SEISMIC DRIFT PERFORMANCE-BASED DESIGN OPTIMIZATION OF REINFORCED CONCRETE BUILDINGS

Transcription

1 3 th World Conference on Earthqake Engneerng Vancover, B.C., Canada Agst -6, 2004 Paper No. 223 SEISMIC DRIFT PERFORMANCE-BASED DESIGN OPTIMIZATION OF REINFORCED CONCRETE BILDINGS Xao-Kang ZO and Chn-Man CHAN 2 SMMARY Performance-based desgn s a modern approach to sesmc engneerng, n whch the desgn am s to delver a strctre capable of meetng certan predctable performance objectves nder dfferent levels of earthqake motons. Performance-based desgn sng nonlnear pshover analyss, whch generally nvolves tedos and ntensve comptatonal effort, s a hghly teratve process to meet desgnerspecfed and code reqrements. Ths paper presents an effectve compter-based technqe that ncorporates pshover analyss together wth nmercal optmzaton procedres to atomate the pshover drft performance desgn of renforced concrete (RC) bldngs. Steel renforcement, as compared wth concrete materals, appears to be the more cost-effectve materal that can be effectvely sed to control drft beyond the occrrence of frst yeldng and to provde the reqred dctlty of RC bldng frameworks. In ths stdy, steel renforcement ratos are taken as desgn varables drng the desgn optmzaton process. sng the prncple of vrtal work, the nonlnear nelastc sesmc drft responses generated by the pshover analyss can be explctly expressed n terms of steel renforcement desgn varables. An Optmalty Crtera technqe s presented n ths paper to solve the explct performancebased sesmc desgn optmzaton problem for RC bldngs. One bldng frame example s presented to llstrate the effectveness and practcalty of the proposed optmal desgn method. INTRODCTION The concept of performance-based desgn has become the ftre drecton of sesmc desgn codes [-3]. In the newly developed performance-based sesmc desgn approach, nonlnear analyss procedres become mportant n dentfyng the patterns and levels of damage to assess a strctre s nelastc behavor and to nderstand the modes of falre of the strctre drng severe sesmc events. Pshover analyss s a smplfed, statc, nonlnear procedre n whch a predefned pattern of earthqake loads s appled ncrementally to framework strctres ntl a plastc collapse mechansm s reached. Ths analyss method generally adopts a lmped-plastcty approach that tracks the spread of nelastcty throgh the formaton of nonlnear plastc hnges at the frame element s ends drng the ncremental loadng process. In general, the determnaton of the satsfactory performance response that flflls both the system level Research Assocate. Department of Cvl Engneerng, Hong Kong nversty of Scence and Technology, Hong Kong, P.R. Chna. E-mal: cezxk@st.hk. 2 Assocate Professor. Department of Cvl Engneerng, Hong Kong nversty of Scence and Technology, Hong Kong, P.R. Chna. E-mal: cecmchan@st.hk.

