DISPLACEMENT-BASED SEISMIC DESIGN OF MDOF CONTINUOUS GIRDER BRIDGES

Transcription

1 3 th World Coferece o Earthquake Egeerg Vacouver, B.C., Caada August -6, 004 Paper No. 05 DISPLACEMENT-BASED SEISMIC DESIGN OF MDOF CONTINUOUS GIRDER BRIDGES X ZHU ad Jawe HUANG SUMMARY The proposed procedure s appled to the trasverse sesmc respose desg of MDOF cotuous grder brdges wth flexble or rgd grders ad reforced cocrete pers o the trasverse drecto of the brdges. The predetermed target dsplacemet ad dsplacemet ductlty of each brdge per to susta a assged damage level are defed as the desg parameters. The ects of trasverse modes of rregular brdges ad the modal mass dstrbuto due to the relatve stffess betwee superstructures ad colums are vestgated. The drect dsplacemet-based desg method (DDBD for MDOF system cludg the fluece of hgher order modes s preseted ad the formula of equvalet dampg of the MDOF equvalet substtute system s also proposed. I order to drectly reflect the olear behavor of the elastc system, the spectral dsplacemet of each mode respodg to modal perod of elastc orgal structural system s determed from the elastc dsplacemet desg spectra related to the ectve system dampg ad ductlty of each mode. The overall dsplaced shape s the obtaed by SRSS method. After determg the target-dsplaced shape of elastc system, the proposed procedure characterzes the MDOF system as a equvalet SDOF system. The system hysteretc dampg ad ectve system mass cosderg the ects of hgher modes are determed. The ectve perod of the equvalet SDOF system s obtaed by eterg the elastc dsplacemet respose spectra wth the system target dsplacemet. The the ectve stffess ad lateral resstg force of the equvalet SDOF system ca be obtaed. The complete desg procedure s outled for the proposed method ad there s o terato ths procedure. The elastc dsplacemet desg spectra compatble wth the spectrum of dyamc amplfcato factor the UBC97 s preseted. The elastc dsplacemet desg spectra based o structural damage performace cludg ductlty factor s also costructed ths paper. Ths s followed wth some desg examples ad the valdato study was performed wth dyamc elastc tme-hstory aalyss usg the ear-source earthquake records to cofrm the proposed method s reasoable. INTRODUCTION The drect dsplacemet-based sesmc desg method has attracted cosderable research the sesmc desg ad retroft of brdges[] due to ts drect relatoshp wth structural deformato demad ad damage performace. Usg the substtute structure model show Fg., Kowalsky, Calv et.al[,3,4] characterzed RC brdge colums ad MDOF cotuous cocrete brdges to the equvalet SDOF substtute structure, ad proposed the dsplacemet-based sesmc desg phlosophy ad comprehesve Professor, Beg Jaotog Uversty, Beg, PR.Cha. E-mal: xzhu@ceter.tu.edu.c Doctor Caddate, Beg Jaotog Uversty. E-mal: teddy003hw@yahoo.com.c

2 desg procedure. Juqg Gog et.al[5] preseted a mproved o-teratve dsplacemet-based sesmc desg method for RC brdge colums usg elastc costat-ductlty dsplacemet respose spectra. Ths method defed the dsplacemet demad ad equvalet ductlty of structure as the desg targets, ad used the elastc stffess of orgal structure as the ectve stffess of the substtute structure, ad the estmated the ectve perod of elastc SDOF system cosstet wth the elastc dsplacemet respose spectra. O the bass of elastc respose spectrum-based approach cludg the ects of hgher modes, ths paper proposed a mproved dsplacemet-based sesmc desg method based o elastc costat-ductlty dsplacemet respose spectra for MDOF cotuous grder brdges, whch s appled to the trasverse respose of brdge structures. The detaled desg procedure, sample brdge desgs for symmetrcal ad asymmetrcal brdges show Fg., ad valdato studes usg dyamc elastc tme-hstory aalyss are llustrated as followg. F F u F Fy αk y K sec K u K y (amode y u (bsecat stffess characterstcs Fg. Substtute Structure Model D (asymmetrcal Brdge (basymmetrcal Brdge Fg. MDOF Cotuous Grder Brdge DISPLACEMENT RESPONSE SPECTRA Elastc Dsplacemet Respose Spectra S de (m Fg.3 Accelerato respose spectra of UBC ξ=0.05 ξ=0.0.6 ξ=0.5 ξ=0.0.4 ξ=0.30 ξ=0.40. ξ= T/s Fg.5 Elastc dsplacemet respose spectra S d (m Dyamc amplfcato factor β S B S C S D S E T /s Fg.4 Dyamcal amplfcato factorβspectra.0.8 =.0,ξ=0.05 =.5,ξ=0.0.6 =.0,ξ=0.3 =3.0,ξ= =4.0,ξ=0.85 =5.0,ξ=0.97. =6.0,ξ= T/s Fg.6 Ielastc dsplacemet respose spectra

