IN2 - A SIMPLE ALTERNATIVE FOR IDA

Transcription

1 3 th World Conference on Erthquke Engineering Vncouver, B.C., Cnd Augut -6, 4 Pper No IN - A SIME ALTERNATIVE FOR IDA Mtjž DOLŠEK nd Peter FAJFAR SUMMARY Simplified ineltic procedure ued in eimic deign nd ement combine the nonliner ttic (puhover) nli nd the repone pectrum pproch. One of uch procedure i the N method, which h been implemented into the Eurocode 8 tndrd. The N method cn be emploed lo imple tool for the determintion of the pproximte ummrized IDA (incrementl dnmic nli) curve. Such nli i clled the incrementl N method (IN). The IN curve cn ubtitute the IDA curve in the probbilitic frmework for eimic deign nd ement of tructure. In the pper, the IN method i ummrized nd pplied to two tet exmple of infilled reinforced concrete (RC) frme, which re chrcterized b ubtntil degrdtion of the trength fter the infill fil. The pproximte ummrized IDA curve, determined b the IN method, nd the dt on diperion due to rndomne in diplcement demnd, determined in previou tud b the uthor, were emploed in the probbilitic rik nli of tet tructure. The reult were compred with the reult obtined uing the exct IDA curve. A fir correltion of reult ugget tht the IN method i vible pproch.. INTRODUCTION Incrementl dnmic nli (IDA) i prmetric nli method for the etimtion of tructurl repone under eimic lod []. A tructurl model i ubjected to multiple level of eimic intenit uing one or more ground motion record. The objective of n IDA tud i the undertnding of tructurl behvior under different level of eimic intenit. IDA i lo ubtntil prt of probbilitic frmework for eimic performnce ement, developed t Stnford [,3]. There i no doubt tht IDA nli provide the mot thorough imge of the eimic behvior of the tructure mong ll nli method preentl vilble. However, it i ver time conuming nd the quetion rie if it i poible to determine IDA curve with le input dt nd with le effort, but till with cceptble ccurc? A poible pproch i to determine eimic demnd for multiple level of eimic intenit with the N method [4], which i bed on puhover nli nd repone pectrum pproch. Such nli, which w emploed in our tud, will be clled the incrementl N method (IN). The IN method i reltivel imple tool for ft determintion of pproximte IDA curve nd m repreent vible pproch for eimic performnce ement, pproprite for exmple for Aitnt, Fcult of Civil nd Geodetic Engineering, Univerit of Ljubljn, Emil: mdolek@ikpir.fgg.uni-lj.i Profeor, Fcult of Civil nd Geodetic Engineering, Univerit of Ljubljn, Emil: pfjfr@ikpir.fgg.uni-lj.i

2 prmetric tudie on influence of different uncertin ground motion nd tructurl input dt. In combintion with predetermined dt on diperion tpicl for pecific tructurl tem, it cn be emploed lo in the probbilitic frmework, demontrted in thi pper. The tructurl tem, ued in our tud, i the infilled reinforced concrete (RC) frme. The puhover curve for n infilled RC frme i chrcterized b ubtntil degrdtion of the trength fter the infill fil. Thu, pecific reduction fctor, developed b the uthor, hve to be ued for the determintion of ineltic pectr. The pproximte IDA curve, determined b the IN method, nd the dt on diperion, determined in previou tud b the uthor, were emploed in the probbilitic rik nli of two tet tructure. The reult were compred with the reult obtined uing the exct IDA curve.. SUMMARY OF THE N METHOD AND ITS EXTENSION TO INFILLED FRAMES The N method combine puhover nli of multi degree-of-freedom (MDOF) model with the repone pectrum nli of n equivlent ingle-degree-of-freedom model (SDOF). The formultion of the method in the ccelertion diplcement formt enble the viul interprettion of the procedure nd of the reltion between the bic quntitie controlling the eimic repone. Detil bout the bic verion of the N method, limited to plnr tructurl model, cn be found in [4]. In the N method, the eimic demnd for the equivlent SDOF tem with period T cn be determined follow: Eltic demnd in term of ccelertion S e nd diplcement S de i determined from the eltic pectrum. The ineltic ccelertion demnd S i equl to the ield ccelertion S, which repreent the ccelertion cpcit of the ineltic tem. The trength reduction fctor R cn be determined the rtio between the ccelertion correponding to the eltic nd ineltic tem. The ductilit demnd µ i then clculted from ineltic pectr, which re defined b the period dependent reltion between reduction fctor nd ductilit ( R µ T reltion), nd the ineltic diplcement demnd S d i computed Sd = ( µ R) Sde. In principle n R µ T reltion cn be ued. A ver imple nd firl ccurte R µ T reltion i bed on the equl diplcement rule in the medium- nd long-period rnge [4]. It h been implemented in Eurocode 8 [5]. The ppliction of the N method cn be extended lo to complex tructurl tem, for exmple to infilled frme, provided tht n pproprite pecific R µ T reltion i known. In order to emplo the N method for infilled RC frme, multi-liner ideliztion of the puhover curve i needed inted of imple elto-pltic ideliztion, in ddition to pecific R µ T reltion. Both modifiction re decribed in [6] nd [7]. The bic chrcteritic of the puhover curve of infilled RC frme i ubtntil decree in trength fter the infill h begun to degrde. The puhover curve cn be modeled with four-liner forcediplcement reltionhip hown in Figure. Four prmeter re needed for the definition of the idelized puhover curve: ield diplcement D nd ield force F, ductilit t the beginning of oftening of infill µ, nd the rtio between the force t which infill completel collpe nd ielding force r u. The R µ T reltion, which cn be ued for infilled frme, i decribed in [6]. The reltion depend on the bic prmeter of the puhover curve nd the corner period of the ccelertion pectrum T C nd T D. T C repreent the corner period between the contnt ccelertion nd contnt velocit prt of the pectrum of the Newmrk-Hll tpe, nd T D repreent the corner period between the contnt velocit nd contnt diplcement prt of the pectrum. A n exmple, n idelized force-diplcement reltion (puhover curve) nd the R µ T reltion for the given puhover curve re hown in Figure.