2 response and element level response reqres a hghly teratve tral-and-error desgn procedre even wth the ad of today s engneerng compter software. It has been recognzed that the nterstory drft performance of a mltstory bldng s an mportant measre of strctral and non-strctral damage of the bldng nder varos levels of earthqake moton [4]. In performance-based desgn, nterstory drft performance has become a prncpal desgn consderaton [,2]. The system performance levels of a mltstory bldng are evalated based on the nterstory drft vales along the heght of the bldng nder dfferent levels of earthqake moton [5]. The control of nterstory drft can also be consdered as a means to provde nform dctlty over all stores of the bldng. A large story drft may reslt n the occrrence of a weak story that may case catastrophc bldng collapse n a sesmc event. Therefore, nform story dctlty over all stores for a mltstory bldng s sally desred n sesmc desgn [6]. Althogh lateral drft performance s a prncpal concern n the sesmc desgn of strctres, economcally desgnng elements of bldng strctres for varos levels of elastc and nelastc lateral drft performance nder mltple levels of earthqake load s generally a rather dffclt and challengng task. Lateral drft desgn reqres the consderaton of a proper dstrbton of the stffness of all strctral elements and, n a severe sesmc event, also the occrrence and redstrbton of plastcty n the strctral elements. Strctral engneers are faced wth the problem of effcently proportonng strctral materals throghot the bldng to lmt the nelastc sesmc drft responses of a strctre. De to the lack of an atomated optmzaton technqe, performance-based sesmc drft desgn s sally carred ot by tral-and-error methods based on ntton and experence. Chan [7] developed an effcent compterbased optmzaton technqe for lateral stffness desgn of tall bldngs. Althogh ths research has reslted n actal applcatons to nmeros notable tall bldng projects n Hong Kong, t shold be noted that the research has been prmarly focsed on the elastc wnd drft performance of tall bldngs. Mch effort s stll needed to extend the crrent optmzaton technqe to nelastc sesmc desgn of mlt-story bldngs. Ths paper presents an effectve optmzaton technqe for the nelastc drft performance desgn of RC bldng frames nder pshover loadng. Attempts have been made to atomate the performancebased sesmc desgn of RC bldngs sng an optmzaton procedre. The qanttes of steel renforcement, the only effectve materal that provdes dctlty to RC bldng frameworks, are consdered as desgn varables n the nelastc sesmc drft optmzaton. Wth carefl trackng of the formaton of plastc hnges, the pshover drft can be explctly expressed n terms of the szng varables sng the prncple of vrtal work. The optmzaton methodology for the solton of the nonlnear sesmc drft desgn of bldngs s fndamentally based on an Optmalty Crtera (OC) approach. A tenstory, two-bay planar frame bldng s then presented to llstrate the detals of the OC optmzaton method for nelastc sesmc drft performance-based desgn. OPTIMAL INELASTIC SEISMIC DESIGN PROBLEM Implct Desgn Optmzaton Problem In sesmc desgn, t s commonly assmed that a bldng behaves lnear-elastcally nder mnor earthqakes and may respond nonlnear-nelastcally when sbjected to moderate and severe earthqakes. nder sch an assmpton, the entre desgn optmzaton process can therefore be decomposed nto two phases [8, 9]. In the frst phase, the strctral concrete cost s mnmzed sbject to elastc drft responses nder mnor earthqake loadng sng elastc response spectrm analyss. In ths phase, concrete member szes are consdered as desgn varables snce the concrete materal plays a more domnant role n mprovng the elastc drft performance of the bldng. Once the optmal strctral member szes are determned at the end of the frst phase of the optmzaton, the steel renforcement qanttes can then be consdered as desgn varables n the second phase. In controllng the nelastc drft responses, steel renforcement s the only effectve materal that provdes dctlty to an RC bldng strctre beyond frst yeldng. In ths second desgn phase, the member szes are kept nchanged and the cost of the steel

