPRWKHUW\SHVRIORDGV
required bar lengths must be determined by calculations. Also, for beams that are part of an ordinary moment frame
LQEXLOGLQJVDVVLJQHGWR6'&%WKHUHTXLUHPHQWVRI$&,PXVWDOVREHVDWLV¿HG
6.7 Deflections
6.7.1 Overview
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
κ௡ଵ Τ3
൒ greater of ൝
κ௡ଶ Τ3
൒ greater of ൝
κ௡ଶ Τ3
κ௡ଷ Τ3
൒ κ௡ଵ Τ4
Note 1
ሺtyp.ሻ
൒ ‫ܣ‬ା௦ଵ /4
Note 2
Note 2 ሺsim.ሻ
‫ܣ‬ା௦ଵ ‫ܣ‬ା௦ଶ ൑ 0.125κ௡ଵ ൒ 6ԣ
൑ 0.125κ௡ଶ ൑ 0.125κ௡ଷ κ௡ଶ κ௡ଵ ‫ܣ‬ା௦ଷ κ௡ଷ Beams Other Than Perimeter Beams
κ௡ଵ Τ3
൒ greater of ൝
κ௡ଶ Τ3
൒ greater of ൝
κ௡ଶ Τ3
κ௡ଷ Τ3
൒ κ௡ଵ Τ4
‫ିܣ‬௦ଵ Note 1
ሺtyp.ሻ
൒ ‫ܣ‬ା௦ଵ /4
൒ 6ԣ
‫ିܣ‬௦ଶ Note 3
‫ିܣ‬௦ଷ Note 3 ሺsim.ሻ
Note 2
Note 2 ሺsim.ሻ
‫ܣ‬ା௦ଵ ‫ܣ‬ା௦ଶ Note 4 ሺtyp.ሻ
൑ 0.125κ௡ଵ ൑ 0.125κ௡ଶ κ௡ଵ κ௡ଶ ‫ܣ‬ା௦ଷ ൑ 0.125κ௡ଷ κ௡ଷ Perimeter Beams
Notes
1. Reinforcement to be anchored to develop ݂௬ at the face of the support. Standard hooks are
shown in this figure.
ା
2. At least the larger of ሺ‫ܣ‬ା
௦ଵ /4 ሻ or ሺ‫ܣ‬௦ଶ /4ሻ, but not less than 2 bars, must be continuous or
spliced with Class B tension lap splices or mechanical or welded splices.
ି
3. At least the larger of ሺ‫ିܣ‬
௦ଵ /6ሻ or ሺ‫ܣ‬௦ଶ /6ሻ, but not less than 2 bars, must be continuous or
spliced with Class B tension lap splices or mechanical or welded splices.
4. Closed stirrups in accordance with ACI Section 25.7.1.6 or hoops must be provided along the
clear span.
5. Where the requirements in Note 2 are not satisfied for beams other than perimeter beams,
closed stirrups in accordance with ACI Section 25.7.1.6 or hoops along the clear span must
be provided.
Figure 6.32 Recommended flexural reinforcement details for beams.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
EDVHGRQFDOFXODWHGGHÀHFWLRQVDQGGHÀHFWLRQOLPLWDWLRQV $&, ,PPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQV
FDQEHFDOFXODWHGIRUDQ\UHLQIRUFHGFRQFUHWHEHDPLQDFFRUGDQFHZLWK$&,UHJDUGOHVVRIZKHWKHUWKHPLQLPXP
GHSWKUHTXLUHPHQWVRI$&,DUHVDWLV¿HGRUQRW+RZHYHUWKHSURYLVLRQVRI$&,PXVWEHXVHGZKHUHPHPEHUVDUHVXSSRUWLQJHOHPHQWVOLNHO\WREHGDPDJHGE\UHODWLYHO\ODUJHGHÀHFWLRQV
&DOFXODWHGGHÀHFWLRQVPXVWEHOHVVWKDQRUHTXDOWRWKHOLPLWLQJGHÀHFWLRQVLQ$&,LQRUGHUWRVDWLVI\VHUYLFHability requirements.
3URFHGXUHVIRUGHWHUPLQLQJGHÀHFWLRQVRIEHDPVDUHJLYHQEHORZ WKHVHSURFHGXUHVFDQDOVREHXVHGWRGHWHUPLQH
GHÀHFWLRQVRIRQHZD\VODEV $OWKRXJKQXPHURXVPHWKRGVDUHDYDLODEOHWRGHWHUPLQHGHÀHFWLRQVLWLVLPSRUWDQWWR
QRWHWKDWWKHVHPHWKRGVFDQRQO\HVWLPDWHGHÀHFWLRQVZLWKLQDQDFFXUDF\UDQJHRIWRSHUFHQW7KLVLVSULPDULO\
due to the variability in the constituent materials of concrete and to tolerances in construction. This range of accuUDF\VKRXOGEHFRQVLGHUHGFDUHIXOO\ZKHQFDOFXODWLQJGHÀHFWLRQVIRUGHÀHFWLRQVHQVLWLYHPHPEHUV
6.7.2 Immediate Deflections
Uncracked Sections
For beams loaded such that maximum bending moment at service loads,
is less than or equal to
the
VHFWLRQLVDVVXPHGWREHXQFUDFNHGDQGWKHLPPHGLDWHGHÀHFWLRQRIWKHEHDPFDQEHFDOFXODWHGXVLQJDQ\HODVWLF
method of analysis. In such cases, the gross moment of inertia, , of the section (neglecting the longitudinal reinIRUFHPHQW FDQEHXVHGLQWKHGHÀHFWLRQFDOFXODWLRQV
The cracking moment,
LVGHWHUPLQHGE\$&,(TXDWLRQ (6.83)
where is the distance from the centroidal axis of the gross section, neglecting longitudinal reinforcement, to the
tension face of the member and the modulus of rupture, LVGHWHUPLQHGE\$&,(TXDWLRQ (6.84)
7KHPRGL¿FDWLRQIDFWRU LVGHWHUPLQHGE\7DEOHRU7DEOHLQWKLVSXEOLFDWLRQ
7KHFUDFNLQJPRPHQWLVPXOWLSOLHGE\WRDFFRXQWIRU UHVWUDLQWWKDWFDQUHGXFHWKHHIIHFWLYHFUDFNLQJPRPHQW
DQG UHGXFHGWHQVLOHVWUHQJWKRIFRQFUHWHGXULQJFRQVWUXFWLRQWKDWFDQOHDGWRFUDFNLQJZKLFKFRXOGVXEVHTXHQWO\
KDYHDQLPSDFWRQVHUYLFHGHÀHFWLRQV
Cracked Sections
2QFHDVHFWLRQFUDFNVWKHJURVVPRPHQWRILQHUWLDFDQQRORQJHUEHXVHGWRGHWHUPLQHLPPHGLDWHGHÀHFWLRQV
, must be used. In general,
is determined by transforming the longituInstead, the cracked moment of inertia,
dinal reinforcing steel into an equivalent area of concrete (see Figure 6.33); the transformation is attained by mulwhere
is the modulus of elasticity of
tiplying the area of reinforcement, , by the modular ratio
WKHUHLQIRUFLQJVWHHO $&, DQG LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWH $&, 7KHFRQFUHWH
IURPWKHH[WUHPHFRPSUHVVLRQ¿EHULVFUDFNHGDQGLV
below the neutral axis, which is located a distance equal to
ignored in the analysis.
Taking moments of areas about the neutral axis results in the following equation:
(6.85)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݇݀
ܾ
݄
݀
N. A.
‫ܣ‬௦ ݊‫ܣ‬௦ ܾ
݊ ൌ ‫ܧ‬௦ /‫ܧ‬௖ Figure 6.33 Cracked transformed section of a rectangular beam with tension reinforcement.
ܾ
݄
݀
݇݀
ܾ
݇݀ ൌ
݀ᇱ ݊‫ܣ‬௦ ܾሺ݇݀ሻଷ
൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ 3
ᇱ
N.A.
݀
‫ܣ‬ᇱ௦ ‫ܫ‬௖௥ ൌ
ܾ
݇݀
ܾ
݄
N.A.
‫ܣ‬௦ ඥ2ܽଵ ݀ ൅ 1 െ 1
ܽଵ
‫ܣ‬௦ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ݊‫ܣ‬௦ ට2ܽଵ ݀ ቀ1 ൅ ܽଶ ݀ ቁ ൅ ሺ1 ൅ ܽଶ ሻଶ െ ሺ1 ൅ ܽଶ ሻ
݀
݇݀ ൌ
ܽଵ
ܾሺ݇݀ሻଷ
‫ܫ‬௖௥ ൌ
൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ ൅ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ሺ݇݀ െ ݀ ᇱ ሻଶ 3
Notes:
݊ ൌ ‫ܧ‬௦ /‫ܧ‬௖ Notes:
‫ܫ‬௚ ൌ ܾ݄ଷ /12
ܽଵ ൌ ܾ/݊‫ܣ‬௦ Notes:
ܽଶ ൌ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ /݊‫ܣ‬௦ Figure 6.34 Cracked section properties for rectangular reinforced concrete beams.
(TXDWLRQ FDQEHVROYHGIRU
:
(6.86)
where
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6-54
݄
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܾ௙ ܾ௙ ‫ݕ‬௧ ݄௕ ݀
݇݀
݇݀ ൌ
N.A.
݊‫ܣ‬௦ ‫ܣ‬௦ ‫ܫ‬௖௥ ൌ
݀ᇱ
݄
ܾ௙ ݇݀ ൌ
݇݀
ܾ௙ ݀
‫ܣ‬ᇱ௦ ‫ݕ‬௧ ൫ܾ௙ െ ܾ௪ ൯݄ଷ ܾ௪ ሺ݇݀ሻଷ
݄ ଶ
൅
൅ ൫ܾ௙ െ ܾ௪ ൯݄ ൬݇݀ െ ൰ ൅ ݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ 12
3
2
ܾ௪ ݄௕ ඥܽଷ ሺ2݀ ൅ ݄ܽସ ሻ ൅ ሺ1 ൅ ܽସ ሻଶ െ ሺ1 ൅ ܽସ ሻ
ܽଷ
N.A.
‫ܣ‬௦ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ݊‫ܣ‬௦ ඥܽଷ ሺ2݀ ൅ ݄ܽସ ൅ 2ܽଶ ݀ ᇱ ሻ ൅ ሺ1 ൅ ܽଶ ൅ ܽସ ሻଶ െ ሺ1 ൅ ܽ2 ൅ ܽ4 ሻ
ܽଷ
‫ܫ‬௖௥ ൌ
൫ܾ௙ െ ܾ௪ ൯݄ଷ ܾ௪ ሺ݇݀ሻଷ
݄ ଶ
൅
൅ ൫ܾ௙ െ ܾ௪ ൯݄ ൬݇݀ െ ൰ 12
3
2
ܾ௪ ൅݊‫ܣ‬௦ ሺ݀ െ ݇݀ሻଶ ൅ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ ሺ݇݀ െ ݀ ᇱ ሻଶ Notes:
Notes:
݊ ൌ ‫ܧ‬௦ Τ‫ܧ‬௖
‫ݕ‬௧ ൌ ൣሺܾ௙ െ ܾ௪ ሻሺ݄௕ ൅ 0.5݄ሻ݄ ൅ 0.5ܾ௪ ሺ݄௕ ൅ ݄ሻଶ ൧/ሾ൫ܾ௙ െ ܾ௪ ൯݄ ൅ ܾ௪ ሺ݄௕ ൅ ݄ሻሿ
ଷ
ଶ
ଷ
ଶ
‫ܫ‬Notes:
௚ ൌ ሾሺܾ௙ െ ܾ௪ ሻ݄ Τ12ሿ ൅ ൫ܾ௙ െ ܾ௪ ൯݄ሺ݄௕ ൅ 0.5݄ െ ‫ݕ‬௧ ሻ ൅ ሾܾ௪ ሺ݄௕ ൅ ݄ሻ Τ12ሿ ൅ ܾ௪ ሺ݄௕ ൅ ݄ሻሾ‫ݕ‬௧ െ 0.5ሺ݄௕ ൅ ݄ሻሿ ܽଶ ൌ ሺ݊ െ 1ሻ‫ܣ‬ᇱ௦ /݊‫ܣ‬௦ ܽଷ ൌ ܾ௪ /݊‫ܣ‬௦ ܽସ ൌ ݄ሺܾ௙ െ ܾ௪ ሻ/݊‫ܣ‬௦ Figure 6.35 Cracked section properties for flanged reinforced concrete beams.
Thus,
is equal to the following:
(6.87)
where
is determined by Equation (6.86).
Similar equations can be derived for rectangular sections with tension and compression reinforcement and for
ÀDQJHGVHFWLRQV&UDFNHGVHFWLRQSURSHUWLHVIRUUHFWDQJXODUDQGÀDQJHGUHLQIRUFHGFRQFUHWHEHDPVDUHJLYHQLQ)LJXUHDQG)LJXUHUHVSHFWLYHO\
Effective Moment of Inertia
It is evident from the preceding discussion that two different moments of inertia would be required for calculating
). To eliminate the need for two moments of inertia, the effective moment
LPPHGLDWHGHÀHFWLRQV WKDWLV and
of inertia, FDQEHXVHGLQOLHXRIDPRUHFRPSUHKHQVLYHDQDO\VLV VHH$&,7DEOH (6.88)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In this equation,
lated.
LVWKHPD[LPXPPRPHQWLQWKHEHDPGXHWRVHUYLFHORDGVDWWKHVWDJHGHÀHFWLRQVDUHFDOFX-
)RUFRQWLQXRXVEHDPVLWLVSHUPLWWHGWRFDOFXODWHLPPHGLDWHGHÀHFWLRQVXVLQJ as the average of the values calcuODWHGE\(TXDWLRQ DWWKHFULWLFDOSRVLWLYHDQGQHJDWLYHPRPHQWVHFWLRQV $&, ,WLVDOVRSHUPLWWHGWR
FDOFXODWHGHÀHFWLRQVXVLQJ at midspan for prismatic beams with simple and continuous spans, and at the support
IRUFDQWLOHYHUV $&, Approximate Immediate Deflections
7KHIROORZLQJHTXDWLRQZKLFKLVLQ$&,&RPPHQWDU\6HFWLRQFDQEHXVHGWRGHWHUPLQHLPPHGLDWH
GHÀHFWLRQ , at the tips of cantilevers and at the midspan of simply supported and continuous members:
(6.89)
In this equation,
is the support bending moment at service loads for cantilevers and the midspan bending moPHQWDWVHUYLFHORDGVIRUVLPSO\VXSSRUWHGDQGFRQWLQXRXVEHDPV9DOXHVRIWKHGHÀHFWLRQFRHI¿FLHQW , are given
in Table 6.19 for beams subjected to uniformly distributed service loads, , and different span conditions.
Table 6.19 Deflection Coefficient, K
Span Condition
K
Cantilever
Simple
Continuous
6.7.3 Time-Dependent Deflections
,QRQHZD\ÀH[XUDOPHPEHUVFUHHSDQGVKULQNDJHGXHWRVXVWDLQHGORDGVFDXVHWLPHGHSHQGHQWGHÀHFWLRQVWKDWFDQ
EHWRWLPHVJUHDWHUWKDQWKHLPPHGLDWHGHÀHFWLRQV1RWDFFRXQWLQJIRUWLPHGHSHQGHQWGHÀHFWLRQVFDQUHVXOWLQD
VLJQL¿FDQWXQGHUHVWLPDWLRQRIWRWDOGHÀHFWLRQ
7LPHGHSHQGHQWGHÀHFWLRQVDUHGHWHUPLQHGE\PXOWLSO\LQJWKHLPPHGLDWHGHÀHFWLRQ
ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
the factor
, due to sustained loads by
(6.90)
In this equation, LVWKHWLPHGHSHQGHQWIDFWRUIRUVXVWDLQHGORDGVJLYHQLQ$&,7DEOH VHH7DEOH is the reinforcement ratio for any compression reinforcement in the section. The term
and
DFFRXQWVIRUWKHHIIHFWRIFRPSUHVVLRQUHLQIRUFHPHQWLQUHGXFLQJWLPHGHSHQGHQWGHÀHFWLRQV,WLVSHUPLWWHGWRFDOculate DWPLGVSDQIRUVLPSOHDQGFRQWLQXRXVVSDQVDQGDWWKHVXSSRUWIRUFDQWLOHYHUV $&, Table 6.20 Time-Dependent Factor for Sustained Loads, ȟ
Sustained Load Duration (Months)
ȟ
3
6
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Only the dead load and a portion of the sustained live load (if applicable) need to be included in the determination
ZKHQFDOFXODWLQJWKHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJH
:
of
(6.91)
6.7.4 Maximum Permissible Calculated Deflections
'HÀHFWLRQVFDOFXODWHGXVLQJWKHPHWKRGVLQ6HFWLRQVDQGRIWKLVSXEOLFDWLRQPXVWEHOHVVWKDQRUHTXDOWR
WKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQVJLYHQLQ$&,7DEOH VHH7DEOH Table 6.21 Maximum Permissible Calculated Deflections
Member
Flat roofs
Deflection to be
Considered
Condition
Not supporting or attached to nonstructural
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Floors
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maximum of
Deflection
Limitation
(1)
,PPHGLDWHGHÀHFWLRQGXHWR
Supporting or attached to nonstructural elements
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Supporting or attached to nonstructural elements
QRWOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV
7KHSDUWRIWKHWRWDOGHÀHFWLRQ
occurring after attachment of
nonstructural elements(2)
(3)
(4)
(1) This limit is not intended to safeguard against ponding. Ponding must be checked by deflection calculations, including (1) additional deflections due to ponded water
and (2) consideration of time-dependent effects of sustained load, camber, construction tolerances, and reliability of provisions for drainage.
(2) The time-dependent deflection is to be determined in accordance with ACI 24.2.4; this deflection is permitted to be reduced by the amount of deflection calculated
to occur before attachment of nonstructural elements. This amount must be calculated based on accepted engineering data relating time-dependent characteristics of
members similar to those being considered.
(3) This limit is permitted to be exceeded if measures are taken to prevent damage to supported or attached elements.
(4) This limit must not exceed the tolerance provided for the nonstructural elements.
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6.8 Design Procedure
The design procedure in Figure 6.36 can be used in the design and detailing of rectangular beam sections with one
OD\HURIORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQW,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVWDEOHQXPEHUVDQG¿JXUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1
EŽ
zĞƐ
A
Figure 6.36 Design procedure for beams.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
EŽ
zĞƐ
1
B
Figure 6.36 Design procedure for beams. (cont.)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
B
EŽ
zĞƐ
Torsion effects can be neglected.
EŽ
zĞƐ
C
EŽ
zĞƐ
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zĞƐ
D
Modify section dimensions.
1
Figure 6.36 Design procedure for beams. (cont.)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
C
zĞƐ
EŽ
No shear reinforcement is required at the
section.
EŽ
zĞƐ
EŽ
zĞƐ
zĞƐ
EŽ
Modify section dimensions.
1
Detail the flexural and shear
reinforcement in accordance with
Sect. 6.6
Figure 6.36 Design procedure for beams. (cont.)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
D
Detail the longitudinal and
transverse reinforcement in
accordance with Sect. 6.6
Figure 6.36 Design procedure for beams. (cont.)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.9 Examples
6.9.1
Example 6.1 – Determination of Beam Size: Building #1 (Framing Option C), Beam is Not Part of the
LFRS, SDC A
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DVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56 VHH)LJXUH $OVRDVVXPHDLQVODELQE\LQFROXPQV
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normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the beam depth based on serviceability requirements
ACI 9.3.1.1
Assume the beam is not supporting or attached to partitions or other construction likely to be damaged by large
GHÀHFWLRQV7KXVGHÀHFWLRQVLQDFFRUGDQFHZLWK$&,QHHGQRWEHFDOFXODWHG
Because the length is the same for all spans, minimum beam depth is based on the support condition of one end
continuous:
Table 6.1
For formwork economy, determine minimum depth for the beams in the east-west direction because the span
lengths are longer than those in the north-south direction:
8VHDLQGHHSEHDPZKLFKLVDGHTXDWHIRUWKHEHDPVLQERWKGLUHFWLRQV VHH([DPSOHIRUWKHGHWHUPLQDWLRQRIWKHPLQLPXPVODEWKLFNQHVVZKLFKGHSHQGVRQWKHEHDPVL]HV Step 2 – Determine the beam width based on strength requirements
Sect. 6.2.2
Assuming
, the required
*UDGHUHLQIRUFHPHQW
and
can be determined by the following equation with
Eq. (6.1)
The largest
along the spans is used to determine
.
The factored bending moments in the beam due to the weight of the slab and the superimposed dead and live loads
DUHGHWHUPLQHGLQ6WHSRI([DPSOHIRUWKLVEHDPVXSSRUWHGWZRZD\VODEXVLQJWKH'LUHFW'HVLJQ0HWKRG VHH
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7DEOH 7KHVHIDFWRUHGEHQGLQJPRPHQWVDUHEDVHGRQ$&,(T E LQWKHEHDPRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQDQGLVHTXDOWRIWNLSV
largest
&RQVHUYDWLYHO\HVWLPDWHWKHIDFWRUHGGHDGORDGRIWKHEHDPZHEDVNLSIW
7KHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVEHDPDQGWKHPD[LPXPIDFWRUHGEHQGLQJPRPHQWGXHWRWKHZHLJKWRIWKHEHDPVWHPDOVRRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQ
Figure 4.3
Total
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Approximate
Therefore,
8QOHVVWKHZLGWKLVUHVWULFWHGIRUDUFKLWHFWXUDOUHDVRQVDLQZLGHEHDPLVLPSUDFWLFDOWRXVH$Q\IXQFWLRQDO
EHDPZLGWKZLGHUWKDQLQVDWLV¿HVVWUHQJWKUHTXLUHPHQWV8VHDLQZLGHEHDPZKLFKLVLQZLGHUWKDQ
WKHLQFROXPQLWIUDPHVLQWR VHH6HFWRIWKLVSXEOLFDWLRQRQJXLGHOLQHVIRUVL]LQJEHDPVIRUHFRQRP\ Check the assumption regarding the weight of the beam web:
8VHDLQE\LQEHDP
6.9.2
Example 6.2 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C),
Beam is Not Part of the LFRS, SDC A, Single Layer of Tension Reinforcement
'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHLQE\LQEHDPDORQJFROXPQOLQHLQ%XLOGLQJ
)UDPLQJ2SWLRQ&DWDW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56 VHH)LJXUH $OVRDVVXPH
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normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the factored bending moments along the span
The factored bending moments in the beam due to the weight of the slab, the superimposed dead load, and the live
ORDGRQWKHVODEDUHGHWHUPLQHGLQ6WHSRI([DPSOHIRUWKLVEHDPVXSSRUWHGWZRZD\VODE VHH7DEOH 7KHVHIDFWRUHGEHQGLQJPRPHQWVDUHEDVHGRQ$&,(T E *LYHQDLQWKLFNVODEWKHIDFWRUHGGHDGORDGRIWKHEHDPZHELVHTXDOWRNLSIW VHH([DPSOH 7KHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVEHDP7KHIDFWRUHGEHQGLQJ
PRPHQWVGXHWRWKHGHDGORDGRIWKHEHDPZHEDUHGHWHUPLQHGXVLQJ)LJXUHDQGDUHJLYHQLQ7DEOHZKHUH
for all spans.
Table 6.22 Factored Bending Moments due to Dead Load of Beam Web (ft-kips)
End Span
Exterior Negative
Positive
Interior Span
First Interior Negative
Positive
Interior Negative
7RWDOGHVLJQEHQGLQJPRPHQWVDUHJLYHQLQ7DEOH VHH7DEOH Table 6.23 Design Bending Moments (ft-kips) for the Beam in Example 6.2
End Span
Exterior Negative
Positive
Interior Span
First Interior Negative
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Positive
Interior Negative
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the effective flange width, bf , of the beam
ACI 6.3.2.1
7KHHIIHFWLYHÀDQJHZLGWK , of the beam is equal to the following:
Figure 6.10
Step 3 – Determine the required flexural reinforcement at the critical sections
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single layer of tension reinforcement:
is determined by the following assuming a
Eq. (6.34)
Eq. (6.33)
where
Figure 6.11
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assuming rectangular section behavior using the largest positive
Eq. (6.49)
where
Because
with
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DWSRVLWLYHPRPHQWVHFWLRQV VHH)LJXUH $VXPPDU\RIWKHUHTXLUHGUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWPLQLPXPUHLQIRUFHPHQWLVUH.
quired at all sections. Also, all sections are tension-controlled because
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.24 Required Flexural Reinforcement for the Beam in Example 6.2
Location
Mu (ft-kips)
Rn (psi)
As (in2)*
161
Exterior negative
End span
Positive
First interior negative
Interior span
*Min.
Positive
Negative
[Eq. (6.35)]
Max.
[Eq. (6.38)]
Step 4 – Select the flexural reinforcement
Sect. 6.6.3
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$&,DQGUHVSHFWLYHO\
• Negative reinforcement
Because
and
distribute the required
within
Figure 6.23
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Minimum number of reinforcing bars
Table 6.9
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).
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Table 6.8
The reinforcement provided in the slab is at least equal to the shrinkage and temperature reinforcement preVFULEHGLQ$&, VHH7DEOHLQ([DPSOH WKLVUHLQIRUFHPHQWVDWLV¿HVWKHUHTXLUHPHQWLQ$&,
IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH
• Positive reinforcement
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6.9.3
Example 6.3 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C),
Beam is Not Part of the LFRS, SDC A
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DW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56 VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWH
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with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the factored shear forces along the span
,WLVGHWHUPLQHGLQ6WHSRI([DPSOHWKDW
for the interior beams in the north-south direction.
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of this publication).
The tributary area that must be used to determine IRUWKHEHDPLQWKLVH[DPSOHLVJLYHQLQ)LJXUHZKLFKLV
WKHVDPHIRUDOOEHDPVDORQJDQ\LQWHULRUFROXPQOLQHLQWKHQRUWKVRXWKGLUHFWLRQ VHH6HFWRIWKLVSXEOLFDWLRQ ͵A
ͶA
ͷA
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ͳͳᇱ Ǧͻԣ
Figure 6.37 Tributary area for shear forces on the beam in Example 6.3.
Determine the factored dead and live loads on the beam:
For simplicity, the uniformly distributed dead load of the beam web is transformed into a triangular distributed
load. The shear force at the face of the support due to the uniformly distributed dead load is equal to
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to
Table 3.3
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VXSSRUWDQGWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUWUHVSHFWLYHO\ VHH)LJXUHDQG7DEOH ,WLVDVVXPHGIRUVLP@Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
݀ ൌ ʹͳǤͷԣ
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ܸ௨ ̷݀ ൌ ͵ͺǤͻ‹’•
ܸ௨ ൌ ߶ߣ—݂௖ᇱ ܾ௪ ݀ ൌ ʹͺǤ͸‹’•
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Figure 6.38 Partial shear diagram for the beam in Example 6.3.
SOLFLW\WKDWWKHWULDQJXODUORDGLVHTXDOWR]HURDWWKHIDFHVRIWKHVXSSRUWVLQVWHDGRIDWWKHFHQWHUVRIWKHFROXPQV
this assumption has a negligible impact on the results.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH
from the face of the support:
Step 2 – Determine the required shear reinforcement
Assuming at least minimum shear reinforcement is provided,
:
with the axial force
can be determined from the following equation
Table 6.4
where
for normalweight concrete.
Table 6.2
Because
, provide minimum shear reinforcement:
Table 6.5
For
:
along the length of the beam.
Maximum spacing
Assuming
,
Figure 6.19
is equal to the following:
Across the width of the beam, maximum spacing
Figure 6.19
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Therefore, 3 stirrup legs must be provided to satisfy maximum spacing requirements across the width of the beam.
8VHVWLUUXSVZLWKOHJVVSDFHGDWLQRQFHQWHU
Stirrups are no longer required at the section where
The length from the face of the sup, can be obtained from the following equation (see Figure 6.38):
port where stirrups are no longer required,
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the face of the column.
6.9.4
Example 6.4 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C),
Beam is Not Part of the LFRS, SDC A
'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUWKHEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DW
DW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56 VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWH
DQG*UDGHUHLQIRUFHPHQW
with
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6-69
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
Step 1 – Determine the flexural reinforcement details
,WLVGHWHUPLQHGLQ([DPSOHWKDWEDUVDUHUHTXLUHGDWWKHQHJDWLYHDQGSRVLWLYHFULWLFDOVHFWLRQVLQWKHEHDP
7KLVQXPEHURIUHLQIRUFLQJEDUVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQG
UHVSHFWLYHO\ VHH6WHSLQ([DPSOH ), the negative reinforce%H\RQGWKHSRLQWRILQÀHFWLRQLQWKHPRPHQWGLDJUDP WKDWLVDWWKHVHFWLRQZKHUH
PHQWLVQRORQJHUUHTXLUHGIRUVWUHQJWK7KHPD[LPXPOHQJWKWRWKHSRLQWRILQÀHFWLRQRFFXUVLQWKHHQGVSDQ VHH
Figure 6.39):
A
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͹ᇱ ǦͲԣ
κ௡ Τʹ ൌ ͳͲǤ͹ͷԢ
ʹᇱ ǦͲԣ
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‫ݔ‬௣௜ ൌ ͶǤ͵ͷԢ
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Figure 6.39 Location of the point of inflection for the beam in Example 6.4
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHQHJDWLYHUHLQIRUFHPHQWPXVWKDYHDQHPEHGPHQWOHQJWKEH\RQGWKHSRLQWRILQÀHFWLRQHTXDOWRWKHJUHDWHURI
the following:
ACI 9.7.3.8.4
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from the face of
WKHVXSSRUW+RZHYHUIURP([DPSOHVWLUUXSV OHJV VSDFHGDWLQRQFHQWHUPXVWEHSURYLGHGDWERWK
HQGVRIWKHEHDPZLWKWKH¿UVWVWLUUXSORFDWHGLQIURPWKHIDFHRIWKHFROXPQ7KLVPHDQVWKHODVWVWLUUXSLVORFDWHG
IURPWKHIDFHRIWKHVXSSRUW7KHUHIRUHSURYLGHDIWLQHPEHGPHQWOHQJWKIRUWKH
WRSEDUV7KLVUHLQIRUFHPHQWLVQRWWHUPLQDWHGLQDWHQVLRQ]RQH VHH)LJXUH DQGFDQEHFRQVHUYDWLYHO\XVHGDW
all support locations.
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ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH
supports. The required lap splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the shear reinforcement details
,WLVGHWHUPLQHGLQ([DPSOHWKDWVWLUUXSVZLWKOHJVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDU
strength requirements. This arrangement consists of (1) a U-stirrup extending across the effective width of the beam
ZLWKGHJUHHKRRNVDWHDFKHQGDQG DFURVVWLHZLWKGHJUHHDQGGHgree hooks at the ends tied to the top and bottom longitudinal reinforcement.
7KHPLQLPXPEHDPZLGWKUHTXLUHGIRUGHYHORSPHQWRIWKHVWLUUXSVLVLQLQ
Table 6.17
5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWV
LQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOEHDPVDORQJWKHLQWHULRUFROXPQOLQHVLQ
the north-south direction that are not part of the LFRS.
2ᇱ -0ᇱᇱ 7ᇱ -0ᇱᇱ A
5-#6 A
5-#6
2ᇱᇱ 9-#3 stirrups @ 10ᇱᇱ ሺtyp. ሻ
2ᇱ -0ᇱᇱ 7ᇱ -0ᇱᇱ 2ᇱ -0ᇱᇱ 7ᇱ -0ᇱᇱ 5-#6 5-#6
21ᇱ -6ᇱᇱ 7ᇱᇱ 5-#6 1Ԣ-5ᇱᇱ Other beams and reinforcement not shown for clarity
#3 stirrups @ 10ԣ 5-#6 2ᇱ -4ᇱᇱ ‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬
Figure 6.40 Reinforcement details for the beam in Examples 6.1 through 6.4.
6.9.5
Example 6.5 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is Not
Part of the LFRS, SDC A
'HWHUPLQHWKHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUWKHEHDPLQWKHHQGVSDQDORQJFROXPQOLQHLQ
%XLOGLQJ)UDPLQJ2SWLRQ&DWDW\SLFDOÀRRUOHYHODVVXPLQJWKHEHDPLVQRWSDUWRIWKH/)56 VHH)LJXUH $VVXPHWKHEHDPLVQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV
DQG*UDGH
DQGSHUFHQWRIWKHOLYHORDGLVVXVWDLQHG$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
reinforcement.
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the service loads and bending moments
The Direct Design Method can be used to determine the bending moments in the beam due to the weight of the slab,
WKHVXSHULPSRVHGGHDGORDGDQGWKHOLYHORDG VHH6WHSVDQGRI([DPSOH 7KHEHQGLQJPRPHQWFRHI¿FLHQWVDWWKHFULWLFDOVHFWLRQVLQWKHEHDPLQWKHHQGVSDQDUHJLYHQLQ7DEOH
Exterior negative:
Positive:
First interior negative:
7KHEHQGLQJPRPHQWVGXHWRWKHZHLJKWRIWKHEHDPVWHPFDQEHGHWHUPLQHGXVLQJWKHVLPSOL¿HGPHWKRGLQ$&,
VHH6WHSLQ([DPSOH Exterior negative:
Positive:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
First interior negative:
7KHVXVWDLQHGSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVDUHWKHIROORZLQJZKHUHSHUFHQWRIWKHOLYHORDGLVVXVtained:
At the exterior support:
$WWKH¿UVWLQWHULRUVXSSRUW
Step 2 – Determine the material properties of the concrete and reinforcing steel
Eq.(6.84)
where
for normalweight concrete.
Table 6.2
ACI Eq. (19.2.2.1b)
ACI 20.2.2.2
Modular ratio
Step 3 – Determine the gross and cracked moments of inertia
(IIHFWLYHÀDQJHZLGWKRIWKHEHDP
Step 2, Example 6.2
ܾ௙ ൌ ͻʹǤͷᇱᇱ ‫ݕ‬௧ ൌ ͳͷǤͶᇱᇱ ݄௕ ൌ ͳ͹ᇱᇱ ݄ ൌ ͹ᇱᇱ • Positive moment section
7KHJURVVVHFWLRQRIWKHEHDPDWWKHSRVLWLYHPRPHQWVHFWLRQLVVKRZQLQ)LJXUH
ܾ௪ ൌ ʹͺᇱᇱ Figure 6.41 Gross section of the beam at a positive moment section.
The gross section properties of the beam are the following:
Figure 6.35
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)URP([DPSOHERWWRPEDUVDUHUHTXLUHG
.
$VVXPLQJDUHFWDQJXODUFRPSUHVVLRQDUHDWKHFUDFNHGVHFWLRQSURSHUWLHVDUHWKHIROORZLQJ VHH)LJXUH ݀ ൌ ʹͳǤͷ ᇱᇱ
ܾ௙ ൌ ͻʹǤͷᇱᇱ ݇݀ ൌ ʹǤ͹ᇱᇱ Figure 6.34
݊‫ܣ‬௦ Figure 6.42 Cracked transformed section of the beam at a positive moment section.
Therefore, the assumption of a rectangular section is correct.
Figure 6.34
• Negative moment section
7KHQHJDWLYHPRPHQWVHFWLRQRIWKHEHDPLVWKHE\LQEHDPZHE)URP([DPSOHWRSEDUV
.
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Figure 6.34
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Determine the effective moment of inertia
The effective moment of inertia,
, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.
• Positive moment section
Therefore,
Therefore,
Therefore,
Eq. (6.88)
• Negative moment section
At the exterior support:
Therefore,
Therefore,
Therefore,
$WWKH¿UVWLQWHULRUVXSSRUW
Therefore,
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6-76
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
Therefore,
• Average effective moment of inertia
Dead load:
ACI 24.2.3.6
Sustained load:
Dead plus live load:
Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads
Assume the loads due to the dead load of the slab, the superimposed dead load, and the live load are uniformly
distributed over the span instead of the actual triangular load distribution. The following equation can be used to
GHWHUPLQHLPPHGLDWHGHÀHFWLRQV
Eq. (6.89)
For continuous members:
Table 6.19
where
and
is the service load moment at midspan. Also assume
• Dead load
• Sustained load
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• Dead plus live load
• Live load
)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQV
WKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ
Table 6.21
Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection
$VVXPLQJDPRQWKGXUDWLRQRIORDGLQJZLWK
:
Eq. (6.90), Table 6.20
7KHWLPHGHSHQGHQWGHÀHFWLRQGXHWRFUHHSDQGVKULQNDJHLVHTXDOWRWKHIROORZLQJ
Eq. (6.91)
Table 6.21
7KHLQE\LQEHDPLVDGHTXDWHIRUGHÀHFWLRQ
6.9.6
Example 6.6 – Determination of Flexural Reinforcement: Beam in Building #1 (Framing Option C),
Beam is Part of the LFRS, SDC A, Multiple Layers of Tension Reinforcement
'HWHUPLQHWKHUHTXLUHGÀH[XUDOUHLQIRUFHPHQWIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ
)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP
DUHOLPLWHGWRLQE\LQ VHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW and the
DQG*UDGHUHLQIRUFHFROXPQVDUHLQE\LQ$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
ment.
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the factored bending moments, Mu along the span
,WFDQEHGHWHUPLQHGXVLQJWKHUHTXLUHPHQWVLQ$&,WKDWDLQWKLFNVODELVDGHTXDWHIRUVHUYLFHDELOLW\
EDVHGRQWKHLQE\LQEHDPV$OVRWKH'LUHFW'HVLJQ0HWKRGFDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWV
LQWKHFROXPQDQGPLGGOHVWULSVRIWKHVODE VHH6HFWRIWKLVSXEOLFDWLRQ @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$FFRUGLQJWRWKH'LUHFW'HVLJQ0HWKRGWKHEHDPVLQWKLVH[DPSOHPXVWUHVLVWSHUFHQWRIWKHFROXPQVWULSEHQGing moments due to the weight of the slab, the superimposed dead load, and the live load.
7KHEHQGLQJPRPHQWVGXHWRZHLJKWRIWKHEHDPVWHPFDQEHGHWHUPLQHGXVLQJWKHVLPSOL¿HGPHWKRGLQ$&,
A summary of the service bending moments due to the dead and live loads at the critical sections of the beam in the
HQGVSDQLVJLYHQLQ7DEOH
Table 6.25 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the
Beam in the End Span in Example 6.6
Exterior Negative
Positive
First Interior Negative
7KHEHQGLQJPRPHQWVLQWKHEHDPGXHWRZLQGORDGVDUHJLYHQLQ7DEOH7KH³SOXVPLQXV´VLJQSUHFHGLQJWKH
WDEXODWHGYDOXHVVLJQL¿HVWKHZLQGORDGVFDQDFWLQERWKWKHQRUWKGLUHFWLRQDQGWKHVRXWKGLUHFWLRQ WKDWLVVLGHVZD\
to the right (SSR) and sidesway to the left (SSL) when looking at the east elevation of the building).
Table 6.26 Bending Moments (ft-kips) due to Wind Loads at the Second-Floor Level for the Beam in the End
Span in Example 6.6
Exterior Negative
Positive
First Interior Negative
ņ
7KHGHVLJQEHQGLQJPRPHQWVIURPWKHJRYHUQLQJORDGFRPELQDWLRQVDUHJLYHQLQ7DEOH
Table 6.27 Design Bending Moments (ft-kips) at the Second-Floor Level for the Beam in the End Span in
Example 6.6
Load Combination
Exterior Negative
Positive
First Interior Negative
SSR
SSL
SSR
SSL
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the effective flange width, bf , of the beam
ACI 6.3.2.1
7KHHIIHFWLYHÀDQJHZLGWK , is equal to the following:
Figure 6.10
Step 3 – Determine the required flexural reinforcement at the critical sections
$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQV ÀDQJHLQWHQVLRQ UHTXLUHG is determined by the following where
for beams with one layer of tension reinforcement:
Eq. (6.34)
Eq. (6.33)
where
Figure 6.11
$WSRVLWLYHPRPHQWFULWLFDOVHFWLRQV ÀDQJHLQFRPSUHVVLRQ GHWHUPLQHWKHGHSWKRIWKHHTXLYDOHQWVWUHVVEORFN ,
LQ7DEOH ZKLFKUHVXOWVLQWKHODUJHVW ):
assuming rectangular section behavior using the largest positive
Eq. (6.49)
where
and
tension reinforcement are provided because the beam is relatively narrow).
Because
ZKHUHLWLVDVVXPHGOD\HUVRI
the section behaves as a rectangular section.
Determine if the reinforcing steel in the inner layer located IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU\LHOGVDVVXPLQJ
DLQFOHDUFRYHUWRVWLUUXSVORQJLWXGLQDOEDUVDQGDLQFOHDUVSDFLQJEHWZHHQWKHOD\HUVRIORQJLWXGLnal reinforcement:
Therefore, the reinforcing steel in the inner layer yields and
FDQEHGHWHUPLQHGXVLQJ(TV DQG ZLWK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VXPPDU\RIWKHUHTXLUHGUHLQIRUFHPHQWLVJLYHQLQ7DEOH,WLVHYLGHQWWKDWDOOVHFWLRQVDUHWHQVLRQFRQWUROOHG
because
Table 6.28 Required Flexural Reinforcement for the Beam in Example 6.6
Location
Mu (ft-kips)
Rn (psi)
As (in2)*
89
Exterior negative
Positive
First interior negative
*Min.
[Eq. (6.35)]
Max.
[Eq. (6.38)]
Step 4 – Select the flexural reinforcement
Sect. 6.6.3
6HOHFWWKHÀH[XUDOUHLQIRUFHPHQWEDVHGRQWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQG
respectively.
• Negative reinforcement
Because
, distribute the required negative
within
.
Figure 6.23
Exterior negative:
:LWKLQFRYHUVWLUUXSV*UDGHUHLQIRUFLQJEDUVDQG
and the maximum center-to-center bar spacing is equal to the following:
Eq. (6.65)
8VHEDUV
Table 6.7
The reinforcement provided in the slab is at least equal to the shrinkage and temperature reinforcement in ACI
VHH7DEOHLQ([DPSOH WKLVUHLQIRUFHPHQWVDWLV¿HVWKHUHTXLUHPHQWLQ$&,IRUDGGLWLRQDOORQJLWXGLQDOUHLQIRUFHPHQWLQWKHRXWHUSRUWLRQVRIWKHÀDQJHZKHUH
First interior negative:
8VHEDUV
Table 6.7
• Positive reinforcement
0LQLPXPQXPEHURIUHLQIRUFLQJEDUVUHTXLUHGZLWKLQDLQZLGWK
Table 6.9
0D[LPXPQXPEHURIEDUVLQDLQZLGHVHFWLRQ
Table 6.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
8VHEDUVLQOD\HUVZLWKEDUVSHUOD\HU
7KHEDUVDUHDOVRDGHTXDWHIRUWKHIWNLSSRVLWLYHPRPHQWDWWKHH[WHULRUVXSSRUWIRU665 VHH7DEOH
Comments.,QVWHDGRIXVLQJEDUVLQOD\HUVIRUWKHSRVLWLYHUHLQIRUFHPHQWEDUVLQDVLQJOHOD\HUFDQEH
ZKLFKVDWLV¿HVPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWV7KHSULPDU\
used
purpose of this example is to illustrate the design procedure with multiple layers of tension reinforcement.
6.9.7
Example 6.7 – Determination of Shear Reinforcement: Beam in Building #1 (Framing Option C),
Beam is Part of the LFRS, SDC A
'HWHUPLQHWKHUHTXLUHGVKHDUUHLQIRUFHPHQWIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ
)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP
DUHOLPLWHGWRLQE\LQ VHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW. Also asDQG*UDGHUHLQIRUFHPHQW
sume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFWSee Example 6.6.
Step 1 – Determine the factored shear forces along the span
It can be determined that
for the interior beams in the north-south direction. Therefore, these beams
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along any interior column line in the north-south direction. In addition to the shear forces due to the gravity loads,
the beam must resist the shear forces due to the wind loads.
Determine the factored dead and live loads on the beam:
For simplicity, the uniformly distributed dead load of the beam web is transformed into a triangular distributed
load. The shear force at the face of the support due to the uniformly distributed dead load is equal to
7KHWULDQJXODUGLVWULEXWHGORDGWKDWSURGXFHVDVKHDUIRUFHRINLSVDWWKHIDFHRIWKHVXSSRUWLVHTXDO
to
7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKH¿UVWLQWHULRUVXSSRUWLQDQHQGVSDQIRUWKHORDGFRPELQDWLRQ
VHH)LJXUHDQG7DEOH ,WLVDVVXPHGIRUVLPSOLFLW\WKDWWKHWULDQJXODUORDGLVHTXDOWR
]HURDWWKHIDFHVRIWKHVXSSRUWVLQVWHDGRIDWWKHFHQWHUVRIWKHFROXPQVWKLVDVVXPSWLRQKDVQHJOLJLEOHLPSDFWRQ
the results.
Table 3.3
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
݀ ൌ ʹͳǤͷԣ
ͻǤͻ‹’•Ȁˆ–
ͳͷǦ͓ͶǦ•–‹””—’•̷ͻᇳ ‘Ǥ Ǥሺ‡ƒǤ‡†ሻ
ʹԣ
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ʹᇱ ǦͲԣ
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۳‫ܖ܉ܘ܁܌ܖ‬
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ܸ௨ ̷݀ ൌ ͷͻǤͻ‹’•
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‫ݔ‬௡௦ ൌ ͳͲǤ͵Ԣ
Figure 6.43 Partial shear diagram for the beam in Example 6.7.
7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the required shear reinforcement
Assuming at least minimum shear reinforcement is provided,
with the axial force
can be determined from the following equation
Table 6.4
where
for normalweight concrete.
Table 6.2
For
Maximum spacing
8VWLUUXSVVSDFHGDW
along the length of the beam.
provides
Figure 6.19
Table 6.6
Table 6.5
Across the width of the beam, maximum spacing
Figure 6.19
)RU8VWLUUXSVZLWKLQFRYHU
8VH8VWLUUXSVVSDFHGDWLQRQFHQWHU
Stirrups are no longer required at the section where
The length from the face of the supFDQEHREWDLQHGIURPWKHIROORZLQJHTXDWLRQ VHH)LJXUH port where stirrups are no longer required,
)RUVLPSOHUGHWDLOLQJXVH8VWLUUXSVVSDFHGDWLQRQFHQWHUDWERWKHQGVRIWKHEHDPZLWKWKH¿UVWVWLUUXS
ORFDWHGLQIURPWKHIDFHRIWKHFROXPQ
Comments. This shear reinforcement arrangement can be used for all beams in all spans along the column lines in
the north-south direction for the frames that are part of the LFRS.
,QVWHDGRIXVLQJ7DEOHWRGHWHUPLQHWKHVL]HDQGVSDFLQJRIWKHVKHDUUHLQIRUFHPHQWXVH&DVHLQ7DEOH$VVXPLQJ8VWLUUXSVWKHUHTXLUHGVSDFLQJLVHTXDOWRWKHIROORZLQJ
$LQVWLUUXSVSDFLQJLVXVHGDWHDFKHQGLQOLHXRIVSHFLI\LQJDXQLIRUPVWLUUXSVSDFLQJDFURVVWKHHQWLUHOHQJWKRI
WKHEHDPEHFDXVHWKHODWWHUUHVXOWVLQDVSDFLQJWKDWLVQRWFRQYHQLHQWWRPHDVXUHLQWKH¿HOG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.9.8
Example 6.8 – Determination of Reinforcement Details: Beam in Building #1 (Framing Option C),
Beam is Part of the LFRS, SDC A
'HWHUPLQHWKHUHTXLUHGUHLQIRUFHPHQWGHWDLOVIRUWKHEHDPLQDQH[WHULRUVSDQDORQJFROXPQOLQHLQ%XLOGLQJ
)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDP
DUHOLPLWHGWRLQE\LQ VHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW. Also asDQG*UDGHUHLQIRUFHPHQW
sume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Determine the flexural reinforcement details
It is determined in Example 6.6 that the following reinforcement is required at the critical sections:
• ([WHULRUQHJDWLYHEDUVZLWKLQLQ
• 3RVLWLYHEDUV OD\HUV
• )LUVWLQWHULRUQHJDWLYHEDUVZLWKLQLQ
7KHSURYLGHGQXPEHURIUHLQIRUFLQJEDUVDWDOOFULWLFDOVHFWLRQVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\ VHH6WHSLQ([DPSOH The top reinforcing bars are made continuous over the entire length of the end span because stirrups are required
over the entire length and the beam is subjected to wind loads.
$&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGQHDUPLGVSDQ$FFRUGLQJWR$&,ZKHUHEDUVRIGLIIHUHQWVL]HDUH
of
lap spliced in tension, the required lap splice length is based on the greater of the tension development length
of the smaller bar.
the larger bar and the tension lap splice length
7KHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVLVGHWHUPLQHGDVIROORZV
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUPRUHWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHQHJDWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
6LPLODUO\IRUWKHEDUV
ACI Table 25.5.2.1
7KHUHIRUHWKHWHQVLRQGHYHORSPHQWOHQJWKRIWKHEDUVJRYHUQ
8VHDIWLQODSVSOLFHOHQJWKIRUWKHWRSUHLQIRUFLQJEDUVLQWKHHQGVSDQ
)RUVLPSOHUGHWDLOLQJWKHERWWRPEDUVDUHPDGHFRQWLQXRXVLQVWHDGRIFXWWLQJRIIDSRUWLRQRIWKHPWKHODWHURI
ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH
supports. The required lap splice length is determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the shear reinforcement details
,WLVGHWHUPLQHGLQ([DPSOHWKDW8VWLUUXSVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDUVWUHQJWK
with
requirements. The U-stirrups extends over the effective width of the beam
GHJUHHKRRNVDWHDFKHQGDQGDUHWLHGWRWKHWRSDQGERWWRPORQJLWXGLQDOUHLQIRUFHPHQW
Minimum
IRUVWLUUXSGHYHORSPHQW
which is equal to the provided
Table 6.17
5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWV
LQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOEHDPVDORQJWKHLQWHULRUFROXPQOLQHVLQ
the north-south direction that are part of the LFRS.
2ᇱ -0ᇱᇱ 2ᇱ -0ᇱᇱ 4-#5 3ᇱ -2ᇱᇱ A
4-#7
4-#8 A
30-#4 U-stirrups @ 9ᇱᇱ 2ᇱᇱ 21ᇱ -6ᇱᇱ Other beams and reinforcement not shown for clarity
7ᇱᇱ 4-#8 1Ԣ-5ᇱᇱ 2ᇱ -2ᇱᇱ #4 U-stirrups @ 9ԣ 4-#7 1ᇱ -0ᇱᇱ ‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬
Figure 6.44 Reinforcement details for the beam in Examples 6.6 through 6.8.
6.9.9
Example 6.9 – Determination of Deflections: Beam in Building #1 (Framing Option C), Beam is Part of
the LFRS, SDC A, Includes Compression Reinforcement
'HWHUPLQHWKHLPPHGLDWHDQGWLPHGHSHQGHQWGHÀHFWLRQVIRUWKHEHDPDORQJFROXPQOLQHLQ%XLOGLQJ)UDPLQJ2SWLRQ&DWWKHVHFRQGÀRRUOHYHODVVXPLQJWKHEHDPLVSDUWRIWKH/)56DQGWKHGLPHQVLRQVRIWKHEHDPDUH
OLPLWHGWRLQE\LQ VHH)LJXUH $VVXPHWKHOLYHORDGDWWKLVÀRRUOHYHOLVHTXDOWROEIW, the beam is
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live load is sustained. Also assume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the service loads and bending moments
7KHVHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVDUHJLYHQLQ7DEOHIRUWKHEHDPLQWKHHQGVSDQ VHH6WHSRI
([DPSOH DQGDUHUHSHDWHGLQ7DEOHIRUFRQYHQLHQFH
Table 6.29 Bending Moments (ft-kips) due to Service Dead and Live Loads at the Second-Floor Level for the
Beam in the End Span in Example 6.9
Exterior Negative
Positive
First Interior Negative
7KHVXVWDLQHGSRVLWLYHDQGQHJDWLYHEHQGLQJPRPHQWVDUHWKHIROORZLQJZKHUHSHUFHQWRIWKHOLYHORDGLVVXVtained:
At the exterior support:
$WWKH¿UVWLQWHULRUVXSSRUW
Step 2 – Determine the material properties of the concrete and reinforcing steel
Eq. (6.84)
where
for normalweight concrete.
Table 6.2
ACI Eq. (19.2.2.1b)
ACI 20.2.2.2
Modular ratio
Step 3 – Determine the gross and cracked moments of inertia
(IIHFWLYHÀDQJHZLGWKRIWKHEHDP
Step 2, Example 6.6
‫ݕ‬௧ ൌ ͳ͹Ǥʹᇱᇱ ܾ௙ ൌ ͹͸Ǥͷᇱᇱ ݄௕ ൌ ͳ͹ᇱᇱ ݄ ൌ ͹ᇱᇱ • Positive moment section
7KHJURVVVHFWLRQRIWKHEHDPDWWKHSRVLWLYHPRPHQWVHFWLRQLVVKRZQLQ)LJXUH
ܾ௪ ൌ ͳʹᇱᇱ Figure 6.45 Gross section of the beam at a positive moment section.
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6-88
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The gross section properties of the beam are the following:
Figure 6.35
)URP([DPSOHERWWRPEDUVDUHUHTXLUHG
.
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Figure 6.46 Cracked transformed section of the beam at a positive moment section.
Therefore, the assumption of a rectangular section is correct.
Figure 6.34
• Negative moment section
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bars are required at the exterior support
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Figure 6.34
At the exterior support:
$WWKH¿UVWH[WHULRUVXSSRUW
Step 4 – Determine the effective moment of inertia
The effective moment of inertia,
, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.
• Positive moment section
Therefore,
Eq. (6.88)
Therefore,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
• Negative moment section
At the exterior support:
Therefore,
Therefore,
Therefore,
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Therefore,
Therefore,
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6-91
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
• Average effective moment of inertia
Dead load:
ACI 24.2.3.6
Sustained load:
Dead plus live load:
Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads
Assume the loads due to the dead load of the slab, the superimposed dead load, and the live load are uniformly
distributed over the span instead of the actual triangular load distribution. The following equation can be used to
GHWHUPLQHLPPHGLDWHGHÀHFWLRQV
Eq. (6.89)
For continuous members:
Table 6.19
where
and
is the service load moment at midspan. Also assume
• Dead load
• Sustained load
• Dead plus live load
• Live load
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Table 6.21
Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection
$VVXPLQJDPRQWKGXUDWLRQRIORDGLQJZLWK
:
Eq. (6.90), Table 6.20
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Eq. (6.91)
Table 6.21
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Step 7 – Recalculate the deflections using compression reinforcement
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• Step 7a – Determine the cracked moments of inertia
Positive moment section
Assuming a rectangular compression area, the cracked section properties are the following:
Figure 6.34
Therefore, the assumption of a rectangular section is correct.
Figure 6.34
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Negative moment section
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Figure 6.34
Figure 6.34
• Step 7b – Determine the effective moment of inertia
Positive moment section
Eq. (6.88)
Negative moment sections
At the exterior support:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$WWKH¿UVWLQWHULRUVXSSRUW
Average effective moment of inertia
Dead load:
ACI 24.2.3.6
Sustained load:
Dead plus live load:
• 6WHSF±'HWHUPLQHWKHLPPHGLDWHGHÀHFWLRQVDQGFKHFNWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQ
for live loads
Dead load:
Sustained load:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Dead plus live load:
Live load:
)RUDÀRRUPHPEHUQRWVXSSRUWLQJRUDWWDFKHGWRQRQVWUXFWXUDOHOHPHQWVOLNHO\WREHGDPDJHGE\ODUJHGHÀHFWLRQVWKHPD[LPXPSHUPLVVLEOHLPPHGLDWHOLYHORDGGHÀHFWLRQLVHTXDOWRWKHIROORZLQJ
Table 6.21
• 6WHSG±'HWHUPLQHWKHWLPHGHSHQGHQWGHÀHFWLRQVDQGFKHFNWKHPD[LPXPSHUPLVVLEOHGHÀHFWLRQ
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ACI 24.2.4.1.2
Eq. (6.90), Table 6.20
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Eq. (6.91)
Table 6.21
7KHLQE\LQEHDPLVQRWDGHTXDWHIRUWLPHGHSHQGHQWGHÀHFWLRQV
Comments.7KHGHÀHFWLRQFDOFXODWLRQVDUHGHWHUPLQHGDVVXPLQJDXQLIRUPO\GLVWULEXWHGORDGHTXDOWRWKHPD[Lmum value of the actual triangular load, which is conservative. As such, it is safe to conclude this beam is adequate
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6.9.10 Example 6.10 – Determination of Joist Size: Joist in Building #2, Joist is Not Part of the LFRS, SDC C
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DQG*UDGH
VHH)LJXUH $VVXPHDLQWKLFNVODE$OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
reinforcement.
