Con crete Frame
De sign Man ual
AASHTO 1997, ACI 318-99, BS 8110-97, BS 8110-85R89
CSAA-23.3-94, CP 65-99, Eurocode 2-92
NZS 3101-95 and UBC 97
For SAP2000
®and ETABS
®Berke ley, Cal i for nia ETABS V10
COPYRIGHT
Copy right ã Com put ers and Struc tures, Inc., 1978-2007. All rights re served.
The CSI Logo®, SAP2000®, and ETABS® are reg is tered trade marks of Com put ers and Struc tures, Inc. SAFETM is trade marks of Com put ers and Struc tures, Inc. Watch & LearnTM is a trade mark of Com put ers and Struc tures, Inc.
The com puter pro grams SAP2000® and ETABS® and all as so ci ated doc u men ta tion are pro pri etary and copy righted prod ucts. World wide rights of own er ship rest with Com put ers and Struc tures, Inc. Unlicensed use of these pro grams or re pro duc tion of doc u men ta tion in any form, with out prior writ ten au tho ri za tion from Com put ers and Struc tures, Inc., is ex plic itly pro hib ited.
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Fur ther in for ma tion and cop ies of this doc u men ta tion may be ob tained from:
Com put ers and Struc tures, Inc. 1995 Uni ver sity Av e nue
Berke ley, Cal i for nia 94704 USA Tel: (510) 649-2200
Fax: (510) 649-2299
E-mail: [email protected] (for gen eral ques tions)
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DISCLAIMER
CON SID ER ABLE TIME, EF FORT AND EX PENSE HAVE GONE INTO THE DE VEL OP MENT AND DOCU MEN TA TION 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 DER STANDS THAT NO WAR -RANTY IS EX PRESSED OR IM PLIED BY THE DE VEL OP ERS OR THE DIS TRIBU TORS ON THE AC CU RACY OR THE RE LI ABIL -ITY OF THE PRO GRAMS.
ETABS AND SAP2000 ARE VERY PRAC TI CAL TOOL FOR THE DE SIGN OF RE IN FORCED CON CRETE STRUC TURES. HOW -EVER, THE USER MUST THOR OUGHLY READ THE MAN UAL AND CLEARLY REC OG NIZE THE AS PECTS OF RE IN FORCED CON CRETE DE SIGN THAT THE PRO GRAM AL GO RITHMS DO NOT AD DRESS.
THE USER MUST EX PLIC ITLY UN DER STAND THE AS SUMP TIONS OF THE PRO GRAMS AND MUST IN DE PEND ENTLY VER -IFY THE RE SULTS.
CHAP TER I In tro duc tion 1
Over view . . . 1
Or ga ni za tion . . . 2
Rec om mended Read ing. . . 3
CHAP TER II De sign Pro cess 5 De sign Load Com bi na tions. . . 6
De sign and Check Sta tions . . . 8
Iden ti fy ing Beams and Col umns . . . 8
De sign of Beams . . . 8
De sign of Col umns . . . 9
De sign of Joints . . . 15
De ter mine the Panel Zone Shear Force. . . 15
De ter mine the Ef fec tive Area of Joint . . . 18
Check Panel Zone Shear Stress. . . 19
Beam/Col umn Flex ural Ca pac ity Ra tios . . . 20
P-D Ef fects. . . 20
El e ment Un sup ported Lengths . . . 21
Spe cial Con sid er ations for Seis mic Loads . . . 23
Choice of In put Units . . . 23
CHAP TER III De sign for AASHTO LRFD 1997 25 De sign Load Com bi na tions . . . 28
Strength Re duc tion Fac tors . . . 29
Col umn De sign . . . 29
Check Col umn Ca pac ity . . . 32
De ter mine Fac tored Mo ments and Forces . . . 32
De ter mine Mo ment Mag ni fi ca tion Fac tors . . . 33
De ter mine Ca pac ity Ra tio . . . 34
De sign Col umn Shear Re in force ment . . . 35
De ter mine Sec tion Forces . . . 36
De ter mine Con crete Shear Ca pac ity . . . 37
De ter mine Re quired Shear Re in force ment . . . 37
Beam De sign . . . 41
De sign Beam Flex ural Re in force ment . . . 41
De ter mine Fac tored Mo ments. . . 41
De ter mine Re quired Flex ural Re in force ment . . . 42
De sign Beam Shear Re in force ment . . . 48
De ter mine Shear Force and Mo ment . . . 48
De ter mine Con crete Shear Ca pac ity . . . 49
De ter mine Re quired Shear Re in force ment . . . 49
CHAP TER IV De sign for ACI 318-99 53 De sign Load Com bi na tions . . . 53
Strength Re duc tion Fac tors . . . 56
Col umn De sign . . . 57
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 57
Check Col umn Ca pac ity . . . 59
De ter mine Fac tored Mo ments and Forces . . . 59
De ter mine Mo ment Mag ni fi ca tion Fac tors . . . 59
De ter mine Ca pac ity Ra tio . . . 61
De sign Col umn Shear Re in force ment . . . 62
De ter mine Sec tion Forces . . . 63
De ter mine Con crete Shear Ca pac ity . . . 64
De ter mine Re quired Shear Re in force ment . . . 66
Beam De sign . . . 66
De sign Beam Flex ural Re in force ment . . . 67
De ter mine Fac tored Mo ments. . . 67
De ter mine Re quired Flex ural Re in force ment . . . 67
De sign Beam Shear Re in force ment . . . 74
De ter mine Shear Force and Mo ment . . . 74
De ter mine Con crete Shear Ca pac ity . . . 76
De ter mine Re quired Shear Re in force ment . . . 76
CHAP TER V De sign for BS 8110-97 79 De sign Load Com bi na tions . . . 79