3 renforcement s mnmzed sbject to desgn constrants on nelastc nterstory drft prodced by the nonlnear pshover analyss. The emphass of ths paper s on the second phase of the desgn optmzaton, the nelastc sesmc drft desgn optmzaton. The detals of the frst phase elastc sesmc drft desgn optmzaton can be fond n the work of Zo [9]. For an RC bldng havng =, 2,, N members and 2N plastc hnges (assmng one hnge at each end of a member), the tenson steel renforcement rato, ρ, and the compresson steel renforcement rato, ρ, for a rectanglar cross secton are taken as desgn varables n the desgn optmzaton, whereas the member szes, B (wdth) and D (depth), are fxed. If the topology of a bldng s strctral system s predefned, the steel renforcement cost of the RC framework s mnmzed as N Mnmze: steel cost = ws ( Ls ρ + Ls ρ ) () = where w s s the cost coeffcent for steel renforcements; and L s and L s are respectvely the lengths of the tenson and compresson steel renforcements for member. Only the longtdnal flexral renforcement of member sectons s consdered as desgn varables n ths stdy, whle the transverse shear renforcement s consdered nvarant nder the assmpton that adeqate shear capacty strength s provded for each member. In the performance-based desgn, t s necessary to check the capacty of a strctre aganst the demand of an earthqake at the performance pont whch s the ntersecton of the pshover capacty and demand spectrm crves. In ths stdy, the nterstory drft responses of a bldng, generated by a specfed earthqake demand, are checked aganst approprate lmts correspondng to a gven performance level. Namely, for a mltstory bldng strctre, the nterstory drft rato cased by pshover loadng shold comply wth the followng reqrement: j j j = d j (2) h j h j where j s the nterstory drft of the jth story; j and j are the respectve story dsplacement of two adjacent j and j- floor levels; h j s the jth story heght; and d j s the specfed nterstory drft rato lmt representng the damage threshold for the jth story. Besdes the consderatons of the nterstory drft responses, local element responses sch as the sectonal plastc rotaton and member strength at the performance pont mst also be checked aganst certan acceptablty lmts. The plastc rotaton, θ, at the hth end secton of a member (where sbscrpt ph h represents one end of a member and h=, 2) shold be checked as θ ph θ p (3) where θ s the plastc rotaton lmt correspondng to a specfc performance level. Once the desgner p determnes the performance levels of the strctre (e.g., Immedate Occpancy, Lfe Safety, Collapse Preventon), the lmtng vales of θ for all members s then determned. nless specfc desgn crtera p are stated, otherwse FEMA-273 [3] and ATC-40 [2] provde gdelnes for estmatng the lmtng vales of plastc rotaton of a flexral member for varos levels of performance crtera of an RC frame. In practcal mltstory bldng strctres, excessve nmber of desgn constrants may create enormos ncreases n comptatonal effort. In order to redce the practcal bldng desgn problem to a manageable sze, the strength desgn of each member s not consdered explctly as a desgn constrant; rather, the strength-based steel renforcement ratos n accordance wth code specfcatons are frst calclated and these vales are then taken as the lower sze bond for each member n the nelastc sesmc drft desgn optmzaton.

4 In addton to the desgn performance constrants on the system-level story drft and element-level sectonal plastc rotaton, the steel renforcement varables are mposed wthn the mnmm and maxmm steel renforcement ratos as L L ρ ρ ρ ; ρ ρ ρ (4a, b) where the sperscrpts L and denote the mnmm and maxmm lmts of the desgn varables, ρ and ρ. In order to facltate a nmercal solton of the drft desgn problem, t s necessary that the mplct story drft constrant (Eq. 2) and the plastc rotaton constrant (Eq. 3) be expressed explctly n terms of the desgn varables, ρ and ρ. Explct Drft Formlaton Based on the nternal element forces and moments of the strctre obtaned from the pshover analyss at the performance pont, the prncple of vrtal work can be employed to express the pshover dsplacement. The pshover story dsplacement, j, at the performance pont ncldes the vrtal work, j, memb, prodced by the strctral members and the vrtal work, j, hnge, generated by the plastc hnges. That s, j = j, memb + j, hnge (5) n whch N C 0j C j C2 j = j, memb( B, D ) = B D B D B D (6) N 2 0 j, hnge = mpjhθ ph (7) = h= In Eq. (6), the dsplacement, j, memb, s expressed n terms of wdth ( B ) and depth ( D ) [7, 9]. Drng the nelastc drft desgn optmzaton process, j, memb s kept nchanged snce B and D of each member secton are fxed. The emphass here s on the dsplacement,, cased by the formaton of j, hnge 0 the plastc hnges. In Eq. (7), m pjh s the vrtal end moment at the locaton of the hth hnge of a member; θ ph s the actal plastc rotaton experenced by the hth plastc hnge, whch s eqal to zero when no plastc hnge s fond. As shown n Fg., the behavor of a plastc hnge s modeled as a blnear crve: the elastc segment, AB, and the hardenng segment, BC, where Pont A corresponds to the nloaded condton, Pont B s the frst yeld moment pont, Pont C s the ltmate moment capacty, whch generally corresponds to the strctral stablty performance level n ATC-40 [2]. Based on the lne segments A-B-C, the plastc rotaton, θ, can be gven as follows p M M y θ p = θ p θ p (8) M M y where θ p s the ltmate plastc rotaton whch can be establshed based on expermental tests or can be obtaned drectly from desgn gdelnes sch as the ATC-40 [2]; M s the appled moment at the locaton of the plastc hnge; M y s the bendng moment at the frst yeldng of the tensle steel; and M s the ltmate moment resstance. Gven the qantty of the steel renforcement sed n a concrete secton, the vales of M and M can then be determned. For smplcty, M can be approxmately related to M y y