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the joist depth based on serviceability requirements
ACI 9.3.1.1
Assume the joist (beam) is not supporting or attached to partitions or other construction likely to be damaged by
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The minimum joist depth is based on the end spans (longest spans) with a support condition of one end continuous:
Table 6.1
8VHDLQGHHSSDQIRUPZKLFKUHVXOWVLQDQRYHUDOOGHSWKHTXDOWR
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the joist width
:KHUHLQZLGHSDQIRUPVDUHXVHGDLQZLGHMRLVWULELVW\SLFDOO\VSHFL¿HGUHVXOWLQJLQDIWFHQWHUWR
FHQWHUVSDFLQJRIWKHULEV VHH)LJXUH 7KHLQZLGWKRFFXUVDWWKHERWWRPRIWKHULEDQGWKHZLGWKW\SLFDOO\
LQFUHDVHVDWDWRVORSHWRWKHXQGHUVLGHRIWKHVODE
7KHDGHTXDF\RIWKHLQZLGHMRLVWULELVFKHFNHGIRUPRPHQWVWUHQJWKLQ([DPSOH
8VHDLQE\LQMRLVW
6.9.11 Example 6.11 – Determination of Flexural Reinforcement: Joist in Building #2, Joist Not Part of
the LFRS, SDC C
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DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the factored bending moments along the span
&KHFNLIWKHOLPLWDWLRQVRIWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,DUHVDWLV¿HGIRUWKLVV\VWHP
• Members are prismatic
• Loads are uniformly distributed
•
• 7KHUHDUHVSDQVZKLFKLVJUHDWHUWKDQVSDQV
•
7KHUHIRUHWKHVLPSOL¿HGPHWKRGRIDQDO\VLVLQ$&,FDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVDQGVKHDU
forces.
Table 3.3
7KHIDFWRUHGEHQGLQJPRPHQWVDUHJLYHQLQ7DEOH
Figure 4.3
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.30 Factored Bending Moments for a Typical Joist in Example 6.11 (ft-kips)
End Span
Exterior Negative
Positive
Interior Span
First Interior Negative
Positive
Step 2 – Determine the effective flange width of the joist
Interior Negative
ACI 6.3.2.1
7KHHIIHFWLYHÀDQJHZLGWK , is equal to the following:
Figure 6.10
Step 3 – Determine the required flexural reinforcement at the critical sections
$WQHJDWLYHPRPHQWFULWLFDOVHFWLRQV ÀDQJHLQWHQVLRQ UHTXLUHG
single layer of tension reinforcement:
is determined by the following assuming a
Eq. (6.34)
Eq. (6.33)
where
Figure 6.11
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assuming rectangular section behavior using the largest positive
Eq. (6.49)
where
Because
with
WKHVHFWLRQEHKDYHVDVDUHFWDQJXODUVHFWLRQDQG(TV DQG FDQEHXVHGWRGHWHUPLQH
VHH)LJXUH A summary of the required reinforcement is given in Table 6.31. All sections are tension-controlled because
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6-98
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.31 Required Flexural Reinforcement for the Joist in Example 6.11
Location
Mu (ft-kips)
Exterior negative
Positive
End span
First interior negative
Interior span
Positive
Negative
*Min.
Rn (psi)
As (in2)*
1.89
[Eq. (6.35)]
Max.
[Eq. (6.38)]
Step 4 – Select the flexural reinforcement
6HOHFWWKHÀH[XUDOUHLQIRUFHPHQWLQDVLQJOHOD\HUEDVHGRQWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ
$&,DQGUHVSHFWLYHO\
• Negative reinforcement
Because
, distribute the required
within
Figure 6.23
Exterior negative:
:LWKLQFRYHU*UDGHUHLQIRUFLQJEDUVDQG
spacing is equal to the following:
the maximum center-to-center bar
Eq. (6.65)
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First interior negative:
5HTXLUHGUHLQIRUFHPHQWZLWKLQWKHLQZLGWK
8VH#LQRQFHQWHU
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• Positive reinforcement
8VHEDUV
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6-99
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VVXPLQJDLQPD[LPXPDJJUHJDWHVL]H
Figure 6.22
Check minimum clear space:
Table 6.7
Comments. The reactions at the ends of the joists in the end spans cause torsional moments on the edge beams. No
DGMXVWPHQWRIPRPHQWVLQDFFRUGDQFHZLWK$&,KDVEHHQFRQVLGHUHGLQWKLVH[DPSOH6HH([DPSOHIRU
the design of the joists considering redistribution of torsional moments.
6.9.12 Example 6.12 – Determination of Shear Reinforcement: Joist in Building #2, Joist is Not Part of
the LFRS, SDC C
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DQG*UDGH
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reinforcement.
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Determine the factored shear forces along the span
7KHVLPSOL¿HGPHWKRGRIDQDO\VLVFDQEHXVHGWRGHWHUPLQHWKHVKHDUIRUFHVIRUWKLVV\VWHP VHH6WHSLQ([DPSOH
7KHPD[LPXPVKHDUIRUFHRFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUVXSSRUW
7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH
IURPWKHIDFHRIWKHWUDQVYHUVHEHDP VHH)LJXUH Step 2 – Determine the required shear reinforcement
Assuming at least minimum shear reinforcement is provided,
with the axial force
can be determined from the following equation
Table 6.4
where
for normalweight concrete.
Table 6.2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For
Maximum spacing
along the length of the joist.
Because of the narrow width of the joist, provide single-leg stirrups.
$VLQJOHOHJVWLUUXSVSDFHGDW
provides
Figure 6.19
Figure 6.16
Table 6.6
Table 6.5
8VHVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHU
Stirrups are no longer required at the section where
The length from the face of the sup, can be obtained from the following equation:
port where stirrups are no longer required,
Eq. (6.58)
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from the face of the transverse beam.
, which is the minimum width of the sloping
Comments. The shear calculations were performed using
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little is gained by doing so.
The reactions at the ends of the joists in the end spans cause torsional moments on the edge beams. No adjustment
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design of the joists considering redistribution of torsional moments.
6.9.13 Example 6.13 – Determination of Reinforcement Details: Joist in Building #2, Joist is Not Part of
the LFRS, SDC C
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DQG*UDGH
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reinforcement.
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Step 1 – Determine the flexural reinforcement details
It is determined in Example 6.11 that the following reinforcement is required at the critical sections:
• ([WHULRUQHJDWLYH#LQRQFHQWHU
• 3RVLWLYHEDUV
• )LUVWLQWHULRUQHJDWLYH#LQRQFHQWHU
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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the joist is subjected to uniformly distributed gravity loads.
)RUVLPSOHUGHWDLOLQJWKHERWWRPEDUVDUHPDGHFRQWLQXRXVLQVWHDGRIFXWWLQJRIIDSRUWLRQRIWKHPWKHODWWHURI
ZKLFKLVVKRZQLQ)LJXUH$FFRUGLQJWR1RWHLQWKDW¿JXUHD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKH
supports. The required lap splice length is determined as follows:
ACI Eq. (25.4.2.4a)
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ACI Table 25.4.2.5
For normalweight concrete,
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For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
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Set
ACI 25.4.2.4
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
)OH[XUDOUHLQIRUFHPHQWGHWDLOVDUHJLYHQLQ)LJXUH
Step 2 – Determine the shear reinforcement details
,WLVGHWHUPLQHGLQ([DPSOHWKDWVLQJOHOHJVWLUUXSVVSDFHGDWLQRQFHQWHUDUHDGHTXDWHIRUVKHDU
VWUHQJWKUHTXLUHPHQWVZKLFKLVEHQWLQWRWKHSUR¿OHVKRZQLQ)LJXUH,WLVDWWDFKHGWRRQHRIWKHERWWRPEDUVRI
the joist and has a standard hook at the top.
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7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,IRUEHDPVRWKHUWKDQSHULPHWHUEHDPVDQGFDQ
be used for all joists.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
10Ԣ-0ԣ
13Ԣ-4ԣ
A
#5 @ 8ԣ
2Ԣ-4½ԣ
#4 @ 10ԣ
Standard hook ሺtyp.ሻ
2ԣ
2-#8
5Ԣ-2ԣ
A
19-#4 stirrups @ 12ԣ ሺtyp.ሻ
2Ԣ-6ԣ
2Ԣ-8ԣ
39Ԣ-10ԣ
4½ԣ
Other reinforcement not shown for clarity
#5 @ 8ԣ
2Ԣ-0ԣ
#4 single-leg stirrup
2-#8
7ԣ
Section A-A
Figure 6.47 Flexural reinforcement details for the joist in Example 6.13.
ʹͺǤͷԣ
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Figure 6.48 Shear reinforcement details for the joist in Example 6.13.
6.9.14 Example 6.14 – Determination of Deflections: Joist in Building #2, Typical Floor, Joist is Not Part of
the LFRS, SDC C
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assume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the service loads and bending moments
7KHVLPSOL¿HGPHWKRGRIDQDO\VLVFDQEHXVHGWRGHWHUPLQHWKHEHQGLQJPRPHQWVIRUWKLVV\VWHP VHH6WHSLQ
([DPSOH $VXPPDU\RIWKHVHUYLFHGHDGDQGOLYHORDGEHQGLQJPRPHQWVLVJLYHQLQ7DEOH
Table 6.32 Service Dead and Live Load Bending Moments (ft-kips) for the Joist in an End Span in Example 6.14
Exterior Negative
Positive
First Interior Negative
The sustained positive and negative bending moments are the following:
At the exterior support:
$WWKH¿UVWLQWHULRUVXSSRUW
Step 2 – Determine the material properties of the concrete and reinforcing steel
Eq. (6.84)
where
for normalweight concrete.
Table 6.2
ACI Eq. (19.2.2.1b)
ACI 20.2.2.2
Modular ratio
Step 3 – Determine the gross and cracked moments of inertia
(IIHFWLYHÀDQJHZLGWKRIWKHMRLVW
Step 2, Example 6.11
• Positive moment section
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 6.49 Gross section of the joist at a positive moment section.
The gross section properties of the joist are the following (not including the sloped portions of the web):
Figure 6.35
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݊‫ܣ‬௦ Figure 6.50 Cracked transformed section of the joist at a positive moment section.
Therefore, the assumption of a rectangular section is correct.
Figure 6.34
• Negative moment section
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Figure 6.34
At the exterior support:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Determine the effective moment of inertia
The effective moment of inertia,
, is determined by Eq. (6.88) for dead, sustained, and dead plus live load cases.
• Positive moment section
Therefore,
Eq. (6.88)
Therefore,
Therefore,
Eq. (6.88)
• Negative moment section
At the exterior support:
Therefore,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
Therefore,
AtWKH¿UVWLQWHULRUVXSSRUW
Therefore,
Therefore,
Therefore,
• Average effective moment of inertia
Dead load:
ACI 24.2.3.6
Sustained load:
Dead plus live load:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 5 – Determine the immediate deflections and check the maximum permissible deflection for live loads
7KHIROORZLQJHTXDWLRQFDQEHXVHGWRGHWHUPLQHLPPHGLDWHGHÀHFWLRQV
Eq. (6.89)
For continuous members:
Table 6.19
where
and
Therefore,
• Dead load
• Sustained load
• Dead plus live load
• Live load
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Table 6.21
Step 6 – Determine the time-dependent deflections and check the maximum permissible deflection
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:
Eq. (6.90), Table 6.20
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Eq. (6.91)
Table 6.21
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included in the analysis.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.9.15 Example 6.15 – Determination of Beam Size: Edge Beam in Building #2, Beam is Not Part of the
LFRS, SDC C
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umns. Also assume normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the beam depth based on serviceability requirements
ACI 9.3.1.1
Assume the beam is not supporting or attached to partitions or other construction likely to be damaged by large
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The minimum beam depth is based on the end spans with a support condition of one end continuous:
Table 6.1
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Step 2 – Determine the beam width
Assuming an initial estimate of
determined by the following equation:
with
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is
Eq. (6.1)
The largest
along the spans is used to determine
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Table 3.3
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6.16).
Exterior spans:
Interior spans:
Maximum
RFFXUVDWWKHIDFHRIWKH¿UVWLQWHULRUFROXPQLQDQHQGVSDQ
Figure 4.3
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Approximate
Therefore,
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6.9.16 Example 6.16 – Determination of Flexural Reinforcement: Edge Beam in Building #2, Beam is Not
Part of the LFRS, SDC C
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reinforcement.
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Step 1 – Determine the factored bending moments along the span
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• Members are prismatic
• Loads are uniformly distributed
•
• 7KHUHDUHVSDQVZKLFKLVJUHDWHUWKDQVSDQV
• All spans have the same length
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forces.
Exterior spans:
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Interior spans:
The factored bending moments are given in Table 6.33.
Figure 4.3
Table 6.33 Factored Bending Moments for an Edge Beam in Example 6.16 (ft-kips)
End Span
Exterior Negative
Positive
Interior Span
First Interior Negative
Positive
Step 2 – Determine the effective flange width of the beam
Interior Negative
ACI 6.3.2.1
)RUDQHGJHEHDPWKHHIIHFWLYHÀDQJHZLGWK , is equal to the following:
Figure 6.10
Step 3 – Determine the required flexural reinforcement at the critical sections
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single layer of tension reinforcement:
is determined by the following assuming a
Eq. (6.34)
Eq. (6.33)
where
Figure 6.11
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in Table 6.33 (which results in the largest ):
assuming rectangular section behavior using the largest positive
Eq. (6.33)
where
Because
with
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Table 6.34 Required Flexural Reinforcement for the Edge Beam in Example 6.16
Location
Mu (ft-kips)
Exterior negative
End span
Positive
First interior negative
Positive
Interior span
*Min.
363.3
Negative
Rn (psi)
As (in2)*
3.68
[Eq. (6.35)]
Max.
[Eq. (6.38)]
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6.9.17 Example 6.17 – Determination of Shear Reinforcement: Edge Beam in Building #2, Beam is Not Part
of the LFRS, SDC C
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reinforcement.
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Step 1 – Determine the factored shear forces along the span
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VHH)LJXUH 7KHFRQGLWLRQVLQ$&,DUHVDWLV¿HGVRWKHFULWLFDOVHFWLRQIRUVKHDUFDQEHWDNHQDGLVWDQFH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the required transverse reinforcement for shear
Assuming at least minimum shear reinforcement is provided,
force
is determined from the following with the axial
Table 6.4
where
for normalweight concrete.
Table 6.2
For
Maximum spacing
along the length of the beam.
Figure 6.19
It is determined in Example 6.18 that torsional effects must be considered for this edge beam. Thus, the shear
reinforcement calculated in this example must be combined with the transverse torsional reinforcement, which is deWHUPLQHGLQ([DPSOH7KHVL]HDQGVSDFLQJRIWKHWRWDOUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWDUHJLYHQLQ([DPSOH
6.19.
6.9.18 Example 6.18 – Determination of Torsion Reinforcement: Edge Beam in Building #2, Second-Floor
Level, Beam is Not Part of the LFRS, SDC C
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reinforcement.
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Step 1 – Determine the factored torsional moments along the span
The wide-module joist system framing into the edge beam produces a uniform torsional load along the length of the
beam. The torsional load is caused by the shear forces and bending moments transferred from the joists to the beam
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considered:
The factored shear and bending moments at the end of the joists per foot width of joists are determined using Figure
The total uniform torsional load on the edge beam is equal to the sum of the torsional loads due to
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6-114
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 6.51 Reactions on the edge beam in Example 6.18.
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face of each end of the beam due to is equal the following:
where the longest clear span is used to calculate
ACI 9.4.4.3
Step 2 – Determine if torsional effects must be considered
Torsion can be neglected where the factored torsional moment from analysis is less than the threshold torsion:
Eq. (6.3)
'HWHUPLQHWKHRYHUKDQJLQJÀDQJHZLGWK , permitted to be used in the calculation of
and
:
Eq. (6.4)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 6.3
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Because
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>$&, E @
Therefore, torsional effects must be considered.
Because the edge beam is part of indeterminate system in which redistribution of internal forces can occur following torsional cracking, the maximum factored torsional moment at the critical section need not exceed the cracking
torsional moment:
ACI 22.7.3.2
This redistribution is equivalent to reducing the factored distributed torsional loading on the edge beam to the following:
Step 3 – Adjust the bending moments and shear forces in the joists in the end spans
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The design bending moments in the joists per foot width of joist after adjustment are obtained at the critical sections
by algebraically combining the bending moments prior to adjustment with those due to redistribution (see Figure
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Figure 6.52 Bending moments in a joist in an end span (ft-kips/ft).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 6.35 Required Flexural Reinforcement for the Joist in the End Span in Example 6.18
Rn (psi)
As (in2)*
Reinforcement
Exterior negative
#Ǝ #Ǝ
Positive
First interior negative
#Ǝ #Ǝ
Location
*Min.
Mu (ft-kips)
[Eq. (6.35)]
Max.
Eq. [(6.38)]
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Assuming at least minimum shear reinforcement is provided,
with the axial force
can be determined from the following equation
Table 6.4
where
for normalweight concrete.
Table 6.2
For
Maximum spacing
along the length of the joist.
Because of the narrow width of the joist, provide single-leg stirrups.
$VLQJOHOHJVWLUUXSVSDFHGDW
provides
Figure 6.19
Figure 6.16
Table 6.6
Table 6.5
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determined prior to redistribution.
which is the same shear reinforcement
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Check the adequacy of the cross-sectional dimensions of the beam
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Figure 6.17
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at the critical section.
at the critical section.
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Eq. (6.31)
Therefore, the cross-sectional dimensions are adequate.
Step 5 – Determine the required transverse reinforcement for torsion
Eq. (6.60)
Step 6 – Determine the required longitudinal reinforcement for torsion
Eq. (6.61)
Eq. (6.62)
Use
The transverse and longitudinal for torsion are combined with the transverse reinforcement required for shear and
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6.9.19 Example 6.19 – Design for Combined Flexure, Shear, and Torsion: Edge Beam in Building #2, Beam is
Not Part of the LFRS, SDC C
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Step 1 – Determine the total transverse reinforcement
7KHWRWDOUHTXLUHGDUHDRIWUDQVYHUVHUHLQIRUFHPHQWSHUVWLUUXSOHJLVHTXDOWRWKDWUHTXLUHGIRUVKHDU ([DPSOH plus that required for torsion (Example 6.18):
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Therefore, the maximum spacing of the transverse reinforcement is equal to the following:
Determine the location where the maximum stirrup spacing can be used.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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the face of the column.
The distance from the face of the support where transverse reinforcement is no longer required for shear is deterPLQHGIURPWKHIROORZLQJHTXDWLRQ VHH)LJXUH Eq. (6.57)
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Figure 6.53 Shear and torsion diagrams in an end span for the edge beam in Example 6.19.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Similarly, the distance from the face of the support where transverse reinforcement is no longer required for torsion
LVGHWHUPLQHGIURPWKHIROORZLQJHTXDWLRQ VHH)LJXUH The transverse reinforcement for torsion must extend a distance
past the location it is no longer required:
ACI 9.7.6.3.2
Therefore, torsion requirements govern
over the entire length of the beam.
Because
closed stirrups are required
Step 2 – Determine the total longitudinal reinforcement
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This reinforcement must be distributed around the perimeter of the
From Step 6 in Example 6.18,
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to each face.
8VHEDUVRQHDFKVLGHIDFHRIWKHEHDP
$VVXPLQJWRSDQGERWWRPORQJLWXGLQDOUHLQIRUFHPHQWZLWKFORVHGVWLUUXSVDQGLQFOHDUFRYHU
Figure 6.28
&KHFNWKHUHTXLUHPHQWRI$&,
Figure 6.28
where the largest spacing of the transverse reinforcement,
is used to check this requirement.
The remaining longitudinal reinforcement for torsion is distributed equally between the top and bottom of the section:
• Exterior negative:
8VHEDUV
• Positive:
8VHEDUV
• First interior negative:
8VHEDUV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The provided number of bars at all locations also satisfy maximum and minimum spacing requirements (see Table
6.8 and Table 6.9, respectively).
6.9.20 Example 6.20 – Determination of Reinforcement Details: Edge Beam in Building #2, Beam is Not Part
of the LFRS, SDC C
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Step 1 – Determine the flexural reinforcement details
It is determined in Example 6.19 that the following reinforcement is required at the critical sections in the end span:
• ([WHULRUQHJDWLYHEDUV
• 3RVLWLYHEDUV
• )LUVWLQWHULRUQHJDWLYHEDUV
$OVREDUVDUHUHTXLUHGRQHDFKVLGHIDFH
7KHSURYLGHGQXPEHURIUHLQIRUFLQJEDUVDWDOOFULWLFDOVHFWLRQVVDWLV¿HVWKHPLQLPXPDQGPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,DQGUHVSHFWLYHO\ VHH6WHSLQ([DPSOH It is determined in Example 6.19 that the transverse torsion reinforcement must be provided over the entire span of
the beam. Consequently, the longitudinal torsional reinforcement must also be provided over the entire span (ACI
7KHOHQJWKVRIWKHUHLQIRUFLQJEDUVLQ)LJXUHDUHQRWDSSOLFDEOHLQWKLVH[DPSOHEHFDXVHRIWRUVLRQ
$OVRWKHWRSEDUVUHTXLUHGDWWKH¿UVWLQWHULRUVXSSRUWDUHPDGHFRQWLQXRXVRYHUWKHHQWLUHVSDQ
$&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHGRYHUWKHVXSSRUWVIRUWKHERWWRPEDUV7KHUHTXLUHGODSVSOLFHOHQJWKLV
determined as follows:
ACI Eq. (25.4.2.4a)
7KHPRGL¿FDWLRQIDFWRUVDUHDVIROORZV
ACI Table 25.4.2.5
For normalweight concrete,
)RU*UDGHUHLQIRUFHPHQW
For uncoated reinforcing bars,
)RUUHLQIRUFLQJEDUV
)RUOHVVWKDQLQRIIUHVKFRQFUHWHFDVWEHORZWKHSRVLWLYHUHLQIRUFHPHQW
Set
ACI 25.4.2.4
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
ACI Table 25.5.2.1
3URYLGHDIWLQODSVSOLFHOHQJWK
Step 2 – Determine the transverse reinforcement details
,WLVGHWHUPLQHGLQ([DPSOHWKDWFORVHGVWLUUXSVPXVWEHSURYLGHGRYHUWKHHQWLUHVSDQRIWKHEHDP
$VVXPLQJRQO\RXWHUVWLUUXSOHJVDUHSURYLGHGFKHFNWKHVSDFLQJDFURVVWKHZLGWKRIWKHEHDP
Figure 6.19
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Based on the calculations in Example 6.19, provide the following closed stirrup arrangements over the length of the
beam:
• FORVHGVWLUUXSVZLWKFURVVWLHVSDFHGDWLQRQFHQWHUZLWKWKH¿UVWFORVHGVWLUUXSORFDWHGLQ
from the face of the column
• FORVHGVWLUUXSVZLWKFURVVWLHVSDFHGDWLQRQFHQWHUZLWKLQWKHFHQWHUSRUWLRQRIWKHEHDP
5HLQIRUFHPHQWGHWDLOVIRUWKHEHDPLQWKHHQGVSDQDUHJLYHQLQ)LJXUH7KHGHWDLOVVDWLVI\WKHVWUXFWXUDOLQWHJULW\UHTXLUHPHQWVLQ$&,IRUSHULPHWHUEHDPVDQGFDQEHXVHGIRUDOOHGJHEHDPV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
2ᇱ -8ᇱᇱ 2ᇱ -4ᇱᇱ 6-#9 5-#9
16-#4 stirrups
@ 5ԣ
A
4ᇱ -0ᇱᇱ 2-#5
15-#4 stirrups @ 11ԣ
ᇱ
16-#4 stirrups
@ 5ԣ
2ᇱᇱ ᇱᇱ
27 -6 4.5ᇱᇱ Other reinforcement not shown for clarity
6-#9 2-#5 2Ԣ-0ᇱᇱ 2ᇱᇱ A
#4 stirrups
5-#9 ᇱ
ᇱᇱ
2 -6 ‫ۯ ܖܗܑܜ܋܍܁‬--‫ۯ‬
Figure 6.54 Reinforcement details for the edge beam in Examples 6.15 through 6.20.
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Chapter 7
COLUMNS
7.1 Overview
7KHSULPDU\IXQFWLRQRIDFROXPQLVWRVXSSRUWD[LDOIRUFHVIURPÀRRUDQGURRIJUDYLW\ORDGVDQGWRWUDQVIHUWKRVH
forces to a structural element below (such as a transfer beam or a foundation system). Columns not part of the
lateral force-resisting system (LFRS) of the building resist primarily axial compression forces due to gravity loads;
bending moments and shear forces from gravity loads must also be resisted, but these reactions are usually relatively
small, especially for interior columns. Columns in moment frames designated to part of the LFRS must be designed
IRUWKHFRPELQHGHIIHFWVRIÀH[XUHDQGD[LDOIRUFHVDQGWKHDFFRPSDQ\LQJVKHDUIRUFHV6OHQGHUQHVVHIIHFWVPD\
need to be considered in the design of a column regardless of whether the column is part of the LFRS or not.
The design and detailing of cast-in-place concrete columns with nonprestressed reinforcement (longitudinal and
WUDQVYHUVH DUHFRYHUHGLQWKLVFKDSWHU3URYLVLRQVIRUFROXPQVDUHJLYHQLQ$&,&KDSWHUZKLFKDUHDSSOLFDEOHWR
members in buildings assigned to Seismic Design Category (SDC) A and B.
7.2 Dimensional Limits
1RPLQLPXPVL]HVDUHH[SOLFLWO\UHTXLUHGLQ$&,IRUFROXPQVLQEXLOGLQJVDVVLJQHGWR6'&$WKURXJK&+RZever, there are some practical constructability issues that need to be considered, which have a direct impact on the
FURVVVHFWLRQDOGLPHQVLRQV,WLVLPSRUWDQWWRHQVXUHWKDWWKHORQJLWXGLQDOEDUVFDQ¿WZLWKLQWKHFURVVVHFWLRQFRQVLGHULQJWKHPLQLPXPFRYHUUHTXLUHPHQWVRI$&,DQGWKHPLQLPXPVSDFLQJUHTXLUHPHQWVRI$&,7KH
FURVVVHFWLRQDOGLPHQVLRQVDQGWKHORQJLWXGLQDOEDUVL]HVPXVWEHVHOHFWHGWRPLQLPL]HUHLQIRUFHPHQWFRQJHVWLRQ
especially at beam-column or slab-column joints. It is also important to verify that the requirements for transverse
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IRUUHFWDQJXODUFROXPQVDQGDWOHDVWDLQGLDPHWHUIRUFLUFXODUFROXPQV
For columns that have cross-sectional dimensions larger than required to resist the factored loads (like those in the
XSSHUÀRRUVRIDEXLOGLQJ LWLVSHUPLWWHGWRFDOFXODWHWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUDWLREDVHGRQWKH
UHTXLUHGFURVVVHFWLRQDODUHDUDWKHUWKDQWKHSURYLGHGFURVVVHFWLRQDODUHD $&, 7KHSURYLGHGDUHDRIORQJLWXGLQDOUHLQIRUFHPHQWLQVXFKFDVHVPXVWQRWEHOHVVWKDQSHUFHQWRIWKHDFWXDOFURVVVHFWLRQDODUHDRIWKHFROumn. This requirement is not applicable to columns that are part of a special moment frame or that must be designed
LQDFFRUGDQFHZLWK$&, FROXPQVQRWSDUWRIWKHVHLVPLFIRUFHUHVLVWLQJV\VWHP Where columns are built integrally with concrete walls, the cross-sectional area to be used in the design of the colwhere and
are the cross-sectional dimensions (in inches)
umn must not exceed an area equal to
RIWKHFROXPQFRUHPHDVXUHGWRWKHRXWVLGHHGJHVRIWKHWUDQVYHUVHUHLQIRUFHPHQW $&,VHH)LJXUH ,QFDVHVZKHUH$&,RUKDYHEHHQDSSOLHGWKHDFWXDOFURVVVHFWLRQDOGLPHQVLRQVRIWKHFROXPQ
must be used in the structural analysis of the building and in the design of any members that interact with the colXPQ $&, 7.3 Required Strength
7.3.1 Analysis Methods
Overview
7KHDQDO\VLVPHWKRGVLQ$&,&KDSWHULQFRQMXQFWLRQZLWKWKHIDFWRUHGORDGFRPELQDWLRQVLQ$&,&KDSWHUDUHWR
EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWK VHH$&,DQGUHVSHFWLYHO\ Axial forces, bending moments, shear forces, and torsional moments on a reinforced concrete column, including the
HIIHFWVRIVOHQGHUQHVVZKHUHDSSOLFDEOH $&, FDQEHGHWHUPLQHGE\WKHIROORZLQJDQDO\VLVPHWKRGVLQ$&,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
൑ ܾଵ ൅ ͵ԣ
ܾଶ ൑ ܾଶ ൅ ͵ԣ
ܾଵ Figure 7.1 Reinforced concrete column built integrally with a concrete wall.
Linear Elastic First-Order Analysis
,QDOLQHDUHODVWLF¿UVWRUGHUDQDO\VLVUHDFWLRQVLQWKHVWUXFWXUDOPHPEHUVDUHGHWHUPLQHGEDVHGRQWKHXQGHIRUPHG
JHRPHWU\RIWKHVWUXFWXUHVOHQGHUQHVV VHFRQGRUGHU HIIHFWVZKLFKFDQKDYHDVLJQL¿FDQWLQÀXHQFHRQWKHGHsign strength of columns, are not explicitly accounted for in the analysis. Where applicable, slenderness effects in
FROXPQVPD\EHGHWHUPLQHGE\WKHPDJQL¿HGPRPHQWPHWKRGLQ$&,XVLQJWKHUHVXOWVIURPWKH¿UVWRUGHU
DQDO\VLV VHH6HFWLRQRIWKLVSXEOLFDWLRQ Linear Elastic Second-Order Analysis
In a linear elastic second-order analysis, it is assumed the structure remains elastic and second-order effects are
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the structure). The effects of cracking along the length of the members and creep must be considered in the analysis
VHH$&, :KHUHDSSOLFDEOHVOHQGHUQHVVHIIHFWVRQFROXPQVPD\EHGHWHUPLQHGE\WKHPRPHQWPDJQL¿FDWLRQPHWKRGIRUQRQVZD\IUDPHVLQ$&,XVLQJWKHUHVXOWVIURPWKHVHFRQGRUGHUDQDO\VLV $&, The method for nonsway frames is applicable because a second-order analysis inherently accounts for the relative
displacements of the ends of the members.
Inelastic Analysis
In an inelastic analysis, nonlinear stress-strain response of the materials in the structure and compatibility of deforPDWLRQVPXVWEHFRQVLGHUHG $&, (TXLOLEULXPPXVWEHVDWLV¿HGLQWKHXQGHIRUPHGFRQ¿JXUDWLRQZKHUH
DQLQHODVWLF¿UVWRUGHUDQDO\VLVLVXVHGRULQWKHGHIRUPHGFRQ¿JXUDWLRQZKHUHDQLQHODVWLFVHFRQGRUGHUDQDO\VLV
is used. It must be demonstrated that the results from an inelastic analysis are in substantial agreement with results
IURPWHVWV $&, /LNHLQWKHFDVHRIDOLQHDUHODVWLFVHFRQGRUGHUDQDO\VLVVOHQGHUQHVVHIIHFWVRQFROXPQV
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results from the inelastic analysis (ACI 6.8.1.3).
Finite Element Analysis
$¿QLWHHOHPHQWDQDO\VLVFRQIRUPLQJWRWKHSURYLVLRQVRI$&,FDQEHXVHGWRDQDO\]HHVVHQWLDOO\DQ\UHLQIRUFHG
concrete structure provided an appropriate model is constructed (ACI 6.9).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHPRPHQWPDJQL¿FDWLRQSURFHGXUHWHQGVWRJLYHPRUHFRQVHUYDWLYHUHVXOWVWKDQWKRVHREWDLQHGIURPDPRUHUHat the ends of the
¿QHGDQDO\VLV5HJDUGOHVVRIWKHDQDO\VLVPHWKRGWKHPD[LPXPIDFWRUHGEHQGLQJPRPHQWV
of any structural
PHPEHUVRIDIUDPHPXVWEHOLPLWHGWRHQVXUHWKHVWUXFWXUHLVVWDEOH$FFRUGLQJWR$&,
PHPEHUREWDLQHGIURPDQDQDO\VLVLQFOXGLQJVHFRQGRUGHUHIIHFWVGXHWRVOHQGHUQHVVPXVWQRWH[FHHGSHUFHQWRI
WKHPRPHQWREWDLQHGIURPD¿UVWRUGHUDQDO\VLV
Section Properties
:KHQSHUIRUPLQJD¿UVWRUGHURUDQHODVWLFVHFRQGRUGHUODWHUDOORDGDQDO\VLVWKHVHFWLRQSURSHUWLHVLQ$&,
may be used in lieu of a more sophisticated analysis to account for member cracking and other effects. The reduced
moments of inertia in ACI Table 6.6.3.1.1(a) and the alternative moments of inertia in ACI Table 6.6.3.1.1(b) for
YDULRXVVWUXFWXUDOPHPEHUVDUHJLYHQLQ7DEOHDQG7DEOHUHVSHFWLYHO\7KHUHGXFHGPRPHQWVRILQHUWLDLQ
7DEOHDUHEDVHGRQDQDQDO\VLVZKHUHVWUHQJWKOHYHO IDFWRUHG ORDGVDUHXVHG)RUDQDQDO\VLVEDVHGRQVHUYLFH
OHYHOORDGVLWLVSHUPLWWHGWRXVHUHGXFHGPRPHQWVRILQHUWLDHTXDOWRWLPHVWKHYDOXHVLQ7DEOHSURYLGHGWKH
moments of inertia do not exceed the gross moments of inertia $&, Table 7.1 Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using StrengthLevel Loads
Member
Moment of Inertia
Columns
:DOOVņXQFUDFNHG
:DOOVņFUDFNHG
Beams
)ODWSODWHVDQGÀDWVODEV
Table 7.2 Alternative Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis
Using Strength-Level or Service-Level Loads
Member
Moment of Inertia*
Columns and walls
%HDPVÀDWSODWHVDQGÀDWVODEV
*For a service-level load analysis, use service-level axial force and moment effects in the equation for columns and walls.
7KHPRUHUH¿QHGHTXDWLRQVIRUUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHLQFOXGHWKH IDFWRUHGD[LDOIRUFH
RQDFROXPQIRUWKHORDGFRPELQDWLRQXQGHUFRQVLGHUDWLRQ QRPLQDOD[LDOVWUHQJWKDW
and factored moment,
These
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equations are applicable for strength-level and service-level loading, even though they are presented in terms of
factored load effects. For service-level analysis, the factored load effects are replaced by the corresponding servicelevel load effects.
Regardless of the equations used to determine moment of inertias, the gross area of the section,
the analysis. Also, the cross-sectional area for calculation of shear deformations is
is to be used in
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,QOLHXRIWKHUHTXLUHPHQWVRI$&,LWLVSHUPLWWHGWRDVVXPHLQD¿UVWRUGHUIDFWRUHGODWHUDOORDGDQDO\VLVWKDW
DOOWKHPHPEHUVLQDVWUXFWXUHKDYHDUHGXFHGPRPHQWRILQHUWLDHTXDOWRSHUFHQWRIWKHJURVVPRPHQWRILQHUWLD
$&, $PRUHGHWDLOHGDQDO\VLVFRQVLGHULQJWKHHIIHFWLYHVWLIIQHVVRIDOOPHPEHUVLVDOVRSHUPLWWHG
7.3.2 Factored Axial Force and Moment
Axial forces and bending moments at the ends of a column are obtained from an analysis of the structure. When
designing a reinforced concrete column, it is not always obvious which load combination for combined factored
and bending moment
governs. Therefore, each applicable factored load combination must be
axial force
FRQVLGHUHGLQGHVLJQ $&, 7.3.3 Slenderness Effects
Overview
&ROXPQVFDQJHQHUDOO\EHFODVVL¿HGDV³VKRUWFROXPQV´DQG³VOHQGHUFROXPQV´7KHWHUP³VKRUWFROXPQ´LVRIWHQ
used to indicate a column with a strength equal to that computed for its cross-section. In such cases, strength can be
represented by an interaction diagram, which is constructed based on the geometric and material properties of the
VHFWLRQ VHH)LJXUHDQG6HFWLRQRIWKLVSXEOLFDWLRQ ,IDFROXPQGRHVQRWGHÀHFWODWHUDOO\DQ\FRPELQDtion of axial force and bending moment falling outside of the interaction curve is assumed to cause failure. This is
commonly referred to as a material failure. For purposes of design, a column has adequate strength when all the
IDFWRUHGFRPELQDWLRQVRID[LDOIRUFHDQGEHQGLQJPRPHQWREWDLQHGIURPDQHODVWLF¿UVWRUGHUDQDO\VLVRIWKHIUDPH
fall within or on the design strength interaction curve.
ܲ݁
ܲȟ
Axial Force, P
Short column
Slender column,
material failure
Slender column,
instability failure
Bending Moment, M
Figure 7.2 Impact of slenderness effects on design strength of columns.
$³VOHQGHUFROXPQ´LVGH¿QHGDVDFROXPQZKRVHVWUHQJWKLVUHGXFHGE\VHFRQGRUGHUGHIRUPDWLRQVGXHWRKRUL]RQWDOGLVSODFHPHQWV6HFRQGDU\HIIHFWVFDXVHGE\KRUL]RQWDOGHÀHFWLRQDUHFRPPRQO\UHIHUUHGWRDV3GHOWDHIIHFWV&RQVLGHUWKHLGHDOL]HGFROXPQLQ)LJXUHZKLFKLVSDUWRIDIUDPH7KHEHQGLQJPRPHQWVDWWKHHQGVRI
where
is the axial force on the column and is the corresponding eccentricity.
the column are equal to
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
an additional (or secondary) moment, which is equal to
Thus, the total moment in the column is equal to the
SOXVWKHPRPHQWFDXVHGE\WKHKRUL]RQWDOGHÀHFWLRQRIWKHPHPEHU
moment caused by the applied loading
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PDWHULDOIDLOXUH´LQWKH¿JXUH ܲ
ܲ
‫ ܯ‬ൌ ܲ݁
‫ ܯ‬ൌ ܲ݁
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Figure 7.3 P-delta effects in a column.
,QWKHFDVHRIYHU\VOHQGHUFROXPQVLWLVSRVVLEOHIRUWKHGHÀHFWLRQ WRLQFUHDVHLQGH¿QLWHO\ZLWKDQLQFUHDVHLQ
the axial force 7KLVW\SHRIIDLOXUHLVNQRZQDV³LQVWDELOLW\IDLOXUH´ZKLFKJHQHUDOO\RFFXUVDWDQD[LDOIRUFHOHVV
WKDQWKDWFRUUHVSRQGLQJWRPDWHULDOIDLOXUHRIWKHVHFWLRQ VHH)LJXUH Slenderness effects can be neglected in columns with slenderness ratios less than or equal to the limiting values in
$&,,QIRUPDWLRQRQKRZWRGHWHUPLQHVOHQGHUQHVVUDWLRVDQGKRZWRGHVLJQIRUWKHHIIHFWVRIVOHQGHUQHVVDUH
covered below.
Columns in Nonsway and Sway Frames
6ZD\LQEXLOGLQJVGXHWRODWHUDORURWKHUIRUFHVFDQKDYHDVXEVWDQWLDOLQÀXHQFHRQVHFRQGRUGHUHIIHFWVLQFROXPQV
It is common for secondary effects to increase with increasing sway. Distinguishing between columns in nonsway
frames and those in sway frames can usually be done by comparing the total lateral stiffness of the columns in a
VWRU\WRWKDWRIWKHEUDFLQJHOHPHQWV0RPHQWIUDPHEXLOGLQJVDUHW\SLFDOO\ODWHUDOO\ÀH[LEOHDQGWKHFROXPQVDUH
more susceptible to secondary effects than those in a frame with structural walls.
Methods to determine whether a column is in a nonsway frame (braced against sidesway) or in a sway frame (not
EUDFHGDJDLQVWVLGHVZD\ DUHJLYHQLQ$&,DQG7KHFRQGLWLRQVIRUFROXPQVLQDQRQVZD\IUDPHDUH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.3 Conditions for Columns in a Nonsway Frame (Braced Against Sidesway)
ACI Section No.
Condition
Bracing elements (for example, moment frames or walls) resisting lateral movement of a
story have a total stiffness
times the gross lateral stiffness of the columns in that story
in the direction of analysis.
Increase in column end moments due to second-order effects
end moments.
SHUFHQWRIWKH¿UVWRUGHU
:KHUHIDFWRUHGORDGVDUHXVHGWRFDOFXODWHWKHODWHUDOGHÀHFWLRQRIDIUDPHVWDELOLW\LQGH[
IRUDVWRU\VDWLV¿HVWKHIROORZLQJHTXDWLRQ
where
total factored vertical load in the story
¿UVWRUGHUUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQGERWWRPRIWKHVWRU\
IDFWRUHGKRUL]RQWDOVWRU\VKHDULQDVWRU\
length of the columns measured center-to-center of the joints in the frame
:KHUHVHUYLFHORDGVDUHXVHGWRFDOFXODWHWKHODWHUDOGHÀHFWLRQRIDIUDPHVWDELOLW\LQGH[
IRUDVWRU\VDWLV¿HVWKHIROORZLQJHTXDWLRQ
where
total service dead and live axial loads in the story
¿UVWRUGHUVHUYLFHOHYHOUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQG
bottom of the story
service-level story shear
in the equation
is the moment in a story due to the axial loads
being
The numerator
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displaced an amount
must correspond to the lateral loading case for which this sum is the greatest. The denominator
in the story,
in the equation for
is the overall moment in a story due to
ȭܲ௨ ȟ௢ κ௖ ܸ௨௦ Figure 7.4 Definition of stability index,
:KHUHWKHFRQGLWLRQVLQ7DEOHDUHQRWPHWWKHFROXPQVDUHFRQVLGHUHGWREHLQDVZD\IUDPH WKDWLVLQDIUDPH
not braced against sidesway). In general, a frame may contain both nonsway and sway columns and stories.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Consideration of Slenderness Effects
Whether slenderness effects need to be considered or not for nonsway and sway frames depends on the slenderness
7KHWHUPVLQWKHVOHQGHUQHVVUDWLRDUHGH¿QHGLQ7DEOH VHH$&, ratio of the column,
Table 7.4 Slenderness Ratio, kℓu / r
Nonsway Frames
Sway Frames
Unsupported length,
LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRU
other members capable of providing lateral support in the direction of analysis.
Radius of gyration,
is permitted to be calculated by the following:
for rectangular columns where is the dimension of the column in
the direction of analysis
for circular columns where is the diameter of the column
is determined based on the stiffness ratio,
at the ends of the column. This ratio is
The effective length factor,
GHWHUPLQHGE\GLYLGLQJWKHVXPRIWKHVWLIIQHVVHVRIWKHFROXPQVDWDMRLQWE\WKHVXPRIWKHVWLIIQHVVHVRIWKHÀH[ural members framing into that joint in the direction of analysis:
(7.1)
In this equation, LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWHZKLFKLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,
LVWKHPRPHQWRILQHUWLDRIWKHFROXPQRUÀH[XUDOPHPEHU is the length of the columns measured center-tocenter of the joints in the frame; and LVWKHOHQJWKRIWKHÀH[XUDOPHPEHUVPHDVXUHGFHQWHUWRFHQWHURIWKHMRLQWV
The stiffness ratio LVGHWHUPLQHGDWERWKWKHWRSDQGERWWRPRIDFROXPQ7KHFKDUWLQ$&,)LJXUH5 D IRU
QRQVZD\IUDPHVDQGLQ$&,)LJXUH5 E IRUVZD\IUDPHVFDQEHXVHGWRJUDSKLFDOO\REWDLQ by drawing a
at the top of the column to the stiffness ratio
at the bottom of the column. The
line from the stiffness ratio
The charts in ACI Figure
value of is obtained from the chart at the location where the line crosses the
5DUHEDVHGRQWKHHTXDWLRQVLQ7DEOH
Table 7.5 Equations for Stiffness Ratio, Ȍ
Nonsway frames
Sway frames
The unsupported length of a column,
LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRURWKHUHOHPHQWVFDSDEOH
RISURYLGLQJODWHUDOVXSSRUWLQWKHGLUHFWLRQRIDQDO\VLV)RUH[DPSOHWKHEHDPLQ)LJXUHSURYLGHVODWHUDOVXSSRUW
which
to the top of the column in the direction parallel to the x-axis, and the unsupported length is equal to
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ݕ‬
ሺκ௨ ሻଵ ሺκ௨ ሻଶ ‫ݔ‬
Figure 7.5 Unsupported column length with beams and slabs.
is the distance from the bottom of the beam to the top of the slab at the bottom of the column. This beam does not
is the distance from
provide lateral support to the column parallel to the y-direction, so the unsupported length
the bottom of the slab at the top of the column to the top of the slab at the bottom of the column.
7KHXQVXSSRUWHGFROXPQOHQJWKIRUFROXPQVZLWKFDSLWDOVRUKDXQFKHVLVLOOXVWUDWHGLQ)LJXUHIRUWKHFDVHRID
is measured from the top of the slab at the bottom of the column to the lower excolumn capital. In this case,
tremity of the capital.
6OHQGHUQHVVOLPLWVDUHJLYHQLQ$&, D IRUFROXPQVQRWEUDFHGDJDLQVWVLGHVZD\DQGLQ$&, E IRU
FROXPQVEUDFHGDJDLQVWVLGHVZD\6OHQGHUQHVVHIIHFWVDUHSHUPLWWHGWREHQHJOHFWHGZKHQWKHHTXDWLRQVLQ7DEOH
DUHVDWLV¿HG
Table 7.6 Limits for Slenderness Effects
Column Bracing Condition
Slenderness Ratios Where Slenderness is Permitted to Be Neglected
Not braced against sidesway (sway)
Braced against sidesway (nonsway)
*Ratio
is negative if a column is bent in single curvature and is positive if a column is bent in double curvature.
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κ௨ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Figure 7.6 Unsupported length of a column with a capital.
Because columns braced against sidesway are more stable when bent in double curvature, the limiting slenderness
ratio is larger than that for columns bent in single curvature.
7KHOLPLWLQJYDOXHVRIWKHVOHQGHUQHVVUDWLRVLQ7DEOHDUHEDVHGRQDSHUFHQWLQFUHDVHLQPRPHQWVGXHWRVOHQderness effects, which is considered to be acceptable.
Minimum column dimensions so that slenderness effects need not be considered as a function of the unsupported
DUHJLYHQLQ7DEOH7KHPLQLPXPGLPHQVLRQVIRUFROXPQVEUDFHGDJDLQVWVLGHVZD\DUHEDVHG
column length,
for columns bent in double curvature where
For columns not braced against sidesway,
on
Tabulated minimum column dimensions are rounded up to
the minimum column dimensions are based on
the next whole even numbers.
Table 7.7 Minimum Column Dimensions to Neglect Slenderness
Unsupported Column
Length, ℓu (ft)
8
9
Minimum Column Dimension (in.)
Column Bracing Condition
Rectangular (c1 )
Circular
Braced against sidesway
8
Not braced against sidesway
Braced against sidesway
8
Not braced against sidesway
Braced against sidesway
Not braced against sidesway
Braced against sidesway
Not braced against sidesway
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Minimum column dimensions to neglect slenderness for columns braced against sidesway are small; the dimensions
DUHDWPRVWHTXDOWRWKHPLQLPXPSUDFWLFDOGLPHQVLRQVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ7KHUHIRUHVOHQGHUness effects typically do not need to be considered for columns braced against sidesway in buildings with typical
VWRU\KHLJKWV,QWDOOHUEXLOGLQJVVKHDUZDOOVRUDFRPELQDWLRQRIVKHDUZDOOVDQGPRPHQWIUDPHVDUHRIWHQXWLOL]HG
which essentially brace the columns against sidesway.
Columns not braced against sidesway most often occur in low-rise buildings without walls where the lateral displacements are usually relatively small. Thus, second-order effects are typically negligible even if column dimenVLRQVDUHVPDOOHUWKDQWKHPLQLPXPGLPHQVLRQVLQ7DEOH
Slenderness effects usually need to be considered where the cross-sectional dimensions of a column are limited due
WRDUFKLWHFWXUDORURWKHUFRQVWUDLQWVDQGRUWKHXQVXSSRUWHGOHQJWKRIWKHFROXPQLVUHODWLYHO\ORQJ
Moment Magnification Method
Overview
7KHPRPHQWPDJQL¿FDWLRQPHWKRGLQ$&,LVDQDSSUR[LPDWHPHWKRGWKDWDFFRXQWVIRUVOHQGHUQHVVHIIHFWVE\
PDJQLI\LQJWKHEHQGLQJPRPHQWVDWWKHHQGVRIDFROXPQREWDLQHGIURPD¿UVWRUGHUDQDO\VLV,QJHQHUDODVOHQGHU
FROXPQPXVWEHGHVLJQHGWRUHVLVWWKHFRPELQHGHIIHFWVIURPIDFWRUHGD[LDOFRPSUHVVLRQIRUFHVDQGPDJQL¿HGEHQGLQJPRPHQWVZKLFKDUHREWDLQHGE\PXOWLSO\LQJWKH¿UVWRUGHUIDFWRUHGEHQGLQJPRPHQWVE\DPRPHQWPDJQL¿HU
that is a function of the factored axial force and the critical buckling load of the column. The column is adequate
ZKHUHDOOWKHIDFWRUHGORDGFRPELQDWLRQVIRUFRPELQHGD[LDOIRUFHDQGPDJQL¿HGEHQGLQJPRPHQWIDOOZLWKLQRURQ
the design strength interaction diagram, which is obtained from a strain compatibility analysis of the section.
Nonsway Frames
3URYLVLRQVIRUVOHQGHUFROXPQVLQQRQVZD\IUDPHVDUHJLYHQLQ$&,&ROXPQVPXVWEHGHVLJQHGIRUFRPELDQGIDFWRUHGPDJQL¿HGPRPHQWV
where
is determined by ACI Equation
nations of factored axial forces,
(7.2)
In this equation,
LVWKHODUJHURI WKHWZRIDFWRUHGFROXPQHQGPRPHQWVIURPD¿UVWRUGHUDQDO\VLVDQG where is the dimension of the column in the direction of analysis
minimum moment
$&, 7KHPDJQL¿FDWLRQIDFWRU
LVGHWHUPLQHGE\$&,(TXDWLRQ (7.3)
The moment factor,
relates the actual moment diagram to an equivalent uniform moment diagram. It is asVXPHGLQWKHPRPHQWPDJQL¿HUPHWKRGWKDWWKHPD[LPXPPRPHQWLVDWRUQHDUWKHPLGKHLJKWRIWKHFROXPQ,IWKH
(that
maximum moment occurs at one end of the column, design is based on an equivalent uniform moment
is assumed to act at both ends of the column). This leads to the same maximum moment at or near midis,
KHLJKWRIWKHFROXPQZKHQPDJQL¿HG
depends on whether transverse loads are applied between the supports of the column or not (see
The value of
7DEOH ,Q$&,(TXDWLRQ D IRUWKHFDVHRIFROXPQVZLWKRXWWUDQVYHUVHORDGVDSSOLHGEHWZHHQVXSSRUWV
is the ratio of the smaller to the larger factored end moment; this ratio is negative if the column is bent in
VLQJOHFXUYDWXUHDQGSRVLWLYHLIWKHFROXPQLVEHQWLQGRXEOHFXUYDWXUH VHH)LJXUH )RUFROXPQVZLWKWUDQVYHUVH
loads applied between supports, it is possible for the maximum moment to occur at a section away from the end of
LQWKLVFDVH>$&,(TXDWLRQ E @
the member; thus,
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.8 Values of Cm
Cm
Column Transverse Load Condition
Transverse loads not applied between supports
Transverse loads applied between supports
ܲ
ܲ
‫ܯ‬ଵ ‫ܯ‬ଵ െͳ ൑ ‫ܯ‬ଵ Τ‫ܯ‬ଶ ൑ Ͳ
Ͳ ൑ ‫ܯ‬ଵ Τ‫ܯ‬ଶ ൑ ͳ
ͳǤͲ ൒ ‫ܥ‬௠ ൒ ͲǤ͸
ͲǤ͸ ൒ ‫ܥ‬௠ ൒ ͲǤʹ
‫ܯ‬ଶ ‫ܯ‬ଶ ܲ
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Figure 7.7 Values of
for columns without transverse loads applied between supports.
, the factored column end moments
and
from analysis are to be used in ACI
In cases where
(TXDWLRQ D WRGHWHUPLQH $&, $OWHUQDWLYHO\ LVSHUPLWWHGWREHWDNHQHTXDOWR
The critical buckling load,
LVGHWHUPLQHGE\$&,(TXDWLRQ (7.4)
7KHHIIHFWLYHÀH[XUDOVWLIIQHVVRIDFROXPQ
LVSHUPLWWHGWREHFDOFXODWHGE\$&,(TXDWLRQ D E RU F >$&,VHH7DEOH@7KHQXPHUDWRUVLQWKHVHHTXDWLRQVUHSUHVHQWWKHVKRUW
term column stiffness. Creep effects due to sustained loads are accounted for by dividing the short-term stiffness by
where
is the ratio of the maximum factored sustained axial load to the maximum factored axial load
associated with the same load combination. This factor is typically equal to the factored axial dead load divided by
the total factored axial load on a column.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.9 Effective Flexural Stiffness, (EI)eff
(EI)eff
ACI Equation No.
D
E
F
*
(see Table 7.2)
$&,(TXDWLRQ D LVDVLPSOL¿HGYHUVLRQRI$&,(TXDWLRQ E DQGGRHVQRWGLUHFWO\DFFRXQWIRU
, which is the moment of
longitudinal reinforcement in the column like the latter equation does through the term
inertia of the longitudinal reinforcement about the centroidal axis of the cross-section in the direction of analysis.
7KXVWKLVHTXDWLRQLVQRWDVDFFXUDWHDV$&,(TXDWLRQ E HVSHFLDOO\LQFROXPQVZLWKJUHDWHUDPRXQWVRI
longitudinal reinforcement.
and large axial load ratios
$&,(TXDWLRQ E ZDVRULJLQDOO\GHULYHGIRUVPDOOHFFHQWULFLW\UDWLRV
and represents the lower limit of the practical range of stiffness values, especially for columns with larger
DPRXQWVRIORQJLWXGLQDOUHLQIRUFHPHQW$&,(TXDWLRQ F XVXDOO\\LHOGVPRUHDFFXUDWHYDOXHVRIWKHHIIHFWLYHÀH[XUDOVWLIIQHVVWKDQWKRVHGHWHUPLQHGE\WKHRWKHUWZRHTXDWLRQVZKHUH in this equation is the moment of
inertia for columns and walls in ACI Table 6.6.3.1.1(b).