De sign Strength . . . 82
Col umn De sign . . . 82
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 83
De ter mine Fac tored Mo ments and Forces . . . 85
De ter mine Ad di tional Mo ments. . . 85
De ter mine Ca pac ity Ra tio . . . 87
De sign Col umn Shear Re in force ment . . . 88
Beam De sign . . . 89
De sign Beam Flex ural Re in force ment . . . 89
De ter mine Fac tored Mo ments. . . 90
De ter mine Re quired Flex ural Re in force ment . . . 90
De sign Beam Shear Re in force ment . . . 95
CHAP TER VI De sign for BS 8110-85 R1989 97 De sign Load Com bi na tions . . . 97
De sign Strength . . . 100
Col umn De sign. . . 100
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 101
Check Col umn Ca pac ity . . . 102
De ter mine Fac tored Mo ments and Forces . . . 103
De ter mine Ad di tional Mo ments . . . 103
De ter mine Ca pac ity Ra tio . . . 105
De sign Col umn Shear Re in force ment. . . 106
Beam De sign . . . 107
De sign Beam Flex ural Re in force ment . . . 107
De ter mine Fac tored Mo ments . . . 108
De ter mine Re quired Flex ural Re in force ment . . . 108
De sign Beam Shear Re in force ment . . . 113
CHAP TER VII De sign for CSAA23.3-94 115 De sign Load Com bi na tions . . . 118
Strength Re duc tion Fac tors . . . 118
Col umn De sign. . . 119
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 119
Check Col umn Ca pac ity . . . 121
De ter mine Fac tored Mo ments and Forces . . . 121
De ter mine Mo ment Mag ni fi ca tion Fac tors . . . 121
De ter mine Ca pac ity Ra tio . . . 124
De sign Col umn Shear Re in force ment. . . 125
De ter mine Sec tion Forces . . . 125
De ter mine Con crete Shear Ca pac ity . . . 127
De ter mine Re quired Shear Re in force ment . . . 128
Beam De sign . . . 131
De sign Beam Flex ural Re in force ment . . . 131
De ter mine Fac tored Mo ments . . . 131
De ter mine Re quired Flex ural Re in force ment . . . 132
De ter mine Shear Force and Mo ment . . . 139
De ter mine Con crete Shear Ca pac ity . . . 141
De ter mine Re quired Shear Re in force ment . . . 142
CHAP TER VIII De sign for CP 65-1999 145 De sign Load Com bi na tions . . . 145
De sign Strength . . . 148
Col umn De sign. . . 148
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 149
Check Col umn Ca pac ity . . . 150
De ter mine Fac tored Mo ments and Forces . . . 151
De ter mine Ad di tional Mo ments . . . 151
De ter mine Ca pac ity Ra tio . . . 153
De sign Col umn Shear Re in force ment. . . 154
Beam De sign . . . 155
De sign Beam Flex ural Re in force ment . . . 155
De ter mine Fac tored Mo ments . . . 156
De ter mine Re quired Flex ural Re in force ment . . . 156
De sign Beam Shear Re in force ment . . . 161
CHAP TER IX De sign for Eurocode 2-92 163 De sign Load Com bi na tions . . . 163
De sign Strength . . . 166
Col umn De sign. . . 167
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 167
Check Col umn Ca pac ity . . . 169
De ter mine Fac tored Mo ments and Forces . . . 169
De ter mine Code To tal Mo ments . . . 169
De ter mine Ca pac ity Ra tio . . . 171
De sign Col umn Shear Re in force ment. . . 172
Beam De sign . . . 176
De sign Beam Flex ural Re in force ment . . . 176
De ter mine Fac tored Mo ments . . . 177
De ter mine Re quired Flex ural Re in force ment . . . 177
De sign Beam Shear Re in force ment . . . 183
CHAP TER X De sign for NZS 3101-95 187 De sign Load Com bi na tions . . . 190
Strength Re duc tion Fac tors . . . 190
Col umn De sign. . . 191
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 191
Check Col umn Ca pac ity . . . 193
De ter mine Mo ment Mag ni fi ca tion Fac tors . . . 194
Dy namic Mo ment Mag ni fi ca tion . . . 196
De ter mine Ca pac ity Ra tio . . . 196
De sign Col umn Shear Re in force ment. . . 197
De ter mine Sec tion Forces . . . 198
De ter mine Con crete Shear Ca pac ity . . . 199
De ter mine Re quired Shear Re in force ment . . . 201
Beam De sign . . . 204
De sign Beam Flex ural Re in force ment . . . 204
De ter mine Fac tored Mo ments . . . 204
De ter mine Re quired Flex ural Re in force ment . . . 205
De sign Beam Shear Re in force ment . . . 211
De ter mine Shear Force and Mo ment . . . 212
De ter mine Con crete Shear Ca pac ity . . . 213
De ter mine Re quired Shear Re in force ment . . . 214
CHAP TER XI De sign for UBC 97 217 De sign Load Com bi na tions . . . 220
Strength Re duc tion Fac tors . . . 221
Col umn De sign. . . 221
Gen er a tion of Bi axial In ter ac tion Sur faces . . . 222
Check Col umn Ca pac ity . . . 224
De ter mine Fac tored Mo ments and Forces . . . 224
