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2 Hajrasoulha I & Doostan A 200 A smplfed model for sesmc response predcton of concentrcally braced frames. Advances n Engneerng Software, 43, A smplfed model for sesmc response predcton of concentrcally braced frames Iman Hajrasoulha a, Alreza Doostan b a Cvl Engneerng Department, The Unversty of Scence & Culture, Tehran, Iran b Department of Mechancal Engneerng, Stanford Unversty, Stanford, CA, USA Abstract Ths paper proposes a smplfed analytcal model for sesmc response predcton of concentrcally braced frames. In the proposed approach, a multstory frame model s reduced to an equvalent shear-buldng one by performng a statc pushover analyss. The conventonal shear-buldng model has been mproved by ntroducng supplementary sprngs to account for flexural dsplacements n addton to shear dsplacements. The adequacy of the modfed model has been verfed by conductng nonlnear dynamc analyss on 5, 0 and 5 story concentrcally braced frames subjected to 5 synthetc earthquae records representng a desgn spectrum. It s shown that the proposed mproved shear-buldng models provde a better estmate of the nonlnear dynamc response of the orgnal framed structures, as compared to the conventonal models. Whle smplfyng the analyss of concentrcally braced frames to a large extend, and thus reducng the computatonal efforts sgnfcantly, the proposed method s accurate enough for practcal applcatons n performance assessment and earthquae-resstant desgn. Keywords: concentrcally braced frames; shear buldngs; non-lnear dynamc analyss; sesmc demands; pushover analyss; cumulatve damage

3 - Introducton Both structural and nonstructural damages observed durng earthquae ground motons are prmarly produced by lateral dsplacements. Thus, the estmaton of lateral dsplacement demands s of sgnfcant mportance n performance-based desgn methods; specally, when damage control s the man quantty of nterest. Most structures experence nelastc deformatons when subjected to severe earthquae ground motons. Therefore, nonlnear behavour of structures should be taen nto account to have accurate estmaton of deformaton demands. Nonlnear tme hstory analyss of a detaled analytcal model s perhaps the best opton for the estmaton of deformaton demands. However, due to many uncertantes assocated wth the ste-specfc exctaton as well as uncertantes n the parameters of analytcal models, n many cases, the effort assocated wth detaled modelng and analyss may not be justfed and feasble. Therefore, t s prudent to have a reduced model, as a smpler analyss tool, to assess the sesmc performance of a frame structure. Constructon of such reduced model s the man goal of the present study. The estmaton of sesmc deformaton demands for mult-degree-of-freedom MDOF structures has been the subject of many studes [-8]. Although those studes dffer n ther approach, they commonly establsh an equvalent sngle-degree-of-freedom SDOF system as the reduced model wth whch the nelastc dsplacement demands of the full model are estmated. Consequently, the nelastc dsplacement demands are converted nto local deformaton demands; ether through multplcatve converson factors, derved from a large number of non-lnear analyses of dfferent types of structural systems, or through buldng specfc relatonshps between global dsplacements and local deformatons developed usng a pushover analyss. These approxmate methods are partcularly ntended to provde rough estmates of maxmum lateral deformatons and are not accurate enough to be a substtute for more detaled analyses, whch are approprate durng the fnal evaluaton of the proposed desgn of a new buldng or durng the detaled evaluaton of exstng buldngs. 2

4 For the purpose of prelmnary desgn and analyss of structures, many studes have been carred out to construct reduced nonlnear models that feature both accuracy and low computatonal cost. Mranda [5, 6] and Mranda et al. [7] have ncorporated a smplfed model of a buldng based on an equvalent contnuum structure consstng of a seres of flexural and shear cantlever beams to estmate deformaton demands n multstory buldngs subjected to earthquaes. Although n that method the effect of nonlnear behavor s consdered by usng some amplfcaton factors, the flexural and shear cantlever beams can only behave n elastc range of vbraton. Some researchers [2, 8, 9] have attempted to develop analytcal models to predct the nelastc sesmc response of renforced concrete shear-wall buldngs, ncludng both the flexural and shear falure modes. La et al. [0] developed a mult-rgd-body theory to analyze the earthquae response of shear-type structures. In that wor, materal non-lnearty can be ncorporated nto the mult-rgd-body dscrete model; however, t s not possble to calculate the nodal dsplacements caused by flexural deformatons, whch n most cases has a consderable contrbuton to the sesmc response of frame-type structures. Among the wde varety of structural models that are used to estmate the non-lnear sesmc response of buldng frames, the conventonal shear buldng model s the most frequently utlzed reduced model. In spte of some of ts drawbacs, the conventonal shear buldng model s wdely used to study the sesmc response of mult-story buldngs manly due to ts excessve smplcty and low computatonal expenses. Ths model has been developed several decades ago and has been successfully employed n prelmnary desgn of many hgh-rse buldngs [- 3]. The relablty of conventonal shear-buldng models to predct non-lnear dynamc response of moment resstance frames s nvestgated by Daz et al. [4]. It has been shown, there, that conventonal shear buldng models overestmate the ductlty demands n the lower stores, as compared wth more accurate frame models. Ths s manly due to nablty of shear buldng models to dstrbute the nelastc deformatons among the members of adjacent stores. To overcome ths ssue, n the present study, the conventonal shear-buldng model has been mproved by ntroducng supplementary sprngs to account for flexural dsplacements n addton to shear drfts. The constructon of such reduced model s based on a statc pushover analyss. Relablty of ths modfed shear-buldng model s nvestgated by conductng nonlnear dynamc 3

