concrete frame design

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Concrete Frame

Design Manual

AASHTO 1997, ACI 318-99, BS 8110-97, BS 8110-85R89

CSAA-23.3-94, CP 65-99, Eurocode 2-92NZS 3101-95 and UBC 97

For SAP2000 and ETABS

Berkeley, California ETABS V10December 2007 SAP2000 V11


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COPYRIGHT

Copyright Computers and Structures, Inc., 1978-2007.

All rights reserved.

The CSI Logo, SAP2000, and ETABS are registered trademarks of

Computers and Structures, Inc. SAFETM

is trademarks of Computers

and Structures, Inc. Watch & LearnTM

is a trademark of Computers and

Structures, Inc.

The computer programs SAP2000 and ETABS and all associated

documentation are proprietary and copyrighted products. Worldwiderights of ownership rest with Computers and Structures, Inc. Unlicensed

use of these programs or reproduction of documentation in any form,

without prior written authorization from Computers and Structures, Inc.,

is explicitly prohibited.

No part of this publication may be reproduced or distributed in any form

or by any means, or stored in a database or retrieval system, without the

prior explicit written permission of the publisher.

Further information and copies of this documentation may be obtainedfrom:

Computers and Structures, Inc.

1995 University Avenue

Berkeley, California 94704 USA

Tel: (510) 649-2200

Fax: (510) 649-2299

E-mail: [email protected] (for general questions)

E-mail: [email protected] (for technical supports questions)

Web: www.csiberkeley.com


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DISCLAIMER

CONSID ER ABLE TIME, EF FORT AND EX PENSE HAVE GONEINTO THE DEVELOPMENT AND DOCUMENTATION OF ETABS

AND SAP2000. THE PRO GRAMS HAVE BEEN THOR OUGHLY

TESTED AND USED. IN US ING THE PRO GRAMS, HOW EVER,

THE USER AC CEPTS AND UN DERSTANDS THAT NO WAR-

RANTY IS EX PRESSED OR IMPLIED BY THE DEVEL OP ERS OR

THE DIS TRIBU TORS ON THE AC CURACY OR THE RE LI ABIL-

ITY OF THE PRO GRAMS.

ETABS AND SAP2000 ARE VERY PRACTICAL TOOL FOR THE

DESIGN OF REINFORCED CONCRETE STRUCTURES. HOW-

EVER, THE USER MUST THOROUGHLY READ THE MANUAL

AND CLEARLY REC OGNIZE THE ASPECTS OF REINFORCED

CONCRETE DE SIGN THAT THE PROGRAM AL GORITHMS DO

NOT ADDRESS.

THE USER MUST EXPLIC ITLY UNDERSTAND THE AS SUMP-

TIONS OF THE PROGRAMS AND MUST IN DEPENDENTLY VER-

IFY THE RESULTS.


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Table of Contents

CHAPTER I Introduction 1Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Recommended Reading. . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAPTER II Design Process 5

Design Load Combinations. . . . . . . . . . . . . . . . . . . . . . . . 6

Design and Check Stations . . . . . . . . . . . . . . . . . . . . . . . . 8

Identifying Beams and Columns . . . . . . . . . . . . . . . . . . . . . 8

Design of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Design of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

Design of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

De termine the Panel Zone Shear Force. . . . . . . . . . . . . . . 15

De termine the Ef fective Area of Joint . . . . . . . . . . . . . . . 18

Check Panel Zone Shear Stress. . . . . . . . . . . . . . . . . . . 19

Beam/Column Flexural Capacity Ratios . . . . . . . . . . . . . . . . 20

P-D Effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Element Unsupported Lengths . . . . . . . . . . . . . . . . . . . . . 21

Special Considerations for Seismic Loads . . . . . . . . . . . . . . . 23

Choice of Input Units . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CHAPTER III Design for AASHTO LRFD 1997 25

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . . 28

Strength Reduction Factors . . . . . . . . . . . . . . . . . . . . . . . 29

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . . 30

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Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 32Deter mine Factored Moments and Forces . . . . . . . . . . 32Deter mine Moment Mag nifi cation Factors . . . . . . . . . . 33Deter mine Capacity Ratio . . . . . . . . . . . . . . . . . . . 34

Design Column Shear Reinforcement . . . . . . . . . . . . . . . 35Deter mine Section Forces . . . . . . . . . . . . . . . . . . . 36Deter mine Concrete Shear Capac ity . . . . . . . . . . . . . 37Deter mine Required Shear Re inforcement . . . . . . . . . . 37

Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . . 41Deter mine Factored Moments. . . . . . . . . . . . . . . . . 41Deter mine Required Flexural Rein forcement . . . . . . . . . 42

Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 48Deter mine Shear Force and Mo ment . . . . . . . . . . . . . 48Deter mine Concrete Shear Capac ity . . . . . . . . . . . . . 49Deter mine Required Shear Re inforcement . . . . . . . . . . 49

CHAPTER IV De sign for ACI 318-99 53


Strength Reduction Factors . . . . . . . . . . . . . . . . . . . . . . . 56

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 59Deter mine Factored Moments and Forces . . . . . . . . . . 59Deter mine Moment Mag nifi cation Factors . . . . . . . . . . 59Deter mine Capacity Ratio . . . . . . . . . . . . . . . . . . . 61

Design Column Shear Reinforcement . . . . . . . . . . . . . . . 62Deter mine Section Forces . . . . . . . . . . . . . . . . . . . 63Deter mine Concrete Shear Capac ity . . . . . . . . . . . . . 64Deter mine Required Shear Re inforcement . . . . . . . . . . 66

Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . . 67Deter mine Factored Moments. . . . . . . . . . . . . . . . . 67Deter mine Required Flexural Rein forcement . . . . . . . . . 67

Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 74Deter mine Shear Force and Mo ment . . . . . . . . . . . . . 74Deter mine Concrete Shear Capac ity . . . . . . . . . . . . . 76

Deter mine Required Shear Re inforcement . . . . . . . . . . 76

CHAPTER V De sign for BS 8110-97 79


Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Column Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . . 84

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De termine Fac tored Moments and Forces . . . . . . . . . . 85De ter mine Ad di tional Moments. . . . . . . . . . . . . . . . 85De ter mine Capac ity Ratio . . . . . . . . . . . . . . . . . . . 87

Design Column Shear Reinforcement . . . . . . . . . . . . . . . 88

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . . 89De termine Fac tored Moments. . . . . . . . . . . . . . . . . 90De termine Required Flex ural Re inforcement . . . . . . . . . 90

Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 95

CHAPTER VI Design for BS 8110-85 R1989 97


Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

Generation of Biaxial Interaction Surfaces . . . . . . . . . . . . 101

Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 102De termine Fac tored Moments and Forces . . . . . . . . . . 103De ter mine Ad di tional Moments . . . . . . . . . . . . . . . 103De ter mine Capac ity Ratio . . . . . . . . . . . . . . . . . . 105

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 106

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . 107De termine Fac tored Moments . . . . . . . . . . . . . . . . 108De termine Required Flex ural Re inforcement . . . . . . . . 108


CHAPTER VII Design for CSAA23.3-94 115

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . 118

Strength Reduction Factors . . . . . . . . . . . . . . . . . . . . . . 118

Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 121De termine Fac tored Moments and Forces . . . . . . . . . . 121De ter mine Mo ment Magni fica tion Fac tors . . . . . . . . . 121De ter mine Capac ity Ratio . . . . . . . . . . . . . . . . . . 124

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 125De termine Sec tion Forces . . . . . . . . . . . . . . . . . . 125De termine Con crete Shear Capacity . . . . . . . . . . . . . 127De termine Required Shear Reinforcement . . . . . . . . . 128

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131



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Deter mine Shear Force and Mo ment . . . . . . . . . . . . 139Deter mine Concrete Shear Capac ity . . . . . . . . . . . . . 141Deter mine Required Shear Re inforcement . . . . . . . . . 142

CHAPTER VIII Design for CP 65-1999 145

Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . 145Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 150Deter mine Factored Moments and Forces . . . . . . . . . . 151Deter mine Ad ditional Mo ments . . . . . . . . . . . . . . . 151Deter mine Capacity Ratio . . . . . . . . . . . . . . . . . . 153


Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . 155Deter mine Factored Moments . . . . . . . . . . . . . . . . 156Deter mine Required Flexural Rein forcement . . . . . . . . 156