De ter mine Mo ment Mag ni fi ca tion Fac tors . . . 224
De ter mine Ca pac ity Ra tio . . . 226
De sign Col umn Shear Re in force ment. . . 227
De ter mine Sec tion Forces . . . 227
De ter mine Con crete Shear Ca pac ity . . . 229
De ter mine Re quired Shear Re in force ment . . . 230
Beam De sign . . . 231
De sign Beam Flex ural Re in force ment . . . 232
De ter mine Fac tored Mo ments . . . 232
De ter mine Re quired Flex ural Re in force ment . . . 232
De sign Beam Shear Re in force ment . . . 239
De ter mine Shear Force and Mo ment . . . 239
De ter mine Con crete Shear Ca pac ity . . . 241
De ter mine Re quired Shear Re in force ment . . . 241
De sign of Joints . . . 242
De ter mine the Panel Zone Shear Force . . . 242
De ter mine the Ef fec tive Area of Joint . . . 243
Check Panel Zone Shear Stress . . . 244
Beam/Col umn Flex ural Ca pac ity Ra tios . . . 244
CHAP TER XII De sign Out put 249 Over view. . . 249
Graph i cal Dis play of De sign In for ma tion . . . 250 Tab u lar Dis play of De sign Out put . . . 252 Mem ber Spe cific In for ma tion . . . 254
Ref er ences 257
Introduction
Overview
The pro gram fea tures pow er ful and com pletely in te grated mod ules for de sign of both steel and re in forced con crete struc tures (CSI 2005a, b). The pro gram pro vides the user with op tions to cre ate, mod ify, ana lyze and de sign struc tural mod els, all from 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 and Dy 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 the stress con di tions, make ap pro pri ate changes, such as mem ber size re vi sions, and up date the de sign with out re- analyzing the struc ture. A sin gle mouse click on an ele ment brings up de tailed de sign in for ma tion. Mem bers can be grouped to gether for de sign pur poses. The out put in both graphi cal and tabu lated for mats can be read 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 cludes the fol low ing de sign codes: U.S. (AASHTO 1997, 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.
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 of ad 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 and shear re in force ment. How ever the user may spec ify the lon gi tu di nal steel, in which case 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 of sta 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 ber over stress. Backup de sign in for ma tion pro duced by the pro gram is also pro vided for 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.
Organization
This 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 of prac tice as in ter preted by and im ple mented in the pro gram. Each chap ter de scribes the de sign load ing com bi na tion, col umn and beam de sign pro ce dures, and other spe cial con sid era tion re quired by the code.
Chap ter III gives a de tailed de scrip tion of the AASHTO LRFD con crete code (AASHTO 1997) as im ple mented in SAP2000 only.
Chap ter IV gives a de tailed de scrip tion of the ACI code (ACI 1999) as im ple -mented in the pro gram.
Chap ter V gives a de tailed de scrip tion of the Brit ish code (BSI 1997) as im ple -mented in the pro gram.
Chap ter VI gives a de tailed de scrip tion of the Brit ish code (BSI 1989) as im ple -mented in the pro gram.
Chap ter VII 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 VIII gives a de tailed de scrip tion of the Singapore code (CP 1999) as im ple -mented in the pro gram.
Chap ter IX gives a de tailed de scrip tion of the Eurocode 2 (CEN 1992) as im ple -mented in the pro gram.
Chap ter X 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 XI 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 XII out lines vari ous as pects of the tabu lar and graphi cal out put from the pro gram re lated to con crete de sign.
Recommended Reading
It is rec om mended that the user read Chap ter II “De sign Process” and one of the ten sub 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 XII 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 ject is pro vided with the pro gram. It is rec om mended that first-time us ers per form the steps of the tu to rial be fore read ing this man ual.
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 1997 Amer i can As so ci a tion of State High way and Trans por ta tion Of fi -cials AASHTO LRFD Bridge De sign Spec i fi ca tions (SAP2000 only), AASHTO LRFD 1997 (AASHTO 1997).
• 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 2002 Amer i can Con crete In sti tute Build ing Code Re quire ments for Struc -tural Con crete, ACI 318-02 (ACI 2002).
