OlderCodes

Concrete Frame Design Manual

ISO SAP072314M8 Rev. 0 Proudly developed in the United States of America July 2014

Concrete Frame Design Manual

ACI31899,BS811097,BS811085R89CSAA23.394,CP6599,Eurocode292

NZS310195andUBC97For SAP2000

Copyright

Copyright Computers & Structures, Inc., 1978-2014 All rights reserved. The CSI Logo and SAP2000 are registered trademarks of Computers & Structures, Inc. Watch & LearnTM is a trademark of Computers & Structures, Inc. The computer program SAP2000 and all associated documentation are proprietary and copyrighted products. Worldwide rights of ownership rest with Computers & Structures, Inc. Unlicensed use of these programs or reproduction of documentation in any form, without prior written authorization from Computers & Structures, Inc., is explicitly pro-hibited. 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 obtained from: Computers & Structures, Inc. www.csiamerica.com [email protected] (for general information) [email protected] (for technical support)

DISCLAIMER

CONSIDERABLE TIME, EFFORT AND EXPENSE HAVE GONE INTO THE DEVELOPMENT AND DOCUMENTATION OF THIS SOFTWARE. HOWEVER, THE USER ACCEPTS AND UNDERSTANDS THAT NO WARRANTY IS EXPRESSED OR IMPLIED BY THE DEVELOPERS OR THE DISTRIBUTORS ON THE ACCURACY OR THE RELIABILITY OF THIS PRODUCT.

THIS PRODUCT IS A PRACTICAL AND POWERFUL TOOL FOR STRUCTURAL DESIGN. HOWEVER, THE USER MUST EXPLICITLY UNDERSTAND THE BASIC ASSUMPTIONS OF THE SOFTWARE MODELING, ANALYSIS, AND DESIGN ALGORITHMS AND COMPENSATE FOR THE ASPECTS THAT ARE NOT ADDRESSED.

THE INFORMATION PRODUCED BY THE SOFTWARE MUST BE CHECKED BY A QUALIFIED AND EXPERIENCED ENGINEER. THE ENGINEER MUST INDEPENDENTLY VERIFY THE RESULTS AND TAKE PROFESSIONAL RESPONSIBILITY FOR THE INFORMATION THAT IS USED.

Ta ble of Con tents

CHAP TER I In tro duc tion 1Over view . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Or ga ni za tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Rec om mended Read ing . . . . . . . . . . . . . . . . . . . . . . . . . 3

CHAP TER II De sign Pro cess 5De sign Load Com bi na tions. . . . . . . . . . . . . . . . . . . . . . . . 6De sign and Check Sta tions . . . . . . . . . . . . . . . . . . . . . . . . 8Iden ti fy ing Beams and Col umns . . . . . . . . . . . . . . . . . . . . . 8De sign of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8De sign of Col umns . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9De sign of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

De ter mine the Panel Zone Shear Force. . . . . . . . . . . . . . . 15De ter mine the Ef fec tive Area of Joint . . . . . . . . . . . . . . . 18Check Panel Zone Shear Stress. . . . . . . . . . . . . . . . . . . 19

Beam/Col umn Flex ural Ca pac ity Ra tios . . . . . . . . . . . . . . . . 20P- Ef fects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20El e ment Un sup ported Lengths . . . . . . . . . . . . . . . . . . . . . 21Spe cial Con sid er ations for Seis mic Loads . . . . . . . . . . . . . . . 23Choice of In put Units . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CHAP TER III De sign for ACI 318-99 25De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . . 25Strength Re duc tion Fac tors . . . . . . . . . . . . . . . . . . . . . . . 28Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . . 29

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Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . . 31 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 31 De ter mine Mo ment Mag ni fi ca tion Fac tors . . . . . . . . . . 31 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . . 33De sign Col umn Shear Re in force ment . . . . . . . . . . . . . . . 34 De ter mine Sec tion Forces . . . . . . . . . . . . . . . . . . . 35 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 36 De ter mine Re quired Shear Re in force ment . . . . . . . . . . 38

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . . 39 De ter mine Fac tored Mo ments. . . . . . . . . . . . . . . . . 39 De ter mine Re quired Flex ural Re in force ment . . . . . . . . . 39De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 46 De ter mine Shear Force and Mo ment . . . . . . . . . . . . . 47 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 48 De ter mine Re quired Shear Re in force ment . . . . . . . . . . 48

CHAP TER IV De sign for BS 8110-97 51De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . . 51De sign Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . . 55Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . . 56 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 57 De ter mine Ad di tional Mo ments. . . . . . . . . . . . . . . . 57 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . . 59De sign Col umn Shear Re in force ment . . . . . . . . . . . . . . . 60

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . . 61 De ter mine Fac tored Mo ments. . . . . . . . . . . . . . . . . 62 De ter mine Re quired Flex ural Re in force ment . . . . . . . . . 62De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 67

CHAP TER V De sign for BS 8110-85 R1989 69De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . . 69De sign Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . . 73Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . . 74 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 75 De ter mine Ad di tional Mo ments. . . . . . . . . . . . . . . . 75 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . . 77De sign Col umn Shear Re in force ment . . . . . . . . . . . . . . . 78

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

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CSI Concrete Design Manual

De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . . 79 De ter mine Fac tored Mo ments. . . . . . . . . . . . . . . . . 80 De ter mine Re quired Flex ural Re in force ment . . . . . . . . . 80De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 85

CHAP TER VI De sign for CSAA23.3-94 87De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . . 90Strength Re duc tion Fac tors . . . . . . . . . . . . . . . . . . . . . . . 90Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . . 91Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . . 93 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 93 De ter mine Mo ment Mag ni fi ca tion Fac tors . . . . . . . . . . 93 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . . 96De sign Col umn Shear Re in force ment . . . . . . . . . . . . . . . 97 De ter mine Sec tion Forces . . . . . . . . . . . . . . . . . . . 97 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 99 De ter mine Re quired Shear Re in force ment . . . . . . . . . 100

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . 103 De ter mine Fac tored Mo ments . . . . . . . . . . . . . . . . 103 De ter mine Re quired Flex ural Re in force ment . . . . . . . . 104De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 112 De ter mine Shear Force and Mo ment . . . . . . . . . . . . 112 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 114 De ter mine Re quired Shear Re in force ment . . . . . . . . . 114

CHAP TER VII De sign for CP 65-1999 119De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . 119De sign Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . 123Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . 124 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 125 De ter mine Ad di tional Mo ments . . . . . . . . . . . . . . . 125 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . 127De sign Col umn Shear Re in force ment. . . . . . . . . . . . . . . 128

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . 129 De ter mine Fac tored Mo ments . . . . . . . . . . . . . . . . 130 De ter mine Re quired Flex ural Re in force ment . . . . . . . . 130De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 135

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Table of Contents

CHAP TER VIII De sign for Eurocode 2-92 137De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . 137De sign Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . 141Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . 143 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 143 De ter mine Code To tal Mo ments . . . . . . . . . . . . . . 143 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . 145De sign Col umn Shear Re in force ment. . . . . . . . . . . . . . . 146

Beam De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150De sign Beam Flex ural Re in force ment . . . . . . . . . . . . . . 150 De ter mine Fac tored Mo ments . . . . . . . . . . . . . . . . 151 De ter mine Re quired Flex ural Re in force ment . . . . . . . . 151De sign Beam Shear Re in force ment . . . . . . . . . . . . . . . . 157

CHAP TER IX De sign for NZS 3101-95 161De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . 164Strength Re duc tion Fac tors . . . . . . . . . . . . . . . . . . . . . . 164Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . 165Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . 167 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 167 De ter mine Mo ment Mag ni fi ca tion Fac tors . . . . . . . . . 168 Dy namic Mo ment Mag ni fi ca tion . . . . . . . . . . . . . . 170 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . 170De sign Col umn Shear Re in force ment. . . . . . . . . . . . . . . 171 De ter mine Sec tion Forces . . . . . . . . . . . . . . . . . . 172 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 173 De ter mine Re quired Shear Re in force ment . . . . . . . . . 175


CHAP TER X De sign for UBC 97 193De sign Load Com bi na tions . . . . . . . . . . . . . . . . . . . . . . 196Strength Re duc tion Fac tors . . . . . . . . . . . . . . . . . . . . . . 197Col umn De sign . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

iv


Gen er a tion of Bi axial In ter ac tion Sur faces . . . . . . . . . . . . 198Check Col umn Ca pac ity . . . . . . . . . . . . . . . . . . . . . 200 De ter mine Fac tored Mo ments and Forces . . . . . . . . . . 200 De ter mine Mo ment Mag ni fi ca tion Fac tors . . . . . . . . . 200 De ter mine Ca pac ity Ra tio . . . . . . . . . . . . . . . . . . 202De sign Col umn Shear Re in force ment. . . . . . . . . . . . . . . 203 De ter mine Sec tion Forces . . . . . . . . . . . . . . . . . . 203 De ter mine Con crete Shear Ca pac ity . . . . . . . . . . . . . 205 De ter mine Re quired Shear Re in force ment . . . . . . . . . 206


De sign of Joints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218De ter mine the Panel Zone Shear Force . . . . . . . . . . . . . . 219De ter mine the Ef fec tive Area of Joint . . . . . . . . . . . . . . 220Check Panel Zone Shear Stress . . . . . . . . . . . . . . . . . . 220

Beam/Col umn Flex ural Ca pac ity Ra tios . . . . . . . . . . . . . . . . 221

CHAP TER XI De sign Out put 225Over view. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225Graph i cal Dis play of De sign In for ma tion . . . . . . . . . . . . . . . 226Tab u lar Dis play of De sign Out put . . . . . . . . . . . . . . . . . . . 228Mem ber Spe cific In for ma tion . . . . . . . . . . . . . . . . . . . . . 230

Ref er ences 233

In dex 237

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Table of Contents

C h a p t e r I

Introduction

OverviewThe pro gram fea tures pow er ful and com pletely in te grated mod ules for de sign ofboth steel and re in forced con crete struc tures (CSI 2005a, b). The pro gram pro videsthe user with op tions to cre ate, mod ify, ana lyze and de sign struc tural mod els, allfrom within the same user in ter face.