CHAPTER IX De sign for Eurocode 2-92 163


Design Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 169Deter mine Factored Moments and Forces . . . . . . . . . . 169Deter mine Code Total Moments . . . . . . . . . . . . . . 169Deter mine Capacity Ratio . . . . . . . . . . . . . . . . . . 171


Beam Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Design Beam Flexural Reinforcement . . . . . . . . . . . . . . 176Deter mine Factored Moments . . . . . . . . . . . . . . . . 177Deter mine Required Flexural Rein forcement . . . . . . . . 177


CHAPTER X De sign for NZS 3101-95 187



Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 193Deter mine Factored Moments and Forces . . . . . . . . . . 193

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De ter mine Mo ment Magni fica tion Fac tors . . . . . . . . . 194Dy namic Mo ment Magni fi cation . . . . . . . . . . . . . . 196De ter mine Capac ity Ratio . . . . . . . . . . . . . . . . . . 196

Design Column Shear Reinforcement. . . . . . . . . . . . . . . 197De termine Sec tion Forces . . . . . . . . . . . . . . . . . . 198De termine Con crete Shear Capacity . . . . . . . . . . . . . 199

De termine Required Shear Reinforcement . . . . . . . . . 201

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204


Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 211De termine Shear Force and Moment . . . . . . . . . . . . 212De termine Con crete Shear Capacity . . . . . . . . . . . . . 213De termine Required Shear Reinforcement . . . . . . . . . 214

CHAPTER XI Design for UBC 97 217Design Load Combinations . . . . . . . . . . . . . . . . . . . . . . 220


Column Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221


Check Column Capacity . . . . . . . . . . . . . . . . . . . . . 224De termine Fac tored Moments and Forces . . . . . . . . . . 224De ter mine Mo ment Magni fica tion Fac tors . . . . . . . . . 224De ter mine Capac ity Ratio . . . . . . . . . . . . . . . . . . 226


De termine Sec tion Forces . . . . . . . . . . . . . . . . . . 227De termine Con crete Shear Capacity . . . . . . . . . . . . . 229De termine Required Shear Reinforcement . . . . . . . . . 230

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231


Design Beam Shear Reinforcement . . . . . . . . . . . . . . . . 239De termine Shear Force and Moment . . . . . . . . . . . . 239De termine Con crete Shear Capacity . . . . . . . . . . . . . 241De termine Required Shear Reinforcement . . . . . . . . . 241

Design of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

De termine the Panel Zone Shear Force . . . . . . . . . . . . . . 242

De termine the Ef fective Area of Joint . . . . . . . . . . . . . . 243

Check Panel Zone Shear Stress . . . . . . . . . . . . . . . . . . 244

Beam/Column Flexural Capacity Ratios . . . . . . . . . . . . . . . . 244

CHAPTER XII Design Output 249

Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

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Graphical Display of Design Information . . . . . . . . . . . . . . . 250

Tabular Display of Design Output . . . . . . . . . . . . . . . . . . . 252

Member Specific Information . . . . . . . . . . . . . . . . . . . . . 254

References 257

Index 261

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C h a p t e r I

Introduction

Overview

The program features powerful and completely integrated modules for design of

both steel and reinforced concrete structures (CSI 2005a, b). The program provides

the user with options to create, modify, analyze and design structural models, all

from within the same user in ter face.

Note: Throughout this manual, use of the term program refers to either ETABS

Integrated Building Design Software or SAP2000 Three Dimensional Static and

Dynamic Analysis and Design Software, unless otherwise noted.

The program provides an interactive environment in which the user can study the

stress conditions, make appropriate changes, such as member size revisions, and

update the design without re-analyzing the structure. A single mouse click on an

element brings up detailed design information. Members can be grouped together

for design purposes. The output in both graphical and tabulated formats can bereadily displayed and printed.

The program is structured to support a wide variety of design codes for the auto-

mated design and check of concrete frame members. This documentation includes

the following design codes: U.S. (AASHTO 1997, ACI 1999, UBC 97), Canadian

(CSA 1994), British (BSI 1997, BSI 1989), European (CEN 1992), New Zealand

(NZS 3101-95) and CP 65 of Singapore.

Overview 1


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The design is based upon a set of user-specified loading combinations. However,

the program provides a set of default load combinations for each design code sup-

ported in program. If the default load combinations are acceptable, no definition of

additional load combinations is required.

In the design of the columns, the program calculates the required longitudinal andshear reinforcement. However the user may specify the longitudinal steel, in which

case a col umn capac ity ratio is reported. The column capacity ratio gives an indica-

tion of the stress condition with respect to the capacity of the column.

Every beam member is designed for flexure and shear at a user-defined number of

stations along the beam span. Torsion design is also available for ACI codes.

The presentation of the output is clear and concise. The information is in a form that

allows the en gi neer to take appropri ate re medial meas ures in the event of member

overstress. Backup design information produced by the program is also provided

for convenient verification of the results.

English as well as SI and MKS metric units can be used to define the model geome-

try and to specify design parameters.

Organization

This manual is organized as follows:

Chapter II outlines various aspects of the concrete design procedures of the pro-gram. This chapter describes the common terminology of concrete design as imple-

mented in the program.

Each of eight subsequent chapters gives a detailed description of a specific code of

practice as interpreted by and imple mented in the pro gram. Each chap ter de scribes

the design loading combination, column and beam design procedures, and other

special consideration required by the code.

Chapter III gives a detailed description of the AASHTO LRFD concrete code

(AASHTO 1997) as implemented in SAP2000 only.

Chapter IV gives a detailed description of the ACI code (ACI 1999) as imple-


Chapter V gives a detailed description of the British code (BSI 1997) as imple-


2 Organization



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Chapter VI gives a detailed description of the British code (BSI 1989) as imple-


Chapter VII gives a detailed description of the Canadian code (CSA 1994) as im-

plemented in the program.

Chapter VIII gives a detailed description of the Singapore code (CP 1999) as imple-


Chapter IX gives a detailed description of the Eurocode 2 (CEN 1992) as imple-


Chapter X gives a detailed description of the New Zealand code (NZS 1997) as im-

plemented in the program.

Chapter XI gives a detailed description of the Uniform Building code (UBC 1997)

as implemented in the program.

Chapter XII outlines various aspects of the tabular and graphical output from the

program related to concrete design.

Recommended Reading

It is rec ommended that the user read Chap ter II De sign Process and one of the ten

subsequent chapters corresponding to the code of interest. Additionally user should

read Design Output in Chapter XII to gain an understanding of program output

related to concrete de sign.

A tutorial manual that provides step-by-step guidance through an example project

is provided with the program. It is recommended that first-time users perform the

steps of the tutorial before reading this manual.

Recommended Reading 3

Chapter I Introduction


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C h a p t e r II

Design Process

This chapter outlines various aspects of the concrete design and design-check pro-

cedures that are used by the program. The concrete design and check may be per-

formed using the program in accordance with one of the following design codes:

The 1997 American Association of State Highway and Transportation Offi-

cials AASHTO LRFD Bridge Design Specifications (SAP2000 only),AASHTO LRFD 1997 (AASHTO 1997).

The 2005 American Concrete Institute Building Code Requirements for Struc-

tural Concrete, ACI 318-05 (ACI 2005).





The 1997 British Standards Institution Structural Use of Concrete, BS 8110-97(BSI 1997).

The 1985 British Standards Institution Structural Use of Concrete, BS 8110-85

R1989 (BSI 1989).

The 1994 Canadian Standards Association Design of Concrete Structures for

Buildings, CSA-A23.3-94 (CSA 1994).

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The 1999 SPRING Singapore Design and construction of Structural use of

Concrete, CP 65 1999 (CP 1999).

The 2004 European Committee for Standardization, Design of Concrete Struc-

tures, EUROCODE 2 (BS EN 2004).

The 1992 European Committee for Standardization, Design of Concrete Struc-tures, EUROCODE 2 (CEN 1992).

The 1995 Standards New Zealand Concrete Structures Standard, NZS 3101-95

(NZS 1995).

International Conference of Building Officials 1997 Uniform Building Code:

Volume 2: Structural Engineering Design Provisions, Chapter 19 "Concrete",

UBC 1997 (ICBO 1997).

Details of the process associated with each of these codes as implemented in the

program are described in the subsequent chapters. This chapter provides a back-

ground common to all of the design codes.