• The 1999 Amer i can Con crete In sti tute Build ing Code Re quire ments for Struc -tural Con crete, ACI 318-99 (ACI 1999).
• 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 8110-85 R1989 (BSI 1989).
• The 1994 Ca na dian Stan dards As so cia tion De sign of Con crete Struc tures for Build ings, CSA- A23.3-94 (CSA 1994).
• The 1999 SPRING Sin ga pore De sign and con struc tion of Struc tural use of Con 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).
• 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 the pro 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 ity with 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 AASHTO code are pre -ceded by the word “AASHTO”. Simi larly,
– Ref er ences to the ACI 31805, ACI 31802 and ACI 31899 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 Combinations
The de sign load com bi na tions are used to determine the vari ous com bi na tions of the 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 -tion fac tors are ap plied to the forces and mo ments ob tained from the as so ci ated load
cases (or anal y sis cases for SAP2000) and are then summed to ob tain the fac tored de sign forces and mo ments for the load com bi na tion.
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 (of type 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 trum cases are not re quired be cause the pro gram auto mati cally takes the min ima to be the nega tive of the maxima for re sponse spec trum cases and the above-de scribed per 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 multival 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 step of 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, and earth quake load or dy namic re sponse spec trum earth quake load, the pro gram has built- in de fault load ing com bi na tions for each de sign code. These are based on the code rec om men da tions and are docu mented for each code in the cor re spond ing chap ters.
For other load ing con di tions in volv ing mov ing load, time his tory, pat tern live loads, sepa rate con sid era tion of roof live load, snow load, and so on., the user must de 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 to be 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 ple lat eral load com bi na tions. Also wind and static earth quake cases pro duce sepa rate load 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 be in cluded or if no other user-de fined com bi na tion is avail able for con crete de sign. If any de fault com bi na tion is in cluded in de sign, then all de fault com bi na tions will
auto mati cally be up dated by the pro gram any time the de sign code is changed or if static or re sponse spec trum load cases are modi fied.
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 the fac 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 of these loads will be as sumed to be zero in any com bi na tion that in cludes them.
Design and Check Stations
For 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 spaced seg 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 fine the 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 each col 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. The ra tio is re ported for the beam flex ural ca pac i ties with re spect to the col umn flex ural ca pac i ties con sid er ing ax ial force ef fects as so ci ated with the weak beam-strong col umn as pect of any beam/col umn in ter sec tion.
Identifying Beams and Columns
In the pro gram all beams and col umns are rep re sented as frame ob jects. But de sign of 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 is of beam or col umn type . If there is any brace ob ject in the frame, the brace el e ment would 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 Beams
In the de sign of con crete beams, in gen eral, the pro gram cal cu lates and re ports the re 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 the
code-spe cific chap ters. The re in force ment re quire ments are cal cu lated at a defined number of sta tions along the beam span.
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 in the 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 tored mo 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 Mu+ and maxi mum nega tive Mu- fac tored mo ment en ve lopes ob tained from all of the load com bi na tions. Nega tive beam mo ments pro 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 be de signed as a rect an gu lar or a T beam. For the de sign of flex ural re in force ment, the beam 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 of load ing com bi na tions at a par tic u lar sta tion due to the beam ma jor shear, the steps in volve the de ter mi na tion of the fac tored shear force, the de ter mi na tion of the 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 Columns
In 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 in terms of a col umn ca pac ity ra tio, which is 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. 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:
• 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 in Fig 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. This step 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 stress dis tri bu tions and some other sim pli fy ing as sump tions. These stress and strain
dis-Figure II-1
tributions and the as sump tions vary from code to code. A typ i cal as sumed strain dis tri bu tion is de scribed in Fig ure II-2.
Here max i mum com pres sion strain is lim ited to ec. For most of the de sign codes, this as sumed dis tri bu tion re mains valid. How ever, the value of ec var ies from code to code. For ex am ple, ec= 0.003 for ACI, UBC and New Zea land codes, and ec= 0.0035 for Ca na dian, Brit ish and Eu ro pean codes. The de tails of the gen er a tion of in ter ac tion sur faces dif fer from code to code. These are de scribed in the chap ters spe cific to the code.
Figure II-2
A typ i cal in ter ac tion sur face is shown in Fig ure II1. 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 these 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 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 of the bar and the al go rithm does not as sume any sim pli fi ca tions in the man ner in which the area of steel is dis trib uted over the crosssec 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 the ax ial ca pac ity are in cor po rated in the in ter ac tion sur faces. The for mu la tion is based con sis tently upon the gen eral prin ci ples of ul ti mate strength de sign, and al lows for rect an gu lar, square or cir cu lar, dou bly sym met ric col umn sec tions. In ad di tion to ax 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 II3 . 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 tion vol ume (P, Mx, and My) 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 tion vol 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 ure sur 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<1) the point lies within the in ter ac tion vol ume and the col -umn ca pac ity is ad e quate.