Note: Through out this man ual, use of the term pro gram re fers to ei ther ETABS

In te grated Build ing De sign Soft ware or SAP2000 Three Di men sional Static andDy namic Anal y sis and De sign Soft ware, un less oth er wise noted.

The pro gram pro vides an in ter ac tive en vi ron ment in which the user can study thestress con di tions, make ap pro pri ate changes, such as mem ber size re vi sions, andup date the de sign with out re- analyzing the struc ture. A sin gle mouse click on anele ment brings up de tailed de sign in for ma tion. Mem bers can be grouped to getherfor de sign pur poses. The out put in both graphi cal and tabu lated for mats can beread ily dis played and printed.

The pro gram is struc tured to sup port a wide va ri ety of de sign codes for the au to -mated de sign and check of con crete frame mem bers. This doc u men ta tion in cludesthe fol low ing de sign codes: U.S. (ACI 1999, UBC 97), Ca na dian (CSA 1994), Brit -ish (BSI 1997, BSI 1989), Eu ro pean (CEN 1992), New Zea land (NZS 3101-95)and CP 65 of Sin ga pore.

Overview 1

The de sign is based upon a set of user- specified load ing com bi na tions. How ever,the pro gram pro vides a set of de fault load com bi na tions for each de sign code sup -ported in program. If the de fault load com bi na tions are ac cept able, no defi ni tion ofad di tional load com bi na tions is re quired.

In the de sign of the col umns, the pro gram cal cu lates the re quired lon gi tu di nal andshear re in force ment. How ever the user may spec ify the lon gi tu di nal steel, in whichcase a col umn ca pac ity ra tio is re ported. The col umn ca pac ity ra tio gives an in di ca -tion of the stress con di tion with re spect to the ca pac ity of the col umn.

Every beam mem ber is de signed for flex ure and shear at a user-de fined number ofsta tions along the beam span. Tor sion de sign is also avail able for ACI codes.

The pres en ta tion of the out put is clear and con cise. The in for ma tion is in a form that al lows the en gi neer to take ap pro pri ate re me dial meas ures in the event of mem berover stress. Backup de sign in for ma tion pro duced by the pro gram is also pro videdfor con ven ient veri fi ca tion of the re sults.

Eng lish as well as SI and MKS met ric units can be used to de fine the model ge ome -try and to spec ify de sign pa rame ters.

OrganizationThis man ual is or ga nized as follows:

Chap ter II out lines vari ous as pects of the con crete de sign pro ce dures of the pro -gram. This chap ter de scribes the com mon ter mi nol ogy of con crete de sign as im ple -mented in the pro gram.

Each of eight sub se quent chap ters gives a de tailed de scrip tion of a spe cific code ofprac tice as in ter preted by and im ple mented in the pro gram. Each chap ter de scribesthe de sign load ing com bi na tion, col umn and beam de sign pro ce dures, and otherspe cial con sid era tion re quired by the code.

Chap ter III gives a de tailed de scrip tion of the ACI code (ACI 1999) as im ple -mented in the pro gram.

Chap ter IV gives a de tailed de scrip tion of the Brit ish code (BSI 1997) as im ple -mented in the pro gram.

Chap ter V gives a de tailed de scrip tion of the Brit ish code (BSI 1989) as im ple -mented in the pro gram.

2 Organization


Chap ter VI gives a de tailed de scrip tion of the Ca na dian code (CSA 1994) as im ple -mented in the pro gram.

Chap ter VII gives a de tailed de scrip tion of the Singapore code (CP 1999) as im ple -mented in the pro gram.

Chap ter VIII gives a de tailed de scrip tion of the Eurocode 2 (CEN 1992) as im ple -mented in the pro gram.

Chap ter IX gives a de tailed de scrip tion of the New Zea land code (NZS 1997) as im -ple mented in the pro gram.

Chap ter X gives a de tailed de scrip tion of the Uni form Build ing code (UBC 1997)as im ple mented in the pro gram.

Chap ter XI out lines vari ous as pects of the tabu lar and graphi cal out put from thepro gram re lated to con crete de sign.

Recommended ReadingIt is rec om mended that the user read Chap ter II De sign Process and one of the tensub se quent chap ters cor re spond ing to the code of in ter est. Additionally user should read De sign Out put in Chap ter XI to gain an un der stand ing of pro gram out put re -lated to con crete de sign.

A tu to rial man ual that pro vides step-by-step guid ance through an ex am ple pro jectis pro vided with the pro gram. It is rec om mended that first-time us ers per form thesteps of the tu to rial be fore read ing this man ual.

Recommended Reading 3

Chapter I Introduction

C h a p t e r II

Design Process

This chap ter out lines vari ous as pects of the con crete de sign and design- check pro -ce dures that are used by the pro gram. The con crete de sign and check may be per -formed us ing the pro gram in ac cor dance with one of the fol low ing de sign codes:

The 2005 Amer i can Con crete In sti tute Build ing Code Re quire ments for Struc -tural Con crete, ACI 318-05 (ACI 2005).



The 1997 Brit ish Stan dards In sti tu tion Struc tural Use of Con crete, BS 8110-97(BSI 1997).

The 1985 Brit ish Stan dards In sti tu tion Struc tural Use of Con crete, BS 8110-85R1989 (BSI 1989).

The 1994 Ca na dian Stan dards As so cia tion De sign of Con crete Struc tures forBuild ings, CSA- A23.3-94 (CSA 1994).

The 1999 SPRING Sin ga pore De sign and con struc tion of Struc tural use ofCon crete, CP 65 1999 (CP 1999).

The 2004 Eu ro pean Com mit tee for Stan dard iza tion, De sign of Con crete Struc -tures, EUROCODE 2 (BS EN 2004).

5

The 1992 Euro pean Com mit tee for Stan dardi za tion, De sign of Con crete Struc -tures, EUROCODE 2 (CEN 1992).

The 1995 Stan dards New Zea land Con crete Struc tures Stan dard, NZS 3101- 95(NZS 1995).

International Con fer ence of Build ing Of fi cials 1997 Uni form Build ing Code:Vol ume 2: Struc tural En gi neer ing De sign Pro vi sions, Chap ter 19 "Con crete",UBC 1997 (ICBO 1997).

De tails of the process as so ci ated with each of these codes as im ple mented in thepro gram are de scribed in the sub se quent chap ters. This chap ter pro vides a back -ground com mon to all of the de sign codes.

In writ ing this man ual it has been as sumed that the user has an en gi neer ing back -ground in the gen eral area of struc tural re in forced con crete de sign and fa mili ar itywith at least one of the above-men tioned de sign codes.

For re fer ring to per ti nent sec tions of the cor re spond ing codes, a unique pre fix is as -signed for each code. For ex am ple, all ref er ences to the ACI code are pre ceded bythe word ACI. Simi larly,

Ref er ences to the ACI 318-05, ACI 318-02 and ACI 318-99 code have the pre -fix of ACI

Ref er ences to the Brit ish code carry the pre fix of BS

Ref er ences to the Ca na dian code carry the pre fix of CSA

Ref er ences to the Eurocode 2 carry the pre fix of EC2

Ref er ences to the New Zea land code carry the pre fix of NZS

Ref er ences to the Singapore code carry the pre fix of CP

Ref er ences to the UBC 1997 code have the pre fix of UBC

Design Load CombinationsThe de sign load com bi na tions are used to determine the vari ous com bi na tions ofthe load cases for which the struc ture is to be de signed/checked. The load com bi na -tion fac tors to be used vary with the se lected de sign code. The load com bi na tionfac tors are ap plied to the forces and mo ments ob tained from the as so ci ated loadcases (or anal y sis cases for SAP2000) and are then summed to ob tain the fac toredde sign forces and mo ments for the load com bi na tion.

6 Design Load Combinations


For multi- valued load com bi na tions in volv ing re sponse spec trum, time his tory,mov ing loads (only ap pli ca ble for SAP2000) and multi- valued com bi na tions (oftype en vel op ing, square- root of the sum of the squares or ab so lute) where any cor -re spon dence be tween in ter act ing quan ti ties is lost, the pro gram auto mati cally pro -duces mul ti ple sub com bi na tions us ing maxima/min ima per mu ta tions of in ter act -ing quan ti ties. Sepa rate com bi na tions with nega tive fac tors for re sponse spec trumcases are not re quired be cause the pro gram auto mati cally takes the min ima to bethe nega tive of the maxima for re sponse spec trum cases and the above-de scribedper mu ta tions gen er ate the re quired sub com bi na tions.

When a de sign com bi na tion in volves only a sin gle multi-val ued case of time his -tory or mov ing load, fur ther op tions are avail able. The pro gram has an op tion to re -quest that time his tory com bi na tions pro duce sub com bi na tions for each time stepof the time his tory. Also an op tion is avail able to re quest that mov ing load com bi -na tions pro duce sub com bi na tions us ing max ima and min ima of each de sign quan -tity but with cor re spond ing val ues of in ter act ing quan ti ties.

For nor mal load ing con di tions in volv ing static dead load, live load, wind load, andearth quake load or dy namic re sponse spec trum earth quake load, the pro gram hasbuilt- in de fault load ing com bi na tions for each de sign code. These are based on thecode rec om men da tions and are docu mented for each code in the cor re spond ingchap ters.

For other load ing con di tions in volv ing mov ing load, time his tory, pat tern liveloads, sepa rate con sid era tion of roof live load, snow load, and so on., the user mustde fine de sign load ing com bi na tions ei ther in lieu of or in ad di tion to the de fault de -sign load ing com bi na tions.