In writing this manual it has been assumed that the user has an engineering back-

ground in the general area of structural reinforced concrete design and familiarity

with at least one of the above-mentioned design codes.

For referring to pertinent sections of the corresponding codes, a unique prefix is as-

signed for each code. For example, all references to the AASHTO code are pre-

ceded by the word AASHTO. Similarly,

References to the ACI 318-05, ACI 318-02 and ACI 318-99 code have the pre-fix of ACI

References to the Brit ish code carry the prefix of BS

References to the Ca na dian code carry the prefix of CSA

References to the Eurocode 2 carry the prefix of EC2

References to the New Zealand code carry the prefix of NZS

References to the Singapore code carry the prefix of CP

References to the UBC 1997 code have the prefix of UBC

Design Load Combinations

The design load combinations are used to determine the various combinations of

the load cases for which the structure is to be designed/checked. The load combina-

tion factors to be used vary with the selected design code. The load combination

factors are applied to the forces and moments obtained from the associated load

6 Design Load Combinations



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cases (or analysis cases for SAP2000) and are then summed to obtain the factored

design forces and moments for the load combination.

For multi-valued load combinations involving response spectrum, time history,

moving loads (only applicable for SAP2000) and multi-valued combinations (of

type enveloping, square-root of the sum of the squares or absolute) where any cor-respondence between interacting quantities is lost, the program automatically pro-

duces multiple sub combinations using maxima/minima permutations of interact-

ing quantities. Separate combinations with negative factors for response spectrum

cases are not re quired because the program automatically takes the minima to be

the negative of the maxima for response spectrum cases and the above-described

permutations generate the required sub combinations.

When a design combination involves only a single multi-valued case of time his-

tory or moving load, further options are available. The program has an option to re-

quest that time history combinations produce sub combinations for each time stepof the time history. Also an option is available to request that moving load combi-

nations produce sub combinations using maxima and minima of each design quan-

tity but with corresponding values of interacting quantities.

For normal loading conditions involving static dead load, live load, wind load, and

earthquake load or dynamic response spectrum earthquake load, the program has

built-in default loading combinations for each design code. These are based on the

code recommendations and are documented for each code in the corresponding

chapters.

For other loading conditions involving moving load, time history, pattern live

loads, separate consideration of roof live load, snow load, and so on., the user must

define design loading combinations either in lieu of or in addition to the default de-

sign loading combinations.

The default load combinations assume all static load cases declared as dead load to

be ad di tive. Similarly, all cases declared as live load are assumed additive. How -

ever, any static load case de clared as wind or earthquake, and any response spec-

trum cases, are assumed to be non-additive with each other and produce multiple

lateral load combinations. Also wind and static earthquake cases produce separateloading combinations with the sense (positive or negative) reversed. If these condi-

tions are not correct, the user must provide the appropriate design combinations.

The default load combinations are included in design if the user requests them to be

included or if no other user-defined combination is available for concrete design. If

any default combination is included in design, then all default combinations will

Design Load Combinations 7

Chapter II Design Process


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automatically be updated by the program any time the design code is changed or if

static or re sponse spec trum load cases are modified.

Live load reduction factors can be applied to the member forces of the live load case

on an element-by-element basis to reduce the contribution of the live load to the

factored loading.

The user is cautioned that if moving load or time history results are not requested to

be re cov ered in the analysis for some or all of the frame members, the ef fects of

these loads will be assumed to be zero in any combination that includes them.

Design and Check Stations

For each load combination, each element is designed or checked at a number of lo-

cations along the length of the element. The locations are based on equally spaced

segments along the clear length of the element. The number of segments in an ele-

ment is requested by the user before the analysis is performed. The user can re fine

the design along the length of an element by requesting more segments.

In ETABS only, when using 1997 UBC design codes, requirements for joint design

at the beam-to-column connections are evaluated at the top most station of each

column. The program also performs a joint shear analy sis at the same station to de-

termine if special considerations are required in any of the joint panel zones. The

ratio is reported for the beam flexural ca paci ties with re spect to the col umn flexural

capacities considering axial force effects associated with the weak beam-strong

column aspect of any beam/column intersection.

Identifying Beams and Columns

In the program all beams and col umns are repre sented as frame objects. But de sign

of beams and col umns re quires separate treatment. Identi fi cation for a concrete ob-

jects is accomplished by specifying that the frame section assigned to the object is

of beam or column type . If there is any brace object in the frame, the brace element

would also be identified as either a beam or a column type based on the section as-signed to the brace object.

Design of Beams

In the design of concrete beams, in general, the program calculates and reports the

required areas of steel for flexure and shear based upon the beam moments, shears,

load combination factors, and other criteria that are described in detail in the

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code-specific chapters. The reinforcement requirements are calculated at a user-

defined number of stations along the beam span.

All of the beams are designed only for major direction flexure, shear and torsion

(torsion is only applicable for ACI design code). Effects due to any axial forces, mi-

nor direction bending, and torsion (except for ACI design code) that may exist inthe beams must be investigated independently by the user.

In designing the flexural reinforcement for the major moment at a particular section

of a particular beam, the steps involve the determination of the maximum factored

moments and the determination of the reinforcing steel. The beam section is de-

signed for the maximum positiveMu+

and maximum negativeMu-

factored moment

envelopes obtained from all of the load combinations. Negative beam moments

produce top steel. In such cases the beam is al ways de signed as a rectan gu lar sec-

tion. Positive beam moments produce bottom steel. In such cases the beam may be

designed as a rectangular or a T beam. For the design of flexural reinforcement, thebeam is first designed as a singly reinforced beam. If the beam section is not ade-

quate, the required compression reinforcement is calculated.

In designing the shear reinforcement for a partic ular beam for a partic ular set of

loading combinations at a particular station due to the beam major shear, the

steps involve the determination of the factored shear force, the determination of

the shear force that can be resisted by con crete, and the deter mination of the rein-

force ment steel re quired to carry the balance.

Special considerations for seismic design are incorporated in ETABS only for ACI,Ca na dian, and New Zealand codes.

Design of Columns

In the design of the columns, the program calculates the required longitudinal steel,

or if the longitudinal steel is specified, the column stress condition is reported in

terms of a column capacity ratio, which is a factor that gives an indication of the

stress condition of the column with respect to the capacity of the column. The de-

sign procedure for the reinforced concrete columns of the structure involves the fol-lowing steps:

Design of Columns 9



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Generate axial force-biaxial moment interaction surfaces for all of the different

con crete section types of the model. A typical in teraction sur face is shown in

Figure II-1.

Check the capacity of each column for the factored axial force and bending mo-

ments obtained from each loading combination at each end of the column. This

step is also used to cal culate the re quired re inforce ment (if none was spec ified)

that will produce a capacity ratio of 1.0.

Design the column shear reinforcement

The generation of the interaction surface is based on the assumed strain and stress

distributions and some other simplifying assumptions. These stress and strain dis-

10 Design of Columns


Figure II-1

A Typical Column Interaction Surface


19/268

tributions and the assumptions vary from code to code. A typical assumed strain

distribution is described in Figure II-2.

Here maximum compression strain is limited to e c . For most of the design codes,this assumed distribution remains valid. However, the value ofe c varies from codeto code. For example, e c = 0.003 for ACI, UBC and New Zea land codes, ande

c= 0.0035 for Canadian, British and European codes. The details of the generation

of interaction sur faces differ from code to code. These are described in the chap ters

spe cific to the code.

Design of Columns 11


Figure II-2

Idealized Strain Distribution for generation of Interaction Surfaces


20/268

A typical interaction surface is shown in Figure II-1. The column capacity interac-

tion volume is numerically described by a series of discrete points that are gener-

ated on the three-dimensional interaction failure surface. The coordinates of these

points are determined by rotating a plane of linear strain in three dimensions on the

section of the column as described in Figure II-2.

The area as so ciated with each rebar is placed at the actual lo cation of the cen ter of

the bar and the algorithm does not assume any simplifications in the manner in

which the area of steel is distributed over the cross-section of the column. The inter-

action algorithm provides corrections to account for the concrete area that is dis-

placed by the reinforcing in the compression zone.

The effects of code-specified strength reduction factors and maximum limit on the

axial capacity are incorporated in the interaction surfaces. The formulation is based

consistently upon the general principles of ultimate strength design, and allows for

rectangular, square or circular, doubly symmetric column sections. In addition toaxial compression and biaxial bending, the formulation allows for axial tension and

biaxial bending considerations, as shown in Figure II-1.

