Evaluation of the Seismic Energy Demand for Asymmetric-Plan Buildings Subjected to Bi- Directional Ground Motions

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1 Evaluaio of he Seismic Eergy Demad for Asymmeric-Pla Buildigs Subjeced o Bi- Direcioal Groud Moios Jui-Liag Li & Keh-Chyua sai Naioal Ceer for Research o Earhquae Egieerig aipei aiwa R.O.C. 9 NZSEE Coferece ABSRAC: his paper proposes a mehod for he evaluaio of he seismic eergy demads of wo-way asymmeric-pla buildigs uder bi-direcioal seismic groud moios. he proposed procedure coducs he modal respose hisory aalyses by usig he hree-degree-of-freedom (3DOF) modal sysems. his modal sysem represes he relaioships of wo roof raslaios versus wo base shears ad roof roaio versus base orque obaied from he modal pushover aalyses. he proposed mehod is compuaioally efficie ad cocepually clear iheried from he advaages of he coveioal modal respose hisory aalysis. he proposed procedure ca be used o esimae he absorbed eergies coribuig from raslaioal ad roaioal moios he sudy shows ha he sigs of modal eccericiy are a deermia for he pla-wise disribuio of he modal absorbed eergy. he accuracy of he proposed procedure was verified by aalyzig he resposes of oe 3-sorey ad oe -sorey wo-way asymmeric-pla buildigs subjeced o bi-direcioal earhquae exciaios. hese aalyical resuls cofirm he relaioships bewee he sigs of he modal eccericiy ad he pla-wise disribuio of he modal absorbed eergy. INRODUCION I addiio o he displaceme duciliy eergy demad is a impora facor i assessig he damage o a srucure uder severe seismic load (Par ad Ag 985). he ipu seismic eergy E i is equal o he summaio of he ieic eergy E viscous dampig eergy E ad absorbed eergy E a. E a cosiss of he recoverable elasic srai eergy E s ad he irrecoverable hysereic eergy E h. E a is very close o E h whe he duciliy facor is larger ha wo (Chou ). I rece years a mehod cosiderig he absorbed eergy resulig from he firs as well as he secod vibraio modes was proposed (Chou ad Uag 3). Recely he coceps of ucoupled modal respose hisory aalysis (UMRHA) ad modal pushover aalysis (MPA) mehods (Chopra ad Goel ) have bee applied o evaluae he eergy respose of symmeric-pla buildigs (Prasah e al.8). hey did apply he ideas of UMRHA ad MPA for ay geeralized dyamic respose r() o eergy resposes of symmerical srucures. However o oe seems o have ever successfully exeded he UMRHA ad MPA coceps (Chopra ad Goel 4) io he sudy of eergy resposes of asymmerical srucures. Prasah e al. (8) suggesed ha he esimaio of he hysereic eergy of buildigs havig a o-symmerical pla is a issue worhy of fuure research. Li ad sai (7 ad 8) exeded he cocep of UMRHA (Chopra ad Goel 4) ad proposed he MDOF modal sysem for asymmerical buildigs. he effeciveess of he proposed MDOF modal sysems for asymmerical buildigs wih differe orsioal-o-laeral frequecy raios has bee exesively verified (Li ad sai 7 ad 8). Neverheless here seems sill o applicaio of he MDOF modal sysem o he sudy of eergy resposes of asymmerical sysems. I oe sudy o oe-sorey asymmeric-pla buildigs (Goel 997) i was poied ou ha differe coclusios for he eergy demad o he siff sides (SS) ad he flexible sides (FS) of asymmericpla buildigs were draw by differe researchers. Some sudies cocluded ha elemes o he SS i code-desiged asymmeric-pla sysems are liely o suffer more damage whereas elemes o he FS are expeced o suffer less or similar damage compared o hose i he correspodig symmeric sysem. However Goel (997) cocluded ha he elemes o he FS could udergo much larger Paper Number