5 analyss on 5, 0 and 5 story concentrcally steel braced frames subjected to 5 dfferent synthetc earthquae records representng a desgn spectrum. It s shown that the proposed modfed shear-buldng models more accurately estmate the nonlnear dynamc response of the correspondng concentrcally braced frames compare to the conventonal shear-buldng models. 2- ng and assumptons In the present study, three steel concentrc braced frames wth 5, 0 and 5 stores have been selected Fg.. The buldngs are assumed to be located on a sol type S D and a sesmcally actve area, zone 4 of the UBC 997 [5] category, wth PGA of 0.44 g. Smple beam to column connectons are consdered to prevent the transmsson of any moment from beams to the supportng columns. The frame members are szed to support gravty and lateral loads determned n accordance wth the mnmum requrements of UBC 997 [5]. In all models, the top story s 25% lghter than the others. IPB, IPE and UNP sectons, accordng to DIN standard, are chosen for columns, beams and bracngs, respectvely. All jont nodes at the same floor were constraned together n the horzontal drecton of the nput ground moton. Once the structural members are sezed, the entre desgn s checed for the code drft lmtatons and f necessary refned to meet the requrements. For the statc and nonlnear dynamc analyss, the computer program Dran-2DX [6] s used. The Raylegh dampng s adopted wth a constant dampng rato 0.05 for the frst few effectve modes. The columns were modelled usng a fbre-type element wth dstrbuted plastcty element 5 n whch the locaton of non-lnearty wthn the elements s computed durng the analyss. The brace members are assumed to have elastc-plastc behavour n tenson and compresson. The yeld capacty n tenson s set equal to the nomnal tensle resstance, whle the yeld capacty n compresson s set equal to 0.28 tmes the nomnal compressve resstance as suggested by Jan et al. [7]. To nvestgate the accuracy of dfferent methods for predcton of sesmc response of concentrcally braced steel frames, ffteen sesmc motons are artfcally generated usng the SIMQKE program [8], havng a close approxmaton to the elastc desgn response spectra of 4

6 UBC 997 [5] wth a PGA of 0.44g. Therefore, these synthetc earthquae records are expected to be representatve of the desgn spectra. The comparsons between artfcally generated spectra and the UBC 997 [5] desgn spectra are shown n Fg Conventonal shear buldng model The conventonal shear buldng model s an assembly of structural members connected along horzontal nterfaces, whch concde wth the floor levels and, therefore, wth the levels where the buldng mass s assumed to be concentrated. These members can only undergo shear deformatons when subjected to lateral forces as shown n Fg. 3. The conventonal shear buldng model has n degrees of freedom where n s the number of stores. The lateral stffness t, yeld strength S and over-strength factor α t of the structural element representng the mechancal propertes of the th floor, are computed on the bass of adequate assumptons regardng the deformed shape of the orgnal frame. To accomplsh ths, a pushover analyss s conducted on the full-model framed structure and the relatonshp between the story shear force V and the total nter-story drft t s extracted. The nonlnear force-dsplacement relatonshp has been replaced wth an dealzed relatonshp to calculate the nomnal story stffness t and effectve yeld strength S of each story as shown n Fg. 4. Lne segments on the dealzed force-dsplacement curve have been located usng an teratve procedure that approxmately balances the area above and below the curve. The nomnal story stffness t s then taen as the secant stffness calculated at a story shear force equal to 60% of the effectve yeld strength of the story [9, 20]. It s well nown that deformaton estmates obtaned from a pushover analyss may be very naccurate for structures n whch hgher vbraton modes have sgnfcant contrbuton to the overall response. Also for stuatons where the resultng story shear forces, caused by the story drfts, are senstve to the appled load pattern the applcaton of the pushover analyss seems questonable [2, 22]. None of the nvarant force dstrbutons can account for the contrbutons of hgher modes to the overall structural response or even the redstrbuton of nerta forces. Ths s due to yeldng of structural components and the resultng changes n the vbraton 5