The capacity check is based on whether the design load points lie inside the interac-

tion volume in a force space, as shown in Figure II-3 . If the point lies inside the vol-

ume, the column capac ity is ad equate, and vice versa. The point in the interaction

volume (P,Mx , andMy ) which is represented by point L is placed in the interac-

tion space as shown in Figure II-3 . If the point lies within the interaction vol ume,

the column capacity is adequate; however, if the point lies outside the interaction

volume, the column is overstressed. As a measure of the stress condition of the col-umn, a capacity ratio is calculated. This ratio is achieved by plotting the point L, de-

fined by P, Mx and My, and determining the location of point C. The point C is de-

fined as the point where the line OL (if extended outwards) will intersect the failure

surface. This point is determined by three-dimensional linear interpolation between

the points that define the failure sur face. The capac ity ratio, CR, is given by the ra-

tio OL OC .

If OL = OC (or CR=1) the point lies on the interaction surface and the column isstressed to ca pacity.

If OL < OC (or CR OC (or CR>1) the point lies outside the interaction volume and the col-

umn is overstressed.




21/268

The ca pac ity ratio is basically a fac tor that gives an in di cation of the stress condi -

tion of the column with respect to the capacity of the column. In other words, if the

axial force and biaxial moment set for which the column is being checked is ampli-

fied by dividing it by the reported capacity ratio, the point defined by the resulting

axial force and biaxial moment set will lie on the failure (or in teraction vol ume)

surface.

The shear reinforcement design procedure for columns is very similar to that for

beams, except that the effect of the axial force on the con crete shear capacity needs

to be considered.

For certain special seis mic cases, the design of col umn for shear is based on the ca -

pacity-shear. The capacity-shear force in a particular direction is calculated from

the mo ment capac ities of the column asso ci ated with the factored ax ial force acting

Design of Columns 13


Figure II-3Geometric Representation of Column Capacity Ratio


22/268

on the column. For each load combination, the factored axial load is calculated, us-

ing the analy sis load cases and the corresponding load combination factors. Then,

the moment capacity of the column in a particular direction under the influence of

the axial force is cal culated, using the uniaxial interaction diagram in the cor re-

sponding direction as shown in Figure II-4.



Figure II-4

Moment Capacity Mu at a Given Axial LoadPu


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Design of Joints

This section is applicable to ETABS only.

To ensure that the beam-column joint of special moment resisting frames possesses

adequate shear strength, the program performs a rational analy sis of the beam-col-umn panel zone to determine the shear forces that are generated in the joint. The

program then checks this against design shear strength.

Only joints having a column below the joint are designed. The material properties

of the joint are assumed to be the same as those of the column below the joint.

The joint analy sis is performed in the major and the minor directions of the column.

The joint design procedure involves the following steps:

Deter mine the panel zone de sign shear force,Vuh

Deter mine the effective area of the joint

Check panel zone shear stress

The fol low ing three sections describe in detail the algorithms associated with these

steps.

Determine the Panel Zone Shear Force

For a particular column direction, major or minor, the free body stress condition of

a typical beam-column intersection is shown in Figure II-5.

The forceVuh

is the hor izon tal panel zone shear force that is to be calculated. The

forces that act on the joint arePu , Vu ,MuL and Mu

R . The forcesPu and Vu are axial

force and shear force, respectively, from the column framing into the top of the

joint. The moments MuL and Mu

R are obtained from the beams framing into the

joint. The joint shear forceVuh is calculated by resolving the moments into Cand T

forces. Noting that T CL L

= and T CR R

= ,

V = T + T - V uh

L R u

The location ofCor Tforces is determined by the direction of the moment. The

magnitude ofCorTforces is conservatively determined using basic principles of

ultimate strength theory, ignoring compression reinforcement as follows. The max-

imum compression,Cmax , and the maximum moment,Mmax , that can be carried by

the beam is calcu lated first.

Design of Joints 15



24/268

C =max 0.85f bdc

M = Cmax maxd

2

Then the Cand Tforces are conservatively determined as follows:

16 Design of Joints


Figure II-5

Beam-Column Joint Analysis


25/268

( )C = T = Cmax

max

1 1- -

abs M

M

The moments and theCand Tforces from beams that frame into the joint in a direc-

tion that is not parallel to the major or minor directions of the column are resolvedalong the direction that is being investigated, thereby contributing force compo-

nents to the analysis. Also Cand Tare cal cu lated for the positive and negative mo -

ments con sid ering the fact that the concrete cover may be different for the direction

of moment.

In the design of special moment resisting concrete frames, the evaluation of the de-

sign shear force is based upon the moment capacities (with reinforcing steel

overstrength factor, a, and no j factors) of the beams framing into the joint,(ACI 21.5.1.1, UBC 1921.5.1.1). The Cand Tforce are based upon these moment

capacities. The column shear forceVu is calculated from the beam mo ment capac i-ties as follows:

V =M + M

Hu

u

L

u

R

See Figure II-6. It should be noted that the points of inflection shown on Figure II-6

are taken as midway between actual lateral support points for the columns. If there

is no column at the top of the joint, the shear force from the top of the column is

taken as zero.

The ef fects of load rever sals, as illus trated in Case 1 and Case 2 ofFigure II-5, are

investigated and the design is based upon the maximum of the joint shears obtained

from the two case

Design of Joints 17



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Determine the Effective Area of Joint

The joint area that resists the shear forces is assumed always to be rectangular in

plan view. The dimensions of the rectangle correspond to the major and minor di-

mensions of the column below the joint, except if the beam framing into the joint is

very narrow. The effective width of the joint area to be used in the calculation is

limited to the width of the beam plus the depth of the column. The area of the joint

18 Design of Joints


Figure II-6

Column Shear Force,Vu


27/268

is assumed not to exceed the area of the column below. The joint area for joint shear

along the major and minor directions is calculated separately (ACI R21.5.3).

It should be noted that if the beam frames into the joint eccentrically, the above as-

sumptions may be unconservative and the user should investigate the acceptability

of the particular joint.

Check Panel Zone Shear Stress

The panel zone shear stress is evaluated by dividing the shear forceVuh by the ef fec-

tive area of the joint and comparing it with the following design shear strengths

(ACI 21.5.3, UBC 1921.5.3) :

v

fc

=

20 for joints confined on all four sides,

15

j

j

,

fc

, for joints confined on three faces or on two opposite faces,12 for all other joints,j fc

,

where j = 0.85 (by default). (ACI 9.3.2.3, UBC 1909.3.2.3, 1909.3.4.1)

A beam that frames into a face of a column at the joint is considered in program to

provide confinement to the joint if at least three-quarters of the face of the joint is

covered by the framing member (ACI 21.5.3.1, UBC 1921.5.3.1).

For light-weight aggregate concrete, the design shear strength of the joint is re-

duced in program to at least three-quarters of that of the normal weight concrete by

replacing the fc

with

{ }min ,,f f fcs factor c c 3 4 (ACI 21.5.3.2, UBC 1921.5.3.2)For joint design, the program reports the joint shear, the joint shear stress, the al-

lowable joint shear stress and a capacity ratio.

Design of Joints 19



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Beam/Column Flexural Capacity Ratios

This section is applicable to ETABS only.

At a particular joint for a particular column direction, major or minor, the program

will calculate the ra tio of the sum of the beam moment capaci ties to the sum of thecolumn moment capacities, (ACI 21.4.2.2, UBC 1921.4.2.2. CSA 21.5.1.2).

M Me g 65

(ACI 21.4.2.2, UBC 1921.4.2.2)

The capacities are calculated with no reinforcing overstrength factor, a , and in-

cluding j factors. The beam capacities are calculated for reversed situations (Cases1 and 2) as illustrated in Figure II-5 and the maximum summation obtained is used.

The moment capacities of beams that frame into the joint in a di rection that is not

parallel to the major or minor direction of the column are resolved along the direc-

tion that is being investigated and the resolved components are added to the sum-

mation.