3 Force F F min r= u F D µ = D R µ =.5 r u =.5 T D/T C=3.33 µ=6 µ=4 µ=3 F min µ= µ=.5 D D Diplcement T/T C ) force-diplcement reltion b) R µ T reltion Figure. The idelized force-diplcement reltion for n infilled frme () nd the R-µ -T reltion (b) for the given force-diplcement reltion 3. IN METHOD Incrementl N (IN) method i reltivel imple nonliner method for determintion of pproximte IDA curve. IN method i, like the IDA nli, prmetric nli method. An IDA curve i determined with nonliner dnmic nle, while ech point of n IN curve (pproximte IDA curve), which correpond to given eimic intenit, i predicted with the N method. All limittion which ppl to the N method [4] ppl lo to IN method. In order to determine n IN curve, firt the ground motion intenit meure nd the demnd meure hve to be elected. The mot pproprite pir of quntitie i the pectrl ccelertion nd the top (roof) diplcement, which llow lo the viuliztion of the procedure (Figure ). Other relevnt quntitie, like mximum tor drift, rottion t the column nd bem end, her force in tructurl element nd in joint, nd tor ccelertion, cn be emploed econdr demnd meure. The re relted to roof diplcement nd cn be uniquel determined if roof diplcement i known. The econdr demnd meure cn be ued, together with the min demnd meure, for performnce ement t different performnce level. Roof diplcement nd other relevnt demnd meure for choen erie of pectrl ccelertion re determined b the N method. Thi tep repreent the min difference in comprion with IDA nli becue the N method i ued for the determintion of eimic repone. Therefore the hpe of the IN curve depend on the ineltic pectr pplied in the N method, which re bed on the reltion between trength reduction fctor, ductilit nd period (the R µ T reltion). If imple R µ T reltion, bed on equl diplcement rule in the medium- nd long-period rnge, i ued, the IN curve i liner for tructure with period higher thn T C nd biliner for tructure with period lower thn T C. A more complex R µ T reltion w propoed b uthor for infilled RC frme [6,7]. In thi ce IN curve i four-liner (Figure ). Conidering the piecewie linerit of the IN curve, onl few point hve to determined in order to obtin the complete N curve. Uull the ineltic pectr, ued in the N method, repreent men pectr nd conequentl the IN curve repreent men curve. More pecificll, the R µ T reltion for infilled frme, ued in thi

4 pper, repreent n ideliztion of the R µ T reltion, clculted for men ductilit given the reduction fctor. The chemtic contruction of the IN curve for SDOF model in ccelertion-diplcement (AD) formt i preented in Figure. The cpcit digrm (multi-liner curve) hown in Figure i chrcteritic for infilled RC frme nd repreent the idelized puhover curve of n equivlent SDOF model. A n exmple, two point ( P nd P ) of the IN curve, correponding to two different ground motion intenitie, re chemticll contructed with the N method. The rdil line from origin nd croing ield point repreent the eltic tem with period T. Eltic eimic demnd in term of eltic pectrl ccelertion ( S e, or S e, ) nd correponding eltic pectrl diplcement ( S de, or S de, ) i determined the interection of thi line with the eltic pectrum for the pproprite ground motion intenit. The ineltic diplcement demnd ( S d, or S d, ) i then determined with the N method. It correpond to the point where the horizontl line, t the ccelertion S, interect the pproprite ineltic pectrum. A point of the IN curve (e.g. the point P nd P ) i defined with the pir: eltic pectrl ccelertion on the Y-xi nd the correponding ineltic diplcement demnd on the X-xi (Figure ). If ineltic diplcement re determined for mn level of eltic pectrl ccelertion, the complete IN curve cn be obtined. Accelertion S e, S e, YP T P IS IS c rv N e I u Cpcit digrm P ES-Eltic pectrum IS-Ineltic pectrum YP-Yield point ES ES S de, S de, S d, S d, Diplcement Figure. Schemtic contruction of n IN curve. 4. IMEMENTATION OF THE IN METHOD IN THE PROBABILISTIC SEISMIC PERFORMANCE ASSESSMENT ANALYSIS The gol of the probbilitic frmework dopted b the Pcific Erthquke Engineering Center [] i to dequtel predict erthquke loe nd/or exceednce of one or more performnce level (or limit tte), which cn onl be predicted probbiliticll. The implet form of the probbilitic frmework i the etimtion of the nnul likelihood of the event tht the demnd exceed the cpcit t choen limit tte, where ll rndom element of the problem re clr vlue [3]. The rndom element of thi frmework re the ground motion intenit meure, the demnd D, nd the cpcit t choen limit tte or performnce level, denoted herein C. The pectrl ccelertion S t the period of the idelized SDOF model i choen to repreent the intenit meure [3]. The hzrd function H( ) define the nnul probbilit tht the pectrl ccelertion S i equl to or more thn the elected level of pectrl ccelertion. Both, the demnd nd the cpcit re here chrcterized in term of top diplcement (ee Section 3) inted of mximum tor drift like in [3]. The top diplcement cpcit m be determined from different cpcit meure like tor drift, ultimte rottion, her trength or

5 other, which cn be etimted from the puhover nli. Thu the IDA nli ield the reltion between the pectrl ccelertion S nd the top diplcement D nd provide the informtion needed to e the tructurl performnce t different limit tte or performnce level. Auming tht ll rndom element re lognormll ditributed, the medin hzrd curve ˆ ( ) medin IDA curve Dˆ ( ) cn be pproximted b the form ˆ k H( ) ko b Dˆ ( ) ( ) H nd the =, () = () nd the nnul likelihood tht the eimic demnd exceed the cpcit t limit tte (performnce level) P cn be expreed ˆ C% k P = H( ) exp ( β + βcr ) b (3) Ĉ where i the pectrl ccelertion correponding to the medin diplcement cpcit Ĉ, β nd β CR re the diperion meure for rndomne in diplcement demnd nd diplcement cpcit, repectivel. The coefficient k nd b re prmeter of the hzrd curve nd IDA curve, repectivel (Eq. () nd ()). Expreion (3) cn be further developed if it i focued on the uncertint of diplcement demnd nd diplcement cpcit nd if men hzrd curve i dopted [3]. In thi ce P itelf become n uncertin quntit nd the x confidence level of P i obtined [3] x C% k k P = H( ) exp ( β + βcr ) exp Kxβ P b, βp ( ) = βdu + βcu (4) b Ĉ where H( ) i the vlue of men hzrd t the pectrl ccelertion correponding to the medin diplcement cpcit, K x i the tndrdized Guin vrite ocited with probbilit x of not being exceeded, β DU i the diperion meure for uncertint of diplcement demnd, nd β CU i the diperion meure for uncertint of diplcement cpcit. The men hzrd curve cn be clculted C ( ) ˆ C H = H( ) exp βh (5) where β H i diperion meure for hzrd. The detiled explntion of the probbilitic frmework nd derivtion cn be found elewhere [3]. To implement the reult of the IN method into the decribed probbilitic frmework, the IDA curve h to be replced with the IN curve. In generl, n IN curve i intended to pproximte ummrized IDA curve (e.g. men or medin) nd i not clculted for ingle ground motion Therefore diperion meure for rndomne in diplcement demnd β nd diplcement cpcit β CR cn not be directl determined from the reult of the IN method. Nonethele the diperion of rndomne in diplcement demnd cn be etimted from the coefficient of vrition for diplcement of the SDOF tem, for which R µ T reltion w determined, while β CR h to be precribed for tpicl tructurl tem in dvnce. It i lo convenient nd prcticl tht diperion meure for uncertint in diplcement demnd β DU nd cpcit β CU re precribed in dvnce. (Note tht it i prcticl tht thee meure re precribed in dvnce lo in the originl probbilitic eimic ement with IDA curve). The determintion of diperion meure for tpicl tructurl tem i not within the cope of thi pper. Additionl tudie re needed to determine the model for diperion or to precribe the pproprite vlue. For the preented exmple diperion meure β CR, β CU nd β DU were rbitrril umed. In the N method ineltic pectr re ued. Uull, the re intended to repreent men pectr nd therefore the reulting IN curve repreent the men curve. In uch ce, the men IN curve h to be trnformed to the medin IN curve. Thi cn be chieved with different method if the diperion i

6 known. For exmple, if the method of moment i pplied, then the tndrd devition for nturl log β cn be clculted β ln( ) = V +, (6) where V i the coefficient of vrition for rndomne in diplcement demnd, nd the medin vlue of the IN curve cn be determined Dˆ ( ) D( ) exp = β (7) where D tnd for men vlue of diplcement determined with the N method. So fr, we hve not et developed the model for the diperion meure for rndomne in diplcement demnd. Bed on our previou reult [6,7,8], n upper limit of V =.7 nd lower limit of.4 cn be umed for coefficient of vrition. The firt vlue i pproprite for hort- period tructure, while the econd vlue i reonble for tructure with moderte nd long period. Uing Eq. (6) nd (7), the medin IN curve cn be etimted. Note tht bed on Eq. (7), the men IN curve i more conervtive (it ield lrger diplcement for given pectrl ccelertion) thn the medin IN curve. For V =.7 nd V =.4, the rtio between the medin nd men vlue mount to.8 nd.93, repectivel. The medin IN curve h to be fitted with the tpe of the function preented in Eq. (). The let qure fit method w pplied to determine prmeter nd b from the medin IN curve. In the ued probbilitic frmework [3] there re no trict guideline for the length of the intervl which h to be ued in the proce of determintion of prmeter nd b. In our previou tud [8] we relized tht mll length of the intervl m cue qulittivel different concluion if the reult for probbilit (Eq. 4) nd the reult obtined with prcticl formt for fet checking [3] re compred. In our nle, the intervl from the ield pectrl ccelertion, determined from the idelized SDOF tem, to the pectrl ccelertion ocited to the diplcement cpcit, w ued in the let qure fit procedure. 5. EXAMES 5. Tet tructure The firt tet tructure i four-tor exiting building (Figure 3), for which the frme hd been deigned to reproduce the deign prctice in Europen nd Mediterrnen countrie bout fort to fift er go [9]. However, it m lo be tpicl of building built more recentl, but without the ppliction of cpcit deign principle (epecill the trong column - wek bem concept), nd without up-to-dte detiling. In uch building oft firt tor effect m occur even though the tructure i uniforml infilled in it elevtion []. The four-tor contemporr building (Figure 3) w deigned ccording to Eurocode 8, highductilit cl tructure []. The deign pek ground ccelertion mounted to.3 g, which reult in be her coefficient equl to.5.

7 .5 5.m 5.m 4.m 6.m Loding 4.m 5.m 5.m m 3.m 3.m 3.m Four-tor exiting building Four-tor contemporr building Figure 3. Tet tructure. For both tet tructure full-cle peudo-dnmic tet hve been performed t the ELSA Lbortor, Ipr [9,]. The technique propoed b the uthor [], which emplo the reult of peudo-dnmic tet, h been ued to contruct the mthemticl model of the tet tructure. All the bem nd column were modeled b perfectl eltic, mle bem element with two nonliner rottionl pring t ech of the two end. The moment rottion reltionhip for ech pring w defined b triliner envelope nd Tked hteretic rule. Ammetric bckbone curve were ued for the bem. In ddition to thee element, imple rottionl connection element were plced between the bem nd joint to model the pinching behvior of the bem. The infill pnel were modeled b equivlent digonl trut, which crr lod onl in compreion. The her-lip hteretic model h been ued for modeling the cclic behvior of the infill pnel. Strength deteriortion w modeled onl for the element repreenting infill, where for the element repreenting RC bem nd column unlimited ductilit w umed. All nonliner nle were performed uing modified verion of the computer progrm AIN-DX [3]. In ll nle 5 % dmping w umed. More detil on the mthemticl model of the tet tructure cn be found elewhere [8,]. 5. Ground motion The eimic loding for the IN nli i defined with the idelized pectrum preented in Figure 4. The prmeter of the idelized Newmrk-Hll tpe pectrum were obtined from the et of ccelerogrm with the procedure explined in [6]. The chrcteritic period T B, T C nd T D of the idelized pectrum re equl to.,.55 nd.76, repectivel. The pectrl mplifiction t the contnt ccelertion rnge mount to.39. For the IDA nli the me et of ccelerogrm [6] w ued. The men pectrum nd it tndrd devition re preented in Figure 4. Note tht the pectrl ccelertion t the initil period of the idelized equivlent SDOF tem w ued the ground motion intenit meure. The period re equl to.9 nd.367 for exiting nd contemporr building, repectivel. Therefore different men pectr re obtined for the two building (Figure 4). Both men pectr fit ver well the

8 idelized trget pectrum, which fcilitte the comprion between the reult of the IDA nd the IN nli. S e /g,mx T=.9 Men pectrum Men pectrum +/- σ Idelized pectrum T() T() ) Exiting building b) Contemporr building Figure 4. The men nd men ± σ of pectr, cled to the me pectrl ccelertion t the period of the idelized equivlent SDOF tem for exiting nd contemporr building, nd the idelized Newmrk-Hll tpe pectrum. Seimic hzrd for centrl Sloveni w emploed in tet exmple. The pectrl ccelertion in the plteu i equl to.6 nd.8 g for hzrd level /475 nd /, repectivel. The prmeter k nd k o, which define the medin hzrd curve (Eq. 7) mount to.45 nd 5.97E-4. For the diperion of the medin hzrd curve β H vlue of.3 w umed. For both exmple the me hzrd curve w ued. 5.3 Seimic demnd Incrementl N method S e /g,mx 5.3. The Puhover curve nd the equivlent SDOF tem The nonliner ttic (puhover) nle were performed with the force pttern, which w clculted from umed inverted tringulr diplcement hpe. Puhover curve for both tet building re hown in Figure 5. It i obviou tht fter certin deformtion infill trt to degrde. After the collpe of infill in firt tor for exiting building nd in the bottom two torie for contemporr building onl frme reit the horizontl loding. Puhover curve re then idelized uing the procedure decribed in [7]. Abolute vlue of F, which i equl to mximum trength (Figure ), i much higher for contemporr building (Tble ). For both building the mximum trength totl weight rtio F W i rther high. It mount to.67 nd.4 for the contemporr nd exiting building, repectivel. The high F W rtio i conequence of rther high trength of infill in comprion with the trength of bre frme. The minimum trength fter the degrdtion of infill F min (Figure ) i lo obtined from the ideliztion of puhover curve (Tble, Figure 5). The rtio between F min nd F (prmeter r u ) mount to.46 nd.6 for exiting nd contemporr building, repectivel. Importnt prmeter obtined from the idelized puhover curve re lo ield diplcement D nd diplcement t the trt of trength degrdtion D (Figure ). The ductilit t the beginning of trength degrdtion µ i clculted the rtio between D nd D. Both T=.367 Men pectrum Men pectrum +/- σ Idelized pectrum

9 prmeter, r nd u µ, re importnt for the determintion of ductilit demnd µ. The vlue of prmeter re ummrized in Tble. The contnt Γ, which i ued for the trnformtion from the MDOF to the equivlent SDOF tem nd vice ver (Eq. (7) in [4]) i obtined from the vector of tor me nd umed diplcement hpe. The equivlent m of the SDOF tem m i lo clculted from the tor me nd normlized diplcement hpe (Eq. 3 in [4]). Yield point of the equivlent SDOF tem ( D, F ) cn be clculted impl b dividing D nd F with the trnformtion contnt Γ. The vlue for both building re preented in Tble. The eltic period of the equivlent SDOF tem T (Eq. 8 in [4]) re for both building mller thn ( T C =.55 (Section 3.3.), therefore the T T C rtio i le then one. The ccelertion t the ield point i clculted the rtio between F nd m. 8 Exiting building 5 Contemporr building Be Sher (kn) 6 4 Idelized puhover curve Ner collpe (% of tor drift) Crcking of infill Top diplcement (cm) Be Sher (kn) 5 Idelized puhover curve 5 Ner collpe (4.5% of tor drift) Crcking of infill 3 Top diplcement (cm) Figure 5. Be her top diplcement reltionhip for exiting nd contemporr building nd idelized puhover curve. Tble. Summr of prmeter defining idelized puhover curve nd equivlent SDOF tem for exiting nd contemporr building. Prmeter Exiting Contemporr Prmeter Exiting Contemporr F (kn) Γ F (kn) min 34 5 m () t 9 36 F W(%) 4 67 F (kn) D (cm) D (cm)..66 D (cm) T () r, µ.46,.87.6,.68 S ( g) u 5.3. The eimic demnd for ingle level of pectrl ccelertion (N method v. nonliner dnmic nli) Knowing the prmeter of the equivlent SDOF tem ( r u, µ, T, S ) nd the prmeter of demnd pectr ( T C, T D, S ( ) e T ), eimic demnd cn be eil computed. A n exmple, we will determine the eimic demnd for exiting building with given reduction fctor R =.8. The reduction fctor repreent the rtio between the eltic pectrl ccelertion t the period of the equivlent SDOF tem S ( ) e T nd the ield ccelertion S. The eltic pectrl ccelertion S ( ) e T i then equl to

10 .8.486g =.87 g. The ductilit demnd µ i clculted from R µ T reltion [6] nd i equl to Multipling the ductilit demnd µ b ield diplcement D =.cm the diplcement demnd of the equivlent SDOF tem S d = 4.55cm i obtined. Finll, top diplcement of exiting building D t = 6.cm i obtined b multipling Γ=.358 nd S d = 4.55cm. Stor drift nd ll locl quntitie of interet re obtined b performing puhover nli with the umed force pttern up to the clculted top diplcement. Mximum drift occur in the firt tor nd mount to.87% of tor height. Note tht ll quntitie obtined with the N method re in fct men vlue (Section 4) [6]. Nonliner dnmic nle were performed with the et of ccelerogrm (Section 5.) ech cled to pectrl ccelertion t the period T =.9 equl to.87 g. The men vlue of the top diplcement mount to 7. cm, which i 3% more thn the top diplcement clculted with the N method. The mximum men vlue of tor drift i obtined in the firt tor, like in the N method. It mount to.6%, bout 5% le thn the tor drift obtined in the N method. The eimic demnd w lo clculted for the contemporr building with the me procedure decribed bove. The top diplcement obtined b the N method mount to.3 cm nd it i bout % mller thn the men top diplcement obtined with nonliner dnmic nle (.6 cm). Mximum tor drift obtined with the N method nd with nonliner dnmic nli re.% nd.9%, repectivel The medin IN curve veru the medin IDA curve nd prmeter nd b The medin IN curve, herein defined the reltionhip between medin top diplcement nd pectrl ccelertion t the period of the equivlent SDOF tem, i determined from the eimic demnd for different level of the eimic intenit. For the lowet eimic intenit vlue of. w choen for the rtio between eltic pectrl ccelertion t the period of equivlent SDOF tem nd ield pectrl ccelertion for prticulr building. The level of the rtio ( S ) e T S, which repreent the reduction fctor R, w then progreivel increed, with the intervl of., until the etimted medin top diplcement cpcit (Section 3.4) w reched. For exmple, ingle point of the medin IN curve i determined from the men top diplcement determined with the N method for given eimic intenit (e.g. 6. cm for exiting building nd umed ( S ) e T S =.8, Section 5.3.). Auming tht the coefficient of vrition for top diplcement i equl to.7 (Section 3) nd uing Eq. (6) nd (7), then β =.63 nd medin top diplcement i equl to 5. cm. In order to demontrte the ccurc of the pproximte pproch, ingle point of the medin IN curve will be compred with ingle point of the medin IDA curve. The men top diplcement nd the coefficient of vrition, both clculted b nonliner dnmic nle for given pectrl ccelertion.87g, re equl to 7. cm (Section 5.3.) nd.77, repectivel. The correponding medin vlue, determined ccording to the method of moment nd ccording to the definition of the medin vlue, mount to 5.5 cm nd 5. cm, repectivel. The lter medin diplcement i the me tht obtined from the IN method. All point of the medin IN curve were determined uing the me coefficient of vrition for rndomne in top diplcement V =.7, while for the medin IDA curve the coefficient of vrition for rndomne in top diplcement, obtined from nonliner dnmic nli, w ued. The vlue re preented in Figure 6. It cn be een tht for both tructure the coefficient of vrition obtined from the IDA nli firt incree with increing ( S ) e T S. Then, it become more or le contnt nd comprble with the umed vlue of the coefficient of vrition ued in IN nli. For comprion

11 lo the coefficient of vrition due to rndomne in mximum drift, clculted with the IDA nli, i preented in Figure 6. For thi coefficient high pek re oberved for both building in the rnge ner to ( S ) e T S =.5. In thi rnge the coefficient of vrition for mximum drift exceed the coefficient of vrition for top diplcement. Thi hppen becue in tht rnge of ( S ) e T S oft tor effect occur for onl one or two ground motion, reulting in much higher mximum drift compred to mximum drift obtined for other ground motion, where the difference in top diplcement re not o lrge. The medin IN nd IDA curve for both building re compred in Figure 7. A fir greement cn be oberved, epecill in the ce of contemporr building. The medin IN curve re lightl on the unfe ide. Finll, medin IDA nd IN curve were fitted uing the procedure decribed in Section 4. The vlue of nd b were obtined (Tble ). The fitted curve mtch the medin IDA nd IN curve ver well it cn be een in Figure 7. Figure 6. Coefficient of vrition V due to rndomne for top diplcement nd mximum drift clculted from IDA nli nd umed vlue for IN nli. S e (g) V Exiting building Exiting building fitted IN IN fitted IDA IDA Top diplcement Mximum drift IN S e/s Cpcit curve IN -.% drift IDA -.% drift Top diplcement (cm) V Contemporr building S e/s Top diplcement (cm) Figure 7. Comprion between medin IN nd IDA curve for exiting nd contemporr building. Fitted curve (Eq. ) re lo hown. S e (g) Contemporr building fitted IN IN fitted IDA IDA IN - 4.5% drift IDA - 5.5% drift

12 5.4 Etimtion of the cpcit of the building tructure In the tet exmple the o-clled ner collpe (NC) limit tte i invetigted. It h been conervtivel umed tht the NC limit tte i reched for the whole tructure, when the NC limit tte i reched for the firt column. For bem nd column, the controlling quntitie re chord rottion nd her force. Sher reitnce nd ultimte chord rottion for tructurl element were determined ccording to Eurocode 8 [4]. Bem column joint were lo checked. However, the comprion of her demnd in joint, obtined from puhover nli, nd the correponding her cpcitie [4], indicted tht joint do not repreent criticl prt of the tet tructure. Further, it w conidered tht the collpe of infill doe not cue the NC limit tte of building. The cpcitie in term of ultimte rottion for column nd bem of exiting building re much lower thn thoe of contemporr building. The minimum clculted vlue for the ultimte chord rottion mount to.5 nd correpond to the trong column in the firt tor of exiting building (Figure 3). Bed on the umption tht NC limit tte of the tructure i reched when the rottion of the firt column exceed the clculted ultimte rottion, the NC limit tte for exiting building occur due to exceeding ultimte rottion in the trong column in the firt tor. Bed on puhover nli, the firt tor drift t the ultimte chord rottion of the trong column in the firt tor i equl to.% of the tor height (5.4 cm), nd the correponding top diplcement i equl to 6.55 cm (Figure 5). At thi deformtion ll infill in the firt tor totll collpe nd infill in the econd tor trt to degrde. The oft tor mechnim i formed in the firt tor t % tor drift, much before the NC limit tte. The her demnd exceed the computed her reitnce before.% mximum tor drift i reched for two column in the firt tor nd t the top of one column in the econd tor. However, experimentl reult [9] ugget tht thi computed reult i too conervtive nd not relitic, therefore it w not tken into ccount in further nle. The experiment topped t.5% firt tor drift to enure the ubequent tet on retrofitted building. At thi drift her crck nd cruhing of cover concrete occurred t the bottom of ome column. According to thee reult we conider tht.% tor drift m be reonbl conervtive vlue defining the NC limit tte. The me drift w lo ued NC limit tte in nonliner dnmic nle. Bed on thee nle, the medin vlue of drift cpcit (.%) i ocited with the medin top diplcement equl to 7.34 cm. The NC limit tte for contemporr building w determined with imilr procedure. The criticl tructurl element i the centrl column (Figure 3). The ultimte chord rottion t the bottom of thi column, clculted ccording to Eurocode 8 (CEN 3), mount to.4. Bed on puhover nli, the correponding tor drift i equl to 4.5% nd the top diplcement i equl to 7.5 cm (Figure 5). Thi deformtion tge w conidered NC limit tte lthough her demnd of ome bem nd column exceed the correponding cpcitie determined ccording to Eurocode 8. Of coure, infill collpe in the firt nd econd tor before 4.5% tor drift i reched. The 4.5% tor drift w lo umed NC limit tte for IDA nli nd it w ocited with the medin top diplcement of 9.9 cm. 5.5 Rik evlution The nnul likelihood ( probbilit ) tht the eimic demnd exceed eimic cpcit w clculted Ĉ with Eq. (3). The pectrl ccelertion t the medin top diplcement cpcit w obtined from Eq. (), uing the prmeter nd b, which were determined decribed in ection (Tble ). The Ĉ medin eimic hzrd t w obtined from Eq. (), with prmeter k o nd k determined in Section 5.. The diperion meure for rndomne in top diplcement cpcit β CR w umed to be.5 for both IDA nd IN nle. The diperion meure for rndomne in diplcement demnd β obtined from the IDA nli mount to.57 nd.64 for exiting nd contemporr building, repectivel. Thee vlue of diperion meure do not differ ignificntl from β =.63, which w

13 ued in the IN nli (Section 4 nd 5.3.). Finll, the nnul probbilit tht demnd exceed NC limit tte P i obtined from Eq. (3). In Tble the reult for P nd P,5 well intermedite reult re preented. P,5 repreent the probbilit of exceednce of NC limit tte in fift er. It i determined from the umed binomil ditribution of n event P ( ) 5,5 = P. A ver good greement cn be oberved between P obtined b IN nd IDA nle. P for contemporr building i bout time mller thn of exiting building. A coniderble probbilit of exceednce of NC limit tte in 5 er (bout 3.8 %) i obtined for exiting building. The probbilit P,5 doe not include uncertint in diplcement demnd β DU nd uncertint in diplcement cpcit β CU. In order to include thee uncertintie, Eq. (4) h to be ued. The probbilit x of exceednce of NC limit tte in fift er with the x=9 % confidence P,5 w determined. For thi nli, diperion meure for uncertint in diplcement demnd nd cpcit β DU nd β CU were umed to be.5 for both IDA nd IN nle. More precie determintion of thee uncertintie i not within the cope of thi pper. The ummr of reult i preented in Tble 3. Similrl in the previou ce without uncertint in diplcement demnd nd cpcit, ver good greement cn be oberved between probbilitie obtined b IN nd IDA nli. A rther high vlue of diperion meure for uncertint in P β P, i obtined. Thi i minl conequence of rther high vlue umed for β CU nd β DU. Note tht ubtntill higher vlue re obtined for probbilitie with high confidence level (e.g. 9 % confidence level) in comprion with the 5 % confidence level. For exmple, x the rtio between 9 nd 5 % confidence level etimte of P,5 mount to bout to.5. Note tht the reult with 5 % confidence level re ner to the reult without uncertint in diplcement demnd nd cpcit. The onl ource of difference i the different repreenttion of hzrd curve. In the ce with uncertintie the men hzrd curve w emploed inted of the medin hzrd curve [3]. According to [5] the tructure i fe enough if there i le thn % chnce in 5 er, with 9 % confidence level, tht the collpe prevention performnce level would be exceeded. If we ppl thi 9 criterion, exiting building i not fe enough ( P,5 mount to bout 7%), while the contemporr building i fe ( P mount to bout %). 9,5 Tble. Summr of rik evlution without uncertintie in diplcement demnd nd cpcit uing medin IN nd IDA curve. Prmeter Exiting building Contemporr building IN IDA IN IDA Ĉ : medin diplcement cpcit (cm) medin drift cpcit (%) prmeter of fitted IN nd IDA curve b Ĉ (g): pectrl cc. correponding to dipl. cpcit ˆ C H ˆ : medin etimte of pectrl ccelertion hzrd 6.E E-4 4.9E E-5 ( ) β : diperion meure of rndomne in dipl. demnd P : nnul probbilit exceednce of limit tte 7.8E E-4 7.8E-5 7.9E-5 P,5 (%): probb. of exceednce limit tte in 5 er

14 Tble 3. Summr of rik evlution with 9 % confidence level uing medin IN nd IDA curve. Exiting building Contemporr building Prmeter IN IDA IN IDA Ĉ H : men etimte of pectrl cc. hzrd 6.99E-4 6.7E E E-5 β P ( ) : diperion meure for uncertint in P P ˆ : medin etimte of P 9.8E-4 9.4E-4 9.6E-5 9.8E-5 9 P : 9% confidence level etimte of P.47E-3.6E-3.9E-4.99E-4 9 P,5 (%) : 9% confidence level etimte of CONCLUSIONS In the pper, the Incrementl N (IN) method, imple lterntive for IDA nli, i introduced. The IN method, which repreent n extenion of the N method, cn be ued for determintion of pproximte ummrized IDA curve (IN curve). The IN curve cn ubtitute the IDA curve in the probbilitic frmework for eimic deign nd ement of tructure. The IN method h been pplied to two tet exmple of infilled reinforced concrete frme, which re chrcterized b ubtntil degrdtion of the trength fter the infill fil. A pecific R µ T reltion, tpicl for infilled frme, nd dt on diperion due to rndomne in diplcement demnd developed b uthor in previou tud, were emploed in the probbilitic rik nli of tet tructure. A reonble ccurc of the IN curve in comprion with the IDA curve h been demontrted for both exmple. It h been lo hown tht the diperion for rndomne in diplcement demnd for the MDOF tem cn be predicted from the diperion for rndomne in ductilit demnd obtined from the ttiticl tud of R µ T reltion. A fir correltion of reult obtined b the pproximte procedure with IN method nd more ccurte nli emploing IDA curve ugget tht the IN method i vible pproch. The tet exmple indicte tht the probbilit of the collpe for building deigned ccording to modern tndrd i bout time mller thn for building built bout 5 er go. It h been lo hown tht high confidence etimte for probbilit of exceeding certin performnce level, which include uncertintie in diplcement demnd nd cpcit in ddition to rndomne, ignificntl incree the probbilit of exceeding thi performnce level compred to the ce in which onl rndomne in diplcement demnd nd cpcit re tken into ccount. 7. ACKNOWLEDGEMENTS The reult preented in thi pper re bed on work continuoul upported b the Minitr for Science nd Technolog of the Republic of Sloveni nd, more recentl, b the Europen Commiion within the 5 th Frmework progrm.

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