The de fault load com bi na tions as sume all static load cases de clared as dead load tobe ad di tive. Simi larly, all cases de clared as live load are as sumed ad di tive. How -ever, any static load case de clared as wind or earth quake, and any re sponse spec -trum cases, are as sumed to be non-ad di tive with each other and pro duce mul ti plelat eral load com bi na tions. Also wind and static earth quake cases pro duce sepa rateload ing com bi na tions with the sense (posi tive or nega tive) re versed. If these con di -tions are not cor rect, the user must pro vide the ap pro pri ate de sign com bi na tions.

The de fault load com bi na tions are in cluded in de sign if the user re quests them to bein cluded or if no other user-de fined com bi na tion is avail able for con crete de sign. Ifany de fault com bi na tion is in cluded in de sign, then all de fault com bi na tions willauto mati cally be up dated by the pro gram any time the de sign code is changed or ifstatic or re sponse spec trum load cases are modi fied.

Design Load Combinations 7

Chapter II Design Process

Live load re duc tion fac tors can be ap plied to the mem ber forces of the live load case on an element- by- element ba sis to re duce the con tri bu tion of the live load to thefac tored load ing.

The user is cau tioned that if mov ing load or time his tory re sults are not re quested to be re cov ered in the analy sis for some or all of the frame mem bers, the ef fects ofthese loads will be as sumed to be zero in any com bi na tion that in cludes them.

Design and Check StationsFor each load com bi na tion, each ele ment is de signed or checked at a number of lo -ca tions along the length of the ele ment. The lo ca tions are based on equally spacedseg ments along the clear length of the ele ment. The number of seg ments in an ele -ment is re quested by the user be fore the analy sis is performed. The user can re finethe de sign along the length of an ele ment by re quest ing more seg ments.

In ETABS only, when us ing 1997 UBC de sign codes, re quire ments for joint de sign at the beam-to-col umn con nec tions are eval u ated at the top most sta tion of eachcol umn. The pro gram also per forms a joint shear anal y sis at the same sta tion to de -ter mine if spe cial con sid er ations are re quired in any of the joint panel zones. Thera tio is re ported for the beam flex ural ca pac i ties with re spect to the col umn flex uralca pac i ties con sid er ing ax ial force ef fects as so ci ated with the weak beam-strongcol umn as pect of any beam/col umn in ter sec tion.

Identifying Beams and ColumnsIn the pro gram all beams and col umns are rep re sented as frame ob jects. But de signof beams and col umns re quires sep a rate treat ment. Iden ti fi ca tion for a con crete ob -jects is ac com plished by spec i fy ing that the frame sec tion as signed to the ob ject isof beam or col umn type . If there is any brace ob ject in the frame, the brace el e mentwould also be iden ti fied as ei ther a beam or a col umn type based on the sec tion as -signed to the brace object.

Design of BeamsIn the de sign of con crete beams, in gen eral, the pro gram cal cu lates and re ports there quired ar eas of steel for flex ure and shear based upon the beam mo ments, shears,load com bi na tion fac tors, and other cri te ria that are de scribed in de tail in thecode-spe cific chap ters. The re in force ment re quire ments are cal cu lated at a user- defined number of sta tions along the beam span.

8 Design and Check Stations


All of the beams are de signed only for ma jor di rec tion flex ure, shear and tor sion(tor sion is only ap pli ca ble for ACI de sign code). Ef fects due to any ax ial forces, mi -nor di rec tion bend ing, and tor sion (ex cept for ACI de sign code) that may ex ist inthe beams must be in ves ti gated in de pend ently by the user.

In de sign ing the flex ural re in force ment for the ma jor mo ment at a par ticu lar sec tion of a par ticu lar beam, the steps in volve the de ter mi na tion of the maxi mum fac toredmo ments and the de ter mi na tion of the re in forc ing steel. The beam sec tion is de -signed for the maxi mum posi tive M u

+ and maxi mum nega tive M u- fac tored mo ment

en ve lopes ob tained from all of the load com bi na tions. Nega tive beam mo mentspro duce top steel. In such cases the beam is al ways de signed as a rec tan gu lar sec -tion. Posi tive beam mo ments pro duce bot tom steel. In such cases the beam may bede signed as a rect an gu lar or a T beam. For the de sign of flex ural re in force ment, thebeam is first de signed as a sin gly re in forced beam. If the beam sec tion is not ad e -quate, the re quired com pres sion re in force ment is cal cu lated.

In de sign ing the shear re in force ment for a par tic u lar beam for a par tic u lar set ofload ing com bi na tions at a par tic u lar sta tion due to the beam ma jor shear, thesteps in volve the de ter mi na tion of the fac tored shear force, the de ter mi na tion ofthe shear force that can be re sisted by con crete, and the de ter mi na tion of the re in -force ment steel re quired to carry the bal ance.

Spe cial con sid era tions for seis mic de sign are in cor po rated in ETABS only for ACI, Ca na dian, and New Zea land codes.

Design of ColumnsIn the de sign of the col umns, the pro gram cal cu lates the re quired lon gi tu di nal steel,or if the lon gi tu di nal steel is spec i fied, the col umn stress con di tion is re ported interms of a col umn ca pac ity ra tio, which is a fac tor that gives an in di ca tion of thestress con di tion of the col umn with re spect to the ca pac ity of the col umn. The de -sign pro ce dure for the re in forced con crete col umns of the struc ture in volves the fol -low ing steps:

Design of Columns 9


Gen er ate ax ial force-bi axial mo ment in ter ac tion sur faces for all of the dif fer ent con crete sec tion types of the model. A typ i cal in ter ac tion sur face is shown inFig ure II-1.

Check the ca pac ity of each col umn for the fac tored ax ial force and bend ing mo -ments ob tained from each load ing com bi na tion at each end of the col umn. Thisstep is also used to cal cu late the re quired re in force ment (if none was spec i fied)that will pro duce a ca pac ity ra tio of 1.0.

De sign the col umn shear re in force ment

The gen er a tion of the in ter ac tion sur face is based on the as sumed strain and stressdis tri bu tions and some other sim pli fy ing as sump tions. These stress and strain dis-

10 Design of Columns


Figure II-1A Typical Column Interaction Surface

tributions and the as sump tions vary from code to code. A typ i cal as sumed straindis tri bu tion is de scribed in Fig ure II-2.

Here max i mum com pres sion strain is lim ited to c . For most of the de sign codes,this as sumed dis tri bu tion re mains valid. How ever, the value of c var ies from codeto code. For ex am ple, c = 0.003 for ACI, UBC and New Zea land codes, and c = 0.0035 for Ca na dian, Brit ish and Eu ro pean codes. The de tails of the gen er a tionof in ter ac tion sur faces dif fer from code to code. These are de scribed in the chap tersspe cific to the code.

Design of Columns 11


Figure II-2Idealized Strain Distribution for generation of Interaction Surfaces

A typ i cal in ter ac tion sur face is shown in Fig ure II-1. The col umn ca pac ity in ter ac -tion vol ume is nu mer i cally de scribed by a se ries of dis crete points that are gen er -ated on the three-di men sional in ter ac tion fail ure sur face. The co or di nates of thesepoints are de ter mined by ro tat ing a plane of lin ear strain in three di men sions on thesec tion of the col umn as de scribed in Fig ure II-2.

The area as so ci ated with each rebar is placed at the ac tual lo ca tion of the cen ter ofthe bar and the al go rithm does not as sume any sim pli fi ca tions in the man ner inwhich the area of steel is dis trib uted over the cross-sec tion of the col umn. The in ter -ac tion al go rithm pro vides cor rec tions to ac count for the con crete area that is dis -placed by the re in forc ing in the com pres sion zone.

The ef fects of code-spec i fied strength re duc tion fac tors and max i mum limit on theax ial ca pac ity are in cor po rated in the in ter ac tion sur faces. The for mu la tion is basedcon sis tently upon the gen eral prin ci ples of ul ti mate strength de sign, and al lows forrect an gu lar, square or cir cu lar, dou bly sym met ric col umn sec tions. In ad di tion toax ial com pres sion and bi axial bend ing, the for mu la tion al lows for ax ial ten sion and bi axial bend ing con sid er ations, as shown in Fig ure II-1.

The ca pac ity check is based on whether the de sign load points lie in side the in ter ac -tion vol ume in a force space, as shown in Fig ure II-3 . If the point lies in side the vol -ume, the col umn ca pac ity is ad e quate, and vice versa. The point in the in ter ac tionvol ume (P, M x , and M y ) which is rep re sented by point L is placed in the in ter ac -tion space as shown in Fig ure II-3 . If the point lies within the in ter ac tion vol ume,the col umn ca pac ity is ad e quate; how ever, if the point lies out side the in ter ac tionvol ume, the col umn is over stressed. As a mea sure of the stress con di tion of the col -umn, a ca pac ity ra tio is cal cu lated. This ra tio is achieved by plot ting the point L, de -fined by P, Mx and My, and de ter min ing the lo ca tion of point C. The point C is de -fined as the point where the line OL (if ex tended out wards) will in ter sect the fail uresur face. This point is de ter mined by three-di men sional lin ear in ter po la tion be tween the points that de fine the fail ure sur face. The ca pac ity ra tio, CR, is given by the ra -tio OL OC .

If OL = OC (or CR=1) the point lies on the in ter ac tion sur face and the col umn is stressed to ca pac ity.

If OL < OC (or CR OC (or CR>1) the point lies out side the in ter ac tion vol ume and the col -umn is over stressed.



The ca pac ity ra tio is ba si cally a fac tor that gives an in di ca tion of the stress con di -tion of the col umn with re spect to the ca pac ity of the col umn. In other words, if theax ial force and bi axial mo ment set for which the col umn is be ing checked is am pli -fied by di vid ing it by the re ported ca pac ity ra tio, the point de fined by the re sult ingax ial force and bi axial mo ment set will lie on the fail ure (or in ter ac tion vol ume)sur face.