2 hysereic eergy demads i a asymmeric-pla sysem ha hose i he correspodig symmeric sysems. he elemes o he SS o he oher had migh o ecessarily experiece ay larger hysereic demads i asymmeric-pla sysems. he above meioed sudies srogly sugges he eed of developig a simple ad effecive mehod for compuig he oal eergy demad of wo-way asymmeric-pla buildigs uder he geeral bidirecioal seismic groud moios. I addiio i appears impora o ideify he ey facors affecig he pla-wise disribuio of eergy demad i he asymmeric-pla buildigs. hus his sudy proposes such a mehod by usig he hree-degree-of-freedom (3DOF) modal sysems (Li ad sai 8). he 3DOF modal sysem ca effecively describes he relaioships of wo roof raslaios versus wo base shears ad roof roaio versus base orque for each vibraio mode of a wo-way asymmeric-pla buildig. I his paper he relaioships bewee he sigs of modal eccericiy ad he red of he pla-wise disribuio of modal absorbed eergy are ideified. Usig wo example buildigs he paper cofirms he effeciveess of usig he sigs of he modal eccericiy i deermiig he red of eergy disribuio. I addiio he proposed mehod ca saisfacorily esimae he eergy demad of asymmeric-pla buildigs subjeced o bi-direcioal earhquae exciaios. HEORECICAL BACKGROUND. hree-degree-of-freedom modal sysems he wo pla axes are he X- ad Z-axis. he Y-axis is upward (opposie o he direcio of graviy). he proporioally damped wo-way asymmeric-pla buildigs wih each floor simulaed as a rigid diaphragm are cosidered i his research. he equaio of moio for a N-sorey wo-way asymmeric-pla buildig subjeced o bi-direcioal groud exciaios is Mu Cu Ku Mι u s u M u M u u gx zu gz s u gx x z gx gz M Mι u z gz i which he displaceme vecor u mode shape mass marix M siffess marix K ifluece vecors x z modal ieria force disribuios s s z ad modal paricipaio facors z are gx z gz () u x u u z u z m M x m x z z I K zx xz x z ι x ι z s M s z z M Mι M x z Mι M z () u u M Whe oly he force gx z gz (Equaio ) becomes Mu u u M ~ N Cu Ku gx z gz 3 is applied o he buildig he equaio of moio i which u is he -h modal displaceme respose ad u u D (3). D is he geeralized modal coordiae. he 3DOF modal sysems have bee proposed for he seismic aalysis of wo-way

3 asymmeric-pla buildigs (Li ad sai 8). he elasic properies of he -h 3DOF modal sysem are as follows: z zx a e e z z m SC S C SC C x z z z m x m z z m z z I θ I z z e e θ θ z S S e C z (4) where C=cos ad S=si. he ielasic properies of he -h 3DOF modal sysem were derived from he hree pushover curves of he origial wo-way asymmeric-pla buildig exered by he -h modal ieria force disribuio. he saed hree pushover curves represeig he wo roof raslaios versus wo base shears ad roof roaio versus base orque relaioships are preseed i ADRS forma ad idealized as hree biliear curves. he yieldig acceleraios ad pos-yieldig siffess raios of hese hree biliear curves are deoed as A y A zy A y z ad. Because A y A zy A y z ad are available he ielasic parameers of he -h 3DOF modal sysem ca be deermied as: M M y A m M A m (5a) y yz zy z y A yi Aymez Azymze (5b) (5c) m I mez mze e z z z m I m ez m z z (5d) mz I mez mze e m I m ez m (5e) where M y M yz M y ad z are he yieldig momes ad he pos-yieldig siffess of he hree roaioal sprigs of he 3DOF modal sysem respecively. hus he oal elasic ad ielasic 3DOF modal properies are all esablished by usig Equaios (4) ad (5).. Modal absorbed eergy of wo-way asymmeric-pla buildigs From Equaio (3) i is foud ha E E E E (6) a i where he icremeal ieic eergy E icremeal dampig eergy E icremeal absorbed eergy E a ad he icremeal ipu eergy E i are E E i Mu u E Cu u E R u Ku ˆ u gx zu gzm u a z z e e e e z u Moreover he oal absorbed eergy of a wo-way asymmeric-pla buildig is (7) 3