7 characterstcs of the structure. Ths problem can be mtgated to some extend by applyng more than one lateral load pattern whch ncludes those that excte elastc hgher mode effects. In ths study, pushover analyses are performed under dfferent lateral load patterns to nvestgate the effects of pre-assumed load pattern on computed mechancal propertes of each story. For all pushover analyses four dfferent vertcal dstrbuton of lateral load are consdered; a vertcal dstrbuton proportonal to the shape of the fundamental mode of vbraton; a trangular dstrbuton accordng to UBC 97 [5]; a unform dstrbuton proportonal to the total mass at each level; and fnally a vertcal dstrbuton proportonal to the values of C vx gven by followng equaton [9, 20]: C vx = n w = x h x wh, where C vx s the vertcal dstrbuton factor, w and h are the weght and heght of the th floor above the base, respectvely. Also, n s the number of stores and s an exponent ncreases from to 2 as perod vares from 0.5 to 2.5 second. The lateral stffness and yeld strength dstrbutons correspondng to each case are compared n Fg. 5 for a 0-story concentrcally braced frame. As shown n ths fgure, mechancal propertes of the stores are rather nsenstve to the predetermned lateral load pattern used for pushover analyses. It s partcularly true f a ratonal lateral load dstrbuton s used. To evaluate the relablty of conventonal shear-buldng models to estmate the dsplacement demands of concentrcally braced frames, tme hstory analyses have been performed on 5, 0 and 5 story full-frame models and ther correspondng conventonal shearbuldng models subjected to 5 synthetc earthquaes. For each sesmc exctaton, the errors n predcton of roof dsplacements, story dsplacements and nter-story drfts have been determned. Subsequently, for each story, the average value of the errors correspondng to 5 6

8 synthetc earthquaes has been calculated. Table summarzes the maxmum errors correspondng to 5, 0 and 5 story concentrcally braced frames. As ndcated n ths table, usng modfed shear buldng models, maxmum errors n estmaton of roof and story dsplacements are small less than 6 percent. However, maxmum roof and story dsplacements are not good ndcators of sesmc performance of a structure as compared wth story drfts. The results presented n Table show that the errors n estmaton of story drfts are much larger 2.5 tmes hgher compared to story dsplacements. Therefore, conventonal shearbuldng models are not relable enough to estmate the maxmum story drfts of concentrcally braced frames for the case of large non-lnear deformatons whch s observed n sever earthquaes. In Fg. 6, the maxmum story dsplacement and maxmum drft dstrbuton of the 0-story frame obtaned usng conventonal shear-buldng models are compared wth the average of actual values for 5 synthetc earthquaes. Ths fgure shows that, on average, conventonal shear buldng models provde reasonable estmates of maxmum roof and story dsplacements; however, estmated story drfts are not accurate enough. The errors are especally large for the case of the maxmum drft estmated at the level of top stores where the estmated drft s 40% hgher than the actual value. Although sesmc forces n top stores may not control the overall desgn of the structure, nter-story drfts at the top floors could govern the sesmc desgn of mult-story frames, especally for hgh-rse buldngs where the hgher mode effects are consderable. As descrbed very brefly, n the present study, the conventonal shear-buldng model has been modfed n order to acheve a better estmaton of nonlnear dynamc response of real framed structures. More detals of such extenson are presented next. 4- Shear and flexural deformatons Recent desgn gudelnes, such as FEMA 273 [9], FEMA 356 [20] and SEAOC Vson 2000 [23], place lmts on acceptable values of response parameters; mplyng that exceedng of these lmts s a volaton of a performance objectve. Among varous response parameters, the nter- 7