• If OL > 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 the ax 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 ing ax 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 for beams, ex cept that the ef fect of the ax ial force on the con crete shear ca pac ity needs to 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 from the mo ment ca pac i ties of the col umn as so ci ated with the fac tored ax ial force act ing
Figure II-3
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 of the 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-4
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 beamcol -umn panel zone to de ter mine the shear forces that are gen er ated in the joint. The pro 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 ties of 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, Vu h
• 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 these steps.
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 of a 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, MuL and MuR. 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 the joint. The mo ments MuL and M
u
R 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 TL =CL and TR =CR,
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. The mag ni tude of C or T forces is con ser va tively de ter mined us ing ba sic prin ci ples of ul ti mate strength the ory, ig nor ing com pres sion re in force ment as fol lows. The max -i mum com pres s-ion, Cmax, and the max i mum mo ment, Mmax, that can be car ried by the beam is cal cu lated first.
Cmax = 0.85 f bdc¢
Mmax = Cmax d
2
Then the C and T forces are con ser va tively de ter mined as fol lows: Fig ure II-5
( )
C = T = Cmax max 1- 1 -æ è ç ç ö ø ÷ ÷ abs M MThe 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 solved along 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 tion of 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 steel overstrength fac tor, a, and no j 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 ment ca 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 + M H u u L u R
See Fig ure II-6. It should be noted that the points of in flec tion shown on Fig ure II-6 are taken as mid way be tween ac tual lat eral sup port points for the col umns. If there is no col umn at the top of the joint, the shear force from the top of the col umn is taken 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, are in ves ti gated and the de sign is based upon the max i mum of the joint shears ob tained from the two case
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 in plan 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 is very nar row. The ef fec tive width of the joint area to be used in the cal cu la tion is lim ited to the width of the beam plus the depth of the col umn. The area of the joint
Figure II-6 Col 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 ity of 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
j j
,
fc¢ , for joints confined on three faces or on two opposite faces,
12j fc¢ for all other joints, ì í ï ï î ï ï ,
where j = 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 to pro vide con fine ment to the joint if at least three-quar ters of the face of the joint is cov ered by the fram ing mem ber (ACI 21.5.3.1, UBC 1921.5.3.1).
For lightweight 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 by re plac ing the fc¢ with
{
}
min fcs factor, fc¢, 3 4 fc¢ (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.
Beam/Column Flexural Capacity Ratios
This 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 gram will cal cu late the ra tio of the sum of the beam mo ment ca pac i ties to the sum of the col umn mo ment ca pac i ties, (ACI 21.4.2.2, UBC 1921.4.2.2. CSA 21.5.1.2).
Me Mg
å
³ 6å
5 (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, a , and in -clud ing j fac tors. The beam ca pac i ties are cal cu lated for re versed sit u a tions (Cases 1 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 not par 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 low the joint. For each load com bi na tion the ax ial force, Pu, in each of the col umns is cal 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 ing ax 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 II4. The mo -ment ca pac i ties of the two col umns are added to give the ca pac ity sum ma tion for the 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 mentRe sist -ing Frames in volv -ing seis mic de sign load com bi na tions only.
P-D Effects
The pro gram de sign process re quires that the analy sis re sults in clude the PD ef -fects. The P-D ef fects are con sid ered dif fer ently for “braced” or “non sway” and “un braced” or “sway” com po nents of mo ments in col umns or frames. For the braced mo ments in col umns, the ef fect of PD 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 that “braced” or “non sway” mo ments are con trib uted from the “dead” or “live” loads.
Whereas, “un braced” or “sway” mo ments are con trib uted from all other types of loads.
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, AASHTO, Ca na dian, and New Zea land codes or using 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-D analy sis is per formed and that the am pli fi ca tion is al ready in cluded in the re sults. The mo ments and forces ob tained from PD 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 New Zea land codes.
Us ers should be aware that the de fault analy sis op tion in the pro gram is turned OFF for P-D ef fect. The user can turn the P-D analy sis ON and set the maxi mum number of it era tions for the analy sis. The de fault number of it era tion for P-D 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 Lengths
To 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 tween sup 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 ENDI and ENDJ of the el e ment. See Fig ure II7. The pro -gram, how ever, al lows us ers to as sign sev eral el e ments to be treated as a sin gle mem ber for de sign. This can be accomplished dif fer ently for ma jor and mi nor bend ing. There fore, ex tra ne ous joints, as shown in Fig ure II8, 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 l22and l33of the el e ments, the pro gram rec og nizes var 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 ing greater 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.
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.
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
Special Considerations for Seismic Loads
The 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 ing frames. The Spe cial mo ment re sist ing frame can pro vide the re quired duc til ity and en ergy dis si pa tion in the non lin ear range of cy clic de for ma tion. The AASHTO code re quires that the con crete frame must be in Zone 1, Zone 2, Zone 3, or Zone 4, where Zone 4 is des ig nated as the zone of se vere earth quake. The Ca na dian code re quires that 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 crete frame must be de signed as ei ther an Or di nary, Elas ti cally re spond ing, frames with Lim ited duc til ity, or Duc tile mo ment re sist ing frame.
Un like the ACI, AASHTO, Ca na dian, and New Zea land codes, the cur rent im ple -men ta tion of the Brit ish code and the Eurocode 2 in the pro gram does not ac count for any spe cial re quire ments for seis mic de sign.
