GLOBAL PROBABILISTIC SEISMIC CAPACITY ANALYSIS OF STRUCTURES THROUGH STOCHASTIC PUSHOVER ANALYSIS

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1 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha GLOBAL PROBABILISTI SEISMI APAITY ANALYSIS OF STRUTURES THROUGH STOHASTI PUSHOVER ANALYSIS Xao-Hu Yu, Da-Gag Lu, Peg-Ya Sog 3, Guag-Yua Wag 4 Graduate Studet, School of vl Egeerg, Harb Isttute of Techology, Harb. ha Professor, School of vl Egeerg, Harb Isttute of Techology, Harb. ha 3 Graduate Studet, School of vl Egeerg, Harb Isttute of Techology, Harb. ha 4 Professor, School of vl Egeerg, Harb Isttute of Techology, Harb. ha Emal: yuxaohu83058@63.com, ludagag@ht.edu.c, sogpegya@sa.com, waggy@ht.edu.c ABSTRAT Probablstc sesmc capacty aalyss (PSA) plays a mportat role the feld of sesmc performace evaluato, sesmc relablty aalyss, sesmc fraglty aalyss, ad sesmc rs aalyss of cvl egeerg structures. I geeral, PSA has two levels of research cotext: global PSA ad local PSA. The local PSA has bee studed by umerous research programs; however, the global PSA has ust recetly bee pad atteto to the feld of earthquae egeerg ad structural egeerg. I ths paper, order to buld global probablstc sesmc capacty model (GPSM) of structures cosderg radom system propertes, a stochastc pushover aalyss (SPA) method s developed, whch combes four ds of approxmatg methods for estmatg statstcal momets of complex radom fucto wth determstc pushover aalyss. The four radom aalyss methods clude mea value frst order secod momet (MVFOSM) approach, Mote arlo smulato (MS) approach, mproved pot estmato approach, ad hou-nowa umercal tegrato approach. The methodology proposed s appled R.. frame structure. A three-bay ad fve-storey plae R.. frame s tae as a example case study, the global sesmc capacty curves of the structure correspodg to four levels of lmt states are derved. It s demostrated by ths example that the approach proposed ths paper s a effcet ad accurate tool for global probablstc sesmc capacty aalyss of structures. KEYWORDS: Stochastc Pushover Aalyss, Global Sesmc apacty, MVFOSM Method, MS Method, Pot Estmato Method, hou-nowa Numercal Itegrato Method. INTODUTION Probablstc sesmc capacty aalyss (PSA) plays a mportat role the feld of sesmc performace evaluato, sesmc relablty aalyss, sesmc fraglty aalyss, ad sesmc rs aalyss of cvl egeerg structures. Recetly, t has bee troduced to the ew-geerato Performace Based Earthquae Egeerg (PBEE) proposed by PEER as oe of ts four buldg blocs (Moehle & Deerle, 004). PSA has two levels of research cotext: global PSA ad local PSA. The local PSA has bee studed by umerous research programs; however, the global PSA has ust recetly bee pad atteto to the feld of earthquae egeerg ad structural egeerg. Actually, global PSA s more dffcult tha local PSA. Frst, the global sesmc capacty of structures has may fluecg factors, such as: structural cofgurato, structural dyamc propertes (stffess ad dampg), geometrc szes of structural elemets, costtutve relatoshps of structural materals, modelg ucertaty of structures, ad so o. Secod, all the fluecg factors metoed above have mult-scale characterstcs. I fact, they ca be classfed to fve scales,.e., materal scale, secto scale, elemet scale, substructure scale ad system scale. The propagato of the ucertates across the dfferet levels of scales maes global PSA very dffcult to mplemet. Thrd, these fluecg factors may be depedet o each other. Forth, the global performace of structures s geerally dscretzed to multple levels. For example, the commo practce earthquae egeerg, the damage states are usually grouped to fve dscrete levels: tact, mor damage, moderate damage, severe damage, ad