9 story drft s consdered as a relable ndcator of damage to nonstructural elements, and s wdely used as a falure crteron because of the smplcty and the convenence assocated wth ts estmaton. Consderng the 2-D frame shown n Fg. 7-a, the axal deformaton of columns results n ncrease of lateral story and nter-story drfts. In each story, the total nter-story drft t s a combnaton of the shear deformaton sh, due to shear flexblty of the story, and the flexural deformaton ax, due to axal flexblty of the lower columns. Hence, nter-story drft can be expressed as: = +. 2 t sh ax Flexural deformaton does not contrbute n the damage mposed to the story, though t may mpar the stablty due to the P- effects. Neglectng the axal deformaton of beams, the shear deformaton for a sngle panel, as shown n Fg. 7-b, s determned by [24], sh = t + H 2L U +U U U where, U 5, U 6, U 2 and U 3 are vertcal dsplacements, as shown n Fg. 7-b. H and L are the heght of the story and the span length, respectvely. The dervaton of Equaton 3 s descrbed n detal n Moghaddam et al. [25]. For mult-span models, the maxmum value of the shear drft n dfferent panels s consdered as the shear story drft. 5- Modfed shear buldng model Lateral deformatons n buldngs are usually a combnaton of lateral shear-type deformatons and lateral flexural-type deformatons. In ordnary shear buldng models, the effect of column axal deformatons s usually neglected. Therefore, t s not possble to calculate the nodal dsplacements caused by flexural deformaton, whle t may have a consderable contrbuton to the sesmc response of most frame-type structures. In the present study, the shear-buldng model has been modfed by ntroducng supplementary sprngs to account for 8

10 flexural dsplacements n addton to shear dsplacements. Accordng to the number of stores, the structure s modeled wth n lumped masses, representng the stores. Only one degree of freedom of translaton n the horzontal drecton s taen nto consderaton and each adjacent mass s connected by two supplementary sprngs as shown n Fg. 8. As shown n ths fgure, the modfed shear-buldng model of a frame condenses all the elements n a story nto two supplementary sprngs, thereby sgnfcantly reduces the number of degrees of freedom. The stffnesses of supplementary sprngs are equal to the shear and bendng stffnesses of each story, respectvely. These stffnesses are determned by enforcng the model to undergo the same dsplacements as those obtaned from a pushover analyss on the orgnal frame model. As shown n Fg. 8, the materal nonlneartes may be ncorporated nto stffness and strength of supplementary sprngs. In Fg. 8, m represents the mass of th floor; and V and S are, respectvely, the total shear force and yeld strength of the th story obtaned from the pushover analyss. t s the nomnal story stffness correspondng to the relatve total drft at th floor t n Fg. 7. sh denotes the shear story stffness correspondng to the relatve shear drft at th floor sh n Fg. 7. ax represents the bendng story stffness correspondng to the flexural deformaton at th floor ax n Fg. 7, and α t, α sh and α ax are over-strength factors for nomnal story stffness, shear story stffness and bendng story stffness at th floor, respectvely. t and α t are determned from a pushover analyss tang nto account the axal deformaton of columns. In ths study, the nonlnear force-dsplacement relatonshp between the story shear force V and the total nter-story drft t has been replaced wth an dealzed blnear relatonshp to calculate the nomnal story stffness t and effectve yeld strength S of each story as shown n Fg. 8. Lne segments on the dealzed force-dsplacement curve have been located usng an teratve procedure that approxmately balanced the area above and below the curve. The nomnal story stffness t s then taen as the secant stffness calculated at a story shear force equal to 60% of the effectve yeld strength of the story [9, 20]. Usng Equaton 3, shear story drft correspondng to each step of pushover analyss can be calculated and consequently sh and α sh are determned. As the transmtted force s equal n two supplementary sprngs, Equaton 2 can be rewrtten as: 9

11 For V S, ax sh t V V V + =. 4 Hence, ax sh t + =. 5 For V > S we have ax ax ax sh sh sh t t t S V S S V S S V S α α α = +. 6 Substtutng Equaton 5 n 6, ax and α ax are obtaned as follows: t sh t sh ax =. 7 [ ] t t sh sh t sh t sh ax α α α α α =. 8 Calculatons show that α ax s almost equal to when columns are desgned to prevent buclng aganst earthquae loads, thus mplyng that the sprng whch represents the axal deformaton always remans n the elastc deformaton range. As wll be descrbed n the sequel, for each frame model, all the requred parameters of the modfed shear-buldng can be determned by performng only one pushover analyss. By consderng P- effects n ths pushover analyss, the modfed model wll be capable to account for P- effects as well. The shear nter-story drft, whch causes damage to the structure, can be separated from the flexural deformaton by usng the modfed shear-buldng model. The modfed shear-buldng model taes nto account both the hgher mode contrbuton to elastc structural response as 0