Choice of Input Units
Eng lish as well as SI and MKS met ric units can be used for in put. But the codes are based on a spe cific sys tem of units. All equa tions and de scrip tions pre sented in the sub se quent chap ters cor re spond to that spe cific sys tem of units un less oth er wise noted. 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 ACI 318-99" cor re spond to inch- pound- second units. How ever, any sys tem of units can be used to de fine and de sign the struc ture in the pro gram.
Design for AASHTO LRFD 1997
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 SAP2000 only when the user se lects the AASHTO LRFD 1997 De -sign Code (AASHTO 1997). 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. But the pro gram pro -vides a set of de fault load com bi na tions that should help satisfy re quire ments for the de sign of most bridge type struc tures.
SAP2000 pro vides op tions to de sign or check mo ment re sist ing frames of Zones 1 (low seis mic ac tiv ity), 2, 3, and 4 (high seis mic ac tiv ity) as re quired for seis mic de -sign pro vi sions. The de tails of the de -sign cri te ria used for the dif fer ent seis mic zones 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 based on Inch- Kip- Second units. For sim plic ity, all equa tions and de scrip tions pre sented in this chap ter cor re spond to Inch- Kip- Second units un less oth er wise noted.
Acv Area of con crete used to de ter mine shear stress, sq- in
Ag Gross area of con crete, sq- in
As Area of ten sion re in force ment, sq- in
As¢ Area of compression re in force ment, sq- in
As required( ) Area of steel re quired for ten sion re in force ment, sq- in
Ast To tal area of col umn lon gi tu di nal re in force ment, sq- in
Av Area of shear re in force ment, sq- in
a Depth of com pres sion block, in
ab Depth of com pres sion block at bal anced condition, in
amax Maxi mum al lowed depth of com pres sion block, in
b Width of mem ber, in
bf Ef fec tive width of flange (T- Beam sec tion), in
bw Width of web (T- Beam sec tion), in
Cm 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, in
cb Depth to neu tral axis at bal anced con di tions, in
d Dis tance from com pres sion face to ten sion re in force ment, in
d¢ Con crete cover to cen ter of re in forc ing, in
ds Thick ness of slab (T- Beam sec tion), in
Ec Modu lus of elas tic ity of con crete, psi
Es Modu lus of elas tic ity of re in force ment, as sumed as 29,000 ksi
fc¢ Speci fied com pres sive strength of con crete, ksi
fy Speci fied yield strength of flex ural re in force ment, ksi
fyh Speci fied yield strength of shear re in force ment, ksi
h Di men sion of col umn, in
Ig 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
Ise Mo ment of in er tia of re in force ment about cen troi dal axis of mem ber cross sec tion, in4
k Ef fec tive length factor
L Clear un sup ported length, in
Table III-1
M1 Smaller fac tored end mo ment in a col umn, kip-in
M2 Larger fac tored end mo ment in a col umn, kip-in
Mc Factored mo ment to be used in design, kip-in
Mb Non sway com po nent of fac tored end mo ment, kip-in
Ms Sway com po nent of fac tored end mo ment, kip-in
Mu Fac tored mo ment at sec tion, kip-in
Mux Fac tored mo ment at sec tion about X-axis, kip-in
Muy Fac tored mo ment at sec tion about Y-axis, kip-in
Pb Ax ial load ca pac ity at bal anced strain con di tions, kip
Pe Euler buck ling strength of col umn, kip
Pmax Maxi mum ax ial load strength al lowed, kip
P0 Ax ial load ca pac ity at zero ec cen tric ity, kip
Pu Fac tored ax ial load at sec tion, kip
r Ra dius of gy ra tion of col umn sec tion, in
Vc Shear re sisted by con crete, kip
VD+L Shear force from span load ing, kip
Vu Fac tored shear force at a sec tion, kip
Vp Shear force com puted from prob able mo ment ca pac ity, kip a Re in forc ing steel over strength fac tor
b is a fac tor in di cat ing the abil ity of di ag o nally cracked con crete to trans mit ten sion
b1 Fac tor for ob tain ing depth of com pres sion block in con crete bd Ab so lute value of ra tio of maxi mum fac tored ax ial dead load to
max i mum fac tored ax ial to tal load
q An an gle of in cli na tion of di ag o nal com pres sive stresses with the lon gi tu di nal axis of beam or column
ds Mo ment mag ni fi ca tion fac tor for sway mo ments
db Mo ment mag ni fi ca tion fac tor for nonsway (braced) mo ments ec Strain in con crete
es Strain in re in forc ing steel j Strength re duc tion fac tor
Table III-1
Design Load Combinations
The de sign load com bi na tions are the vari ous com bi na tions of the pre scribed load cases for which the struc ture is to be checked. There are more types of loads speci -fied in the code than are con sid ered in the cur rent im ple men ta tion of the de fault load com bi na tions. How ever, the user has full con trol of the defi ni tion of loads and load com bi na tions.
There are six types of dead loads: dead load of struc tural com po nents and non struc -tural at tach ments (DC), down drag (DD), dead load of wear ing sur face and utili ties (DW), hori zon tal earth pres sure load (EH), ver ti cal earth pres sure load (EV), and earth sur charge load (ES). Each type of dead load case re quires a sep a rate load fac -tor.
There are six types of live loads: ve hicu lar live load (LL), ve hicu lar dy namic load al low ance (IM), ve hicu lar cen trifu gal force (CE), ve hicu lar brak ing force (BR), pe -des trian live load (PL), and live load sur charge (LS). All of these load cases re quire the same fac tor and do not need to be treated sepa rately.
If the struc ture is sub jected to struc tural dead load (DL), live load (LL), wind load (WL), and earth quake loads (EL), and con sid er ing that wind and earth quake forces are re versi ble, the fol low ing de fault load com bi na tions have been con sid ered for Strength and Ex treme 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 de fault de sign load com bi na tions in SAP2000 when ever the AASHTO LRFD 1997 code is used. The user is ex pected to de fine the other load com bi na tions as nec es sary.
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 the fac tored load ing.
Strength Reduction Factors
The strength re duc tion fac tors, j, are ap plied on the nom i nal strength to ob tain the de sign strength pro vided by a mem ber. The j fac tors for flex ure, ax ial force, shear, and tor sion are as fol lows:
j = 0.90 for flex ure, (AASHTO 5.5.4.2.1) j = 0.90 for ax ial ten sion, (AASHTO 5.5.4.2.1) j = 0.90 for ax ial ten sion and flex ure, (AASHTO 5.5.4.2.1) j = 0.90 for shear and tor sion, (AASHTO 5.5.4.2.1) j = 0.75 for ax ial com pres sion only, or ax ial com pres sion and flex ure, and
(AASHTO 5.5.4.2.1) j = 0.50 for ax ial com pres sion only, or ax ial com pres sion and flex ure in
seis mic Zones 3 and 4. (AASHTO 5.5.4.2.3, 5.10.11.4.1b) The value of j in volv ing ax ial com pres sion and flex ure var ies from 0.75 to 0.9 based on the ax ial load. For low val ues of ax ial load, j is in creased lin early from 0.75 to 0.9 as the ax ial load de creases from 0.1 f Ac¢ g to zero (AASHTO 5.5.4.2.1).
For seis mic de sign in Zones 3 and 4, the value of j in volv ing ax ial com pres sion and flex ure var ies from 0.5 to 0.9 based on the ax ial load. For low val ues of ax ial load, j is in creased lin early 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 in volv ing ax ial ten sion, j is al
-ways 0.9 (AASHTO 5.5.4.2.1).
Column Design
The 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 gram checks the col umn ca pac ity. How ever, if the area of re in forc ing is not pro vided by the 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 volves the 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 ent con crete sec tion types of the model. A typ i cal bi axial in ter ac tion sur face is shown in Fig ure II-1. When the steel is un de fined, the pro gram gen er ates the in ter ac tion sur faces for the range of al low able re in force ment ra tio (Ast Ag) ¾
0.135 fc¢ fy to 0.08 for mo ment re sist ing frames (AASHTO 5.7.4.2) and 0.01
to 0.06 for duc tile mo ment re sist ing frames in seis mic Zones 3 and 4 (AASHTO 5.10.11.4.1a).
• 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 as 1.0 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 with these 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 is shown in Fig ure II-1.
The co or di nates of these 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 strain dia gram lim its the maxi mum con crete strain, ec, at the ex trem ity of the sec tion to 0.003 (AASHTO 5.7.2.1).
The for mu la tion is based con sis tently on the gen eral prin ci ples of ul ti mate strength de sign (AASHTO 5.7), and al lows for any dou bly sym met ric rec tan gu lar, square, or cir cu lar col umn sec tion.
The stress in the steel is given by the prod uct of the steel strain and the steel modu -lus of elas tic ity, esE , and is lim ited to the yield stress of the steel, fs y (AASHTO 5.7.2.1). The area as so ci ated with each re in forc ing bar is as sumed to be placed at the ac tual lo 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 (AASHTO 5.7.2.1), with a stress value of 0.85 fc¢ (AASHTO 5.7.2.2). See Figure III-1. The
depth of the stress block is b1c, where
b1 =0.85 0.05- (f ¢ -4)
c , (AASHTO 5.7.2.2)
0.65£b1 £0.85, and (AASHTO 5.7.2.2) The limit of fc¢ is taken to be 10 ksi for all seis mic re gions:
fc¢ £ 10 ksi. (AASHTO 5.1, 5.4.2.1)
The limit of fy is taken to be 75 ksi for all frames:
fy £ 75 ksi. (AASHTO 5.4.3.1)
The in ter ac tion al go rithm pro vides a 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, j, are in cluded in the gen era tion of the in ter ac tion sur faces. The maxi mum com pres sive ax ial load is lim ited to Pmax, where
Figure III-1
Pmax= 0.85 j[ 0.85 f A - Ac¢( g st) + f Ay st] spi ral col umn, (AASHTO 5.7.4.4)
Pmax= 0.80j[ 0.85 fc¢(A - Ag st) + f Ay st] tied col umn. (AASHTO 5.7.4.4) The value of j in volv ing ax ial com pres sion and flex ure var ies from 0.75 to 0.9 based on the ax ial load. For low val ues of ax ial load, j is in creased lin early from 0.75 to 0.9 as the ax ial load de creases from 0.1 f Ac¢ g to zero (AASHTO 5.5.4.2.1). For seis mic de sign in Zones 3 and 4, the value of j involving ax ial com pres sion and flex ure var ies from 0.5 to 0.9 based on the ax ial load. For low val ues of ax ial load, j is in creased lin early 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 in volv ing ax ial ten sion, j is al ways 0.9 (AASHTO 5.5.4.2.1).