3 The approprate groud respose cocets C a ad C v are defed ad the ear-source amplfcato factors N a ad N v are troduced the accelerato respose spectra show Fg.3 of UBC97[6] order to reflect the ear-source ects o the structural respose. The spectrum of dyamcal amplfcato factor respodg to four dfferet sol profle types of Zoe 4 the UBC97 s preseted Fg.4, ad the elastc dsplacemet desg spectra uder the dfferet a g of desg peak acceleratos compatble wth the spectrum of dyamcal amplfcato factor s costructed usg equato (: T ( S de = ηa g β T ( π where T s the atural perod of the SDOF structure; η s the modfed cocet of dampg, whch s expressed as η =.07 ( ξ 0 order to cosder the ects of dampg property o the structural dsplacemet respose. I ths study, the elastc dsplacemet respose spectrum related to S D sol type uder the desg peak accelerato 0.4g for varous levels of vscous dampg s geerated show Fg.5. Ielastc Dsplacemet Respose Spectra based o Damage Performace I ths study, the Park-Ag[7] model was employed for the assessmet of sesmc damage whch s expressed as a lear combato of the damage caused by maxmum elastc deformato ad the cumulatve damage resulted from repeated cyclc respose, ad the dual-damage crtero matched the expermet result well. Park ad Ag suggested the total damage could be represeted terms of a damage dex DI as followg: δ E H E H DI = + β = + β ( δ u Pyδ u u Pyδ y u Ag et al.(993[8] suggested the followg detaled classfcato to descrbe the structural damage performace: DI<0. No damage or localzed mor crackg 0. DI<0.5 Mor damage-lght crackg throughout 0.5 DI<0.40 Moderate damage-severe crackg, localzed spallg 0.40 DI<.0 Severe damage-crushg of cocrete, reforcemet exposed DI>.0 Collapsed A formula of equvalet ductlty cocet was preseted by Farfar et.al[9] uder Park-Ag dualdamage crtero as: + 4DIβγ u = (3 βγ E m where γ H / =, γ value s desgated to 0.8, β value s fxed to 0.5, ad the ultmate ductlty ωδ cocet u resulted from the mootoc loadg expermet s assged to be 0 ths paper. Equato (3 dcates that the equvalet ductlty cocet s drectly related to the structural damage dex DI. The stregth reducto factors R [0], drectly related to the equvalet ductlty cocet resulted from the damage dex DI uder the well-kow Park-Ag dual-damage crtero, had bee studed. By usg the obtaed R, the costat-ductlty elastc dsplacemet respose spectra show Fg.6 were costructed based o the elastc dsplacemet spectra show Fg.5. The relatoshp betwee those spectra was expressed as:

4 Normalzed modal mass(% S = (4 d S de R THE SIMPLIFIED DISPLACEMENT-BASED DESIGN PROCEDURE M ode 40 M ode Stffess of deck( K 0 T(s M ode M ode Stffess of deck( K 0 (a (b Fg7 Effects of varety of deck stffess o ormalzed modal mass ad atural modal perods For geeral regular brdge structures, the trasverse momet-resstg stffess of deck s very large so that the fudametal modal ects resultg from trasverse vbrato of the superstructure plays the predomat role durg the total respose of structures. Wth the decrease of the trasverse stffess of the superstructure, the trasverse vbrato respose of brdge structures s ot oly cotrolled by the frst modal ect predomatly resulted from superstructure respose, but also by the multple modes ects cotrbuted from the trasverse vbrato of the system. Especally for the asymmetrcal brdge show Fg.