4 E a Ea Ea R u Ku ˆ E a E E E E E a a ax az a u (8) where E a E ad E are a a E a V D Ea VzDz Ea D (9) ad V V z ad are he resorig force of he 3DOF modal sysem. E a E a ad E a are he -h modal absorbed eergy of he wo-way asymmeric-pla buildig resulig from he X- Z- raslaioal ad Y-roaioal deformaios respecively. E ax E az ad E a are he fracios of he oal absorbed eergy iduced by raslaioal ad roaioal deformaios respecively. Equaio (8) idicaes ha he oal absorbed eergy of a asymmeric-pla buildig is approximaely equal o he summaio of he modal absorbed eergy E a which ca be decomposed io hree pars ad E a. hus he oal absorbed eergy E a is a combiaio of absorbed eergies E ax E az ad E a resulig from hree ypes of deformaio. Sice he higher modes usually remai i a elasic sae oly he firs few 3DOF modal sysems eed o be cosidered. herefore our proposed mehod is much more efficie ha he compuaio iesive ad ime cosumig oliear RHA for he whole buildig. he oher advaage of he proposed mehod is ha he causes of duraio-relaed seismic damages of asymmeric-pla buildigs ca be furher udersood by looig a he fracios of oal absorbed eergy resulig from differe ypes of deformaio..3 he relaioships of he sigs of modal eccericiy ad he red of he pla-wise disribuio of modal absorbed eergy By comparig asymmeric-pla buildigs wih he correspodig ucoupled symmeric-pla buildigs i is appare ha eccericiy is he oly facor causig he ueve pla-wise disribuio of absorbed eergy. Hece he effecs of sigs of modal eccericiy i.e. e ad e z o pla-wise disribuio of modal absorbed eergy are ivesigaed i his paper. he FS of a 3DOF modal sysem is a he ed of he rigid beam wih lumped mass. he oher ed of ha rigid beam which is coeced o he op of he colum is he SS for a 3DOF modal sysem. A posiive roaio of he rigid beam of a 3DOF modal sysem wih posiive eccericiy e z resuls i a posiive icreme of X-raslaio a he lumped mass i.e. he FS i he X-direcio. hus he eergy demad o he FS i he X-direcio of ha 3DOF modal sysem icreases due o he posiive roaio of he rigid beam. A he same ime he eergy demad o he SS i he X-direcio of ha 3DOF modal sysem decreases. Similarly we discuss a 3DOF modal sysem wih posiive eccericiy e as followig. A posiive roaio of he rigid beam of a 3DOF modal sysem wih posiive eccericiy e resuls i a egaive icreme of he Z-raslaio a he lumped mass i.e. he FS i he Z-direcio. hus he eergy demad o he FS i he Z-direcio of ha 3DOF modal sysem decreases due o he posiive roaio of he rigid beam. A he same ime he eergy demad o he SS i he Z-direcio of ha 3DOF modal sysem icreases. Based o he foregoig he red of he pla-wise disribuio of modal absorbed eergy for asymmeric-pla buildigs ca be prediced by usig he sigs of he modal eccericiies. For example he modal absorbed eergy of he -h vibraio mode wih posiive e ad e z will cocerae more o he SS ad FS i he Z- ad X-direcio respecively compared wih heir couerpar i each direcio. he sigs of he modal eccericiies are deoed as sig(e ) ad sig(e z ) i he remaider of his paper. able shows he side of asymmeric-pla buildigs wih he larger modal absorbed eergy demad amog he wo sides i each horizoal direcio for all combiaios of sig(e ) ad sig(e z ). As a whole he pla-wise disribuio of oal absorbed eergy depeds upo he exe of he coribuio of each vibraio mode o he oal absorbed eergy. a E E a 4