Check Column Capacity
The col umn ca pac ity is checked for each load ing com bi na tion at each check sta tion of 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 and the speci fied load com bi na tion fac tors to give P Mu, ux,andMuy.
• 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 mine
whether 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 the iden ti fi ca tion of the in di vid ual col umn as "braced" or "un braced."
The fol low ing three sec tions de scribe in de tail the al go rithms as so ci ated with these steps.
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 the cor re spond ing load fac tors to all the load cases, giv ing P Mu, ux,andMuy. The com -puted mo ments are fur ther am pli fied by us ing “Mo ment Mag ni fi ca tion Fac tors” to al low for sta bil ity ef fects.
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), ds, and for non sway or braced (in di vid ual col umn sta bil ity ef fect), dns. Also the mo 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.
The pro gram as sumes that a P-D anal y sis has been per formed in SAP2000 and, there fore, mo ment mag ni fi ca tion fac tors for mo ments caus ing sidesway are taken as unity (AASHTO 4.5.3). For the P-D anal y sis the load should cor re spond to a load com bi na tion of (1.25 dead load + 1.35 live load) /j, where j is the re sis tance fac tor for ax ial com pres sion, which is taken as 0.75 for seis mic Zones 1 and 2 and as 0.5 for seis mic Zones 3 and 4 by de fault (AASHTO 5.5.4.2.1). See also White and Hajjar (1991).
The mo ment ob tained from analy sis is sepa rated into two com po nents: the sway (Ms) and the non sway (Mb) com po nents. The non- sway or braced com po nents, which are iden ti fied by “b” sub scripts, are pre domi nantly caused by grav ity load. The sway com po nents are iden ti fied by “s” sub scripts. The sway mo ments are pre -domi nantly caused by lat eral loads and are re lated to the cause of side sway. For in di vid ual col umns or column- members in a floor, the mag ni fied mo ments about two axes at any sta tion of a col umn can be ob tained as
M =dbMb +dsMs . (AASHTO 4.5.3.2.2b) The fac tor ds is the mo ment mag ni fi ca tion fac tor for mo ments caus ing sidesway. This fac tor is taken as 1 be cause the com po nent mo ments Ms and Mb are ob tained from a “sec ond or der elas tic (P-D) anal y sis.”
The non sway mo ment mag ni fi ca tion fac tor, db, as so ci ated with the ma jor or mi nor di rec tion of the col umn is given by (AASHTO 4.5.3.2.2b),
d j b m u e C P P = 1.0 1 -³ , where (AASHTO 4.5.3.2.2b) P = EI kL e p2 2 ( ) , (AASHTO 4.5.3.2.2b)
k is taken as 1, how ever SAP2000 al lows the user to over ride this value
EI is as so ci ated with a par ticu lar col umn di rec tion given by: EI = E I + c g d 2.5 1 b , (AASHTO 5.7.4.3)
bd = maximum factored dead load moment
maximum factored total load moment , and (AASHTO 5.7.4.3)
C = + M M m a b 0.6 0.4 ³0.4 . (AASHTO 4.5.3.2.2b)
Ma and Mb are the mo ments at the ends of the col umn, and Mb is nu meri cally larger than Ma. Ma Mb is posi tive for sin gle cur va ture bend ing and nega tive for dou ble cur va ture bend ing. The above ex pres sion of Cm is valid if there is no trans verse load ap plied be tween the sup ports and the mem ber is braced against sidesway. If trans verse load is pres ent on the span, or the length is over writ ten, or for any other case, Cm =1 . Cm can be 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, db, must be a posi tive number and greater than one. There fore Pu must be less than jPe. If Pu is found to be greater than or equal to jPe, a fail ure con di tion is de clared.
The above cal cu la tions use the un sup ported lengths 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 lengths be 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 user can ex plic itly spec ify val ues of dsanddb.
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 ment mag ni fi ca tion fac tors are ap plied to the fac tored loads to ob tain P Mu, ux, andMuy. The point (P Mu, ux,Muy) is then placed in the in ter ac tion space shown as point L in Fig ure II-3. If the point lies within the in ter ac tion vol ume, the col umn ca pac ity is ade quate; how ever, if the point lies out side the in ter ac tion vol ume, the col umn is over stressed.