(b, the crease of per stffess ad the great dfferece of the stffess betwee adacet pers have salet ect o the trasverse vbrato of structures. The ects of varety of deck stffess o the frst two ormalzed modal masses ad atural modal perods of trasverse vbrato for the asymmetrcal brdge are see Fg.7(a(b, whch K 0 s the orgal trasverse momet-resstg stffess of deck. Wth the crease of the deck stffess, the frst ormalzed modal mass of trasverse vbrato creases gradually, ad the ects of the hgher modes decrease accordgly. If the trasverse momet-resstg stffess of deck s large eough, cosderato of the basc modal ect s sutable the dsplacemetbased desg procedure, whle t must cosder multple mode ects for the brdge structures of flexble superstructure. The proposed dsplacemet-based sesmc desg procedure for MDOF brdge structures s summarly descrbed as followg: frstly to perform modal aalyss for the elastc MDOF system, the to calculate the modal parameters such as the modal shape, modal partcpato factors ad correspodg modal perods of the orgal elastc MDOF system. The key step the complete desg procedure s to determe the spectral dsplacemet of each mode for pers by eterg the elastc dsplacemet respose spectra wth referece to the equvalet ductlty, equvalet vscous dampg ad modal perods of each mode. The ext mportat step s to obta the dsplacemet at top of each per usg SRSS method to cosder the multple mode ects, ad the to determe the target dsplacemet patter of MDOF system the trasverse drecto accordg to the calculated dsplacemet at top of each per ad the prescrbed damage lmt states for pers. Havg establshed the approach for determg the targetdsplaced shape, the ext step s to characterze the MDOF system of mult-spa brdge as a equvalet SDOF system. The varous system propertes of the equvalet SDOF system such as system target dsplacemet, system hysteretc dampg ad ectve system mass must be defed. After the equvalet SDOF system has bee detfed, the ectve perod of the substtute structure s obtaed by eterg the elastc dsplacemet respose spectra show Fg.6 wth the system target dsplacemet wth referece to the equvalet ductlty ad system hysteretc dampg, ad the the ectve stffess at maxmum respose ad base shear force of the equvalet SDOF system ca be obtaed. The last step s to perform the lateral force dstrbuto, the cross secto desg ad the capacty check for pers. Usg the same desg procedure for the symmetrcal ad asymmetrcal brdges show Fg., the mportat dfferece betwee them s that the proposed desg procedure ca obta the ratoal desg results by cosderg ect of the frst mode for symmetrcal brdge, whle t should cosder multple mode ects for asymmetrcal brdge. The detaled desg process s descrbed as followg: At frst, the target dsplacemet crtera of each bet ca be specfed as[4]:

5 0 = drft h (5 where, drft meas the desg drft rato determed by damage cotrol lmt states, ths paper t s value s fxed to %; h meas the heght of the th per. Accordg to the lear elastc system of MDOF brdge structures uder the cosstet oscllato of sesmc groud moto, the equato of moto s wrtte the form: [ M ]{ X& ( t } + [ C] { X& ( t } + [ K ]{ X ( t } = [ M ]{} u& g (6 Equato(6ca be decoupled usg modal aalyss procedure: q & + ξ ω q& + ω q = γ u& g ( =,L (7 where, u g s the dsplacemet of the put groud moto; ω ad ξ are the atural frequecy ad dampg rato of the th mode respectvely; γ represets the th modal partcpato factor as: T mφ {}[ φ M ]{} = γ = = (8 T {}[ φ M ]{} φ m φ where the dex represets the bet umber ad the dex represets the mode umber, m s the equvalet mass of th per, adφ s the modal fucto for bet ad mode. s the dsplacemet quatty at the top of per assocated wth bet ad mode, ad the ca be expressed as: = φ γ Sd ( T, ξ (9 equato (9, the elastc spectral dsplacemet Sd ( T, ξ for mode s obtaed from the elastc dsplacemet respose spectra Fg.6 by cosderg the th modal perod, equvalet ductlty cocet ad equvalet system dampg value ξ for mode. It s oted that T s correspodg to the th modal perod of the orgal elastc MDOF system take as the substtute structure the desg procedure, whch ca be easly