5 able. he relaioships of he sigs of modal eccericiies ad he pla-wise disribuio of modal absorbed eergy. Combiaio Sides wih larger eergy demad sig(e ) sig(e z ) Z-dir. X-dir. + + SS FS + - SS SS - + FS FS - - FS SS 3 NUMERICAL VALIDAION 3. he seleced srucural sysem groud moios ad basic assumpios he example buildig is a 3-sorey wo-way asymmeric-pla buildigs. he 3-sorey buildig wide ad 3@3.5m all. he laeral force resisig sysems of he example buildig is he perimeer mome frame show i Figures. Each floor is simulaed as a rigid diaphragm. he CR is a he geomeric ceer ad he CM is eccerically locaed wih same eccericiy raios i he X- ad Z-direcios ad equal o 5% for he 3-sorey buildigs. he maerial used i he 3-sorey buildig is A36 seel which is simulaed as a biliear maerial wih a yieldig sress equal o.45 5 N/m Youg s modulus E=. 8 N/m ad 5% pos-yieldig siffess raio. he member sizes of he colums ad beams used i he 3-sorey buildig are H ad H569 respecively. he mass ad he mass mome of ieria for each floor of he 3-sorey buildig are 5 g ad g-m respecively. Rayleigh dampig wih 5% dampig raios for he firs wo vibraio modes of he example buildig were assumed. he /5 se of hisorical groud moio records cosidered for buildigs locaed i Los Ageles i he SAC Seel Projec was used for he aalysis of he 3-sorey buildig. For each hisorical earhquae oe pair of groud moios recorded o he same sie was seleced from ha se. here are a oal of four loadig cases for he example buildig i.e. LA/ LA3/4 LA7/8 ad LA9/3. Z X A B C 5m 5m 5m 3.5m 3 A 5m B 5m C C C C 5m 3 CR.5m.5m CM 3.5m 3.5m C C C C C C (a) Figure. (a) he ypical floor pla ad (b) he elevaio of he 3-sorey buildig. 3. Discussio of Aalyical resuls : H569 C: H he modal properies of he 3DOF modal sysems for he example buildig are lised i ables. Based o he sigs of modal eccericiy (ables (a) ad (b)) he prediced sides of hese example buildigs wih larger modal absorbed eergy demad for he firs five vibraio modes are lised i ables (c). Figure shows he disribuio of he posiive ad egaive accumulaive plasic hige roaios of he example buildig subjeced o he modal ieria force. By comparig he sizes of he circles show i Figure he side wih he larger modal absorbed eergy demad amog he wo sides i each horizoal direcio of he example buildigs is readily evide. he sides wih he larger (b) 5

6 modal absorbed eergy demad for he firs five vibraio modes show i Figure are cosise wih hose lised i able which were prediced by usig he sigs of modal eccericiy. ha is o say he red of he pla-wise disribuio of he modal absorbed eergy prediced by usig he sigs of modal eccericiy is validaed. able. he sides of he floor pla of he 3-sorey buildig wih larger modal absorbed eergy demad. Mode combiaio sides wih larger eergy demad sig(e ) sig(e z ) Z-dir. X-dir. - + FS FS - - FS SS SS SS FS FS FS SS posiive A B C egaive sadard circle =.5 rad. Figure. he disribuio of he posiive ad egaive accumulaive plasic hige roaios. he esimaed absorbed eergy obaied by usig he proposed mehod ad he exac absorbed eergy obaied by usig oliear RHA of he whole buildig are deoed as E a3dof ad E arha respecively. he exac absorbed eergy of he correspodig symmeric-pla