7KHIDFWRULQWKHGHQRPLQDWRURI(TXDWLRQ LVDVWLIIQHVVUHGXFWLRQIDFWRUVLPLODUWRWKHRQHGHULYHGIRUWKH
UHGXFHGPRPHQWRILQHUWLDZKLFKLVHTXDOWR$GGLWLRQDOLQIRUPDWLRQRQWKHGHYHORSPHQWRIWKLVVWLIIQHVVIDFWRULVJLYHQLQ$&,5
Sway Frames
3URYLVLRQVIRUVOHQGHUFROXPQVLQVZD\IUDPHVDUHJLYHQLQ$&,&ROXPQVPXVWEHGHVLJQHGIRUFRPELQDand factored total moments
and
at the ends of the column, which are the
tions of factored axial forces
summation of the following:
and
due to loads causing no appreciable sidesway (like gravity
1. 8
QPDJQL¿HGIDFWRUHGPRPHQWV
loads), and
and
due to loads causing appreciable sidesway (like wind and
0DJQL¿HGIDFWRUHGPRPHQWV
HDUWKTXDNHORDGV >VHH$&,(TXDWLRQV D DQG E @
Therefore,
(7.5)
(7.6)
The moments
DUHGHWHUPLQHGIURPD¿UVWRUGHUHODVWLFDQDO\VLVRIWKHVWUXFWXUHXVLQJUHGXFHGVHFWLRQSURSHUWLHVRIWKHPHPEHUVLQDFFRUGDQFHZLWK$&,7DEOH D >VHH7DEOH@
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHPRPHQWPDJQL¿FDWLRQIDFWRU
for sway frames, is determined by either a linear elastic second-order elastic
DQDO\VLV VHH6HFWLRQRIWKLVSXEOLFDWLRQ RUE\$&,(TXDWLRQ D RU E >$&,@
(7.7)
(7.8)
(TXDWLRQ LVWKHVROXWLRQRIDQLQ¿QLWHVHULHVUHSUHVHQWLQJWKHLWHUDWLYH3GHOWDDQDO\VLVDQGFORVHO\SUHGLFWVWKH
(TXDWLRQ RUDOLQHDUHODVWLFVHFRQGRUGHU
magnitude of second-order moments in a sway frame where
HODVWLFDQDO\VLVLQDFFRUGDQFHZLWK$&,PXVWEHXVHGLQFDVHVZKHUH FDOFXODWHGE\(TXDWLRQ H[FHHGV
7KHVWDELOLW\LQGH[ LVGHWHUPLQHGE\$&,(TXDWLRQ >VHH7DEOH@DQGLVEDVHGRQGHÀHFWLRQV
REWDLQHGIURPD¿UVWRUGHUHODVWLFDQDO\VLVXVLQJUHGXFHGVHFWLRQSURSHUWLHVRIWKHPHPEHUV
(TXDWLRQ ZDVSDUWRIWKHPRPHQWPDJQL¿HUPHWKRGWKDWDSSHDUHGLQSUHYLRXVHGLWLRQVRI$&,7KHIDFare summed over the entire story, and the critical buckling loads,
are summed over the
tored axial loads,
used in the
sway-resisting columns in that story. Like in the case of nonsway frames, the effective stiffness
which is the ratio of the maximum
calculation of LVGHWHUPLQHGE\$&,)RUVZD\IUDPHVWKHWHUP
factored sustained story shear to the maximum factored story shear, is used in the denominator of the equations of
instead of
Note that
LVQRUPDOO\HTXDOWR]HUREHFDXVHZLQGDQGVHLVPLFORDGVDUHQRWVXVWDLQHG
loads. An example of a sustained lateral load is lateral earth pressure.
7KHPDJQL¿HGPRPHQWVDWWKHHQGVRIDFROXPQPXVWEHFRQVLGHUHGLQWKHGHVLJQRIWKHEHDPVIUDPLQJLQWRWKH
FROXPQV $&, %HDPVWLIIQHVVSOD\VDNH\UROHLQWKHVWDELOLW\RIFROXPQVDQGWKLVSURYLVLRQHQVXUHVWKH
EHDPVZLOOKDYHDGHTXDWHVWUHQJWKWRUHVLVWWKHPDJQL¿HGFROXPQPRPHQWV
It is possible for the maximum moment in a slender column in a sway frame to occur at a section away from its
HQGV,WLVSHUPLWWHGWRPDJQLI\VXFKPRPHQWVXVLQJWKHSURYLVLRQVIRUQRQVZD\IUDPHVLQ$&,ZKHUH
and
GHWHUPLQHGE\(TXDWLRQV DQG >$&,@
calculated using the total moments
is
7.3.4 Required Shear Strength for Columns in Buildings Assigned to Seismic Design Category B
For certain columns in buildings assigned to SDC B that are part of an ordinary moment frame resisting seismic
ORDGHIIHFWVWKHVKHDUVWUHQJWKUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HGLQDGGLWLRQWRDOORWKHUDSSOLFDEOH
LVOHVVWKDQRUHTXDOWR¿YHWLPHVWKHSODQ
requirements. In cases where the unsupported length of a column,
in the direction of analysis, the design shear strength of the column must be taken as
dimension of the column,
the lesser of the following:
associated with the development of nominal moment strengths,
of the column at each
• The shear force,
UHVWUDLQHGHQGRIWKHXQVXSSRUWHGOHQJWKGXHWRUHYHUVHFXUYDWXUHEHQGLQJ VHH)LJXUH ,QWKH¿JXUH
DUHWKHQRPLQDOÀH[XUDOVWUHQJWKVDWWKHWRSDQGERWWRPRIWKHFROXPQUHVSHFWLYHO\7KHVHÀH[XUDO
and
consistent with the direction of analysis resulting
strengths must be calculated for the factored axial force,
LQWKHKLJKHVWÀH[XUDOVWUHQJWK6LGHVZD\WRWKHULJKWDQGVLGHVZD\WRWKHOHIWPXVWERWKEHFRQVLGHUHG
• 7KHPD[LPXPVKHDUIRUFHREWDLQHGIURPWKHGHVLJQORDGFRPELQDWLRQVRI$&,&KDSWHULQFOXGLQJWKHHDUWKwith
substituted for
in the load combinations. The term
is the overstrength factor
quake effect,
IRURUGLQDU\PRPHQWIUDPHVRIUHLQIRUFHGFRQFUHWHZKLFKLVHTXDOWR VHH$6&(6(,7DEOH The provisions for the design shear strength are intended to provide additional capacity to resist shear in relatively
squat columns, which are vulnerable to shear failure under earthquake loading.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ܯ‬௡௧ ܿଵ ܸ௨ ܸ௨ ൌ
‫ܯ‬௡௧ ൅ ‫ܯ‬௡௕
κ௨
κ௨ ൑ ͷܿଵ ܸ௨ ‫ܯ‬௡௕ Figure 7.8 Design shear strength of reinforced concrete columns in buildings assigned to SDC B.
7.4 Design Strength
7.4.1 General
)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\
VHFWLRQLQDFROXPQ $&, (7.9)
(7.10)
(7.11)
(7.12)
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ
IDFWRUVSHUWLQHQWWRWKHGHVLJQRIUHLQIRUFHGFRQFUHWHFROXPQVLVJLYHQLQ7DEOH7KHVWUDLQXVHGWRGH¿QHD
LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHUHLQIRUFHPHQW divided by the
compression-controlled section,
ZKLFKFDQEHWDNHQDVSVLIRUDQ\JUDGHRIUHLQIRUFHmodulus of elasticity of the reinforcing steel,
PHQW $&, Table 7.10 Strength Reduction Factors, Ҁ
Strength Reduction Factor, Ҁ
Net Tensile Strain, İt
Classification
Compression-controlled
Type of Transverse Reinforcement
Spirals Conforming to ACI 25.7.3
Other
Transition*
Tension-controlled
*Sections classified as transition are permitted to use
corresponding to compression-controlled sections.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUFROXPQVVXEMHFWHGSULPDULO\WRXQLD[LDOFRPSUHVVLRQIRUFHV(TXDWLRQ LVDSSOLFDEOH)RUFROXPQVVXEMHFWHG
WRFRPELQHGPRPHQWDQGD[LDOIRUFHV(TXDWLRQV DQG PXVWERWKEHVDWLV¿HGIRUDOODSSOLFDEOHIDFWRUHG
ORDGFRPELQDWLRQV7KHVKHDUDQGWRUVLRQDOVWUHQJWKHTXDWLRQV>(TXDWLRQV DQG UHVSHFWLYHO\@PXVWEH
VDWLV¿HGIRUDQ\FROXPQ
combined
Methods to determine the nominal strengths
DQGRIWKLVSXEOLFDWLRQUHVSHFWLYHO\
and
DUHJLYHQLQ6HFWLRQV
7.4.2 Nominal Axial Strength
Nominal Axial Compressive Strength
The nominal axial compressive strength,
of a nonslender, reinforced concrete column is determined in accorGDQFHZLWK$&,)RUQRQSUHVWUHVVHGFROXPQVZLWKWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRI WLHVFRQIRUPLQJWR$&,DQG VSLUDOVFRQIRUPLQJWR$&, VHH)LJXUH must be less than or equal to
which is determined using the equations in ACI Table
the maximum nominal axial compressive strength,
DQGWKHQRPLQDOD[LDOVWUHQJWKDW]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ (a) tied column
(b) spiral column
Figure 7.9 Reinforced concrete columns. (a) With tied transverse reinforcement.
(b) With spiral transverse reinforcement.
• For columns with ties:
(7.13)
• For columns with spirals:
(7.14)
These equations account for any accidental eccentricities that may exist in a column that were not accounted for in
DQDO\VLV,QWKHFDVHRIWLHGFROXPQVWKHHFFHQWULFLW\LVDSSUR[LPDWHO\SHUFHQWRIWKHFROXPQGHSWKZKLOHIRU
VSLUDOFROXPQVWKHHFFHQWULFLW\LVDSSUR[LPDWHO\SHUFHQW
WKHYDOXHRIWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHORQJLWXGLQDOUHLQIRUFHPHQW
is limited
When calculating
WRSVLHYHQWKRXJK for the longitudinal reinforcement used in the column may be larger than that (ACI
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Nominal Axial Tensile Strength
The nominal axial tensile strength of a nonprestressed, reinforced concrete column,
must be less than or equal
ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ to the maximum nominal axial tensile strength,
(7.15)
It is evident that the entire tensile force must be resisted by the longitudinal reinforcement in a column.
7.4.3 Nominal Strength of Columns Subjected to Moment and Axial Forces
Overview
The nominal strength of a reinforced concrete column subjected to both moment and axial force is determined using
HTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ$&,ZKLFKDUHVXPPDUL]HGLQ7DEOH
Table 7.11 Design Assumptions for Concrete and Nonprestressed Reinforcement
Material
Assumptions
1. 0D[LPXPVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHULVDVVXPHGHTXDOWR
7HQVLOHVWUHQJWKRIFRQFUHWHLVQHJOHFWHGLQÀH[XUDODQGD[LDOVWUHQJWKFDOFXODWLRQV
3. The relationship between concrete compressive stress and strain is to be represented by
DUHFWDQJXODUWUDSH]RLGDOSDUDEROLFRURWKHUVKDSHWKDWUHVXOWVLQSUHGLFWLRQRIVWUHQJWK
in substantial agreement with results of comprehensive tests.
7KHHTXLYDOHQWUHFWDQJXODUFRQFUHWHVWUHVVGLVWULEXWLRQLQDFFRUGDQFHZLWK$&,
WKURXJKVDWLV¿HVWKHUHTXLUHPHQWRI$&,IRUWKHUHODWLRQVKLSEHWZHHQ
concrete compressive stress and strain.
Concrete
is to be assumed uniformly distributed over an equivalent
A concrete stress of
FRPSUHVVLRQ]RQHERXQGHGE\WKHHGJHVRIWKHFURVVVHFWLRQDQGDOLQHSDUDOOHOWRWKH
neutral axis located a distance IURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQZKHUH
7KHGLVWDQFHIURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQWRWKHQHXWUDOD[LV is to be
measured perpendicular to the neutral axis.
Values of
DUHDVIROORZV $&,7DEOH •
•
•
Nonprestressed
reinforcement
1. Deformed reinforcement used to resist tensile or compressive forces must conform to
$&,
The stress-strain relationship and the modulus of elasticity of the deformed reinforcement
DUHWREHLGHDOL]HGLQDFFRUGDQFHZLWK$&,DQG
Rectangular Sections
7KHJHQHUDOSULQFLSOHVDQGDVVXPSWLRQVRIWKHVWUHQJWKGHVLJQPHWKRGLQ7DEOHDUHDSSOLHGWRWKHUHFWDQJXODU
UHLQIRUFHGFRQFUHWHFROXPQLQ)LJXUH)RUSXUSRVHVRIGLVFXVVLRQLWLVDVVXPHGWKHORQJLWXGLQDOUHLQIRUFHPHQW
LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ
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7-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ͲǤͲͲ͵
ߝ௦ଵ ݀ଷ ݄
݀ଶ ܿ
‫ܣ‬௦ଵ ‫ܣ‬௦ଶ ‫ܣ‬௦ଷ ߝ௦ଶ ߝ௦ଷ ൌ ߝ௧ ܽ ൌ ߚଵ ܿ
݀ଵ ͲǤͺͷ݂௖ᇱ ‫ܨ‬௦ଵ ‫ܥ‬
‫ܨ‬௦ଶ ‫ܨ‬௦ଷ ܾ
Figure 7.10 Rectangular reinforced concrete column subjected to moment and axial compression.
The nominal axial strength,
DQGWKHQRPLQDOÀH[XUDOVWUHQJWK
RIWKHFROXPQLQ)LJXUHFDQEHREXVLQJWKHHTXDWLRQVLQ)LJXUHZKHUH is the net tensile strain in the extreme layer
tained for a given strain,
of tension reinforcement at nominal strength, excluding strains due to creep, shrinkage, and temperature.
Circular Sections
7KHJHQHUDOSULQFLSOHVDQGDVVXPSWLRQVRIWKHVWUHQJWKGHVLJQPHWKRGLQ7DEOHDUHDSSOLHGWRWKHFLUFXODU
UHLQIRUFHGFRQFUHWHFROXPQLQ)LJXUH)RUSXUSRVHVRIGLVFXVVLRQLWLVDVVXPHGWKHORQJLWXGLQDOUHLQIRUFHPHQW
LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ)RUFLUFXODUVHFWLRQVWKHVKDSHRIWKH
FRPSUHVVLRQ]RQHLVUHODWHGWRDVHJPHQWRIDFLUFOH VHH)LJXUH DQGWKHQRPLQDOÀH[XUDOVWUHQJWK
The nominal axial strength,
XVLQJWKHHTXDWLRQVLQ)LJXUH
tained for a given strain,
RIWKHFROXPQLQ)LJXUHFDQEHRE-
Interaction Diagrams
An interaction diagram for a reinforced concrete column is a collection of
and
values determined for a
series of strain distributions using the methods above for rectangular and circular sections. Nominal strengths for a
given strain distribution represent a single point on the interaction diagram. Such diagrams are useful in establishing
the adequacy of a column subjected to a combination of axial forces and bending moments.
Nominal and design strength interaction diagrams for a tied, rectangular reinforced concrete column with a symmetULFDOGLVWULEXWLRQRIORQJLWXGLQDOUHLQIRUFHPHQWDUHJLYHQLQ)LJXUH7KHSRUWLRQVRIWKHGLDJUDPVFRUUHVSRQGLQJ
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WKH¿JXUH
The design strength for combined moment and axial forces must be greater than or equal to the corresponding
UHTXLUHGVWUHQJWKVWRVDWLVI\VWUHQJWKUHTXLUHPHQWV>VHH(TXDWLRQV DQG @'HVLJQVWUHQJWKYDOXHVDUHREWDLQHGE\PXOWLSO\LQJWKHQRPLQDOVWUHQJWKYDOXHVZLWKWKHVWUHQJWKUHGXFWLRQIDFWRUVLQ$&, VHH6HFWLRQ
of this publication).
Factored axial force-bending moment combinations falling on or within the boundaries of the design strength interaction diagram can be safely carried by the column. This column is adequate for the combination denoted by Point 1
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Interaction diagrams about the major axis and the minor axis of a rectangular column can be different; this must be
taken into consideration when designing the column. For circular reinforced concrete columns with 6 longitudinal
EDUVRUOHVVWKHGHVLJQÀH[XUDOVWUHQJWKGHSHQGVRQWKHEDUDUUDQJHPHQWDQGWKHFROXPQPXVWEHFKHFNHGXVLQJWKH
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7-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿൌ
ͲǤͲͲ͵݀‫ݐ‬
ߝ‫ ݐ‬൅ ͲǤͲͲ͵
ͲǤ͸ͷ ൑ ߚͳ ൌ ͲǤͺͷ െ
ͲǤͲͷሺ݂ܿԢ െ ͶǡͲͲͲሻ
൑ ͲǤͺͷ
ͳǡͲͲͲ
ܽ ൌ ߚͳ ܿ ൑ ݄
‫ ܥ‬ൌ ͲǤͺͷ݂ܿԢ ܾܽ
݅ ൌ ͳ
ߝ‫ ݅ݏ‬ൌ
ͲǤͲͲ͵ሺܿ െ ݀݅ ሻ
ܿ
݂‫ ݅ݏ‬ൌ ‫ ݅ݏߝ ݏܧ‬
Figure 7.11 Combined flexural and axial strength of a rectangular reinforced concrete column.
DUUDQJHPHQWWKDWJLYHVPLQLPXPGHVLJQÀH[XUDOVWUHQJWK8WLOL]LQJRUPRUHORQJLWXGLQDOEDUVHVVHQWLDOO\HOLPLQDWHVWKHQHHGWRGHWHUPLQHGLIIHUHQWGHVLJQÀH[XUDOVWUHQJWKVEDVHGRQEDUDUUDQJHPHQW
5HIHUHQFHFRQWDLQVGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKHIROORZLQJ
• 7LHGUHFWDQJXODUFROXPQVUDQJLQJLQVL]HIURPWRLQLQFOXVLYH
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7-18
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
•ȁ݂‫ ݅ݏ‬ȁ ൐ ݂‫ ݕ‬ǫ
EŽ
zĞƐ
݂‫ ݅ݏ‬ൌ േ݂‫ ݕ‬
݂‫ ݅ݏ‬ൌ ‫ ݅ݏߝ ݏܧ‬
•݀݅ ൐ ܽǫ
EŽ
‫ ݅ݏܨ‬ൌ ሺ݂‫ ݅ݏ‬െ ͲǤͺͷ݂ܿԢ ሻ‫ ݅ݏܣ‬
zĞƐ
‫ ݅ݏܨ‬ൌ ݂‫ ݅ݏܣ ݅ݏ‬
݅ ൌ ݅ ൅ ͳ
•݅ ൐ ݊ǫ
EŽ
zĞƐ
݊
ܲ݊ ൌ ‫ ܥ‬൅ ෍ ‫ ݅ݏܨ‬
݅ൌͳ
݊
‫ ݊ܯ‬ൌ ͲǤͷ‫ܥ‬ሺ݄ െ ܽሻ ൅ ෍ ‫ ݅ݏܨ‬ሺͲǤͷ݄ െ ݀݅ ሻ
݅ൌͳ
Figure 7.11 (cont.) Combined flexural and axial strength of a rectangular reinforced concrete column.
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7-19
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿ
‫ܣ‬௦ଷ ߝ௦ଶ ݀ହ ݄
݀ସ ݀ଷ ‫ܣ‬௦ଶ ݀ଶ ‫ܣ‬௦ଵ ͲǤͲͲ͵
ߝ௦ଵ ܽ ൌ ߚଵ ܿ
݀ଵ ͲǤͺͷ݂௖ᇱ ‫ܨ‬௦ଷ ߝ௦ଷ ߝ௦ସ ‫ܣ‬௦ସ ‫ܣ‬௦ହ ‫ܨ‬௦ଵ ‫ܥ‬
‫ܨ‬௦ଶ ‫ܨ‬௦ସ ‫ܨ‬௦ହ ߝ௦ହ ൌ ߝ௧ Figure 7.12 Circular reinforced concrete column subjected to moment and axial compression.
‫ݕ‬ത
݄Ȁʹ
ܽ
‫ ܣ‬ൌ ݄ଶ ሺߠ െ •‹ ߠሻΤͺ
ߠȀʹ ߠȀʹ
ʹܽ
൰
݄
݄Ȁʹ
ߠ ൌ ʹ ‘•ିଵ ൬ͳ െ
‫ݕ‬ത ൌ
ʹ݄ •‹ଷ ሺߠ Τʹሻ
͵ሺߠ െ •‹ ߠሻ
Figure 7.13 Section properties of the compression zone for a circular reinforced concrete column.
Biaxial Loading
Overview
Biaxial bending of columns occurs where loading causes bending simultaneously about both principal axes. This
is often encountered in corner columns. The strength of an axially loaded column subjected to bending moments
DERXWERWKSULQFLSDOD[HVFDQEHUHSUHVHQWHGE\DELD[LDOVWUHQJWKLQWHUDFWLRQVXUIDFH VHH)LJXUH 7KLVVXUIDFH
is formed by a series of uniaxial interaction diagrams drawn radially from the vertical axis. Intermediate interaction
diagrams between the angle HTXDOWR]HURGHJUHHV XQLD[LDOEHQGLQJDERXWWKHxD[LV DQGGHJUHHV XQLD[LDO
bending about the yD[LV DUHREWDLQHGE\YDU\LQJWKHDQJOHRIWKHQHXWUDOD[LVIRUDVVXPHGVWUDLQFRQ¿JXUDWLRQV
A biaxial strength interaction surface is constructed by performing a series of involved and time-consuming strain
compatibility analyses. For example, the strains and forces for a rectangular column subjected to an axial compresand biaxial bending
DUHJLYHQLQ)LJXUH'HVLJQLQJDFROXPQIRU
sion force
WKLVORDGLQJFRQGLWLRQFDQEHDYHU\OHQJWK\SURFHVVZLWKRXWWKHXVHRIDFRPSXWHUSURJUDP+RZHYHUFRQVHUYDWLYH
results for columns subjected to biaxial bending can be obtained using some approximate methods.
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7-20
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿൌ
ͲǤͲͲ͵݀‫ݐ‬
ߝ‫ ݐ‬൅ ͲǤͲͲ͵
ͲǤ͸ͷ ൑ ߚͳ ൌ ͲǤͺͷ െ
ͲǤͲͷሺ݂ܿԢ െ ͶǡͲͲͲሻ
൑ ͲǤͺͷ
ͳǡͲͲͲ
ܽ ൌ ߚͳ ܿ ൑ ݄
ߠ ൌ ʹ ‘• െͳ ൬ͳ െ
‫ݕ‬ത ൌ
‫ܥ‬ൌ
ʹܽ
൰
݄
ʹ݄•‹͵ ሺߠΤʹሻ
͵ሺߠ െ •‹ ߠሻ
ͲǤͺͷ݂ܿԢ ݄ʹ
ሺߠ െ •‹ ߠሻ
ͺ
݅ ൌ ͳ
ߝ‫ ݅ݏ‬ൌ
ͲǤͲͲ͵ሺܿ െ ݀݅ ሻ
ܿ
݂‫ ݅ݏ‬ൌ ‫ ݅ݏߝ ݏܧ‬
Figure 7.14 Combined flexural and axial strength of a circular reinforced concrete column.
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7-21
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
•ȁ݂‫ ݅ݏ‬ȁ ൐ ݂‫ ݕ‬ǫ
EŽ
zĞƐ
݂‫ ݅ݏ‬ൌ േ݂‫ ݕ‬
݂‫ ݅ݏ‬ൌ ‫ ݅ݏߝ ݏܧ‬
•݀݅ ൐ ܽǫ
EŽ
‫ ݅ݏܨ‬ൌ ሺ݂‫ ݅ݏ‬െ ͲǤͺͷ݂ܿԢ ሻ‫ ݅ݏܣ‬
zĞƐ
‫ ݅ݏܨ‬ൌ ݂‫ ݅ݏܣ ݅ݏ‬
݅ ൌ ݅ ൅ ͳ
•݅ ൐ ݊ǫ
EŽ
zĞƐ
݊
ܲ݊ ൌ ‫ ܥ‬൅ ෍ ‫ ݅ݏܨ‬
݅ൌͳ
݊
‫ ݊ܯ‬ൌ ‫ݕܥ‬ത ൅ ෍ ‫ ݅ݏܨ‬ሺͲǤͷ݄ െ ݀݅ ሻ
݅ൌͳ
Figure 7.14 (cont.) Combined flexural and axial strength of a circular reinforced concrete column.
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7-22
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܲ௢ ‫ܔ܉ܑܠۯ‬۴‫܍܋ܚܗ‬
ܲ௡ǡ௠௔௫ ൌ ͲǤͺͲܲ௢ ܲ௡ ǡ ‫ܯ‬௡ ߶ܲ௢ ༄
ܲ௨ ǡ ‫ܯ‬௨ ߶ܲ௡ǡ௠௔௫ ߶ܲ௡ ǡ ߶‫ܯ‬௡ ‘’”‡••‹‘Ǧ
‘–”‘ŽŽ‡†
༃
ܲ௨ ǡ ‫ܯ‬௨ ƒŽƒ ‡† ƒ‹Ž—”‡
”ƒ•‹–‹‘
‡•‹‘Ǧ
‘–”‘ŽŽ‡†
۰‫ܜܖ܍ܕܗۻ܏ܖܑ܌ܖ܍‬
Figure 7.15 Nominal and design strength interaction diagrams.
ܲ
ܲ௢ Žƒ‡‘ˆ ‘•–ƒ–ܲ௡ ‫ܯ‬௡௬ ‫ܯ‬௡௫ ‘ƒ† ‘–‘—”
ܲ௡ ‫ܯ‬௡ ‫ܯ‬௬ ߠ
Žƒ‡‘ˆ ‘•–ƒ–ߠ
‫ܯ‬௫ Figure 7.16 Biaxial strength interaction surface.
$SSUR[LPDWH0HWKRGIRU&ROXPQVZLWK(TXDO%HQGLQJ0RPHQW&DSDFLWLHV$ERXW%RWK3ULQFLSDO$[HV
For circular columns or for square columns with the same longitudinal reinforcement on each face, the uniaxial
is the same as that about the y-axis,
$KRUL]RQWDOVOLFHWKURXJKWKH
bending capacity about the x-axis,
LVVKRZQLQ)LJXUH
biaxial interaction surface of a square column at an axial compression force
where the dash-dot line represents the nominal moment strengths and the solid line represents the design moment
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7-23
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ݕ‬
༃
݁௬ ༄
ܲ௨ ݁௫ ݄
Žƒ•–‹ ‡–”‘‹†
‫ݔ‬
༆
༅
ܾ
‫ܯ‬௨௫ ൌ ܲ௨ ݁௬ ‫ܯ‬௨௬ ൌ ܲ௨ ݁௫ Figure 7.17 Rectangular column subjected to biaxial bending.
‫ݕ‬
‫ܯ‬௡௢௬ ‫ܯ‬௡ ߶‫ܯ‬௡௢௬ ߶‫ܯ‬௡ Approx.
߶‫ܯ‬௡
൅ 1
൫‫ܯ‬௨௫ , ‫ܯ‬௨௬ ൯
߶‫ܯ‬௡௢௫ ‫ݔ‬
‫ܯ‬௡௢௫ ߶‫ܯ‬௢௡௫ ൌ ߶‫ܯ‬௡௢௬ Figure 7.18 Biaxial capacity approximation for circular columns and square columns
with the same longitudinal reinforcement on each face.
strengths. The moments
and
computer software or design aid.
can be determined using the methods described above or by any other
This line
7KHVWUDLJKWGDVKHGOLQHLQ)LJXUHFRQQHFWVWKHXQLD[LDOGHVLJQPRPHQWVWUHQJWKV
always lies inside the actual biaxial capacity curve, so it can be used as a conservative (but not overly conservative)
representation of the actual capacity diagram. Therefore, a circular column or a square column with the same longiis adequate for any combinatudinal reinforcement on each face subjected to a factored axial compression force
and
that lies within the approximate biaxial capacity surface (an example of an adequate bending
tion of
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7-24
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The approximate capacity surface can be represented by the following equation:
(7.16)
(TXDWLRQ FDQEHUHZULWWHQDVIROORZV
Because
(7.17)
Therefore, for a given
a column is adequate where
Reciprocal Load Method
The Reciprocal Load Method provides a simple and conservative estimate of the strength of a column under biaxial
corresponding to eccentricities about both prinORDGLQJFRQGLWLRQV 5HIHUHQFH 7KHQRPLQDOD[LDOVWUHQJWK
cipal axes of a column is calculated by the following equation:
(7.18)
where
nominal axial strength when the column is subjected to the uniaxial moment
nominal axial strength when the column is subjected to the uniaxial moment
QRPLQDOD[LDOVWUHQJWKDW]HURHFFHQWULFLW\
Determination of the items needed to calculate
is relatively straightforward. The nominal axial strengths
can be obtained from uniaxial nominal strength strain compatibility analyses, and
is determined by ACI
and
(TXDWLRQ is obtained by multiplying
GHWHUPLQHGE\(TXDWLRQ ZLWKWKHDSSURSULThe design axial strength
ate strength reduction factor based on the strain in the reinforcing bars farthest from the compression face of the
is greater than or equal to the factored axial
column. The column is adequate where the design axial strength
force
7KH5HFLSURFDO/RDG0HWKRGSURGXFHVUHDVRQDEO\DFFXUDWHUHVXOWVZKHQÀH[XUHGRHVQRWJRYHUQWKHGHVLJQRID
and
are greater than the axial strength corresponding to balanced failure. In cases
column, that is, when
ZKHUHÀH[XUHJRYHUQVDQRWKHUPHWKRGOLNHWKH/RDG&RQWRXU0HWKRGGLVFXVVHGEHORZVKRXOGEHXWLOL]HG
Load Contour Method
)RUFROXPQVZLWKXQHTXDOEHQGLQJPRPHQWFDSDFLWLHVDERXWWKHSULQFLSDOD[HVDQGZKHUHÀH[XUHJRYHUQVWKHGHVLJQ
(that is, where axial forces are relatively small), the Load Contour Method can be used to determine biaxial strength
5HIHUHQFH 7KHORDGFRQWRXU WKDWLVWKHKRUL]RQWDOVOLFHWKURXJKWKHELD[LDOLQWHUDFWLRQVXUIDFH FRUUHVSRQGLQJWR
can be approximated by the following nondimensional interaction equation:
a nominal axial compression force,
(7.19)
In this equation,
the column, and
ly. The exponents
and
are the nominal biaxial moments that occur simultaneously on
and
are the nominal uniaxial moment strengths about the x-axis and y-axis, respectiveand are a function of the amount, distribution, and location of the longitudinal reinforcement
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7-25
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
in the column; the dimensions of the column; and the strength and elastic properties of the concrete and reinforcing
Interaction curves for variVWHHO,WLVGHPRQVWUDWHGLQ5HIHUHQFHWKDWLWLVUHDVRQDEO\DFFXUDWHWRDVVXPH
DUHJLYHQLQ)LJXUH,QFOXGHGLVWKHVWUDLJKWOLQHFRUUHVSRQGLQJWR
which results in the
ous
following interaction equation:
(7.20)
This equation always yields conservative results because it underestimates the capacity of the column. The coland the applied bending moments,
and
umn is adequate for the applied axial force,
that lie within the approximate biaxial capacity surface, that is, where the following equation is
VDWLV¿HG
(7.21)
The PCA Load Contour Method is an extension of the Load Contour Method and information on this method can be
found in Reference 18.
Slenderness effects must also be considered in the design of columns subjected to biaxial loading, where applicable.
A review of several methods to evaluate slenderness effects in columns subjected to axial compression forces and
biaxial bending is given in Reference 19.
ϭ͘Ϭ
ࡹ࢔࢟ Ȁࡹ࢔࢕࢟ Ϭ͘ϴ
Ϭ͘ϲ
Ϭ͘ϰ
Ϭ͘Ϯ
Ϭ͘Ϭ
Ϭ͘Ϭ
Ϭ͘Ϯ
Ϭ͘ϰ
Ϭ͘ϲ
Ϭ͘ϴ
ϭ͘Ϭ
ࡹ࢔࢞ Τࡹ࢔࢕࢞ Figure 7.19 Interaction curves for the Load Contour Method.
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7-26
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.4.4 Nominal Shear Strength
Overview
The nominal one-way shear strength,
DWDVHFWLRQLQDFROXPQLVGHWHUPLQHGLQDFFRUGDQFHZLWK$&, $&,
(7.22)
where
is the nominal shear strength provided by concrete and
reinforcement.
is the nominal shear strength provided by shear
Nominal Shear Strength Provided by Concrete
For nonprestressed members, LVGHWHUPLQHGE\WKHDSSOLFDEOHHTXDWLRQLQ$&,7DEOH $&, 7KH
HTXDWLRQVLQWKDWWDEOHDUHJLYHQLQ7DEOHDORQJZLWKWKHPD[LPXPSHUPLWWHGQRPLQDOVKHDUVWUHQJWK
VSHFL¿HGLQ$&,
Table 7.12 Nominal Shear Strength Provided by Concrete, Vc
Vc**
Area of Shear Reinforcement, Av*
*Minimum area of shear reinforcement,
**
(ACI 22.5.5.1.2)
, is determined by ACI 10.6.2.2
The minimum area of shear reinforcement in a column,
(TXDWLRQ EHORZ@
LVGHWHUPLQHGLQDFFRUGDQFHZLWK$&,>VHH
7KHPRGL¿FDWLRQIDFWRU UHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ7DEOH
EDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[ VHH$&, Table 7.13 Values of
Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
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7-27
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.14 Values of
Based on Composition of Aggregates
Concrete
All-lightweight
/LJKWZHLJKW¿QHEOHQG
Sand-lightweight
Sand-lightweight, coarse blend
Composition of Aggregates
)LQH$670&
&RDUVH$670&
)LQH&RPELQDWLRQRI$670&DQG&
&RDUVH$670&
Fine: ASTM C33
&RDUVH$670&
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&
and ASTM C33
Ȝ
WR(1)
WR (1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate.
(2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength
attributed to concrete in members without shear reinforcement does not increase in direct proportion with member
GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(7.23)
,WLVHYLGHQWIURP(TXDWLRQ WKDW
LVOHVVWKDQIRUPHPEHUVZLWK
The term
LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by
where
is the
width of a rectangular column perpendicular to the direction of analysis or the diameter of a circular column [ACI
E @7KHHIIHFWLYHGHSWK can be obtained from a strain compatibility analysis for each applicable load
combination; in the case of circular columns, it is permitted to take DVSHUFHQWRIWKHGLDPHWHURIWKHFROXPQ
>$&, D @$FFRUGLQJWR$&,5 may be taken as the sum of the areas of the longitudinal reinIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU
$FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken
, and is to be taken as negative for tenas positive for compression forces acting on the gross area of the column,
sion forces.
used to calculate DUHOLPLWHGWRSVL $&,WKHH[FHSWLRQLQ$&,LVQRWDSValues of
is primarily due to the fact that there is a lack of test data and practical
plicable to columns). This limitation on
H[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL
In rare occasions where axial tension acts on a column,
FDQEHWDNHQDV]HUR
7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH
FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@
(7.24)
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7-28
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Nominal Shear Strength Provided by Shear Reinforcement
7KHQRPLQDOVKHDUVWUHQJWKRIWKHWUDQVYHUVHUHLQIRUFHPHQWLVGHWHUPLQHGE\$&,(TXDWLRQ (7.25)
where s is the center-to-center spacing of ties or the spiral pitch. For columns with rectilinear ties and crossties,
is equal to the area of all the ties and crossties within the spacing LQWKHGLUHFWLRQRIDQDO\VLV $&, ,Q
is equal to two times the area of the tie or spiral within the spacing (ACI
the case of circular ties or spirals,
A minimum area of shear reinforcement,
must be provided at sections where
$&, (7.26)
)RUFROXPQVVXEMHFWHGWRUHODWLYHO\VPDOOIDFWRUHGVKHDUIRUFHVWKHFRQFUHWHDORQHPD\EHVXI¿FLHQWWRVDWLVI\VKHDU
is typically calculated using
strength requirements. When more than just the concrete shear strength is needed,
WKHUHTXLUHPHQWVLQ$&,DQGIRUWUDQVYHUVHEDUVL]HDQGVSDFLQJ,IVKHDUVWUHQJWKUHTXLUHPHQWV
DUHQRWVDWLV¿HGEDVHGRQWKRVHUHTXLUHPHQWVWKHDPRXQWDQGRUVSDFLQJRIWKHWUDQVYHUVHUHLQIRUFHPHQWPXVWEH
adjusted appropriately, or the compressive strength of the concrete can be increased, although the latter is generally
QRWWKHPRVWHIIHFWLYHZD\WRLQFUHDVHVKHDUVWUHQJWK,QFDVHVZKHUHWKHVKHDUGHPDQGLVYHU\ODUJHWKHVL]HRIWKH
column may need to be increased.
Biaxial Shear Strength
Reinforced concrete columns may be subjected to the effects from biaxial shear forces (this commonly occurs in
corner columns). For symmetrically reinforced circular columns, the nominal one-way shear strength is the same
along any axis of the member, and shear strength can be evaluated based on the requirements outlined above using
the resultant of the shear forces acting on the section.
For rectangular sections, it is not practical to determine the nominal shear strength based on the direction of the
UHVXOWDQWVKHDUIRUFH$VDQDOWHUQDWLYHWKHELD[LDOVKHDUIRUFHUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG
and
are calculated in the orthogonal x and y directions of
Required shear stresses
and
are the factored shear forces in the x and y directions, respectively,
is the crossthe section where
sectional dimension of the column perpendicular to the direction of analysis, and is the distance from the extreme
FRPSUHVVLRQ¿EHUWRWKHFHQWURLGRIWKHORQJLWXGLQDOWHQVLRQUHLQIRUFHPHQWLQWKHGLUHFWLRQRIDQDO\VLV VHH)LJXUH
and
in the x and y directions are determined by dividing the nominal shear strength,
Nominal shear stresses
FDOFXODWHGLQDFFRUGDQFHZLWK(TXDWLRQ E\WKHDSSOLFDEOH and in each direction.
$FFRUGLQJWR$&,LQWHUDFWLRQRIRUWKRJRQDOVKHDUIRUFHVQHHGQRWEHFRQVLGHUHGZKHUHRQH RUERWK RI
WKHIROORZLQJHTXDWLRQVLVVDWLV¿HG
(7.27)
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7-29
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿଶ ݀௬ ܿଵ ܸ௨ǡ௫ ‫ݒ‬௨ǡ௫ ൌ
ܸ௨ǡ௫
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Figure 7.20 Determination of biaxial shear stresses on a rectangular reinforced concrete column.
(7.28)
In cases where
and
WLRQ PXVWEHVDWLV¿HG
, then interaction effects must be considered and ACI Equa-
(7.29)
:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHFURVVVHFWLRQDOGLPHQVLRQVRIWKHFROXPQWKHDPRXQWRIVKHDUUHLQIRUFHment, or both must be increased accordingly. The compressive strength of the concrete can also be increased but
doing so does has a smaller impact than increasing the other items.
7.4.5 Nominal Torsional Strength
Columns in building structures are rarely subjected to torsional moments. In cases where such effects must be conVLGHUHGWKHSURYLVLRQVLQ$&,&KDSWHUPXVWEHVDWLV¿HG $&, 7KHVHUHTXLUHPHQWVDUHFRYHUHGLQ6HFWLRQV
DQGRIWKLVSXEOLFDWLRQIRUEHDPV
7.5 Reinforcement Limits
7.5.1 Longitudinal Reinforcement
For columns supporting only gravity loads or that are part of an ordinary or intermediate moment frame, the miniDUHJLYHQLQ$&,ZKLFKPXVWEHVDWLV¿HG
mum and maximum areas of longitudinal reinforcement,
regardless of the type of transverse reinforcement used in the column:
•
•
where
is the gross cross-sectional area of the column.
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7-30
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The 1 percent lower limit is meant to provide resistance to any bending moments not accounted for in the analysis,
such as those caused by construction tolerances or misalignments. It is also meant to help reduce creep and shrinkage
LQWKHFRQFUHWHXQGHUVXVWDLQHGFRPSUHVVLRQVWUHVVHV7KHXSSHUOLPLWVKHOSPLQLPL]HFRQJHVWLRQDQGWKHGHYHORSLVJHQHUDOO\PRUHHI¿FLHQW
PHQWRIKLJKHUVKHDUVWUHVVHVDOWKRXJKDWRSHUFHQWUHLQIRUFHPHQWUDWLR
DQGFRVWHIIHFWLYH VHH5HIHUHQFH $WORFDWLRQVZKHUHODSVSOLFHVDUHXVHGWKHORQJLWXGLQDOUHLQIRUFHPHQWUDWLRPD\
EHHTXDOWRWZRWLPHVWKDWDWVHFWLRQVDZD\IURPWKHODSVSOLFHDQGWKDWUDWLRPXVWEHOHVVWKDQRUHTXDOWR
$VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQDFROXPQLVSHUPLWWHGWREHGHVLJQHGRIVXI¿FLHQWVL]HWRFDUU\WKH
required factored loads and additional concrete is permitted to be added to the section without having to increase
WKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWWRVDWLVI\WKHSHUFHQWOLPLWLQ$&, $&, ,WLVDVVXPHG
WKHDGGLWLRQDOFRQFUHWHGRHVQRWFDUU\DQ\ORDGKRZHYHUWKHDFWXDOFROXPQVL]HPXVWEHLQFOXGHGLQWKHVWUXFWXUDO
analysis to correctly account for its actual stiffness. The columns in the upper stories of a building where the factored loads are usually relatively small can typically be designed using a reinforcement ratio less than 1 percent
based on this provision.
7.5.2 Shear Reinforcement
0LQLPXPVKHDUUHLQIRUFHPHQWUHTXLUHPHQWVIRUFROXPQVDUHJLYHQLQ$&,DQG(TXDWLRQ )RUFROXPQV
is equal to the area of all the ties and crossties within the spacing in the
with rectilinear ties and crossties,
is equal to two times the area of the
GLUHFWLRQRIDQDO\VLV $&, ,QWKHFDVHRIFLUFXODUWLHVRUVSLUDOV
tie or spiral within the spacing $&, 7.6 Sizing the Cross-Section
7.6.1 Axial Compression
For columns subjected primarily to axial compression forces, the cross-sectional dimension of a column can be
equal to the design axial load strength,
obtained by setting the total factored axial compression force,
JLYHQLQ$&,7DEOHIRUFROXPQVZLWKWLHVFRQIRUPLQJWR$&,DQGZLWKVSLUDOVFRQIRUPLQJWR$&,
>VHH(TXDWLRQV DQG LQWKLVSXEOLFDWLRQ@
• For tied columns:
(7.30)
• For spiral columns:
(7.31)
where
$SUHOLPLQDU\FROXPQVL]HFDQEHREWDLQHGIURPWKHVHHTXDWLRQVIRUDJLYHQ assuming a value of
QRWHGSUHYLRXVO\VKRXOGEHLQWKHUDQJHRIWRSHUFHQWWRDFKLHYHRYHUDOOHFRQRP\
which as
7KHFKDUWVLQ$SSHQGL[%RI5HIHUHQFHFDQEHXVHGWRTXLFNO\GHWHUPLQHDSUHOLPLQDU\VL]HRIDQRQVOHQGHUWLHG
with the following:
square column for a given factored axial compression force
• *UDGHRU*UDGHORQJLWXGLQDOUHLQIRUFHPHQW
• &RQFUHWHFRPSUHVVLYHVWUHQJWKVIURPSVLWRSVLDQG
• /RQJLWXGLQDOUHLQIRUFHPHQWUDWLRVIURPWRSHUFHQW
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7-31
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The required gross column area,
LVDOVRSURYLGHGLQWKHFKDUWVZKLFKFDQEHXVHGWRREWDLQSUHOLPLQDU\VL]HVIRU
UHFWDQJXODURUFLUFXODUWLHGFROXPQV$PRGL¿FDWLRQIDFWRULVJLYHQLQWKHDSSHQGL[WRDFTXLUHSUHOLPLQDU\VL]HVIRU
spiral columns.
7KHXVHRIKLJKVWUHQJWKORQJLWXGLQDOUHLQIRUFHPHQWFDQKDYHDQLPSDFWRQWKHUHTXLUHGVL]HRIDFROXPQDOWKRXJK
WKHLPSDFWLVJHQHUDOO\QRWVLJQL¿FDQWEHFDXVHWKHFRQFUHWHLVUHVLVWLQJPRVWRIWKHFRPSUHVVLRQIRUFH+RZHYHU
when high-strength longitudinal reinforcement is combined with high-strength concrete, less congestion in the structure and more usable space in the building may be possible.
7.6.2 Combined Moment and Axial Force
Columns that are part of the LFRS in a moment frame resisting the effects from lateral forces are typically subjected
to combined moment and axial forces. Because of the inherent complexity of this situation, a preliminary column
VL]HLVRIWHQGHWHUPLQHGEDVHGRQD[LDOFRPSUHVVLRQIRUFHVIURPJUDYLW\ORDGVRQO\DQGWKHQDGMXVWHGODWHULIUHquired, to account for the effects due to the bending moments.
$VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQ$SSHQGL[&LQ5HIHUHQFHFDQEHXVHGWRIDFLOLWDWHVHOHFWLRQRID
SUHOLPLQDU\FROXPQVL]HIRUFROXPQVVXEMHFWHGWRFRPELQHGPRPHQWDQGD[LDOIRUFHV
7.6.3 Slenderness Effects
Minimum column dimensions so that slenderness effects need not be considered as a function of the unsupported
DUHJLYHQLQ7DEOHRIWKLVSXEOLFDWLRQ
column length,
7.7 Determination of Required Reinforcement
7.7.1 Required Longitudinal Reinforcement
Axial Compression
For nonslender reinforced concrete columns subjected to primarily uniaxial compression forces, the required area
FDQEHREWDLQHGE\XVLQJWKHDSSURSULDWHHTXDWLRQVLQ$&,7DEOHIRU
of longitudinal reinforcement,
and are known, the following equation can be used to determine the
tied and spiral columns. Where
longitudinal reinforcement ratio
(7.32)
In this equation, the strength reduction, LVHTXDOWRIRUWLHGFROXPQVDQGIRUVSLUDOFROXPQVDQGWKH
term LVHTXDOWRIRUWLHGFROXPQVDQGIRUVSLUDOFROXPQV$VQRWHGSUHYLRXVO\RYHUDOOHFRQRP\LVW\SLFDOO\DFKLHYHGE\XVLQJDORQJLWXGLQDOUHLQIRUFHPHQWUDWLREHWZHHQDSSUR[LPDWHO\DQG
The charts in Appendix B of Reference 16 can be used to determine
for nonslender, tied and spiral columns.
Combined Moment and Axial Force
The required longitudinal reinforcement can be acquired from an interaction diagram for columns subjected to combined moment and axial force.
The design strength interaction diagrams in Appendix C of Reference 16 can be used to quickly obtain the longituGLQDOUHLQIRUFHPHQWIRUDZLGHUDQJHRIUHFWDQJXODUDQGFLUFXODUFROXPQVL]HVORQJLWXGLQDOEDUFRQ¿JXUDWLRQVDQG
material properties.
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7-32
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.7.2 Required Shear Reinforcement
For a given
the spacing of the transverse reinforcement in a column,
HTXDWLRQZKLFKLVEDVHGRQ(TXDWLRQ can be determined by the following
(7.33)
0D[LPXPVSDFLQJRIVKHDUUHLQIRUFHPHQWLQFROXPQVLVJLYHQLQ$&,DQGLVVXPPDUL]HGLQ7DEOH
Table 7.15 Maximum Spacing of Shear Reinforcement in Columns
Vs (lb)
For a given
Maximum Spacing, s (in.)
can be determined from the following equation:
(7.34)
For columns with rectilinear ties and crossties,
is equal to the area of all the ties and crossties within the spacing
LQWKHGLUHFWLRQRIDQDO\VLV $&, ,QWKHFDVHRIFLUFXODUWLHVRUVSLUDOV
is equal to two times the
area of the tie or spiral within the spacing $&, 7.8 Reinforcement Detailing
7.8.1 Concrete Cover
5HLQIRUFLQJEDUVDUHSODFHGLQFROXPQVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU
HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&, $&, )RUFROXPQVZLWKWLHVDQGVSLUDOV
concrete cover is measured from the surface of the concrete to the outer edge of the transverse reinforcement (see
)LJXUHIRUDUHFWDQJXODUFROXPQZLWKWLHVDQGFURVVWLHV 7KHPLQLPXPFRYHUWRWKHWUDQVYHUVHUHLQIRUFHPHQW
LVHTXDOWRLQIRUUHLQIRUFLQJEDUVLQFROXPQVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG $&,7DEOH
‘˜‡”
Figure 7.21 Concrete cover for columns.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.8.2 Minimum Number of Longitudinal Bars
7KHPLQLPXPQXPEHURIORQJLWXGLQDOEDUVWKDWPXVWEHSURYLGHGLQDFROXPQLVJLYHQLQ$&,DQGLVEDVHG
RQWKHW\SHRIWUDQVYHUVHUHLQIRUFHPHQW VHH7DEOH Table 7.16 Minimum Number of Longitudinal Bars in a Reinforced Concrete Column
Type of Transverse Reinforcement
Minimum Number of Longitudinal Bars
Triangular ties
3
Rectangular or circular ties
Spirals
6
7.8.3 Spacing of Longitudinal Bars
0LQLPXPFOHDUVSDFLQJEHWZHHQWKHORQJLWXGLQDOEDUVLQDFROXPQPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&,
$&, 7KHSXUSRVHRIWKHVHUHTXLUHPHQWVLVWRHQVXUHFRQFUHWHFDQHDVLO\ÀRZEHWZHHQWKHORQJLWXGLQDO
bars. The minimum clear spacing,
between longitudinal bars at any section in a column, including at locations
ZKHUHODSVSOLFHVDUHXVHGLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ VHH$&,DQG)LJXUH (7.35)
In this equation,
is the diameter of the longitudinal bars in the column and
coarse aggregate in the concrete mix.
LVWKHQRPLQDOPD[LPXPVL]HRI
‫ݏ‬௠௜௡ GE ‫ۓ‬
ۖ
ͳǤͷ‹Ǥ
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Figure 7.22 Minimum clear spacing requirements for longitudinal bars in columns.
Minimum face dimensions of rectangular, tied columns with bearing splices, normal lap splices, and tangential lap
VSOLFHVEDVHGRQWKHPLQLPXPFOHDUVSDFLQJUHTXLUHPHQWVRI$&,DUHJLYHQLQ7DEOH VHH)LJXUHIRU
WKHVSOLFHW\SHV 7KHIDFHGLPHQVLRQVDUHURXQGHGXSWRWKHQHDUHVWLQFKDQGDUHEDVHGRQD LQFOHDUFRYHU
WRWLHVIRUWKURXJKORQJLWXGLQDOEDUVDQGWLHVIRUDQGORQJLWXGLQDOEDUVDQG DPD[LPXPDJJUHJDWHVL]HRILQ
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ƒ ‡†‹‡•‹‘ǡܾ
ƒ ‡†‹‡•‹‘ǡܾ
ܿ
ƒ ‡†‹‡•‹‘ǡܾ
ܿ
ܿ
݀௧௜௘ ‫ݏ‬௠௜௡ ݀௧௜௘ ݀௕ ݀௧௜௘ ‫ݏ‬௠௜௡ ݀௕ ‫ݏ‬௠௜௡ (a)
(b)
݀௕ (c)
Figure 7.23 Column splices for columns with longitudinal bars arranged in a rectangular pattern.
(a) Bearing. (b) Normal lap. (c) Tangential lap.
Table 7.17 Minimum Face Dimensions (Inches) of Rectangular, Tied Columns
Number of Bars Per Face
Splice Type
Bearing
Normal
Lap
Tangential
Lap
Bar Size
2
3
4
5
6
7
8
9
10
9
11
13
18
9
16
18
19
8
13
18
8
11
19
31
9
18
31
9
13
16
31
38
19
36
18
8
16
18
8
11
13
9
11
13
16
18
9
19
13
18
33
36
11
18
33
36
13
38
16
38
61
8
13
16
19
8
11
8
18
31
9
16
19
33
9
13
33
19
11
16
31
36
18
36
38
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHPLQLPXPIDFHGLPHQVLRQVIRUDQ\UHFWDQJXODUWLHGFROXPQ
7KHLWHPVLQ7DEOHIRUQRUPDOODSVSOLFHVDUHGH¿QHGLQ)LJXUH
Table 7.18 Equations to Determine Minimum Face Dimensions (Inches) of Rectangular, Tied Columns
Minimum Face Dimension, b*
Splice Type
Bearing
where
Normal Lap
Tangential Lap
‫ݏ‬ҧ
ܽത
ܿ
݀௧௜௘ ݀௕ Ȁʹ
ܽത
ߠ
ܾത
‫ݏ‬௠௜௡ ݀௕ ሺ–›’Ǥ ሻ
ܽത ൌ ݀௕ Ȁξʹ
ߠ ൌ ƒ” •‹ሾܾതȀሺ‫ݏ‬௠௜௡ ൅ ݀௕ ሻሿ
ܾത ൌ ሺͳ െ ͳȀξʹሻ݀௕ ‫ݏ‬ҧ ൌ ሺ‫ݏ‬௠௜௡ ൅ ݀௕ ሻ ‘• ߠ ൅ ܽത
ͳǤͷ‹Ǥ
‫ݏ‬௠௜௡ ൌ ‰”‡ƒ–‡”‘ˆ ͳǤͷ݀௕ ‫۔‬
ۖ
‫ە‬ሺͶΤ͵ሻ ݀௔௚௚
‫ۓ‬
ۖ
Figure 7.24 Items for normal lap splices.
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7-36
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݀௕ ሺ–›’Ǥሻ
݀௕ ሺ–›’Ǥሻ
ܿ
ܿ
݀௧௜௘ ‘”݀௦ ݀௧௜௘ ‘”݀௦ ‫ݏ‬ҧ
‫ݏ‬௠௜௡ ‫ݏ‬ҧ
‫ݏ‬௠௜௡ ߠ
ߠ
ߠ
ߠ
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ͳ ͵͸Ͳ
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ܿ
ͳ ͵͸Ͳ
൬
൰
ʹ ݊
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‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕ ‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕ (a)
(b)
݀௕ ሺ–›’Ǥሻ
ܿ
”݀௦ ‫ݏ‬ҧ
‫ݏ‬௠௜௡ ߠ
ߠ
݄Ȁʹ
ߠ ൌ ʹƒ” •‹ ൤
݀௕
൨
ܿ
݊ െ ʹሺܿ ൅ ݀௧௜௘ ሻ െ ݀௕
‫ݏ‬ҧ ൌ ‫ݏ‬௠௜௡ ൅ ݀௕ (c)
Figure 7.25 Column splices for columns with longitudinal bars arranged in a circular pattern.
(a) Bearing. (b) Normal lap. (c) Tangential lap.
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7-37
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The maximum number of longitudinal bars in a column where the bars are arranged in a circle with bearing splices,
QRUPDOODSVSOLFHVDQGWDQJHQWLDOODSVSOLFHVLVJLYHQLQ7DEOH VHH)LJXUHIRUWKHVSOLFHW\SHVZKHUH
is the spiral diameter). In addition to satisfying the minimum clear spacing requirements
is the tie diameter and
RI$&,WKHUHTXLUHPHQWVIRUPLQLPXPDQGPD[LPXPDUHDRIORQJLWXGLQDOUHLQIRUFHPHQWLQ$&,
DUHDOVRVDWLV¿HGDWVHFWLRQVDZD\IURPODSVSOLFHORFDWLRQVZKHUHDOOWKHEDUVDUHVSOLFHG+RZHYHULQPDQ\FDVHV
HLWKHUWKHSURYLGHGUHLQIRUFHPHQWUDWLRLVPXFKODUJHUWKDQWKHUHFRPPHQGHGPD[LPXPYDOXHRISHUFHQWRUWKHUH
is an inordinately large number of smaller longitudinal bars. The number of bars have been rounded down to the
QHDUHVWZKROHQXPEHUDQGDUHEDVHGRQDLQFOHDUFRYHUWRWLHVRUVSLUDOV
Table 7.19 Maximum Number of Longitudinal Bars in Columns with Longitudinal Bars Arranged in a Circle
Bar Size
Splice Type
Bearing
Normal
Lap
h (in.)