5 as M =.M y [2]. M M M M y B C A θ y θ p θ p Fg.. Moment-rotaton crve For the explct problem formlaton, t s necessary that the plastc rotaton, θ p, be accrately expressed n terms of the desgn varables (.e., ρ and ρ ' ). Frthermore, a good formlaton shold reflect accrately the change n the plastc rotaton, θ, de to a change n the desgn varables drng the p optmzaton reszng process. In other words, any change n the desgn varables, ρ and ρ, drng the nelastc optmzaton process reqres a correspondng pdate on the vales of M and M y. θ D d ρ' ρ B d' d' d kd f y d' f c f s Netral axs f s Bdρ' 0.5 f c Bkd f y Bdρ M y Secton Stress Resltant force Fg. 2. Doble renforced member secton at frst yeld In pshover analyss, moment hnges are assmed and are generally assgned to the two ends of each beam or colmn. By the force eqlbrm shown n Fg. 2, where f c s the stress at the extreme compresson concrete fber, f s s the stress n the compresson steel, f y s the yeld strength of the tenson steel, and d s the effectve depth, whch s eqal to the dstance from the extreme compresson fber to the centrod of the tenson steel, M y for a moment hnge can be expressed n terms of desgn varables, ρ and ρ, as kd M y = 0.5 f c Bkd( d') + f y Bd( d d')ρ (9) 3 where k s the netral axs depth factor at the frst yeld and t s gven as 2 2 d' k = ( ρ + ρ ) nsc + 2( ρ + ρ ) nsc ( ρ + ρ ) nsc (0) d

6 E n whch n s sc = where E c and E s are the modl of elastcty of the concrete and of the steel, Ec respectvely. To take nto accont the change n θ p de to a change n ρ and ρ whle mantanng an nstantaneosly fxed vale of M, a second-order Taylor seres approxmaton for evalatng the vale of θ s gven as p 2 θ θ θ ( ρ) = p p p θ p ( ρ ρ ) + 0 ( ρ ρ ) () ρ= ρ ρ= ρ 2 ρ ρ ρ 2 ρ = where the tenson steel rato, ρ, s consdered as the major desgn varable; for smplcty, the compresson steel rato, ρ, s assmed to be lnearly related to ρ for beams and to be the same as ρ for colmns. Gven the explct expresson of M y as a fncton of ρ from Eq. (9), the gradent, 2 θ p θ, and the second-order term, p, can be analytcally calclated from Eq. (8). 2 ρ ρ By sbstttng the explct plastc rotaton, (ρ), gven n Eq. () nto Eq. (7), the pshover dsplacement, θ p j, n Eq. (5) can also be explctly expressed n terms of the desgn varable, Plastc Rotaton Constrant and Szng Constrant In ths desgn optmzaton, when the plastc rotaton, θ p, s to be modfed wth changes n the desgn varable, ρ, t s necessary to make sre that θ p does not exceed the specfed threshold of plastc rotaton, θ p, for each specfed performance level. Moreover, n order to prevent drastc changes n the nternal element force and moment redstrbton de to the changes n the desgn varables resltng n flctaton of solton convergence drng the pshover reanalyss and desgn optmzaton processes, t s necessary that each plastc hnge reman plastc once t appears drng the reszng teraton of the desgn varables. Frthermore, to mantan the accracy of the Taylor approxmaton of the pshover dsplacement n Eq. (), t s necessary to ensre that the varaton of ρ for the members wth plastc hnges be restrcted wthn a relatvely small range. It s fond from Fg. that, n order to mantan the relatonshp of 0 θ, the nternal moment, p θ p M, leadng to the occrrence of a plastc hnge mst satsfy the followng condton: M M (2) y M As a reslt, based on Eq. (2), the lower and pper bonds of ρ for each plastc hnge can be nstantaneosly establshed drng the OC teratve reszng process. It shold be noted that the proper establshment of the lower and pper bonds of ρ not only lmts the changes n the steel renforcement desgn varables, bt also t satsfes the local performance-based constrants on the control of the local plastc rotaton at the ends of members. Explct Desgn Problem Formlaton pon establshng the explct nelastc drft formlaton, the optmzaton problem of mnmzng the steel constrcton cost of a mltstory RC bldng can be explctly wrtten n terms of the desgn varable, ρ, as Mnmze: N = = ρ. F( ρ ) w s ρ (3)

7 sbject to: where N N g j ( ρ ) = j 0 + α + d j = ( ρ ρ ) α 2 ( ρ ρ ) ρ ρ h j = 2 = j =, 2,..., N ) (4) ( j ρ ρ ρ =, 2,..., N ) (5) L = j = 2 0 ρ = ρ ρ h= ( θ 0 ph α m jh 0 (6a) ρ = ρ ρ 2 j ph = = 2 2 θ 0 α 2 0 m jh 0 (6b) 2 ρ = ρ 2 ρ = ρ ρ h= ρ In Eq. (3), w s s the cost coeffcent for the steel renforcement, ρ. Eq. (4) defnes the set of sesmc nterstory drft performance constrants nder specfed earthqake grond motons. Eq. (5) defnes the L szng constrants for the steel renforcement, where ρ and ρ correspond to the lower and pper sze bonds specfed for the tensle steel renforcement varable, ρ, and they shold be pdated after each nonlnear pshover analyss. Once the desgn optmzaton problem s explctly expressed n terms of desgn varables, the next task s to adopt a stable method for solvng the problem. The OC approach s proposed to solve the explct nelastc drft optmzaton problem Eqs. (3)-(5). When sng the OC technqe, a set of necessary optmalty condtons for the desgn are derved and then a recrsve algorthm s appled to resze the strctre to satsfy the optmalty condtons and ths ndrectly optmze the strctre. Frther detals of the OC technqe can be fond n the reference by Zo [9]. Intal Preprocessor Althogh the OC method does not mpose any restrctons on the ntal vales of the desgn varables, the rate of convergence of the OC process depends on the ntal desgn vales. To speed p the convergence of the OC process, t s mportant to begn wth a reasonably good startng desgn. One effectve scalng approach was proposed by Chan [0]. In sch an approach, the desgn optmzaton wth a sngle drft constrant s frst consdered and a smple closed form solton for the problem s derved analytcally and exploted as an ntal preprocessor for the teratve OC process. The advantage of ths approach s that a reasonable ntal desgn can be qckly establshed based on a representatve sngle drft constrant. Experence ndcates that the ntal preprocessor can generally lead to steady and rapd solton convergence of the mltple nelastc drft optmzaton problem. OVERALL DESIGN OPTIMIZATION PROCEDRE The overall desgn optmzaton procedre for lmtng lateral elastc and nelastc drfts of a renforced concrete bldng strctre s lsted as follows. () Establsh an ntal desgn wth optmal member szes, whch can be obtaned from the elastc sesmc desgn optmzaton by mnmzng the concrete cost of an RC strctre sbjected to a mnor earthqake loadng sng the elastc response spectrm analyss method [9]. (2) Determne the desgn spectrm, correspondng to a severe earthqake event, that wll be sed n the nonlnear pshover analyss. (3) Condct a statc vrtal load analyss to obtan the member nternal forces that wll be sed n formlatng nelastc drft responses by employng the prncple of vrtal work.

8 (4) Based on the optmal member sze, determne the mnmm and maxmm sze bonds of the steel renforcement ratos, ρ and ρ, n accordance wth the strength-based code reqrements. (5) Apply the ntal preprocessor based on a representatve sngle drft constrant to establsh a reasonable startng set of steel renforcement desgn varables for the mltple drft constraned optmzaton. (6) Carry ot the nonlnear pshover analyss sng commercally avalable software to determne the performance pont of the strctre and the assocated nelastc drft responses of the strctre at the performance pont. (7) Track down the locatons of the plastc hnges, establsh the nstantaneos lower and pper bond move lmts of ρ for those members wth plastc hnges based on Eq. (2) and determne the vales of the frst-order and second-order dervatves of the drft responses sng Eqs. (6a) and (6b). (8) Establsh the explct nterstory drft constrants sng a second-order Taylor seres approxmaton and formlate the explct desgn problem, Eqs. (3)-(5). (9) Apply the recrsve OC optmzaton algorthm to resze all steel renforcement desgn varables and to dentfy the actve nelastc drft constrants. (0) Check convergence of the steel cost and the nelastc drft performance of the strctre. Termnate wth the optmm desgn f the solton convergence s fond; otherwse, retrn to Step 6. ILLSTRATIVE EXAMPLE A ten-story, two-bay planar frame s sed to llstrate the proposed optmal desgn method. The geometry of the example s gven n Fg. 3. Concrete wth the cylnder strength of 20MPa and steel renforcement wth the yeld strength, f y, of 335MPa are sed for all members. The loads consdered n the pshover analyss are lateral sesmc loads and vertcal gravty loads. Whle the lateral loads are ncrementally appled, the gravty loads are mantaned to be nchanged drng the nonlnear pshover analyss process. A nformly dstrbted gravty load of 30kN/m s to be appled to the beams of each story. Intal member szes of the framework sed to commence the nelastc desgn optmzaton are shown n Table 3. Intal steel renforcement ratos are frst calclated based on the strength reqrements of the RC members n accordance wth the Chnese sesmc desgn code []. Sch strength-based renforcement ratos are taken ntally as the lower bonds for the nelastc desgn optmzaton. The pper sze bonds of the steel renforcement ratos are assmed to be 6.0% for colmns and 4.0% for beams. For smplcty, symmetrcal arrangement of the steel renforcement of each member s assmed sch that ρ = ρ. Flexral moment hnges are assgned to the end locatons of the beams and colmns and the ltmate plastc hnge rotaton, θ p, s assmed to be 0.02 radan The 5% damped desgn spectrm wth an ntal peak acceleraton of.4g accordng to the Chnese sesmc desgn code [] s modfed by the spectral redcton method n the pshover analyss of ths example. A typcal nt constrcton cost of the steel renforcement (ncldng the costs of the steel materal and the labor) of S$950/tonne s assmed. Interstory drft constrants are consdered wth an assmed allowable nterstory drft rato lmt of /00. The ntal preprocessor, n whch analytcal optmzaton wth the top dsplacement constrant alone s consdered, s appled before the mltple nelastc nterstory drft constraned optmzaton s nvoked. The desgn process s deemed to converge when the dfference n the strctre costs for two sccessve desgn cycles s wthn 0.5% and when the dfference between the actve nterstory drft vale and ts allowable lmt at the performance pont s wthn 0.5%.

9 q=30kn/m 3.0*0=30m C B C2 5m B2 C3 5m Fg. 3 A Ten-Story, Two-Bay Frame The optmal desgn hstory of the example s presented n Fg. 4. The ntal preprocessng s frst completed and the mltple nelastc nterstory drft constraned optmzaton s then commenced. It s fond that there s a relatvely large ncrease of 4% n the steel cost from the ntal S$693 to S$93 after the ntal preprocessng wth only the top dsplacement constrant. However, the optmal desgn process wth the mltple drft constrant converges slowly bt steadly wthn desgn cycles, wth only a slght dfference of 2% n the steel cost from S$93 to the fnal S$978. Relatvely slow, bt steady, convergence s fond de to the need for mantanng a small change n the steel renforcement ratos drng the nelastc desgn optmzaton process Steel renforcement cost (S$) Desgn hstory Fg. 4 Desgn Hstory of Steel Renforcement Costs