2 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha collapse. It s dffcult to clarfy the lmt values of dfferet performace levels. Ffth, the global sesmc capacty of structures behaves olearty, especally durg the stages of severe damage ad collapse. Last, t usually eeds umercal aalyss method global PSA. However, the global sesmc capacty of structures s a hghly mplct fucto of the basc radom varables. Therefore, t s dffcult to derve the statstcs of the radom global sesmc capacty of structures. To overcome the dffcultes metoed above, ths paper develops a stochastc pushover aalyss method combg four ds of statstcal momet estmatg methods,.e., MVFOSM method, MS method, pot estmato method ad hou-nowa umercal tegrato method, wth determstc pushover aalyss approach drve by a ew force cotrollg method. The proposed methodology s the appled R.. frame structures. A three-bay ad fve-storey plae R.. frame s tae as a example case study, the probablstc models ad ther correspodg fraglty curves of the global sesmc capactes of the structure correspodg to four levels of lmt states are derved.. APPROXIMATE METHODS FOR ESTIMATING STATISTIAL MOMENTS OF OMPLEX RANDOM FUNTIONS.. Overvew Let us deote by the global sesmc capacty of structures, e.g., the maxmum ter-storey dsplacemet agle (ISDA). It s a olear fucto of the basc radom varables X: g( X ) g( X, X,, X ) (.) Geerally, g(x) s a complex ad mplct fucto of X. The th cetral momet of s obtaed by + + E g ( ) g ( x,, x ) f ( x,, x ) dx dx X X (.) where, E [] s the expectato operato, fx ( x,, x ) s the ot probablty desty fucto (JPDF) of X. The mea value ad varace σ of are E[ g( X )] (.3) ( ( ) [ ( )]) ( ) [ ( )] σ E g E g E g E g X X X X (.4) Ufortuately, due to the mplct ad complex characterstcs of g(x), t s dffcult to drectly solve the quadrature Eqs. (.) to (.3). Alteratvely, some approxmate methods for estmatg statstcal momets of complex fuctos are troduced. The approxmato strateges ca be dvded to three categores: () Approxmate aalytcal methods. The Taylor seres expaso of the fucto s the most frequetly used. The well-ow mea value frst order secod momet (MVFOSM) method belogs to ths category, whch the fucto s expaded aroud the mea value pot of the basc radom varables wth frst order seres fucto, ad t oly eeds the frst secod momets formato of the basc radom varables to propagate the ucertaty. () Numercal smulato methods. Mote arlo smulato (MS) method ad ts varous varace-reducto techques perhaps are the most rgorously employed. (3) Numercal tegrato methods. The pot estmato method orgally proposed by Roseblueth (975) ad the umercal tegrato approach developed by hou ad Nowa (988) eter to ths category... MVFOSM Method I MVFOSM method, the radom fucto s learly expaded at the mea-value pot of the basc varables: g( X) g( Μ) + g T Μ ( X Μ ) (.5) T where, M [ X,,, ] X X s the mea value vector of X, Σ ρσσ s the covarace matrx of X, T deotes the traspose of a matrx. The, we ca get the frst two momets approxmato of g( X ): g( M ) (.6a)

3 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha T σ M M g Σ g (.6b).3. MS Method Mote arlo smulato ca be used to estmate the mea value ad stadard devato of a radom fucto. () ( ) Assume that a sample set of put vectors has bee geerated, say { x,, x }. The usual estmators of the frst two momets of respectvely read: () ˆ g ( ) x (.7a) ( ) ˆ σ ( ) ˆ g x (.7b).4. Pot Estmato Method (PEM) Pot estmato method (PEM) was proposed by Roseblueth (975) to approxmate the lower-order momets of fuctos of radom varables. It s a specal case of umercal quadrature based o orthogoal polyomals. For ormal varables, t correspods to Gauss-Hermte quadrature. hao ad Oo (000) troduced a ew pot estmato method based o Roseblatt trasformato (Hohebchler & Racwtz, 98) whch the umercal quadrature s completed stadard ormal space. Ufortuately, Roseblatt trasformato caot deal wth the case of radom varables wth gve margal dstrbutos ad correlato formato. We here troduce Nataf trasformato (Lu & Der Kuregha, 986) to hao-oo pot estmato method to replace Roseblatt trasformato. For a sgle-varable fucto g( X), the th cetral momet of the fucto s estmated by m w M g T ( x ) (.8) π where, x s the Gauss-Hermt tegrato pot,.e., estmatg pot; w s the correspodg weght. For a mult-varable fucto g( X ), two fucto approxmato approaches are used: g ( X ) (.9) g ( X ) ( ) + (.0) whch, g( ) g(,,,, ) (.) N ( u) ( u) (,,,,, +,, ) (.) g T G G u u u u u u where, T N () deotes verse Nataf trasformato; represets the vector whch all the radom varables tae ther mea values; u represets the vector whch oly u s a radom varable, whle other varables tae the correspodg trasformed values of ther mea values stadard ormal space; u ( ) s the th elemet of the trasformed vector u who correspods the vector stadard ormal space u ; G u u s the formulato of radom fucto g( x ) stadard ormal space based o Nataf ( ) g[ T N ( )] trasformato. Based o the product-rule as show Eq. (.9), the mea value ad the th cetral momet of the fucto ca be estmated: (.3a)