12 well as the effects of materal non-lnearty; therefore, t represents the behavor of frame models more realstcally as compared to the conventonal shear-buldng model. To nvestgate the relablty of the proposed modfed model n estmatng the sesmc response parameters of concentrcally braced frames, non-lnear tme hstory analyses have been performed for 5, 0 and 5 story frames and ther correspondng modfed shear-buldng models subjected to 5 synthetc earthquaes. It s shown n Fg.6 that the modfed model s capable to estmate the nonlnear sesmc response of the 0 story concentrcally braced frame more accurately compare to the conventonal shear-buldng model. Average of the dsplacement demands for 5, 0 and 5 story frame models and ther correspondng modfed shear buldng models are compared n Fg. 9. Ths Fgure ndcates that on average, modfed shear-buldng models are capable to predct story dsplacement, total nter-story drft and shear nter-story drft of concentrcally braced frames very accurately. For each synthetc exctaton, the errors n predcton of dsplacement demands between the modfed shear-buldng model analyss and the orgnal frame are determned. Consequently, the average of these errors s calculated for every story. Maxmum errors correspondng to 5, 0 and 5 story frames are summarzed n Table. It s shown that maxmum errors assocated wth the modfed shear buldng model are sgnfcantly less than the correspondng values for the conventonal shear-buldng model, partcularly for story drfts where the errors are almost one thrd of those estmated by conventonal models. The errors are slghtly larger for predcton of drft than for estmaton of dsplacement. However, for modfed shear buldng models, the maxmum errors n all response quanttes are only a few percent less than 6%. Based on the above dscusson, dsplacement demands estmated by modfed shearbuldng models proved to be good representatves of those obtaned based on typcal nonlnear frame models of the same structure. Next, t s nvestgated how the errors n dsplacement demands obtaned by modfed shear-buldng models vary wth the deformaton demands mposed by the ground moton and n partcular wth the degree to whch the system deforms beyond ts elastc lmt. For ths purpose, dsplacement demands for the 0-story frame

13 model and ts correspondng modfed shear-buldng model are obtaned for ground motons of dfferent ntensty. These exctatons are scaled El Centro 940 ground motons wth scalng coeffcents 0.5, 0.25, 0.5, 0.75,.0,.5, 2.0, 2.5, and 3.0. For each exctaton, the errors n response quanttes obtaned by the modfed shear-buldng model compared to the correspondng orgnal frame response quanttes are determned. Fg. 0 summarzes the maxmum errors n dsplacement demands estmated by modfed shear-buldng models as a functon of ground moton ntensty, ndcated by the ground moton scale coeffcent, and maxmum story ductlty. One can observe from ths fgure that these errors are larger n story drfts compared to story dsplacements; however, maxmum errors are less than 20% even for very ntense ground motons. Ths s further llustrated n Fg. 0 that the errors are almost ndependent to the ground moton ntensty and maxmum story ductlty. Therefore, t can be concluded that the modfed shear-buldng model estmates the sesmc response of buldngs experenced hgh nelastc deformatons.e. story ductlty more than 0 wth the same degree of accuracy as t predcts the response of elastc systems. The same observatons have been made wth other models and under dfferent ground motons. As mentoned before, the behavor of modfed shear buldng model s dealzed by a blnear force-dsplacement curve. For the concentrcally braced frames, the nomnal story stffness n the equvalent modfed shear buldng model s very close to the ntal tangent stffness of the typcal full-frame model. Therefore, modfed shear buldng model has a good capablty to estmate the natural perods of the correspondng full-frame model. The close predcton of the natural perods n full frame models and ther correspondng modfed shear buldng models for 5, 0 and 5 story braced frames are llustrated n Table 2. It s shown that usng modfed shear-buldng model, the perod of the frst three vbraton modes agree very well wth the natural perods of the full-frame model. Ths s partcularly true for the fundamental perod st mode where the predcted values are almost dentcal wth the actual values. Total computatonal tme for 5, 0 and 5 story braced frames and ther correspondng modfed shear-buldng model under 5 synthetc earthquaes are compared n Table 2. As t s llustrated, the relatvely small number of degrees of freedom for modfed shear-buldng model 2