The shear re in force ment de sign pro ce dure for col umns is very sim i lar to that forbeams, ex cept that the ef fect of the ax ial force on the con crete shear ca pac ity needsto be con sid ered.

For cer tain spe cial seis mic cases, the de sign of col umn for shear is based on the ca -pac ity-shear. The ca pac ity-shear force in a par tic u lar di rec tion is cal cu lated fromthe mo ment ca pac i ties of the col umn as so ci ated with the fac tored ax ial force act ing

Design of Columns 13


Figure II-3Geometric Representation of Column Capacity Ratio

on the col umn. For each load com bi na tion, the fac tored ax ial load is cal cu lated, us -ing the anal y sis load cases and the cor re spond ing load com bi na tion fac tors. Then,the mo ment ca pac ity of the col umn in a par tic u lar di rec tion un der the in flu ence ofthe ax ial force is cal cu lated, us ing the uni ax ial in ter ac tion di a gram in the cor re -spond ing di rec tion as shown in Fig ure II-4.



Figure II-4Moment Capacity M u at a Given Axial Load Pu

Design of Joints This sec tion is ap pli ca ble to ETABS only.

To en sure that the beam-col umn joint of spe cial mo ment re sist ing frames pos sesses ad e quate shear strength, the pro gram per forms a ra tio nal anal y sis of the beam-col -umn panel zone to de ter mine the shear forces that are gen er ated in the joint. Thepro gram then checks this against de sign shear strength.

Only joints hav ing a col umn be low the joint are de signed. The ma te rial prop er tiesof the joint are as sumed to be the same as those of the col umn be low the joint.

The joint anal y sis is performed in the ma jor and the mi nor di rec tions of the col umn. The joint de sign pro ce dure in volves the fol low ing steps:

De ter mine the panel zone de sign shear force, Vuh

De ter mine the ef fec tive area of the joint

Check panel zone shear stress

The fol low ing three sec tions de scribe in de tail the al go rithms as so ci ated with thesesteps.

De ter mine the Panel Zone Shear Force

For a par tic u lar col umn di rec tion, ma jor or mi nor, the free body stress con di tion ofa typ i cal beam-col umn in ter sec tion is shown in Fig ure II-5.

The force Vuh is the hor i zon tal panel zone shear force that is to be cal cu lated. The

forces that act on the joint are Pu , Vu , M uL and M u

R . The forces Pu and Vu are ax ial

force and shear force, re spec tively, from the col umn fram ing into the top of thejoint. The mo ments M u

L and M uR are ob tained from the beams fram ing into the

joint. The joint shear force Vuh is cal cu lated by re solv ing the mo ments into C and T

forces. Not ing that T CL L= and T CR R= ,

V = T + T - Vuh

L R u

The lo ca tion of C or T forces is de ter mined by the di rec tion of the mo ment. Themag ni tude of C or T forces is con ser va tively de ter mined us ing ba sic prin ci ples oful ti mate strength the ory, ig nor ing com pres sion re in force ment as fol lows. The max -i mum com pres sion, Cmax , and the max i mum mo ment, M max , that can be car ried bythe beam is cal cu lated first.

Design of Joints 15


C = max 0.85 f bdc

M = Cmax maxd

2

Then the C and T forces are con ser va tively de ter mined as fol lows:

16 Design of Joints


Fig ure II-5Beam- Column Joint Analy sis

( )C = T = C max

max

1 1

abs M

M

The mo ments and the C and T forces from beams that frame into the joint in a di rec -tion that is not par al lel to the ma jor or mi nor di rec tions of the col umn are re solvedalong the di rec tion that is be ing in ves ti gated, thereby con trib ut ing force com po -nents to the anal y sis. Also C and T are cal cu lated for the pos i tive and neg a tive mo -ments con sid er ing the fact that the con crete cover may be dif fer ent for the di rec tionof mo ment.

In the de sign of spe cial mo ment re sist ing con crete frames, the eval u a tion of the de -sign shear force is based upon the mo ment ca pac i ties (with re in forc ing steeloverstrength fac tor, , and no fac tors) of the beams fram ing into the joint,(ACI 21.5.1.1, UBC 1921.5.1.1). The C and T force are based upon these mo mentca pac i ties. The col umn shear force Vu is cal cu lated from the beam mo ment ca pac i -ties as fol lows:

V = M + MHu

uL

uR

See Fig ure II-6. It should be noted that the points of in flec tion shown on Fig ure II-6are taken as mid way be tween ac tual lat eral sup port points for the col umns. If thereis no col umn at the top of the joint, the shear force from the top of the col umn istaken as zero.

The ef fects of load re ver sals, as il lus trated in Case 1 and Case 2 of Fig ure II-5, arein ves ti gated and the de sign is based upon the max i mum of the joint shears ob tainedfrom the two case

Design of Joints 17


De ter mine the Ef fec tive Area of Joint

The joint area that re sists the shear forces is as sumed al ways to be rect an gu lar inplan view. The di men sions of the rect an gle cor re spond to the ma jor and mi nor di -men sions of the col umn be low the joint, ex cept if the beam fram ing into the joint isvery nar row. The ef fec tive width of the joint area to be used in the cal cu la tion islim ited to the width of the beam plus the depth of the col umn. The area of the joint

18 Design of Joints


Figure II-6Col umn Shear Force, Vu

is as sumed not to ex ceed the area of the col umn be low. The joint area for joint shear along the ma jor and mi nor di rec tions is cal cu lated sep a rately (ACI R21.5.3).

It should be noted that if the beam frames into the joint ec cen tri cally, the above as -sump tions may be unconservative and the user should in ves ti gate the ac cept abil ityof the par tic u lar joint.

Check Panel Zone Shear Stress

The panel zone shear stress is eval u ated by di vid ing the shear force Vuh by the ef fec -

tive area of the joint and com par ing it with the fol low ing de sign shear strengths(ACI 21.5.3, UBC 1921.5.3) :

v

fc=

20 for joints confined on all four sides,15

,

fc , for joints confined on three faces or on two opposite faces,

12 for all other joints, fc

,

where = 0.85 (by de fault). (ACI 9.3.2.3, UBC 1909.3.2.3, 1909.3.4.1)

A beam that frames into a face of a col umn at the joint is con sid ered in program topro vide con fine ment to the joint if at least three-quar ters of the face of the joint iscov ered by the fram ing mem ber (ACI 21.5.3.1, UBC 1921.5.3.1).

For light-weight ag gre gate con crete, the de sign shear strength of the joint is re -duced in pro gram to at least three-quar ters of that of the nor mal weight con crete byre plac ing the fc

with

{ }min ,,f f fcs factor c c 3 4 (ACI 21.5.3.2, UBC 1921.5.3.2)For joint de sign, the pro gram re ports the joint shear, the joint shear stress, the al -low able joint shear stress and a ca pac ity ra tio.

Design of Joints 19


Beam/Column Flexural Capacity RatiosThis sec tion is ap pli ca ble to ETABS only.

At a par tic u lar joint for a par tic u lar col umn di rec tion, ma jor or mi nor, the pro gramwill cal cu late the ra tio of the sum of the beam mo ment ca pac i ties to the sum of thecol umn mo ment ca pac i ties, (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 ca pac i ties are cal cu lated with no re in forc ing overstrength fac tor, , and in -clud ing fac tors. The beam ca pac i ties are cal cu lated for re versed sit u a tions (Cases1 and 2) as il lus trated in Fig ure II-5 and the max i mum sum ma tion ob tained is used.

The mo ment ca pac i ties of beams that frame into the joint in a di rec tion that is notpar al lel to the ma jor or mi nor di rec tion of the col umn are re solved along the di rec -tion that is be ing in ves ti gated and the re solved com po nents are added to the sum -ma tion.

The col umn ca pac ity sum ma tion in cludes the col umn above and the col umn be lowthe joint. For each load com bi na tion the ax ial force, Pu , in each of the col umns iscal cu lated from the ETABS anal y sis load com bi na tions. For each load com bi na -tion, the mo ment ca pac ity of each col umn un der the in flu ence of the cor re spond ingax ial load Pu is then de ter mined sep a rately for the ma jor and mi nor di rec tions of the col umn, us ing the uni ax ial col umn in ter ac tion di a gram: see Fig ure II-4. The mo -ment ca pac i ties of the two col umns are added to give the ca pac ity sum ma tion forthe cor re spond ing load com bi na tion. The max i mum ca pac ity sum ma tions ob tained from all of the load com bi na tions are used for the beam/col umn ca pac ity ra tio.

The beam/col umn flex ural ca pac ity ra tios are re ported for Spe cial Mo ment-Re sist -ing Frames in volv ing seis mic de sign load com bi na tions only.

P- EffectsThe pro gram de sign process re quires that the analy sis re sults in clude the P- ef -fects. The P- ef fects are con sid ered dif fer ently for braced or non sway andun braced or sway com po nents of mo ments in col umns or frames. For thebraced mo ments in col umns, the ef fect of P- is lim ited to "in di vid ual mem ber sta -bil ity." For un braced com po nents,"lat eral drift ef fects" should be con sid ered in ad -di tion to in di vid ual mem ber sta bil ity ef fect. In the pro gram, it is as sumed thatbraced or non sway mo ments are con trib uted from the dead or live loads.

20 Beam/Column Flexural Capacity Ratios


Whereas, un braced or sway mo ments are con trib uted from all other types ofloads.

For the in di vid ual mem ber sta bil ity ef fects, the mo ments are mag ni fied using mo -ment mag ni fi ca tion fac tors, as in the ACI, Ca na dian, and New Zea land codes orusing ad di tional mo ments, as in the Brit ish and Euro pean codes.

For lat eral drift ef fects, the pro gram as sumes that the P- analy sis is per formedand that the am pli fi ca tion is al ready in cluded in the re sults. The mo ments andforces ob tained from P- analy sis are fur ther am pli fied for in di vid ual col umn sta -bil ity ef fect if re quired by the gov ern ing code, as in the ACI, Ca na dian, and NewZea land codes.