obtaed from the prevously performed modal aalyss. The equvalet ductlty cocet ad equvalet system dampg value ξ for mode ca be expressed as: Qφ h ξ = w ξ = w w = = = Q ( ( Q = M a + M b M am b (0 6 EI = k = kφ k where w s the weghtg factor for bet ad mode, M a ad M b are the member edg momets of th per respectvely determed by the tal structural aalyss, ξ s the equvalet vscous dampg rato of the th per[]: ( α α ξ = ( π where s the equvalet ductlty cocet of the th per, α s the post-yeldg stffess rato assumed to be Through respose spectrum aalyss cosderg hgher mode ects, the overall dsplaced shape the trasverse drecto of the MDOF brdge structure s the obtaed by a approprate combato of modes such as SRSS or CQC method ad cosderg the target dsplacemet crtera of each bet. The dsplacemet patter of each bet s determed by equato(: = m = (

6 After determg the overall target dsplacemet shape of MDOF brdge structures, the MDOF system ca be characterzed as a equvalet SDOF system by defg the respodg system target dsplacemet, system hysteretc dampg ad system mass. (System target dsplacemet Accordg to the requremet that the work doe by the equvalet SDOF system must be the same as that doe by the MDOF system, the system target dsplacemet ca be defed as[4]: m = = (3 m = (System hysteretc dampg ad equvalet system ductlty Shbata ad Soze[] suggested that system hysteretc dampg ca be weghted accordg to flexural stra eergy for the substtute approach. The system hysteretc dampg ξ ad equvalet ductlty of the equvalet SDOF system ca be obtaed by equato (4. Q h ξ = w ξ = w w = = = k = Q ( ( Q = M a + M b M am b (4 6 EI k I equato (4, w s the weghtg factor of the th per whch s assumed to be ot varable durg the sesmc respose of the MDOF system ths paper, so the weghtg factor of each bet ad the system hysteretc dampg are the determed through the tal structural aalyss. The equvalet ductlty cocet ca also be calculated through the same procedure by cosderg the ductlty capacty of each bet ad the yeldg mechasm of overall brdge structures accordg to Eurocode 8. (3Effectve system mass The ectve system mass s defed by the requremet of work equvalece betwee the MDOF system ad respodg equvalet SDOF system[4], ad the t ca be obtaed by equato (5 where s the target dsplacemet for th per ad m s the equvalet mass of th per. = M = m (5 Sce the overall dsplaced shape of the brdge structures the trasverse drecto, whch s cossted of the target dsplacemet of each bet, was establshed wth referece to multple modes of respose, the ectve system mass of equvalet SDOF system cosders the ects of multple modes of respose. (4Effectve system stffess ad force dstrbuto After determg the equvalet SDOF system, the ectve perod of the substtute structure s obtaed by eterg the costat-ductlty elastc dsplacemet respose spectra Fg.6 accordg to the system target dsplacemet, equvalet system ductlty ad system hysteretc dampg. The ectve stffess ad base shear force for the equvalet SDOF system are the obtaed as followg: 4π M K = F = K (6 T After establshg the total desg force of the equvalet SDOF system, the colum desg forces of the respodg MDOF system are the determed wth equato (7 suggested by Kowalsky[3] where h s the heght of th colum. h F = F (7 h = Oce the colum lateral forces are obtaed wth equato (7, the colum momets are easly calculated, the colum sectos are the desged accordg to the momet capacty, ad the the