14 results n sgnfcant computatonal savngs, whle mantanng the accuracy, as compared to the correspondng frame model. Accordng to the results, total computatonal tme for modfed shear-buldng models are less than 4% of those based on typcal frame models. 6- Cumulatve damage The pea shear story drft may not always be the best performance crteron for performance base desgn as t occasonally fals n predctng the state of structural damage n earthquaes. To nvestgate the extent of cumulatve damage, the damage crteron proposed by Ba et al. [26] based on the classcal low-cycle fatgue approach has been adopted. The story nelastc shear deformaton s chosen as the basc damage quantty, and the cumulatve damage ndex after N excursons of plastc deformaton s calculated as: D N δ = j= δy pj c 9 Where D s the cumulatve damage ndex at th story, rangng from 0 for undamaged to for severely damaged stores, δ pj s the plastc deformaton of th story n j th excurson, δ y s the nomnal yeld deformaton, and c s a parameter that accounts for the effect of magntude of plastc deformaton taen to be.5 [27]. To assess the damage experenced by the whole structure, the global damage ndex s obtaned as a weghted average of the damage ndces at the story levels, wth the energy dsspated beng the weghtng functon gven by: D g = n D W p, = n W p = 0 where D g s the global damage ndex, W p s the energy dsspated at th story, D s the damage ndex at th story, and n s the number of stores. 3

15 Usng ths equaton, the global damage ndex has been calculated for 5, 0 and 5 story concentrcally braced frames and ther correspondng modfed shear-buldng models subjected to 5 synthetc earthquaes. As an example, the global damage ndex of 0-story frame obtaned by modfed shear-buldng model s compared wth those obtaned by full-frame model n Fg.. The results suggest that, from low level less than 20% to thgh level more than 70% of damage ntensty, the global damage experenced by the concentrcally braced frames can be estmated utlzng modfed shear-buldng models up to an acceptable accuracy for practcal applcatons. Estmaton of pea nelastc deformaton demands s a ey component of any performancebased procedure for earthquae-resstant desgn of new structures or for sesmc performance evaluaton of exstng structures. The modfed shear buldng models proved to be capable to account for contrbuton of several modes of vbraton, P- effects and characterstcs of the ground motons. Therefore, evaluatng the deformaton demands and cumulatve damages usng modfed shear-buldng models s demonstrated to be reasonably close to those of the full-frame models. Ths maes t an approprate model to be utlzed n sesmc performancebased desgn softwares. In practcal applcatons, due to sgnfcantly low computatonal efforts assocated wth the proposed modfed shear-buldng model, one can possbly consder more desgn alternatves and earthquae ground motons as opposed to desgns based on the fullframe model. Therefore, the modfed shear-buldng model can be effcently used for optmum sesmc desgn of structures where many nonlnear dynamc analyses would be requred to get to the optmum soluton [25]. 7- Conclusons. It s shown that, n general, conventonal shear buldng models provde accurate estmates of maxmum roof and story dsplacements of concentrcally braced frames; but are not able to provde good estmates of nter-story drfts. Whle the maxmum errors n the estmaton of maxmum roof and story dsplacements are usually less than 4

16 5%, they are partcularly large for the maxmum drft at top stores where the estmated drft could be more than 40% hgher than the actual value. 2. The conventonal shear-buldng model has been modfed by ntroducng supplementary sprngs to account for flexural dsplacements n addton to shear drfts. It s shown that the accuracy of modfed shear buldng models to predct story dsplacements and pea nter-story drfts s sgnfcantly hgher than conventonal models. 3. It s shown that the modfed shear-buldng model s not senstve to the ground moton ntensty and maxmum story ductlty; and therefore, could be utlzed to estmates the sesmc response of concentrcally braced frames from elastc to hghly nelastc range of behavour. The results ndcate that the proposed model s also capable to estmate the global damage experenced by the concentrcally braced frames from low less than 20% to hgh more than 70% level of damage ntensty. References [] Q X, Moelhe JP. Dsplacement desgn approach for renforced concrete structures subjected to earthquaes. Report No. UBC/EERC-9/02. Unversty of Calforna, Earthquae Engneerng Research Center, Bereley, CA; 99. [2] Fajfar P, Gaspersc P. The N2 method for the sesmc damage analyss of RC buldngs, Earthquae Engneerng and Structural Dynamcs 996; 25: [3] Senevratna GDPK, Krawnler H. Evaluaton of nelastc MDOF effects for sesmc desgn. John A. Blume Earthquae Engneerng Center Report No. 20, Department of Cvl Engneerng, Stanford Unversty; 997. [4] Guptas A, Krawnler H. Estmaton of sesmc drft demands for frame structures. Earthquae Engneerng and Structural Dynamcs 2000; 2999:

17 [5] Mranda E. Estmaton of maxmum nter-story drft demands n dsplacement-based desgn. In: Fajfar P, Krawnler H, Proc. Worshop on Sesmc Desgn Methodologes for the Next Generaton of Codes. Balema; Rotterdam; 997. p [6] Mranda E. Approxmate sesmc lateral deformaton demands n multstory buldngs. Journal of Structural Engneerng 999; 254: [7] Mranda E, Reyes CJ. Approxmate lateral drft demands n multstory buldngs wth nonunform stffness, Journal of Structural Engneerng 2002; 287: [8] Sad M, Sozen MA. Smple Nonlnear Sesmc Analyss of RC Structures, Journal of Structural Dvson, ASCE 98; 075: [9] Hdalgo PA, Jordan RM, Martnez MP. An analytcal model to predct the nelastc sesmc behavor of shear-wall, renforced concrete structures. Engneerng Structures 2002; 24: [0] La M, L Y, Zhang Ch. Analyss method of mult-rgd-body model for earthquae responses of shear-type structure. In: WCEE 0 th Conference; 992, p [] Penzen J. Elasto-plastc response of dealzed multy-story structures subjected to a strong moton earthquae. In: WCEE 2 nd Conference; 960. [2] Hsada T, Naagawa K, Izum M. Earthquae response of dealzed twenty story buldngs havng varous elasto-plastc propertes. In: WCEE 3 rd Conference; 965. [3] Veletsos AS, Vann WP. Response of ground-excted elasto-plastc systems, ASCE 97; 974: [4] Daz O, Mendoza E, Esteva L. Sesmc ductlty demands predcted by alternate models of buldng frames, Earthquae Spectra 994; 03:

18 [5] UBC. Structural engneerng desgn provsons. In: Unform Buldng Code. Internatonal Conference of Buldng Offcals, vol. 2; 997. [6] Praash V, Powell GH, Flppou, FC. DRAIN-2DX: Base program user gude. Report No. UCB/SEMM-92/29; 992. [7] Jan AK, Goel SC, Hanson RD. Hysteretc cycles of axally loaded steel members. Journal of Structural Dvson, ASCE 980; 068: [8] Vanmare EH. SIMQKE: A Program for Artfcal Moton Generaton. Cvl Engneerng Department, Massachusetts Insttute of Technology; 976. [9] FEMA 273. NEHRP gudelnes for the sesmc rehabltaton of buldngs. Federal Emergency Management Agency; 997. [20] FEMA 356, Prestandard and commentary for the sesmc rehabltaton of buldngs. Washngton, DC: Federal Emergency Management Agency; [2] Krawnler H, Senevratna GDPK. Pros and cons of a pushover analyss of sesmc performance evaluaton. Engneerng Structures 998; 204 6: [22] Guptas A, Krawnler H. Estmaton of sesmc drft demands for frames structures, Earthquae Engneerng and Structural Dynamcs 2000; 29 99: [23] SEAOC. Vson 2000, performance based sesmc engneerng for buldngs. Sacramento, CA: Structural Engneers Assocaton of Calforna; 995. [24] Bertero VV, Anderson JC, Krawnler H, Mranda E. Desgn gudelnes for ductlty and drft lmts. Report No. UCB/EERC-9/5. Unversty of Calforna, Earthquae Engneerng Center, Bereley, CA; 99. 7

19 [25] Moghaddam H, Hajrasoulha I, Doostan A. Optmum sesmc desgn of concentrcally braced steel frames: concepts and desgn procedures. Journal of Constructonal Steel Research 2005; 62: [26] Ba SW, Lee DG, Krawnler H. A smplfed model for sesmc response predcton of steel frame structures. In: The 9 th word Conference on Earthquae Engneerng, Japan, Vol. V.; 988. [27] Krawnler H, Zohre M. Cumulatve damage n steel structures subjected to earthquae ground motons. Computers and Structures 983; 6:

20 Lst of symbols The followng symbols are used n ths paper: α ax = Over-strength factors for bendng story stffness at th floor α sh = Over-strength factors for shear story stffness at th floor α t = Over-strength factors for nomnal story stffness at th floor δ pj = Plastc deformaton of th story n j th excurson δ y = Nomnal yeld deformaton of th story t = Total nter-story drft sh = Shear nter-story drft ax = Flexural nter-story drft C vx = Vertcal dstrbuton factor for lateral loads c = Parameter that accounts for the effect of magntude of plastc deformaton D g = Global damage ndex D = Cumulatve damage ndex at th story H = Heght of the story h = Heght of th story = Postve number as a power t = Nomnal story stffness of th story 9

21 ax = Bendng story stffness of th story sh = Shear story stffness of th story L = Span length N = Number of plastc excursons n = Number of stores S = Shear yeld strength of th story V = Total shear force of th story U = Horzontal dsplacement at the bottom lne of the panel U 2, U 3 = Vertcal dsplacements at the bottom lne of the panel U 4 = Horzontal dsplacement at the top lne of the panel U 5, U 6 = Vertcal dsplacements at the top lne of the panel w = Weght of th story W p = Energy dsspated at th story 20

22 3m = 5 3m = 30 3m = 45 6m = 30 m 6m = 30 m 6m = 30 m Fg.. Typcal geometry of concentrc braced frames.4 Pseudo Acceleraton g UBC 97 Ave. of 5 Sm. Eq Perod Sec Fg. 2. UBC desgn spectrum and average response spectra of 5 synthetc earthquaes 5% dampng 2

23 F M Fg. 3. Conventonal shear-buldng model Base Shear V S α t K t 0.6S K t y t Dsplacement Fg. 4. Idealzed force-dsplacement curves 22

24 Story a Cvx Equaton Unform Trangular Frst Mode Effectve Stffness Ton.f/m b Story Cvx Equaton Unform Trangular Frst Mode Strength Ton.f Fg. 5. The effect of vertcal dstrbuton of lateral loads on computed mechancal propertes; a Story stffness, b Story strength 23

25 0 9 a Story Modfed Shear-Buldng Conventonal Shear-Buldng Frame Story Drft cm 0 9 b Story Modfed Shear-Buldng Conventonal Shear-Buldng Frame Story Dsplacement cm Fg. 6. Comparson of frame model, conventonal shear-buldng model and modfed shear-buldng model for 0-story braced frame, Average of 5 synthetc earthquaes; a Story drft, b Story dsplacement 24

26 ax sh U 5 U 6 U 4 U 4 H U 2 U 3 U U t = ax + sh L a b Fg. 7. a Defntons of total nter-story drft t, shear nter-story drft sh and the effect of axal flexblty of columns ax, b Dsplacement components of a sngle panel. m n- m n ax n sh n ax n- sh n- m ax sh V V V S α t t S α sh sh S α ax ax t sh ax t sh ax Fg. 8. Usng push over analyss to defne equvalent modfed shear-buldng model 25

27 Story Shear Frame 3 5 Total Story Drft Cm Story Shear Frame 3 5 Shear Story Drft Cm Story Shear Frame Max Drft Cm Story Shear Frame Total Story Drft Cm Story Shear Frame 3 5 Shear Story Drft Cm Story Shear Frame Max Drft Cm Story 9 7 Story 9 7 Shear Story Shear 5 Frame 5 Shear 3 Frame 3 3 Frame 3 5 Total Story Drft Cm Shear Story Drft Cm Max Drft Cm Fg. 9. Comparson of the full-frame model and the correspondng modfed shear-buldng model for 5, 0 and 5-story braced frames, Average of 5 synthetc earthquaes 26

28 40% Roof Dsplacement a 40% Roof Dsplacement b 30% Story Dsplacement Story Drft 30% Story Dsplacement Story Drft Error % 20% Error % 20% 0% 0% 0% Ground Moton Multpler 0% Maxmum Story Ductlty Fg. 0. Errors n dsplacement demands obtaned by modfed shear-buldng models as a functon of a ground moton ntensty; b maxmum story ductlty, 0-story model subjected to El Centro 940 Fg.. Comparson of the global damage ndex of 0-story frame obtaned by modfed shearbuldng model and full-frame model subjected to 5 synthetc earthquaes 27

29 Table. Maxmum errors n estmated dsplacement demands usng conventonal and modfed shearbuldng models, Average of 5 synthetc earthquaes Max error n roof dsplacement % Max error n story dsplacement % Max error n story drft % 5-Story 0-Story 5-Story Conventonal 7.5% 7.5% 20.4% Modfed 3.3% 4.% 8.4% Conventonal 2.0% 5.6% 45.9% Modfed 6.9% 9.6% 6.% Conventonal 6.3% 5.% 38.6% Modfed 3.9% 7.8%.3% 28

30 Table 2. Natural perods and total computatonal tme for full-frame model and the correspondng modfed shear-buldng model 5-Story 0-Story 5-Story Frame Modfed Shear-Buldng Frame Modfed Shear-Buldng Frame Modfed Shear-Buldng Perod sec st Mode nd Mode rd Mode Total Computatonal Tme sec