Us ers should be aware that the de fault analy sis op tion in the pro gram is turned OFFfor P- ef fect. The user can turn the P- analy sis ON and set the maxi mum numberof it era tions for the analy sis. The de fault number of it era tion for P- analy sis is 1.For more in for ma tion, re fer to the CSI Analy sis Ref er ence man ual (CSI 2005c).

Element Unsupported LengthsTo ac count for col umn slen der ness ef fects, the col umn un sup ported lengths are re -quired. The two un sup ported lengths are l33 and l22 . These are the lengths be tweensup port points of the ele ment in the cor re spond ing di rec tions. The length l33 cor re -sponds to in sta bil ity about the 3-3 axis (ma jor axis), and l22 cor re sponds to in sta bil -ity about the 2-2 axis (mi nor axis).

Normally, the un sup ported el e ment length is equal to the length of the el e ment, i.e.,the dis tance be tween END-I and END-J of the el e ment. See Fig ure II-7. The pro -gram, how ever, al lows us ers to as sign sev eral el e ments to be treated as a sin glemem ber for de sign. This can be accomplished dif fer ently for ma jor and mi norbend ing. There fore, ex tra ne ous joints, as shown in Fig ure II-8, that af fect the un -sup ported length of an el e ment are au to mat i cally taken into con sid er ation.

In de ter min ing the val ues for l22 and l33 of the el e ments, the pro gram rec og nizesvar i ous as pects of the struc ture that have an ef fect on these lengths, such as mem ber con nec tiv ity, di a phragm con straints and sup port points. The pro gram au to mat i -cally lo cates the el e ment sup port points and eval u ates the cor re spond ing un sup -ported el e ment length.

There fore, the un sup ported length of a col umn may ac tu ally be eval u ated as be inggreater than the cor re spond ing el e ment length. If the beam frames into only one di -rec tion of the col umn, the beam is as sumed to give lat eral sup port only in that di rec -tion.

Element Unsupported Lengths 21


The user has op tions to spec ify the un sup ported lengths of the el e ments on an el e -ment-by-el e ment ba sis.

22 Element Unsupported Lengths


l33

l22

Eleme

nt Axis

ENDI

ENDJ

3

2

1

Figure II-7Axes of Bending and Unsupported Length

Figure II-8Unsupported Lengths and Interior Nodes

Special Considerations for Seismic LoadsThe ACI code im poses a spe cial duc til ity re quire ment for frames in seis mic re gions by spec i fy ing frames as Or di nary, In ter me di ate, or Spe cial mo ment re sist ingframes. The Spe cial mo ment re sist ing frame can pro vide the re quired duc til ity anden ergy dis si pa tion in the non lin ear range of cy clic de for ma tion. The UBC code re -quires that the con crete frame must be in Zone 1, Zone 2, Zone 3, or Zone 4, whereZone 4 is des ig nated as the zone of se vere earth quake. The Ca na dian code re quiresthat the con crete frame must be de signed as ei ther an Or di nary, Nom i nal, or Duc tile mo ment re sist ing frame. The New Zea land code also re quires that the con creteframe must be de signed as ei ther an Or di nary, Elas ti cally re spond ing, frames withLim ited duc til ity, or Duc tile mo ment re sist ing frame.

Un like the ACI, Ca na dian, and New Zea land codes, the cur rent im ple men ta tion ofthe Brit ish code and the Eurocode 2 in the pro gram does not ac count for any spe cialre quire ments for seis mic de sign.

Choice of Input UnitsEng lish as well as SI and MKS met ric units can be used for in put. But the codes arebased on a spe cific sys tem of units. All equa tions and de scrip tions pre sented in thesub se quent chap ters cor re spond to that spe cific sys tem of units un less oth er wisenoted. For ex am ple, the ACI code is pub lished in inch- pound- second units. By de -fault, all equa tions and de scrip tions pre sented in the chap ter "De sign for ACI318-99" cor re spond to inch- pound- second units. How ever, any sys tem of units canbe used to de fine and de sign the struc ture in the pro gram.

Special Considerations for Seismic Loads 23


C h a p t e r III

Design for ACI 318-99

This chap ter de scribes in de tail the vari ous as pects of the con crete de sign pro ce dure that is used by the pro gram when the user se lects the ACI 318-99 De sign Code(ACI 1999). Vari ous no ta tions used in this chap ter are listed in Table III-1.

The de sign is based on user- specified load ing com bi na tions. The pro gram pro vides a set of de fault load com bi na tions that should sat isfy re quire ments for the de sign of most build ing type struc tures.

The pro gram pro vides op tions to de sign or check Or di nary, In ter me di ate (mod er ate seis mic risk ar eas), and Spe cial (high seis mic risk ar eas) mo ment re sist ing framesas re quired for seis mic de sign pro vi sions. The de tails of the de sign cri te ria used forthe dif fer ent fram ing sys tems are de scribed in the fol low ing sec tions.

Eng lish as well as SI and MKS met ric units can be used for in put. The code is basedon Inch- Pound- Second units. For sim plic ity, all equa tions and de scrip tions pre -sented in this chap ter cor re spond to Inch- Pound- Second units un less oth er wisenoted.

Design Load CombinationsThe de sign load com bi na tions are the vari ous com bi na tions of the pre scribed loadcases for which the struc ture is to be checked. For the ACI 318-99 code, if a struc -


26 Design Load Combinations


Acv Area of con crete used to de ter mine shear stress, sq- inAg Gross area of con crete, sq- inA s Area of ten sion re in force ment, sq- inA s

Area of compression re in force ment, sq- in

A s required( ) Area of steel re quired for ten sion re in force ment, sq- inA st To tal area of col umn lon gi tu di nal re in force ment, sq- inAv Area of shear re in force ment, sq- ina Depth of com pres sion block, inab Depth of com pres sion block at bal anced condition, inb Width of mem ber, inbf Ef fec tive width of flange (T- Beam sec tion), inbw Width of web (T- Beam sec tion), inCm Co ef fi cient, de pend ent upon col umn cur va ture, used to cal cu late mo -

ment mag ni fi ca tion fac tor

c Depth to neu tral axis, incb Depth to neu tral axis at bal anced con di tions, ind Dis tance from com pres sion face to ten sion re in force ment, ind Con crete cover to cen ter of re in forc ing, ind s Thick ness of slab (T- Beam sec tion), inEc Modu lus of elas tic ity of con crete, psiE s Modu lus of elas tic ity of re in force ment, as sumed as 29,000,000 psifc

Speci fied com pres sive strength of con crete, psi

fy Speci fied yield strength of flex ural re in force ment, psifys Speci fied yield strength of shear re in force ment, psih Di men sion of col umn, inIg Mo ment of in er tia of gross con crete sec tion about cen troi dal axis,

ne glect ing re in force ment, in4

I se Mo ment of in er tia of re in force ment about cen troi dal axis ofmem ber cross sec tion, in4

k Ef fec tive length factorL Clear un sup ported length, in

Table III-1List of Symbols Used in the ACI code


Chapter III Design for ACI 318-99

M1 Smaller fac tored end mo ment in a col umn, lb- inM 2 Larger fac tored end mo ment in a col umn, lb- inM c Factored mo ment to be used in design, lb- inM ns Non sway com po nent of fac tored end mo ment, lb-inM s Sway com po nent of fac tored end mo ment, lb-inM u Fac tored mo ment at section, lb- inM ux Fac tored mo ment at sec tion about X-axis, lb- inM uy Fac tored mo ment at sec tion about Y-axis, lb- inPb Ax ial load ca pac ity at bal anced strain con di tions, lbPc Criti cal buck ling strength of col umn, lbPmax Maxi mum ax ial load strength al lowed, lbP0 Ax ial load ca pac ity at zero ec cen tric ity, lbPu Fac tored ax ial load at sec tion, lbr Ra dius of gy ra tion of col umn sec tion, inVc Shear re sisted by con crete, lbVE Shear force caused by earth quake loads, lbVD L+ Shear force from span load ing, lbVu Fac tored shear force at a sec tion, lbVp Shear force com puted from prob able mo ment ca pac ity, lb Re in forc ing steel over strength fac tor1 Fac tor for ob tain ing depth of com pres sion block in con creted Ab so lute value of ra tio of maxi mum fac tored ax ial dead load to maxi -

mum fac tored ax ial to tal load s Mo ment mag ni fi ca tion fac tor for sway mo ments ns Mo ment mag ni fi ca tion fac tor for nonsway moments c Strain in con crete s Strain in re in forc ing steel Strength re duc tion fac tor

Table III-1List of Symbols Used in the ACI code (continued)

ture is sub jected to dead load (DL) and live load (LL) only, the stress check mayneed only one load com bi na tion, namely 1.4 DL + 1.7 LL (ACI 9.2.1). How ever, inad di tion to the dead and live loads, if the struc ture is sub jected to wind (WL) andearth quake (EL) loads, and con sid er ing that wind and earth quake forces are re -versi ble, then the fol low ing load com bi na tions must be con sid ered (ACI 9.2).

1.4 DL1.4 DL + 1.7 LL (ACI 9.2.1)

0.9 DL 1.3 WL 0.75 (1.4 DL + 1.7 LL 1.7 WL) (ACI 9.2.2)

0.9 DL 1.3 * 1.1 EL 0.75 (1.4 DL + 1.7 LL 1.7 * 1.1 EL) (ACI 9.2.3)

These are also the de fault de sign load com bi na tions in the pro gram when ever theACI 318-99 code is used.

Live load re duc tion fac tors can be ap plied to the mem ber forces of the live loadcon di tion on an element- by- element ba sis to re duce the con tri bu tion of the live load to the fac tored load ing.

Strength Reduction FactorsThe strength re duc tion fac tors, , are ap plied on the nom i nal strength to ob tain thede sign strength pro vided by a mem ber. The fac tors for flex ure, ax ial force, shear, and tor sion are as fol lows:

= 0.90 for flex ure, (ACI 9.3.2.1)

= 0.90 for ax ial ten sion, (ACI 9.3.2.2)

= 0.90 for ax ial ten sion and flex ure, (ACI 9.3.2.2)

= 0.75 for ax ial com pres sion, and ax ial com pres sion and flex ure (spi rally re in forced col umn), (ACI 9.3.2.2)

= 0.70 for ax ial com pres sion, and ax ial com pres sion and flex ure (tied col umn), and (ACI 9.3.2.2)

= 0.85 for shear and tor sion. (ACI 9.3.2.3)

28 Strength Reduction Factors


Column DesignThe user may de fine the ge ome try of the re in forc ing bar con figu ra tion of each con -crete col umn sec tion. If the area of re in forc ing is pro vided by the user, the pro gramchecks the col umn ca pac ity. How ever, if the area of re in forc ing is not pro vided bythe user, the pro gram cal cu lates the amount of re in forc ing re quired for the col umn.The de sign pro ce dure for the re in forced con crete col umns of the struc ture in volvesthe fol low ing steps:

Gen er ate ax ial force/bi axial mo ment in ter ac tion sur faces for all of the dif fer entcon crete sec tion types of the model. A typ i cal bi axial in ter ac tion sur face isshown in Fig ure II-1. When the steel is un de fined, the pro gram gen er ates thein ter ac tion sur faces for the range of al low able re in force ment 1 to 8 per centfor Or di nary and In ter me di ate mo ment re sist ing frames (ACI 10.9.1) and 1 to 6 per cent for Spe cial mo ment re sist ing frames (ACI 21.4.3.1).

Cal cu late the ca pac ity ra tio or the re quired re in forc ing area for the fac tored ax -ial force and bi ax ial (or uni ax ial) bend ing mo ments ob tained from each load ing com bi na tion at each sta tion of the col umn. The tar get ca pac ity ra tio is taken asone when cal cu lat ing the re quired re in forc ing area.

De sign the col umn shear re in force ment.

The fol low ing three sub sec tions de scribe in de tail the al go rithms as so ci ated withthese steps.

Generation of Biaxial Interaction Surfaces

The col umn ca pac ity in ter ac tion vol ume is nu meri cally de scribed by a se ries of dis -crete points that are gen er ated on the three- dimensional in ter ac tion fail ure sur face.In ad di tion to ax ial com pres sion and bi ax ial bend ing, the for mu la tion al lows for ax -ial ten sion and bi ax ial bend ing con sid era tions. A typi cal in ter ac tion dia gram isshown in Fig ure II-1.

The co or di nates of those points are de ter mined by ro tat ing a plane of lin ear strain in three di men sions on the sec tion of the col umn. See Fig ure II-2. The lin ear straindia gram lim its the maxi mum con crete strain, c , at the ex trem ity of the sec tionto 0.003 (ACI 10.2.3).

The for mu la tion is based con sis tently on the gen eral prin ci ples of ul ti mate strengthde sign (ACI 10.3), and al lows for any dou bly sym met ric rec tan gu lar, square, or cir -cu lar col umn sec tion.

Column Design 29


The stress in the steel is given by the prod uct of the steel strain and the steel modu -lus of elas tic ity, s sE , and is lim ited to the yield stress of the steel, fy (ACI 10.2.4).The area as so ci ated with each re in forc ing bar is as sumed to be placed at the ac tuallo ca tion of the cen ter of the bar and the al go rithm does not as sume any fur ther sim -pli fi ca tions in the man ner in which the area of steel is dis trib uted over the cross sec -tion of the col umn, such as an equiva lent steel tube or cyl in der. See Figure III-1.

The con crete com pres sion stress block is as sumed to be rec tan gu lar, with a stressvalue of 0.85 fc

(ACI 10.2.7.1). See Figure III-1. The in ter ac tion al go rithm pro -

vides cor rec tion to ac count for the con crete area that is dis placed by the re in force -ment in the com pres sion zone.

The ef fects of the strength re duc tion fac tor, , are in cluded in the gen era tion of thein ter ac tion sur faces. The maxi mum com pres sive ax ial load is lim ited to Pn(max) ,where

P = f A - A + f Ac g st y stn(max) 0.85 [0.85 ( ) ] spi ral col umn, (ACI 10.3.5.1)

P = f A - A f Ac g st y stn(max) 0.80 [ 0.85 ( ) + ] tied col umn, (ACI 10.3.5.2)

= 0.70 for tied col umns, and

30 Column Design


Figure III-1Idealization of Stress and Strain Distribution in a Column Section

= 0.75 for spi rally re in forced col umns.

The value of used in the in ter ac tion di a gram var ies from min to 0.9 based on theax ial load. For low val ues of ax ial load, is in creased lin early from min to 0.9 asthe ax ial load de creases from the smaller of Pb or 0.1 f Ac g

to zero, where Pb is the

ax ial force at the bal anced con di tion. In cases in volv ing ax ial ten sion, is al ways0.9 (ACI 9.3.2.2).

Check Column Capacity

The col umn ca pac ity is checked for each load ing com bi na tion at each check sta tionof each col umn. In check ing a par ticu lar col umn for a par ticu lar load ing com bi na -tion at a par ticu lar sta tion, the pro gram uses the fol low ing steps:

De ter mine the fac tored mo ments and forces from the analy sis load cases andthe speci fied load com bi na tion fac tors to give P M Mu ux uy, ,and .

De ter mine the mo ment mag ni fi ca tion fac tors for the col umn mo ments.

Ap ply the mo ment mag ni fi ca tion fac tors to the fac tored mo ments. De ter minewhether the point, de fined by the re sult ing ax ial load and bi ax ial mo ment set,lies within the in ter ac tion vol ume.

The fac tored mo ments and cor re spond ing mag ni fi ca tion fac tors de pend on theiden ti fi ca tion of the in di vid ual col umn as ei ther sway or non-sway.


Determine Factored Moments and Forces

The fac tored loads for a par ticu lar load com bi na tion are ob tained by ap ply ing thecor re spond ing load fac tors to all the load cases, giv ing P M Mu ux uy, ,and . The fac -tored mo ments are fur ther in creased for nonsway col umns, if re quired, to ob tainmini mum ec cen trici ties of (0.6 0.03+ h) inches, where h is the di men sion of thecol umn in the cor re spond ing di rec tion (ACI 10.12.3.2).

Determine Moment Magnification Factors

The mo ment mag ni fi ca tion fac tors are cal cu lated sepa rately for sway (over all sta -bil ity ef fect), s and for nonsway (in di vid ual col umn sta bil ity ef fect), ns . Also themo ment mag ni fi ca tion fac tors in the ma jor and mi nor di rec tions are in gen eral dif -fer ent.

Column Design 31


The pro gram as sumes that a P- anal y sis has been per formed in the pro gram and,there fore, mo ment mag ni fi ca tion fac tors for mo ments caus ing sidesway are takenas unity (ACI 10.10.2). For the P- anal y sis the load should cor re spond to a loadcom bi na tion of 0.75 (1.4 dead load + 1.7 live load)/ , where is the understrengthfac tor for sta bil ity which is taken as 0.75 (ACI 10.12.3). See also White and Hajjar(1991).

The mo ment ob tained from analy sis is sepa rated into two com po nents: the sway ( )M s and the nonsway (M ns ) com po nents. The nonsway com po nents, which areiden ti fied by ns sub scripts, are pre domi nantly caused by grav ity load. The swaycom po nents are iden ti fied by the s sub scripts. The sway mo ments are pre domi -nantly caused by lat eral loads, and are re lated to the cause of sidesway.

For in di vid ual col umns or column- members in a floor, the mag ni fied mo mentsabout two axes at any sta tion of a col umn can be ob tained as

M M Mns s s= + . (ACI 10.13.3)

The fac tor s is the mo ment mag ni fi ca tion fac tor for mo ments caus ing side sway.The mo ment mag ni fi ca tion fac tors for sway mo ments, s , is taken as 1 be cause thecom po nent mo ments M s and M ns are ob tained from a sec ond or der elas tic (P-)analy sis (ACI R10.13).

The com puted mo ments are fur ther am pli fied for in di vid ual col umn sta bil ity ef fect(ACI 10.13.5) by the non sway mo ment mag ni fi ca tion fac tor, ns , as fol lows:

M Mc ns= 2 , where (ACI 10.12.3)

M c is the fac tored mo ment to be used in de sign, and

M 2 is the larger fac tored and am pli fied end mo ment.

The nonsway mo ment mag ni fi ca tion fac tor, ns , as so ci ated with the ma jor or mi -nor di rec tion of the col umn is given by (ACI 10.12.3)

ns mu

c

C

P

P

=

0.75

1.01

, where

P = EI

klc

u

pi 2

2( ) ,

k is con ser va tively taken as 1, how ever the pro gram al lows the user to over ridethis value, and

32 Column Design


EI is as so ci ated with a par ticu lar col umn di rec tion given by:

EI = E I

+ c g

d

0.41 ,

d = maximum factored axial dead loadmaximum factored axial total load , and

C = + M

Mm

a

b

0.6 0.4 0.4 . (ACI 10.12.3.1)

M a and M b are the mo ments at the ends of the col umn, and M b is nu meri callylarger than M a . M Ma b is posi tive for sin gle cur va ture bend ing and nega tivefor dou ble cur va ture bend ing. The pre ced ing ex pres sion of Cm is valid if thereis no trans verse load ap plied be tween the sup ports. If trans verse load is pres enton the span, or the length is over writ ten, or for any other case, Cm = 1 . Cm canbe over writ ten by the user on an el e ment-by-el e ment ba sis.

The mag ni fi ca tion fac tor, ns , must be a posi tive number and greater than one.There fore, Pu must be less than 0.75Pc . If Pu is found to be greater than or equal to 0.75Pc , a fail ure con di tion is de clared.

The pre ced ing cal cu la tions use the un sup ported length of the col umn. The two un -sup ported lengths are l22 and l33 , cor re spond ing to in sta bil ity in the mi nor and ma -jor di rec tions of the ele ment, re spec tively. See Figure II-7. These are the lengthsbe tween the sup port points of the ele ment in the cor re spond ing di rec tions.

If the pro gram as sump tions are not sat is fac tory for a par ticu lar mem ber, the usercan ex plic itly spec ify val ues of s nsand .

Determine Capacity Ratio

As a meas ure of the stress con di tion of the col umn, a ca pac ity ra tio is cal cu lated.The ca pac ity ra tio is ba si cally a fac tor that gives an in di ca tion of the stress con di -tion of the col umn with re spect to the ca pac ity of the col umn.

Be fore en ter ing the in ter ac tion dia gram to check the col umn ca pac ity, the mo mentmag ni fi ca tion fac tors are ap plied to the fac tored loads to ob tain P M Mu ux uy, , and .The point (P M Mu ux uy, , ) is then placed in the in ter ac tion space shown as point L inFig ure II-3. If the point lies within the in ter ac tion vol ume, the col umn ca pac ity isade quate; how ever, if the point lies out side the in ter ac tion vol ume, the col umn isover stressed.

Column Design 33


This ca pac ity ra tio is achieved by plot ting the point L and de ter min ing the lo ca tionof point C. The point C is de fined as the point where the line OL (if ex tended out -wards) will in ter sect the fail ure sur face. This point is de ter mined by three- dimensional lin ear in ter po la tion be tween the points that de fine the fail ure sur face.

See Fig ure II-3. The ca pac ity ra tio, CR, is given by the ra tio OLOC

.

If OL = OC (or CR=1) the point lies on the in ter ac tion sur face and the col umn is stressed to ca pac ity.

If OL < OC (or CR OC (or CR>1) the point lies out side the in ter ac tion vol ume and the col -umn is over stressed.

The maxi mum of all the val ues of CR cal cu lated from each load com bi na tion is re -ported for each check sta tion of the col umn along with the con trol ling, P M Mu ux uy, , and set and as so ci ated load com bi na tion number.

If the re in forc ing area is not de fined, the pro gram com putes the re in force ment thatwill give an in ter ac tion ra tio of unity.

Design Column Shear Reinforcement

The shear re in force ment is de signed for each load ing com bi na tion in the ma jor andmi nor di rec tions of the col umn. In de sign ing the shear re in forc ing for a par ticu larcol umn for a par ticu lar load ing com bi na tion due to shear forces in a par ticu lar di -rec tion, the pro gram uses the fol low ing steps:

De ter mine the fac tored forces act ing on the sec tion, Pu and Vu . Note that Pu isneeded for the cal cu la tion of Vc .

De ter mine the shear force, Vc , that can be re sisted by con crete alone.

Cal cu late the re in force ment steel re quired to carry the bal ance.

For Spe cial and In ter me di ate mo ment re sist ing frames (duc tile frames), the shearde sign of the col umns is also based upon the prob able and nomi nal mo ment ca paci -ties of the mem bers, re spec tively, in ad di tion to the fac tored mo ments. Ef fects ofthe ax ial forces on the col umn mo ment ca paci ties are in cluded in the for mu la tion.


34 Column Design


Determine Section Forces

In the de sign of the col umn shear re in force ment of an Or di nary mo ment re -sist ing con crete frame, the forces for a par ticu lar load com bi na tion, namely,the col umn ax ial force, Pu , and the col umn shear force, Vu , in a par ticu lar di rec -tion are ob tained by fac tor ing the pro gram analy sis load cases with the cor re -spond ing load com bi na tion fac tors.

In the shear de sign of Spe cial mo ment re sist ing frames (seis mic de sign), thefol low ing are checked in ad di tion to the re quire ment for the Or di nary mo mentre sist ing frames. In the de sign of Spe cial mo ment re sist ing con crete frames, the de sign shear force in a col umn, Vu , in a par ticu lar di rec tion is also cal cu latedfrom the prob able mo ment ca paci ties of the col umn as so ci ated with the fac -tored ax ial force act ing on the col umn.

For each load com bi na tion, the fac tored ax ial load, Pu , is cal cu lated. Then, theposi tive and nega tive mo ment ca paci ties, M u

+ and M u , of the col umn in a par -

ticu lar di rec tion un der the in flu ence of the ax ial force Pu are cal cu lated us ingthe uni ax ial in ter ac tion dia gram in the cor re spond ing di rec tion. The de signshear force, Vu , is then given by (ACI 21.4.5.1)

V V + Vu p D+ L= (ACI 21.4.5.1)

where, Vp is the shear force ob tained by ap ply ing the cal cu lated prob able ul ti -mate mo ment ca paci ties at the two ends of the col umn act ing in two op po sitedi rec tions. There fore, Vp is the maxi mum of VP1 and VP2 , where

V = M + M

LP

I-

J+

1 , and

V = M + M

LP

I+

J-

2, where

M I+ , M I

= Posi tive and nega tive mo ment ca paci ties at end I of

the col umn us ing a steel yield stress value of fy and no fac tors ( =1.0),

M J+ , M J

= Pos i tive and neg a tive mo ment ca pac i ties at end J of

the col umn us ing a steel yield stress value of fy and no fac tors ( =1.0), and

L = Clear span of col umn.

Column Design 35


For Spe cial mo ment re sist ing frames, is taken as 1.25 (ACI R21.4.5.1). VD L+is the con tri bu tion of shear force from the in- span dis tri bu tion of grav ity loads.For most of the col umns, it is zero.

For In ter me di ate mo ment re sist ing frames, the shear ca pac ity of the col umnis also checked for the de sign nom i nal shear based on the nom i nal mo ment ca -pac i ties at the ends and the fac tored grav ity loads, in ad di tion to the check re -quired for Or di nary mo ment re sist ing frames. The de sign shear force is taken to be the min i mum of that based on the nom i nal ( =1.0) mo ment ca pac ity andfac tored shear force. The pro ce dure for cal cu lat ing nom i nal mo ment ca pac ityis the same as that for com put ing the prob a ble mo ment ca pac ity for Spe cial mo -ment re sist ing frames, ex cept that is taken equal to 1 rather than 1.25 (ACIR21.10). The fac tored shear forces are based on the spec i fied load fac tors, ex -cept the earth quake load fac tors are dou bled (ACI 21.10.3).

Determine Concrete Shear Capacity

Given the de sign force set Pu and Vu , the shear force car ried by the con crete, Vc , iscal cu lated as fol lows:

If the col umn is sub jected to ax ial com pres sion, i.e., Pu is posi tive,

V = f + P

A Ac c

u

g

cv2 12000

, (ACI 11.3.1.2)

where,

fc 100 psi, and (ACI 11.1.2)

V f + P

A Ac c

u

g

cv

3.5 1500

. (ACI 11.3.2.2)

The term P

Au

g

must have psi units. Acv is the ef fec tive shear area, which is shown

shaded in Figure III-2.

If the col umn is sub jected to ax ial ten sion, Pu is nega tive,

V = f + P

A A c c

u

g

cv2 1500

0

(ACI 11.3.2.3)

36 Column Design


For Spe cial mo ment re sist ing con crete frame de sign, Vc is set to zero if thefac tored ax ial com pres sive force, Pu , in clud ing the earth quake ef fect, is small ( )P f A /u c g max , com pres sion re in force ment is re quired (ACI 10.3.3) and is cal cu -lated as fol lows:

The com pres sive force de vel oped in con crete alone is given by

C f bac=0.85 max , and (ACI 10.2.7.1)

the mo ment re sisted by con crete com pres sion and ten sile steel is

M C da

uc =

max

2 .

There fore the mo ment re sisted by com pres sion steel and ten sile steel is

M M Mus u uc= .

So the re quired com pres sion steel is given by

AM

f d ds

us

s

= ( )

, where

f Ec d

cs s =

0.003 . (ACI 10.2.4)

The re quired ten sile steel for bal anc ing the com pres sion in con crete is

AM

f da

suc

y

1

2

=

max , and

the ten sile steel for bal anc ing the com pres sion in steel is given by

Beam Design 41


AM

f d ds

us

y

2 = ( )

.

There fore, the to tal ten sile re in force ment A A As s s= +1 2 , and to tal com -pres sion re in force ment is A s

. A s is to be placed at bot tom and A s is to be

placed at top if M u is posi tive, and vice versa if M u is nega tive.

Design for T-Beam

In de sign ing for a fac tored nega tive mo ment, M u , (i.e., de sign ing top steel), the cal -cu la tion of the steel area is ex actly the same as de scribed for a Rect an gu lar beam,i.e., no T- Beam data is to be used. See Figure III-4. If M u > 0 , the depth of the com -pres sion block is given by

a d d M

f bu

c f

=

2 2

0.85 .

The maxi mum al lowed depth of the com pres sion block is given by

a cbmax = 0.751 . If a d s , the sub se quent cal cu la tions for A s are ex actly the same as pre vi -

ously de fined for the rect an gu lar sec tion de sign. How ever, in this case thewidth of(ACI 10.2.7.1)

42 Beam Design


the com pres sion flange is taken as the width of the beam for analy sis. Whethercom pres sion re in force ment is re quired de pends on whether a a> max .

If a d s> , cal cu la tion for A s is accomplished in two parts. The first part is forbal anc ing the com pres sive force from the flange, C f , and the sec ond part is forbal anc ing the com pres sive force from the web, Cw , as shown in Figure III-4. C fis given by

C f b b df c f w s= 0.85 ( ) .

There fore, A = C

fs

f

y

1 and the por tion of M u that is re sisted by the flange is

given by

M = C d d

uf fs

2 .

Again, the value for is 0.90. There fore, the bal ance of the mo ment, M u to becar ried by the web is given by

M = M Muw u uf .

Beam Design 43


Figure III-4Design of a T-Beam Section

The web is a rec tan gu lar sec tion of di men sions bw and d, for which the de signdepth of the com pres sion block is re cal cu lated as

a d dM

f buw

c w

12 2=

0.85 .

If a a1 max , the area of ten sile steel re in force ment is given by

A M

f d a

suw

y

2

1

2

=

, and

A A As s s= +1 2 .

This steel is to be placed at the bot tom of the T- beam.

If a a1 > max , com pres sion re in force ment is re quired (ACI 10.3.3) and iscal cu lated as fol lows:

The com pres sive force in web con crete alone is given by

C f bac=0.85 max . (ACI 10.2.7.1)

There fore the mo ment re sisted by con crete web and ten sile steel is

M C da

uc =

max

2 , and

the mo ment re sisted by com pres sion steel and ten sile steel is

M M Mus uw uc= .

There fore, the com pres sion steel is com puted as

AM

f d ds

us

s

= ( )

, where

f Ec d

cs s =

0.003 . (ACI 10.2.4)

The ten sile steel for bal anc ing com pres sion in web con crete is

44 Beam Design


AM

f da

suc

y

2

2

=

max , and

the ten sile steel for bal anc ing com pres sion in steel is

AM

f d ds

us

y

3 = ( )

.

The to tal ten sile re in force ment A A A As s s s= + +1 2 3 , and to tal com pres -sion re in force ment is A s

. A s is to be placed at bot tom and A s is to be

placed at top.

Minimum Tensile Reinforcement

The mini mum flex ural ten sile steel pro vided in a rec tan gu lar sec tion in an Or di nary mo ment re sist ing frame is given by the mini mum of the two fol low ing lim its:

Af

fb d

fb ds

c

y

w

y

w

max and3 200

or (ACI 10.5.1)

A As s required43

( ). (ACI 10.5.3)

Special Consideration for Seismic Design

For Spe cial mo ment re sist ing con crete frames (seis mic de sign), the beam de signsat is fies the fol low ing ad di tional con di tions (see also Table III-2 for com pre hen -sive list ing) :

The mini mum lon gi tu di nal re in force ment shall be pro vided at both the top andbot tom. Any of the top and bot tom re in force ment shall not be less than A s min( )(ACI 21.3.2.1).

Af

fb d

fb dc

y

w

y

ws(min)

max and3 200

or (ACI 10.5.1)

A A s requireds(min) 43

( ) . (ACI 10.5.3)

The beam flex ural steel is lim ited to a maxi mum given by

Beam Design 45


A b ds w 0.025 . (ACI 21.3.2.1)

At any end (sup port) of the beam, the beam pos i tive mo ment ca pac ity (i.e., as -so ci ated with the bot tom steel) would not be less than one-half of the beam neg -a tive mo ment ca pac ity (i.e., as so ci ated with the top steel) at that end (ACI21.3.2.2).

Nei ther the nega tive mo ment ca pac ity nor the posi tive mo ment ca pac ity at anyof the sec tions within the beam would be less than one-quarter of the maxi mum of posi tive or nega tive mo ment ca paci ties of any of the beam end (sup port) sta -tions (ACI 21.3.2.2).

For In ter me di ate mo ment re sist ing con crete frames (seis mic de sign), the beam de -sign would sat isfy the fol low ing con di tions:

At any sup port of the beam, the beam posi tive mo ment ca pac ity would not beless than one-third of the beam nega tive mo ment ca pac ity at that end (ACI21.10.4.1).

Nei ther the nega tive mo ment ca pac ity nor the posi tive mo ment ca pac ity at anyof the sec tions within the beam would be less than one-fifth of the maxi mum ofposi tive or nega tive mo ment ca paci ties of any of the beam end (sup port) sta -tions (ACI 21.10.4.1).

Design Beam Shear Reinforcement

The shear re in force ment is de signed for each load com bi na tion at a user-de finednumber of sta tions along the beam span. In de sign ing the shear re in force ment for apar ticu lar beam for a par ticu lar load ing com bi na tion at a par ticu lar sta tion due tothe beam ma jor shear, the pro gram uses the fol low ing steps:

De ter mine the fac tored shear force, Vu .

De ter mine the shear force, Vc , that can be re sisted by the con crete.

De ter mine the re in force ment steel re quired to carry the bal ance.

For Spe cial and In ter me di ate mo ment re sist ing frames (duc tile frames), the shearde sign of the beams is also based on the prob able and nomi nal mo ment ca paci tiesof the mem bers, re spec tively, in ad di tion to the fac tored load de sign.


46 Beam Design


Determine Shear Force and Moment

In the de sign of the beam shear re in force ment of an Or di nary mo ment re sist -ing con crete frame, the shear forces and mo ments for a par ticu lar load com bi -na tion at a par ticu lar beam sec tion are ob tained by fac tor ing the as so ci atedshear forces and mo ments with the cor re spond ing load com bi na tion fac tors.

In the de sign of Spe cial mo ment re sist ing con crete frames (seis mic de sign),the shear ca pac ity of the beam is also checked for the prob a ble shear due to theprob a ble mo ment ca pac i ties at the ends and the fac tored grav ity load. Thischeck is per formed in ad di tion to the de sign check re quired for Or di nary mo -ment re sist ing frames. The shear force, Vu , is cal cu lated from the prob a ble mo -ment ca pac i ties of each end of the beam and the grav ity shear forces. The pro -ce dure for cal cu lat ing the de sign shear force in a beam from prob a ble mo mentca pac ity is the same as that de scribed for a col umn in the Col umn De sign sec -tion on page 35. See also Table III-2 for de tails.

The de sign shear force Vu is then given by (ACI 21.3.4.1)

V V + Vu p D+ L= (ACI 21.3.4.1)

where, Vp is the shear force ob tained by ap ply ing the cal cu lated prob able ul ti -mate mo ment ca paci ties at the two ends of the beams act ing in two op po site di -rec tions. There fore, Vp is the maxi mum of VP1 and VP2 , where

V = M + M

LP

I-

J+

1 , and

V = M + M

LP

I+

J-

2 , where

M I = Mo ment ca pac ity at end I, with top steel

in ten sion, us ing a steel yield stress value of fy and no fac tors ( =1.0),

M J+ = Mo ment ca pac ity at end J, with bot tom

steel in ten sion, us ing a steel yield stress value of fy and no fac tors ( =1.0),

M I+ = Mo ment ca pac ity at end I, with bot tom

steel in ten sion, us ing a steel yield stress value of fy and no fac tors ( =1.0),

Beam Design 47


MJ = Mo ment ca pac ity at end J, with top steel

in ten sion, us ing a steel yield stress value of fy and no fac tors ( =1.0), and

L = Clear span of beam.

For Spe cial mo ment re sist ing frames, is taken as 1.25 (ACI R21.3.4.1). VD L+is the con tri bu tion of shear force from the in- span dis tri bu tion of grav ity loads.

For In ter me di ate mo ment re sist ing frames, the shear ca pac ity of the beam isalso checked for the de sign nom i nal shear based on the nom i nal mo mentcapacities at the ends and the fac tored grav ity loads, in ad di tion to the check re -quired for Or di nary mo ment re sist ing frames. The de sign shear force in beamsis taken to be the min i mum of that based on the nom i nal mo ment ca pac ity andfac tored shear force. The pro ce dure for cal cu lat ing nom i nal ( =1.0) mo mentca pac ity is the same as that for com put ing the prob a ble mo ment ca pac ity forSpe cial mo ment re sist ing frames, ex cept that is taken equal to 1 rather than1.25 (ACI R21.10). The fac tored shear forces are based on the spec i fied loadfac tors, ex cept the earth quake load fac tors are dou bled (ACI 21.10.3). Thecom pu ta tion of the de sign shear force in a beam of an In ter me di ate mo mentre sist ing frame is also the same as that for col umns, which is de scribed ear -lier on page 36. See also Table III-2 for de tails.

Determine Concrete Shear Capacity

The al low able con crete shear ca pac ity is given by

V = f b dc c w2 . (ACI 11.3.1.1)

For Spe cial mo ment re sist ing frame con crete de sign, Vc is set to zero if both the fac -tored ax ial com pres sive force, in clud ing the earth quake ef fect Pu , is less than f A /c g

20, and the shear force con tri bu tion from earth quake VE is more than half of

the to tal maxi mum shear force over the length of the mem ber Vu (i.e. V VE u 0.5 )(ACI 21.3.4.2).

Determine Required Shear Reinforcement

Given V Vu cand , the re quired shear re in force ment in area/unit length is cal cu latedas

A V / V s

f dv

u c

ys

=( )

. (ACI 11.5.6.2)

48 Beam Design


The shear force re sisted by steel is lim ited by

( )V / V f bdu c c 8 , (ACI 11.5.6.9)

where, , the strength re duc tion fac tor, is 0.85 (ACI 9.3.2.3). The maxi mum of allthe cal cu lated Av val ues, ob tained from each load com bi na tion, is re ported alongwith the con trol ling shear force and as so ci ated load com bi na tion number.

The beam shear re in force ment re quire ments dis played by the pro gram are basedpurely upon shear strength con sid era tions. Any mini mum stir rup re quire ments tosat isfy spac ing and volu met ric con sid era tions must be in ves ti gated in de pend entlyof the pro gram by the user.

Beam Design 49


Beam Design 50


Type ofCheck/Design

Ordinary Moment

Resisting Frames(non-Seismic)

Intermediate Moment

Resisting Frames (Seismic)

Special Moment

Resisting Frames (Seismic)

ColumnCheck(interaction)

NLDa Combinations NLDa Combinations NLDa Combinations

ColumnDesign(Interaction)

NLDa Combinations 1% < < 8%

NLD

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