4 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha K E K K E g ( ) X (.3b) K Based o the o-product rule as show Eq. (.0), the mea value ad varace of the fucto ca be estmated: I Eqs. (.3) ad (.4), of sgle-varable fucto. ad ( ) + (.4a) σ σ z (.4b) σ are mea value ad stadard devato of G by usg pot-estmato.5. hou-nowa Numercal Itegrato Method (NIM) For the depedet stadard ormal radom vector (,,, ), hou ad Nowa (988) proposed a specal umercal tegrato method for drectly estmatg the statstcal momets of radom fucto g(). The tegrato pots ( z, z,, z) ad ther correspodg weghts w are lsted Table.. Table. Estmatg pots ad weghts for hou-nowa umercal tegrato approach Itegrato pots Weght factors ( z,... z ) Pot umber N + (,0,0,0, 0) ( + )( ) (,,0,0, 0) ( + ) ( + )( ) ( ) ( ) 3 (,,,0, 0) 3 ( + ) ( + ) + (,,, ) ( ) ( )( ) w w + w + w + w + ( + ) ( + ) + + (,,, ) w + ( ) ( )( ) + + (,0,0,0, 0) w w + + (0,,0,0, 0) w w + + (0,0,0,0, ) w w + (0,0,...,0) w + 4 * + ( ± +,0,...,0) w ( + ) + + * ( ±, ±,...,0) w ( + ) * Pots clude all possble permutatos coordates

5 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha Usg the abscssas ad weghts Table., the th cetral momet of radom fucto g() s estmated by ( ) m E g (,,, ) wg z, z,, z (.5) For radom fucto g( X ) geeral radom space, the verse Nataf trasformato X T N ( ) s appled to result, ( ) m E g X X X w g TN z TN z TN z (,,, ) ( ), ( ),, ( ) (.6) Based o Eq. (.6), the mea value ad varace of ca be obtaed by Eq. (.3) ad Eq. (.4). 3. STOHASTI PUSHOVER ANALYSIS ONSIDERING RANDOM SYSTEM PROPERTIES I earthquae egeerg practce, the global performace of structures s usually dscretzed to multple levels. For example, fve levels of damage states are usually adopted,.e., tact, mor damage, moderate damage, severe damage, ad collapse. However, how to characterze the lmt values of cosecutve damage states has bee remag a dffcult problem due to the large ucertaty modelg ad statstcal aalyss. I ths paper, the covetoal pushover aalyss s mproved to determe the global capacty lmtg values (,,3,4). A ew force cotrol techque s troduced to set up the relatoshps betwee (,,3,4) ad the total base shear V, by addg lateral load to the structure creasgly, utl V s equal to the correspodg levels of the base shear V (,,3,4), as show Fgure. V V 4 V 3 V V mor damage moderate damage severe damage collapse tact 3 4 Fgure Setch dagram of force-cotrolled pushover aalyss I hese sesmc desg code of buldgs (GB500-00), three levels of sesmc fortfcato for buldg structures are specfed, whch ca be summarzed as: do ot be damaged uder mor earthquae, ca be repared uder moderate earthquae, ad do ot collapse uder maor earthquae. The moderate earthquae s the basc testy I 0 of the desgated ste, whose exceedace probablty 50yrs s 0%. The exceedace probablty 50yrs of the mor earthquae s 63.%, whose earthquae testy s I d -.55, where I d s the fortfcato testy. The exceedace probablty 50yrs of the maor earthquae s %, whose earthquae testy s I d +. The testes of mor, moderate ad maor earthquaes ca be drectly used to cotrol the lateral force added 3 to the structure correspodg to, ad 4. To preset the lateral force correspodg to, aother level of earthquae fortfcato testy, amed as sub-moderate earthquae, or frequet earthquae, s

6 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha troduced ths paper, whose testy taes the mea value of mor earthquae testy ad moderate earthquae testy: [( Id.55) + Id] I Id 0.8 (3.) Based o hese sesmc desg code of buldgs (GB500-00), the relatoshps betwee the lateral sesmc factors α (,,3,4) ad (,,3,4) are lsted Table 3.. Earthquae levels mor earthquae frequet earthquae moderate earthquae maor earthquae Table 3. Relatoshps betwee I d.55 d d I, α ad α I 7 I 8 I 9 d I I I d I d d Usg α (,,3,4) Table, the lateral sesmc force o the th floor for each sesmc level ca be determed by αf F, (,,3,4) (,..., ) (3.) α3 where, F s the lateral sesmc force o the th floor uder fortfcato testy I d, s the umber of structural storeys. The cotrollg lateral base shear V (,,3,4) s obtaed by V F, (,,3,4) (3.3) The pushover aalyss uder the lateral load F s carred o utl V V (,,3,4), ad the (,,3,4) ca be obtaed from the pushover curve. V Logarthm ormal dstrbuto Meda Pushover curve V Fgure Setch dagram of stochastc pushover aalyss

7 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha The above-metoed procedure ca oly derve the determstc lmtg values of the global sesmc capacty of structures. Due to the radom system propertes, the lmtg values of the global sesmc capacty of structures behaves radom dsperso ature, as show Fgure. To cosder radom structural propertes pushover aalyss, we here propose a stochastc pushover aalyss approach, whch combes four approxmate radom aalyss methods troduced secto, amely, MVFOSM method, MS method, pot estmato method ad hou-nowa umercal tegrato method, wth the force-cotrolled pushover aalyss method stated above. I ths ew approach, the global capacty measure of structures s assumed to be a fucto of basc radom varables. Each tme that the fucto s evaluated durg the estmatg the statstcal momets of, the force-cotrolled pushover aalyss s ru for oe tme. For stochastc pushover aalyss based o MVFOSM method, because of Eq. (.6b), t eeds fte elemet respose sestvty aalyss. The fte elemet relablty aalyss module of OpeSees (Hauaas & Kuregha, 007) s utlzed to compute the statstcal momets of by drect dfferetal method of respose sestvty aalyss. For stochastc pushover aalyss based o MS method, the radom samples of the basc varables are frstly geerated, ad the, the radom samples of fte elemet models of structures are obtaed. The force cotrolled pushover aalyss s the appled for each structural sample, ad the statstcal momets of (,,3,4) are estmated by Eq. (.8). For stochastc pushover aalyss based o pot estmato method,we should choose Gauss-Hermt estmatg pots ad weghts at frst, ad the, the structural samples are obtaed accordg to the strategy of PEM. The force cotrolled pushover aalyss s carred o for each structural sample. The statstcal momets of (,,3,4) are the approxmated by product rule Eq. (.3) ad o-product rule Eq. (.4), respectvely. For stochastc pushover aalyss based o hou-nowa umercal tegrato approach, we should also select the estmatg pots ad weghts accordg to Table. at frst, ad the, the structural samples are obtaed accordg to the strategy of ths approach. The force cotrolled pushover aalyss s aga tae for each structural sample, ad the statstcal momets of (,,3,4) are the estmated by Eq. (.6). 4. PROBABILISTI MODEL OF GLOBAL SEISMI APAITY OF STRUTURES The probablstc sesmc capacty model of structures s defed as a codtoal probablty of by sesmc demad D exceedg the capacty, gve the specfc value of sesmc demad: F ( d ) P D D d (4.) where, D s the global sesmc demad parameter wth the same ut as capacty parameter, e.g., the maxmum ISDA; d s the gve value of demad parameter. The model Eq. (4.) s also termed as sesmc fraglty of structures. It has bee supported by umerous research programs that the fraglty ca be modeled by a logormal cumulatve dstrbuto fucto: l d λ F ( d ) Φ (4.) ζ where, λ ad ζ are mea value ad stadard devato of l( ), respectvely. They are related to mea value ad stadard devato of as

8 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha λ ζ l + δ ( δ ) (4.3) l + (4.4) where, ad δ are mea value ad coeffcet of varato of, respectvely. The meda value m of s the expoet of λ : m exp( λ ) (4.5) Therefore, the probablstc model of sesmc capacty ca be re-wrtte as a smpler formulato: l( d / m ) F ( d ) Φ (4.6) ζ 5 APPLIATION OF THE METHODOLOGY IN A R.. FRAME STRUTURE A three-bay ad fve-storey R.. frame structure, as show Fgure 3, s tae as the example ths case study. It s desged accordg to hese sesmc desg code of buldgs (GB500-00) wth fortfcato testy I d 8. F5 F4 F3 F F 0.KN/m 3.3KN/m 3.3KN/m 3.3KN/m 3.3KN/m 3.3m 3.3m 3.3m 3.3m 4.4m 6.0m.4m 6.0m Fgure 3 Three-bay ad fve-storey R.. frame For smplcty, oly the radom materal propertes are cosdered, whch clude sx basc radom varables: yeld stregth f y ad tal elastc modulus E of steel; compressve stregth f c, crushg stregth f cr, stra ε c at compressve stregth, ad straε cu at crushg stregth of cocrete. The statstcs are show Table 5.. All varables are assumed to follow ormal dstrbuto, ad to be depedet o each other. Table 5. Statstcs of basc radom varables Dstrbuto parameters Dstrbuto parameters Varables Varables Mea value Std Mea value Std f (N/mm ) f (N/mm ) y E (N/mm ) ε c f (N/mm ) c εcu cr