10 Table Intal and Fnal Steel Renforcement Ratos Element Story Member Intal member szes Steel ratos type level grop Wdth Depth Intal Optmal (mm) (mm) (%) (%) 9th~0th C,C C th C,C C Colmn 7th C,C C th C,C C th C,C C th C,C C rd C,C C nd C,C C st C,C C th~0th B,B th B,B Beam 7th B,B th B,B th B,B th B,B rd B,B nd B,B st B,B Table presents the ntal and optmal steel renforcement ratos. Intally, the startng desgn wth strength-based steel renforcement s fond to be nfeasble n terms of the assmed allowable nterstory drft lmt. After the optmzaton, the steel renforcement ratos of the beams greatly ncrease partclarly n the lower levels of the strctre, whle those of colmns are fond wth lttle changes. Fg. 5 presents the ntal and fnal performance ponts respectvely. The performance pont P of the ntal strctre has a spectral acceleraton capacty of 0.068g and a spectral dsplacement capacty of 0.262m. The optmzed strctre correspondng to the fnal performance pont P2 has a spectral acceleraton capacty of 0.086g and a spectral dsplacement of 0.2m. Sch a reslt of shftng the spectral dsplacement from 0.262m to 0.2m ndcates that, for the optmzed nelastc frame, the nelastc lateral load resstance has been enhanced throgh optmal reszng of the steel renforcement by the OC procedre. Also, shftng the ltmate spectral acceleraton capacty from 0.068g to 0.085g ndcates that the optmzed strctre attracts an ncrease n the sesmc loadng acton and therefore, reqres the strctre to be stffened. The OC procedre developed s fond to be able to atomatcally drve from any ntal performance pont to the fnal performance pont resltng n the mnmm cost desgn.

11 Spectral Acceleraton, Sa (g) Fnal demand Intal demand Fnal capacty Intal capacty Fnal performance pont (0.2, 0.086g) P2 P Intal performance pont (0.262, 0.068g) Spectral Dsplacement, Sd (m) Fg. 5 Comparson of Performance Ponts 0 8 Drft rato lmt=/00 Intal drft rato Fnal drft rato Story level Interstory drft rato Fg. 6 Intal and Fnal Interstory Drft Ratos The ntal and fnal nterstory drft ratos are shown n Fg. 6. The ntal nterstory drft constrants at the second throgh the eghth floors are fond to volate sbstantally the allowable nterstory drft rato lmt of /00, resltng n the occrrence of the weak stores on the these floor levels of the bldng. However, these pshover nterstory drft constrants are fond to be close to and wthn the allowable vales after the optmzaton, ndcatng that a rather nform nterstory drft dstrbton over the heght of the bldng has been acheved and the occrrence of weak story has been prevented at the optmm performance pont.

12 Nmber of plastc hnges Total nmber B-IO IO-LS LS-CP of plastc hnges Intal desgn Fnal desgn (a) Nmber of Plastc Hnges A M B IO C CP LS B-IO: IO-LS: LS-CP: θ (b) Intal Desgn (c) Fnal Desgn Fg. 7 Intal and Fnal Plastc Hnge Dstrbton Fg. 7(a) ncldes a table showng the nmber of plastc hnges at three dfferent performance states. Fgs. 7(b)-7(c) show the ntal and fnal plastc hnge dstrbtons nder the pshover loadng at the performance pont of the strctre. No plastc hnge rotaton s fond to exceed the specfed threshold of plastc rotaton. As shown n Fg. 7(b), the rotatons of twenty plastc hnges of the ntal desgn are fond to be located between the LS-CP state. However, after the optmzaton, most of the plastc hnges are fond to be n the B-IO and IO-LS states and only one hnge s n the LS-CP state, as can be observed from the optmzed framework n Fg. 7(c). Frthermore, the nterstory drfts along the heght of the bldng are also fond to be almost all flly constraned at the optmm, resltng n a rather lnear deflected profle of the nelastc desgn. Sch a reslt frther ndcates that the optmzaton method developed can atomatcally resze the steel renforcements of all members to attan a nform dctlty demand along the heght of the mltstory bldng.

13 CONCLSIONS It has been demonstrated that steel renforcement plays a sgnfcant role n controllng the lateral drft beyond frst yeldng and n provdng dctlty to an RC bldng framework. sng the prncple of vrtal work and the Taylor seres approxmaton, the nelastc performance-based sesmc desgn problem has been explctly expressed n terms of the steel renforcement desgn varables. It s demonstrated that the OC desgn method s able to mprove atomatcally and gradally a performance-based nterstory drft desgn to attan optmal performance. Also, ths OC desgn method developed s able to atomatcally shft any ntal performance pont to acheve the fnal optmal performance pont. However, the restrctve move lmt mposed on the steel renforcement desgn varables s necessary to ensre a smooth and steady convergence of the nelastc drft desgn process. At optmm, a nform lateral drft or dctlty demand over all stores of the bldng wth the mnmm cost s acheved. It s also beleved that ths optmzaton methodology provdes a powerfl compter-based technqe for performance-based desgn of mltstory RC bldng strctres. The proposed optmzaton methodology provdes a good bass for more comprehensve performance-based optmzaton of strctres as more accrate nonlnear pshover procedres takng nto the hgher mode effects are developed and mltple levels of performance crtera and desgn objectves are to be smltaneosly consdered. ACKNOWLEDGMENTS The athors are gratefl for the fnancal spport provded by the Research Grants Concl of Hong Kong nder Project No. HKST6249/00E. REFERENCES. SEAOC Vson 2000 Commttee. Performance Based Sesmc Engneerng of Bldngs, Part 2: Conceptal Framework. Sacramento, Calforna: Strctral Engneers Assocaton of Calforna, ATC-40. Sesmc Evalaton and Retroft of Concrete Bldngs. Volme, ATC-40 Report. Redwood Cty, Calforna: Appled Technology Concl, FEMA. NEHRP Gdelnes for the Sesmc Rehabltaton of Bldngs. Washngton, D.C., SA: Bldng Sesmc Safety Concl for the Federal Emergency Management Agency (Report No. FEMA 273), Moehle JP, Mahn SA. Observatons on the Behavor of Renforced Concrete Bldngs Drng Earthqakes. Ghosh SK, Edtor. Earthqake-Resstant Concrete Strctres Inelastc Response and Desgn. Amercan Concrete Insttte SP-27, Ghobarah A, Aly NM, El-Attar M. Performance Level Crtera and Evalaton. Fajfar P, Krawnkler H, Edtors. Proceedngs of the nternatonal workshop on Sesmc Desgn Methodologes for the Next Generaton of Codes, Slovena, Chopra AK. Dynamcs of Strctres: theory and applcatons to earthqake engneerng. Englewood Clffs, New Jersey, Chan CM. Optmal Lateral Stffness Desgn of Tall Bldngs of Mxed Steel and Concrete Constrcton. Jornal of Strctral Desgn of Tall Bldngs 200; 0: Zo XK, Chan CM. Optmal Drft Performance Desgn for Nonlnear Pshover Response of Concrete Strctres. WCSMO-4: Proceedngs of the Forth Wold Congress of Strctral and Mltdscplnary Optmzaton. Jne 4-8, Dalan, Chna, Zo XK. Optmal Sesmc Performance-Based Desgn of Renforced Concrete Bldngs. Ph.D. Dssertaton, Hong Kong nversty of Scence and Technology, 2002.

14 0. Chan CM. How to Optmze Tall Steel Bldng Frameworks. Gde to Strctral Optmzaton, ASCE Manals and Reports on Engneerng Practce No.90, Amercan Socety of Cvl Engneers, 997; Natonal Standard of the People s Repblc of Chna. Chnese Code for Sesmc Desgn Bldngs (GBJ-89). Bejng, Chna: New World Press, 994.