buildig is deoed as E arha. he compariso of E a3dof ad E arha is illusraed i Figure 3. Figure 3 idicaes ha he seismic eergy demads of wo-way asymmeric-pla buildigs uder bi-direcioal seismic groud moios esimaed by usig he proposed mehod are saisfacory. Moreover he values of E arha /E arha which rage bewee.96 ad.47 idicae ha he oal seismic eergy demad of a asymmeric-pla buildig is o ecessarily larger ha ha of he correspodig symmeric-pla buildig. However due o he ueve pla-wise disribuio of he absorbed eergy of asymmeric-pla buildigs hose buildigs may sill be more vulerable o earhquaes ha he correspodig symmeric-pla buildigs. he absorbed eergy of he 3-sorey example buildig resulig from he secod loadig case is much larger ha ha resulig from oher loadig cases. A ivesigaio of he PSA specra of hese seleced groud moios shows ha he LA4 groud moio which was used for he secod loadig case has sigificaly larger specral values a he periods of he firs hree vibraio modes ha hose of he oher seleced groud moios. he raios of he modal absorbed eergy E a o he oal absorbed eergy E a3dof are illusraed i Figure 3(b). Figure 3(b) show ha he hird vibraio mode of he 3-sorey example buildig is elasic or almos elasic uder hese four cases of groud moios. he fourh ad he fifh vibraio modes of he 3-sorey example buildig oly sigificaly coribue o he oal absorbed eergy i he firs case. Figure 3(b) also shows ha he secod vibraio mode is he mai coribuor o he oal absorbed eergy i he secod ad he hird cases. However he firs ad he secod vibraio mode coribue abou equally o he oal absorbed eergy 6

7 Absorbed Eergy (N-m) Absorbed Eergy (N-m) Absorbed Eergy (N-m) i he fourh case. he disribuios of he posiive ad egaive accumulaive plasic hige roaios of he example buildig are illusraed i Figure 3(c). Due o he domia coribuio of he secod vibraio mode o he oal absorbed eergy i he secod ad hird cases for he 3-sorey example buildig he plasic hige roaios of hose wo cases are coceraed o he FS ad SS i he Z- ad X-direcio respecively. I he fourh case for he 3-sorey example buildig he plasic higes are coceraed o he FS i boh he Z- ad X-direcio because he raios of he modal absorbed eergy of he firs ad secod vibraio modes o he oal absorbed eergy are abou he same. (a) Ea3DOF EaRHA 3 4 Case No. (b) Ea Ea Case Case Case 3 Case 4 Ea 7. Ea 43.5 Case No. Ea3DOF EaRHA Ea 9.3 Ea 9. Ea4 Ea 88.5 Ea.5 Ea 56.4 Ea 43.6 Case No. 3 A B C 3 4 posiive Z Y X egaive (c) sadard circle =. rad. for Case =. rad. for ohers Figure 3(a) he esimaed ad exac oal absorbed eergy E a3dof ad E arha of he example buildig. 3(b) he raios of he modal absorbed eergy E a o he oal absorbed eergy E a3dof of he example buildig. 3(c) he disribuios of he posiive ad egaive accumulaive plasic hige roaios for he example buildig uder he exciaios of he seleced groud moios. (a) 8 Case Ea Ea Ea Case 4 - Ea Ea Ea Case Ea Ea Ea Case Ea Ea Ea Case Case Case 3 Case 4 Ea.5 Eax 3. Ea 3. Eax 3. Ea.8 Eax. Ea.7 Eax 44.8 (b) Eaz 46.4 Eaz 39.7 Figure 4. (a) he modal absorbed eergy of he example buildig (b) he raios of absorbed eergy resulig from differe ypes of deformaio o he oal absorbed eergy E a3dof of he example buildig. Eaz 54.9 Eaz 3.6 7

8 he modal absorbed eergy iduced by he differe ypes of deformaio i.e. E a Ea Ea are show i Figure 4(a). he raios of he oal absorbed eergy iduced by he differe ypes of deformaio i.e. E ax Eazad E a o he oal absorbed eergy E a3dof of he example buildig are show i Figure 4(b). Figure 4(b) shows ha he majoriy of he absorbed eergy for he 3-sorey buildig i cases o 3 was iduced by he Z-raslaioal deformaio. However he major absorbed eergy for he 3-sorey example buildig i case 4 was iduced by he X-raslaioal deformaio. For all four cases he absorbed eergy resulig from roaioal deformaio aes o 3 % of he oal absorbed eergy of he buildig. 4 CONCLUSIONS he seismic behaviour of wo-way asymmeric-pla buildigs uder bi-direcioal groud moios are much more complicaed ha hose of symmeric-pla buildigs uder sigle-direcioal groud moios. Excep he compuaio-iesive oliear RHA mehod mos of he simplified mehods for he esimaio of seismic eergy demad were developed for symmeric-pla buildigs. his sudy proposes a aleraive mehod o evaluae he seismic eergy demad of wo-way asymmeric-pla buildigs uder bi-direcioal groud moios. he followig coclusios ca be draw from his sudy:. he proposed mehod performs he modal respose hisory aalyses by usig he 3DOF modal sysems. his paper cofirms ha he absorbed eergy of wo-way asymmeric-pla buildigs uder bi-direcioal groud moios ca be saisfacorily evaluaed by usig he proposed mehod. Usig his mehod he absorbed eergies resulig from differe ypes of moio i ay mode ca be separaed. he assessme of he absorbed eergies resulig from differe ypes of moio provides a complee ew udersadig of he effecs of buildig s orsioal respose o he disribuio of oliear deformaioal demad i srucural members.. I is cofirmed ha he modal absorbed eergy coceraig more o flexible side or siff side is deermied by he sigs of modal eccericiy. For a 3DOF modal sysem wih posiive modal eccericiies e z ad e he modal absorbed eergy is more coceraed o he FS ad SS i he X- ad Z-direcio respecively. O he corary for a 3DOF modal sysem wih egaive modal eccericiies e z ad e he modal absorbed eergy is more coceraed o he SS ad FS i he X- ad Z-direcio respecively. herefore he pla-wise disribuio of he oal absorbed eergy for a asymmeric-pla buildig ca be evaluaed from he sigs of modal eccericiies e ad e z of he domia mode. Uder a differe earhquae he pla-wise disribuio of he oal absorbed eergy could be alered depedig o he domia mode. REFERENCES: Par Y.J. ad Ag A.H.S Mechaisic seismic damage model for reiforced cocree. Joural of Srucural Egieerig ASCE (4): Chou C.C... A eergy based seismic evaluaio procedure for mome resisig frame srucures. Ph.D. hesis Deparme of Srucural Egieerig Uiversiy of Califoria Sa Diego La Jolla CA U.S.A. Chou C.C.. ad Uag C.M. 3. A procedure for evaluaig seismic eergy demad of framed srucures. Earhquae Egieerig ad Srucural Dyamics 3: Chopra A.K. Goel R.K.. A modal pushover aalysis procedure for esimaig seismic demads for buildigs. Earhquae Egieerig ad Srucural Dyamics 3: Chopra A.K. ad Goel R.K. 4. A modal pushover aalysis procedure o esimae seismic demads for usymmeric-pla buildigs. Earhquae Egieerig ad Srucural Dyamics 33: Prasah. Ghosh S. ad Collis K.R. 8. Esimaio of hysereic eergy demad usig coceps of modal pushover aalysis. Earhquae Egieerig ad Srucural Dyamics 37: Li J.L. ad sai K.C. 7. Simplified seismic aalysis of asymmeric buildig sysems. Earhquae Egieerig ad Srucural Dyamics 36: Li J.L. ad sai K.C. 8. Seismic aalysis of wo-way asymmeric buildig sysems uder bi-direcioal seismic groud moios Earhquae Egieerig ad Srucural Dyamics 37: Goel R.K Seismic respose of asymmeric sysems: eergy-based approach. Joural of Srucural Egieerig ASCE 3 ():