7 logtudal ad trasverse reforcemet are desged to resst the respodg desg momet ad target dsplacemet. EXAMPLE BRIDGE DESIGNS The proposed dsplacemet-based sesmc desg procedure was evaluated by applyg t to the symmetrcal ad asymmetrcal MDOF cotuous brdges show Fg. used prevously referece [3]. Each brdge had four 50m spas ad detcal per cross-sectos, dfferg oly the heghts of the three colums. The yeld stress value of steel was assged to be 455Mpa, ad cocrete compressve stregth was 35Mpa, ad the elastc modulus of all the cocrete members was 9.6Gpa. The brdge deck was assumed to be ped at the abutmets, ad all the pers were assumed to be fully fxed at the base ad ped at the deck. The sesmc exctato has bee appled smultaeously to the bases of all pers, eglectg ay ect of o-sychrosm exctato. Smoothed elastc ad elastc dsplacemet respose spectra are show Fg.5 ad Fg.6 respodg to a maxmum groud accelerato of 0.4g o subsol class D defed the UBC97. The secto propertes of two brdges are lsted Table ad the detaled desgs of both brdges are descrbed below. Table. Secto propertes of symmetrcal ad asymmetrcal brdges Secto Propertes Symmetrcal Brdges(7++7 Asymmetrcal Brdges(4+7+ Deck Per Deck Per Area(m I xx (m I yy (m Weght(t/m Symmetrcal Brdge Desg I the desg of the symmetrcal brdge, the dameter of each crcular per was assumed to be.5m, ad the tal target dsplaced shape was determed by specfyg a desred drft rato of % each of the pers, or dsplacemets of 0.07, 0. ad 0.07m respectvely at the top of the pers. Accordg to the symmetrcal brdge, the ductlty capacty of each per was assumed to be performed well durg the sesmc respose. I ths paper the damage dex DI of each per was assged to be 0.55 wth referece to the prevously preseted desg lmt states uder the specfed desg groud moto. The equvalet ductlty cocet of each per s 4.0 calculated wth equato (3 accordg to the respodg damage dex DI, ad the equvalet vscous dampg of each per s 0.85 calculated wth equato (. Sce the symmetrcal brdge respods predomatly the frst mode, the smplfed desg procedure ca obta the ratoal desg results by cosderg ect of the frst mode the trasverse drecto. The detaled desg results are summarzed Table. Table. Summary of Desg Results for Symmetrcal Brdge Bet No. Per Per 3 Per 4 h (m M a,m b M a =.73, M b =9.67 M a3 =5.6, M b3 =4.8 M a4 =.73, M b4 =9.67 w m (t μ ξ Modal aalyss T =0.83s,γ =.33,ξ =0.85,μ =4.0,sd =0.077m Δ (m Equvalet SDOF System ξ =0.85,μ =4.0,Δ =0.077m,M =688*0 3 kg,t =0.83s, K =96733KN/m, F =7448KN F (KN M (KN.m ρ (%

8 Asymmetrcal Brdge Desg I the desg of the asymmetrcal brdge, the dameter of each crcular per was assumed to be.0m, ad the tal target dsplaced shape was determed by specfyg a desred drft rato of % each of the pers, or dsplacemets of 0.4, 0.07 ad 0.m respectvely at the top of the pers. Accordg to the asymmetrcal brdge, the ductlty capacty of each per could t be smultaeously performed well durg the sesmc respose, so t s mpossble to form the plastc hge at the bottom of each per at the same tme. Geerally the frst plastc hge was geerated at the bottom of the shortest oe of per3, whle the per4 maybe sustaed elastc utl the collapse of the whole brdge structure. I ths paper the damage dex DI was assged to be 0.4, 0.55 ad 0. respectvely for per, per3 ad per4 wth referece to the prevously preseted desg lmt states uder the specfed desg groud moto. The equvalet ductlty cocets are.0, 4.0 ad.0 for per, per3 ad per4 calculated wth equato (3 accordg to the respodg damage dex DI, ad the equvalet vscous dampg are 0.43, 0.85 ad 0.05 for per, per3 ad per4 calculated wth equato (. Ulke the desg of the symmetrcal brdge, the smplfed desg procedure ca obta the ratoal desg results by cosderg ect of the frst two modes the trasverse drecto. The detaled desg results are summarzed Table 3. Table 3. Summary of Desg Results for Asymmetrcal Brdge Bet No. Per Per 3 Per 4 h (m M a,m b M a =5.49, M b =9. M a3 =3.6, M b3 =.9 M a4 =3.85, M b4 =4.4 w m (t μ ξ Modal aalyss Frst modal: T =0.89s,γ =0.57,ξ =0.043,μ =.0,sd =0.96m Secod modal: T =0.68s,γ =.36,ξ =0.07,μ =.93,sd =0.076m Δ (m Equvalet SDOF System ξ =0.06,μ =.09,Δ =0.06m,M =589*0 3 kg,t =0.90s, K =7744KN/m, F =855KN F (KN M (KN.m ρ (% Verfcato of Desg Approach wth Tme-hstory Aalyss I order to verfy the accuracy of the smplfed dsplacemet-based desg procedure terms of meetg the desg dsplacemet ad the specfed damage levels, the equvalet SDOF systems respodg to the