The column capacity summation includes the column above and the column below

the joint. For each load combination the axial force,Pu , in each of the columns is

calculated from the ETABS analysis load combinations. For each load combina-

tion, the moment capacity of each column under the influence of the corresponding

axial loadPu is then determined separately for the major and minor directions of the

column, using the uniaxial column interaction diagram: see Figure II-4. The mo-ment capacities of the two columns are added to give the capacity summation for

the corresponding load combination. The maximum capacity summations obtained

from all of the load combinations are used for the beam/column capacity ratio.

The beam/column flexural ca pacity ratios are reported for Spe cial Moment-Resist-

ing Frames involving seismic design load combinations only.

P-D EffectsThe program design process requires that the analysis results include the P-D ef-fects. The P-D effects are con sid ered differ ently for braced or nonsway andunbraced or sway components of moments in columns or frames. For the

braced moments in columns, the effect of P-D is limited to "in di vidual member sta-bility." For unbraced components,"lateral drift effects" should be considered in ad-

dition to in di vidual member stability ef fect. In the program, it is assumed that

braced or non sway moments are contributed from the dead or live loads.

20 Beam/Column Flexural Capacity Ratios



29/268

Whereas, un braced or sway moments are contributed from all other types of

loads.

For the individual member stability effects, the moments are magnified using mo-

ment magni fication factors, as in the ACI, AASHTO, Ca na dian, and New Zea land

codes or using additional moments, as in the British and European codes.

For lateral drift effects, the program assumes that the P-D analysis is performedand that the amplification is already in cluded in the results. The moments and

forces obtained from P-D analy sis are further amplified for individual column sta-bility effect if required by the governing code, as in the ACI, Canadian, and New

Zealand codes.

Users should be aware that the default analy sis option in the program is turned OFF

for P-D effect. The user can turn the P-D analy sis ON and set the maximum numberof iterations for the analysis. The default number of iteration for P-D analy sis is 1.For more information, refer to the CSI Analysis Reference manual (CSI 2005c).

Element Unsupported Lengths

To account for column slenderness effects, the column unsupported lengths are re-

quired. The two unsupported lengths are l33

and l22

. These are the lengths be tween

support points of the element in the corresponding directions. The length l33

corre-

sponds to instability about the 3-3 axis (major axis), and l22

corresponds to instabil-

ity about the 2-2 axis (minor axis).

Normally, the unsupported element length is equal to the length of the element, i.e.,

the distance between END-I and END-J of the element. See Figure II-7. The pro-

gram, however, allows users to as sign several elements to be treated as a single

member for design. This can be accomplished differently for major and minor

bending. Therefore, extraneous joints, as shown in Figure II-8, that affect the un-

supported length of an element are automatically taken into consideration.

In determining the values for l22

and l33

of the elements, the program recognizes

various aspects of the structure that have an effect on these lengths, such as member

connectivity, diaphragm constraints and support points. The program automati-

cally locates the element support points and evaluates the corresponding unsup-

ported element length.

Therefore, the unsupported length of a column may actually be evaluated as being

greater than the corresponding element length. If the beam frames into only one di-

rection of the column, the beam is assumed to give lateral support only in that direc-

tion.

Element Unsupported Lengths 21



30/268

The user has options to specify the unsupported lengths of the elements on an ele-

ment-by-element basis.

22 Element Unsupported Lengths


l33

l22

Elem

ent A

xis

END

I

END

J

3

2

1

Figure II-7

Axes of Bending and Unsupported Length

Figure II-8

Unsupported Lengths and Interior Nodes


31/268

Special Considerations for Seismic Loads

The ACI code imposes a special ductil ity require ment for frames in seis mic regions

by specifying frames as Ordinary, Intermediate, or Special moment resisting

frames. The Special mo ment resist ing frame can provide the re quired ductility and

energy dissipation in the nonlinear range of cyclic deformation. The AASHTOcode requires that the concrete frame must be in Zone 1, Zone 2, Zone 3, or Zone 4,

where Zone 4 is designated as the zone of severe earthquake. The Canadian code

requires that the concrete frame must be designed as ei ther an Ordi nary, Nomi nal,

or Ductile mo ment resisting frame. The New Zealand code also re quires that the

con crete frame must be designed as either an Ordinary, Elas ti cally re spond ing,

frames with Limited ductility, or Ductile moment resisting frame.

Unlike the ACI, AASHTO, Canadian, and New Zealand codes, the current imple-

mentation of the British code and the Eurocode 2 in the program does not account

for any special requirements for seismic design.

Choice of Input Units

English as well as SI and MKS metric units can be used for input. But the codes are

based on a specific system of units. All equations and descriptions presented in the

subsequent chapters correspond to that specific system of units unless otherwise

noted. For example, the ACI code is published in inch-pound-second units. By de-fault, all equations and descriptions presented in the chapter "Design for ACI

318-99" correspond to inch-pound-second units. However, any system of units can

be used to define and design the structure in the program.

Special Considerations for Seismic Loads23



32/268

C h a p t e r III

Design for AASHTO LRFD 1997

This chapter describes in detail the various aspects of the concrete design procedure

that is used by SAP2000 only when the user selects the AASHTO LRFD 1997 De-

sign Code (AASHTO 1997). Various notations used in this chapter are listed in

Table III-1.

The design is based on user-specified loading combinations. But the program pro-vides a set of default load combinations that should help satisfy requirements for

the design of most bridge type structures.

SAP2000 provides options to design or check moment resisting frames of Zones 1

(low seismic activity), 2, 3, and 4 (high seismic activ ity) as re quired for seismic de-

sign provisions. The details of the design criteria used for the different seismic

zones are described in the following sections.

English as well as SI and MKS metric units can be used for input. The code is based

on Inch-Kip-Second units. For simplicity, all equations and descriptions presentedin this chapter correspond to Inch-Kip-Second units unless otherwise noted.

25


33/26826


Acv Area of con crete used to determine shear stress, sq-in

Ag Gross area of concrete, sq- in

As Area of tension reinforcement, sq-in

As

Area of compression reinforcement, sq-inA

s required( )Area of steel required for tension reinforcement, sq-in

Ast Total area of column longitudinal reinforcement, sq-in

Av Area of shear rein force ment, sq-in

a Depth of compression block, in

ab

Depth of compression block at balanced condition, in

amax Maximum allowed depth of compression block, in

b Width of member, in

bf Ef fective width of flange (T- Beam section), inbw Width of web (T-Beam section), in

Cm Coefficient, dependent upon column curvature,

used to calculate moment magnification factor

c Depth to neutral axis, in

cb

Depth to neutral axis at balanced conditions, in

d Distance from compression face to tension reinforcement, in

d Concrete cover to center of reinforcing, inds Thickness of slab (T-Beam section), in

Ec Modulus of elasticity of concrete, psi

Es Modulus of elasticity of reinforcement, assumed as 29,000 ksi

fc Specified compressive strength of concrete, ksi

fy Specified yield strength of flexural reinforcement, ksi

fyh

Specified yield strength of shear reinforcement, ksi

h Dimension of column, in

Ig Moment of inertia of gross concrete section about centroidal axis,

neglecting reinforcement, in4

Ise Moment of inertia of reinforcement about centroidal axis ofmember cross section, in

4

k Effective length factor

L Clear unsupported length, in

Table III-1

List of Symbols Used in the AASHTO code


34/268 27

Chapter III Design for AASHTO LRFD 1997

M1

Smaller factored end moment in a column, kip-in

M2

Larger factored end moment in a column, kip-in

Mc Factored moment to be used in design, kip-in

Mb Nonsway component of factored end moment, kip-inMs Sway component of factored end moment, kip-in

Mu Factored moment at section, kip-in

Mux Factored moment at section about X-axis, kip-in

Muy Factored moment at section about Y-axis, kip-in

Pb

Axial load capacity at balanced strain conditions, kip

Pe Euler buckling strength of column, kip

Pmax Maximum axial load strength allowed, kip

P0 Axial load capacity at zero eccentricity, kipPu Factored ax ial load at sec tion, kip

r Radius of gyration of column section, in

Vc Shear resisted by con crete, kip

VD L+ Shear force from span loading, kip

Vu Factored shear force at a section, kip

Vp Shear force computed from probable moment capacity, kip

a Reinforcing steel overstrength factorb is a fac tor indi cating the abil ity of diagonally cracked con crete

to transmit tension

b1

Factor for obtaining depth of compression block in concrete

bd

Absolute value of ratio of maximum factored axial dead load to

maximum factored axial total load

q An angle of inclination of diagonal compressive stresses with thelongitudinal axis of beam or column

ds Moment magnification factor for sway momentsd

bMoment magnification factor for nonsway (braced) moments

e c Strain in concretees Strain in reinforcing steelj Strength reduction factor

Table III-1

List of Symbols Used in the AASHTO code (continued)


35/268

Design Load Combinations

The design load combinations are the various combinations of the prescribed load

cases for which the structure is to be checked. There are more types of loads speci-

fied in the code than are considered in the current implementation of the default

load combinations. However, the user has full control of the definition of loads and

load combinations.

There are six types of dead loads: dead load of structural components and nonstruc-

tural attachments (DC), downdrag (DD), dead load of wearing surface and utilities

(DW), horizon tal earth pres sure load (EH), ver tical earth pressure load (EV), and

earth sur charge load (ES). Each type of dead load case requires a sep arate load fac-

tor.

There are six types of live loads: vehicular live load (LL), vehicular dynamic load

allow ance (IM), ve hicular centrifugal force (CE), ve hicular braking force (BR), pe-destrian live load (PL), and live load surcharge (LS). All of these load cases require

the same factor and do not need to be treated separately.

If the structure is subjected to structural dead load (DL), live load (LL), wind load

(WL), and earthquake loads (EL), and considering that wind and earthquake forces

are reversible, the following default load combinations have been considered for

Strength and Extreme Event limit states (AASHTO 3.4.1).

1.50 DL (Strength-IV)

1.25 DL + 1.75 LL (Strength-I)

0.90 DL 1.4 WL (Strength-III)1.25 DL 1.4 WL (Strength-III)1.25 DL + 1.35 LL 0.40 WL (Strength-V)

0.90 DL 1.0 EL (Extreme-I)1.25 DL + 0.5 LL 1.0 EL (Extreme-I)

These are also the default design load combinations in SAP2000 whenever the

AASHTO LRFD 1997 code is used. The user is expected to define the other load

combinations as necessary.

Live load reduction factors can be applied to the member forces of the live load case

on an element-by-element basis to reduce the contribution of the live load to the

factored loading.

28 Design Load Combinations



36/268

Strength Reduction Factors

The strength reduction factors,j , are applied on the nominal strength to obtain thedesign strength provided by a member. Thej factors for flex ure, axial force, shear,and torsion are as follows:

j = 0.90 for flexure, (AASHTO 5.5.4.2.1)

j = 0.90 for axial tension, (AASHTO 5.5.4.2.1)

j = 0.90 for axial tension and flexure, (AASHTO 5.5.4.2.1)

j = 0.90 for shear and torsion, (AASHTO 5.5.4.2.1)

j = 0.75 for axial compression only, or axial compression and flexure, and(AASHTO 5.5.4.2.1)

j = 0.50 for axial compression only, or axial compression and flexure inseismic Zones 3 and 4. (AASHTO 5.5.4.2.3, 5.10.11.4.1b)

The value ofj involving axial compression and flexure varies from 0.75 to 0.9based on the axial load. For low values of axial load, j is in creased linearly from0.75 to 0.9 as the axial load decreases from 0.1 f Ac g

to zero (AASHTO 5.5.4.2.1).

For seismic design in Zones 3 and 4, the value of j involving axial compressionand flexure varies from 0.5 to 0.9 based on the axial load. For low values of axial

load, j is in creased linearly from 0.5 to 0.9 as the axial load de creases from

0.2 f Ac g to zero (AASHTO 5.10.11.4.1b). In cases involving axial tension,j is al-ways 0.9 (AASHTO 5.5.4.2.1).

Column Design

The user may define the geometry of the reinforcing bar configuration of each con-

crete column section. If the area of reinforcing is provided by the user, the program

checks the column capacity. However, if the area of reinforcing is not provided by

the user, the program calculates the amount of reinforcing required for the column.

The design procedure for the reinforced concrete columns of the structure involvesthe following steps:

Generate axial force/biaxial moment interaction surfaces for all of the different

con crete section types of the model. A typ ical bi axial interaction sur face is

shown in Figure II-1. When the steel is undefined, the program generates the

interaction surfaces for the range of allowable reinforcement ratio (A Ast g)

Strength Reduction Factors 29



37/268

0.135f fc y to 0.08 for moment resisting frames (AASHTO 5.7.4.2) and 0.01

to 0.06 for ductile moment resisting frames in seismic Zones 3 and 4

(AASHTO 5.10.11.4.1a).

Calculate the capac ity ratio or the required re in forc ing area for the factored ax-

ial force and biaxial (or uniaxial) bending moments obtained from each loadingcombi na tion at each sta tion of the column. The tar get ca pac ity ratio is taken as

1.0 when calculating the required reinforcing area.

Design the column shear reinforcement.

The following three subsections describe in detail the algorithms associated with

these steps.

Generation of Biaxial Interaction Surfaces

The col umn ca pacity in teraction volume is nu merically de scribed by a series of dis-

crete points that are gen erated on the three-dimensional interaction fail ure surface.

In addition to axial compression and biaxial bending, the formulation allows for ax-

ial tension and biaxial bending considerations. A typical interaction diagram is

shown in Figure II-1.

The coordinates of these points are determined by rotating a plane of linear strain in

three dimensions on the section of the column. See Figure II-2. The linear strain

diagram limits the maximum concrete strain, e c , at the extremity of the sectionto 0.003 (AASHTO 5.7.2.1).

The formulation is based consistently on the general principles of ultimate strength

design (AASHTO 5.7), and allows for any doubly symmetric rectangular, square,

or circular column section.

The stress in the steel is given by the product of the steel strain and the steel modu-

lus of elasticity, es sE , and is limited to the yield stress of the steel, fy (AASHTO5.7.2.1). The area as sociated with each rein forc ing bar is as sumed to be placed at

the actual location of the center of the bar and the algorithm does not assume any

further simplifications in the manner in which the area of steel is distributed over

the cross section of the column, such as an equivalent steel tube or cylinder. SeeFigure III-1.

30 Column Design



38/268

The concrete compression stress block is assumed to be rectangular (AASHTO

5.7.2.1), with a stress value of0.85fc

(AASHTO 5.7.2.2). See Figure III-1. The

depth of the stress block is b1 c, whereb

1= - -0.85 0.05 4( )fc , (AASHTO 5.7.2.2)

0.65 0.85 b1

, and (AASHTO 5.7.2.2)

The limit offc is taken to be 10 ksi for all seismic regions:

fc 10 ksi. (AASHTO 5.1, 5.4.2.1)

The limit offy is taken to be 75 ksi for all frames:

fy 75 ksi. (AASHTO 5.4.3.1)

The interaction algorithm provides a correction to account for the concrete area that

is displaced by the reinforcement in the compression zone.

The effects of the strength reduction factor, j , are included in the generation of theinteraction surfaces. The maximum compressive axial load is limited toPmax , where

Column Design 31


Figure III-1

Idealization of Stress and Strain Distribution in a Column Section


39/268

P = f A - A + f Ac g st y st max 0.85 [0.85 ( ) ]j spiral column, (AASHTO 5.7.4.4)

P = f A - A f Ac g st y st max 0.80 [ 0.85 ( ) + ]j tied column. (AASHTO 5.7.4.4)

The value ofj involving axial compression and flexure varies from 0.75 to 0.9

based on the axial load. For low values of axial load, j is in creased lin early from0.75 to 0.9 as the axial load decreases from 0.1 f Ac g

to zero (AASHTO 5.5.4.2.1).

For seismic design in Zones 3 and 4, the value ofj involving axial compression andflexure varies from 0.5 to 0.9 based on the axial load. For low values of axial load,jis in creased linearly from 0.5 to 0.9 as the ax ial load de creases from 0.2 f Ac g

to

zero (AASHTO 5.10.11.4.1b). In cases involving axial tension, j is always 0.9(AASHTO 5.5.4.2.1).

Check Column Capacity

The col umn ca pacity is checked for each loading combi na tion at each check station

of each column. In checking a particular column for a particular loading combina-

tion at a particular station, the program uses the following steps:

Determine the factored moments and forces from the analysis load cases and

the specified load combination factors to giveP M Mu ux uy, ,and .

Determine the moment magnification factors for the column moments.

Apply the moment magnification factors to the factored moments. Determine

whether the point, defined by the resulting axial load and biaxial moment set,

lies within the interaction vol ume.

The factored moments and corresponding magnification factors depend on the

identification of the individual column as "braced" or "unbraced."

The following three sections describe in detail the algorithms associated with these

steps.

Determine Factored Moments and Forces

The factored loads for a particular load combination are obtained by applying thecorresponding load factors to all the load cases, givingP M Mu ux uy, ,and . The com-

puted moments are further amplified by using Moment Magnification Factors to

allow for stability effects.

32 Column Design



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Determine Moment Magnification Factors

The moment magnification factors are calculated separately for sway (overall sta-

bility effect), ds , and for nonsway or braced (individual column stability effect),d ns . Also the moment magnification factors in the major and minor directions are in

general different.

The program assumes that a P-D analy sis has been performed in SAP2000 and,therefore, moment magnification factors for moments causing sidesway are taken

as unity (AASHTO 4.5.3). For the P-D analy sis the load should correspond to aload combination of (1.25 dead load + 1.35 live load) /j , where j is the resistancefactor for axial compression, which is taken as 0.75 for seismic Zones 1 and 2 and

as 0.5 for seismic Zones 3 and 4 by default (AASHTO 5.5.4.2.1). See also White

and Hajjar (1991).

The moment obtained from analysis is separated into two components: the sway( )Ms and the nonsway ( )Mb components. The non-sway or braced components,

which are identified by b subscripts, are predominantly caused by gravity load.

The sway components are identified by s sub scripts. The sway moments are pre -

domi nantly caused by lateral loads and are related to the cause of side sway.

For individual columns or column-members in a floor, the magnified moments

about two axes at any station of a column can be obtained as

M M Mb b s s

= +d d . (AASHTO 4.5.3.2.2b)

The factords is the moment magnification factor for moments causing sidesway.This factor is taken as 1 because the component momentsMs andMb are obtained

from a second or der elas tic (P-D) analysis.

The nonsway moment magnification factor, db, associated with the major or minor

direction of the column is given by (AASHTO 4.5.3.2.2b),

d

j

b

m

u

e

C

P

P

= 1.0

1 - , where (AASHTO 4.5.3.2.2b)

P =EI

kLe

p 2

2( )

, (AASHTO 4.5.3.2.2b)

k is taken as 1, however SAP2000 allows the user to override this value

(AASHTO 4.6.2.5, 5.7.4.3), and

Column Design 33



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EI is associated with a particular column direction given by:

EI =E I

+

c g

d

2.5

1 b, (AASHTO 5.7.4.3)

bd

= maximum factored dead load momentmaximum factored total load moment

, and (AASHTO 5.7.4.3)

C = +M

Mm

a

b

0.6 0.4 0.4 . (AASHTO 4.5.3.2.2b)

Ma andMb are the moments at the ends of the column, andMb is nu merically

larger thanMa .M Ma b is positive for single curvature bending and negative

for double curvature bending. The above expression ofCm is valid if there is no

transverse load applied between the supports and the member is braced against

sidesway. If transverse load is present on the span, or the length is overwritten,

or for any other case, Cm =1 . Cm can be overwritten by the user on an ele-ment-by-element basis.

The magnification factor, db, must be a positive number and greater than one.

ThereforePu must be less than jPe . IfPu is found to be greater than or equal to jPe ,a failure con dition is declared.

The above calculations use the unsupported lengths of the column. The two unsup-

ported lengths are l22

and l33

corresponding to instability in the minor and major di-

rections of the element, respectively. See Figure II-7. These are the lengths be tween

the support points of the element in the corresponding directions.

If the program assumptions are not satisfactory for a particular member, the user

can explicitly specify values ofd ds band .

Determine Capacity Ratio

As a measure of the stress condi tion of the col umn, a ca pacity ratio is calculated.

The capacity ratio is ba sically a factor that gives an indication of the stress condi-

tion of the column with respect to the capacity of the column.

Before entering the inter action dia gram to check the column capacity, the mo ment

magnification factors are applied to the factored loads to obtainP M Mu ux uy, ,and .

The point (P M Mu ux uy, , ) is then placed in the interaction space shown as point L in

Figure II-3. If the point lies within the in teraction vol ume, the col umn capacity is

adequate; however, if the point lies outside the interaction volume, the column is

overstressed.

34 Column Design



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This capacity ratio is achieved by plotting the point L and determining the location

of point C. Point C is defined as the point where the line OL (if extended outwards)

will intersect the failure surface. This point is determined by three-dimensional lin-

ear interpo lation between the points that define the failure surface. See Figure II-3 .

The ca pacity ratio, CR, is given by the ra tio

OL

OC.

If OL = OC (or CR=1) the point lies on the interaction surface and the column is

stressed to capac ity.

If OL < OC (or CR OC (or CR>1) the point lies outside the interaction volume and the col-

umn is overstressed.

The maximum of all the values of CR calculated from each load combination is re-ported for each check station of the column along with the controlling

P M Mu ux uy, ,and set and associated load combination number.

If the reinforcing area is not defined, the program computes the reinforcement that

will give an interaction ratio of unity.

Design Column Shear Reinforcement

The shear reinforcement is designed for each loading combination in the major and

minor directions of the column. In designing the shear reinforcing for a particularcolumn for a particular loading combination due to shear forces in a particular di-

rection, the program uses the following steps:

Deter mine the factored forces acting on the section,Mu , Pu and Vu . Note that

Mu andPu are needed for the calculation of v c .

Deter mine the shear stress, v c , that can be resisted by con crete alone.

Calculate the reinforcement steel re quired to carry the balance.

For moment resisting frames in seismic Zones 3 and 4, the shear design of the col-umns is also based on the overstrength moment capacities of the members in addi-

tion to the factored moments (AASHTO 3.10.9.4.3). Effects of the axial forces on

the column moment capacities are included in the formulation.

The fol low ing three sections describe in detail the algorithms associated with these

steps.

Column Design 35



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Determine Section Forces

In the design of the column shear reinforcement of moment resisting concrete

frames in seis mic Zones 1 and 2, the forces for a particular load combination,

namely, the column axial force, Pu , and the column shear force, Vu , in a par-

ticular direction are obtained by factoring the SAP2000 analy sis load caseswith the corresponding load combination factors.

In the shear design of moment resisting frames in seismic Zones 3 and 4 (seis-

mic design), the following are checked in addition to the requirement for the

moment resisting frames in Zones 1 and 2. In the design of moment resisting

concrete frames in seismic Zones 3 and 4, the design shear force in a column,

Vu , in a particular direction is also calculated from the overstrength moment ca-

paci ties of the column associated with the factored axial force act ing on the col-

umn (AASHTO 3.10.9.4.3).

For each load combination, the factored axial load,Pu , is cal culated. Then, thepositive and negative moment capacities,Mu

+ andMu- , of the column in a par-

ticular direction under the influence of the axial forcePu is calculated using the

uniaxial interaction diagram in the corresponding direction. The design shear

force,Vu , is then given by (AASHTO 3.10.9.4.3)

V V + V u p D+ L= (AASHTO 3.10.9.4.3)

where, Vp is the shear force obtained by apply ing the calculated overstrength

ultimate moment capacities at the two ends of the column acting in two oppo-

site directions. Therefore,Vp is the maximum ofVP1 and VP2 , where

V =M + M

LP

I

-

J

+

1, and

V =M + M

LP

I

+

J

-

2, where

MI

+ ,MI

- = Positive and negative moment capaci ties at end I of the col-

umn obtained by multiplying the nominal resistance by 1.3,

MJ

+ ,MJ

- = Positive and negative moment capaci ties at end J of the col-

umn obtained by multiplying the nominal resistance by 1.3,

and

L = Clear span of column.

VD L+ is the contribution of shear force from the in-span distribution of gravity

loads. For most of the columns, it is zero. See also Table III-2 for details.

36 Column Design



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Determine Concrete Shear Capacity

Given the design force set Mu , Pu andVu , the shear capacity provided by the con-

crete alone, v c , is cal cu lated as follows:

For designing moment resist ing concrete frames in any seismic zone, v c is setto:

v = fc c0.0316 b

, (AASHTO 5.8.3.3)

where, b is a factor indi cating the abil ity of diago nally cracked con crete totransmit tension. It is a function of stress condition and its approximate value is

2.0 (AASHTO 5.8.3.4.1). It is computed from AASHTO Table 5.8.3.4.2-1

through an iterative procedure.

For designing moment re sist ing con crete frames in seismic Zones 3 and 4, v c istaken as that given earlier (AASHTO 5.8.3.3) if compressive axial force is high

( )P f Au c g>0.1 (AASHTO 5.10.11.4.1c). v c is taken as zero if axial force is

tensile. v c is linearly interpolated between zero and that given by AASHTO

5.8.3.3, if the factored axial compressive force,Pu , including the earthquake ef-

fect is small ( )P f Au c g0.1 (AASHTO 5.10.11.4.1c). This provision is ap-

plied to all locations of the column irrespective of whether it is in the end region

or not.

Determine Required Shear Reinforcement

The average shear stress is computed for a rectan gu lar section as,

v =V

b d

u

w

.

For other types of sections b dw is replaced by Acv , the effective shear area

which is shown in Figure III-2.

The average shear stress, v, is limited to a maximum limit, v max , given by

v fcmax = 0.25 . (AASHTO 5.8.3.3)

The shear reinforcement per unit spacing is computed as follows:

If ( )v v c j 2 ,A

s

v = 0 , (AASHTO 5.8.2.4)

Column Design 37



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else if ( ) ( )j j qv v v f c c c2 < + 0.0316 cot ,A

s

f b

f

v c w

yh

=0.0316

, (AASHTO 5.8.2.5)

else if ( )j q jv f v vc c+ < 0.0316 cot max ,A

s

v v b

f

v c w

yh

=-( )

cot

j

j q, (AASHTO 5.8.3.3)

else if v v> max ,

a fail ure con di tion is de clared. (AASHTO 5.8.3.3)

Hereq is an angle of inclination of diagonal compressive stresses. It is a function ofcurrent stress condition and its approximate value is 45 degree (AASHTO

5.8.3.4.1). It is computed from AASHTO Table 5.8.3.4.2-1 through an iterative

procedure. Here the default value of j , the strength reduction factor, is 0.90(AASHTO 5.5.4.2.1).

The limit offyh

is taken to be 60 ksi for all frames:

fyh

60 ksi. (AASHTO 5.8.2.8)

The limit offc is taken to be 10 ksi for all seismic regions:

fc 10 ksi. (AASHTO 5.1, 5.4.2.1)

In calculating the shear reinforcement for seismic moment resisting frames in

Zones 3 and 4, a limit was imposed on the fys as

f fys y . (AASHTO 5.10.11.4.1d)

The maxi mum of all the calculatedA sv values, obtained from each load combina-

tion, is reported for the major and minor directions of the column along with the

controlling shear force and associated load combination number.

For all columns and at any station, the minimum area of transverse circular hoop re-

inforcement is imposed as follows:

A

s

A

A

f

f

hv g

c

c

yh

core -

0.45 14

(AASHTO 5.7.4.6)

38 Column Design



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In po tential plastic hinge lo cations, as described later, of seismic moment resisting

frames in Zones 2, 3 and 4, the minimum area of circular hoops and transverse stir-

rups is imposed as follows:

A

s

P

f A

f

f

hv u

c g

c

yh

core

+

0.16 0.5

1.25

4 (Hoops) (AASHTO 5.10.11.4.1d)

A

s

A

A

f

fhv

g

c

c

yh

core -

0.30 1 (Stirrups) (AASHTO 5.10.11.4.1d)

A

s

P

f A

f

fhv u

c g

c

yh

core +

0.12 0.51.25

(Stirrups) (AASHTO 5.10.11.4.1d)

In the preceding equations, the following limits are imposed:

0.51.25

+

P

f A

u

c g

10. (AASHTO 5.10.11.4.1d)

f fyh y

(AASHTO 5.10.11.4.1d)

For the definition of the potential plastic hinge, it is assumed in the current version

of the program that any beam and column segment near the joint is a potential plas-

tic hinge. The length of the plastic hinge,Lhinge

, in a column is taken as follows:

{ }L h b lhinge = max , , ,6 18 (AASHTO 5.10.11.4.1c)

The column shear reinforcement requirements reported by the program are based

purely on these considerations. Any other minimum stirrup requirements to satisfy

spacing and/or volumetric requirements must be investigated independently of the

program by the user.

Column Design 39



47/268

40 Column Design


Figure III-2

Shear Stress Area,Acv


48/268

Beam Design

In the design of concrete beams, SAP2000 calculates and reports the required areas

of steel for flexure and shear based on the beam moments, shears, load combination

factors, and other criteria de scribed in the text that follows. The reinforcement re-

quirements are calculated at a user-defined number of check/design stations along

the beam span.

All of the beams are designed for major direction flexure and shear only. Effects

due to any axial forces, minor direction bending, and torsion that may exist in the

beams must be investigated independently by the user.

The beam design procedure involves the following steps:

Design beam flexural reinforcement

Design beam shear reinforcement

Design Beam Flexural Reinforcement

The beam top and bottom flexural steel is designed at check/design stations along

the beam span. In designing the flexural reinforcement for the major moment for a

particular beam for a particular section, the following steps are involved:

Determine the maximum factored moments

Determine the reinforcing steel

Determine Factored Moments

In the design of flexural reinforcement of moment resisting concrete beams, the

factored moments for each load combi nation at a par ticu lar beam sec tion are ob-

tained by factoring the corresponding moments for different load cases with the

corresponding load factors. The beam section is then designed for the maximum

positiveMu+ and maximum negativeMu

- factored moments obtained from all of the

load combinations.

Negative beam moments produce top steel. In such cases the beam is always de-

signed as a rectangular section. Positive beam moments produce bottom steel. In

such cases the beam may be designed as a Rect an gu lar or a T beam.

Beam Design 41



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Determine Required Flexural Reinforcement

In the flexural reinforcement design process, the program calculates both the ten-

sion and compression reinforcement. Compression reinforcement is added when

the applied design moment exceeds the maximum moment capacity of a singly re-

inforced section. The user has the option of avoiding the compression reinforce-ment by increasing the effective depth, the width, or the grade of concrete.

The design procedure is based on the simplified rectangular stress block, as shown

in Figure III-3 (AASHTO 5.7). Furthermore, it is assumed that the maximum depth

of compression block is limited to 0.42d(AASHTO 5.7.3.3.1). When the applied

moment exceeds the moment capacity at this limiting condition, the area of com-

pression reinforcement is calculated on the assumption that the additional moment

will be carried by compression and additional tension reinforcement.

In designing the beam flexural reinforcement, the following limits are imposed onthe steel ten sile strength and the concrete compressive strength:

fc 10 ksi. (AASHTO 5.1, 5.4.2.1)

fy 75 ksi. (AASHTO 5.4.3.1)

The design procedure used by SAP2000, for both rectangular and flanged sections

(L and T beams), is summarized below. All the beams are designed for major di rec-

tion flexure and shear only.

Design for Rectangular Beam

In designing for a factored negative or positive moment,Mu , (i.e., designing top or

bottom steel) the depth of the compression block, a, (see Figure III-3) is computed

as,

a d dM

f b

u

c

= - -

22

0.85 j, (AASHTO 5.7.2.1

where, the value ofj is 0.90 (AASHTO 5.5.4.2.1) in the preceding equation and inthe equation that follow. Also b1

and cb

are calculated as fol lows:

( )b1 = 0.85 0.05- -fc 4 , 0.65 0.85 b1 , (AASHTO 5.7.2.2)

c =E

E + fd =

+ fd

b

c s

c s y y

e

e87

87. (AASHTO 5.7.2.1)

42 Beam Design



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The maximum allowed depth of the compression block is given by

{ }a c dbmax = min ,b1 0.42 . (AASHTO 5.7.2.2, 5.7.3.3.1) If a a max (AASHTO 5.7.3.3.1), the area of tensile steel reinforce ment is

then given by

AM

f da

s

u

y

=-

j

2

.

This steel is to be placed at the bottom ifMu is positive, or at the top ifMu isnegative.

If a a> max (AASHTO 5.7.3.3.1), compression reinforcement is required and iscalculated as follows:

The compressive force developed in concrete alone is given by

Beam Design 43


Figure III-3

Design of Rectangular Beam Section


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C f bac=

0.85 max , and (AASHTO 5.7.2.2)

the moment resisted by concrete compression and bottom steel is

M C d auc = - max

2 j .

There fore the mo ment resisted

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