#5
#6
#7
#8
#9
#10
#11
#14
#18
9
9
8
6
ņ
13
11
11
9
8
ņ
16
16
13
11
9
18
19
18
16
11
8
19
18
16
6
18
16
11
18
16
13
9
31
18
16
11
33
31
13
38
33
19
38
36
36
38
16
38
36
38
18
36
19
38
61
36
31
8
6
6
ņ
ņ
ņ
ņ
11
9
8
ņ
ņ
16
13
11
9
6
ņ
18
16
13
11
9
8
6
ņ
19
16
13
11
18
16
13
9
18
13
8
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.19 Maximum Number of Longitudinal Bars in Columns with Longitudinal Bars Arranged in a Circle (cont.)
Bar Size
Splice Type
Normal
Lap
(cont.)
Tangential
Lap
h (in.)
#5
#6
#7
#8
#9
#10
#11
#14
#18
19
13
9
33
19
38
33
31
16
11
38
36
33
18
36
38
36
31
19
13
38
38
36
31
16
38
33
31
18
33
19
39
8
6
6
ņ
ņ
9
8
6
6
ņ
16
11
9
8
6
ņ
18
13
11
8
6
13
11
9
19
18
16
13
11
8
18
13
11
9
6
18
16
13
18
16
11
8
19
9
31
18
16
13
33
18
36
19
16
11
38
38
31
36
33
31
18
13
38
33
19
39
36
38
16
*Indicates the tabulated value is not applicable to columns where a minimum of 6 longitudinal bars are required.
**Indicates the longitudinal reinforcement ratio is less than 1 percent.
A dash (ņ) indicates the maximum number of longitudinal bars is less than 4.
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7-39
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHHTXDWLRQVLQ7DEOHFDQEHXVHGWRGHWHUPLQHWKHPD[LPXPQXPEHURIORQJLWXGLQDOEDUVIRUDQ\FROXPQ
where the longitudinal reinforcement is arranged in a circle.
Table 7.20 Equations to Determine the Maximum Number of Longitudinal Bars in a Column with Longitudinal Bars Arranged in a Circle
Maximum Number of Longitudinal Bars, n*
Splice Type
Bearing
Normal Lap
Lap Splice Length
Tangential Lap
Offset Bent
Longitudinal Bar (typ.)
max. 6
1
Figure 7.26 Offset bent longitudinal reinforcement in a column.
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7-40
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.8.4 Offset Bent Longitudinal Reinforcement
Offset bent longitudinal reinforcement is typically provided at locations where the longitudinal bars are spliced in a
FROXPQ VHH)LJXUH $FFRUGLQJWR$&,WKHVORSHRIWKHLQFOLQHGSRUWLRQRIDQRIIVHWEHQWEDUUHODWLYH
to the longitudinal axis of the column must not exceed 1 to 6, and the portions of the bar above and below the offset
must be parallel to the axis of the column.
In cases where the face of a column is offset by 3 in. or more, offset bent longitudinal reinforcement is not permitted
$&, ,QVWHDGWKHRIIVHWORQJLWXGLQDOUHLQIRUFLQJEDUVRQHDFKIDFHPXVWEHODSVSOLFHGZLWKVHSDUDWHGRZHO
EDUV6KRZQLQ)LJXUHLVWKHFDVHZKHUHVHSDUDWHGRZHOEDUVPXVWEHSURYLGHGDWDORFDWLRQZKHUHDFKDQJHLQ
column cross-section occurs.
൒ ͵ԣ
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Figure 7.27 Separate dowel bars at offset column faces.
7.8.5 Splices of Longitudinal Reinforcement
Overview
6SOLFHUHTXLUHPHQWVIRUORQJLWXGLQDOUHLQIRUFHPHQWLQFROXPQVDUHJLYHQLQ$&,IRUODSVSOLFHVPHFKDQLFDO
splices, butt-welded splices, and end-bearing splices. Splice lengths must be provided that satisfy the factored load
combinations on a column. Where the stress in all the longitudinal bars due to factored loads is compressive, all the
splice types listed above may be used. All but end-bearing splices are permitted when the longitudinal bar stress is
WHQVLOH5HTXLUHPHQWVIRUVSOLFHVDUHJLYHQLQ$&, $&, Lap Splices
Lap splices, which are the most popular and usually the most economical type of splices used in columns, are permitted to occur immediately above the top of the slab for columns that are not part of special moment frames (see
)LJXUH 7KLVLVWKHSUHIHUUHGORFDWLRQIRUHDVHRIFRQVWUXFWLRQ7KHORQJLWXGLQDOEDUVIURPWKHFROXPQEHORZ
extend above the slab a distance greater than or equal to the required lap splice length, and the longitudinal bars in
WKHFROXPQDERYHDUHWLHGWRWKHVHEDUVDIWHUWKHÀRRUEHORZKDVEHHQFRQVWUXFWHG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
/DSVSOLFHVDUHQRWSHUPLWWHGIRUEDUVODUJHUWKDQEDUVH[FHSWLQWKHFDVHRIFRPSUHVVLRQODSVSOLFHVZKHUH
RUEDUVDUHSHUPLWWHGWREHVSOLFHGWRRUVPDOOHUEDUV $&,DQG The type of lap splice that must be used (compressive or tensile) depends on the stress in the longitudinal bars due
WRWKHIDFWRUHGORDGFRPELQDWLRQV VHH)LJXUH 5HTXLUHGODSVSOLFHOHQJWKVIRUWKHWKUHH]RQHVLQGLFDWHGLQ
)LJXUHDUHJLYHQLQ7DEOH
‫ܔ܉ܑܠۯ‬۴‫܍܋ܚܗ‬
݂௦ ൌ Ͳ
݂௦ ൌ ͲǤͷ݂௬ ‫ͳ܍ܖܗ܈‬
ͳ
ʹ
‫ʹ܍ܖܗ܈‬
‫͵܍ܖܗ܈‬
͵
۰‫ܜܖ܍ܕܗۻ܏ܖܑ܌ܖ܍‬
Figure 7.28 Splice requirements for reinforced concrete columns.
݄ଶ ݄ଵ ‹”‡ –‹‘‘ˆƒƒŽ›•‹•
͵‫ܣ‬௧௜௘ ൒ ͲǤͲͲͳͷ݄ଶ ‫ݏ‬
‫ܣ‬௧௜௘ ൌ ƒ”‡ƒ‘ˆ–‹‡„ƒ”
‹”‡ –‹‘‘ˆƒƒŽ›•‹•
ʹ‫ܣ‬௧௜௘ ൒ ͲǤͲͲͳͷ݄ଵ ‫ݏ‬
Figure 7.29 Application of ACI 10.7.5.2.1(a).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.21 Required Lap Splices Lengths
Zone
Stress in
Longitudinal
Bars
1
All longitudinal bars are in
compression
3
Stress in the
longitudinal
bars on the
tension face
is tensile and
Stress in the
longitudinal
bars on the
tension face
is tensile and
Minimum Lap Splice Length
Class A tension lap splice
Notes
ACI
Section No.
±
±
±
Class B tension lap splice length
Class B tension lap splice length
Notes:
1. For tied columns where ties throughout the lap splice length have an effective area
in both directions,
is permitted to be multiplied by 0.83. Tie legs perpendicular to are considered in calculating the effective tie area [ACI 10.7.5.2.1(a); see
Figure 7.29]. For spiral columns where spirals satisfying ACI 25.7.3 are provided throughout the lap splice length,
is permitted
to be multiplied by 0.75 [ACI 10.7.5.2.1(b)].
must be increased by one-third [ACI 25.5.5.1].
2. For
3. Compression lap splices must not be used for bars larger than #11 except as permitted in ACI 25.5.5.3 [ACI 25.5.5.2].
4. Compression lap splices of #14 and #18 bars to #11 or smaller bars are permitted in accordance with ACI 25.5.5.4 (see Note 5)
[ACI 25.5.5.3].
5. Where bars of different size are lap spliced in compression,
is the longer of (a)
of the larger bar calculated by ACI 25.4.9.1
of the smaller bar calculated by ACI 25.5.5.1 [ACI 25.5.5.4].
(see below) and (b)
where
(see Table 7.22)
6.
7.
Reduction of development length in accordance with ACI 25.4.10.1 is not permitted in calculating lap splice lengths (ACI
25.5.1.4).
8. Requirements for Class A and Class B tension lap splices are given in ACI Table 10.7.5.2.2 (see Table 7.23).
is determined in accordance with ACI 25.4.2.1(a).
9. Tension development length
is the greater of (a)
of the larger bar and (b)
of the smaller bar
10. Where bars of different size are lap spliced together,
(ACI 25.5.2.2).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.22 Modification Factors for Deformed Bars in Compression
Modification Factor
Condition
Value of Factor
Lightweight,
Lightweight concrete
Normalweight concrete
&RQ¿QLQJUHLQIRUFHPHQW
Longitudinal reinforcement enclosed in the following:
1. A spiral
$
FLUFXODUFRQWLQXRXVO\ZRXQGWLHZLWK
and a pitch
WLHVLQDFFRUGDQFHZLWK$&,VSDFHG
on center
+
RRSVLQDFFRUGDQFHZLWK$&,VSDFHG
on center
Other
Table 7.23 Requirements for Class A and Class B Tension Lap Splices
Stress in the longitudinal
bars on the tension face
*Tension development length
Splice Details
Splice Type
There are
percent of the longitudinal bars spliced at any section and the
lap splices on adjacent bars are staggered
by at least VHH)LJXUH
Class A
Other
Class B
All cases
Class B
Tension Lap Splice Length, ℓst*
is determined in accordance with ACI 25.4.2.1(a).
3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHOopment length, LVGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU
UHTXLUHPHQWVDUHFRYHUHG¿UVW
Method 1 – ACI 25.4.2.4
The development length in tension of a deformed reinforcing bar,
LVGHWHUPLQHGE\$&,(TXDWLRQ D (7.36)
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7-44
Stagger distance ൒ κௗ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Class A lap
splice length
Clear space to determine κௗ
݊௕,௦௣௟௜௖௘ௗ ൑ 0.5݊௕,௧௢௧௔௟ Figure 7.30 Class A Tension Lap Splice
7KHIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOHWKHVHIDFWRUVDUHJLYHQLQ7DEOHDORQJZLWK
must be taken less than or equal
GH¿QLWLRQVIRUWKHRWKHUWHUPVLQ(TXDWLRQ 7KHFRQ¿QLQJWHUP
WRLQ(TXDWLRQ >$&,@
Table 7.24 Factors for Development of Deformed Bars in Tension
Factor
Lightweight,
Reinforcement
grade,
Epoxy,
6L]H
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
*UDGHRU*UDGH
*UDGH
*UDGH
1.3
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG
or clear
reinforcement with clear cover
spacing
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHG
reinforcement for all other conditions
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQforcement
DQGODUJHUEDUV
DQGVPDOOHUEDUV
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.24 Factors for Development of Deformed Bars in Tension (cont.)
Factor
Casting
position,
Condition
Value of Factor
0RUHWKDQLQRIIUHVKFRQFUHWHSODFHG
EHORZKRUL]RQWDOUHLQIRUFHPHQW
1.3
Other
All
Lesser of:
1. The distance from the center of a longitudinal bar to the nearest concrete surface.
2
QHKDOIWKHFHQWHUWRFHQWHUVSDFLQJRI
the longitudinal bars being developed (see
)LJXUH Spacing or
cover
dimension,
Transverse
reinforcement
index,
where:
total cross-sectional area of all
1.
transverse reinforcement within a spacing
that crosses the potential plane of splitting through the longitudinal reinforcement
being developed
center-to-center spacing of the trans verse reinforcement
number of bars being developed
3.
along the plane of splitting
All
*The product
need not exceed 1.7.
**It is permitted to conservatively use
if transverse reinforcement is present or required. For reinforcing bars with
center, transverse reinforcement must be provided along the development length such that
(ACI 25.4.2.2).
spaced closer than 6.0 in. on
Method 2 – ACI 25.4.2.3
7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHTwo sets of spacing and cover cases are given in ACI Table
VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿଵ ݀௕ ܿଶ ‫ݏ‬
ܿଵ
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ۖ
ܿ௕ ൌ Ž‡••‡”‘ˆ ܿଶ ‫ ۔‬
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‫ݏە‬Ȁʹ
Figure 7.31 Spacing or cover dimension, cb.
Table 7.25 Tension Development Length, ℓd, for Deformed Bars in Accordance with ACI 25.4.2.3
Spacing and Cover
#6 and Smaller Bars
#7 and Larger Bars
Condition 1
1. Clear spacing of bars being developed or lap
spliced
&OHDUFRYHU
3. Stirrups or ties throughout
not less than the
required minimum
Case 1
&RQGLWLRQ
1. Clear spacing of bars being developed or lap
spliced
&OHDUFRYHU
&DVH
Other conditions
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
End-Bearing Splices
End-bearing splices, which are also referred to as compression-only mechanical splices, are permitted to be used to
splice the longitudinal bars in a column where the force in all the bars is compressive for all load combinations (ACI
&RPSUHVVLRQIRUFHVDUHWUDQVPLWWHGIURPRQHORQJLWXGLQDOEDUWRDQRWKHUE\EHDULQJRIVTXDUHFXWHQGV
KHOGLQFRQFHQWULFFRQWDFWE\DVXLWDEOHSURSULHWDU\GHYLFH7ROHUDQFHVIRUWKHEDUHQGVDUHJLYHQLQ$&,
End-bearing splices are permitted to be used only in columns where closed ties, closed stirrups, spirals, or hoops are
SURYLGHG $&, 7KLVHQVXUHVPLQLPXPVKHDUUHVLVWDQFHLVSURYLGHGLQWKHPHPEHU
$FFRUGLQJWR$&,HQGEHDULQJVSOLFHVPXVWKDYHDWHQVLOHVWUHQJWKRIDWOHDVWSHUFHQWRI of the continuing longitudinal reinforcement area on each face of a column. This is typically achieved by either staggering the
location of the end-bearing splices or adding additional longitudinal reinforcement through the splice locations.
Additional information and various types of end-bearing splices can be found in Reference 9.
Mechanical and Welded Splices
0HFKDQLFDORUZHOGHGVSOLFHVDUHSHUPLWWHGLQFROXPQVDQGPXVWPHHWWKHUHTXLUHPHQWVRI$&,7KHVHW\SHV
of splices may be used in both tension and compression.
A variety of proprietary mechanical devices are available that can be used to splice longitudinal reinforcing bars in
DFROXPQ VHH5HIHUHQFH 7KHVHW\SHVRIVSOLFHVDUHFRPPRQO\VSHFL¿HGDWORFDWLRQVZKHUHLQRUGLQDWHO\ORQJODS
splices would be required or where lap splices would cause congestion. Mechanical splices must be able to develop
LQWHQVLRQRUFRPSUHVVLRQDWOHDVWSHUFHQWRIWKH\LHOGVWUHQJWKRIWKHORQJLWXGLQDOEDUV
of the longitudinal bars. These types of splices are primarily
Welded splices must also be able to develop
LQWHQGHGIRUEDUVDQGODUJHU:HOGLQJRIUHLQIRUFLQJEDUVPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&,
0HFKDQLFDORUZHOGHGVSOLFHVLQFROXPQVQHHGQRWEHVWDJJHUHG $&, 7.8.6 Transverse Reinforcement
Overview
The main purposes of transverse reinforcement in columns are to provide lateral support of the longitudinal reinIRUFHPHQWDQGWRSURYLGHVKHDUUHVLVWDQFHZKHUHUHTXLUHG7KHPD[LPXPVSDFLQJUHTXLUHPHQWVLQ$&,IRU
WLHVDQGLQ$&,IRUVSLUDOVDUHPHDQWWRDFKLHYHWKHVHSXUSRVHV$OVRLQFOXGHGLQWKHVHVHFWLRQVDUHPLQLPXP
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7KHW\SHVRIWUDQVYHUVHUHLQIRUFHPHQWW\SLFDOO\XVHGLQFROXPQVDUH WLHV UHFWLOLQHDURUFLUFXODU VSLUDOVRU
KRRSV5HTXLUHPHQWVIRUVL]HDQGVSDFLQJRIWLHVDQGVSLUDOVDUHFRYHUHGEHORZ
Tie Reinforcement
5HTXLUHPHQWVIRUFROXPQVZLWKWLHUHLQIRUFHPHQWDUHJLYHQLQ$&,DQG7LHVFRQVLVWRIDFORVHGORRS
RIGHIRUPHGUHLQIRUFLQJEDUV>VHH)LJXUH D IRUUHFWLOLQHDUWLHV@0LQLPXPLQVLGHEHQGGLDPHWHUVDQGVWDQGDUG
KRRNJHRPHWU\IRUWLHVDUHJLYHQLQ$&,7DEOH,QIRUPDWLRQIRUGHJUHHKRRNVLVLQFOXGHGLQWKDWWDEOHEXW
VXFKKRRNVDUHQRWUHFRPPHQGHGPDLQO\GXHWRWKHGLI¿FXOWLHVDVVRFLDWHGZLWKSODFLQJWKHPLQWKH¿HOG
7KHUHTXLUHPHQWVIRUPLQLPXPDQGPD[LPXPWLHVSDFLQJ $&, DQGPLQLPXPWLHEDUVL]HEDVHGRQWKH
VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ $&, DUHJLYHQLQ)LJXUH7DEXODWHGYDOXHVRIWKHPD[LPXPFHQWHUWRFHQWHUWLHVSDFLQJDUHDOVRSURYLGHGLQ)LJXUHIRUYDULRXVFROXPQORQJLWXGLQDOEDUVL]HVWKH
smallest cross-sectional dimension of the column must be compared to the appropriate tabulated value and the lesser
of the two is to be used for the tie spacing.
7KHSURYLVLRQVLQ$&,SHUWDLQWRUHFWLOLQHDUWLHFRQ¿JXUDWLRQVDQGWKHPD[LPXPFOHDUVSDFLQJSHUPLWWHG
between laterally supported longitudinal bars in a column. Corner bars and every other longitudinal bar in the sec@Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܿଵ ܿଶ ܿଵ ൏ ܿଶ ݀௕ሺ௖௢௟௨௠௡ሻ ݀௕ሺ௧௜௘ሻ ͳ͸݀௕ሺ௖௢௟௨௠௡ሻ
‫ ݏ‬൑ Ž‡ƒ•–‘ˆ Ͷͺ݀௕ሺ௧௜௘ሻ ‫۔‬
ۖ
‫ܿ ە‬ଵ ൒ ሺͶΤ͵ሻ݀௔௚௚ ൅ ݀௕ሺ௧௜௘ሻ ‫ۓ‬
ۖ
൑ ‫ݏ‬Ȁʹ
‹‡ƒ”
͓͵
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ƒšǤ࢙ሺ‹Ǥሻȗ
͓ͷ
ͳͲ
͓͸
ͳʹ
͓͹
ͳͶ
͓ͺ
ͳ͸
͓ͻ
ͳͺ
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ʹͶ
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ʹͶ
ȗƒšǤ‫ ݏ‬൑ ܿଵ —•–ƒŽ•‘„‡ ‘•‹†‡”‡†
Figure 7.32 Tie requirements for reinforced concrete columns.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
WLRQPXVWKDYHODWHUDOVXSSRUWSURYLGHGE\WKHFRUQHURIDWLHWKDWKDVDKRRNRIQRWPRUHWKDQGHJUHHVZKHUH
the clear spacing between the longitudinal bars on each side is less than or equal to 6 in. This is illustrated in FigXUH D IRUWKHFDVHRIDVTXDUHWLHGFROXPQZLWKORQJLWXGLQDOEDUV%HFDXVHWKHFOHDUVSDFLQJEHWZHHQWKH
EDUVLVOHVVWKDQLQFURVVWLHVZLWKGHJUHHDQGGHJUHHKRRNVRQHDFKHQGDUHSURYLGHGDURXQGHYHU\RWKHU
ORQJLWXGLQDOEDU7KHFOHDUVSDFHEHWZHHQORQJLWXGLQDOEDUVLVJUHDWHUWKDQLQIRUWKHFROXPQLQ)LJXUH E ,Q
WKLVFDVHFURVVWLHVDUHUHTXLUHGDURXQGHYHU\ORQJLWXGLQDOEDULQDFFRUGDQFHZLWK$&, E >VHH$&,)LJXUH
5D@
൑ ͸ԣሺ–›’Ǥሻ
൐ ͸ԣሺ–›’Ǥሻ
”‘••–‹‡ሺ–›’Ǥሻ
ሺƒሻ
”‘••–‹‡ሺ–›’Ǥሻ
ሺ„ሻ
Figure 7.33 Lateral support requirements for longitudinal bars in tied columns.
$GGLWLRQDOUHTXLUHPHQWVIRUODWHUDOVXSSRUWRIORQJLWXGLQDOEDUVZLWKWLHVDUHJLYHQLQ$&,7KHVHUHTXLUHPHQWVDUHJLYHQLQ)LJXUH
Circular ties may be used in cases where the longitudinal bars are located around the perimeter of a circle, like in
FLUFXODURUVTXDUHFROXPQV $&, $QFKRUDJHUHTXLUHPHQWVIRUFLUFXODUWLHVDUHJLYHQLQ$&, VHH
$&,)LJXUH5 Spiral Reinforcement
5HTXLUHPHQWVIRUFROXPQVZLWKVSLUDOUHLQIRUFHPHQWDUHJLYHQLQ$&,DQG6WDQGDUGVSLUDOVL]HV
DUHWRDQGWKHFOHDUVSDFLQJEHWZHHQFRQVHFXWLYHWXUQVRQDVSLUDOPXVWQRWH[FHHGLQRUEHOHVVWKDQWKH
used in the concrete mix.
JUHDWHURILQRU WLPHVWKHGLDPHWHURIWKHODUJHVWDJJUHJDWHVL]H
The minimum volumetric spiral reinforcement ratio,
LVJLYHQE\$&,(TXDWLRQ (7.37)
In this equation,
is the area of the column core enclosed by the spiral, which is equal to
where
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7-50
is
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.2ሻ
Offset bent bars ሺACI 10.7.4ሻ
Additional ties within 6ԣ of bend
ሺACI 10.7.6.4ሻ
Lap splice
Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ
൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.1ሻ
Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ
൑ ‫ݏ‬Τ2 ሺACI 10.7.6.2.1ሻ
൑ 3ᇳ ሺACI 10.7.6.2.2ሻ
Additional ties within 6ԣ of bend
ሺACI 10.7.6.4ሻ
Beams on all four sides of column
Tie spacing ‫ ݏ‬ሺACI 25.7.2.1; see Figure 7.32ሻ
Figure 7.34 Tie and splice details for reinforced concrete columns.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‘ ”‡–‡ ‘”‡
’‹”ƒŽ„ƒ”
’‹”ƒŽ’‹– Šǡ‫ݏ‬
‫ܦ‬௖௛ ‘˜‡”ሺ–›’Ǥ ሻ
݄
Figure 7.35 Spiral reinforcement.
The volumetric spiral reinforcement ratio,
is equal to the following:
(7.38)
where
is the area of the spiral reinforcing bar and
monly referred to as the spiral pitch).
An equation for
is the center-to-center spacing of the spirals (which is com-
can be obtained by substituting Equation (7.38) into Equation (7.37):
(7.39)
The limitations in ACI 25.7.3.1 must also be considered when determining
Recommended standard spirals for circular columns with various reinforcement yield strengths and concrete compressive strengths are given in Table 7.26.
Table 7.26 Recommended Standard Spirals for Circular Columns
fyt (psi)
Iүc (psi)
Column Size (in.)
Spiral Size and Pitch
12 – 24
#3 @Ǝ
26 – 48
#3 @Ǝ
12 – 16
#4 @Ǝ
18 – 48
#4 @Ǝ
10,000
20 – 48
#5 @Ǝ
14,000
36 – 48
#5 @Ǝ
4,000
60,000
7,000
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.26 Recommended Standard Spirals for Circular Columns (cont.)
fyt (psi)
Iүc (psi)
Column Size (in.)
Spiral Size and Pitch
±
#Ǝ
±
#Ǝ
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
±
#Ǝ
6SLUDOVPXVWEHDQFKRUHGDWHDFKHQGE\SURYLGLQJRQHDQGRQHKDOIH[WUDWXUQVRIWKHVSLUDOEDU $&,VHH
$&,)LJXUH5 5HTXLUHPHQWVIRUVSLUDOVSOLFHVDUHJLYHQLQ$&,0HFKDQLFDORUZHOGHGVSOLFHVLQDFFRUGDQFHZLWK$&,
Lap splice lengths must
DUHSHUPLWWHGDVDUHODSVSOLFHVLQDFFRUGDQFHZLWK$&,IRU
EHWKHJUHDWHURILQRUWKHODSOHQJWKGHWHUPLQHGE\$&,7DEOHZKLFKLVHTXDOWRDPXOWLSOHRIWKHGLDPHWHURIWKHVSLUDOEDU/DSVSOLFHOHQJWKVDUHJLYHQLQ$&,7DEOHIRUXQFRDWHGEDUVZLWKRXWKRRNVDQGIRU
coated bars with and without hooks and are based on HTXDOWRSVLUHJDUGOHVVRIWKHDFWXDO\LHOGVWUHQJWK
RIWKHEDU>VHH$&, E @
• 8QFRDWHGRU]LQFFRDWHGJDOYDQL]HGGHIRUPHGEDUVZLWKRXWKRRNV/DSOHQJWK
• (SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGGHIRUPHGEDUVZLWKRXWKRRNV/DSOHQJWK
• (SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGGHIRUPHGEDUVZLWKDVWDQGDUGKRRNFRQIRUPLQJWR$&,
Lap length
Requirements for lateral support of longitudinal bars using spirals at the top and bottom of a story are given in ACI
DQGDUHVXPPDUL]HGLQ)LJXUH
7.9 Connections to Foundations
7.9.1 Overview
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reinforcement. Tension forces must be transferred entirely by reinforcement, which may consist of extended longituGLQDOEDUVGRZHOVDQFKRUEROWVRUPHFKDQLFDOFRQQHFWRUV+RUL]RQWDOIRUFHVDUHWUDQVIHUUHGXVLQJWKHVKHDUIULFWLRQ
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
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™Š‡”‡„‡ƒ•†‘‘–ˆ”ƒ‡‹–‘
ƒŽŽ•‹†‡•‘ˆ–Š‡ ‘Ž—
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”‡‹ˆ‘” ‡‡–‹–Š‡‡„‡”•—’’‘”–‡†ƒ„‘˜‡
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ʹ݄
‘Ž— ƒ’‹–ƒŽ
š–‡†•’‹”ƒŽ–‘–Š‡Ž‡˜‡Žƒ–™Š‹ Š–Š‡†‹ƒ‡–‡”‘”
™‹†–Š‘ˆ–Š‡ ƒ’‹–ƒŽ‹•–™‹ ‡–Šƒ–‘ˆ–Š‡ ‘Ž—
Figure 7.36 Lateral support of longitudinal bars using spirals.
7.9.2 Vertical Transfer
Compression
:KHUHYHUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGWRDIRXQGDWLRQEHDULQJVWUHQJWKUHTXLUHPHQWVRI$&,PXVW
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must be less than or equal to the design bearing strength,
where the nominal bearing strength,
force,
LVJLYHQLQ$&,7DEOH
(7.40)
In this equation,
LVHTXDOWRIRUEHDULQJ $&,7DEOH DQG
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7-54
is the gross area of the column.
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUEHDULQJRQDIRXQGDWLRQZLGHURQDOOVLGHVWKDQWKHORDGHGDUHDWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG
(7.41)
The term
is the area of the lower base of the largest frustrum of a pyramid, cone, or tapered wedge contained
and having side slopes of 1 vertical to
wholly within the foundation and having for its upper base the loaded area
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Figure 7.37 Determination of areas for bearing strength.
Where
the excess compression stress from the column must be transferred by reinforcement to the founis determined by the following equation:
dation. The required area of interface reinforcement,
(7.42)
In this equation,
is the gross area of the column. The minimum amount of interface reinforcement,
$&, be provided even where
must
Dowel bars emanating from the foundation are the type of interface reinforcement commonly used because of ease
of construction. The dowel bars are set in the foundation prior to casting the foundation concrete and are subsequently spliced to the column longitudinal bars. Dowels should be positioned so as not to interfere with the longitudinal bars or the tie hooks in the column.
,OOXVWUDWHGLQ)LJXUHDUHGRZHOEDUVDFURVVWKHLQWHUIDFHEHWZHHQDUHLQIRUFHGFRQFUHWHFROXPQDQGVSUHDGIRRWing where all the longitudinal bars in the column are in compression for all factored load combinations (Zone 1 in
deter)LJXUH 7KHGRZHOEDUVPXVWH[WHQGLQWRWKHIRRWLQJDWOHDVWDFRPSUHVVLRQGHYHORSPHQWOHQJWK
PLQHGLQDFFRUGDQFHZLWK$&, VHH6HFWLRQRIWKLVSXEOLFDWLRQ ,WLVQRWUHTXLUHGWRSURYLGHGRZHO
bars for all the longitudinal bars in such cases.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
൒ κௗ௖ ‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ –ƒ†ƒ”†Š‘‘
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Figure 7.38 Dowel bars at the interface of a reinforced concrete column and footing
where all the longitudinal bars in the column are in compression.
Standard hooks are typically provided at the ends of the dowel bars, which are tied to the reinforcing bars in the
footing; this detail is far more economical than terminating the dowel bars above the footing reinforcing bars. The
hooked portion of the dowels cannot be considered effective for developing the dowels bars in compression (ACI
7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKH
footing:
(7.43)
In this equation, LVWKHUDGLXVRIWKHGRZHOEDUEHQG VHH$&,7DEOHZKLFKFRQWDLQVPLQLPXPLQVLGHEHQG
GLDPHWHUVIRUEDUVZLWKVWDQGDUGGHJUHHKRRNV :KHUH(TXDWLRQ LVQRWVDWLV¿HGHLWKHUDJUHDWHUQXPEHURI
smaller dowel bars can be used (if possible) or the depth of the footing must be increased.
The information above pertaining to the development of dowel bars in spread footings is also applicable to the
development of dowel bars in pile caps. For columns supported on drilled piers or caissons, the dowel bars in the
drilled pier or caisson usually do not have hooks at the ends.
The dowel bars must also be fully developed in the column; this is typically achieved by lap splicing the dowel bars
WRWKHORQJLWXGLQDOEDUVLQWKHFROXPQ VHH)LJXUH :KHUHWKHGRZHOEDUVDUHWKHVDPHVL]HDVWKHFROXPQEDUV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
WKHPLQLPXPFRPSUHVVLRQODSVSOLFHOHQJWKLVXVHG VHH7DEOH :KHUHWKHGRZHOEDUVDUHGLIIHUHQWLQVL]HWKDQ
WKHFROXPQEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWKPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,/DSVSOLFHVRI
DQGORQJLWXGLQDOEDUVLQFRPSUHVVLRQIRUDOOIDFWRUHGORDGFRPELQDWLRQVZLWKDQGVPDOOHUGRZHOEDUVDUH
SHUPLWWHGSURYLGHGWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG $&, Tension
Tension forces transferred from a column to a foundation must be resisted entirely by reinforcement across the inWHUIDFH $&, E DQG DQGGRZHOEDUVPXVWEHSURYLGHGIRUDOOWKHORQJLWXGLQDOEDUVLQWKHFROXPQ
,QVXFKFDVHVIDFWRUHGORDGFRPELQDWLRQVIDOOZLWKLQ=RQHVRURIWKHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDP VHH
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determined in acstandard hooks at the ends of the dowel bars with the development length of the hooked bar,
FRUGDQFHZLWK$&, VHH)LJXUH ͵ԣ Ž‡ƒ”
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Figure 7.39 Dowel bars at the interface of a reinforced concrete column and footing
where the longitudinal bars in the column are in compression and tension.
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7-57
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(7.44)
7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRN VHH)LJXUHRIWKLV
SXEOLFDWLRQ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOH VHH7DEOH Table 7.27 Modification Factors for Development of Hooked Bars in Tension
Modification Factor
Lightweight,
Epoxy,
&RQ¿QLQJUHLQIRUFHPHQW
Location,
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
or
Other
)RUDQGVPDOOHUKRRNHGEDUV
1. terminating inside a column core with side cover
normal to the plane of the hook
or
ZLWKVLGHFRYHUQRUPDOWRWKHSODQHRIWKHKRRN
Other
Concrete strength,
1.6
7KHFRQ¿QLQJUHLQIRUFHPHQWIDFWRU
LVW\SLFDOO\HTXDOWRIRUKRRNHGGRZHOEDUVLQIRRWLQJVEHFDXVHFRQ¿Qing reinforcement is usually not provided in such cases.
7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKHIRRWLQJ FRQVLGering the hooked portion of the dowel bars can be developed in tension):
(7.45)
:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHGHSWKRIWKHIRRWLQJXVXDOO\PXVWEHLQFUHDVHG
For development of the dowel bars into the column, a tension lap splice or a mechanical connection in accordance
ZLWK$&,RUUHVSHFWLYHO\PXVWEHSURYLGHGEHWZHHQWKHGRZHOEDUVDQGWKHORQJLWXGLQDOEDUV,QWKH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.9.3 Horizontal Transfer
7KHVKHDUIULFWLRQPHWKRGRI$&,LVSHUPLWWHGWREHXVHGWRGHWHUPLQHWKHQRPLQDOVKHDUVWUHQJWK
at the
FRQWDFWVXUIDFHEHWZHHQWKHFROXPQDQGWKHIRXQGDWLRQ $&, 7KHUHTXLUHGDUHDRIUHLQIRUFHPHQW
across the interface between the column and the foundation is determined by the following equation, which is apSOLFDEOHWRVKHDUIULFWLRQUHLQIRUFHPHQWSHUSHQGLFXODUWRWKHLQWHUIDFH $&, (7.46)
In this equation,
is the factored shear force due to the lateral force effects at the interface, the strength reduction
factor LVHTXDOWRDQG LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP$&,7DEOH VHH7DEOH
Table 7.28 Coefficient of Friction, μ
Contact Surface Condition
μ
Concrete placed monolithically
Concrete placed against hardened concrete that is clean, free of laitance,
and intentionally roughened to a full amplitude of approximately ¼ in.
Concrete placed against hardened concrete that is clean, free of laitance,
and not intentionally roughened
8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOH VHH7DEOH 7KHWHUP is the area of
For example, where a column is supported by a footing,
is equal to the gross area of the
concrete resisting
column. Where the concrete strengths of the column and the foundation are different, the smaller of the two must be
XVHGLQWKHVHHTXDWLRQV $&, Table 7.29 Maximum Vn Across the Assumed Shear Plane
Contact Surface Condition
Maximum Vn = Vu / Ҁ*
Normalweight concrete placed monolithically or placed against
hardened concrete intentionally roughened to a full amplitude of
approximately ¼ in.
Other cases
*
area of concrete resisting
The required amount of shear friction reinforcement,
LVSHUPLWWHGWREHGHWHUPLQHGE\(TXDWLRQ EDVHG
where
is a permanent net compression force transmitted across the ason a net force equal to
VXPHGVKHDUSODQH $&, $FFRUGLQJWR$&,WKHDUHDRIUHLQIRUFHPHQWUHTXLUHGWRUHVLVWDQHW
factored tension force across the assumed shear plane is to be added to the area of reinforcement required for shear
friction that crosses the assumed shear plane. For columns transmitting a bending moment across the shear plane,
WKHÀH[XUDOFRPSUHVVLRQDQGWHQVLRQIRUFHVDUHLQHTXLOLEULXPDQGGRQRWFKDQJHWKHUHVXOWDQWFRPSUHVVLRQIRUFH
acting across the shear plane or the shear-friction resistance. Thus, it is not necessary to provide additional
UHLQIRUFHPHQWWRUHVLVWWKHÀH[XUDOWHQVLRQVWUHVVHVXQOHVVWKHUHTXLUHGÀH[XUDOWHQVLRQUHLQIRUFHPHQWH[FHHGVWKH
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7-59
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
EŽ
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A
Figure 7.40 Design procedure for columns.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
A
Provide dowel bars to satisfy the
requirements in ACI 16.3 for
connections to foundations
ሺSect. 7.9ሻ
Figure 7.40 Design procedure for columns (cont.).
Full tension anchorage of the shear-friction reinforcement must be provided into the foundation and into the column. The lengths of these bars are determined in the same way as those for vertical transfer where tension forces are
present.
The area of the dowel bars is usually determined initially based on the requirements for vertical transfer. That area is
FRPSDUHGZLWKWKHDUHDUHTXLUHGIRUKRUL]RQWDOWUDQVIHUDQGWKHODUJHURIWKHWZRLVSURYLGHGDWWKHLQWHUIDFH
7.10 Design Procedure
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DQG¿JXUHQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQLQWKLVFKDSWHUFDQEHIRXQG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.11 Examples
7.11.1
Example 7.1 – Determination of Preliminary Column Size: Building #1 (Framing Option B),
Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJDUHFWDQJXODUWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the factored axial forces
Table 3.3
Roof:
Floors:
This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible.
$VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOH
Table 7.30 Summary of Axial Forces for Column C3
Load Case
Axial Force (kips)
Dead
Roof live
11.8
Live
Load Combination
$&,(T D
$&,(T E
661.6
$&,(T F
Step 2 – Determine the gross area of the column
Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement
a tied column is the following using the largest LQ7DEOH
the required gross area of
Eq. (7.30)
With a 1 percent reinforcement ratio, an 18-in. square column is adequate
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7-62
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG
column.
is equal to
ZKLFKFRUUHVSRQGVWRDLQVTXDUH
An 18-in. square column is selected.
Comments.,WLVVKRZQLQ([DPSOHWKDWDQLQWKLFNVODELVDGHTXDWHIRUVHUYLFHDELOLW\DQGVKHDUVWUHQJWK
UHTXLUHPHQWVEDVHGRQLQVTXDUHFROXPQV7KHUHTXLUHGVODEWKLFNQHVVEDVHGRQLQFROXPQVLVHTXDOWR
,WFDQEHVKRZQWKDWWZRZD\VKHDUUHTXLUHPHQWVDUHQRWVDWLV¿HGZLWK
LQFROXPQVDQGDQLQWKLFNVODE7KHLQFROXPQVFDQEHXVHGKRZHYHULIVKHDUUHLQIRUFHPHQWLQWKHVODE
is provided.
7.11.2
Example 7.2 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),
Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
%DVVXPLQJDQLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH
DQG*UDGHUHLQIRUFHPHQW
with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the required area of longitudinal reinforcement
Based on an 18-in. square tied column, the required reinforcement ratio,
is equal to the following:
Eq. (7.32)
6HOHFWEDUV
Step 2 – Check the minimum number of longitudinal bars and the minimum face dimension of the column
For rectangular, tied columns:
Minimum number of longitudinal bars with rectangular ties
Minimum face dimension
IRUEDUVSHUIDFHDQGQRUPDOODSVSOLFHV
Table 7.16
Table 7.17
8VHEDUV
7.11.3
Example 7.3 – Determination of Transverse Reinforcement: Building #1 (Framing Option B),
Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%
DVVXPLQJDQLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH
DQG*UDGHUHLQIRUFHPHQW
with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Determine the required tie bar size
ACI 25.7.2.2
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$&,
)RUORQJLWXGLQDOEDUVXVHWLHV
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7-63
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the required tie bar spacing
ACI 25.7.2.1
Because the shear force due to the factored gravity loads is negligible, the required tie bar spacing is determined usLQJ$&,
8VHWLHVVSDFHGDWLQRQFHQWHUWKLVPDWFKHVWKHYDOXHLQ)LJXUH
7.11.4
Example 7.4 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing
Option B), Rectangular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily
Axial Forces
'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%
spread footing (see Figure 1.1). Also
assuming an 18-in. square tied column supported by a
and
DVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Check the bearing loads on the column and footing
ACI 16.3.3.4
• Bearing strength of the column:
Eq. (7.40)
• Bearing strength of the footing:
Eq. (7.41)
Footing thickness
+RUL]RQWDOSURMHFWLRQIRUDVORSH
Figure 7.37
Projected length
Therefore,
Step 2 – Determine the required interface reinforcement
ACI 16.3.4.1
Because the design bearing strengths are adequate for the column and footing, provide minimum area of reinforcement across the interface:
3URYLGHGRZHOEDUV
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7-64
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Check the development of the dowel bars in the footing
Determine the compression development length,
RIWKHGRZHOEDUV
Table 7.21, Note 5
Table 7.22
Table 7.13
Therefore,
$VVXPLQJWZROD\HUVRIEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJWKLFNQHVV
for the development of the dowel bars:
Eq. (7.43)
Because the minimum
LVOHVVWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQWKHKRRNHGGRZHOEDUV
can be fully developed in the footing for compression.
Step 4 – Determine the development length of the dowel bars in the column
The dowel bars are lap spliced to the longitudinal reinforcement in the column. Because the dowel bars are smaller
must be greater than or equal to the
in diameter than the longitudinal bars, the compression lap splice length,
larger of the following:
1. Development length in compression,
RIWKHORQJLWXGLQDOEDUV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Compression lap splice length,
RIWKHGRZHOEDUV
Table 7.21
3URYLGHDODSVSOLFHOHQJWKRIIWLQ
1HJOLJLEOHKRUL]RQDOIRUFHVDUHWUDQVIHUUHGIURPWKHFROXPQWRWKHIRRWLQJVRKRUL]RQWDOIRUFHWUDQVIHULVQRWLQYHVWLJDWHG
5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH
ͳԢǦ͸ԣ
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Figure 7.41 Reinforcement details for column C3 in Examples 7.1 through 7.4.
7.11.5
Example 7.5 – Determination of Preliminary Column Size: Building #1 (Framing Option B),
Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJWKHFROXPQLVFLUFXODUZLWKWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIWLHV VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODE
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
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7-66
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the factored axial forces
Table 3.3
Roof:
Floors:
This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible.
$VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHLQ([DPSOH
Step 2 – Determine the gross area of the column
Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement
a tied column is the following using the largest LQ7DEOH
the required gross area of
Eq. (7.30)
:LWKDSHUFHQWUHLQIRUFHPHQWUDWLRDLQFLUFXODUFROXPQLVDGHTXDWH
)RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG
LQFLUFXODUFROXPQ
is equal to
which corresponds approximately to a
$LQFLUFXODUFROXPQLVVHOHFWHG
7.11.6
Example 7.6 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),
Circular, Tied Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
%DVVXPLQJDLQFLUFXODUWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH
DQG*UDGHUHLQIRUFHPHQW
with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the required area of longitudinal reinforcement
%DVHGRQDLQFLUFXODUWLHGFROXPQWKHUHTXLUHGUHLQIRUFHPHQWUDWLR
is equal to the following:
Eq. (7.32)
6HOHFWEDUV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Check the minimum and maximum number of longitudinal bars
For circular, tied columns:
Minimum number of longitudinal bars with circular ties
Table 7.16
0D[LPXPQXPEHURIEDUV
Table 7.19
IRUDLQFLUFXODUFROXPQDQGQRUPDOODSVSOLFHV
8VHEDUV
Comments.,WFDQEHVKRZQWKDWWKHVL]HDQGVSDFLQJRIWKHFLUFXODUWLHVDQGWKHQXPEHUDQGVL]HRIWKHGRZHOEDUV
IRUWKHLQFLUFXODUWLHGFROXPQDUHWKHVDPHDVWKRVHGHWHUPLQHGLQ([DPSOHVDQGIRUWKHLQVTXDUH
WLHGFROXPQ VHH)LJXUH ͳԢǦͺԣ
ͶǦ͓ͻ
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Figure 7.42 Reinforcement details for column C3 in Examples 7.5 and 7.6.
7.11.7
Example 7.7 – Determination of Preliminary Column Size: Building #1 (Framing Option B), Circular,
Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHDSUHOLPLQDU\VL]HIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVVXPLQJDFLUFXODU
and
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the factored axial forces
Table 3.3
Roof:
Floors:
This column is not part of the LFRS, and the bending moments due to gravity load effects are negligible.
$VXPPDU\RIWKHD[LDOIRUFHVIRUFROXPQ&LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHLQ([DPSOH
Step 2 – Determine the gross area of the column
Assuming a 1 percent reinforcement ratio for the longitudinal reinforcement
a spiral column is the following using the largest LQ7DEOH
the required gross area of
Eq. (7.31)
With a 1 percent reinforcement ratio, a 19-in. circular column is adequate
)RUDSHUFHQWUHLQIRUFHPHQWUDWLRWKHUHTXLUHG
18-in. circular column.
is equal to
which corresponds approximately to an
An 18-in. circular column is selected.
7.11.8
Example 7.8 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),
Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
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crete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the required area of longitudinal reinforcement
Based on an 18-in. circular spiral column, the required reinforcement ratio,
is equal to the following:
Eq. (7.32)
6HOHFWEDUV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Check the minimum and maximum number of longitudinal bars
For circular, spiral columns:
Minimum number of longitudinal bars with spirals
Table 7.16
0D[LPXPQXPEHURIEDUV
Table 7.19
for an 18-in. circular column and normal lap splices
8VHEDUV
7.11.9
Example 7.9 – Determination of Transverse Reinforcement: Building #1 (Framing Option B), Circular,
Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily Axial Forces
'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%
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DQG*UDGHUHLQIRUFHPHQW
with
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Step 1 – Determine the required spiral bar size
ACI 25.7.3.2
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6SLUDOEDUVPXVWKDYHDGLDPHWHURIDWOHDVWLQVRXVHDVSLUDO
Step 2 – Determine the required spiral bar spacing
ACI 25.7.3.1, 25.7.3.3
Because the shear force due to the factored gravity loads is negligible, the required spiral bar spacing is determined
XVLQJ$&,
Eq. (7.39)
8VHDVSLUDODWDLQSLWFKWKLVPDWFKHVWKHYDOXHLQ7DEOH
7.11.10 Example 7.10 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing
Option B), Circular, Spiral Column is Not Part of the LFRS, SDC A, Column Subjected to Primarily
Axial Forces
'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%DVspread footing (see Figure 1.1). Also
suming an 18-in. circular spiral column supported by a
and
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*UDGHUHLQIRUFHPHQW
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7-70
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Check the bearing loads on the column and footing
ACI 16.3.3.4
• Bearing strength of the column:
Eq. (7.40)
Therefore, transfer the excess compression force by interface reinforcement:
Eq. (7.42)
• Bearing strength of the footing:
Eq. (7.41)
Footing thickness
+RUL]RQWDOSURMHFWLRQIRUDVORSH
Projected diameter of cone
Therefore,
Step 2 – Determine the required interface reinforcement
ACI 16.3.4.1
It is determined in Step 1 that the required area of interface reinforcement is
3URYLGHGRZHOEDUV
Step 3 – Check the development of the dowel bars in the footing
Determine the compression development length,
RIWKHGRZHOEDUV
Table 7.21, Note 5
(the dowel bars are not enclosed by any transverse reinforcement in the footing)
Table 7.22
Table 7.13
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VVXPLQJWZROD\HUVRIEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJWKLFNQHVV
IRUWKHGHYHORSPHQWRIWKHGRZHOEDUV
Eq. (7.43)
Because the minimum
LVOHVVWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQWKHKRRNHGGRZHOEDUV
can be fully developed in the footing for compression.
Step 4 – Determine the development length of the dowel bars in the column
7KHGRZHOEDUVDUHODSVSOLFHGWRWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQ%HFDXVHWKHGRZHOEDUVDUH
must be greater than or equal to the
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following:
Table 7.21
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Figure 7.43 Reinforcement details for column C3 in Examples 7.7 through 7.10.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For spiral columns,
LVSHUPLWWHGWREHPXOWLSOLHGE\
ACI 10.7.5.2.1(b)
Provide a lap splice length of 1 ft-6 in.
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7.11.11 Example 7.11 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1
(Framing Option B), Rectangular, Tied Column, Grade 60 Longitudinal Reinforcement
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Figure 7.44 Reinforced concrete column in Example 7.11.
Nominal and design axial forces and bending moments are determined at the following points:
A. Pure axial compression
B. Balanced point
C. Stress in reinforcement farthest from compression face
D. Stress in reinforcement farthest from compression face
E. Pure bending
Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for
all the other points. In the calculations, it is assumed compression is positive.
A. Pure axial compression
The nominal axial compressive strength for a tied column is determined by the following:
Eq. (7.13)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The section is compression-controlled, so
Table 7.10
B. Balanced point
Step 1 – Determine the neutral axis depth
Table 7.11
%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQ
(Note: ACI
in the reinforcement farthest from the compression face is
IRU*UDGHGHIRUPHGUHLQIRUFHPHQWEXWWKDWYDOXHLVQRWXVHGLQWKLVH[DPSOH 7KH
SHUPLWV
neutral axis depth, is determined by the following equation:
Figure 7.10
Step 2 – Determine the depth of the equivalent stress block
Table 7.11
Step 3 – Determine the resultant compression force
Step 4 – Determine the strain in the reinforcement
Layer 1:
/D\HU
Step 5 – Determine the stress in the reinforcement
Layer 1:
/D\HU
Step 6 – Determine the force in the reinforcement
Layer 1
/D\HU
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 7 – Determine the nominal and design axial strengths
Step 8 – Determine the nominal and design flexural strengths
C. Stress in reinforcement farthest from the compression face = 0
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
Table 7.31 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
D. Stress in reinforcement farthest from the compression face = 0.5fy
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.32 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0.5fy
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
E. Pure bending
There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
1,400
1,200
Axial Force ሺkipsሻሻ
1,000
݂௦ ൌ 0
800
݂௦ ൌ 0.5݂௬
600
400
200
0
0
100
200
300
400
Bending Moment ሺft-kipsሻ
Figure 7.45 Nominal and design strength interaction diagrams for the column in Example 7.11.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.33 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
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7.11.12 Example 7.12 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1
(Framing Option B), Rectangular, Tied Column, Grade 100 Longitudinal Reinforcement
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WKURXJK7KHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUVDQGDFOHDUFRYHURILQLVSURYLGHGWRWLHV
DQG*UDGHUHLQIRUFHPHQW
VHH)LJXUH $VVXPHQRUPDOZHLJKWFRQFUHWHZLWK
Nominal and design axial forces and bending moments are determined at the following points:
A. Pure axial compression
B. Balanced point
C. Stress in reinforcement farthest from compression face
D. Stress in reinforcement farthest from compression face
E. Pure bending
Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for
all the other points. In the calculations, it is assumed compression is positive.
A. Pure axial compression
The nominal axial compressive strength for a tied column is determined by the following:
Eq. (7.13)
Note that
is used in the equation for
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limits WRNVLIRUPHPEHUVVXEMHFWHGWRSXUHD[LDOFRPSUHVVLRQ
The section is compression-controlled, so
Table 7.10
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7-77
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
B. Balanced point
Step 1 – Determine the neutral axis depth
Table 7.11
%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQLQ
The neutral
the reinforcement farthest from the compression face is
axis depth, is determined by the following equation:
Figure 7.10
Step 2 – Determine the depth of the equivalent stress block
Table 7.11
Step 3 – Determine the resultant compression force
Step 4 – Determine the strain in the reinforcement
Layer 1:
/D\HU
Step 5 – Determine the stress in the reinforcement
Layer 1:
/D\HU
Step 6 – Determine the force in the reinforcement
Layer 1
/D\HU
Step 7 – Determine the nominal and design axial strengths
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7-78
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 8 – Determine the nominal and design flexural strengths
C. Stress in reinforcement farthest from the compression face = 0
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
Table 7.34 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
D. Stress in reinforcement farthest from the compression face = 0.5fy
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.35 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0.5fy
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
E. Pure bending
There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
1,600
Point A
1,400
Axial Force ሺkipsሻሻ
1,200
݂௦ ൌ 0
1,000
800
600
݂௦ ൌ 0.5݂௬ 400
200
0
0
100
200
300
400
Bending Moment ሺft-kipsሻ
Figure 7.46 Nominal and design strength interaction diagrams for the column in Example 7.12.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.36 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
7KHQRPLQDODQGGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH3RLQW$RQWKHQRPLwhich corresponds
nal strength interaction diagram represents the maximum nominal axial force
WRWKHNVLOLPLWIRUSXUHD[LDOFRPSUHVVLRQLQ$&,$KRUL]RQWDOOLQHFRQQHFWVWKDWSRLQWWRWKHSRLQW
and
where
7.11.13 Example 7.13 – Construction of Nominal and Design Strength Interaction Diagrams: Building #1
(Framing Option B), Circular, Tied Column, Grade 60 Longitudinal Reinforcement
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Figure 7.47 Reinforced concrete column in Example 7.13.
Nominal and design axial forces and bending moments are determined at the following points:
A. Pure axial compression
B. Balanced point
C. Stress in reinforcement farthest from compression face
D. Stress in reinforcement farthest from compression face
E. Pure bending
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7-81
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Calculations are provided for pure axial compression and the balanced point; a summary of the results is given for
all the other points. In the calculations, it is assumed compression is positive.
A. Pure axial compression
The nominal axial compressive strength for a tied column is determined by the following:
Eq. (7.13)
where
The section is compression-controlled, so
Table 7.10
B. Balanced point
Step 1 – Determine the neutral axis depth
Table 7.11
%DODQFHGIDLOXUHRIWKHVHFWLRQRFFXUVZKHQWKHPD[LPXPVWUDLQLQWKHFRQFUHWHLVHTXDOWRDQGWKHVWUDLQ
(Note: ACI
in the reinforcement farthest from the compression face is
IRU*UDGHGHIRUPHGUHLQIRUFHPHQWEXWWKDWYDOXHLVQRWXVHGLQWKLVH[DPSOH 7KH
SHUPLWV
neutral axis depth, is determined by the following equation:
Figure 7.12
Step 2 – Determine the depth of the equivalent stress block
Table 7.11
Step 3 – Determine the section properties of the compression zone
Figure 7.13
Step 4 – Determine the resultant compression force
Step 5 – Determine the strain in the reinforcement
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Layer 1:
/D\HU
Layer 3:
Step 6 – Determine the stress in the reinforcement
Layer 1:
/D\HU
Layer 3:
Step 7 – Determine the force in the reinforcement
Layer 1
/D\HU
Layer 3
Step 8 – Determine the nominal and design axial strengths
Step 9 – Determine the nominal and design flexural strengths
C. Stress in reinforcement farthest from the compression face = 0
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7-83
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Figure 7.13
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
Table 7.37 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
D. Stress in reinforcement farthest from the compression face = 0.5fy
Figure 7.13
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.38 Summary of Reinforcement Strains, Stresses, and Forces for Stress in Reinforcement Farthest
from Compression Face = 0.5fy
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
E. Pure bending
There is no closed-form solution to determine the depth of the neutral axis for the case of pure bending. From trialand-error, it is found that
Figure 7.13
7KHVWUDLQVVWUHVVHVDQGIRUFHVDWHDFKOD\HURIUHLQIRUFHPHQWDUHJLYHQLQ7DEOH
Table 7.39 Summary of Reinforcement Strains, Stresses, and Forces for Pure Bending
dsi (in.)
İsi
fsi (ksi)
Fsi (kips)
36.6
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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1,400
1,200
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Axial Force ሺkipsሻሻ
1,000
800
݂௦ ൌ 0.5݂௬ 600
400
200
0
0
50
100
150
200
250
300
Bending Moment ሺft-kipsሻ
Figure 7.48 Nominal and design strength interaction diagrams for the column in Example 7.13.
7.11.14 Example 7.14 – Determination of Preliminary Column Size: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Column Subjected to Uniaxial Bending
and Axial Forces
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WKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDUHFWDQJXODUWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDLQWKLFNVODE
DQG*UDGHUHLQIRUFHPHQW
E\LQEHDPVQRUPDOZHLJKWFRQFUHWHZLWK
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the factored axial forces and bending moments
Roof:
Floors:
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7-86
Table 3.3
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
This column is part of the LFRS and the bending moments due to gravity load effects are small compared to the
bending moments due to wind load effects in the north-south direction, so the former are not considered.
$VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ&LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW 665 DQGVLGHVZD\WRWKHOHIW 66/ Table 7.40 Summary of Axial Forces and Bending Moments for Column C1
Bending Moment (ft-kips)
Load Case
Axial Force (kips)
Top
Bottom
Dead
Roof live
Live
Wind
Load Combination
$&,(T D
$&,(T E
63.9
63.9
$&,(T F
SSR
SSL
SSR
SSL
SSR
SSL
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Step 2 – Determine the gross area of the column
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a tied column is the following using the largest LQ7DEOH
the required gross area of
Eq. (7.30)
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part of a nonsway or sway frame and whether slenderness effects need to be considered or not and (3) preliminary
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.11.15 Example 7.15 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A
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FROXPQVDUHLQVTXDUH VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKW
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concrete with
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Step 1 – Determine the total loads in the first story
• Roof:
• 7\SLFDOÀRRU
The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI
(T G SURGXFHVWKHODUJHVWORDG
Step 2 – Determine the stability index for the first story
An elastic analysis of the structure was performed with the building subjected to the north-south wind forces in
7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRU
the columns, beams, and slabs. The following results were obtained from the analysis:
The stability index,
is determined from the following equation:
Table 7.3
Because
WKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPH VHH$&, @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.11.16 Example 7.16 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame
&KHFNLIVOHQGHUQHVVHIIHFWVPXVWEHFRQVLGHUHGIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
&DVVXPLQJWKHFROXPQLVLQVTXDUH VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVQRQVZD\LQWKHQRUWKVRXWKGLUHFWLRQ7KHUHIRUH
VOHQGHUQHVVHIIHFWVFDQEHQHJOHFWHGZKHQWKHIROORZLQJHTXDWLRQLVVDWLV¿HG
Table 7.6
For all load combinations including wind,
Table 7.40
The column is bent in double curvature for all load combinations; therefore, slenderness can be neglected where the
IROORZLQJLVVDWLV¿HG
Assuming the effective length factor,
7DEOH LVHTXDOWRWKH ZKLFKLVWKHPD[LPXPYDOXHIRUQRQVZD\IUDPHVVHH
Therefore, slenderness effects need not be considered.
7.11.17 Example 7.17 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to
Uniaxial Bending and Axial Forces
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
&DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
Step 1 – Determine the required area of longitudinal reinforcement
Eq. (7.32)
%DVHGRQDLQVTXDUHWLHGFROXPQWKHUHTXLUHGUHLQIRUFHPHQWUDWLR
is equal to the following for the largest
IDFWRUHGD[LDOIRUFHLQ7DEOHZKLFKRFFXUVZLWKRXWDQ\DSSUHFLDEOHEHQGLQJPRPHQWV
Therefore, use the minimum required
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6HOHFWEDUV
Step 2 – Check if the selected longitudinal reinforcement is adequate for all load combinations
7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVFROXPQUHLQIRUFHGZLWKEDUVLVJLYHQLQ)LJXUH$OVR
VKRZQLQWKH¿JXUHDUHORDGFRPELQDWLRQSRLQWVIRUWKHIDFWRUHGD[LDOIRUFHVDQGEHQGLQJPRPHQWVLQ7DEOH
900
800
݂௦ ൌ 0
Axial Force ሺkipsሻሻ
700
600
݂௦ ൌ 0.5݂௬ 500
300 200
400
100
0
0
50
100
150
200
250
300
350
Bending Moment ሺft-kipsሻ
Figure 7.49 Design strength interaction diagram for column C1.
,WLVHYLGHQWIURPWKH¿JXUHWKDWWKHFROXPQLVDGHTXDWHIRUDOOIDFWRUHGORDGFRPELQDWLRQV
Step 3 – Check the minimum number of longitudinal bars and the minimum face dimension of the column
For rectangular, tied columns:
Minimum number of longitudinal bars with rectangular ties
Minimum face dimension
IRUEDUVSHUIDFHDQGQRUPDOODSVSOLFHV
Table 7.16
Table 7.17
8VHEDUV
Comments.,WLVHYLGHQWIURP)LJXUHWKDWWKHLQFROXPQZLWKEDUVSRVVHVVHVVLJQL¿FDQWO\PRUH
VWUHQJWKWKDQUHTXLUHGEDVHGRQWKHIDFWRUHGORDGFRPELQDWLRQV+RZHYHUDVQRWHGLQ([DPSOHLQFROXPQV
are needed primarily for lateral stiffness of the moment frames. Furthermore, the column is also part of the LFRS in
the east-west direction and must also be designed for combined gravity and wind load effects for wind in that direcWLRQ'HFUHDVLQJWKHDPRXQWRIORQJLWXGLQDOUHLQIRUFHPHQWEHORZSHUFHQWLQDFFRUGDQFHZLWK$&,LVQRW
done in this example.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.11.18 Example 7.18 – Determination of Transverse Reinforcement: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to
Uniaxial Bending and Axial Forces
'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
&DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
Step 1 – Determine the required tie bar size
ACI 25.7.2.2
7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&,
)RUORQJLWXGLQDOEDUVXVHWLHV
Step 2 – Determine the required tie bar spacing
ACI 25.7.2.1
7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR
determine if they are adequate for shear strength requirements.
7U\WLHVVSDFHGDWLQRQFHQWHU
Step 3 – Check the adequacy of the tie bar size and spacing for shear
ACI 22.5
Maximum factored shear force,
LVREWDLQHGIURP$&,(TV G DQG I EDVHGRQWKHIDFWRUHGEHQGLQJ
PRPHQWGXHWRWKHZLQGORDGHIIHFWVDQGLVHTXDOWRNLSV VHH7DEOHDQG)LJXUH Check if the design shear strength of the concrete,
LVSURYLGHGRUQRW VHH7DEOH alone is adequate, which is determined based on whether
Eq. (7.26)
The minimum area of shear reinforcement is greater than
SURYLGHGE\WKHWLHVVSDFHG
18 in. on center. Because LVUHODWLYHO\VPDOOLWLVQRWQHFHVVDU\WRLQFUHDVHWKHWLHEDUVL]HDQGRUGHFUHDVHWKH
is determined based on
spacing, so
Table 7.12
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܲ௨ ‫ܯ‬௨ǡ௧௢௣ ൌ ʹͲǤͶˆ–Ǧ‹’•
κ௨ ൌ ͳͶǤͲ െ ሾʹͶǤͲȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͵ǤͲˆ–
ܸ௨ ܸ௨ ൌ
ʹͲǤͶ ൅ ͸͵Ǥͻ
ൌ ͸Ǥͷ‹’•
ͳ͵ǤͲ
ܸ௨ ‫ܯ‬௨ǡ௕௢௧௧௢௠ ൌ ͸͵Ǥͻˆ–Ǧ‹’•
ܲ௨ Figure 7.50 Factored shear force for column C1.
Eq. (7.23)
Table 7.13
WKHUHDUHEDUVORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWK
IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU Use minimum
FRUUHVSRQGLQJWR$&,(T I EHFDXVHWKLVUHVXOWVLQPLQLPXP
ACI 22.5.5.1.2
Also,
so minimum shear reinforcement is not required.
7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU
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7-92
ACI 10.6.2.1
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7.11.19 Example 7.19 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing
Option C), Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected
to Uniaxial Bending and Axial Forces
'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ&DVspread footing (see Figure 1.1). Also assume
VXPLQJDLQVTXDUHWLHGFROXPQVXSSRUWHGE\D
DQG*UDGH
DLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
reinforcement.
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
Step 1 – Check the bearing stresses on the column and footing
ACI 16.3.3.4
Bearing strength of the column.
is determined for pure compression using the factored axial force determined by ACI
Factored bearing stress,
(TV E DQGIRUFRPSUHVVLRQSOXVEHQGLQJXVLQJWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\
$&,(TV G DQG I >VHH7DEOH@
• $&,(T E • $&,(T G • $&,(T I Design bearing strength:
Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the
VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQVLRQIRUFHV
are adequately transferred from the column into the footing.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Bearing strength of the footing.
There is no need to check the bearing strength of the footing because interface reinforcement must be provided due
WRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(T I Step 2 – Determine the required interface reinforcement
ACI 16.3.4.1
7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHVWKDWERWK
the compression and tension forces are adequately transferred through the interface.
Check minimum area of reinforcement across the interface:
Step 3 – Check the development of the dowel bars in the footing
,WLVHYLGHQWIURP)LJXUHWKDWWKHVWUHVVLQWKHORQJLWXGLQDOUHLQIRUFHPHQWIDUWKHVWIURPWKHH[WUHPHFRPSUHVVLRQ
IDFHLVWHQVLOHIRURQHRIWKHORDGFRPELQDWLRQV WKDWLVWKHORDGFRPELQDWLRQIDOOVZLWKLQ=RQHRIWKHLQWHUDFWLRQ
GLDJUDPVHH)LJXUH 7KHUHIRUHWKHGRZHOEDUVPXVWEHGHYHORSHGIRUWHQVLRQLQWRWKHIRRWLQJ DQGLQWRWKH
column).
$VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSPHQW
RIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN
length,
Eq. (7.44)
Table 7.27
$VVXPLQJWZROD\HUVRIÀH[XUDOEDUVLQWKHIRRWLQJZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPIRRWLQJ
thickness for the development of the dowel bars:
Eq. (7.45)
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7-94
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The minimum
LVJUHDWHUWKDQWKHSURYLGHGIRRWLQJWKLFNQHVVRILQ7KXVLQFUHDVHWKHIRRWLQJ
WKLFNQHVVWRLQWRDFFRPPRGDWHGHYHORSPHQWRIWKHGRZHOEDUV
Step 4 – Determine the development length of the dowel bars in the column
The dowel bars must be lap spliced to the longitudinal reinforcement in the column using a tension lap splice.
Because all the dowel bars are spliced at the same location, a Class B tension lap splice is required (ACI Table
Development length in tension,
RIWKHEDUV
Eq. (7.36)
Table 7.24
Set
ACI 25.4.2.4
Therefore,
3URYLGHDODSVSOLFHOHQJWKRIIWLQ
Step 5 – Check horizontal force transfer
,WLVGHWHUPLQHGLQ([DPSOHWKDW
LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGIRRWLQJ7KH
required area of shear-friction reinforcement is determined as follows assuming the surface between the column and
footing has not been intentionally roughened:
Eq. (7.46), Table 7.28
7KHGRZHOEDUVSURYLGHV
across the interface, which is greater than the required amount of
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7-95
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Check the upper shear limit:
Table 7.29
5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH
ͳԢǦͺԣ
ͳԢǦͺԣ
ͶǦ͓ͻ
ͶԢǦʹԣ
ͶǦ͓ͻ†‘™‡Ž„ƒ”•
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͵ԢǦͲԣ
͓͵–‹‡•̷ͳԢǦ͸ԣ‘Ǥ Ǥ
Figure 7.51 Reinforcement details for column C1 in Examples 7.14 through 7.19.
7.11.20 Example 7.20 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to Biaxial
Bending and Axial Forces
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ(LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
&DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEQRUPDOZHLJKWFRQFUHWH
DQG*UDGHUHLQIRUFHPHQW
with
'HVLJQGDWDDUHJLYHQLQ6HFW
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7-96
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the required area of longitudinal reinforcement
This corner column is subjected to axial forces and biaxial bending, so there is no direct way of determining the
required area of longitudinal reinforcement.
Therefore, initially try the minimum required
6HOHFWEDUV
Step 2 – Check if the selected longitudinal reinforcement is adequate for all load combinations
An elastic analysis of the structure was performed with the building subjected to the north-south and east-west wind
IRUFHVLQ7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHFULWLFDOZLQGORDGLQJIRUDFRUQHUFROXPQRFFXUVZKHQSHUFHQWRIWKHZLQGIRUFHVLQERWKGLUHFWLRQVDUHDSSOLHGVLPXOWDQHRXVO\WRWKHVWUXFWXUH /RDG&DVH
LQ$6&(6(,)LJ 7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRUWKHFROXPQVEHDPVDQGVODEV
in the analysis.
$VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ(LQWKH¿UVWVWRU\LVJLYHQLQ7DEOHIRUVLGHsway to the right (SSR) and sidesway to the left (SSL) in both principal directions.
Table 7.41 Summary of Axial Forces and Bending Moments for Column E1
Bending Moment (ft-kips)
Axial Force
(kips)
Load Case
North-South
Top
East-West
Bottom
Top
ņ
ņ
Dead
Roof live
Live
38.3
6.8
11.9
$&,(T D
38.8
$&,(T E
ņ
Bottom
ņ
Wind
Load Combination
$&,(T F
$&,(T G
$&,(T I
SSR
SSL
SSR
SSL
SSR
SSL
9.1
,WLVGHWHUPLQHGLQ([DPSOHWKDWWKH¿UVWVWRU\RIWKLVEXLOGLQJLVQRQVZD\,WFDQDOVREHGHWHUPLQHGWKDWVOHQderness effects need not be considered.
@Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
:LWKORQJLWXGLQDOEDUVWKLVFROXPQKDVWKHVDPHEHQGLQJPRPHQWFDSDFLW\DERXWERWKSULQFLSDOD[HV7KXV
WKHDSSUR[LPDWHPHWKRGGHSLFWHGLQ)LJXUHFDQEHXVHGWRGHWHUPLQHWKHDGHTXDF\RIWKHFROXPQIRUHDFKORDG
FRPELQDWLRQLQ7DEOH
To illustrate the design procedure, the column will be checked for the factored axial force and bending moments
REWDLQHGIURP$&,(T G IRUVLGHVZD\WRWKHULJKW
WKHXQLD[LDOGHVLJQPRPHQWVWUHQJWKVDUHWKHIROORZLQJ VHH)LJXUH At an axial force
900
800
Axial Force ሺkipsሻ
700
600
500
400
300
ܲ௨ ൌ 204.3 kips
200
100
0
0
50
100
150
200
Bending Moment ሺft-kipsሻ
250
300
350
߶‫ܯ‬௡௢ ேௌ ൌ ߶‫ܯ‬௡௢ ாௐ
ൌ 274.0 ft-kips
Figure 7.52 Uniaxial design strength interaction diagram for column E1.
The approximate biaxial strength diagram corresponding to
LVJLYHQLQ)LJXUH,WLVHYLGHQW
IURPWKH¿JXUHWKDWWKHFROXPQLVDGHTXDWHIRUWKHDSSOLHGIDFWRUHGELD[LDOPRPHQWV
It can be determined in a similar fashion that the column is adequate for all the other load combinations.
8VHORQJLWXGLQDOEDUV
Comments. )RUFRPSDULVRQSXUSRVHV5HIHUHQFHZDVXVHGWRFKHFNWKHDGHTXDF\RIWKLVFROXPQ$VH[SHFWHG
the column was found to be adequate.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ሺ߶‫ܯ‬௡௢ ሻாௐ ൌ ʹ͹ͶǤͲˆ–Ǧ‹’•
ܲ௨ ൌ ʹͲͶǤ͵‹’•
ሺ‫ܯ‬௨ ሻேௌ ൌ ʹͳǤ͹ˆ–Ǧ‹’•
ሺ‫ܯ‬௨ ሻாௐ ൌ ͶͺǤ͹ˆ–Ǧ‹’•
н
н ሺ‫ܯ‬௨ ሻேௌ ൌ ͶͶǤʹˆ–Ǧ‹’•
ሺ‫ܯ‬௨ ሻாௐ ൌ ͵ͶǤͺˆ–Ǧ‹’•
ሺ߶‫ܯ‬௡௢ ሻேௌ ൌ ʹ͹ͶǤͲˆ–Ǧ‹’•
Figure 7.53 Approximate biaxial strength diagram for column E1.
7.11.21 Example 7.21 – Determination of Transverse Reinforcement: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to
Biaxial Bending and Axial Forces
'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ(LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ&
DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the required tie bar size
ACI 25.7.2.2
7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&,
)RUORQJLWXGLQDOEDUVXVHWLHV
Step 2 – Determine the required tie bar spacing
ACI 25.7.2.1
7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR
determine if they are adequate for shear strength requirements.
7U\WLHVVSDFHGDWLQRQFHQWHU
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7-99
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Check the adequacy of the tie bar size and spacing for shear
ACI 22.5
Maximum factored shear force,
is obtained from ACI Eq. (5.3.1d) based on the factored bending moments at
the top and bottom of the column (see Table 7.41):
• North-south wind and SSL:
• East-west wind and SSR:
Check if the design shear strength of the concrete,
is provided or not (see Table 7.12).
alone is adequate, which is determined based on whether
Eq. (7.26)
The minimum area of shear reinforcement is greater than Av = 0.15 in.2 provided by the #3 ties spaced 18 in. on cenis relatively small in both directions, it is not necessary to increase the tie bar size and/or decrease
ter. Because
is determined based on
the spacing, so
Table 7.12
Eq. (7.23)
where
Table 7.13
(there are 2-#9 bars located more than two-thirds of the overall member depth
from the extreme compression fiber).
• North-south wind:
Use axial force
which corresponds to ACI Eq. (5.3.1d) and sidesway to the left.
ACI 22.5.5.1.2
7-100
@Seismicisolation
@Seismicisolation
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• East-west wind:
Use axial force
ZKLFKFRUUHVSRQGVWR$&,(T G DQGVLGHVZD\WRWKHULJKW
ACI 22.5.5.1.2
Check if interaction of shear forces acting along both the north-south and east-west directions must be considered
VHH$&, %HFDXVHWKHUDWLRVDUHOHVVWKDQLQWHUDFWLRQRIVKHDUIRUFHVDUHSHUPLWWHGWREHQHJOHFWHG
7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU
7.11.22 Example 7.22 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option C),
Rectangular, Tied Column is Part of the LFRS, SDC A, Nonsway Frame, Column Subjected to
Uniaxial Bending and Axial Forces, Slenderness Effects
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ&LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
&DVVXPLQJDLQVTXDUHWLHGFROXPQPXVWEHXVHGIRUWKLVFROXPQRQO\GXHWRDUFKLWHFWXUDOUHDVRQV VHH)LJXUH
and
$OVRDVVXPHDLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKWFRQFUHWHZLWK
*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the factored axial forces
Table 3.3
Roof:
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7-101
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Floors:
This column is part of the LFRS for wind in the north-south direction and the bending moments due to gravity load
effects are negligible.
$VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ&LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW 665 DQGVLGHVZD\WRWKHOHIW 66/ Table 7.42 Summary of Axial Forces and Bending Moments for Column C1
Bending Moment (ft-kips)
Load Case
Axial Force (kips)
Top
Bottom
Dead
Roof live
Live
Wind
Load Combination
$&,(T D
$&,(T E
SSR
SSL
SSR
SSL
SSR
SSL
$&,(T F
$&,(T G
$&,(T I
6.3
6.3
Step 2 – Check if slenderness effects must be considered
,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVQRQVZD\ ZKLFKLVVWLOOYDOLGHYHQWKRXJKWKHVL]H
of this one column has been reduced). Therefore, slenderness effects can be neglected when the following equation
LVVDWLV¿HG
Table 7.6
For all load combinations including wind,
Table 7.42
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7-102
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The column is bent in double curvature for all load combinations; therefore, slenderness can be neglected where the
IROORZLQJLVVDWLV¿HG
Assuming the effective length factor,
7DEOH LVHTXDOWRWKH ZKLFKLVWKHPD[LPXPYDOXHIRUQRQVZD\IUDPHVVHH
Therefore, slenderness effects must be considered.
Step 3 – Determine the magnified moments
0DJQL¿HGPRPHQWV
tion:
ACI 6.6.4.5
for each load combination are determined for this nonsway frame by the following equaEq. (7.2)
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,WLVDVVXPHGWKDWWKHFROXPQLVUHLQIRUFHGZLWKORQJLWXGLQDOEDUV
DQGWLHV7KH
of
results for sidesway to the right and sidesway to the left are equal because the gravity bending moments are equal to
]HUR
Table 7.43 Magnified Moments for Column C1
(EI)eff
M2,min
Mc
(ft-kips)
(ft-kips)
16.8
16.8
ȕdns
(kip-in2)
Pc (kips)
Cm
į
ACI Eq.
F
ACI Eq.
G
ACI Eq.
I
Load Combination
,WLVHYLGHQWIURP7DEOHWKDW
in all load combinations, which means the structural system is stable and
is greater than
in all load
QHHGQRWEHUHYLVHG $&, $OVRWKHPLQLPXPPRPHQW
FRPELQDWLRQV VHH7DEOH 6DPSOHFDOFXODWLRQVIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(T G DUHDVIROORZV
ACI Eq. (19.2.2.1a)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(IIHFWLYHÀH[XUDOVWLIIQHVV
used in this example:
FDQEHFDOFXODWHGE\DQ\RIWKHHTXDWLRQVLQ7DEOH$&,(T E LV
The moment of inertia of the reinforcement about the centroidal axis of the column,
is determined by multiplying the area of the reinforcing bars by the square of the distance from the centroid of the reinforcing bars to the
centroid of the column (the moment of inertia of a reinforcing bar about its centroidal axis is negligible):
Eq. (7.4)
For the column bent in double curvature:
Table 7.8
400
350
Axial Force ሺkipsሻሻ
300
݂௦ ൌ 0
250
200
݂௦ ൌ 0.5݂௬
150
100
50
0
0
25
50
75
Bending Moment ሺft-kipsሻ
Figure 7.54 Design strength interaction diagram for column C1.
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7-104
100
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Eq. (7.3)
Step 4 – Check the adequacy of the assumed longitudinal reinforcement
7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUFROXPQ&LVJLYHQLQ)LJXUH$OVRVKRZQLQWKH¿JXUHDUHWKH
load combination points for the factored axial forces LQ7DEOHDQGWKHFRUUHVSRQGLQJIDFWRUHGEHQGLQJPRLQ7DEOHZKHUHDSSOLFDEOH%HFDXVHDOOORDGFRPELQDWLRQVIDOOZLWKLQWKHGHVLJQVWUHQJWKLQWHUDFWLRQ
ments
diagram, the column is adequate.
8VHORQJLWXGLQDOEDUV
7.11.23 Example 7.23 – Determination of Nonsway or Sway Frame: Building #1 (Framing Option B),
Rectangular, Tied Column is Part of the LFRS, SDC A
'HWHUPLQHZKHWKHUWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%LVDQRQVZD\RUVZD\IUDPHDVVXPLQJDOOWKH
FROXPQVDUHLQVTXDUH VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVQRUPDOZHLJKW
DQG*UDGHUHLQIRUFHPHQW
concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the total loads in the first story
Roof:
7\SLFDOÀRRU
The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI
(T G SURGXFHVWKHODUJHVWORDG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 2 – Determine the stability index for the first story
An elastic analysis of the structure was performed with the building subjected to the north-south wind forces in
7DEOHDVVXPLQJWKHFROXPQVDUH¿[HGDWWKHEDVH7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHZHUHXVHGIRU
the columns, beams, and slabs. The following results were obtained from the analysis:
The stability index,
is determined from the following equation:
Table 7.3
Because
WKHIUDPHLQWKH¿UVWVWRU\LVDVZD\IUDPH VHH$&, 7.11.24 Example 7.24 – Check if Slenderness Effects Must be Considered: Building #1 (Framing Option B),
Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame
&KHFNLIVOHQGHUQHVVHIIHFWVPXVWEHFRQVLGHUHGIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%
DVVXPLQJWKHFROXPQLVLQVTXDUH VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
,WLVGHWHUPLQHGLQ([DPSOHWKDWWKHIUDPHLQWKH¿UVWVWRU\LVDVZD\IUDPH7KHUHIRUHVOHQGHUQHVVHIIHFWVFDQ
EHQHJOHFWHGZKHQWKHIROORZLQJHTXDWLRQLVVDWLV¿HG
Table 7.6
The effective length factor, LVGHWHUPLQHGIURP$&,)LJ5IRUVZD\IUDPHV,QWKHDQDO\VLVWKHFROXPQVDUH
At the top of the column,
is determined using the
DVVXPHGWREH¿[HGDWWKHEDVHVRVWLIIQHVVUDWLR
UDWLRRIWKHVWLIIQHVVRIWKHFROXPQVWRWKHVWLIIQHVVRIWKHEHDPV7KHUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHDUH
used in calculating the stiffnesses.
ACI Eq. (19.2.2.1a)
&ROXPQEHORZWKHVHFRQGÀRRUOHYHO
&ROXPQDERYHWKHVHFRQGÀRRUOHYHO
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
Eq. (7.1)
)URPWKHDOLJQPHQWFKDUWLQ$&,)LJ5IRUVZD\IUDPHVZLWK
HTXDOWR
and
is approximately
Therefore, slenderness effects must be considered.
7.11.25 Example 7.25 – Determination of Longitudinal Reinforcement: Building #1 (Framing Option B),
Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to Uniaxial
Bending and Axial Forces, Slenderness Effects
'HWHUPLQHWKHUHTXLUHGORQJLWXGLQDOUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
%DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVDQG
Step 1 – Determine the factored axial forces and bending moments
Roof:
Floors:
This column is part of the LFRS.
$VXPPDU\RIWKHD[LDOIRUFHVDQGEHQGLQJPRPHQWVIRUFROXPQ'LQWKH¿UVWVWRU\IURPDQDQDO\VLVRIWKHVWUXFWXUHLVJLYHQLQ7DEOHIRUVLGHVZD\WRWKHULJKW 665 DQGVLGHVZD\WRWKHOHIW 66/ @Seismicisolation
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.44 Summary of Axial Forces and Bending Moments for Column D1
Bending Moment (ft-kips)
Load Case
Axial Force (kips)
Top
Bottom
Dead
Roof live
Live
$&,(T D
3.9
$&,(T E
316.9
ņ
ņ
Wind
Load Combination
$&,(T F
SSR
SSL
SSR
SSL
SSR
SSL
$&,(T G
$&,(T I
Step 2 – Determine the total moments including slenderness effects
The total moments,
ACI 6.6.4.6
at the end of the column are determined by the following equation:
Eq. (7.6)
These moments are used because they are greater than the
PRPHQWVGHWHUPLQHGE\(T In this example, the nonsway moments
are due to gravity load effects and the sway moments
ZLQGORDGHIIHFWV7KHVHPRPHQWVDUHJLYHQLQ7DEOHIRUDOOORDGFRPELQDWLRQV
are due to
Table 7.45 Summary of Nonsway and Sway Moments (ft-kips) for Column D1
M1
Load Combination
M2
M1ns
M2ns
M1s
M2s
$&,(T D
3.9
3.9
ņ
ņ
$&,(T E
ņ
ņ
ņ
ņ
116.1
$&,(T F
SSR
SSL
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.45 Summary of Nonsway and Sway Moments (ft-kips) for Column D1 (cont.)
M1
Load Combination
M2
M1ns
SSL
SSL
M2s
69.1
69.1
SSR
$&,(T I
M1s
SSR
$&,(T G
M2ns
7KHIROORZLQJHTXDWLRQLVXVHGWRGHWHUPLQHWKHPRPHQWPDJQL¿FDWLRQIDFWRU
for sway frames:
Eq. (7.7)
The stability index,
LVGHWHUPLQHGE\$&,(T IRUHDFKORDGFRPELQDWLRQWKDWLQFOXGHV
$VXPPDU\RIWKHVOHQGHUQHVVFDOFXODWLRQVIRUFROXPQ'LVJLYHQLQ7DEOH
Table 7.46 Summary of Slenderness Calculations for Column D1
Ȉ3u
ǻo
Vus
(kips)
(in.)
(kips)
$&,(T D
ņ
ņ
$&,(T E
ņ
Q
įs
ņ
ņ
ņ
ņ
ņ
ņ
ņ
ņ
ņ
ņ
ņ
ņ
SSR
1.11
SSL
1.11
SSR
SSL
SSR
SSL
Load Combination
$&,(T F
$&,(T G
$&,(T I
M2
(ft-kips)
6DPSOHFDOFXODWLRQVIRUWKHORDGFRPELQDWLRQLQ$&,(T G IRUVLGHVZD\WRWKHULJKWDUHDVIROORZV
Example 7.23, Step 1
Example 7.23, Step 2
Example 7.23, Step 2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
,WFDQEHGHWHUPLQHGEDVHGRQWKHUHTXLUHPHQWVLQ$&,WKDWPD[LPXPPRPHQWVRFFXUDWWKHHQGVRIWKH
columns.
Step 3 – Determine the required area of longitudinal reinforcement
A 1 percent reinforcement ratio is initially selected for this column. Therefore,
1,400
1,200
݂௦ ൌ 0
Axial Force ሺkipsሻሻ
1,000
800
݂௦ ൌ 0.5݂௬
600
400
200
0
0
100
200
300
400
500
600
700
Bending Moment ሺft-kipsሻ
Figure 7.55 Design strength interaction diagram for column D1.
7U\EDUV
7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUFROXPQ'UHLQIRUFHGZLWKEDUVLVJLYHQLQ)LJXUH$OVR
VKRZQLQWKH¿JXUHDUHWKHORDGFRPELQDWLRQSRLQWVIRUIDFWRUHGD[LDOIRUFHV LQ7DEOHDQGWKHFRUUHVSRQGLQ7DEOH7KHFROXPQLVDGHTXDWHEHFDXVHDOOWKHORDGFRPELQDWLRQSRLQWVIDOO
ing total bending moments
within the design strength interaction diagram.
8VHEDUV
7.11.26 Example 7.26 – Determination of Transverse Reinforcement: Building #1 (Framing Option B),
Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected to Uniaxial
Bending and Axial Forces, Slenderness Effects
'HWHUPLQHWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ
%DVVXPLQJDLQVTXDUHWLHGFROXPQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPV
DQG*UDGHUHLQIRUFHPHQW
normalweight concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the required tie bar size
ACI 25.7.2.2
7KHUHTXLUHGWLHEDUVL]HLVGHWHUPLQHGXVLQJ$&,
)RUORQJLWXGLQDOEDUVXVHWLHV
Step 2 – Determine the required tie bar spacing
ACI 25.7.2.1
7KHUHTXLUHGWLHEDUVSDFLQJLVGHWHUPLQHGXVLQJ$&,WKHWLHEDUVL]HDQGVSDFLQJZLOOWKHQEHFKHFNHGWR
determine if they are adequate for shear strength requirements.
7U\WLHVVSDFHGDWLQRQFHQWHU
Step 3 – Check the adequacy of the tie bar size and spacing for shear
ACI 22.5
Maximum factored shear force,
LVREWDLQHGIURP$&,(T G IRUVLGHVZD\WRWKHULJKWDQGPXVWLQFOXGHWKH
HIIHFWVIURPVOHQGHUQHVV VHH7DEOHVDQG Check if the design shear strength of the concrete,
LVSURYLGHGRUQRW VHH7DEOH alone is adequate, which is determined based on whether
Eq. (7.26)
The minimum area of shear reinforcement is greater than
LQRQFHQWHU'HWHUPLQH based on
SURYLGHGE\WKHWLHVVSDFHG
Table 7.12
Eq. (7.23)
where
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 7.13
WKHUHDUHEDUVORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWK
IURPWKHH[WUHPHFRPSUHVVLRQ¿EHU Use
ZKLFKFRUUHVSRQGVWR$&,(T G ACI 22.5.5.1.2
Also,
VRPLQLPXPVKHDUUHLQIRUFHPHQWQHHGQRWEHSURYLGHG $&, 7KHUHIRUHXVHWLHVVSDFHGDWLQRQFHQWHU
7.11.27 Example 7.27 – Determination of Dowel Reinforcement at the Foundation: Building #1 (Framing
Option B), Rectangular, Tied Column is Part of the LFRS, SDC A, Sway Frame, Column Subjected
to Uniaxial Bending and Axial Forces, Slenderness Effects
'HWHUPLQHWKHUHTXLUHGGRZHOUHLQIRUFHPHQWIRUFROXPQ'LQWKH¿UVWVWRU\RI%XLOGLQJ)UDPLQJ2SWLRQ%
DVVXPLQJDLQVTXDUHWLHGFROXPQVXSSRUWHGE\DPDWIRXQGDWLRQZLWKDWKLFNQHVVRIIWLQ VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
DQG*UDGHUHLQIRUFHPHQW
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOHVWKURXJK
Step 1 – Check the bearing stresses on the column and mat foundation
ACI 16.3.3.4
Bearing strength of the column.
Factored bearing stress,
is determined for compression plus bending using the factored axial forces and bending
PRPHQWVGHWHUPLQHGE\$&,(TV E DQG G >VHH7DEOHVDQG@
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7-112
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• $&,(T G Design bearing strength:
Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the
VL]HRIWKHORQJLWXGLQDOEDUVLQWKHFROXPQ7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQVLRQIRUFHV
are adequately transferred from the column into the footing.
Bearing strength of the mat foundation.
There is no need to check the bearing strength of the mat foundation because interface reinforcement must be proYLGHGGXHWRWKHQHWWHQVLRQVWUHVVIURPWKHIDFWRUHGD[LDOIRUFHDQGEHQGLQJPRPHQWGHWHUPLQHGE\$&,(T G Step 2 – Determine the required interface reinforcement
ACI 16.3.4.1
7U\GRZHOEDUV7KLVUHLQIRUFHPHQWPDWFKHVWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHFROXPQDQGHQVXUHVWKDW
both the compression and tension forces are adequately transferred through the interface.
Check minimum area of reinforcement across the interface:
Step 3 – Check the development of the dowel bars in the footing
,WLVHYLGHQWIURP)LJXUHWKDWWKHVWUHVVLQWKHORQJLWXGLQDOUHLQIRUFHPHQWIDUWKHVWIURPWKHH[WUHPHFRPSUHVVLRQIDFHLVWHQVLOHIRUDQXPEHURIORDGFRPELQDWLRQV WKDWLVWKHORDGFRPELQDWLRQVIDOOZLWKLQ=RQHVDQGRIWKH
LQWHUDFWLRQGLDJUDPVHH)LJXUH 7KHUHIRUHWKHGRZHOEDUVPXVWEHGHYHORSHGIRUWHQVLRQLQWRWKHIRRWLQJ DQG
into the column).
$VWDQGDUGGHJUHHKRRNZLOOEHSURYLGHGDWWKHHQGVRIWKHGRZHOEDUV'HWHUPLQHWKHWHQVLRQGHYHORSPHQW
RIWKHGRZHOEDUVWHUPLQDWLQJLQDKRRN
length,
Eq. (7.44)
Table 7.27
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$VVXPLQJWZROD\HUVRIEDUVLQWKHPDWIRXQGDWLRQZLWKDFOHDUFRYHURILQGHWHUPLQHWKHPLQLPXPPDW
thickness for the development of the dowel bars:
Eq. (7.45)
The minimum
LVOHVVWKDQWKHSURYLGHGPDWWKLFNQHVVRILQ7KHUHIRUHWKHGRZHOEDUVFDQEH
adequately developed in the mat.
Step 4 – Determine the development length of the dowel bars in the column
The dowel bars must be lap spliced to the longitudinal reinforcement in the column using a tension lap splice.
Because all the dowel bars are spliced at the same location, a Class B tension lap splice is required (ACI Table
Development length in tension,
RIWKHEDUV
Eq. (7.36)
Table 7.24
Set
ACI 25.4.2.4
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7-114
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
3URYLGHDODSVSOLFHOHQJWKRIIWLQ
Step 5 – Check horizontal force transfer
,WLVGHWHUPLQHGLQ([DPSOHWKDW
LVWUDQVIHUUHGKRUL]RQWDOO\EHWZHHQWKHFROXPQDQGPDWIRXQdation. The required area of shear-friction reinforcement is determined as follows assuming the surface between the
column and the mat has not been intentionally roughened:
Eq. (7.46), Table 7.28
7KHGRZHOEDUVSURYLGHV
across the interface, which is greater than the required amount of
Check the upper shear limit:
Table 7.29
5HLQIRUFHPHQWGHWDLOVIRUWKLVFROXPQDUHJLYHQLQ)LJXUH
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7-115
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ʹԢǦͲԣ
ʹԢǦͲԣ
ͶǦ͓ͳͳ
ͷԢǦͻԣ
ͶǦ͓ͳͳ†‘™‡Ž„ƒ”•
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Figure 7.56 Reinforcement details for column D1 in Examples 7.23 through 7.27.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 8
WALLS
8.1 Overview
$ZDOOLVGH¿QHGDVDYHUWLFDOVWUXFWXUDOHOHPHQWZLWKDKRUL]RQWDOOHQJWKWRWKLFNQHVVUDWLRJUHDWHUWKDQZKLFK
LVGHVLJQHGWRUHVLVWD[LDOIRUFHVODWHUDOIRUFHVRUERWK $&, 6WUXFWXUDOZDOOVDUHGHVLJQHGWRUHVLVWWKHHIIHFWV
IURPYHUWLFDOIRUFHVDQGODWHUDOIRUFHVLQWKHSODQHRIWKHZDOO WKDWLVLQWKHGLUHFWLRQSDUDOOHOWRWKHKRUL]RQWDO
length of the wall) and are commonly referred to as shear walls.
$FFRUGLQJWR5HIHUHQFHDORDGEHDULQJ RUEHDULQJ FRQFUHWHZDOOLVDZDOOVXSSRUWLQJPRUHWKDQSRXQGVSHU
linear foot of vertical load in addition to its own weight. A nonload-bearing (or, nonbearing) wall is any wall that is
not a load-bearing wall; such walls support essentially their own weight and possibly out-of-plane loads like those
from wind.
$[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVDUHUHVLVWHGE\RQHOD\HURUWZROD\HUVRIYHUWLFDODQGKRUL]RQWDO
reinforcement in a wall.
The design and detailing of cast-in-place concrete walls with nonprestressed reinforcement are covered in this chapter. Provisions for walls are given in ACI Chapter 11 and are applicable to walls in buildings assigned to Seismic
'HVLJQ&DWHJRU\ 6'& $%DQG&&DQWLOHYHUUHWDLQLQJZDOOVDUHQRWFRYHUHGKHUHVHH5HIHUHQFHIRUFRPSUHKHQVLYHGHVLJQDQGGHWDLOLQJUHTXLUHPHQWVIRUWKHVHW\SHVRIZDOOV6SHFL¿FGHVLJQUHFRPPHQGDWLRQVIRUFDVWLQ
place walls constructed with insulating concrete forms are given in the references in ACI R11.1.6.
8.2 Design Limits
8.2.1 Minimum Wall Thickness
Minimum thicknesses of bearing walls, nonbearing walls, and exterior basement and foundation walls are given in
ACI 11.3.1.1 (see Figure 8.1 for bearing and nonbearing walls). These minimum thicknesses need not be applied to
EHDULQJZDOOVDQGH[WHULRUEDVHPHQWDQGIRXQGDWLRQZDOOVGHVLJQHGE\WKHSURYLVLRQVLQ$&,RUDQDO\]HGE\
the alternative analysis method in ACI 11.8.
8.2.2 Intersecting Elements
For cast-in-place concrete walls where the factored axial force,
exceeds
the portion of the wall within
of the wall
WKHWKLFNQHVVRIWKHÀRRUV\VWHPPXVWKDYHDVSHFL¿HGFRQFUHWHFRPSUHVVLYHVWUHQJWKRIDWOHDVW
$&, 7KHIDFWRUUHÀHFWVWKHUHGXFHGFRQ¿QHPHQWLQÀRRUZDOOMRLQWVFRPSDUHGZLWKÀRRUFROXPQ
joints for members subjected to gravity loads.
8.3 Required Strength
8.3.1 Analysis Methods
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EHXVHGWRFDOFXODWHUHTXLUHGVWUHQJWK VHH$&,DQGUHVSHFWLYHO\ $[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVFDQEHGHWHUPLQHGE\DQ\RIWKHDQDO\VLVPHWKRGVLQ$&,
:KHQSHUIRUPLQJD¿UVWRUGHURUDQHODVWLFVHFRQGRUGHUODWHUDOORDGDQDO\VLVWKHVHFWLRQSURSHUWLHVLQ$&,
may be used in lieu of a more sophisticated analysis to account for member cracking and other effects. The reduced
moments of inertia in ACI Table 6.6.3.1.1(a) and the alternative moments of inertia in ACI Table 6.6.3.1.1(b) for
YDULRXVVWUXFWXUDOPHPEHUVDUHJLYHQLQ7DEOHDQG7DEOHUHVSHFWLYHO\7KHUHGXFHGPRPHQWVRILQHUWLDLQ
Table 8.1 are based on an analysis where strength-level (factored) loads are used. For an analysis based on serviceOHYHOORDGVLWLVSHUPLWWHGWRXVHUHGXFHGPRPHQWVRILQHUWLDHTXDOWRWLPHVWKHYDOXHVLQ7DEOHSURYLGHGWKH
moments of inertia do not exceed the gross moments of inertia $&, @Seismicisolation
@Seismicisolation
8-1
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
κ௨ κ௨ ݄
κ௪ κ௪ ݄ ൒ greater of
‫ ۓ‬lesser of ൝
‫۔‬
‫ە‬4ᇳ
κ௪ Τ25
κ௨ Τ25 ݄ ൒ greater of
۰‫ כܔܔ܉ܟ ܏ܖܑܚ܉܍‬
‫ ۓ‬lesser of ൝
‫۔‬
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κ௪ Τ30
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*Applicable to walls designed by ACI 11.5.3 only
Figure 8.1 Minimum thicknesses for bearing and nonbearing walls.
Table 8.1 Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using
Strength-Level Loads
Member
Moment of Inertia
Columns
Walls — uncracked
Walls — cracked
Beams
)ODWSODWHVDQGÀDWVODEV
Table 8.2 Alternative Moments of Inertia to Use in a Linear Elastic First-Order or Second-Order Analysis Using Strength-Level or Service-Level Loads
Member
Moment of Inertia*
Columns and walls
%HDPVÀDWSODWHVDQGÀDWVODEV
*For a service-level load analysis, use service-level axial force and moment effects in the equation for columns and walls.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHPRUHUH¿QHGHTXDWLRQVIRUUHGXFHGPRPHQWVRILQHUWLDLQ7DEOHLQFOXGHWKH IDFWRUHGD[LDOIRUFH
RQDZDOOIRUWKHORDGFRPELQDWLRQXQGHUFRQVLGHUDWLRQ QRPLQDOD[LDOVWUHQJWKDW
and factored moment,
]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ DQG DUHDRIORQJLWXGLQDO YHUWLFDO UHLQIRUFHPHQW
These equations are applicable for strength-level and service-level loading, even though they are presented in
terms of factored load effects. For service-level analysis, the factored load effects are replaced by the corresponding
service-level load effects.
Regardless of the equations used to determine moments of inertia, the gross area of the section,
the analysis. Also, the cross-sectional area for calculation of shear deformations in a wall is
is to be used in
,QOLHXRIWKHUHTXLUHPHQWVRI$&,LWLVSHUPLWWHGWRDVVXPHLQD¿UVWRUGHUIDFWRUHGODWHUDOORDGDQDO\VLVWKDW
DOOWKHPHPEHUVLQDVWUXFWXUHKDYHDUHGXFHGPRPHQWRILQHUWLDHTXDOWRSHUFHQWRIWKHJURVVPRPHQWRILQHUWLD
$&, $PRUHGHWDLOHGDQDO\VLVFRQVLGHULQJWKHHIIHFWLYHVWLIIQHVVRIDOOPHPEHUVLVDOVRSHUPLWWHG
8.3.2 Factored Axial Force, Moment, and Shear
:DOOVPXVWEHGHVLJQHGIRUIDFWRUHGD[LDOIRUFHVEHQGLQJPRPHQWVDQGVKHDUIRUFHVGXHWRLQSODQHDQGRURXW
of-plane forces, including the effects of slenderness where appropriate, for each applicable load combination (ACI
DQG ,QSODQHDQGRXWRISODQHIRUFHVDQGWKHLUHIIHFWVRQDEHDULQJZDOODUHLOOXVWUDWHGLQ
)LJXUHDQG)LJXUHUHVSHFWLYHO\
Axial force, ܲ௨
In-plane forces
In-plane shear force, ܸ௨
In-plane bending moment, ‫ܯ‬௨
Figure 8.2 In-plane forces on a bearing wall.
8.3.3 Slenderness Effects
Overview
Like columns, walls must be designed for the effects of slenderness where applicable. Determining whether slenderness effects need to be considered or not depends on whether the frame is nonsway or sway. According to ACI
DIUDPHLVFRQVLGHUHGWREHQRQVZD\ZKHUHWKHVWDELOLW\LQGH[ LVOHVVWKDQZKHUH is determined
E\$&,(TXDWLRQ (8.1)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Axial force, ܲ௨
Axial force, ܲ௨
Eccentricity, ݁
Out-of-plane forces
Out-of-plane shear force, ܸ௨
Out-of-plane bending moment, ‫ܯ‬௨
Out-of-plane bending moment, ‫ܯ‬௨
Figure 8.3 Out-of-plane forces on a bearing wall.
where Ȉ3u = total factored vertical load in the story
ǻo ¿UVWRUGHUUHODWLYHODWHUDOGHÀHFWLRQEHWZHHQWKHWRSDQGERWWRPRIWKHVWRU\
Vus IDFWRUHGKRUL]RQWDOVWRU\VKHDULQDVWRU\
ℓc = length of the walls measured center-to-center of the joints in the frame
More often than not, the frame in the direction of analysis can be considered nonsway where structural walls are
used to resist the lateral forces in that direction.
Whether slenderness effects need to be considered or not for nonsway and sway frames depends on the slenderness
LQWKHGLUHFWLRQRIDQDO\VLV7KHWHUPVLQWKHVOHQGHUQHVVUDWLRDUHGH¿QHGLQ7DEOH VHH
ratio of the wall,
$&, Table 8.3 Slenderness Ratio, kℓu / r
Nonsway Frames
Sway Frames
Unsupported length,
LVWKHFOHDUGLVWDQFHEHWZHHQÀRRUVODEVEHDPVRURWKHU
members capable of providing lateral support in the direction of analysis.
Radius of gyration,
is permitted to be calculated by the following:
1.
for rectangular walls where is the dimension of the wall in the
direction of analysis (that is, either for slenderness about the minor axis or
for slenderness about the major axis)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
6OHQGHUQHVVHIIHFWVDUHSHUPLWWHGWREHQHJOHFWHGZKHQWKHHTXDWLRQVLQ7DEOHDUHVDWLV¿HG $&, Table 8.4 Limits for Slenderness Effects
Wall Bracing Condition
Slenderness Ratios Where Slenderness is Permitted to Be Neglected
Not braced against sidesway (sway)
Braced against sidesway (nonsway)
*Ratio
is negative if a wall is bent in single curvature and is positive if a wall is bent in double curvature.
,WLVPRUHOLNHO\WKHRXWRISODQHEHQGLQJPRPHQWVLQDZDOOZRXOGQHHGWREHPDJQL¿HGWKDQWKHLQSODQHEHQGLQJ
moments: In the direction parallel to the length of the wall, slenderness effects can usually be neglected because the
radius of gyration of the wall, in that direction is relatively large, which results in a slenderness ratio less than
WKHOLPLWVLQ7DEOH,QWKHGLUHFWLRQSDUDOOHOWRWKHWKLFNQHVVRIWKHZDOOWKHVOHQGHUQHVVUDWLRLVPRUHOLNHO\WREH
greater than the prescribed slenderness limits, which means factored bending moments about the minor axis of the
ZDOOZRXOGQHHGWREHPDJQL¿HG
7KHIROORZLQJPHWKRGVDUHSHUPLWWHGWREHXVHGWRDFFRXQWIRUVOHQGHUQHVVHIIHFWV $&, ƒ 0RPHQWPDJQL¿FDWLRQPHWKRG $&,
ƒ /LQHDUHODVWLFVHFRQGRUGHUDQDO\VLV $&,
ƒ Inelastic analysis (ACI 6.8)
ƒ Alternative method for out-of-plane slender wall analysis (ACI 11.8)
7KHPRPHQWPDJQL¿FDWLRQPHWKRGDQGWKHDOWHUQDWLYHPHWKRGDUHFRYHUHGEHORZ,QIRUPDWLRQRQWKHOLQHDUHODVWLF
VHFRQGRUGHUDQDO\VLVDQGWKHLQHODVWLFDQDO\VLVLVJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ
Moment Magnification Method
'HWDLOVRIWKHPRPHQWPDJQL¿FDWLRQPHWKRGDUHJLYHQLQ6HFWLRQRIWKLVSXEOLFDWLRQ:KHQGHWHUPLQLQJWKH
E\$&,(TXDWLRQ WKHHIIHFWLYHVWLIIQHVV
determined by ACI Equation
critical buckling load,
F LVXVHGIRUZDOOV
(8.2)
LVWKHPRGXOXVRIHODVWLFLW\RIWKHFRQFUHWHGHWHUPLQHGLQDFFRUGDQFHZLWK$&, is the
In this equation,
is the ratio of
PRPHQWRILQHUWLDRIWKHZDOOGHWHUPLQHGE\WKHHTXDWLRQIRUFROXPQVDQGZDOOVLQ7DEOHDQG
the maximum factored sustained axial load to the maximum factored axial load associated with the same load combination, which is typically equal to the factored axial dead load divided by the total factored axial load on a wall.
As noted previously, frames are typically nonsway when structural walls are part of the frame; in such cases, the
GHWHUPLQHGE\$&,(TXDWLRQ ZKHUHDSSOLFDEOH
ZDOOLVWREHGHVLJQHGXVLQJWKHPDJQL¿HGPRPHQWV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Alternative Method for Out-of-Plane Slender Wall Analysis
Limitations
The design procedure given in ACI 11.8 can be used for the out-of-plane design of slender, reinforced concrete
ZDOOVSURYLGHGWKHIROORZLQJOLPLWDWLRQVDUHVDWLV¿HG $&, 1. The cross-section of the wall is constant over the entire height.
The wall is tension-controlled for out-of-plane moment effects.
is greater than or equal to the cracking moment,
3. 7KHGHVLJQÀH[XUDOVWUHQJWKRIWKHZDOO
and
VHH$&, where
The factored axial force,
at the mid-height section of the wall is less than or equal to
7KHFDOFXODWHGRXWRISODQHGHÀHFWLRQGXHWRVHUYLFHORDGV
including P-delta effects, is less than or equal
where
is the length of the wall measured center-to-center of the joints.
to
:KHQRQHRUPRUHRIWKHVH¿YHOLPLWDWLRQVDUHQRWVDWLV¿HGWKLVPHWKRGFDQQRWEHXVHG)RUH[DPSOHZDOOVZLWK
relatively large window and door openings are not considered to have a constant cross-section over their full height,
VRWKHDOWHUQDWLYHPHWKRGLVQRWDSSOLFDEOHEHFDXVHWKH¿UVWOLPLWDWLRQLVQRWVDWLV¿HG
Modeling
7KHIROORZLQJPRGHOLQJUHTXLUHPHQWVPXVWEHVDWLV¿HGZKHQXVLQJWKHDOWHUQDWLYHPHWKRG $&, 1. 7
KHZDOOPXVWEHDQDO\]HGDVDVLPSO\VXSSRUWHGD[LDOO\ORDGHGPHPEHUVXEMHFWHGWRDXQLIRUPO\GLVWULEXWHG
ODWHUDOORDGZLWKWKHPD[LPXPPRPHQWDQGGHÀHFWLRQRFFXUULQJDWWKHPLGKHLJKWRIWKHZDOO VHH)LJXUH applied to the wall above any section are assumed to be distributed over a
Concentrated gravity loads,
ZLGWKHTXDOWRWKHEHDULQJZLGWKSOXVDZLGWKRQHDFKVLGHWKDWLQFUHDVHVDWDVORSHRIYHUWLFDOWRKRUL]RQWDO
EXWQRWH[WHQGLQJEH\RQGWKHVSDFLQJRIWKHFRQFHQWUDWHGORDGVRUWKHHGJHVRIWKHZDOOSDQHO VHH)LJXUH ܲ௨௚ ȟ௨ κ௖ Ȁʹ
‫ݓ‬௨ Figure 8.4 Wall analysis in accordance with ACI 11.8.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܵ
κ௖ Ȁʹ
ܵ
ʹ ʹ
ͳ
ͳ
‫ܤ‬
ሺ–›’Ǥ ሻ
κ௖ Ȁʹ
‫ܧ‬
‫ݓ‬ଵ ‫ݓ‬ଶ ‫ݓ‬ଶ ‫ݓ‬ଵ ൌ ‫ ܤ‬൅ ሺκ௖ ΤͶሻ ൅ ‫ ܧ‬൑ ܵ
‫ݓ‬ଶ ൌ ‫ ܤ‬൅ ሺκ௖ Τʹሻ ൑ ܵ
Figure 8.5 Distribution of concentrated loads in the alternative design method for walls.
Factored Moment
7KHIROORZLQJVWUHQJWKHTXDWLRQPXVWEHVDWLV¿HGDWWKHPLGKHLJKWRIDZDOO
(8.3)
Two methods to determine
which includes the effects due to slenderness, are given in ACI 11.8.3: (1) by iteraWLYHFDOFXODWLRQDQG E\GLUHFWFDOFXODWLRQ
Iterative Calculation Method
In the iterative calculation method, the factored moment,
FRQVLVWVRIWZRSDUWV>$&,(TXDWLRQ D @
(8.4)
The moment
LVWKHPD[LPXPIDFWRUHGPRPHQWDWWKHPLGKHLJKWRIWKHZDOOGXHWR D ODWHUDOORDGVDQGRU E applied at an eccentricity from the centroid of the wall. Note that
does not include
vertical factored loads,
3GHOWDHIIHFWV7KHGHÀHFWLRQ LVWKHWRWDOGHÀHFWLRQDWWKHPLGKHLJKWRIWKHZDOOGXHWRWKHIDFWRUHGORDGV7KLV
GHÀHFWLRQLVGHWHUPLQHGE\$&,(TXDWLRQ E (8.5)
In this equation,
is the length of the wall, measured center-to-center of joints, and
is the moment of inertia of
the cracked wall section transformed to concrete, which is determined by ACI Equation (11.8.3.1c):
(8.6)
In this equation,
is the modulus of elasticity of the reinforcing steel,
is the modulus of elasticity of the conis the area of vertical reinforcement in the wall, LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWR
crete,
LVWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LVDQG
is the length of
the centroid of
must be taken greater than or equal to 6 in this equation.
the wall. The ratio
In the strength design method, the neutral axis depth, is related to the depth of the equivalent rectangular stress
In general, is the force in the tension reinforcement
divided by the equivalent comblock, by
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
pressive stress
dinal reinforcement,
times the width of the section. In the alternative design method, an effective area of longituis used in the determination of and , and is calculated by the following equation:
(8.7)
This effective area of longitudinal reinforcement is used in the determination of
LQ(TXDWLRQ @
>VHHWKH¿UVWWHUPLQSDUHQWKHVHV
Therefore, the depth of the equivalent stress block for bending about the minor axis of the wall can be determined
by the following equation:
(8.8)
$OVRWKHGLVWDQFHIURPWKHH[WUHPHFRPSUHVVLRQ¿EHUWRWKHQHXWUDOD[LVLVWKHIROORZLQJ
(8.9)
,WLVHYLGHQWIURP(TXDWLRQ WKDWWKHGHÀHFWLRQ
is a function of
+RZHYHU
is a function of
ZKLFKLVDSSDUHQWIURP(TXDWLRQ 7KLVFOHDUO\LOOXVWUDWHVWKHLWHUDWLYHQDWXUHLQKHUHQWWRWKLVPHWKRG7KXV
and then performing several calculation iterations until convergence
can be determined by assuming a value of
occurs.
Direct Calculation Method
In the direct calculation method,
is determined by ACI Equation (11.8.3.1d):
(8.10)
Regardless of which method is used to determine
is determined by the following equation:
(8.11)
2XWRI3ODQH'HÀHFWLRQ
,QDGGLWLRQWRVDWLVI\LQJVWUHQJWKUHTXLUHPHQWVWKHIROORZLQJGHÀHFWLRQUHTXLUHPHQWPXVWEHVDWLV¿HG
ZKLFKLVWKH¿IWKOLPLWDWLRQLQ$&,
7KHPD[LPXPGHÀHFWLRQGXHWRVHUYLFHORDGV
which includes second-order effects, depends on the magnitude
>$&,(TXDWLRQ @
of the service load moment,
(8.12)
The terms
respectively.
and
DUHWKH¿UVWRUGHUVHUYLFHOHYHOEHQGLQJPRPHQWDQGD[LDOORDGDWWKHPLGKHLJKWRIWKHZDOO
exceeds two-thirds of the cracking moment,
2XWRISODQHGHÀHFWLRQVLQFUHDVHUDSLGO\ZKHQ
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8-8
Thus,
is
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
(8.13)
(8.14)
In these equations,
and
WKHQRPLQDOÀH[XUDOVWUHQJWK
E @
DUHWKHPLGKHLJKWGHÀHFWLRQVFRUUHVSRQGLQJWRWKHFUDFNLQJPRPHQW
and
UHVSHFWLYHO\DQGDUHGHWHUPLQHGDVIROORZV>VHH$&,(TXDWLRQV D DQG
(8.15)
(8.16)
Because
is a function of
>VHH(TXDWLRQ RU @DQG
is a function of
>VHH(TXDWLRQ @
A value of
is obtained by iteration, that is, an initial value of
is asthere is no closed-form solution for
sumed and calculations are performed until convergence occurs.
6HUYLFHOHYHOORDGFRPELQDWLRQVDUHQRWJLYHQLQ$&,&KDSWHUEXWDUHGLVFXVVHGLQ$SSHQGL[&&RI5HIHUHQFH
3. For structures subjected to wind, the following service-level load combination is recommended for calculating
VHUYLFHOHYHOODWHUDOGHÀHFWLRQVRIZDOOVDQGIUDPHV
(8.17)
where
are the effects due to wind forces obtained from serviceability wind speeds, which are given in Figures
&&&&&&DQG&&IRU\HDU\HDU\HDUDQG\HDUPHDQUHFXUUHQFHLQWHUYDOVUHVSHFWLYHO\8VLQJZLQGVSHHGVEDVHGRQPRUHWKDQD\HDUPHDQUHFXUUHQFHLQWHUYDOLQFKHFNLQJODWHUDOGHÀHFWLRQVLV
excessively conservative.
For slender walls subjected to strength-level earthquake effects,
HYDOXDWHVHUYLFHOHYHOODWHUDOGHÀHFWLRQV
the following load combination can be used to
(8.18)
8.4 Design Strength
8.4.1 General
)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHIROORZLQJHTXDWLRQVPXVWEHVDWLV¿HGDWDQ\
VHFWLRQLQDZDOO $&, (8.19)
(8.20)
(8.21)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ
IDFWRUVSHUWLQHQWWRWKHGHVLJQRIUHLQIRUFHGFRQFUHWHZDOOVLVJLYHQLQ7DEOHIRUWUDQVYHUVHUHLQIRUFHPHQWRWKHU
LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWK
WKDQVSLUDOV7KHVWUDLQXVHGWRGH¿QHDFRPSUHVVLRQFRQWUROOHGVHFWLRQ
divided by the modulus of elasticity of the reinforcing steel,
which can be taken as
of the reinforcement,
SVLIRUDQ\JUDGHRIUHLQIRUFHPHQW $&, Table 8.5 Strength Reduction Factors, Ҁ
Net Tensile Strain, İt
Classification
Strength Reduction Factor, Ҁ*
Compression-controlled
Transition**
Tension-controlled
* Applicable to transverse reinforcement other than spirals.
**Sections classified as transition are permitted to use corresponding to compression-controlled sections.
For bearing walls subjected primarily to uniaxial compression forces, Equation (8.19) is applicable. For bearing
ZDOOVVXEMHFWHGWRFRPELQHGD[LDOIRUFHVDQGEHQGLQJPRPHQW LQSODQHRURXWRISODQH (TXDWLRQV DQG PXVWERWKEHVDWLV¿HGIRUDOODSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQV $&, $OWHUQDWLYHO\EHDULQJZDOOV
VXEMHFWHGWRD[LDOIRUFHVDQGRXWRISODQHEHQGLQJPRPHQWVDUHSHUPLWWHGWREHGHVLJQHGXVLQJWKHVLPSOL¿HGGHVLJQ
PHWKRGLQ$&,SURYLGHGWKHOLPLWDWLRQVRIWKDWPHWKRGDUHVDWLV¿HG VHH6HFWLRQRIWKLVSXEOLFDWLRQ )RUQRQEHDULQJZDOOV(TXDWLRQ PXVWEHVDWLV¿HGZKHUH
VWUHQJWKUHTXLUHPHQWVRI$&, $&, LVGHWHUPLQHGLQDFFRUGDQFHZLWKWKHÀH[XUDO
(TXDWLRQ PXVWEHVDWLV¿HGIRUERWKLQSODQHDQGRXWRISODQHVKHDUIRUFHVRQDZDOO
Methods to determine the nominal strengths
DQGUHVSHFWLYHO\
combined
and
DUHJLYHQLQ6HFWLRQV
8.4.2 Nominal Axial Strength
Nominal Axial Compressive Strength
The nominal axial compressive strength,
of a nonslender, reinforced concrete wall is determined in accordance
must be less than or equal to the maximum nominal
ZLWK$&,7KHQRPLQDOD[LDOFRPSUHVVLYHVWUHQJWK
ZKLFKLVGHWHUPLQHGXVLQJ$&,7DEOHDQGWKHQRPLQDOD[LDOVWUHQJWKDW
axial compressive strength,
]HURHFFHQWULFLW\ GHWHUPLQHGE\$&,(TXDWLRQ (8.22)
When calculating
WKHYDOXHRIWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHORQJLWXGLQDOUHLQIRUFHPHQW
is limited to
SVLHYHQWKRXJK for the longitudinal (vertical) reinforcement used in the wall may be larger than that (ACI
Nominal Axial Tensile Strength
The nominal axial tensile strength of a nonprestressed, reinforced concrete wall,
must be less than or equal to
ZKLFKLVGHWHUPLQHGE\$&,(TXDWLRQ the maximum nominal axial tensile strength,
(8.23)
It is evident that the entire tensile force must be resisted by the longitudinal reinforcement in a wall.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
8.4.3 Nominal Strength of Walls Subjected to Moment and Axial Forces
Overview
The nominal strength of a reinforced concrete wall subjected to both moment and axial force is determined using
HTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ$&,ZKLFKDUHVXPPDUL]HGLQ7DEOH
Table 8.6 Design Assumptions for Concrete and Nonprestressed Reinforcement
Material
Assumptions
1. 0D[LPXPVWUDLQLQWKHH[WUHPHFRQFUHWHFRPSUHVVLRQ¿EHULVDVVXPHGHTXDOWR
7HQVLOHVWUHQJWKRIFRQFUHWHLVQHJOHFWHGLQÀH[XUDODQGD[LDOVWUHQJWKFDOFXODWLRQV
3. The relationship between concrete compressive stress and strain is to be represented by a
UHFWDQJXODUWUDSH]RLGDOSDUDEROLFRURWKHUVKDSHWKDWUHVXOWVLQSUHGLFWLRQRIVWUHQJWKLQ
substantial agreement with results of comprehensive tests.
7KHHTXLYDOHQWUHFWDQJXODUFRQFUHWHVWUHVVGLVWULEXWLRQLQDFFRUGDQFHZLWK$&,
WKURXJKVDWLV¿HVWKHUHTXLUHPHQWRI$&,IRUWKHUHODWLRQVKLSEHWZHHQ
concrete compressive stress and strain.
Concrete
is to be assumed uniformly distributed over an equivalent
A concrete stress of
FRPSUHVVLRQ]RQHERXQGHGE\WKHHGJHVRIWKHFURVVVHFWLRQDQGDOLQHSDUDOOHOWRWKH
neutral axis located a distance IURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQZKHUH
.
7KHGLVWDQFHIURPWKH¿EHURIPD[LPXPFRPSUHVVLYHVWUDLQWRWKHQHXWUDOD[LV is to be
measured perpendicular to the neutral axis.
Values of
are as follows:
•
•
•
Nonprestressed
reinforcement
1. Deformed reinforcement used to resist tensile or compressive forces must conform to
$&,
The stress-strain relationship and the modulus of elasticity of the deformed reinforcePHQWDUHWREHLGHDOL]HGLQDFFRUGDQFHZLWK$&,DQG
Rectangular Sections
The general principles and assumptions of the strength design method in Table 8.6 are applied to the rectangular reinforced concrete wall given in Figure 8.6. For purposes of discussion, it is assumed the longitudinal reinforcement
LQOD\HULVORFDWHGFORVHVWWRWKHH[WUHPHFRPSUHVVLRQ¿EHURIWKHVHFWLRQ
7KHHTXDWLRQVLQ)LJXUHRIWKLVSXEOLFDWLRQIRUUHFWDQJXODUFROXPQVHFWLRQVFDQEHXVHGWRGHWHUPLQHWKH
DQGWKHQRPLQDOÀH[XUDOVWUHQJWK
of the rectangular wall in Figure 8.6 for a given
nominal axial strength,
which is the net tensile strain in the extreme layer of tension reinforcement at nominal strength, excluding
strain,
strains due to creep, shrinkage, and temperature.
I-, T-, and L-Shaped Sections
)RUZDOOVZLWKÀDQJHVZKHUHWKHÀDQJHVDUHLQFRPSUHVVLRQWKHQRPLQDOD[LDODQGÀH[XUDOVWUHQJWKVGHSHQGRQ
whether the depth of the equivalent stress block, LVOHVVWKDQRUJUHDWHUWKDQWKHÀDQJHWKLFNQHVV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ͲǤͺͷ݂௖ᇱ ݀ଵ ݄
‫ܨ‬௦ଵ ‫ܥ‬
ߝ௦ଶ ‫ܨ‬௦ଶ ߝ௦ଷ ‫ܨ‬௦ଷ ݀଺ ݀ହ ݀ସ ݀ଷ ܿ
݀ଶ ߝ௦ଵ ܽ ൌ ߚଵ ܿ
ͲǤͲͲ͵
κ௪ ݀଻ ߝ௦ସ ߝ௦ହ ߝ௦଺ ߝ௦଻ ൌ ߝ௧ ‫ܨ‬௦ସ ‫ܨ‬௦ହ ‫ܨ‬௦଺ ‫ܨ‬௦଻ Figure 8.6 Reinforced concrete wall subjected to moment and axial compression.
Where LVOHVVWKDQRUHTXDOWRWKHÀDQJHWKLFNQHVV
IRUDUHFWDQJXODUZDOOVHFWLRQ
and
can be determined using the equations in Figure
are determined based on a T- or L-shaped compression
Where LVJUHDWHUWKDQWKHÀDQJHWKLFNQHVV and
]RQH&RQVLGHUWKH,VKDSHGZDOOVHFWLRQLQ)LJXUHZKHUHWKHHTXLYDOHQWVWUHVVEORFNIDOOVLQWKHZHE7KHUHVXOcan be determined by the following equation:
tant compression force,
‫ݐ‬௙ (8.24)
κ௪ ܽ
ܾ௙ ݄
Figure 8.7 An I-shaped wall where the equivalent stress block falls in the web.
The strains, stresses, and forces in the longitudinal reinforcement and
are determined using the expressions in
)LJXUH7KHQRPLQDOÀH[XUDOVWUHQJWKLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ
(8.25)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ZKHUHWKHGLVWDQFHIURPWKHFHQWURLGRIWKHZDOOWRWKHFHQWURLGRIWKHFRPSUHVVLRQ]RQH
ing:
is equal to the follow-
(8.26)
6LPLODUH[SUHVVLRQVFDQEHGHULYHGIRU/VKDSHGFRPSUHVVLRQ]RQHV
,WLVDVVXPHGLQWKHDERYHGLVFXVVLRQWKDWWKHHQWLUHZLGWKRIWKHÀDQJHLVHIIHFWLYHLQUHVLVWLQJWKHFRPELQHGD[LDO
IRUFHDQGEHQGLQJPRPHQW$OWKRXJKQRWVSHFL¿FDOO\UHTXLUHGLQ$&,&KDSWHULWPD\EHZDUUDQWHGWRXVHD
ÀDQJHZLGWKOHVVWKDQWKHDFWXDOÀDQJHZLGWKLQWKHDQDO\VLV$PHWKRGWRGHWHUPLQHWKHSRUWLRQRIWKHÀDQJHFRQVLGHUHGWREHHIIHFWLYHIRUZDOOVVXEMHFWHGWRHDUWKTXDNHHIIHFWVLVJLYHQLQ$&,WKDWHIIHFWLYHÀDQJHZLGWK
FDQEHXVHGIRUDQ\ZDOOZLWKÀDQJHVLQOLHXRISHUIRUPLQJDPRUHUH¿QHGDQDO\VLV$QRWKHURSWLRQLVWRFRPSOHWHO\
and thickness
GLVUHJDUGWKHÀDQJHVDQGWRGHVLJQWKHZDOODVDUHFWDQJXODUVHFWLRQRIOHQJWK
Simplified Design Method
7KHVLPSOL¿HGGHVLJQPHWKRGLQ$&,FDQEHXVHGWRGHWHUPLQHWKHQRPLQDOD[LDOFRPSUHVVLYHVWUHQJWK
RIDZDOOZKHUHWKHIROORZLQJWZROLPLWDWLRQVDUHVDWLV¿HG
1. The wall has a solid, rectangular cross-section.
The resultant of all applicable factored loads falls within the middle third of the wall thickness.
acting at an eccentricity,
The second limitation is illustrated in Figure 8.8 for the case of a factored axial force,
less than or equal to
In general, the total eccentricity caused by all applicable factored load effects, including those from lateral loads, must be less than or equal to
ܲ௨ ݁ ൑ ݄Ȁ͸
݄
݄Ȁ͸
Figure 8.8 Eccentricity limitation in the simplified design method.
$&,(TXDWLRQ FDQEHXVHGWRGHWHUPLQH
ZKHQERWKOLPLWDWLRQVDUHVDWLV¿HG
(8.27)
In this equation,
7DEOH is the gross area of the wall and
LVWKHHIIHFWLYHOHQJWKIDFWRUJLYHQLQ$&,7DEOH VHH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.7 Effective Length Factor for Walls Designed by the Simplified Design Method
k
Boundary Conditions
Walls braced top and bottom
against lateral translation
Restrained against rotation at one or both ends
(top, bottom, or both)
Unrestrained against rotation at both ends
Walls not braced against lateral translation
A value of HTXDOWRLPSOLHVWKHHQGRIWKHZDOOLVDWWDFKHGWRDPHPEHUZLWKDÀH[XUDOVWLIIQHVVDWOHDVWHTXDO
to that of the wall in the direction of analysis. A value of HTXDOWRZRXOGEHDSSOLFDEOHIRUH[DPSOHWRIUHHVWDQGLQJ FDQWLOHYHU ZDOOVRUWRZDOOVFRQQHFWHGWRGLDSKUDJPVWKDWXQGHUJRVLJQL¿FDQWGHÀHFWLRQVZKHQVXEMHFWHG
to lateral loads.
(TXDWLRQ WDNHVLQWRFRQVLGHUDWLRQERWKHFFHQWULFLW\DQGVOHQGHUQHVVHIIHFWV,QRUGHUWRVDWLVI\VWUHQJWKUHmust be greater than or equal to the factored axial force,
where
quirements, the design strength,
for compression-controlled sections.
,WLVLQKHUHQWO\DVVXPHGLQWKHVLPSOL¿HGGHVLJQPHWKRGWKDWWKHZDOOVDUHGHVLJQHGFRQVLGHULQJ as a concentric
axial compression force. As such, it is best suited for relatively short walls subjected to vertical loads. Because the
the application of this method becomes very limited when lateral loads need to
eccentricity must not exceed
be considered.
8.4.4 Nominal Shear Strength
In-Plane Shear
For walls subjected to in-plane shear forces, the nominal shear strength,
ZLWK$&,DQG $&, , at a section is determined in accordance
(8.28)
The term
is the nominal shear strength of the concrete and
WUDQVYHUVH KRUL]RQWDO UHLQIRUFHPHQWLQWKHZDOO
is the nominal shear strength of the
7KHFRHI¿FLHQW GH¿QHVWKHUHODWLYHFRQWULEXWLRQRIWKHFRQFUHWHVWUHQJWKWRWKHQRPLQDOVKHDUVWUHQJWKDQGLV
determined by the following:
(8.29)
For walls where a net tension force is determined for the entire wall section,
>$&,@
is calculated by ACI Equation
(8.30)
The axial tension force,
is negative and has the units of pounds.
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8-14
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHPRGL¿FDWLRQIDFWRU UHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWRQRUPDOweight concrete of the same compressive strength is given in Table 8.8 based on equilibrium density and in Table
EDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[WXUH VHH$&, Table 8.8 Values of Ȝ Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
Table 8.9 Values of Ȝ Based on Composition of Aggregates
Concrete
All-lightweight
/LJKWZHLJKW¿QHEOHQG
Sand-lightweight
Sand-lightweight, coarse blend
Composition of Aggregates
)LQH$670&
&RDUVH$670&
)LQH&RPELQDWLRQRI$670&
and C33
&RDUVH$670&
Fine: ASTM C33
&RDUVH$670&
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&
and ASTM C33
Ȝ
WR(1)
WR (1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute
volume of fine aggregate.
(2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute
volume of aggregate.
The term LVWKHUDWLRRIWKHGLVWULEXWHGWUDQVYHUVH KRUL]RQWDO UHLQIRUFHPHQWWRWKHJURVVDUHDRIWKHFRQFUHWHDUHD
perpendicular to that reinforcement.
is the gross area of the cross-section of the wall resisting the shear force. For a rectangular wall with
The area
Illustrated in Figure 8.9 is the area
WREHXVHGLQ(TXDWLRQ ZKHUHWKHHQGVRIWKH
no openings,
ZDOODUHHQODUJHGRUIUDPHLQWRFROXPQV WKLVLVRIWHQUHIHUUHGWRDVD³EDUEHOO´ZDOOEHFDXVHRILWVVKDSH For walls with one or more openings, the area of the opening(s) must not be included in
(see Figure 8.9).
For walls with
WKHVWUXWDQGWLHPHWKRGLQ$&,&KDSWHUPD\EHXVHGWRGHVLJQIRULQSODQHVKHDU
IRUFHVLQVWHDGRIWKHPHWKRGGHVFULEHGDERYH $&, Out-of-Plane Shear
For walls subjected to out-of-plane shear forces, the nominal shear strength,
is determined by the one-way shear
VWUHQJWKUHTXLUHPHQWVLQ$&, $&, %HFDXVHVKHDUUHLQIRUFHPHQWLVJHQHUDOO\QRWSURYLGHGWRVDWLVI\
where the nominal shear strength of the concrete,
is deterone-way out-of-plane shear requirements,
PLQHGE\WKHIROORZLQJHTXDWLRQIURP$&,7DEOHZKHUH
(8.31)
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8-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
κ௪ ‫ܣ‬௖௩ ൌ ݄κ௪ κ௪ ݄
κ௢௣௘௡ ‫ܣ‬௖௩ ൌ ݄ሺκ௪ െ κ௢௣௘௡ ሻ
Figure 8.9 Determination of Acv
7KHVL]HHIIHFWPRGL¿FDWLRQIDFWRU , accounts for the phenomenon indicated in test results that the shear strength
attributed to concrete in members without shear reinforcement does not increase in direct proportion with member
GHSWK7KLVIDFWRULVGHWHUPLQHGE\$&,(TXDWLRQ >$&,@
(8.32)
,WLVHYLGHQWIURP(TXDWLRQ WKDW
LVOHVVWKDQIRUPHPEHUVZLWK
The term
LVHTXDOWRWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW , at the section divided by
where
is commonly
WDNHQDVRQHIRRW VHH)LJXUH $FFRUGLQJWR$&,5 may be taken as the sum of the areas of the lonJLWXGLQDOÀH[XUDOUHLQIRUFHPHQWORFDWHGPRUHWKDQWZRWKLUGVRIWKHRYHUDOOPHPEHUGHSWKDZD\IURPWKHH[WUHPH
FRPSUHVVLRQ¿EHU)RUW\SLFDOZDOOVHFWLRQV is the area of longitudinal tension reinforcement within the one-foot
GHVLJQVWULS VHH)LJXUHIRUWKHFDVHRIDZDOOZLWKWZROD\HUVRIORQJLWXGLQDOUHLQIRUFHPHQW $FFRUGLQJWRWKH¿UVW1RWHLQ$&,7DEOHWKHD[LDOIRUFH , which has the units of pounds, is to be taken
, and is to be taken as negative for tension
as positive for compression forces acting on the gross area of the wall,
$&, forces. Also,
Values of
used to calculate DUHOLPLWHGWRSVL WKHH[FHSWLRQLQ$&,LVQRWDSSOLFDEOHWRZDOOV
VHH$&, 7KLVOLPLWDWLRQRQ is primarily due to the fact that there is a lack of test data and practical
H[SHULHQFHZLWKFRQFUHWHKDYLQJFRPSUHVVLYHVWUHQJWKVJUHDWHUWKDQSVL
7RPLQLPL]HWKHOLNHOLKRRGRIGLDJRQDOFRPSUHVVLRQIDLOXUHLQWKHFRQFUHWHDQGWROLPLWWKHH[WHQWRIFUDFNLQJWKH
FURVVVHFWLRQDOGLPHQVLRQVRIDVHFWLRQPXVWEHVHOHFWHGWRVDWLVI\$&,(TXDWLRQ >$&,@
(8.33)
:KHUHRXWRISODQHVKHDUVWUHQJWKUHTXLUHPHQWVDUHQRWVDWLV¿HGWKHWKLFNQHVVRIWKHZDOOPXVWW\SLFDOO\EHLQcreased. Increasing the compressive strength of the concrete is usually not as effective as increasing
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8-16
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
ͳʹԣ
κ௪ ܸ௨ ‫ܣ‬௦ ݀
ߩ௪ ൌ
‫ܣ‬௦
ͳʹ݀
Figure 8.10 Out-of-plane shear force.
8.5 Reinforcement Limits
%RWKORQJLWXGLQDO YHUWLFDO DQGWUDQVYHUVH KRUL]RQWDO UHLQIRUFHPHQWPXVWEHSURYLGHGLQDUHLQIRUFHGFRQFUHWH
wall. Minimum longitudinal and transverse reinforcement requirements are given in ACI Table 11.6.1 and are sumand
The longitudinal
PDUL]HGLQ7DEOHIRUGHIRUPHGUHLQIRUFLQJEDUVZLWK
corresponds to the vertical bars in the wall and is equal to the area of the vertical reinforcreinforcement ratio,
ing bars divided by the gross area of the wall perpendicular to those bars, that is, the thickness of the wall times
FRUUHVSRQGVWRWKHKRUL]RQWDOEDUVLQWKH
the length of the wall. Similarly, the transverse reinforcement ratio,
ZDOODQGLVHTXDOWRWKHDUHDRIWKHKRUL]RQWDOUHLQIRUFLQJEDUVGLYLGHGE\WKHJURVVDUHDRIWKHZDOOSHUSHQGLFXODUWR
those bars, that is, the thickness of the wall times the height of the wall. For regularly spaced reinforcing bars, the
reinforcement ratios can be determined on the basis of the applicable bar spacing instead of on the overall length or
height of the wall.
Table 8.10 Minimum Reinforcement for Walls in Accordance with ACI Table 11.6.1
Bar Size
Minimum longitudinal, ȡℓ
Minimum transverse, ȡt
WKHUHLQIRUFHPHQWUHTXLUHPHQWVLQ$&,PXVWEHVDWLV¿HG7KHVHUHTXLUHPHQWV
Where
along with the minimum reinforcement requirements of ACI 11.6.1, are given in Figure 8.11.
8.6 Determining the Wall Thickness
Because structural walls in buildings are typically located around elevator and stair openings, the length of a wall
LVRIWHQGLFWDWHGE\DUFKLWHFWXUDOUHTXLUHPHQWV7KHWKLFNQHVVRIDZDOOLQFRQMXQFWLRQZLWKWKHVSHFL¿HGOHQJWK
DQGPDWHULDOSURSHUWLHVQHHGVWREHGHWHUPLQHGVRWKDWVWUHQJWK D[LDOIRUFHÀH[XUHDQGVKHDU DQGVHUYLFHDELOLW\
UHTXLUHPHQWVDUHVDWLV¿HG
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8-17
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ۓ‬
ۖ
18ᇳ ‫۔‬
ۖ
‫ە‬κ௪ Τ3‫כ‬
‫ ݏ‬൑ lesser of
3݄
ߩκ ߩ௧ 3݄
‫ۓ‬
ۖ
‫ ݏ‬൑ lesser of 18ᇳ ‫۔‬
ۖ
‫ە‬κ௪ Τ5‫כ‬
* Applicable where shear reinforcement
is required for in-plane shear strength
All other reinforcement not shown for clarity
κ௪ Ž‡˜ƒ–‹‘
Longitudinal reinforcement, ࣋र
Longitudinal reinforcement,
ܸ௨ ൑ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩
൑ #5 bars
൐ #5 bars
0.0012
0.0015
Transverse reinforcement, ࢚࣋
Transverse reinforcement,
ܸ௨ ൐ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩
0.0025 ൅ 0.5ሺ2.5 െ ݄௪ Τκ௪ ሻሺߩ௧ െ 0.0025ሻ ൒ 0.0025‫ככ‬
ܸ௨ ൑ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩
൑ #5 bars
൐ #5 bars
0.0020
0.0025
ܸ௨ ൐ 0.5߶ߙ௖ ߣඥ݂௖ᇱ ‫ܣ‬௖௩
0.0025
**ߩκ need not exceed ߩ௧ required by ACI 11.5.4.3
Figure 8.11 Minimum reinforcement requirements for cast-in-place reinforced concrete walls.
)RUZDOOVZKHUHVKHDUVWUHQJWKUHTXLUHPHQWVJRYHUQ(TXDWLRQ FDQEHXVHGWRGHWHUPLQHDPLQLPXP
on a maximum in-plane shear force,
based
(8.34)
where
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8-18
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
)RUZDOOVVDWLVI\LQJWKHOLPLWDWLRQVRIWKHVLPSOL¿HGGHVLJQPHWKRGLQ$&,WKHIROORZLQJHTXDWLRQFDQEH
used to determine minimum >VHH(TXDWLRQ @
(8.35)
where
for compression-controlled sections.
5HODWLYHO\WDOODQGRUVOHQGHUZDOOVFDQEHJRYHUQHGE\DFRPELQDWLRQRID[LDOIRUFHDQGLQSODQHEHQGLQJPRPHQWV
$ZDOOWKLFNQHVVDQGORQJLWXGLQDOZDOOUHLQIRUFHPHQW VL]HDQGVSDFLQJ PXVWEHVHOHFWHGVRWKDWDQLQWHUDFWLRQGLDgram can be constructed. Because there is no closed-form solution for multiple factored load cases, iterations must
and
PXVWEHVDWLV¿HGIRUDOO
EHSHUIRUPHGXQWLODOOGHVLJQVWUHQJWKFULWHULDDUHVDWLV¿HGWKDWLV
factored load combinations.
8.7 Determination of Required Reinforcement
8.7.1 Longitudinal Reinforcement
$VQRWHGLQ6HFWLRQRIWKLVSXEOLFDWLRQWKHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQDZDOOPXVWEH
VHOHFWHGVRWKDWDOOGHVLJQVWUHQJWKFULWHULDDUHVDWLV¿HGIRUFRPELQHGÀH[XUHDQGD[LDOIRUFH1RFORVHGIRUPVROXfor a given set of factored load combinations; it is usually determined by
tion exists to determine the required
SHUIRUPLQJDQXPEHURILWHUDWLRQVZKHUHDWOHDVWWKHPLQLPXPUHLQIRUFHPHQWSUHVFULEHGLQ$&,DQG
DUHXVHGLQWKH¿UVWLWHUDWLRQ
Minimum longitudinal reinforcement per foot length of wall as a function of wall thickness for cases where
is given in Table 8.11
IRUEDUVDQGVPDOOHUVHH7DEOH
DQG)LJXUH $OVRLQFOXGHGLQWKHWDEOHLVVXJJHVWHGUHLQIRUFLQJEDUVL]HDQGVSDFLQJ7ZROD\HUVRIUHLQIRUFHPHQWPXVWEHSURYLGHGZKHUHWKHWKLFNQHVVRIWKHZDOOLVJUHDWHUWKDQLQH[FHSWIRUVLQJOHVWRU\EDVHPHQW
ZDOOVDQGFDQWLOHYHUUHWDLQLQJZDOOV $&, (DFKOD\HURIUHLQIRUFHPHQWVKRXOGEHSODFHGQHDUHDFKIDFH
—
Table 8.11 Minimum Required Longitudinal Reinforcement for Walls where Vu”‫׋‬ťc Ȝ¥ f'c Acv
Wall Thickness, h (in.)
As (in.2/ft)
Suggested Reinforcement
6
#Ǝ
8
#Ǝ
#Ǝ
#Ǝ
#Ǝ
16*
#Ǝ
18*
#Ǝ
*Two layers of reinforcement are required.
It is common for the walls in a low-rise building to have minimum longitudinal reinforcement over the entire height
of the wall.
WKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG0LQLPXP
Where
assumlongitudinal reinforcement per foot length of wall as a function of wall thickness where
LVJLYHQLQ7DEOH
see Figure 8.11).
ing
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8-19
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
—
Table 8.12 Minimum Required Longitudinal Reinforcement for Walls where Vu!‫׋‬ťc Ȝ¥ f'c Acv
Wall Thickness, h (in.)
As (in.2/ft)
Suggested Reinforcement
6
#Ǝ
8
#Ǝ
#Ǝ
#Ǝ
#Ǝ
16*
#Ǝ
18*
#Ǝ
*Two layers of reinforcement are required.
8.7.2 Transverse Reinforcement
The required amount of transverse reinforcement,
depends on the maximum factored in-plane shear force,
ZKLFKDFWVRQWKHZDOO(TXDWLRQ FDQEHVROYHGIRU
(8.36)
Where
minimum
determined by ACI 11.6.1 must be provided. Minimum transverse reinis given in Table 8.13
forcement per foot length of wall as a function of wall thickness where
IRUEDUVDQGVPDOOHUVHH7DEOHDQG)LJXUH —
Table 8.13 Minimum Required Transverse Reinforcement for Walls where Vu”‫׋‬ťc Ȝ¥ f'c Acv
Wall Thickness, h (in.)
As (in.2/ft)
Suggested Reinforcement
6
#Ǝ
8
#Ǝ
#Ǝ
#Ǝ
#Ǝ
16*
#Ǝ
18*
#Ǝ
*Two layers of reinforcement are required.
Where
WKHPLQLPXPUHLQIRUFHPHQWUHTXLUHPHQWVRI$&,PXVWEHVDWLV¿HG0LQLPXP
is given
transverse reinforcement per foot length of wall as a function of wall thickness where
see Figure 8.11).
LQ7DEOH WKDWLV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
—
Table 8.14 Minimum Required Transverse Reinforcement for Walls where Vu!‫׋‬ťc Ȝ¥ f'c Acv
Wall Thickness, h (in.)
As (in.2/ft)
Suggested Reinforcement
6
#Ǝ
8
#Ǝ
#Ǝ
#Ǝ
#Ǝ
16*
#Ǝ
18*
#Ǝ
*Two layers of reinforcement are required.
8.8 Reinforcement Detailing
8.8.1 Concrete Cover
5HLQIRUFLQJEDUVDUHSODFHGLQZDOOVZLWKDPLQLPXPFRQFUHWHFRYHUWRSURWHFWWKHPIURPZHDWKHU¿UHDQGRWKHU
HIIHFWV0LQLPXPFRYHUUHTXLUHPHQWVDUHJLYHQLQ$&, $&, &RQFUHWHFRYHUIRUZDOOVLVPHDsured from the surface of the concrete to the outer edge of the layer of reinforcement closest to the wall surface (see
)LJXUHIRUDZDOOZKHUHWKHWUDQVYHUVHUHLQIRUFHPHQWLVFORVHVWWRWKHFRQFUHWHVXUIDFH 7KHPLQLPXPFRYHULV
HTXDOWRLQIRUDQGVPDOOHUUHLQIRUFLQJEDUVLQZDOOVQRWH[SRVHGWRZHDWKHURULQFRQWDFWZLWKWKHJURXQG
$&,7DEOH )RUDQGEDUVWKHPLQLPXPFRYHULVHTXDOLQ
݄
‘˜‡”
”ƒ•˜‡”•‡
”‡‹ˆ‘” ‡‡–
Figure 8.12 Concrete cover for walls.
8.8.2 Splices of Reinforcement
Overview
Splice lengths for longitudinal and transverse reinforcement in walls must be in accordance with the provisions in
$&, $&, /DSVSOLFHVDQGPHFKDQLFDOVSOLFHVDUHFRPPRQO\XVHGLQZDOOV)RUWKHORQJLWXGLQDOEDUV
compression or tension splice lengths must be provided that satisfy the factored load combinations resisted by the
wall. Tension lap and mechanical splices are used for the transverse reinforcement.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Lap Splices
Lap splices are the most popular and usually the most economical type of splices used in walls. The longitudinal
reinforcement in a wall is permitted to be lap spliced immediately above the top of the slab (see Figure 8.13). This
is the preferred location for ease of construction: The longitudinal bars from the wall below extend above the slab a
distance that is greater than or equal to the required lap splice length, and the longitudinal bars in the wall above are
WLHGWRWKHVHEDUVDIWHUWKHÀRRUEHORZKDVEHHQFRQVWUXFWHG
ƒ’•’Ž‹ ‡Ž‡‰–Š
Ȁ Ž‘‘”
Žƒ„”‡‹ˆ‘” ‡‡–‘–
•Š‘™ˆ‘” Žƒ”‹–›
ƒŽŽŽ‡˜ƒ–‹‘
Figure 8.13 Location of lap splices for longitudinal bars in a wall.
/DSVSOLFHVDUHQRWSHUPLWWHGIRUEDUVODUJHUWKDQEDUVH[FHSWLQWKHFDVHRIFRPSUHVVLRQODSVSOLFHVZKHUH
RUEDUVDUHSHUPLWWHGWREHVSOLFHGWRRUVPDOOHUEDUV $&,DQG The type of lap splice that must be used (compressive or tensile) depends on the stress in the longitudinal bars due
WRWKHIDFWRUHGORDGFRPELQDWLRQV VHH)LJXUHRIWKLVSXEOLFDWLRQZKLFKLVDOVRDSSOLFDEOHWRZDOOV 5HTXLUHG
ODSVSOLFHOHQJWKVIRUWKHWKUHH]RQHVLQGLFDWHGLQ)LJXUHDUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.15 Required Lap Splices Lengths
Zone
Stress in
Longitudinal
Bars
1
All longitudinal
bars are in
compression
3
Stress in the
longitudinal
bars on the
tension face
is tensile and
Stress in the
longitudinal
bars on the
tension face
is tensile and
Notes
ACI
Section No.
±
Class B tension lap splice length
±
Class B tension lap splice length
±
Minimum Lap Splice Length
Class A tension lap splice
Notes:
1. For
must be increased by (4/3) [ACI 25.5.5.1].
2. Compression lap splices must not be used for bars larger than #11 except as permitted in ACI 25.5.5.3 [ACI 25.5.5.2].
3. Compression lap splices of #14 and #18 bars to #11 or smaller bars are permitted in accordance with ACI 25.5.5.4 (see Note 4) [ACI 25.5.5.3].
4. Where bars of different size are lap spliced in compression,
is the longer of (a)
of the larger bar and (b)
of the smaller bar where
is
calculated by ACI 25.4.9.1 [ACI 25.5.5.4]:
where
5.
6.
7.
8.
9.
(see Table 8.16)
Reduction of development length in accordance with ACI 25.4.10.1 is not permitted in calculating lap splice lengths (ACI 25.5.1.4).
Requirements for Class A and Class B tension lap splices are given in ACI Table 10.7.5.2.2 (see Table 8.17).
Tension development length
is determined in accordance with ACI 25.4.2.1(a).
Where bars of different size are lap spliced together,
is the greater of (a)
of the larger bar and (b)
of the smaller bar (ACI 25.5.2.2).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.16 Modification Factors for Deformed Bars in Compression
Modification Factor
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
Longitudinal reinforcement enclosed in the following:
1. A spiral
A circular continuously wound tie with
and a pitch
3. WLHVLQDFFRUGDQFHZLWK$&,VSDFHG
on center
on center
+RRSVLQDFFRUGDQFHZLWK$&,VSDFHG
Other
Lightweight,
&RQ¿QLQJ
reinforcement,
Table 8.17 Requirements for Class A and Class B Tension Lap Splices
Stress in the
longitudinal
bars on the
tension face
Splice Details
Splice Type
There are
percent of the longitudinal bars
spliced at any section and the lap splices on
adjacent bars are staggered by at least
Class A
Other
Class B
All cases
Class B
*Tension development length
Tension Lap Splice Length, ℓst*
is determined in accordance with ACI 25.4.2.1(a).
3URYLVLRQVIRUWKHGHYHORSPHQWRIGHIRUPHGUHLQIRUFLQJEDUVLQWHQVLRQDUHJLYHQLQ$&,7KHWHQVLRQGHYHOopment length, LVGHWHUPLQHGXVLQJWKHSURYLVLRQVRI$&,RULQFRQMXQFWLRQZLWKWKHPRGL¿FDWLRQIDFWRUVLQ$&,7KHUHTXLUHPHQWVRI$&,DUHEDVHGRQWKRVHLQ$&,VRWKHODWWHU
UHTXLUHPHQWVDUHFRYHUHG¿UVW
Method 1 – ACI 25.4.2.4
The development length in tension of a deformed reinforcing bar,
LVGHWHUPLQHGE\$&,(TXDWLRQ D (8.37)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Table 8.18 Factors for Development of Deformed Bars in Tension
Factor
Lightweight,
Reinforcement
grade,
Epoxy,
6L]H
Casting
position,
Spacing
or cover
dimension,
Transverse
reinforcement
index,
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
*UDGHRU*UDGH
*UDGH
*UDGH
1.3
(SR[\FRDWHGRU]LQFDQGHSR[\GXDO
coated reinforcement with clear cover
or clear spacing
(SR[\FRDWHGRU]LQFDQGHSR[\GXDO
coated reinforcement for all other
conditions
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG reinforcement
DQGODUJHUEDUV
DQGVPDOOHUEDUV
0RUHWKDQLQRIIUHVKFRQFUHWH
SODFHGEHORZKRUL]RQWDOUHLQIRUFHPHQW
1.3
Other
All
Lesser of:
1. The distance from the center of a longitudinal
bar to the nearest concrete surface.
One-half the center-to-center spacing of the lonJLWXGLQDOEDUVEHLQJGHYHORSHG VHH)LJXUH
for the case of the longitudinal bars in a wall
being developed).
where:
total cross-sectional area of all transverse
1.
reinforcement within a spacing that crosses
the potential plane of splitting through the longitudinal reinforcement being developed
center-to-center spacing of the transverse
reinforcement
number of bars being developed along the
3.
plane of splitting
All
*The product
need not exceed 1.7.
**It is permitted to conservatively use
if transverse (confining) reinforcement is present or required. For reinforcing bars with
than 6.0 in. on center, transverse reinforcement must be provided along the development length such that
(ACI 25.4.2.2).
spaced closer
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
‫ݏ‬
ܿͳ ൌ ‘˜‡” ൅ ܾ݀ǡ‫ ݏ݊ܽݎݐ‬൅ ൫ܾ݀ǡ݈‫ ݃݊݋‬Τʹ൯
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Figure 8.14 Spacing or cover dimension, cb.
Method 2 – ACI 25.4.2.3
7KHPHWKRGJLYHQLQ$&,WRGHWHUPLQH LVEDVHGRQWKHUHTXLUHPHQWVJLYHQLQ$&,DQGSUHTwo sets of spacing and cover cases are given in ACI Table
VHOHFWHGYDOXHVRIWKHFRQ¿QLQJWHUP
VHH7DEOH 7KHPRGL¿FDWLRQVIDFWRUVLQWKHVHHTXDWLRQVDUHREWDLQHGIURP$&,7DEOH VHH
Table 8.18).
Table 8.19 Tension Development Length, ℓd, for Deformed Bars in Accordance with ACI 25.4.2.3
Spacing and Cover
Case 1
#6 and Smaller Bars
#7 and Larger Bars
Condition 1
1. Clear spacing of bars being developed or lap
spliced
Clear cover
3. Stirrups or ties throughout
not less than
minimum required
&RQGLWLRQ
1. Clear spacing of bars being developed or lap
spliced
Clear cover
&DVH
Other conditions
Mechanical and Welded Splices
0HFKDQLFDORUZHOGHGVSOLFHVDUHSHUPLWWHGLQZDOOVDQGPXVWPHHWWKHUHTXLUHPHQWVRI$&,7KHVHW\SHVRI
splices may be used in both tension and compression.
A variety of proprietary mechanical devices are available that can be used to splice longitudinal reinforcing bars in
DZDOO VHH5HIHUHQFH 7KHVHW\SHVRIVSOLFHVDUHFRPPRQO\VSHFL¿HGDWORFDWLRQVZKHUHLQRUGLQDWHO\ORQJODS
splices would be required or where lap splices would cause congestion. Mechanical splices must be able to develop
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Welded splices must also be able to develop
of the longitudinal bars. These types of splices are primarily
LQWHQGHGIRUEDUVDQGODUJHU:HOGLQJRIUHLQIRUFLQJEDUVPXVWFRQIRUPWRWKHUHTXLUHPHQWVRI$&,
8.8.3 Spacing of Longitudinal Reinforcement
0D[LPXPVSDFLQJRIORQJLWXGLQDOUHLQIRUFHPHQWLQZDOOVLVJLYHQLQ$&, VHH)LJXUH (8.38)
The
limit in Equation (8.38) is applicable where shear reinforcement is required for in-plane strength.
8.8.4 Spacing of Transverse Reinforcement
0D[LPXPVSDFLQJRIWUDQVYHUVHUHLQIRUFHPHQWLQZDOOVLVJLYHQLQ$&, VHH)LJXUH (8.39)
The
limit in Equation (8.39) is applicable where shear reinforcement is required for in-plane strength.
8.8.5 Lateral Support of Longitudinal Reinforcement
Longitudinal reinforcement in a wall must be laterally supported with transverse ties where the following two condiWLRQVDUHVDWLV¿HG $&, 1. The longitudinal reinforcement is required to resist compression (that is, the longitudinal reinforcement is
required as compression reinforcement).
is greater than
where
is the gross cross-sectional area
The area of longitudinal reinforcement,
of the wall.
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൝
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Figure 8.15 Transverse ties in a wall in accordance with ACI 11.7.4.1.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHVL]HDQGH[WHQWRIWKHUHTXLUHGWUDQVYHUVHUHLQIRUFHPHQWLVQRWJLYHQLQ$&,6LPLODUWRFROXPQVLWLV
UHFRPPHQGHGWRVSHFLI\WUDQVYHUVHWLHVIRUDQGVPDOOHUORQJLWXGLQDOEDUVLQWKHZDOODQGWUDQVYHUVHWLHVIRU
larger longitudinal bars. The transverse ties must engage all longitudinal bars subjected to compression considering all
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$QH[DPSOHRIDZDOOZKHUHFURVVWLHVDUHSURYLGHGDVWKHUHTXLUHGWUDQVYHUVHWLHVLVJLYHQLQ)LJXUH7KLVLVWKH
SUHIHUUHGW\SHRIWUDQVYHUVHUHLQIRUFHPHQWIRUZDOOVZLWKOD\HUVRIORQJLWXGLQDOUHLQIRUFHPHQWFRPSDUHGWRUHFWDQgular ties that enclose four of the longitudinal bars in a wall.
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‘–‡•
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”‡‹ˆ‘” ‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ–‡ƒ Š•‹†‡‘ˆ–Š‡‘’‡‹‰Ǥ
͵Ǥ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†–”ƒ•˜‡”•‡™ƒŽŽ
”‡‹ˆ‘” ‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ––Š‡–‘’ƒ†„‘––‘‘ˆ–Š‡‘’‡‹‰Ǥ
Figure 8.16 Recommended reinforcement details around a door opening in a
reinforced concrete wall with two layers of reinforcement.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
8.8.6 Reinforcement Around Openings
)RUZDOOVZLWKZLQGRZGRRUDQGVLPLODUO\VL]HGRSHQLQJVWKHIROORZLQJDGGLWLRQDOUHLQIRUFHPHQWPXVWEHSURYLGHG
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The additional reinforcement must be anchored to develop in tension at the corners of the openings. These bars,
which are commonly referred to as trim bars, essentially replace the reinforcement interrupted by the opening.
More than the minimum amounts of trim bars may need to be provided based on an analysis of the wall with the
opening(s). A recommended detail for the reinforcement around a typical door opening is given in Figure 8.16 for a
ZDOOZLWKOD\HUVRIUHLQIRUFHPHQW$VLPLODUGHWDLOIRUZLQGRZDQGRWKHUZDOORSHQLQJVLVJLYHQLQ)LJXUH
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”‡‹ˆ‘” ‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ–‡ƒ Š•‹†‡‘ˆ–Š‡‘’‡‹‰Ǥ
͵Ǥ ”‘˜‹†‡ƒ–Ž‡ƒ•–‘‡ǦŠƒŽˆ‘ˆ–Š‡–‘–ƒŽƒ”‡ƒ‘ˆ‹–‡””—’–‡†–”ƒ•˜‡”•‡™ƒŽŽ
”‡‹ˆ‘” ‡‡–‘”ƒ‹‹—‘ˆʹǦ͓ͷ„ƒ”•ƒ––Š‡–‘’ƒ†„‘––‘‘ˆ–Š‡‘’‡‹‰Ǥ
Figure 8.17 Recommended reinforcement details around an opening in a
reinforced concrete wall with two layers of reinforcement.
of the longitudinal and transverse
The preferred reference point for measuring the tension development length,
WULPEDUVLVWKHFRUQHURIWKHRSHQLQJEHFDXVHLWLVD¿[HGSRLQW VHH)LJXUHRU)LJXUH 0HDVXULQJ
from a longitudinal or transverse trim bar can be done, but the development length may end up being too short if the
perpendicular trim bars adjacent to the opening are not at their intended locations.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For openings occurring close to the top of a wall or where openings are stacked on top of each other, the typical
straight diagonal bar detail may not be possible. In order to properly develop the bars, either a standard hook needs
to be provided at one end of the bar or the bar can be bent, as shown in Figure 8.18 for the case of stacked openings.
κௗ௛ –Š‡””‡‹ˆ‘” ‡‡–‘–•Š‘™ˆ‘” Žƒ”‹–›
Figure 8.18 Diagonal bar development at stacked wall openings.
8.9 Connections to Foundations
8.9.1 Overview
9HUWLFDODQGKRUL]RQWDOIRUFHVPXVWEHWUDQVIHUUHGDWWKHLQWHUIDFHEHWZHHQDZDOODQGDIRXQGDWLRQLQDFFRUGDQFH
ZLWK$&, $&, 9HUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGE\EHDULQJRQWKHFRQFUHWHRUE\DFRPELnation of bearing and interface reinforcement. Tension forces must be transferred entirely by reinforcement, which
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8.9.2 Vertical Transfer
Compression
:KHUHYHUWLFDOFRPSUHVVLRQIRUFHVDUHWUDQVIHUUHGWRDIRXQGDWLRQEHDULQJVWUHQJWKUHTXLUHPHQWVRI$&,PXVW
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must be less than or equal to the design bearing strength,
where the nominal bearing strength,
is given
LQ$&,7DEOH
(8.40)
In this equation,
LVHTXDOWRIRUEHDULQJ $&,7DEOH DQG
is the gross area of the wall.
)RUEHDULQJRQDIRXQGDWLRQZLGHURQDOOVLGHVWKDQWKHORDGHGDUHDWKHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HG
(8.41)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
The term
is the area of the lower base of the largest frustrum of a pyramid, cone, or tapered wedge contained
and having side slopes of 1 vertical to
wholly within the foundation and having for its upper base the loaded area
are depicted in Figure 8.19 for the case of a wall supported by a reinforced concrete
KRUL]RQWDO$UHDV and
foundation.
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κ௪ ‫ܤ‬
ܾ
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Figure 8.19 Determination of areas for bearing strength.
the excess compression stress from the wall must be transferred by reinforcement to the foundaWhere
is determined by the following equation:
tion. The required area of interface reinforcement,
(8.42)
In this equation,
LVWKHPLQLPXPORQJLWXGLQDOUHLQIRUFHPHQWUDWLRVDWLVI\LQJ$&, $&,VHH
must be provided even where
7DEOH 7KHPLQLPXPDPRXQWRILQWHUIDFHUHLQIRUFHPHQW
Dowel bars emanating from the foundation are the type of interface reinforcement commonly used due to ease of
FRQVWUXFWLRQ7KHGRZHOEDUVZKLFKDUHW\SLFDOO\RIWKHVDPHVL]HDQGVSDFLQJRIWKHORQJLWXGLQDOUHLQIRUFHPHQWLQ
the wall, are set in the foundation prior to placing the foundation concrete and are subsequently spliced to the longitudinal bars.
,OOXVWUDWHGLQ)LJXUHDUHGRZHOEDUVDFURVVWKHLQWHUIDFHEHWZHHQDUHLQIRUFHGFRQFUHWHZDOODQGIRXQGDWLRQ
where all the longitudinal bars in the wall are in compression for all factored load combinations. The dowel bars
determined in accordance with ACI
must extend into the foundation at least a compression development length,
VHH6HFWLRQRIWKLVSXEOLFDWLRQ 6WDQGDUGKRRNVDUHW\SLFDOO\SURYLGHGDWWKHHQGVRIWKHGRZHOEDUVZKLFKDUHWLHGWRWKHÀH[XUDOUHLQIRUFLQJEDUV
LQWKHIRXQGDWLRQWKLVGHWDLOLVIDUPRUHHFRQRPLFDOWKDQWHUPLQDWLQJWKHGRZHOEDUVDERYHWKHÀH[XUDOUHLQIRUFLQJ
bars. The hooked portion of the dowels cannot be considered effective for developing the dowels bars in compresVLRQ $&, 7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHOoped into the foundation:
(8.43)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
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͵ԣ Ž‡ƒ”
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‘’”‡••‹‘Žƒ’
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‘™‡Ž„ƒ”•ǡሺ݀௕ ሻௗ Figure 8.20 Dowel bars at the interface of a reinforced concrete wall and foundation
where all the longitudinal bars in the wall are in compression.
In this equation, LVWKHUDGLXVRIWKHGRZHOEDUEHQG VHH$&,7DEOHZKLFKFRQWDLQVPLQLPXPLQVLGHEHQG
GLDPHWHUVIRUEDUVZLWKVWDQGDUGGHJUHHKRRNV :KHUH(TXDWLRQ LVQRWVDWLV¿HGRQHW\SLFDOZD\WRHQVXUH
adequate development is to increase the thickness of the foundation,
The dowel bars must also be fully developed in the wall; this is typically achieved by lap splicing the dowel bars to
WKHORQJLWXGLQDOEDUVLQWKHZDOO VHH)LJXUH :KHUHWKHGRZHOEDUVDUHWKHVDPHVL]HDVWKHZDOOORQJLWXGLQDO
EDUVWKHPLQLPXPFRPSUHVVLRQODSVSOLFHOHQJWKLVXVHG VHH7DEOH :KHUHWKHGRZHOEDUVDUHODUJHULQGLDPHWHUWKDQWKHZDOOORQJLWXGLQDOEDUVWKHFRPSUHVVLRQODSVSOLFHOHQJWKPXVWVDWLVI\WKHUHTXLUHPHQWVRI$&,
/DSVSOLFHVRIDQGORQJLWXGLQDOEDUVLQFRPSUHVVLRQIRUDOOIDFWRUHGORDGFRPELQDWLRQVZLWKDQGVPDOOHU
GRZHOEDUVDUHSHUPLWWHGSURYLGHGWKHUHTXLUHPHQWVRI$&,DUHVDWLV¿HG $&, Tension
Tension forces transferred from a wall to a foundation must be resisted entirely by reinforcement across the interIDFH>$&, E DQG@
7HQVLOHDQFKRUDJHRIWKHGRZHOEDUVLQWRDIRXQGDWLRQLVW\SLFDOO\DFFRPSOLVKHGE\SURYLGLQJGHJUHHVWDQGDUG
determined in accordance
hooks at the ends of the dowel bars with the development length of the hooked bar,
ZLWK$&, VHH)LJXUH (8.44)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
݄
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൒ κௗ௛ Žƒ••–‡•‹‘
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Figure 8.21 Dowel bars at the interface of a reinforced concrete wall and foundation
where the longitudinal bars in the wall are in compression and tension.
7KLVGHYHORSPHQWOHQJWKLVPHDVXUHGIURPWKHFULWLFDOVHFWLRQWRWKHRXWVLGHIDFHRIWKHKRRN VHH)LJXUHRIWKLV
SXEOLFDWLRQ 7KHPRGL¿FDWLRQIDFWRUVLQ(TXDWLRQ DUHJLYHQLQ$&,7DEOH VHH7DEOH Table 8.20 Modification Factors for Development of Hooked Bars in Tension
Modification Factor
Lightweight,
Epoxy,
&RQ¿QLQJ
reinforcement,
Condition
Value of Factor
Lightweight concrete
Normalweight concrete
(SR[\FRDWHGRU]LQFDQGHSR[\GXDOFRDWHGUHLQIRUFHPHQW
8QFRDWHGRU]LQFFRDWHG JDOYDQL]HG UHLQIRUFHPHQW
)RUDQGVPDOOHUEDUVZLWK
Other
or
1.6
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.20 Modification Factors for Development of Hooked Bars in Tension (cont.)
Modification Factor
Location,
Condition
Value of Factor
)RUDQGVPDOOHUKRRNHGEDUV
1. terminating inside a column core with side cover normal to the
plane of the hook
or
with side cover normal to the plane of the hook
Other
Concrete
strength,
7KHFRQ¿QLQJUHLQIRUFHPHQWIDFWRU
is typically equal to 1.6 for hooked dowel bars in foundations because con¿QLQJUHLQIRUFHPHQWLVXVXDOO\QRWSURYLGHGLQVXFKFDVHV
7KHIROORZLQJHTXDWLRQPXVWEHVDWLV¿HGWRHQVXUHWKDWWKHGRZHOEDUVDUHDGHTXDWHO\GHYHORSHGLQWRWKHIRXQGDWLRQ
(considering that the hooked portion of the dowel bars can be developed in tension):
(8.45)
:KHUH(TXDWLRQ LVQRWVDWLV¿HGWKHGHSWKRIWKHIRXQGDWLRQXVXDOO\PXVWEHLQFUHDVHG
For development of the dowel bars into the wall, a tension lap splice or a mechanical connection in accordance with
$&,RUUHVSHFWLYHO\PXVWEHSURYLGHGEHWZHHQWKHGRZHOEDUVDQGWKHORQJLWXGLQDOEDUV,QWKHFDVH
RIODSVSOLFHVD&ODVV%WHQVLRQODSVSOLFHLVUHTXLUHG VHH7DEOHDQG)LJXUH 8.9.3 Horizontal Transfer
7KHVKHDUIULFWLRQPHWKRGRI$&,LVSHUPLWWHGWREHXVHGWRGHWHUPLQHWKHQRPLQDOVKHDUVWUHQJWK
at the
across
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the interface is determined by the following equation, which is applicable to shear-friction reinforcement perpenGLFXODUWRWKHLQWHUIDFH $&, (8.46)
In this equation,
is the factored shear force due to the lateral force effects at the interface (in-plane or out-ofplane), the strength reduction factor LVHTXDOWRDQG LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP
$&,7DEOH VHH7DEOH Table 8.21 Coefficient of Friction, μ
Contact Surface Condition
Concrete placed monolithically
Concrete placed against hardened concrete that is clean, free of laitance,
and intentionally roughened to a full amplitude of approximately ¼ in.
Concrete placed against hardened concrete that is clean, free of laitance,
and not intentionally roughened
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8-34
μ
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOH VHH7DEOH 7KHWHUP is the area of
which is equal to the gross cross-sectional area of the wall. Where the concrete strengths of
concrete resisting
WKHZDOODQGWKHIRXQGDWLRQDUHGLIIHUHQWWKHVPDOOHURIWKHWZRPXVWEHXVHGLQWKHVHHTXDWLRQV $&, Table 8.22 Maximum Vn Across the Assumed Shear Plane
Contact Surface Condition
Maximum Vn = Vu / Ҁ*
Normalweight concrete placed monolithically or placed
against hardened concrete intentionally roughened
to a full amplitude of approximately ¼ in.
Other cases
*
area of concrete resisting
Full tension anchorage of the shear-friction reinforcement must be provided into the foundation and into the wall. The
lengths of these bars are determined in the same way as those for vertical transfer where tension forces are present.
The area of the dowel bars is usually determined initially based on the requirements for vertical transfer. That area is
FRPSDUHGZLWKWKHDUHDUHTXLUHGIRUKRUL]RQWDOWUDQVIHUDQGWKHODUJHURIWKHWZRLVSURYLGHGDWWKHLQWHUIDFH
8.10 Design Procedure
7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHZDOOVVXEMHFWHG
WRD[LDOFRPSUHVVLRQIRUFHVDQGXQLD[LDOEHQGLQJ LQSODQHRURXWRISODQH ,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQ
QXPEHUVWDEOHQXPEHUVDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQLQWKLVFKDSWHUFDQEHIRXQG
8.11 Examples
8.11.1 Example 8.1 – Design of Reinforced Concrete Wall: Building #2, Interior Wall is Not Part of the SFRS,
Simplified Design Method
Design a solid, interior reinforced concrete wall that is not part of the seismic force-resisting system (SFRS) in the
¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJD[LDOIRUFHVZKLFKDUHDVVXPHGWRDFWDWDLQHFFHQWULFLW\
and
Assume normalZLWKUHVSHFWWRWKHFHQWURLGRIWKHZDOO VHH)LJXUH DQG*UDGHUHLQIRUFHPHQW
weight concrete with
Step 1 – Determine the factored axial force
Table 3.3
The following load combination governs:
Step 2 – Determine a trial wall thickness
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Provide dowel bars to satisfy the
requirements in ACI 16.3 for
connections to foundations
ሺSect. 8.9ሻ
Figure 8.22 Design procedure for walls.
A minimum wall thickness can be determined by the following equation:
Eq. (8.35)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ܲ஽ ǡ ܲ௅ κ௖ ൌ ͳͺǤͲᇱ ͳᇳ Figure 8.23 Interior bearing wall of Example 8.1.
Assuming the wall is braced at the top and bottom against lateral translation and unrestrained against rotation at
both ends:
Table 8.7
Therefore,
For a bearing wall, the minimum thickness is equal to the following:
Figure 8.1
Try
Step 3 – Determine if the Simplified Design Method can be used
ACI 11.5.3
&KHFNWKHOLPLWDWLRQVRIWKH6LPSOL¿HG'HVLJQ0HWKRG
1. The wall is solid and rectangular.
Eccentricity
Figure 8.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Determine the design strength of the wall
Eq. (8.27)
Step 5 – Determine the minimum reinforcement in the wall
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Minimum longitudinal reinforcement
Table 8.10
3URYLGHDVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU
Minimum transverse reinforcement
3URYLGHDVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU
Maximum spacing
Figure 8.11
8.11.2 Example 8.2 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS,
Moment Magnification Method, Out-of-Plane Forces
'HVLJQDVROLGH[WHULRUUHLQIRUFHGFRQFUHWHZDOOWKDWLVQRWSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJIRUFHV VHH)LJXUH (it can be determined that the effects due to out-of-plane seismic effects in accordance with
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DQG*UDGHUHLQIRUFHPHQWDQGWKHIUDPHLQWKH¿UVWVWRU\LV
Assume normalweight concrete with
nonsway. Also, the gravity loads are assumed to act concentrically on the wall.
Step 1 – Determine a trial wall thickness
There is no closed-form solution to determine
for a wall subjected to these types of loads.
A trial wall thickness of 8 in. is selected. The adequacy of this wall thickness will be checked in the following steps.
Step 2 – Determine the factored load combinations
Table 3.3
Weight of the wall at mid-height
Total dead load
The gravity loads do not cause any bending moments in the wall because they act through the centroid of the wall.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
κ௨ ൌ ͳͺǤͲ െ ሾʹͺǤͷȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͸Ǥͺᇱ ܲ஽ ǡ ܲ௅ ‫ݓ‬ௐ Figure 8.24 Exterior bearing wall of Example 8.2.
Assuming the wall is pinned at the top and bottom, the bending moment at the mid-height of the wall due to the
uniformly distributed wind load is equal to the following:
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Table 8.23 Summary of Axial Forces and Bending Moments for the Wall in Example 8.2
Load Case
Axial Force (kips/ft)
Bending Moment (in.-kips/ft)
Dead
Live
Wind
Load Combination
$&,(T D
$&,(T E
$&,(T G
$&,(T I
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Determine if slenderness effects need to be considered
%HFDXVHWKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPHVOHQGHUQHVVHIIHFWVQHHGQRWEHFRQVLGHUHGZKHUH$&,(TV
E DQG F DUHVDWLV¿HG VHH7DEOHRIWKLVSXEOLFDWLRQ Therefore, slenderness effects must be considered in the design of this wall.
Step 4 – Determine the magnified moments in the wall using the moment magnification method
ACI 6.6.4
7KHPRPHQWPDJQL¿FDWLRQPHWKRGLVFRYHUHGLQGHWDLOLQ6HFWRIWKLVSXEOLFDWLRQ
)RUQRQVZD\IUDPHVIDFWRUHGPDJQL¿HGPRPHQWV
DUHGHWHUPLQHGE\(T where
LVWKHIDFWRUHGPLGKHLJKWEHQGLQJPRPHQWLQWKHZDOODQGWKHPDJQL¿FDWLRQIDFWRU
(T Because there are transverse loads between the supports,
The critical buckling load,
LVGHWHUPLQHGE\(T The effective stiffness,
is determined by the following equation:
is determined by
ACI 6.6.4.5.3(b)
Eq. (8.2)
Table 8.2
It is evident that the area of longitudinal reinforcement in the wall,
is needed to calculate
purposes, the minimum amount of longitudinal reinforcement is assumed:
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For calculation
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7U\DVLQJOHOD\HURIEDUVVSDFHGDWLQRQFHQWHU
checked in the following steps.
The adequacy of this reinforcement will be
&DOFXODWLRQVDUHSURYLGHGIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(T G LQ7DEOH
For a 1-foot design strip:
ACI Eq. (22.4.2.2)
ACI Eq. (19.2.2.1a)
Eq. (8.2)
Therefore,
Determine the minimum factored bending moment,
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HIIHFWVDUHJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.24 Summary of Results from the Moment Magnification Method for the Wall in Example 8.2
Load Combination
ȕdns
I
(EI)eff x 10 3
Pc
(in.4)
(kip-in.2)
(kips)
Pu
Mu
(kips)
(in.-kips)
į
Step 5 – Check the adequacy of the section for combined flexure and axial forces
Mc
(in.-kips)
ACI 22.4
The lower portion of the design strength interaction diagram for a 1-foot-wide section of the wall reinforced with
EDUVVSDFHGDWLQRQFHQWHULVVKRZQLQ)LJXUH%HFDXVHDOOORDGFRPELQDWLRQVIDOOZLWKLQWKHGHVLJQ
strength interaction diagram, the wall is adequate.
80
70
Axial Force ሺkipsሻሻ
60
50
40
30
20
10
0
0
25
50
75
100
125
150
175
200
Bending Moment ሺin.-kipsሻ
Figure 8.25 Design strength interaction diagram for the wall in Example 8.2.
Step 6 – Check the out-of-plane shear strength requirements
The required shear strength,
is determined based on the design wind force:
The design shear strength,
is equal to the design shear strength of the concrete,
ACI 22.5
Eq. (8.31)
Eq. (8.32)
for normalweight concrete
Table 8.8
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
The axial force
smaller value of
FRUUHVSRQGLQJWR$&,(T I LVXVHGLQGHWHUPLQLQJ
because it results in a
than that determined using
FRUUHVSRQGLQJWR$&,(T G The wall is adequate for out-of-plane shear.
8.11.3 Example 8.3 – Design of Reinforced Concrete Wall: Building #2, Exterior Wall is Not Part of the SFRS,
Alternative Method for Out-of-Plane Forces
'HVLJQDVROLGH[WHULRUUHLQIRUFHGFRQFUHWHZDOOWKDWLVQRWSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJVXEMHFWHGWRWKHIROORZLQJIRUFHV VHH)LJXUH (it can be determined that the effects due to out-of-plane seismic effects in accordance with
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ܲ஽ ǡ ܲ௅ κ௨ ൌ ͳͺǤͲ െ ሾʹͺǤͷȀሺʹ ൈ ͳʹሻሿ ൌ ͳ͸Ǥͺᇱ ݁ ൌ ʹǤͲԣ
‫ݓ‬ௐ Figure 8.26 Exterior bearing wall in Example 8.3.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Assume normalweight concrete with
DQG*UDGHUHLQIRUFHPHQW$OVRDVVXPHDQHFFHQWULFLW\RI
in. between the dead and live load axial forces and the centroid of the wall.
Step 1 – Determine a trial wall thickness
There is no closed-form solution to determine
for a wall subjected to these types of loads.
A trial wall thickness of 8 in. is selected. The adequacy of this wall thickness will be checked in the following steps.
Step 2 – Determine the factored load combinations at the mid-height of the wall
Weight of the wall from top to mid-height
Service dead load moment
mid-height)
(only the axial dead load causes a bending moment at
Service live load moment
Wind load moment
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Table 8.25 Summary of Axial Forces and Bending Moments for the Wall in Example 8.3
Load Case
Axial Force (kips/ft)
Bending Moment (in.-kips/ft)
Dead
Live
Wind
Load Combination
$&,(T D
3.9
$&,(T E
$&,(T G
3.8
$&,(T I
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Determine the factored load combinations at the mid-height of the wall, including slenderness effects
8VLQJWKHGLUHFWFDOFXODWLRQPHWKRGRIWKHDOWHUQDWLYHPHWKRG>$&, E @WKHWRWDOIDFWRUHGPRPHQW
which includes slenderness effects, is determined by the following equation:
Eq. (8.10)
where
DUHWKHIDFWRUHGPRPHQWVLQWKHZDOODWPLGKHLJKW VHH7DEOH A summary of the factored axial forces and bending moments, which include slenderness effects, is given in Table
Table 8.26 Summary of Factored Axial Forces and Bending Moments, Including Slenderness Effects, for the
Wall in Example 8.3
Load Combination
Pu
Mua
Ase,w
c
Icr
Mu
(kips)
(in.-kips)
(in.2)
(in.)
(in.4)
(in.-kips)
3.9
31.6
3.8
&DOFXODWLRQVDUHSURYLGHGIRUWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(T G Maximum longitudinal bar spacing
Figure 8.11
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Eq. (8.7)
Eq. (8.9)
ACI 20.2.2.2
Check if the section is tension-controlled:
ACI Table 21.2.2
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore, the section is tension-controlled. It can be determined that this wall section is tension-controlled for all
ORDGFRPELQDWLRQV7KLVVDWLV¿HVWKHVHFRQGOLPLWDWLRQLQ$&,
ACI Eq. (19.2.2.1a)
ACI 11.8.3.1
Eq. (8.6)
Therefore,
Similar calculations can be performed for the other load combinations.
Step 4 – Determine the cracking moment
ACI Eq. (19.2.3.1)
ACI Eq. (24.2.3.5)
Step 5 – Determine the design flexural strength of the wall and check the adequacy of the wall
7KHGHVLJQÀH[XUDOVWUHQJWKRIWKHZDOOLVGHWHUPLQHGE\WKHIROORZLQJHTXDWLRQ
Eq. (8.11)
,WLVGHWHUPLQHGLQ6WHSWKDWWKHZDOOVHFWLRQLVWHQVLRQFRQWUROOHGIRUDOOORDGFRPELQDWLRQVVR
7DEOH (see
so the wall is adequate
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IRUDOOORDGFRPELQDWLRQVZKLFKVDWLV¿HVWKHWKLUGOLPLWDWLRQLQ$&,
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11.8.1.1.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.27 Summary of Design Flexural Strengths for the Wall in Example 8.3
Load Combination
Ase,w
c
a
ҀMn
Mu
(in.2)
(in.)
(in.2)
(in.-kips)
(in.-kips)
68.9
66.9
63.1
Step 6 – Check the maximum axial force at the mid-height of the wall section
According to the fourth limitation in ACI 11.8.1.1,
at the mid-height of the wall must be less than or equal to
,WLVHYLGHQWWKDWWKLVOLPLWDWLRQLVVDWLV¿HGIRUDOOORDGFRPELQDWLRQV VHH7DEOH Step 7 – Determine the maximum service-level at the mid-height of the wall section
Because there is no closed-form solution for
for the initial iteration:
The maximum service-level bending moment,
assume
Also assume the following value of
at the mid-height section of the wall is equal to the following:
Eq. (8.17)
where the strength-level bending moment due to wind loads is divided by 1.6 to convert it to a service-level bending
moment.
7KHGHÀHFWLRQ
is determined by the following equation:
Eq. (8.15)
Therefore,
Determine
from the following equation:
Eq. (8.12)
Because it has been assumed that
determine
from the following equation:
Eq. (8.13)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
This value of
is the same as the value initially assumed for
so no additional iterations are required.
Check the initial assumption:
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Step 8 – Check the out-of-plane shear strength requirements
ACI 22.5
The required shear strength,
LVGHWHUPLQHGEDVHGRQWKHORDGFRPELQDWLRQFRUUHVSRQGLQJWR$&,(T G The design shear strength,
is equal to the design shear strength of the concrete,
Eq. (8.31)
Eq. (8.32)
for normalweight concrete
Table 8.8
Therefore,
The wall is adequate for out-of-plane shear.
8.11.4 Example 8.4 – Determination of Trial Wall Thickness of Reinforced Concrete Wall: Building #2, SDC C,
Interior Wall is Part of the SFRS
Determine a trial wall thickness for the interior reinforced concrete wall along column line F that is part of the
6)56LQWKH¿UVWVWRU\RI%XLOGLQJFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQ VHH)LJXUH $VVXPH
and Grade 60 reinforcement.
normalweight concrete with
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Solution
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RIWKLVSXEOLFDWLRQWKDWZLQGORDGHIIHFWVDUHOHVVWKDQWKRVHIURPVHLVPLFHIIHFWV
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
There is no closed-form solution to determine h for a wall subjected to these combined load effects. However, a trial
wall thickness can be obtained based on shear strength requirements:
Eq. (8.34)
)URP7DEOHWKHEDVHVKHDUGXHWRVHLVPLFHIIHFWVLVHTXDOWRNLSV7KHZDOOVDORQJFROXPQOLQHV%DQG
)HDFKUHVLVWSHUFHQWRIWKLVEDVHVKHDUVRWKHIDFWRUHGVKHDUIRUFH LQHDFKZDOOLQWKH¿UVWVWRU\LVHTXDOWR
NLSV
Eq. (8.29)
for normalweight concrete
$VVXPLQJ
Table 8.8
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#2, SDC C, Interior Wall is Part of the SFRS
'HVLJQWKHLQWHULRUUHLQIRUFHGFRQFUHWHZDOODORQJFROXPQOLQH)WKDWLVSDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJIRUFRPELQHGÀH[XUHDQGD[LDOIRUFHVFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQ VHH)LJXUH and Grade
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60 reinforcement.
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Step 1 – Determine the factored load combinations
Table 3.3
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each level of the building and a moment of inertia equal to
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UHVLVWHGE\WKHIWZDOOZHEVLQWKHGLUHFWLRQRIDQDO\VLVVRQRHIIHFWLYHÀDQJHZLGWKVDUHFRQVLGHUHGLQGHVLJQ
that is, all the lateral load effects are assigned to the webs.
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Table 8.28 Summary of Axial Forces and Bending Moments for the Wall in Example 8.4
Load Case
Axial Force (kips)
Bending Moment (ft-kips)
Dead
0
Roof live
ņ
Live
0
Seismic
0
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 8.28 Summary of Axial Forces and Bending Moments for the Wall in Example 8.4 (cont.)
Load Case
Axial Force (kips)
Bending Moment (ft-kips)
$&,(T D
$&,(T E
$&,(T F
$&,(T H
$&,(T J
Load Combination
Further explanation is needed on how the load factors in the load combinations that include
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$6&(6(, FDQEHWDNHQDVIRUVWUXFWXUHVDVVLJQHGWR6'&&WKHUHIRUH
Therefore, the load combination becomes the following:
are obtained. In ACI
According to
,Q$&,(T J WKHHIIHFWVRIJUDYLW\DQGVHLVPLFJURXQGPRWLRQFRXQWHUDFWVR
and the load combination becomes the following:
Step 2 – Determine if the first story is nonsway or sway
The stability index,
LVXVHGWRGHWHUPLQHZKHWKHUWKH¿UVWVWRU\LVQRQVZD\RUVZD\
7KHIROORZLQJUHVXOWVDUHREWDLQHGIURPWKHDQDO\VLVLQWKH¿UVWVWRU\
Total
Total
The total factored load must correspond to the lateral loading case for which it is a maximum. In this example, ACI
(T H SURGXFHVWKHODUJHVWORDG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
7KHUHIRUHWKHIUDPHLQWKH¿UVWVWRU\LVDQRQVZD\IUDPH
Step 3 – Determine if slenderness effects must be considered
%HFDXVHWKH¿UVWVWRU\IUDPHLVDQRQVZD\IUDPHVOHQGHUQHVVHIIHFWVQHHGQRWEHFRQVLGHUHGZKHUHWKHIROORZLQJ
HTXDWLRQLVVDWLV¿HG
Table 8.4
7KHPRPHQWDWWKHWRSRIWKHZDOOLQWKH¿UVWVWRU\LVHTXDOWRIWNLSVDQGWKHZDOOLVEHQWLQVLQJOHFXUYDture. Therefore,
Therefore,
Thus, slenderness effects need not be considered in the design of this wall in the direction of analysis.
Step 4 – Determine if the wall is adequate for combined flexure and axial forces
An estimate of the longitudinal reinforcement in the wall is needed in order to determine if the section is adequate
RUQRWIRUWKHIDFWRUHGORDGFRPELQDWLRQVLQ7DEOH
The minimum amount of longitudinal reinforcement depends on the magnitude of
(see Figure 8.11).
with respect to the limit
From the analysis,
Eq. (8.29)
Preliminary analysis indicates that minimum transverse reinforcement can be used (that is,
8.11 and Example 8.6).
see Figure
Therefore,
Minimum
Figure 8.11
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Check the adequacy of the wall using minimum longitudinal reinforcement:
Maximum longitudinal bar spacing
Figure 8.11
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7KHGHVLJQVWUHQJWKLQWHUDFWLRQGLDJUDPIRUWKLVZDOOUHLQIRUFHGZLWKOD\HUVRIEDUVVSDFHGDWLQRQFHQWHU
LVJLYHQLQ)LJXUH,WZDVFRQVWUXFWHGEDVHGRQHTXLOLEULXPVWUDLQFRPSDWLELOLW\DQGWKHGHVLJQDVVXPSWLRQVLQ
$&, VHH6HFWLRQDQG)LJXUHRIWKLVSXEOLFDWLRQRQKRZWRFRQVWUXFWLQWHUDFWLRQGLDJUDPVIRUUHFWDQJXODUVHFWLRQV 7KHORDGFRPELQDWLRQVIURP7DEOHDUHDOVRJLYHQLQWKH¿JXUH%HFDXVHDOOORDGFRPELQDWLRQ
points fall within the design strength interaction diagram, the wall is adequate.
15,000
݂௦ ൌ 0
Axial Force ሺkipsሻሻ
12,000
݂௦ ൌ 0.5݂௬ 9,000
6,000
3,000
0
0
15,000
30,000
45,000
60,000
75,000
90,000
105,000
Bending Moment ሺft-kipsሻ
Figure 8.27 Design strength interaction diagram for the wall in Example 8.5
8.11.6 Example 8.6 – Design of Reinforced Concrete Wall for Shear Forces: Building #2, SDC C, Interior Wall
is Part of the SFRS
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DQG*UDGHUHLQIRUFHPHQW
concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 1 – Determine the transverse reinforcement
An estimate of the transverse reinforcement in the wall is needed in order to determine if the section is adequate or
not for the factored shear force
The minimum amount of transverse reinforcement depends on the magnitude of
(see Figure 8.11).
with respect to the limit
Eq. (8.29)
Therefore, minimum
Figure 8.11
Maximum transverse bar spacing
Figure 8.11
7U\OD\HUVRIEDUVVSDFHGDWLQRQFHQWHU
Step 2 – Check the adequacy of the section
Eq. (8.28)
7KHUHIRUHWKHVHFWLRQLVDGHTXDWHIRUVKHDUXVLQJWUDQVYHUVHUHLQIRUFHPHQWFRQVLVWLQJRIOD\HUVRIEDUVVSDFHG
at 18 in. on center.
8.11.7 Example 8.7 – Determination of Dowel Reinforcement at the Foundation of a Reinforced Concrete
Wall: Building #2, SDC C, Interior Wall is Part of the SFRS
Determine the required dowel reinforcement for the interior reinforced concrete wall along column line F that is
SDUWRIWKH6)56LQWKH¿UVWVWRU\RI%XLOGLQJFRQVLGHULQJVHLVPLFIRUFHVLQWKHHDVWZHVWGLUHFWLRQDVVXPLQJ
WKHZDOOLVVXSSRUWHGE\DIWWKLFNPDWIRXQGDWLRQ VHH)LJXUH $OVRDVVXPHQRUPDOZHLJKWFRQFUHWHZLWK
and
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Step 1 – Check the bearing stresses on the wall and mat foundation
ACI 16.3.3.4
Bearing strength of the wall.
Factored bearing stress,
is determined for compression and for compression and bending using the factored axial
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
• $&,(T E • $&,(T H Tension forces cannot be transferred through bearing at the interface, so dowel bars will be provided that match the
VL]HDQGVSDFLQJRIWKHORQJLWXGLQDOEDUVLQWKHZDOO7KLVUHLQIRUFHPHQWHQVXUHVWKDWERWKWKHFRPSUHVVLRQDQGWHQsion forces are adequately transferred from the wall into the mat foundation.
Bearing strength of the mat foundation.
There is no need to check the bearing strength of the mat foundation because dowel bars will be provided that match
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Step 2 – Determine the required interface reinforcement
ACI 16.3.4.1
)URP([DPSOHWKHORQJLWXGLQDOUHLQIRUFHPHQWLQWKHZDOOLVOD\HUVRIEDUVVSDFHGDWLQRQFHQWHUDWWKLV
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Check minimum area of reinforcement across the interface:
ACI 16.3.4.2
7KHORQJLWXGLQDOUHLQIRUFHPHQWUDWLRRILVXVHGWRGHWHUPLQHWKHPLQLPXPDUHDRILQWHUIDFHUHLQIRUFHPHQWLQ
DFFRUGDQFHZLWK$&,EHFDXVHORQJLWXGLQDOEDUV *UDGH DUHXVHGLQWKHZDOO VHH$&,7DEOH Step 3 – Check horizontal force transfer
)URP([DPSOHWKHIDFWRUHGVKHDUIRUFHDWWKHEDVHRIWKHZDOOLVHTXDOWRNLSV7KLVIRUFHLVWUDQVIHUUHG
KRUL]RQWDOO\EHWZHHQWKHZDOODQGWKHIRXQGDWLRQ7KHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVGHWHUPLQHGDV
follows assuming the surface between the wall and the foundation has not been intentionally roughened:
Eq. (8.46)
7KLVDUHDRIUHLQIRUFHPHQWLVODUJHUWKDQWKHDUHDRIUHLQIRUFHPHQWSURYLGHGE\WKHGRZHOEDUVWKDWPDWFKWKHVL]H
and spacing of the longitudinal reinforcement in the wall, which is equal to
7KHUHIRUHWU\GRZHOEDUVVSDFHGDWLQRQFHQWHU
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Check the upper shear limit:
Table 8.22
8VHGRZHOEDUVVSDFHGDWLQRQFHQWHU
Step 4 – Check the development of the dowel bars in the mat foundation
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face is tensile for the load combinations that include seismic effects. Therefore, the dowel bars must be developed
for tension into the mat (and into the wall).
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length,
Eq. (8.44)
Table 8.20
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WKLFNQHVVIRUWKHGHYHORSPHQWRIWKHGRZHOEDUV
Eq. (8.45)
The minimum
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Step 5 – Determine the development length of the dowel bars in the wall
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Development length in tension,
RIWKHGRZHOEDUV
Eq. (8.37)
Table 8.18
Figure 8.14
Set
ACI 25.4.2.4
Therefore,
Tension lap splice length,
RIWKHORQJLWXGLQDOEDUV
Eq. (8.37)
Table 8.18
Figure 8.14
Set
ACI 25.4.2.4
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Therefore,
3URYLGHDODSVSOLFHOHQJWKRIIWLQ
5HLQIRUFHPHQWGHWDLOVIRUWKLVZDOODUHJLYHQLQ)LJXUH
ͳᇱ ǦͲԣ
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Figure 8.28 Reinforcement details for the wall in Examples 8.4 through 8.7.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Chapter 9
DIAPHRAGMS
9.1 Overview
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Diaphragms must support and transfer gravity loads to columns, walls, and other supporting elements in a building.
The weight of the structure, superimposed dead loads, and live loads are common out-of-plane gravity loads applied
to the surface of a diaphragm. Roof diaphragms must also be able to support effects due to rain, snow, and wind, to
name a few.
Lateral forces from wind, earthquakes, and soil pressure are common types of forces transferred from a diaphragm
to the lateral force-resisting system (LFRS) in a building. The load path from the application of the force on the diaphragm to the vertical elements of the LFRS depends on several factors, including the type of applied force and the
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axial forces in a diaphragm. The connections between the diaphragm and the vertical elements of the LFRS are very
important; these connections must be properly designed and detailed otherwise there is no mechanism for proper
load transfer to the LFRS.
Collectors, which are part of a diaphragm and are sometimes referred to as drag struts, are required where the LFRS
does not extend the full depth of a diaphragm or where there is a discontinuity in the LFRS. These elements are parallel to the applied force and their main functions are to collect and transfer diaphragm shear forces to the vertical
elements of the LFRS, distribute forces within the diaphragm, or both.
In general, diaphragms must be designed and detailed for the combined effects due to in-plane and out-of-plane load
effects. Reinforcement is provided to resist the effects from in-plane and out-of-plane axial forces, bending moments, and shear forces.
The design and detailing of cast-in-place diaphragms with nonprestressed reinforcement are covered in this chapter.
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Seismic Design Category (SDC) A, B, and C.
9.2 Minimum Diaphragm Thickness
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forces, and axial forces due to wind, seismic, and other applicable lateral forces.
For buildings assigned to SDC A, B, or C, the minimum diaphragm thickness must be the largest of the following
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two-way slabs.
Thickness based on out-of-plane strength requirements.
(3) Thickness based on in-plane strength requirements.
6HUYLFHDELOLW\UHTXLUHPHQWVIRURQHZD\VODEVDQGWZRZD\VODEVDUHFRYHUHGLQ6HFWLRQVDQGRIWKLVSXEOLFDtion, respectively. Methods are given in those sections that can be used to determine minimum slab thicknesses that
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
slab thickness determined based on serviceability requirements. Two-way shear requirements may have an impact
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WKDWWZRZD\VKHDUUHTXLUHPHQWVDUHVDWLV¿HG,QOLHXRILQFUHDVLQJWKHVODEWKLFNQHVVRUFROXPQVL]HWZRZD\VKHDU
UHTXLUHPHQWVFDQEHVDWLV¿HGE\SURYLGLQJVKHDUUHLQIRUFHPHQW VHH6HFWLRQRIWKLVSXEOLFDWLRQ A slab thickness based on serviceability requirements is usually adequate to satisfy in-plane strength requirements.
In-plane moments are resisted by chord reinforcement placed perpendicular to the direction of the lateral force. In
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It is common for the design shear strength of the concrete alone to satisfy in-plane shear strength requirements of a
diaphragm.
Fire-resistance requirements of the governing building code must also be considered when selecting a minimum
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9.3 Required Strength
9.3.1 General
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simultaneous effects from out-of-plane and in-plane loading. For example, collectors are commonly designed for
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plane forces in the diaphragm.
9.3.2 Diaphragm Design Forces
Overview
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)LJXUH5 (a) Diaphragm in-plane forces due to lateral loads acting on the building
(b) Diaphragm transfer forces
(c) Connection forces between the diaphragm and vertical framing or nonstructural elements
(d) Forces resulting from bracing vertical or sloped building elements
(e) Diaphragm out-of-plane forces due to gravity and other loads applied to the surface of the diaphragm
In-Plane Forces due to Lateral Loads
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at level in a building is equal to the total
this publication. In the case of wind loads, the resultant wind force,
at that level (which is sum of the windward pressure,
and the leeward pressure, ) times the
wind pressure,
times the tributary story height,
(see Figure 9.1). This force is applied at the centroid of
width of the building,
the face of the building in the direction of analysis.
is applied at the center of mass (CM) of the diaphragm at level because it is an
The seismic design force,
LQHUWLDOIRUFH VHH)LJXUH )RUULJLGGLDSKUDJPV VHH6HFWLRQRIWKLVSXEOLFDWLRQIRUFODVVL¿FDWLRQRIGLDphragms), the diaphragm translates and rotates about the center of rigidity (CR).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ܮ‬
‫ܤ‬
‫݌‬௫ ‫ܤ‬Τʹ
ܹ௫ ൌ ‫݌‬௫ ‫݄ܤ‬௫ Figure 9.1 Resultant wind force acting on a diaphragm.
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Figure 9.2 Seismic force acting on a rigid diaphragm.
For buildings with one or more basement levels, lateral forces are generated by soil pressure bearing against the
basement walls (see Figure 9.3). The magnitude of the soil pressure is usually obtained from a geotechnical investigation. In cases where the results of such an investigation are not available, the lateral soil loads in IBC Table
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9.3 that a portion of the soil pressure is transferred to the foundation system and a portion is transferred to the reinforced concrete slab, which acts as a diaphragm. The diaphragm must be designed for the in-plane effects due to
the soil pressure (plus any other applicable in-plane load effects) and must transfer the forces to the basement walls
parallel to the direction of the soil pressure.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‡‹ˆ‘” ‡† ‘ ”‡–‡•Žƒ„
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Figure 9.3 Distribution of at-rest soil pressure on a basement wall.
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In-Plane Forces due to Transfer Forces
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The podium slab, in turn, transfers all the in-plane forces to the basement walls parallel to the direction of analysis.
‹ƒ’Š”ƒ‰ˆ‘” ‡
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Figure 9.4 Transfer forces to a podium slab.
Connection Forces Between the Diaphragm and Vertical Framing or Nonstructural Elements
Nonstructural components, such as cladding, is commonly attached to reinforced concrete slabs over the height of a
building. These components transfer wind pressures, inertial forces from earthquake shaking, or both, to the diaphragm through the connections between the two.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
”‡ ƒ•– ‘ ”‡–‡™ƒŽŽ’ƒ‡Ž
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Figure 9.5 Wind forces from a nonstructural element transferred to a diaphragm.
,OOXVWUDWHGLQ)LJXUHLVDJHQHULFWLHEDFNFRQQHFWLRQEHWZHHQDSUHFDVWFRQFUHWHZDOOSDQHODQGDUHLQIRUFHGFRQcrete slab, which is representative of the many types of connections between cladding and diaphragms. Component
and cladding (C&C) wind pressure normal to the surface of the panel is transferred from the precast concrete wall
panel to the diaphragm via the connector assembly. The diaphragm must be designed for this force in combination
with all other applicable forces.
A similar force transfer to a diaphragm occurs for nonstructural wall panels subjected to out-of-plane seismic forces.
Forces Resulting from Bracing Vertical or Sloped Building Elements
Diaphragms must be designed for forces resulting from bracing vertical structural elements in a building. For example, in the case of a building frame system, it is assumed that all the lateral forces are resisted by the shear walls.
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when subjected to lateral forces (see Figure 9.6). In an actual building, the diaphragms connect the columns and
shear walls together, which means these elements must displace the same amount at each level. Thus, the resulting
transfer forces caused by displacement compatibility between the columns that are not part of the LFRS and the
shear walls must be resisted by the diaphragms.
Diaphragms must also be designed to resist forces from any sloped structural members in a building. An inclined
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gravity and lateral forces, where applicable, into the reinforced concrete slab (diaphragm) in cases where there are
no other local structural elements to counteract these forces.
Out-of-Plane Forces
One of the main functions of a diaphragm is to support and transfer gravity loads to the supporting structural members in a building. The weight of the structure, superimposed dead loads, and live loads are typical out-of-plane
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are also applicable, as are vertical accelerations due to seismic effects.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‹ƒ’Š”ƒ‰ሺ–›’Ǥሻ
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Figure 9.6 Diaphragm force transfer due to displacement compatibility.
Collector Design Forces
Where vertical elements of the LFRS do not extend the full depth of a diaphragm, collectors are needed to transfer
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several factors, a collector can be part of the slab or can be a beam.
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loads (axial tension and compression forces caused by in-plane shear transfer in the diaphragm due to wind forces,
seismic forces, or both). The in-plane diaphragm forces are determined using the methods presented in Section 9.3.3
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collector axial forces in diaphragms subjected to wind and seismic forces in buildings assigned to SDC A, B, or C.
9.3.3 Diaphragm Modeling and Analysis
Overview
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Some of the general analysis requirements in ACI Chapter 6 are applicable to diaphragms. For example, the provisions for elastic analysis in ACI 6.6.1 through 6.6.3 can be applied because diaphragms are designed to remain
essentially elastic when subjected to in-plane wind and seismic forces.
In-plane stiffness modeling and analysis methods for diaphragms are covered below.
In-Plane Stiffness Modeling
It is permitted to use any set of reasonable and consistent assumptions for in-plane diaphragm stiffness (ACI
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are contingent on the in-plane stiffness of the diaphragm.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
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to that of the vertical elements of the LFRS. For purposes of analysis, rigid diaphragms displace and rotate as a rigid
body when subjected to lateral forces, and the vertical elements of the LFRS move together accordingly (displacement compatibility). Also, lateral forces are distributed to the vertical elements of the LFRS in proportion to their
relative rigidities (stiffnesses) and their location with respect to the center of rigidity (CR), which is the stiffness
centroid within a diaphragm.
,QFRQWUDVWWKHLQSODQHGHÀHFWLRQRIDÀH[LEOHGLDSKUDJPLVUHODWLYHO\ODUJHFRPSDUHGWRWKDWRIWKHYHUWLFDOHOHments of the LFRS, and the distribution of lateral forces to the vertical elements of the LFRS is independent of their
relative rigidities. In such cases, lateral forces are distributed based on the tributary mass of the diaphragm to the
vertical elements of the LFRS. For diaphragms of uniform material and weight, lateral forces can be distributed by
tributary areas. Flexible diaphragms do not undergo rigid body rotation like rigid diaphragms.
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• :KHQVXEMHFWHGWRODWHUDOVHLVPLFIRUFHV $6&(6(, and
(a) Span-to depth ratio
(b) 6WUXFWXUHKDVQRQHRIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH
݄
When determining the span-to-depth ratio, the span is equal to the distance between lines of lateral resistance (such
DVZDOOVDQGIUDPHV LQWKHGLUHFWLRQRIDQDO\VLV VHH)LJXUH 7KHRYHUDOOGHSWKRIWKHGLDSKUDJPLQWKHGLUHFWLRQ
of analysis is used to determine the span-to-depth ratio.
κଵ κଶ κଵ
’ƒ
ൌ ‡’–Š ݄
’ƒ
κଶ
ൌ ‡’–Š
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Figure 9.7 Definition of span-to-depth ratio for a diaphragm.
7KHUHLQIRUFHGFRQFUHWHGLDSKUDJPLOOXVWUDWHGLQ$&,)LJXUH5DPD\QRWEHFRQVLGHUHGULJLGLQWKHGLUHFwhere is the distance between the
tion of the lateral force because of its relatively large span-to-depth ratio,
walls at each end, which have been designated to be part of the LFRS (the interior moment frames are not part of
the LFRS), and is the depth of the diaphragm in the direction of analysis. Similarly, ramps in parking structures
PD\QRWEHFODVVL¿HGDVULJLGEHFDXVHRIUHODWLYHO\ORQJVSDQWRGHSWKUDWLRVZKLFKDUHFRPPRQLQVXFKVWUXFWXUHV
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9-7
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$FFRUGLQJWR$6&(6(,DGLDSKUDJPLVSHUPLWWHGWREHLGHDOL]HGDVÀH[LEOHZKHUH
LVWKHPD[LPXPLQSODQHGHÀHFWLRQRIWKHGLDSKUDJPDQG
is
>$6&(6(,(TXDWLRQ @7KHWHUP
the average drift of adjoining vertical elements of the LFRS over the story below the diaphragm under consideration
ZKHQVXEMHFWHGWRWKHLQSODQHIRUFH$ULJLGGLDSKUDJPLVGH¿QHGLQ,%&EDVHGRQGHÀHFWLRQVDQGGULIWV
as well: A diaphragm is rigid for the purpose of distribution of story shear and torsional moment when the lateral
is less than or equal to two times the average story drift,
deformation of the diaphragm,
)RUGLDSKUDJPVWKDWFDQQRWEHLGHDOL]HGDVHLWKHUULJLGRUÀH[LEOHWKHLQSODQHVWLIIQHVVRIWKHGLDSKUDJPPXVWEH
explicitly accounted for in the analysis. A three-dimensional analysis considering the stiffnesses of the diaphragms
and the elements of the LFRS usually provides the most accurate distribution of forces in such cases.
7RDFFRXQWIRUFUDFNLQJLQUHLQIRUFHGFRQFUHWHGLDSKUDJPVDVWLIIQHVVPRGL¿HUVKRXOGEHDSSOLHGWRWKHJURVV
in-plane stiffness of the diaphragm. In the case of wind forces, the structure is typically assumed to respond in the
HODVWLFUDQJHVRXVLQJSHUFHQWRIWKHJURVVLQSODQHPRPHQWRILQHUWLDRIWKHGLDSKUDJPZRXOGEHDSSURSULDWH
VHH$&, )RUVHLVPLFIRUFHVLWKDVEHHQVKRZQWKDWVWLIIQHVVPRGL¿HUVW\SLFDOO\IDOOLQWKHUDQJHRI
WR
Analysis Methods
Once the diaphragm forces have been determined and the type of diaphragm has been established based on in-plane
VWLIIQHVV ULJLGVHPLULJLGRUÀH[LEOH WKHGLDSKUDJPDQGDQ\FROOHFWRUVPXVWEHDQDO\]HGIRUWKHHIIHFWVGXHWRLQ
plane forces in combination with out-of-plane forces.
$FFRUGLQJWR$&,LQSODQHGHVLJQPRPHQWVVKHDUIRUFHVDQGD[LDOIRUFHVLQGLDSKUDJPVPXVWEHGHWHUPLQHGXVLQJDQ\PHWKRGWKDWVDWLV¿HVHTXLOLEULXPDQGWKHGHVLJQERXQGDU\FRQGLWLRQV0HWKRGVRIDQDO\VLVEDVHG
on the following models are permitted to be used:
• $ULJLGGLDSKUDJPPRGHOZKHUHGLDSKUDJPVFDQEHLGHDOL]HGDVULJLG
• $ÀH[LEOHGLDSKUDJPPRGHOZKHUHGLDSKUDJPVFDQEHLGHDOL]HGDVÀH[LEOH
• A bounding analysis in which the design values are the envelope of values obtained by assuming upper
bound and lower bound in-plane stiffnesses for the diaphragm in two or more separate analyses.
• $¿QLWHHOHPHQWPRGHOFRQVLGHULQJGLDSKUDJPÀH[LELOLW\
• $VWUXWDQGWLHPRGHOLQDFFRUGDQFHZLWK$&,
The applicability of these models and other pertinent information are given in Table 9.1.
Table 9.1 Diaphragm Models
Model
Applicability
Notes
Rigid
diaphragm
• For lateral wind forces, reinforced concrete slabs
ZLWKDVSDQWRGHSWKUDWLRRIRUOHVV
• For lateral seismic forces, when the following two
FRQGLWLRQVDUHVDWLV¿HG UHLQIRUFHGFRQFUHWH
VODEVZLWKDVSDQWRGHSWKUDWLRRIRUOHVVDQG VWUXFWXUHSRVVHVVHVQRQHRIWKHKRUL]RQWDOLUUHJXODULWLHVLQ$6&(6(,7DEOH
Beam models are commonly used to
determine diaphragm internal forces in
buildings without major irregularities
DQGRUWUDQVIHUIRUFHV
Flexible
diaphragm
$GLDSKUDJPLVSHUPLWWHGWREHLGHDOL]HGDVÀH[LEOH
ZKHUHWKHUDWLRRIWKHPD[LPXPLQSODQHGHÀHFWLRQ
and the average story drift,
of the diaphragm,
LVJUHDWHUWKDQ
Beam models are commonly used to
determine diaphragm internal forces in
buildings without major irregularities
DQGRUWUDQVIHUIRUFHV
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.1 Diaphragm Models (cont.)
Model
Applicability
Notes
Suitable for semirigid diaphragms
Diaphragm internal forces are taken as
the envelope of values from the following two analyses: (1) rigid diaphragm on
ÀH[LEOHVXSSRUWVDQG ÀH[LEOHGLDphragm on rigid supports
Finite
element
Suitable for any diaphragm
This model is especially useful for
irregularly shaped diaphragms and diaSKUDJPVZLWKODUJHWUDQVIHUIRUFHVDQG
or openings
Strut-and-tie
Suitable for any diaphragm
Model should include consideration of
force reversals that may occur under
design load combinations
Bounding
analysis
%HDPPRGHOVDUHZLGHO\XVHGWRGHWHUPLQHLQWHUQDOIRUFHVLQULJLGDQGÀH[LEOHGLDSKUDJPVLQEXLOGLQJVZLWKRXW
PDMRULUUHJXODULWLHVDQGRUWUDQVIHUIRUFHV'LDSKUDJPVDUHWUHDWHGDVEHDPVWKDWVSDQEHWZHHQWKHYHUWLFDOHOHPHQWV
RIWKH/)56ZKLFKDUHLGHDOL]HGDVHLWKHUULJLGRUÀH[LEOH VSULQJ VXSSRUWV%HDPPRGHOVIRUULJLGGLDSKUDJPVDUH
covered in detail below.
Equivalent Beam Model with Rigid Supports
This model is often used for low-rise buildings with regular geometries where two or more lines of lateral forceresisting systems are provided in a given direction that have approximately the same relative rigidity.
Diaphragm Forces
Illustrated in Figure 9.8 is a reinforced concrete diaphragm and a LFRS consisting of structural (shear) walls (it is
assumed that the perimeter frames are not part of the LFRS). In the direction of analysis, the diaphragm is modeled
The walls, which have the same in-plane stiffness,
as a beam whose depth is equal to the full diaphragm depth,
are assumed to be rigid supports in the direction of analysis. This method can also be used where walls are at the
perimeter of the building or where more than two walls are in the direction of analysis (which means the equivalent
beam is continuous). Because the walls do not extend the full depth of the diaphragm, collectors are needed to collect the shear from the diaphragm and to transfer it to the walls.
in Figure 9.8 is uniformly distributed over the width of the diaphragm. The reacThe equivalent in-plane load,
are the diaphragm forces transferred to the vertical elements of the LFRS, and
WLRQVLQZDOOVDQG and
are obtained from statics. It is permitted to assume that the diaphragm resists in-plane moment and axial force in
DFFRUGDQFHZLWK$&,DQGZKLFKLQFOXGHVWKHDVVXPSWLRQWKDWVWUDLQVYDU\OLQHDUO\RYHUWKHGHSWKRIWKH
diaphragm. Shear and moment diagrams for the diaphragm are also given in Figure 9.8.
,QWKLVPRGHOURRIDQGÀRRUV\VWHPVDFWDVWKHZHERIWKHHTXLYDOHQWEHDPZKLFKUHVLVWVWKHGHVLJQVKHDUIRUFHV
must be uniform over the depth )RUZDOO
)RUGLDSKUDJPVDQDO\]HGE\WKLVPHWKRGWKHVKHDUÀRZ
)RUZDOOWKHPD[LPXPXQLIRUPVKHDUÀRZLVHTXDOWR
which must be less than
RUHTXDOWRWKHGHVLJQVKHDUVWUHQJWKRIWKHGLDSKUDJP6KHDUÀRZLVWUDQVIHUUHGIURPWKHGLDSKUDJPWRWKHYHUWLFDO
elements of the LFRS by shear friction.
7KHGLDSKUDJPERXQGDULHVSHUSHQGLFXODUWRWKHODWHUDOIRUFHDFWDVWKHÀDQJHVRIWKHHTXLYDOHQWEHDP FRPPRQO\
referred to as chords) and resist the tension and compression forces induced in the diaphragm due to bending. These
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9-9
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‫ܤ‬
ࢀ࢛ ᬇ
Š‘”†ˆ‘” ‡
ሺ–›’Ǥሻ
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ܴଵ ܴଶ ‹ˆ‘”•Š‡ƒ”ˆŽ‘™ǡ
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‫ݓ‬௨ ܴଶ ܸ௨ ܸ௨ǡ௠௔௫ ‫ܯ‬௨ ‫ܯ‬௨ǡ௠௔௫ Figure 9.8 Equivalent beam model with rigid supports.
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9-10
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
forces can be determined by dividing the maximum bending moment in the diaphragm by the distance between the
forces (that is, the moment arm):
(9.1)
In this equation, LVWKHSHUSHQGLFXODUGLVWDQFHEHWZHHQWKHFKRUGIRUFHVZKLFKLVFRPPRQO\WDNHQDVSHUFHQW
). Reinforcement must be provided
of the total diaphragm depth in the direction of analysis (in this case,
in the chords to resist the tension forces generated by this bending moment. Because wind and seismic forces can
act in any direction, chord reinforcement is required at both boundaries of the diaphragm perpendicular to the direction of analysis.
Collector Forces
A reinforced concrete beam or a portion of a reinforced concrete slab in line with the LFRS can be used as a collector. In cases where the beam or slab has the same width as the vertical element of the LFRS it frames in to (for
example, the width of the collector is equal to the thickness of the structural wall), all the collected forces are
WUDQVIHUUHGGLUHFWO\LQWRWKHPHPEHURIWKH/)56DWLWVHQGV7KLVLVLOOXVWUDWHGLQ)LJXUHIRUZDOOZKLFKLV
part of the LFRS of the building in Figure 9.8. The uniform factored shear force in the diaphragm at this location,
is equal to the reaction in the wall,
divided by the depth of the diaphragm in the direction of analysis,
The net factored
Similarly, the uniform factored shear force in the wall along its length is equal to
shear force at any point is equal to the difference between the unit factored shear forces in the wall and diaphragm at
that point.
ܴଶ
‫ܮ‬
ܴଶ
‫ܮ‬
κଷ Collector ሺtyp.ሻ
ܴଶ
κଶ
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κଵ ܴଶ
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Unit Shear Forces
Net Shear Forces
Collector Axial Forces
Figure 9.9 Unit shear forces, net shear forces, and collector forces in a diaphragm.
The axial forces in the collectors are determined by summing the areas in the net shear force diagram. At the edges
RIWKHGLDSKUDJPWKHD[LDOIRUFHLQWKHFROOHFWRUVLVHTXDOWR]HUR 1RWH,QFDVHVZKHUHZLQGDQGRUVHLVPLFIRUFHV
IURPSHULPHWHUHOHPHQWVVXFKDVFODGGLQJPXVWEHFROOHFWHGWKHIRUFHVDWWKHHQGVRIDFROOHFWRUPD\QRWEH]HUR The axial force increases linearly as shear is transferred from the diaphragm to the collectors. At the end of the wall
from the edge of the diaphragm, the axial tension force in the collector is equal to
located a distance
at a distance
from the edge of the
At the other end of the wall, the axial compression force is equal to
diaphragm. Because wind and seismic forces can act in any direction, the axial forces in the collectors change from
compression to tension and from tension to compression, so a collector must be designed for the most critical effects. Similar force diagrams can be obtained for wall 1.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Where a collector is assumed to be wider than the member of the LFRS element it frames in to, a part of the collector force is transferred directly into the member of the LFRS at its ends and a part is transferred through shearIULFWLRQDORQJWKHOHQJWKRIWKHYHUWLFDOHOHPHQWRIWKH/)56 VHH$&,)LJXUH5IRUWKHFDVHRIDZDOOORFDWHG
at the edge of a diaphragm). Wider collectors are usually required in buildings where it is not practical or possible
(from a design or constructability perspective) to provide collector elements concentric with the vertical elements of
the LFRS. An example of where wider collectors may be necessary is in a building with relatively thin walls where
only a narrow strip of slab is available to resist relatively large collector forces.
adjacent to the shear wall or frame is used to resist the collector forces. Requirements
An effective slab width,
DUHQRWFRYHUHGLQWKHFRGHVHFWLRQVRI$&,RULQ$6&(6(,5HIHUHQFHUHFRPPHQGVXVLQJ
for
equal to the width of the vertical element of the LFRS plus a width on either side of the vertical element equal to
RQHKDOIWKHFRQWDFWOHQJWKEHWZHHQWKHGLDSKUDJPDQGWKHYHUWLFDOHOHPHQW VHH)LJXUHIRUWKHHIIHFWLYHFROOHF$&,5UHFRPPHQGVDQHIIHFWLYHVODEZLGWKHTXDOWREDVLWRUZLGWKIRUZDOOLQ)LJXUHZKHUH
FDOO\WKHVDPHDVWKDWLQ5HIHUHQFH
ܾ௘௙௙ ݁
ܾ௘௙௙ ൌ ‫ ݐ‬൅ ሺκଶ Τʹሻ
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κଶ ሺܶ௩ ൅ ‫ܥ‬௩ ሻΤκଶ κଵ ܴଶ Ȁ‫ܮ‬
‫ܥ‬௩ ‫ܥ‬ௗ Figure 9.10 Effective slab width for collectors wider than the vertical elements of the LFRS.
$OVRLOOXVWUDWHGLQ)LJXUHLVWKHIRUFHWUDQVIHUWRZDOO7KHWHQVLRQDQGFRPSUHVVLRQIRUFHVWUDQVIHUUHGGLUHFWO\
respectively. The tension force,
is resisted by reinforcement in
LQWRWKHHQGVRIZDOODUHGHVLJQDWHG and
is resisted by the concrete. The remainthe wall parallel to the direction of analysis and the compression force,
and
respectively, are transferred through shear along
which is the
ing tension and compression forces
this reinforceOHQJWKRIZDOO6KHDUIULFWLRQUHLQIRUFHPHQWLVUHTXLUHGWRUHVLVWWKHWRWDOVKHDUIRUFH
PHQWLVSODFHGSHUSHQGLFXODUWRWKHZDOO VHH$&,)LJXUH5 act at an eccentricity with respect to the centerline of wall
,WLVHYLGHQWIURP)LJXUHWKDWWKHIRUFHV and
,ILWLVDVVXPHGWKHUHLQIRUFHPHQWLQWKHFROOHFWRULVXQLIRUPO\GLVWULEXWHGLQWKHGLUHFWLRQRIDQDO\VLVZLWKLQ
and the in-plane bending
OLNHWKDWVKRZQLQ$&,)LJXUH5 WKHQWKHHFFHQWULFLW\ can be taken as
moment resulting from this eccentricity and the portion of the collector force not transferred directly into the end of
WKHZDOOPXVWEHFRQVLGHUHGLQGHVLJQ VHH6HFWLRQRIWKLVSXEOLFDWLRQ In general, the section of the collector concentric with the vertical element of the LFRS should be designed for a
reasonable portion of the total collector force considering overall design and construction limitations. Reinforcing
bar congestion at the ends of a shear wall, for example, is a practical limitation to consider when selecting the portion of the total tension force transferred directly into the end of a wall. In some cases, the total collector force will
need to be resisted entirely by the shear in the slab adjacent to the vertical element of the LFRS.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Corrected Equivalent Beam Method with Spring Supports
This model is best suited for buildings with rigid diaphragms and lateral force-resisting systems with different stiffnesses. Effects of torsion are automatically accounted for in this method.
Diaphragm Forces and Center of Rigidity
7KH¿UVWVWHSLQGHWHUPLQLQJWKHLQWHUQDOLQSODQHIRUFHVLQDGLDSKUDJPEDVHGRQWKHFRUUHFWHGHTXLYDOHQWEHDP
model is to obtain the diaphragm forces transferred to the vertical elements of the LFRS. In multistory buildings,
a three-dimensional model of the building is usually constructed with the appropriate lateral forces applied at each
level over the height of the building. The forces in the vertical elements of the LFRS are obtained from an analysis
of the structure using this model. In low-rise buildings with rigid diaphragms, the stiffnesses of the elements of the
/)56FDQEHREWDLQHGXVLQJDSSUR[LPDWHDQDO\VHVZKLFKGRQRWUHVXOWLQDVLJQL¿FDQWORVVRIDFFXUDF\0HWKRGVWR
GHWHUPLQHDSSUR[LPDWHLQSODQHVWLIIQHVVHVIRUZDOOVDQGIUDPHVDUHJLYHQLQ5HIHUHQFH2WKHUZLVHVWLIIQHVVHV
FDQEHREWDLQHGIURPDPRUHUH¿QHGDQDO\VLV
:KHQXWLOL]LQJDQDSSUR[LPDWHPHWKRGRIDQDO\VLVWRGHWHUPLQHWKHIRUFHVLQWKHYHUWLFDOHOHPHQWVRIWKH/)56
WKHORFDWLRQRIWKHFHQWHURIULJLGLW\ &5 QHHGVWREHGHWHUPLQHG7KH&5LVWKHSRLQWRQDÀRRUURRIOHYHOZKHUH
the equivalent story stiffness is assumed to be located. For buildings with rigid diaphragms, application of a lateral
force through the CR produces only rigid body displacement of the story. Displacement and rotation occur where
the lateral force is applied at any other point. The following equations can be used to locate the CR in the x-direction
and y-direction
from a selected origin (see Figure 9.11 where the origin is selected at the centroid of the
column located at the bottom left of the framing system):
(9.2)
(9.3)
where
in-plane lateral stiffness of lateral-force-resisting element i in the y-direction
in-plane lateral stiffness of lateral-force-resisting element i in the x-direction
distance in the x-direction from the origin to the centroid of lateral-force-resisting element i
distance in the y-direction from the origin to the centroid of lateral-force-resisting element i
All the elements of the LFRS parallel to the y- and xGLUHFWLRQVRIDQDO\VHVDUHLQFOXGHGLQ(TXDWLRQV DQG respectively; it is commonly assumed that out-of-plane resistance of a lateral-force-resisting element is negligible
compared to its in-plane resistance. Thus, when determining the location of the CR in the x-direction by Equation
RQO\WKHVWLIIQHVVHVRIZDOOVDQGDUHFRQVLGHUHG1RWHWKDWDSRUWLRQRIZDOO WKDWLVDQHIIHFWLYHÀDQJH
width) may be included when determining the stiffness of wall 1. Similarly, when determining the location of the
CR in the yGLUHFWLRQE\(TXDWLRQ RQO\WKHVWLIIQHVVRIZDOO LQFOXGLQJDQHIIHFWLYHÀDQJHZLGWKRIZDOOLI
GHVLUHG DQGZDOODUHFRQVLGHUHG
which is
Once the location of the CR has been determined, the following equation can be used to determine
resisted by element i considering both direct and torsional shear forces:
the portion of the total story shear,
(9.4)
where
perpendicular distance from the centroid of element i to the CR parallel to the x-axis
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ᬈ
‫ݕ‬ସ ‫ݕ‬௖௥ ᬆ
‫ݕ‬ଷ ᬅ
ᬇ
‫ݔ‬ଵ ‫ݔ‬௖௥ ‫ݔ‬ଶ Figure 9.11 Location of center of rigidity (CR).
perpendicular distance from the centroid of element i to the CR parallel to the y-axis
perpendicular distance between
and the CR
Similarly, the following equation can be used to determine
which is the portion of the total story shear,
resisted by element i considering both direct and torsional shear forces:
(9.5)
where
perpendicular distance between
and the CR.
7KH¿UVWDQGVHFRQGWHUPVLQWKHVHHTXDWLRQVUHSUHVHQWWKHGLUHFWVKHDUIRUFHVDQGWKHWRUVLRQDOVKHDUIRUFHVWUDQVferred to the vertical elements of the LFRS, respectively.
Once the diaphragm forces transferred to the vertical elements of the LFRS have been obtained using either an
DSSUR[LPDWHRUPRUHUH¿QHGDQDO\VLVDVRXWOLQHGDERYHWKHVHFRQGVWHSLVWRGHWHUPLQHDQHTXLYDOHQWLQSODQH
distributed load on the diaphragm that is in equilibrium with these forces (reactions). The distributed load is usuat each end of
DOO\WUDSH]RLGDOZKLFKDFFRXQWVIRUDQ\WRUVLRQDOPRPHQWV VHH)LJXUH 7KHORDGV and
the equivalent beam can be obtained by using the equations for force equilibrium and moment equilibrium and then
and
solving these two equations for the two unknowns
)RUWKHULJLGGLDSKUDJPLQ)LJXUHWKHIRUFHDQGPRPHQWHTXDWLRQVRIHTXLOLEULXPLQWKHGLUHFWLRQRIDQDO\VLV
are the following where moments are summed about the left edge of the diaphragm:
(9.6)
(9.7)
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9-14
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
ᬉ
‫ܮ‬
ᬈ
ᬅ
ᬇ
ᬆ
ᬊ
ᬋ
ܾଶ ܾଵ ܾଷ ܸ௬ ܴଵ ܴଶ ܴଷ ‫ݓ‬ଵ ‫ݓ‬ଶ ܸ௨ǡ௠௔௫ Š‡ƒ”
‘‡–
‫ܯ‬௨ǡ௠௔௫ Figure 9.12 Equivalent distributed load, shear diagram, and moment diagram for a rigid diaphragm.
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9-15
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHUHVXOWDQWIRUFHRIWKHWUDSH]RLGDOGLVWULEXWHGORDGLVHTXDOWRWKHDSSOLHGODWHUDOIRUFHDWWKDWOHYHOREWDLQHGIURP
DQDO\VLVZKLFKLVUHÀHFWHGLQ(TXDWLRQ 7KHPRPHQWFDXVHGE\WKHIRUFHVLQWKHHOHPHQWVRIWKH/)56LQWKH
GLUHFWLRQSHUSHQGLFXODUWRWKHLQSODQHIRUFHLVRIWHQLJQRUHGLQRYHUDOOKRUL]RQWDOIRUFHGLVWULEXWLRQLWFDQEHLQFRUSRUDWHGLQWRWKHWUDSH]RLGDOORDGLIGHVLUHG
and
KDYHEHHQGHWHUPLQHGWKH¿QDOVWHSLVWRFRQVWUXFWVKHDUDQGPRPHQWGLDJUDPVIRUWKHGLDSKUDJP
Once
VHH)LJXUH 7KHVKHDUGLDJUDPLVXVHGLQ FKHFNLQJWKHGHVLJQVKHDUVWUHQJWKRIWKHGLDSKUDJP GHVLJQing the connections of the diaphragm to the vertical elements of the LFRS, and (3) determining the axial compression and tension forces in the collectors, if any. As noted previously, the moment diagram is used in determining the
tension and compression forces in the chords, the former of which is needed to calculate the required area of chord
reinforcement near the edges of the diaphragm.
Diaphragms with Openings
$FFRUGLQJWR$&,WKHHIIHFWVRIVODERSHQLQJVDQGVODEYRLGVPXVWEHFRQVLGHUHGLQWKHDQDO\VLVDQGGHVLJQ
of diaphragms.
The corrected equivalent beam method can also be used for diaphragms with relatively large openings. For purposes
of analysis, the diaphragm segments (commonly referred to as subdiaphragms) above and below the opening can be
LGHDOL]HGDVEHDPV¿[HGDWHDFKHQG,WLVDVVXPHGWKDWWKHFROOHFWRUHOHPHQWRQRQHVLGHRIWKHRSHQLQJFROOHFWVWKH
uniform diaphragm shear on that side and transfers it to the subdiaphragms above and below the opening in proportion to their relative stiffness or mass. The collector on the other side of the opening then collects the shear from the
subdiaphragms and transfers it to the portion of the diaphragm on that side of the opening. Thus, the loading on a
subdiaphragm is based on the total applied force at that level and the relative stiffness or mass of the subdiaphragm.
Secondary chord forces occur in each subdiaphragm due to local bending caused by this loading.
ᬆ
ᬇ
κଶ ᬅ
‫ܣ‬ଵ κ௢௣௘௡ ‫ܮ‬
κଵ 7KHGLDSKUDJPGHSLFWHGLQ)LJXUHLVWKHVDPHDVWKHRQHLQ)LJXUHEXWZLWKDUHODWLYHO\ODUJHRSHQLQJLQLW
and
ZKLFKFDQEHREWDLQHGIURPWKHRYHUDOOWUDSH]RLThe forces at each edge of the opening are designated
dal force distribution.
‫ܣ‬ଶ ܾଶ ܾଵ ܴଵ ܾଷ ܴଶ ܴଷ ܾ௢௣௘௡ ‫ݓ‬ଵ ‫ݓ‬௅ ‫ݓ‬ோ Figure 9.13 Diaphragm with a relatively large opening.
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‫ݓ‬ଶ Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
When a diaphragm is subjected to wind forces, the forces on the subdiaphragms above and below the opening can
be approximately determined using the in-plane stiffness ratios of these subdiaphragms. The mass of the subdiaphragms is used to determine the forces when a diaphragm is subjected to seismic forces. The stiffness and mass raand
DUHJLYHQLQ7DEOH7KHPDVVUDWLRVLQ7DEOHIRUXVH
tios for the top and bottom subdiaphragms,
and
where it is assumed that the diaphragm
with seismic forces are based on the areas of the subdiaphragms,
has the same thickness and material properties everywhere.
ܴ௧ǡ௅ ܾ௢௣௘௡ ܴ௧ǡோ ି
‫ܯ‬௨ǡ௧ǡ௅
ି
‫ܯ‬௨ǡ௧ǡோ
‫ݓ‬௧ǡ௅ ൌ ݂௧௢௣ ‫ݓ‬௅ ‫ݓ‬௧ǡோ ൌ ݂௧௢௣ ‫ݓ‬ோ ା
‫ܯ‬௨ǡ௧
‘‡–
ି
‫ܯ‬௨ǡ௧ǡ௅
ି
‫ܯ‬௨ǡ௧ǡோ
ሺƒሻ ‘’•—„†‹ƒ’Š”ƒ‰
ܴ௕ǡ௅ ܾ௢௣௘௡ ܴ௕ǡோ ି
‫ܯ‬௨ǡ௕ǡ௅
ି
‫ܯ‬௨ǡ௕ǡோ
‫ݓ‬௕ǡ௅ ൌ ݂௕௢௧ ‫ݓ‬௅ ‫ݓ‬௕ǡோ ൌ ݂௕௢௧ ‫ݓ‬ோ ା
‫ܯ‬௨ǡ௕
‘‡–
ି
‫ܯ‬௨ǡ௕ǡ௅
ሺ„ሻ‘––‘•—„†‹ƒ’Š”ƒ‰
ି
‫ܯ‬௨ǡ௕ǡோ
Figure 9.14 Free-body diagrams and moment diagrams for the subdiaphragms in Figure 9.13.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.2 Stiffness and Mass Ratios for Diaphragms with Openings
In-Plane Force
ftop
fbot
Wind
Seismic
)UHHERG\GLDJUDPVRIWKHWRSDQGERWWRPVXEGLDSKUDJPVDUHJLYHQLQ)LJXUHDVQRWHGSUHYLRXVO\WKHHQGVRI
WKHVHHOHPHQWVDUHDVVXPHGWREH¿[HG7KHPRPHQWGLDJUDPVLQWKH¿JXUHFDQEHREWDLQHGIURPVWDWLFVZKHUHWKH
and
GHWHUPLQHGIURP7DEOHDUHXVHGWRREWDLQWKHIRUFHVDWHDFKHQGRIWKHVXEGLDSKUDJP
appropriate
and compression
chord forces are determined using Equation
The secondary tension
is the maximum positive moment
in each subdiaphragm.
(9.1) where
The larger of the following two tension chord forces is used to determine the required area of chord reinforcement
is the bending moment in the diaphragm at the center of the opening (see
at the edges of the diaphragm where
)LJXUH (9.8)
ࢀ࢛ǡ࢕࢖ࢋ࢔ ᬆ
ᬇ
ࢀ࢛ǡ૛ ࡯࢛ǡ૛ κଶ ᬅ
࡯࢛ǡ૚ κ௢௣௘௡ ‫ܮ‬
κଵ ࢀ࢛ǡ૚ ܾଵ ࡯࢛ǡ࢕࢖ࢋ࢔ ܾଶ ܾଷ ‘‡–
‫ܯ‬௨ǡ௠௔௫ ‫ܯ‬௨ Figure 9.15 Determination of chord forces in a diaphragm with an opening.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Secondary chord forces also develop at the corners of the opening due to the negative moments at these locations.
along the bottom edge of the opening (that is, at the
)RUWKHGLDSKUDJPLQ)LJXUHWKHWHQVLRQFKRUGIRUFH
and
Reinforcing bars
top of the bottom subdiaphragm) is equal to the larger of
must be provided along the entire top and bottom faces of the opening to resist the larger of the tension forces due to
lateral forces acting in either direction.
9.4 Design Strength
9.4.1 General
)RUHDFKDSSOLFDEOHIDFWRUHGORDGFRPELQDWLRQLQ$&,7DEOHWKHDSSOLFDEOHGHVLJQVWUHQJWKV
of diaphragms and connections must be greater than or equal to the applicable required strengths, $&, $V
noted previously, interaction between load effects must be considered in design.
Strength reduction factors, DUHGHWHUPLQHGLQDFFRUGDQFHZLWK$&,$VXPPDU\RIWKHVWUHQJWKUHGXFWLRQ
factors pertinent to the design of diaphragms and collectors subject to moment, axial force, or combined moment
DQGD[LDOIRUFHLVJLYHQLQ7DEOHIRUPHPEHUVZLWKRXWVSLUDOWUDQVYHUVHUHLQIRUFHPHQW7KHVWUDLQXVHGWRGH¿QH
LVHTXDOWRWKHVSHFL¿HG\LHOGVWUHQJWKRIWKHUHLQIRUFHPHQW divided by the
a compression-controlled section,
ZKLFKFDQEHWDNHQDVSVLIRUDQ\JUDGHRIUHLQIRUFHmodulus of elasticity of the reinforcing steel,
PHQW $&, Table 9.3 Strength Reduction Factors, ‫׋‬
Net Tensile Strain, İt
Classification
Strength Reduction Factor, Ҁ*
Compression-controlled
Transition**
Tension-controlled
*Applicable to transverse reinforcement other than spirals.
**Sections classified as transition are permitted to use corresponding to compression-controlled sections.
For diaphragms and collectors subjected to shear forces,
$&,7DEOH The design strength of a diaphragm depends on the type of model used to determine the internal force distribution
(see Section 9.3.3 of this publication for permitted analysis models). Design strength requirements based on the
PRGHOW\SHDUHJLYHQLQ$&, VHH7DEOH Table 9.4 Design Strength Requirements for Diaphragms
Model
Design Strength Requirements
Beam
$&,WKURXJK
Strut-and-tie
$&,
Finite element
$&,&KDSWHU
Alternative
$&, G
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
9.4.2 Nominal Moment and Axial Force Strength
A diaphragm modeled as a beam is permitted to be designed for in-plane moment and axial force using the assumpWLRQVLQ$&, ÀH[XUDOVWUHQJWK DQG D[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWK ZKLFKDUH
WKHVDPHDVVXPSWLRQVXVHGLQWKHGHVLJQRIUHLQIRUFHGFRQFUHWHEHDPVFROXPQVDQGZDOOV $&, 7KLV
includes the assumption that strains vary linearly over the depth of the diaphragm when it is subjected to in-plane
forces.
$FFRUGLQJWR$&,QRQSUHVWUHVVHGUHLQIRUFHPHQWDQGPHFKDQLFDOFRQQHFWRUVXVHGWRUHVLVWWHQVLRQFKRUG
IRUFHVLQDGLDSKUDJPDUHSHUPLWWHGWREHORFDWHGZLWKLQ]RQHVH[WHQGLQJIURPWKHWHQVLRQHGJHRIWKHGLDSKUDJPD
GLVWDQFHHTXDOWRSHUFHQWRIWKHGHSWKRIWKHGLDSKUDJPLQWKHGLUHFWLRQRIDQDO\VLV VHH$&,)LJXUH5 ,WLVDVVXPHGWKDWE\SODFLQJWKHFKRUGUHLQIRUFHPHQWZLWKLQWKHVH]RQHVDQHVVHQWLDOO\XQLIRUPVKHDUÀRZRYHUWKH
depth of the diaphragm occurs, just like the case where the chord reinforcing bars are concentrated along the edges
of the diaphragm. Where the depth of the diaphragm changes along its span, it is permitted to develop chord reinIRUFHPHQWLQWRDGMDFHQWVHFWLRQVHYHQZKHUHWKHUHLQIRUFHPHQWIDOOVRXWVLGHWKHSHUFHQWGHSWKOLPLWRIWKHDGMDcent section.
9.4.3 Nominal Shear Strength
Nominal Shear Strength of Diaphragms
Nominal shear strength,
for cast-in-place reinforced concrete diaphragms is determined by ACI Equation
(9.9)
where
gross area of the diaphragm (diaphragm thickness times width in the direction of analysis)
PRGL¿FDWLRQIDFWRUDFFRXQWLQJIRUWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWH
ratio of distributed slab transverse reinforcement to gross concrete area
7KHPRGL¿FDWLRQIDFWRU WKDWUHÀHFWVWKHUHGXFHGPHFKDQLFDOSURSHUWLHVRIOLJKWZHLJKWFRQFUHWHUHODWLYHWR
QRUPDOZHLJKWFRQFUHWHRIWKHVDPHFRPSUHVVLYHVWUHQJWKLVJLYHQLQ7DEOHEDVHGRQHTXLOLEULXPGHQVLW\DQGLQ
7DEOHEDVHGRQFRPSRVLWLRQRIDJJUHJDWHVLQWKHFRQFUHWHPL[ VHH$&, Table 9.5 Values of Ȝ Based on Equilibrium Density, wc
Ȝ
Equilibrium Density, wc
Table 9.6 Values of Ȝ Based on Composition of Aggregates
Concrete
Composition of Aggregates
Ȝ
All-lightweight
)LQH$670&
&RDUVH$670&
/LJKWZHLJKW¿QHEOHQG
)LQH&RPELQDWLRQRI$670&DQG&
&RDUVH$670&
WR(1)
(table continued on next page)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.6 Values of Ȝ Based on Composition of Aggregates (cont.)
Concrete
Composition of Aggregates
Ȝ
Sand-lightweight
Fine: ASTM C33
&RDUVH$670&
Sand-lightweight, coarse blend
Fine: ASTM C33
&RDUVH&RPELQDWLRQRI$670&
and ASTM C33
WR (1) Linear interpolation from 0.75 to 0.85 is permitted based on the absolute volume of normalweight fine aggregate as a fraction of the total absolute volume of fine aggregate.
(2) Linear interpolation from 0.85 to 1.0 is permitted based on the absolute volume of normalweight coarse aggregate as a fraction of the total absolute volume of aggregate.
The transverse reinforcement ratio,
is determined using the slab reinforcement parallel to the in-plane shear
by Equation (9.9), it is conservative to check the design shear strength requirements asforce. When calculating
LQWKLVHTXDWLRQLVOLPLWHGWRSVL $&, suming LVHTXDOWR]HUR$OVR
The factored shear force,
must not exceed the maximum design shear strength determined by ACI Equation
(9.10)
Shear Transfer
6KHDUWUDQVIHUEHWZHHQGLDSKUDJPVFROOHFWRUVDQGYHUWLFDOHOHPHQWVRIWKH/)56PXVWEHVDWLV¿HGXVLQJWKHVKHDU
IULFWLRQSURYLVLRQVRI$&,RUE\PHFKDQLFDOFRQQHFWRUVRUGRZHOV $&, ,QFDVWLQSODFHFRQVWUXFWLRQ
VKHDUWUDQVIHUUHTXLUHPHQWVPXVWW\SLFDOO\EHVDWLV¿HGDWFRQVWUXFWLRQMRLQWVEHWZHHQWKHHOHPHQWV
Where shear-friction provisions are used, the nominal shear strength across the assumed shear plane is determined
E\$&,(TXDWLRQ (9.11)
In this equation, LVWKHFRHI¿FLHQWRIIULFWLRQZKLFKLVREWDLQHGIURP$&,7DEOH VHH7DEOH DQG
is the area of shear-friction reinforcement crossing the shear plane with the reinforcing bars oriented perpendicular
to that plane.
Table 9.7 Coefficient of Friction, μ
Contact Surface Condition
ȝ
Concrete placed monolithically
Concrete placed against hardened concrete that is clean, free of laitance, and intentionally
roughened to a full amplitude of approximately ¼ in.
Concrete placed against hardened concrete that is clean, free of laitance, and not
intentionally roughened
8SSHUOLPLWVRQVKHDUIULFWLRQVWUHQJWKDUHJLYHQLQ$&,7DEOH VHH7DEOH 7KHWHUP is the area of
which is equal to the gross cross-sectional area of the diaphragm in the direction of analysis.
concrete resisting
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9-21
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.8 Maximum Vn Across the Assumed Shear Plane
Contact Surface Condition
Maximum Vn = Vu / Ҁ*
Normalweight concrete placed monolithically or placed
against hardened concrete intentionally roughened
to a full amplitude of approximately ¼ in.
Other cases
*
area of concrete resisting
9.4.4 Collectors
Collectors are to be designed as tension members, compression members, or both in accordance with the design proYLVLRQVIRUD[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWKLQ$&, $&, ,QFDVHVZKHUHFROOHFWRUVDUHVXEMHFWHGWRUHODWLYHO\VLJQL¿FDQWHIIHFWVIURPJUDYLW\IRUFHVLQDGGLWLRQWRWKRVHIURPWHQVLRQDQGFRPSUHVsion axial forces due to in-plane lateral forces, design strength interaction diagrams are often constructed to check
the adequacy of a collector for all applicable load combinations, just like for columns. Unlike typical columns, both
the axial compression and tension parts of the interaction diagram are generally needed for the design of collectors.
,QDGGLWLRQWRWKHVWUHQJWKUHTXLUHPHQWVQRWHGDERYHWKHVKHDUVWUHQJWKUHTXLUHPHQWVLQ$&,IRURQHZD\VODEV
DQGLQ$&,IRUWZRZD\VODEVPXVWEHVDWLV¿HGZKHUHDSRUWLRQRIDVODELVXVHGDVDFROOHFWRU:KHUHEHDPV
DUHXWLOL]HGDVFROOHFWRUVWKHVWUHQJWKUHTXLUHPHQWVIRUVKHDURUIRUFRPELQHGVKHDUDQGWRUVLRQLQ$&,DQG
PXVWDOVREHVDWLV¿HG
9.5 Reinforcement Limits
The area of reinforcement in diaphragms must be at least equal to the area corresponding to shrinkage and tem $&, WKLVLVDOVRWKHPLQLPXPDUHD
SHUDWXUHUHLQIRUFHPHQWJLYHQLQ$&,ZKLFKLVHTXDOWR
RIÀH[XUDOUHLQIRUFHPHQWIRURQHZD\VODEV VHH$&,DQG )RUWZRZD\VODEVWKHPLQLPXPÀH[XUDO
H[FHSWLQWKHHIIHFWLYHVODEZLGWKVSHFL¿HGLQ$&,ZKHUHWKHPLQLreinforcement is also equal to
DQGWKHDUHDGHWHUPLQHGE\$&,(TXDWLRQ LQ
mum area of reinforcement is equal to the larger of
FDVHVZKHUHWKHIDFWRUHGWZRZD\VKHDUVWUHVVH[FHHGVWKHYDOXHJLYHQLQ$&,
Reinforcement required to resist in-plane forces in diaphragms must be in addition to the reinforcement required to
UHVLVWRWKHUORDGHIIHFWVVXFKDVWKRVHIURPJUDYLW\ $&, 6KULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQWKRZever, is permitted to also resist diaphragm in-plane forces.
9.6 Determination of Required Reinforcement
9.6.1 Chord Reinforcement
The maximum tension chord force,
which is determined by Equation (9.1), must be resisted by reinforcement
must be less
perpendicular to the direction of the applied in-plane force. At any location within the diaphragm,
than or equal to the design tension strength of the chord reinforcement:
(9.12)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
In this equation,
for reinforcing bars in tension and
(TXDWLRQ FDQEHXVHGWRGHWHUPLQH
For a given
is the required area of chord reinforcement.
(9.13)
For diaphragms with openings, the maximum tension chord force is determined by Equation (9.8) and that force
DWWKHHGJHVRIWKHGLDSKUDJP VHH)LJXUH 7KHUHTXLUHGDUHD
is used in Equation (9.13) to determine
of chord reinforcement along the edges of the opening perpendicular to the direction of analysis can also be deterand
VHH)LJXUHDQG6HFWLRQ
mined by Equation (9.13) using the subdiaphragm tension chord forces
9.3.3 of this publication).
Because wind and seismic forces can act in any direction, chord reinforcement is required along all edges of diaphragms and openings.
For diaphragms in buildings assigned to Seismic Design Category (SDC) A, B, or C, diaphragm chords subjected to
FRPSUHVVLRQIRUFHVQHHGQRWEHGHVLJQHGDQGGHWDLOHGDVFROXPQV $&,5 +RZHYHULQFDVHVZKHUHGLDphragm boundaries (edges of a diaphragm or edges of an opening in a diaphragm) are subjected to relatively large
compression forces compared to the axial compression strength of the boundary element, transverse reinforcement,
VXFKDVKRRSVVKRXOGEHXVHGWRFRQ¿QHWKHFRQFUHWH6XFKWUDQVYHUVHUHLQIRUFHPHQWLVXVXDOO\IHDVLEOHZKHUHWKHUHDUH
beams along the diaphragm boundaries; without beams, the hoops must be developed in the slab, which is not always
possible, especially for relatively thin slabs. In such cases, a thicker slab may be required if beams are not an option.
9.6.2 Diaphragm Shear Reinforcement
7KHIROORZLQJUHTXLUHPHQWVPXVWEHVDWLV¿HGIRUGHVLJQVKHDUVWUHQJWKRIGLDSKUDJPV VHH$&,DQG6HFWLRQ
RIWKLVSXEOLFDWLRQ (9.14)
Required shear strength,
due to in-plane forces is determined using one of the models in Section 9.3.3 of this
publication. For diaphragms modeled as beams with chord reinforcement concentrated near it edges, the factored
is uniformly distributed over the depth of the diaphragm (see Figure 9.8).
VKHDUÀRZ
The shear reinforcement ratio,
is equal to the area of the uniformly distributed slab reinforcement oriented parallel to the shear force divided by the gross area of the slab perpendicular to that reinforcement. Equation (9.9) can be
solved for the required
(9.15)
,WLVHYLGHQWIURP(TXDWLRQ WKDWVKHDUUHLQIRUFHPHQWLVQRWUHTXLUHGZKHUH
Providing a
slab thickness adequate for serviceability or, where applicable, for two-way shear is usually adequate for in-plane
shear strength as well.
:KHUHLQSODQHVKHDUUHLQIRUFHPHQWLVUHTXLUHGLWLVXVXDOO\FRPELQHGZLWKWKHÀH[XUDOUHLQIRUFHPHQWLQWKHVODELQ
WKHGLUHFWLRQRIDQDO\VLV ZKHUHWZROD\HUVRIÀH[XUDOUHLQIRUFHPHQWDUHSURYLGHGLQDVODEWKHFRPELQDWLRQXVXDOO\
occurs with the bottom reinforcement). Thus, the total area of reinforcement in such cases is equal to the area of
VKHDUUHLQIRUFHPHQWGHWHUPLQHGE\(TXDWLRQ SOXVWKHDUHDRIÀH[XUDOUHLQIRUFHPHQW&RPELQLQJWKHUHLQIRUFHPHQWLQWKLVIDVKLRQJHQHUDOO\UHVXOWVLQVLPSOHUGHWDLOLQJDQGPRUHHI¿FLHQWSODFHPHQWRIWKHUHLQIRUFLQJEDUV
LQWKH¿HOG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
9.6.3 Shear Transfer Reinforcement
Shear transfer reinforcement must be provided between diaphragms, collectors, and vertical elements of the LFRS.
In cast-in-place construction, shear transfer reinforcement is provided mainly at construction joints between the elewhich depends on the overall layout of the elements,
ments. The required area of shear-friction reinforcement,
the magnitude of the factored shear force at the location of interest, the width of the collector with respect to the
width of the vertical element of the LFRS it frames into, and the construction sequence, can be determined by Equation (9.11):
(9.16)
In this equation,
is the factored shear force at the interface of interest, is the length of the interface of interest,
has the
and LVGHWHUPLQHGE\7DEOHFRQVLGHULQJWKHFRQWDFWVXUIDFHFRQGLWLRQDWWKHLQWHUIDFH1RWHWKDW
units of square inches per unit length in Equation (9.16).
Shear transfer reinforcement is typically required at the interface between the diaphragm and the vertical elements
of the LFRS and at the interface between the diaphragm and the collectors (if any). The equations in Table 9.9,
which are based on Equation (9.16), can be used to determine the required area of shear-friction reinforcement at
these locations.
Table 9.9 Required Shear Transfer Reinforcement in Diaphragms
Case
Shear Transfer Interface
1
Between the diaphragm and the
vertical elements of the LFRS
Between the diaphragm and the
collectors
Required Avf
In Case 1,
is equal to the reaction
in the vertical element of the LFRS where there are no collectors or
where a collector is required and the collector has the same width as the vertical element of the LFRS it frames in
(see Section 9.3.3 of this publication).
to. Where a collector is wider than the vertical element,
is the length of the vertical element of the LFRS in the direction of analysis (for example, the length
Also,
of the structural wall or the total length of the moment-resisting frames minus the length of any openings in the diaor
dependphragm adjacent to these elements). For buildings assigned to SDC A, B, or C, is equal to
LQJRQWKHFRQWDFWVXUIDFHFRQGLWLRQSURYLGHGDWWKHLQWHUIDFH VHH7DEOH for concrete
,Q&DVH is the overall depth of the diaphragm in the direction of analysis and is equal to
placed monolithically because the collector is either part of the slab or, where beams are used as collectors, the
beams and slab are placed at the same time (that is, there is no cold joint between these two elements).
7KHFRQVWUXFWLRQPHWKRGKDVDQLPSDFWRQWKHGHWHUPLQDWLRQRIVKHDUWUDQVIHUUHLQIRUFHPHQW7KH¿UVWW\SHRI
FRQVWUXFWLRQPHWKRGLVLOOXVWUDWHGLQ)LJXUHZKHUHVWUXFWXUDOZDOOVDUHXWLOL]HGDVWKHYHUWLFDOHOHPHQWVRIWKH
/)56,QWKLVPHWKRGWKHZDOOEHORZLVFRQVWUXFWHG¿UVWWRWKHXQGHUVLGHRIWKHVODEWKHVODEDQGWKHQWKHZDOO
above are subsequently constructed in that order. Cold joints typically occur at the top and bottom surfaces of the
The shear-friction reinforcement in Figure 9.16 consists of dowel
slab (diaphragm) along the length of the wall,
is calculated by the
bars connecting the slab to the wall below. In this case, the required area of the dowel bars,
equation in Table 9.9 for Case 1:
7KHVHFRQGW\SHRIFRQVWUXFWLRQPHWKRGLVVKRZQLQ)LJXUH,QWKLVFDVHWKHZDOOLVFRQVWUXFWHGDKHDGRIWKH
slab (for example, where slip forms are used to construct the wall). A cold joint occurs at the face of the wall and
shear transfer reinforcement must be provided at that joint. Proprietary form saver systems are commonly used in
WKLVW\SHRIFRQVWUXFWLRQPDLQO\IRUHFRQRPLFUHDVRQV7KHVKHDUIULFWLRQUHLQIRUFHPHQWLQ)LJXUHFRQVLVWVRI
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
‘Ž†Œ‘‹–
‘™‡Ž„ƒ”•ǣ ‫ܣ‬௩௙ ൒
ܸ௨ଵ Τκ௪
߶ߤ݂௬
൒ κ ሺ–›’Ǥ ሻ
—’’‘”–„ƒ”
‘Ž†Œ‘‹–
–”— –—”ƒŽ™ƒŽŽ
Figure 9.16 Shear transfer between a diaphragm and a structural wall
utilizing dowel bars – Construction Method 1.
‘”•ƒ˜‡”ƒ••‡„Ž›Ȃ†‡ˆ‘”‡†
„ƒ”™‹–ŠƒͻͲǦ†‡‰Š‘‘
‘™‡Ž„ƒ”•ǣ ‫ܣ‬௩௙ ൒
ܸ௨ଵ Τκ௪
߶ߤ݂௬
‘Ž†Œ‘‹–
Figure 9.17 Shear transfer between a diaphragm and a structural wall
utilizing dowel bars – Construction Method 2.
dowel bars, each of which requires its own form saver. The required area of the dowel bars,
equation in Table 9.9 for Case 1:
is calculated by the
0HWKRGVXWLOL]LQJVKULQNDJHDQGWHPSHUDWXUHUHLQIRUFHPHQWRUWKHERWWRPUHLQIRUFHPHQWLQDVODEDVVKHDUIULFWLRQ
UHLQIRUFHPHQWDUHJLYHQLQ5HIHUHQFH
7KHDERYHGLVFXVVLRQLVHTXDOO\YDOLGIRUPRPHQWUHVLVWLQJIUDPHVWKDWDUHXWLOL]HGDVSDUWRIWKH/)56LQVWHDGRI
or in combination with, structural walls.
9.6.4 Reinforcement Due to Eccentricity of Collector Forces
For collectors wider than the width of the vertical elements of the LFRS that they frame in to, an in-plane bending
and
which act
moment is generated in a diaphragm adjacent to the vertical elements due in part to the forces
DWDQHFFHQWULFLW\IURPWKHFHQWHUOLQHRIWKHYHUWLFDOHOHPHQWV VHH)LJXUH $PHWKRGWRGHWHUPLQHWKHUHTXLUHG
UHLQIRUFHPHQWWRUHVLVWWKLVEHQGLQJPRPHQWLVJLYHQEHORZZKLFKLVEDVHGRQWKHPHWKRGJLYHQLQ5HIHUHQFH
Illustrated in Figure 9.18 is a portion of a diaphragm with a structural wall that is part of the LFRS located at its
edge and a collector wider than the thickness of the wall. The internal forces acting on the free-body diagram of the
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
GLDSKUDJPDGMDFHQWWRWKHZDOODUHLQGLFDWHGLQWKH¿JXUHEDVHGRQVRPHVLPSOLI\LQJFRQVHUYDWLYHDVVXPSWLRQV7KH
due to these internal forces can be approximated by the following equation:
eccentric bending moment,
(9.17)
݁
‫ݐ‬Τʹ
ܶௗ ܶ௩ ܸ௦ κ௪ ‫ݐ‬
ܸ௨ ‫ܯ‬௘ ܸ௦ ‫ܥ‬ௗ ‫ܥ‬௩ ܾ௘௙௙ Figure 9.18 Internal forces in a diaphragm where the collector is wider
than the vertical element of the LFRS.
In this equation,
is the shear strength of the diaphragm based on only the reinforcing bars in the slab (that is,
). The nominal strength of the concrete,
is not included because tension forces are present, which
makes
The area of tension reinforcement,
required to resist
can be determined by the following equation:
(9.18)
where
for reinforcing bars in tension. This reinforcement is placed perpendicular to the face of the wall at
both ends and must be developed into the slab and into the wall (see Figure 9.19).
κ௪ ‫ܣ‬௦ሺ‡ ሻ ሺ–›’Ǥሻ
–Š‡””‡‹ˆ‘” ‡‡–‘–
•Š‘™ˆ‘” Žƒ”‹–›
Figure 9.19 Required tension reinforcement where the collector is wider
than the vertical element of the LFRS.
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9-26
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
9.6.5 Collector Reinforcement
Overview
In general, collectors must be designed as tension members, compression members, or both in accordance with
WKHSURYLVLRQVRI$&,IRUPHPEHUVVXEMHFWHGWRD[LDOVWUHQJWKRUFRPELQHGÀH[XUDODQGD[LDOVWUHQJWK $&,
7KHW\SHVDQGDPRXQWRIUHLQIRUFHPHQWDQGWKHGHWDLOLQJUHTXLUHPHQWVWKDWPXVWEHVDWLV¿HGGHSHQGPRVWO\RQ WKHW\SHRIFROOHFWRU WKDWLVDFROOHFWRUWKDWLVDSRUWLRQRIWKHVODERULVDEHDP DQG WKHZLGWKRIWKHFROOHFWRU
with respect to the width of the vertical element of the LFRS that it frames in to.
Slabs
:KHUHVODEVDUHXWLOL]HGDVFROOHFWRUVWKHIROORZLQJHTXDWLRQFDQEHXVHGWRGHWHUPLQHWKHUHTXLUHGDUHDRIORQJLWXin a collector to resist the factored axal tension force,
dinal reinforcement,
(9.19)
where
IRUUHLQIRUFLQJEDUVLQWHQVLRQ7KLVUHLQIRUFHPHQWLVSURYLGHGLQDGGLWLRQWRWKHÀH[XUDOUHLQIRUFHment in the slab required for gravity loads. A typical detail where the collector is the same width as the vertical elePHQWRIWKH/)56LVJLYHQLQ)LJXUH
‘ŽŽ‡ –‘””‡‹ˆ‘” ‡‡–ǡ‫ܣ‬௦ሺ ‘ŽŽ‡ –‘”ሻ Figure 9.20 Longitudinal reinforcement in a collector that has the same width
as the vertical element of the LFRS.
For the case of axial compression forces,
WKHVHFWLRQDW]HURHFFHQWULFLW\
must be less than or equal to the design axial compression strength of
(9.20)
In this equation,
IRUFRPSUHVVLRQFRQWUROOHGVHFWLRQV $&,7DEOH is equal to the thickness of
is the area of longitudinal reinforcement in
The design strength
the slab times the width of the collector, and
requirements for axial compression rarely govern.
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9-27
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
For slabs wider than the vertical element of the LFRS, the total area of longitudinal tension reinforcement,
required to resist
can also be determined by Equation (9.19). The axial tension force
and the corin the width of the vertical element of the LFRS is selected
responding required area of tension reinforcement,
FRQVLGHULQJGHVLJQDQGFRQVWUXFWLRQOLPLWDWLRQV2QFHWKHVL]HDQGQXPEHURIUHLQIRUFLQJEDUVDUHFKRVHQEDVHGRQ
the area of reinforcement,
required in the effective slab width outside the width of the vertical elements
minus the area of longitudinal reinforcement provided in the width of the vertical elements of
is equal to
are usually uniformly distributed over the effective width of
the LFRS. The reinforcing bars corresponding to
WKHFROOHFWRU VHH)LJXUH ܾ௘௙௙ ‫ܣ‬௦ሺௗሻ ‫ܣ‬௦ሺ௩ሻ ‫ܣ‬௦ሺ ‘ŽŽ‡ –‘”ሻ ൌ ‫ܣ‬௦ሺௗሻ ൅ ‫ܣ‬௦ሺ௩ሻ Figure 9.21 Longitudinal reinforcement in a collector wider than the vertical element of the LFRS.
)RUFROOHFWRUVZLGHUWKDQWKHYHUWLFDOHOHPHQWRIWKH/)56WKHGHVLJQD[LDOVWUHQJWKGH¿QHGLQ(TXDWLRQ PXVW
and
using the appropriate
and
for each segment.
be checked for
Beams
%HDPVGHVLJQDWHGDVFROOHFWRUHOHPHQWVPXVWEHGHVLJQHGIRUWKHFRPELQHGHIIHFWVIURPÀH[XUHVKHDUWRUVLRQD[LDO
compression forces, and axial tension forces due to gravity and lateral loads. Applicable design and detailing proviVLRQVLQ$&,&KDSWHUPXVWEHVDWLV¿HGDORQJZLWKWKHDSSOLFDEOHGHWDLOLQJUHTXLUHPHQWV
/RQJLWXGLQDOUHLQIRUFHPHQWPXVWEHGHWHUPLQHGIRUWKHFRPELQHGHIIHFWVGXHWRÀH[XUHWRUVLRQDQGD[LDOFRPSUHVsion and tension forces, and transverse reinforcement must be determined for combined effects due to shear and
torsion, where applicable. Generally, a design strength interaction diagram is constructed, which includes both the
axial compression and tension portions, to determine whether the collector is adequate for all the combined factored
ÀH[XUHDQGD[LDOORDGHIIHFWV7KHLQIRUPDWLRQSURYLGHGDERYHIRUVODEVZLGHUWKDQWKHYHUWLFDOHOHPHQWVRIWKH
LFRS is also applicable to beams.
9.7 Reinforcement Detailing
Requirements for concrete cover, development lengths, splices, bar spacing, and reinforcement detailing for diaSKUDJPVDQGFROOHFWRUVDUHJLYHQLQ$&,$VXPPDU\RIWKHVHUHTXLUHPHQWVLVJLYHQLQ7DEOH
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.10 Reinforcement Detailing Requirements for Diaphragms and Collectors
Requirement
ACI Section No.
Concrete cover
Development lengths
Splices
Spacing
Reinforcement detailing
Minimum
Maximum
One-way slabs
Two-way slabs
At critical sections in diaphragms and collectors, tension and compression forces in the reinforcement must be
DGHTXDWHO\GHYHORSHGRQHDFKVLGHRIWKRVHVHFWLRQV $&, 7HQVLRQUHLQIRUFHPHQWPXVWH[WHQGEH\RQGWKH
determined in
point at which it is no longer required a distance equal to at least the tension development length,
DFFRUGDQFHZLWK$&,H[FHSWDWGLDSKUDJPHGJHVDQGDWH[SDQVLRQMRLQWV $&, Longitudinal reinforcement in collectors must extend into the vertical elements of the LFRS a length equal to at
OHDVWWKHJUHDWHURIWKHWZROHQJWKVJLYHQLQ$&,7KLVUHTXLUHPHQWLVLOOXVWUDWHGLQ$&,)LJXUH5
$GGLWLRQDOLQIRUPDWLRQRQGHWDLOLQJUHTXLUHPHQWVIRUGLDSKUDJPVDQGFROOHFWRUVLVJLYHQLQ5HIHUHQFH
9.8 Design Procedure
7KHGHVLJQSURFHGXUHLQ)LJXUHFDQEHXVHGLQWKHGHVLJQDQGGHWDLOLQJRIUHLQIRUFHGFRQFUHWHGLDSKUDJPV
,QFOXGHGLQWKH¿JXUHDUHWKHVHFWLRQQXPEHUVDQGHTXDWLRQQXPEHUVZKHUHVSHFL¿FLQIRUPDWLRQRQWKDWWRSLFLQWKLV
FKDSWHUFDQEHIRXQG&RPSUHKHQVLYHÀRZFKDUWVDVVRFLDWHGZLWKHDFKRIWKHVWHSVLQ)LJXUHDUHJLYHQLQ5HIHUHQFH
9.9 Examples
9.9.1
Example 9.1 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A,
Collectors Not Required
'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQG
IRUFHVLQWKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDOOWKHIUDPHVDORQJFROXPQOLQHVDQGDUHSDUWRIWKH/)56 VHH
)LJXUH $OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVDQGE\LQFROXPQV
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the diaphragm design in-plane forces
)URP7DEOHLQ([DPSOHWKHZLQGIRUFHLQWKHQRUWKVRXWKGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR
kips. This force is applied at the centroid of the building face at this level.
Step 2 – Determine the classification of the diaphragm
Sect. 9.3.3
For forces in the north-south direction, the span-to-depth ratio is equal to
%HFDXVHWKHVSDQWRGHSWKUDWLRLVOHVVWKDQWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG
ASCE/SEI 26.2
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9-29
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Determine the diaphragm thickness
ሺSect. 9.2ሻ
Determine the diaphragm design forces
ሺSect. 9.3.2ሻ
Determine the classification of the diaphragm,
select the diaphragm model, and determine the
diaphragm internal forces
ሺSect. 9.3.3ሻ
Determine the combined load effects
ሺACI Table 5.3.1ሻ
Determine the chord reinforcement
ሾEq. ሺ9.13ሻሿ
Determine the diaphragm shear reinforcement
ሾEq. ሺ9.15ሻሿ
Determine the shear transfer reinforcement
ሾEq. ሺ9.16ሻሿ
Determine the reinforcement due to eccentricity
of collector forces ሺwhere applicableሻ
ሾEq. ሺ9.18ሻሿ
Determine the collector reinforcement
ሺSect. 9.6.5ሻ
Figure 9.22 Design procedure for diaphragms.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 3 – Select the diaphragm model
Table 9.1
Because this building has no irregularities and is not subjected to any transfer forces, and because the frames along
FROXPQOLQHVDQGKDYHWKHVDPHODWHUDOVWLIIQHVV WKDWLVWKHVL]HVRIWKHEHDPVDQGFROXPQVDUHWKHVDPHLQERWK
frames), an equivalent beam model with rigid supports is selected to determine the internal forces for this diaphragm.
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
ሺtyp.ሻ
E
A
‫ܥ‬௨ ൌ 10.8 kips
ᇱ
23 -6
ሺtyp.ሻ
ᇳ
D
25.7
ൌ 0.27 kips/ft
94.0
ሺ‫ݒ‬௨ ሻ଻ ൌ
25.7
ൌ 0.27 kips/ft
94.0
N
A
ሺ‫ݒ‬௨ ሻଵ ൌ
A
C
B
961.9
ൌ 10.8 kips
0.95 ൈ 94.0
A
ܶ௨ ൌ
A
‫ݓ‬ௌ ൌ 0.342 kipsΤft
150.0Ԣ
ሺܴଵ ሻ௬ ൌ 25.7 kips
ሺܴ଻ ሻ௬ ൌ 25.7 kips
25.7
‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ
961.9
25.7
‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬-‫ܛܘܑܓ‬ሻ
Figure 9.23 Equivalent beam model for the diaphragm in Example 9.1 for wind in the south direction.
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9-31
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Step 4 – Determine the diaphragm internal forces
Sect. 9.3.3
7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ)LJXUHIRUZLQGLQWKH
VRXWKGLUHFWLRQ%HFDXVHWKHPRPHQWUHVLVWLQJIUDPHVRQFROXPQOLQHVDQGDUHLGHQWLFDOWKH&5LVORFDWHG
from either end of the diaphragm, which coincides with the location where the wind force is
applied at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an
equivalent uniformly distributed wind load equal to
• 5HDFWLRQVLQHDFKRIWKHPRPHQWUHVLVWLQJIUDPHVDORQJFROXPQOLQHVDQG
7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH
• Chord forces at the edges of the diaphragm perpendicular to the direction of analysis
Eq. (9.1)
• Maximum unit shear forces in the diaphragm
Collectors are not required because the moment-resisting frames extend the entire depth of the diaphragm in the
direction of analysis.
9.9.2
Example 9.2 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A,
Collectors Not Required
'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQGIRUFHVLQWKHQRUWKVRXWKGLUHFWLRQDVVXPLQJDOOWKHIUDPHVDORQJFROXPQOLQHVDQGDUHSDUWRIWKH
DQG*UDGH
/)56 VHH)LJXUH $OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKWFRQFUHWHZLWK
reinforcement.
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the chord reinforcement
From Example 9.1,
Eq. (9.13)
$WFROXPQOLQHV$DQG(SURYLGHEDUORFDWHGMXVWRXWVLGHRIWKHFURVVVHFWLRQRIWKHEHDPVLQWKHPRPHQW
IUDPHV VHH)LJXUH Step 2 – Determine the diaphragm shear reinforcement
7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHVDQGDQGDUHHTXDOWRNLSVIW VHH([DPSOH
9.1).
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Žƒ„–‘’”‡‹ˆ‘” ‡‡–
‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘” ‡‡–
ͺΦԣ
ͳǦ͓Ͷ Š‘”†„ƒ”
ʹԢǦͲԣ
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‡ƒ–”ƒ•˜‡”•‡”‡‹ˆ‘” ‡‡–
‡ƒˆŽ‡š—”ƒŽ”‡‹ˆ‘” ‡‡–
ʹԢǦͶԣ
Figure 9.24 Location of chord reinforcement in the diaphragm in Example 9.2 for
wind forces in the north-south direction.
Check shear strength requirements assuming
Eq. (9.14)
Therefore, no shear reinforcement is required to satisfy shear strength requirements.
Step 3 – Determine the shear transfer reinforcement
Only shear transfer reinforcement between the diaphragm and the moment-resisting frames is required:
Eq. (9.16)
A value of
for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPH VHH7DEOH %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO
reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the momentUHVLVWLQJIUDPHVDORQJFROXPQOLQHVDQG
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
9.9.3
Example 9.3 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option B), SDC A,
Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
'HWHUPLQHWKHGLDSKUDJPIRUFHVLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2SWLRQ%IRUZLQG
forces in the north-south direction assuming only the frames between column lines B and D along column lines 1
DQGDUHSDUWRIWKH/)56 VHH)LJXUH ,WLVDVVXPHGWKDWWKHZLGWKRIWKHFROOHFWRUV EHDPV DUHHTXDOWRWKH
ZLGWKVRIWKHEHDPVLQWKHPRPHQWUHVLVWLQJIUDPHV$OVRDVVXPHDQLQWKLFNVODEE\LQEHDPVDQG
E\LQFROXPQV
'HVLJQGDWDDUHJLYHQLQ6HFW
Step 1 – Determine the diaphragm design in-plane forces
)URP7DEOHLQ([DPSOHWKHZLQGIRUFHLQWKHQRUWKVRXWKGLUHFWLRQDWWKHVHFRQGÀRRUOHYHOLVHTXDOWR
kips. This force is applied at the centroid of the building face at this level.
Step 2 – Determine the classification of the diaphragm
Sect. 9.3.3
For forces in the north-south direction, the span-to-depth ratio is equal to
%HFDXVHWKHVSDQWRGHSWKUDWLRLVOHVVWKDQWKHGLDSKUDJPFDQEHFODVVL¿HGDVULJLG
Step 3 – Select the diaphragm model
ASCE/SEI 26.2
Table 9.1
Because this building has no irregularities and is not subjected to any transfer forces, and because the frames
EHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQGKDYHWKHVDPHODWHUDOVWLIIQHVV WKDWLVWKHVL]HVRIWKH
beams and columns are the same in both frames), an equivalent beam model with rigid supports is selected to determine the internal forces for this diaphragm.
Step 4 – Determine the diaphragm internal forces
Sect. 9.3.3
7KHHTXLYDOHQWEHDPPRGHOIRUWKHGLDSKUDJPDWWKHVHFRQGÀRRUOHYHOLVJLYHQLQ)LJXUHIRUZLQGLQWKHVRXWK
GLUHFWLRQ%HFDXVHWKHPRPHQWUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQGDUH
from either end of the diaphragm, which coincides with the location
identical, the CR is located
where the wind force is applied at this level. Thus, there is no inherent torsional moment, which means the diaphragm is subjected to an equivalent uniformly distributed wind load equal to
• 5HDFWLRQVLQHDFKRIWKHPRPHQWUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQG
7KHVKHDUDQGPRPHQWGLDJUDPVDUHDOVRJLYHQLQ)LJXUH
• Chord forces at the edges of the diaphragm perpendicular to the direction of analysis
Eq. (9.1)
• Maximum unit shear forces in the moment-resisting frames between column lines B and D
• Maximum unit shear force in the diaphragm
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1A
2A
3A
4A
5A
ᇱ
6A
7A
ᇳ
A
25 -0
ሺtyp.ሻ
E
‫ܥ‬௨ ൌ 10.8 kips
Collector
A
Collector
23ᇱ -6ᇳ
ሺtyp.ሻ
D
ሺ‫ݒ‬௨ ሻଵ ൌ
25.7
ൌ 0.27 kips/ft
94.0
ሺ‫ݒ‬௨ ሻ଻ ൌ
ሺ‫ݒ‬௨,ி ሻଵ ൌ
25.7
ൌ 0.55 kips/ft
47.0
ሺ‫ݒ‬௨,ி ሻ଻ ൌ
25.7
ൌ 0.27 kips/ft
94.0
A
N
C
A
25.7
ൌ 0.55 kips/ft
47.0
B
Collector
Collector
961.9
ൌ 10.8 kips
0.95 ൈ 94.0
A
ܶ௨ ൌ
A
‫ݓ‬ௌ ൌ 0.342 kipsΤft
150.0Ԣ
ሺܴ଻ ሻ௬ ൌ 25.7 kips
ሺܴଵ ሻ௬ ൌ 25.7 kips
25.7
‫ ܚ܉܍ܐ܁‬ሺ‫ܛܘܑܓ‬ሻ
961.9
25.7
‫ ܜܖ܍ܕܗۻ‬ሺ܎‫ܜ‬-‫ܛܘܑܓ‬ሻ
Figure 9.25 Equivalent beam model for the diaphragm in Example 9.3 for wind in the south direction.
• Collector forces
Collectors are required because the moment-resisting frames do not extend the entire depth of the diaphragm in
the direction of analysis.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
7KHXQLWVKHDUIRUFHVQHWVKHDUIRUFHVDQGFROOHFWRUIRUFHVDORQJFROXPQOLQHDUHJLYHQLQ)LJXUHZKHUH
the collectors are the beams between column lines A and B and between column lines D and E. Note that in FigXUHWKHIRUFHVDUHVKRZQZLWKPRUHVLJQL¿FDQW¿JXUHVDIWHUWKHGHFLPDOSODFHWKDQDUHXVHGLQWKHDERYH
calculations. The main reason for this is to demonstrate that the maximum axial force in the collector is equal
to the same value regardless of which end of the net shear force diagram is used to calculate it. Using a smaller
QXPEHURIVLJQL¿FDQW¿JXUHVUHVXOWVLQWZRVOLJKWO\GLIIHUHQWYDOXHVRIWKHPD[LPXPFROOHFWRUIRUFHZKLFKLV
due to due roundoff only.
1A
2A
3A
4A
5A
A
25ᇱ -0ᇳ
0.273 kips/ftሺtyp.ሻ
6A
7A
0.273 kips/ft
E
‫ܥ‬௨ ൌ 10.8 kips
Collector
Collector
A
0.546 kips/ft
0.273 kips/ft
23ᇱ -6ᇳ
ሺtyp.ሻ
D
ሺ‫ݒ‬௨ ሻଵ ൌ
25.65
ൌ 0.273 kips/ft
94.0
ሺ‫ݒ‬௨ ሻ଻ ൌ
25.8
ൌ 0.27 kips/ft
96.0
ሺ‫ݒ‬௨,ி ሻଵ ൌ
25.65
ൌ 0.546 kips/ft
47.0
ሺ‫ݒ‬௨,ி ሻ଻ ൌ
A
N
6.4 kips
C
A
25.8
ൌ 0.53 kips/ft
49.0
B
6.4 kips
Collector
A
Collector
A
0.273 kips/ft
‫ ܚ܉܍ܐ܁ ܜܑܖ܃‬۴‫ܛ܍܋ܚܗ‬
‫ ܚ܉܍ܐ܁ ܜ܍ۼ‬۴‫ܛ܍܋ܚܗ‬
۱‫ ܚܗܜ܋܍ܔܔܗ‬۴‫ܛ܍܋ܚܗ‬
0 34
k Τf forces in the diaphragm in Example 9.3.
Figure 9.26 Unit shear forces, net shear forces, and
collector
9.9.4
Example 9.4 – Determination of Diaphragm Reinforcement: Building #1 (Framing Option B), SDC A,
Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
'HWHUPLQHWKHUHTXLUHGGLDSKUDJPUHLQIRUFHPHQWLQWKHVODEDWWKHVHFRQGÀRRUOHYHORI%XLOGLQJ)UDPLQJ2Stion B for wind forces in the north-south direction assuming only the frames between column lines B and D along
FROXPQOLQHVDQGDUHSDUWRIWKH/)56 VHH)LJXUH ,WLVDVVXPHGWKDWWKHZLGWKRIWKHFROOHFWRUV EHDPV DUH
HTXDOWRWKHZLGWKVRIWKHEHDPVLQWKHPRPHQWUHVLVWLQJIUDPHV$OVRDVVXPHDQLQWKLFNVODEQRUPDOZHLJKW
DQG*UDGHUHLQIRUFHPHQW
concrete with
'HVLJQGDWDDUHJLYHQLQ6HFW6HH([DPSOH
Step 1 – Determine the chord reinforcement
From Example 9.3,
Eq. (9.13)
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
$WFROXPQOLQHV$DQG(SURYLGHEDUORFDWHGMXVWRXWVLGHRIWKHFURVVVHFWLRQRIWKHEHDPVLQWKHPRPHQW
IUDPHV VHH)LJXUH Step 2 – Determine the diaphragm shear reinforcement
7KHODUJHVWIDFWRUHGXQLWVKHDUIRUFHVRFFXUDORQJFROXPQOLQHVDQGDQGDUHHTXDOWRNLSVIW VHH([DPSOH Check shear strength requirements assuming
Eq. (9.14)
Therefore, no shear reinforcement is required to satisfy shear strength requirements.
Step 3 – Determine the shear transfer reinforcement
Shear transfer reinforcement between the diaphragm and the moment-resisting frames between column lines B and D:
Eq. (9.16)
A value of
for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPH VHH7DEOH %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO
reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the momentUHVLVWLQJIUDPHVEHWZHHQFROXPQOLQHV%DQG'DORQJFROXPQOLQHVDQG
Shear transfer reinforcement between the diaphragm and the collectors:
Eq. (9.16)
A value of
for monolithic concrete is used because the concrete for the slab and beams in the momentUHVLVWLQJIUDPHVDUHSODFHGDWWKHVDPHWLPH VHH7DEOH %HFDXVHWKHUHTXLUHGDUHDRIVKHDUIULFWLRQUHLQIRUFHPHQWLVYHU\VPDOOLWLVVDIHWRDVVXPHWKDWWKHERWWRPÀH[XUDO
reinforcement in the slab can be used as the shear transfer reinforcement between the diaphragm and the collectors.
Step 4 – Determine the collector reinforcement
The collector beams must be designed for the combined effects due to gravity loads and the axial force due to wind
ORDGVZKLFKLVREWDLQHGIURP)LJXUH
A summary of the axial forces, bending moments, and shear forces for a typical collector beam is given in Table 9.11.
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Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
Table 9.11 Summary of Axial Forces, Bending Moments, and Shear Forces for a Typical Collector Beam in
Example 9.4
Axial Force
(kips)
Load Case
Dead
Live
Bending Moment (ft-kips)
Exterior
Negative
Positive
First Interior
Negative
Shear
(kips)
63.8
ņ
Wind
ņ
ņ
ņ
Load Combination
$&,(T D
89.3
36.3
$&,(T E
$&,(T G
$&,(T I
The collector beams are also subjected to a uniform torsional load from the slab. Because the torsional moments can
be redistributed, the beam can be designed for the compatibility torsional moment (see Sect. 6.3.1 of this publication).
The longitudinal reinforcement in the beam is determined based on the factored axial forces, bending moments,
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factored load combinations in Table 9.11. It is evident that the longitudinal reinforcement is adequate because all the
factored load combination points fall within the diagram.
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Figure 9.27 Required reinforcement for the collector beam in Example 9.4.
@Seismicisolation
@Seismicisolation
9-38
Design Guide on the ACI 318 Building Code Requirements for Structural Concrete
1,500
1,250
Axial Force ሺkipsሻሻ
1,000
750
500
250
0
0
100
200
300
400
500
600
700
-250
-500
Bending Moment ሺft-kipsሻ
Figure 9.28 Design strength interaction diagram for the collector beam in Example 9.4.
9.9.5
Example 9.5 – Determination of Diaphragm In-Plane Forces: Building #1 (Framing Option C), SDC A,
Collectors Required, Collector Width the Same as the Width of the Vertical Elements of the LFRS
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forces in the north-south direction assuming the lateral forces are resisted by the following momen