9 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha The values of F,(,,3, 4;,,3, 4,5) ad V (,,3,4) are lsted Table 5.. Table 5. Lateral cotrollg load F ad correspodg base shear F /KN F F F 3 F 4 F 5 V /KN Stochastc pushover aalyss wth four radom aalyss methods are appled to estmate the statstcs of the global drft capacty of the structure, the computed results are show Table 5.3. Table 5.3 Statstcs of the global drft capactes Methods Performace levels σ m ζ MS PEM (product rule ) PEM (No-product rule ) hou-nowa umercal tegrato approach MVFOSM Mor damage Moderate damage Severe damage ollapse Mor damage Moderate damage Severe damage ollapse Mor damage Moderate damage Severe damage ollapse Mor damage Moderate damage Severe damage ollapse Mor damage Moderate damage Severe damage ollapse From Table 5.3, we ca see that the results of stochastc pushover aalyss method based o dfferet approxmatg methods are approachg to each other; however, there exst some cosstece the results of collapse capactes, sce the structure has goe to a hghly olear rage. Usg the statstcs Table 5.3 ad Eq. (4.) or (4.6), the global sesmc capacty fraglty curves ca be

10 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha obtaed, as show Fgure 4. From ths fgure, we ca assess the falure probablty of dfferet performace levels uder a gve sesmc demad level > Mor damge > moderate damage > severe damage > collapse Falure probablty Itact Moderate damage severe damage ollapse Mor damage ONLUSION max Fgure 4 Fraglty curves of global sesmc capactes of the structure (MS method) Ths paper proposes a stochastc pushover aalyss method for probablstc sesmc capacty aalyss of structures, whch combes four ds of methods for estmatg statstcal momets of radom fucto, amely MVFOSM method, MS method, pot estmato method (PEM) ad hou-nowa umercal tegrato method (NIM), wth a force-cotrolled pushover aalyss method, The developed methodology s appled R.. frame structures. A three-bay ad fve-storey plae R.. frame s tae as a example case study, the global sesmc capacty curves of the structure correspodg to four levels of lmt states are derved. It s demostrated by ths example that the approach proposed ths paper s a effcet ad accurate tool for global probablstc sesmc capacty aalyss of structures. 7. AKNOWLEDGEMENTS The support of Natoal Scece Foudato of ha through proects (Grat No , , ) s greatly apprecated. Ay opos, fdgs, ad coclusos or recommedatos expressed ths materal are those of the authors ad do ot ecessarly reflect those of the Natoal Scece Foudato of ha. REFERENES Hauaas, T., ad Der Kuregha, A. (007). Methods ad obect-oreted software for FE relablty ad sestvty aalyss wth applcato to a brdge structure. ASE Joural of omputg vl Egeerg :3, Hohebchler, M., ad Racwtz, R. (98). No-ormal depedet vectors structural safety. ASE Joural of Egeerg Mechacs 07:6, Lu, P.L., ad Der Kuregha, A. (986). Multvarate dstrbuto models wth prescrbed margal ad covaraces. Probablstc Egeerg Mechacs :, 05-.

11 The 4 th World oferece o Earthquae Egeerg October -7, 008, Beg, ha Moehle, J., ad Deerle, G.G. (004). A framewor methodology for performace-based earthquae egeerg. Proceedgs of the 3th World oferece o Earthquae Egeerg, Vacouver, aada, 6 August, (D-ROM). Natoal Stadard of ha P.R. (00). Sesmc Desg ode of Buldgs (GB500-00). Buldg Idustry Press of ha, Beg. Roseblueth, E. (975). Pot estmates for probablty momets. Proceedgs of the Natoal Academy of Scece 7:0, hao, Y.G., ad Oo, T. (000). New pot estmates for probablty momets. ASE Joural of Egeerg Mechacs 6:4, hou, J.H., ad Nowa, A.S. (988). Itegrato formulas to evaluate fuctos of radom varables. Structural Safety 5:4,