symmetrcal ad asymmetrcal brdges were subected to dyamc elastc tme-hstory aalyss. The aalyss was performed usg eght ear-feld accelerato tme hstores o subsol class D defed the UBC97 of whch the peak values were all adusted to 0.4g order to compare the o-lear tme-hstory aalyss results wth the desred desg dsplacemets resultg from the preseted dsplacemet-based desg procedure ths paper. Table 4. Tme-hstory Aalyss Results of Equvalet SDOF System Earthquakes PGA(g DAF D sym (m D asym (m ImperalValleyElcetro# ImperalValleyElcetro# ImperalValleyElcetro# ImperalValleyHotvllePostOffce# ImperalValleyMeloladOFF# CapmedcoPetrola# Kobetakarazuka# LomapretaGlroyHstorcBLDG# Average Value 0.45 /

9 As show Table 4, the calculated average maxmum dsplacemets from the o-lear aalyss of the equvalet SDOF systems for symmetrcal ad asymmetrcal brdges are m ad 0.4m respectvely. Compared to the respectvely desred desg dsplacemets of the equvalet SDOF systems, whch are 0.077m show Table ad 0.06m show Table 3, the calculated average maxmum dsplacemets from the o-lear aalyss agree well wth the desred desg dsplacemets respectvely. CONCLUDING REMARKS The dsplacemet-based sesmc desg offers the ablty to cotrol explctly the dsplacemet demad each member ad to cosder structural damage drectly related to dsplacemet demads respodg to dfferet dsplacemet lmt states. I ths study, the elastc dsplacemet desg spectra cosderg earfeld ects were preseted, ad the elastc dsplacemet desg spectra based o structural damage performace are also costructed. Ths paper troduced the respose spectrum-based approach to cosder the ects of hgher modes o the structural respose of MDOF cotuous brdges o the trasverse drecto uder earthquake exctato. A mproved dsplacemet-based sesmc desg method for MDOF brdge structures based o costat-ductlty elastc dsplacemet desg spectra s preseted. I the desg procedure, there s o terato ad the specfed desg obectve ductlty ca be drectly realzed. I ths complete desg process for two example brdges, the smplfed desg procedure ca obta the ratoal desg results by cosderg ect of the frst mode for symmetrcal brdge, whle t must cosder ects of the frst two modes for asymmetrcal brdge to obta the ratoal desg results the trasverse drecto. Sce the o-lear dyamc tme-hstory aalyss results for the equvalet SDOF system agree well wth the desred desg results respectvely, t dcates the smplfed dsplacemet-based sesmc desg method for MDOF brdge structures s soud. ACKNOWLEDGMENTS Ths study s sposored by Natoal Nature Scece Foudatos of Cha Grat No ad No Ths support s gratefully ackowledged. REFERENCES. M.J.N.Prestley, F.Sebler ad G.M.Calv, Sesmc Desg ad Retroft of Brdges. New York: Joh Wly & Sos, M.J.Kowalsky. M.J.N.Prestley, G.A.Macrae. Dsplacemet-based Desg of RC Brdge Colums Sesmc Regos. Earthquake Egeerg ad Structural Dyamcs 995; 4: M.J.Kowalsky. A Dsplacemet-based Approach for the Sesmc Desg of Cotuous Cocrete Brdges. Earthquake Egeerg ad Structural Dyamcs 00; 3: G.M.Calv, G.R.Kgsley. Dsplacemet-based Sesmc Desg of Mult-degree-of-freedom Brdge Structures. Earthquake Egeerg ad Dyamc Structures 995; 4: Juqg Gog, X Zhu. Dsplacemet-based sesmc desg method for RC brdge colums. Cha Joural of Hghway ad Trasport 00; 4(4: 4-46( Chese. 6. Uform Buldg Code. Iteratoal Coferece of Buldg Offcals, Whtter, CA, Y.J.Park, A.H-S.Ag. Mechastc Sesmc Damage Model for Reforced Cocrete. Joural of Structural Egeerg 985; (4: A.H-S.Ag, W.J.Km, S.B.Km. Damage estmato of exstg brdge structures. Structural Egeerg Natural Hazards Mtgato: Professo of ASCE Structures Cogress, 993, Irve CA, Vol.: P.Fafar. Equvalet ductlty factors, takg to accout low-cycle fatgue. Earthquake Egeerg ad Structural Dyamcs 99; : X Zhu, Yogu N. Stregth reducto factor based o structural damage performace. Cha-US Mlleary Symposum o Earthquake Egeerg, Beg, A.Shbata, M.Soze. Substtute structure method for sesmc desg R/C. Joural of Structural Egeerg 988; 4: