COMPONENT-LEVEL AND SYSTEM-LEVEL SENSITIVITY STUDY FOR EARTHQUAKE LOSS ESTIMATION

Transcription

1 13 th World Conference on Earthquae Engneerng Vancouver, B.C., Canada August 1-6, 2004 Paper No COMPONEN-LEVEL AND SYSEM-LEVEL SENSIIVIY SUDY FOR EARHQUAKE LOSS ESIMAION Hesameddn ASLANI 1, Eduardo MIRANDA 2 SUMMARY A probablstc methodology s presented to estmate economc losses resultng from earthquae damage n buldngs. he methodology allows the dentfcaton of dfferent sources that contrbute to economc losses n buldngs both at the component-level and at the system-level. Senstvty analyses are then performed to dentfy and quantfy those that contrbute the most to the overall uncertanty n loss estmates. he expected annual loss and the mean annual frequency of exceedance of a certan dollar loss are computed both at the component-level, for a slab-column connecton, and at the system-level. he methodology s llustrated by applyng t to a seven-story renforced concrete buldng. INRODUCION Recent earthquaes n the Unted States have hghlghted the mportance of controllng damage n buldngs n order to lmt economc losses. Desgnng structures n whch economc losses are controlled requres reasonable estmatons of losses n earthquaes wth dfferent levels of ntensty. Dfferent approaches can be used to estmate economc losses n a buldng. In the proposed approach losses are estmated as a functon of the damage n ndvdual structural and nonstructural components. As part of the research beng conducted at the Pacfc Earthquae Engneerng Research (PEER) center a methodology s developed to estmate economc losses n buldngs. he methodology s developed on the bass of the PEER s probablstc framewor. One prmary advantage of the proposed methodology s ts transparency n dentfyng the sources of uncertanty that contrbute to probablty parameters of economc losses both at the component-level and at the system-level. Once dfferent sources of uncertanty are dentfed, senstvty analyses can be performed to quantfy whch sources are sgnfcant and whch ones are not sgnfcant. hs nformaton can then be used to ntroduce smplfcatons n the procedure of loss estmaton. Presented n ths paper s a senstvty study on dfferent sources of uncertantes that affect economc losses at the component-level and at the system-level n buldngs. Frst, we develop the formulaton to estmate the expected annual loss, EAL, and the mean annual frequency, MAF, of exceedng a certan 1 Ph.D. student, Stanford Unversty, Stanford, CA, USA. Emal: haslan@stanford.edu 2 Assstant Professor, Stanford Unversty, Stanford, CA, USA. Emal: emranda@stanford.edu

2 dollar loss n an ndvdual component. he methodology s then llustrated by applyng t to a renforced concrete slab-column connecton. Senstvty studes are performed on dfferent sources of uncertanty to dentfy the sgnfcance of each source n the EAL and MAF of the loss of the case study component. he results show that the modelng uncertanty n the sesmc hazard s the most mportant source of uncertanty to estmate EAL of a slab-column connecton. he MAF of the component s manly senstve to the uncertanty n the repar cost of the component. Smlarly, at the system-level, the formulaton to estmate the EAL and MAF of the loss of the system s developed. he loss estmaton methodology s then appled to an exstng seven-story renforced concrete buldng. he effects of dfferent sources of uncertanty on the estmaton of the system EAL are nvestgated. For the case of MAF the effects of correlaton between losses n ndvdual components are nvestgated. Our studes show that the EAL of the buldng s manly senstve to the uncertanty n the damage estmaton. For the MAF of the system, we found that the effect of correlaton s sgnfcant for low probablty events,.e. global collapse, and can be gnored for hgh-probablty events. he later result s promsng snce t allows for sgnfcant smplfcatons n the loss estmaton of buldngs when damage control s the man sesmc performance objectve. LOSS ESIMAION A HE COMPONEN-LEVEL Estmaton of the expected annual loss (EAL) of a component Usng the total probablty theorem, the average annual loss n an ndvdual component, E [ L ] can be computed as E [ L ] = E [ L IM ] dν ( IM ) 0 (1) where dν (IM) s the dervatve of the sesmc hazard curve evaluated by performng a probablstc sesmc hazard analyss at the ste as a functon of a ground moton ntensty measure, IM. E [ L IM ] s the expected loss n a component condtoned on IM. L s a random varable that represents the loss n the th component normalzed by the orgnal cost of that component. For example, a realzaton of L equal to 0.7 means that the loss n the component s 70% of ts orgnal cost. Smlarly to Eq. (1), the expected loss n the th component for a gven scenaro can be computed as where d P ( EDP IM ) component, EDP, condtoned on IM. [ ] E [ L IM ] = E [ L EDP ] d P ( EDP IM ) (2) 0 s the probablty densty functon of the engneerng demand parameter n the th E L EDP s the average loss n component as a functon of the level of sesmc demand n that component, EDP. P s the structural response parameter whch s closely correlated wth the sesmc damage n the component. For example, damage n almost all types of structural components, such as columns and slab-column connectons s closely correlated wth the level of nterstory drft rato, IDR, n the component. For some of the non-structural components, however, pea floor acceleraton, PFA, s the prmary deformaton parameter whch s closely correlated wth the sesmc damage n some nonstructural components. he deformaton parameter, EDP, n the ( EDP IM ) Dfferent approaches can be used to estmate the probablty densty functon of EDP condtoned on IM. One feasble way s through response hstory smulatons for a sute of earthquae ground motons scaled to certan levels of ntensty, Mranda and Aslan [1].

3 Usng the total probablty theorem for dscrete random varables, we can expand E [ L EDP ] as follows E m [ L EDP ] = E [ L DM = dm ] P ( DM = dm EDP ) = 1 where ( DM dm EDP ) component when the structure s subjected to a deformaton level equal to edp. E [ L DM = dm ] P = s the probablty of beng n the th damage state, DM, of the th s the expected loss n the th component gven that the component s n ts th damage state. he summaton s on all the possble damage states, m, that a component can experence before loosng ts vertcal carryng capacty. he probablty of beng n a certan damage state, DM, condtoned on EDP, P ( DM ) estmated as ( DM dm EDP ) = P ( DM > dm EDP ) P ( DM dm EDP ) EDP (3), can be P = +1 > (4) +1 > are the probablty of exceedng +1th and th damage states, respectvely, condtoned on EDP, and are nown as fraglty functons for those damage states. Damage states n a component are defned based on the requred courses of acton to repar a sesmcally damaged component, Aslan and Mranda [2]. where functons P ( DM > dm EDP ) and P ( DM dm EDP ) Estmaton of the mean annual frequency of exceedng a certan level of loss he approach used to derve Equatons (1) through (4) for the average expected loss of a component can be appled to estmate the mean annual frequency of exceedance of the loss n an ndvdual component, by smply replacng the expected values wth probablty dstrbuton of the component loss n a gven damage state m [ L > l ] = P [ L > l DM = dm ] P ( DM = dm EDP ) d P ( EDP IM ) dν ( IM ) = ν (5) where P [ L l DM = dm ] > s the probablty of exceedng a certan level of loss gven that component s n the th damage state. If the loss curve of the component has been developed, usng Eq. (5), an alternatve approach to estmate the expected annual loss of the component nstead of usng Eqs. (1) - (4) s to compute the area underneath the loss curve E [ L ] = [ L > l ] d l ν (6) 0 Loss estmaton of a renforced concrete slab-column connecton o exemplfy the formulaton of loss estmaton at the component-level, we have appled t to a renforced concrete slab-column connecton located n the thrd story of a testbed structure. he testbed s a seven - story renforced concrete structure. It was desgned n 1965 and bult n he structural system of the buldng conssts of moment-resstng permeter frames and nteror gravty-resstng frames (flat slabs and columns). he structure s nomnally symmetrc wth the excepton of an nfll wall n the frst floor of the north frame of the buldng. A detaled descrpton of the testbed buldng has been presented n Brownng et al. [3].

4 Four basc ngredents are requred to evaluate Eqs. (1) (4): (1) sesmc hazard curve at the ste,ν (IM); (2) probablty dstrbuton of EDP condtoned on IM, P( EDP IM )(3) fraglty functons correspondng to dfferent damage states that a component can experence as a functon of the level of EDP n that component, P ( DM > dm EDP ); and (4) loss functons correspondng to the cost of repar of the component n a gven damage state, P [ L > l DM = dm ]. Fgures 1 through 4 presents each of the above basc ngredents for loss estmaton of a renforced concrete slab-column connecton n the testbed buldng. Fgure 1 presents the sesmc hazard curve at the ste of the testbed structure. In ths study the spectral dsplacement of a lnear elastc sngle-degree-of-freedom system evaluated at the frst perod of vbraton of the mult-degree-of-freedom model of the structure, S d, s selected as the ground moton ntensty measure. Informaton shown n Fgure 1 s used to estmate dν (IM) n Eqs. (1) and (5). Fgure 2 presents the varatons of the medan nterstory drft rato along the heght of the buldng at dfferent levels of ntensty. he results n Fgure 2 are obtaned usng a seres of non-lnear tme hstory analyses of the testbed structure, Mranda [1]. Correspondng to the medan response at each floor level and each level of ntensty, shown n Fgure 2, a probablty dstrbuton exsts. An example s presented for the IDR n the thrd story when the structure s subjected to a spectral dsplacement of 20 cm. he nformaton presented n Fgure 2 s used to estmate d P (EDP IM) n Eqs. (2) and (5). Fraglty functons correspondng to dfferent damage states of a slab-column connecton are shown n Fgure 3. Four damage states are defned for ths component. he frst damage state, DM 1, occurs when the small levels of cracng are observed n the connecton. he requred repar acton s cosmetc repars such as pastng and pantng. he next damage state, DM 2, s defned when the cracs are wde enough to requre epoxy njecton as the repar acton. he thrd damage state, DM 3, s defned when the component has experenced a punchng shear falure. At ths stage the damaged concrete needs to be removed and addtonal renforcement may be requred before pourng new concrete. he last damage state, DM 4, corresponds to the loss of vertcal carryng capacty, LVCC, n the component. At ths damage state the component collapses. As a smplfyng assumpton n ths study we assume that loosng the vertcal carryng capacty at the component level results n a system falure. Informaton presented n Fgure 3 s used to estmate P ( DM EDP) n Eqs. (3) and (5). ν ( IM ) 1.E+00 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 1.E-06 1.E Fg. 1. Sesmc hazard curve at the ste of the case study buldng. Story Sd = 2.5 cm Sd = 10 cm Sd = 20 cm Sd = 30 cm Medan IDR Fg. 2. Varatons of the medan nterstory drft rato (IDR) along the buldng heght at dfferent levels of ntensty. Sd = 40 cm Sd = 50 cm

5 P ( DM EDP ) Slab-Column connecton P ( L DM ) Slab - Column Connectons DM1 DM2 DM3 DM EDP [IDR] Fg. 3. Fraglty functons for dfferent damage states of a slab-column connecton located n the thrd story of the buldng. DM1 DM2 DM Normalzed Loss (L ) Fg. 4. Loss functons for a slab-column connecton. Fraglty functons developed for slab-column connectons are based on expermental research on ths type of component. A summary of the expermental data used to develop these fraglty functons s presented n Aslan and Mranda [2]. Further, we assume that the fraglty functons follow a cumulatve lognormal dstrbuton. hs assumpton ntroduces some approxmaton n estmatng the EAL at the componentlevel and system-level. Fgure 4 presents the loss functons assocated wth the repar cost for each of the frst three damage states of a slab-column connecton. Loss functons are calculated by temzng the tass requred to repar a component at dfferent damage states. Informaton presented n Fgure 4 s used to estmate E [L DM ] n Eq. (3) and P [ L > l DM ] n Eq. (5). E [ L EDP ] Slab-Column connecton σ [ L EDP ] Slab-Column connecton EDP [ IDR ] EDP [ IDR ] Fg. 5. Varatons of the mean, E [ L EDP ], and the standard devaton of the loss, σ [ L EDP ], for a slab-column connecton located n the thrd story of the buldng wth changes n the nterstory drft rato, IDR, n that component.

6 E [ L IM ] Slab-Column connecton σ [ L IM ] Slab-Column connecton Fg. 6. Varatons of the mean, E [ L IM ], and the standard devaton of the loss, σ [ L IM ], for a slab-column connecton located n the thrd story of the buldng wth changes n the level of sesmc hazard ntensty, IM. Informaton presented n Fgures 1 to 4 are used n Eqs. (1) to (4) to estmate the average economc losses n a slab-column connecton. Beyond the expected loss, varatons of the standard devaton of the loss for an ndvdual component are nvestgated usng the same approach presented n Eqs. (1) to (4). Fgure 5 presents the varatons of the mean, E [ L EDP ], and the standard devaton of the loss, σ [ L EDP ], wth changes n the nterstory drft rato, IDR, for the slab-column connecton located at the thrd story of the testbed structure. As shown n the fgure the loss n the component starts at around % drft. At around 2% drft the slope of both mean and standard devaton of the loss becomes almost 0, caused by the relatvely large demand requred to experence the thrd damage state compared to the drft level at whch the second damage state occurs, Fgure 3. Integratng results presented n Fgure 5 over all possble EDP s, Eq. (2), we can now estmate the mean, E [ L IM ], and the standard devaton of the loss, σ [ L IM ], for a gven scenaro, IM. Fgure 6 presents the varatons of E [ L IM ] and σ [ L IM ] for the slab-column connecton located n the thrd story of the testbed structure. As can be seen n the fgure, loss n the component ntates at IM = 5 cm, whereas the standard devaton of the loss starts at around 2.5 cm. Comparng the ntaton of standard devaton of the loss wth the expected loss shows that standard the coeffcent of varaton (c.o.v.) at small levels of ntensty s sgnfcantly larger than 1 snce σ [ L IM ] approaches toward 0 more slowly than E [ L IM ]. hs observaton s of hgh mportance snce the ntaton of loss n a component plays a sgnfcant role both n component expected annual loss and the whole buldng (system) expected annual loss. At IM = 20 cm the average loss n the component s around 40% of the component orgnal cost. At ths level the standard devaton of the loss s sgnfcant, leadng to c.o.v. s around. he man reason of the relatvely large values of the c.o.v. at ths level and n general n ths example s the sgnfcant uncertanty for the repar costs, uncertanty at DV DM level. Later, n the senstvty analyss t s shown how ths source of uncertanty domnates the results of estmatng the mean annual frequency of exceedng a certan level of loss.

7 he results from Fgure 6 can be ntegrated together wth the sesmc hazard curve at the ste to estmate the mean annual frequency of exceedance of the loss n the component, Eq. (1). he curve labeled uncertanty at all levels n Fgure 7 presents the loss curve for the slab-column connecton example. Shown n the fgure s the annual rate of exceedng a certan level of loss n the component. For example, the mean annual frequency of exceedng 50% of the component orgnal cost s 05. Senstvty study on the EAL and MAF of loss of a slab-column connecton A man advantage of the formulaton presented to evaluate the expected annual loss of a component, Eqs. (1) - (4), s ts transparency n dentfyng dfferent sources of uncertanty that effect EAL. wo dfferent sources of uncertanty are nvestgated: uncertanty stemmed from the estmaton of the sesmc response at dfferent levels of ntensty, P ( EDP IM); and uncertanty stemmed from the ncurred sesmc damage n the component caused by the mposed sesmc demand, P ( DM EDP ). o nvestgate the effects of each of the above sources of uncertanty n estmaton of the EAL, we have estmated the changes n average annual loss by assumng uncertanty at one level and certanty n another level. For example, we estmated the EAL of the component when there s no uncertanty at DM EDP level but there s uncertanty at EDP IM level and compared t to the case when uncertantes at both levels are taen nto account. Our observatons show that for the type of uncertantes consdered n ths study and for ths component the uncertanty at DM EDP level s more sgnfcant than the one of EDP IM n estmatng the EAL. ν [ L > l ] 1.E-01 1.E-02 1.E-03 1.E-04 1.E-05 Slab-column connecton 1.E Uncertanty at all levels No uncertanty at DV DM level No uncertanty at DM EDP level No uncertanty at EDP IM level Uncertanty at IM level L Fg. 7. Effects of dfferent sources of uncertanty on the estmaton of the component loss curve. For the case of MAF of the loss at the component-level three sources of uncertanty are nvestgated. he frst two sources of uncertanty are the same as what consdered n EAL senstvty analyss. he thrd source s the uncertanty correspondng to the DV DM level. he results of the senstvty analyss on the loss curve of the slab-column connecton are presented n Fgure 7. As shown n the fgure the domnatng source of uncertanty n ths case s the one correspondng to that assocated wth repar costs of the component. Effects of the sesmc hazard uncertanty he senstvty of the EAL and MAF to the assumptons made to estmate the sesmc hazard at the ste s nvestgated. he uncertanty n the sesmc hazard s accounted for by computng the 84 th percentle hazard curve, Jalayer [4]. Our observatons show a 50% ncrease n the expected annual loss of the component because of the modelng uncertanty at the sesmc hazard level. he changes n the loss curve of the slab-column connecton are shown n Fgure 7.

8 LOSS ESIMAION A HE SYSEM-LEVEL Estmaton of the expected annual loss of the system he EAL of the system can be expanded as E [ L ] = E [ L IM ] dν ( IM ) where E [ L IM ] s the average loss of the system condtoned on IM. he scenaro-based expected loss of the system can be computed as E 0 (7) [ L IM ] E [ L IM,NC][ P ( C IM )] E [ L C] P ( C IM ) = 1 (8) + where [ L IM,NC] collapse, E [ L C] s the expected loss of the system when the structure collapses, ( C IM ) E s the expected loss of the system condtoned on IM when the structure does not probablty that structure collapses n a gven earthquae scenaro. he non-collapse system expected loss condtoned on IM, [ L IM,NC] summng over all component losses n the buldng. E [ L IM,NC] = a E [ L IM ] n = 1 where a s the orgnal cost of the th component, and [ L IM ] P s the E, can be estmated by smply E s estmated from Eq. (2). he summaton s over all non-rugged components n the buldng, n. he probablty of collapse consdered n ths study taes nto account the collapse caused by the loss of vertcal carryng capacty, LVCC. o estmate P ( C IM ), we assume, Aslan [4], that no re-dstrbuton of vertcal loads occurs at the system-level and the probablty of collapse due to LVCC would be equal to the largest probablty of any ndvdual structural element that can loose ts vertcal carryng capacty (9) E [ L IM,NC ] $ 10 M $ 8 M $ 6 M $ 4 M $ 2 M E [ L C ] $ 10 M $ 8 M $ 6 M $ 4 M $ 2 M $ 0 M $ 0 M IM [ S d (cm) ] Fg. 8. Varaton of the system expected loss wth changes n the level of ntensty, IM, for both cases of occurrence and non-occurrence of the structural collapse.

9 where ( LVCC IM ) P ( C IM ) max [ P ( LVCC IM )] = (10) s P s the probablty of losng the vertcal carryng capacty n the th component condtoned on IM and s computed as P ( LVCC IM ) = P ( LVCC EDP ) d P ( EDP IM ) (11) 0 where ( LVCC EDP ) that t s subjected to a deformaton level equal to edp. ( LVCC ) P s the probablty of the th component losng ts vertcal carryng capacty gven P EDP s computed from fraglty surfaces, Aslan and Mranda [5], developed for LVCC damage states on the bass of expermental studes on structural components. In a fraglty surface the mean and standard devaton of EDP correspondng to the LVCC damage state are evaluated as a functon of a new parameter, α, whch allows the ncorporaton of addtonal nformaton. he parameter α can ncorporate nformaton on the element (e.g., geometry, detalng, etc.), the loadng and or a combnaton of the two. he probablty of exceedng the LVCC damage state s then estmated as a functon of the level of EDP n the component but also as a functon of the parameter α. Estmaton of expected annual loss for the renforced concrete testbed structure he formulaton presented n the prevous secton s appled to the seven-story testbed structure. All structural and non-structural components n the buldng are carefully dentfed. For each component fraglty functons and loss functons correspondng to dfferent damage states n the component are developed, Aslan and Mranda [2], aghav and Mranda [6]. Further, for certan damage states of structural components fraglty surfaces are developed for a relable estmate of that damage state. Fgure 8 presents the expected loss of the system as a functon of the level of ntensty n the buldng both for non-collapse and collapse cases. he average loss of the system for the case of non-collapse starts at around 2.5 cm lnear spectral dsplacement. he man source of loosng money at such small levels of ntensty stems from the facts that the loss n non-structural components ntates at low levels of P ( DM EDP,α ) EDP [ IDR ] 2 α [ V g / V 0 ] Fg. 9. Fraglty surface for loss of vertcal carryng capacty (LVCC) n slab-column connecton as a functon of the IDR and α. P ( C IM ) IM [ S d (cm) ] Fg. 10. Probablty of system collapse caused by loosng the vertcal carryng capacty at the component level.

10 ntensty and the fraglty functons are assumed lognormal. If the structure collapses at any level of ntensty the loss of the system s equal to the total cost of the buldng, as shown on the rght-hand sde graph of Fgure 8. o estmate the expected loss at the system level usng Eq. (8), we need to estmate the probablty of collapse as a functon of the level of ntensty n the buldng. For the testbed buldng we assume that the most probable mode of collapse s losng the vertcal carryng capacty. o capture ths mode we used fraglty surfaces of dfferent structural components n the buldng. Fgure 9 presents an example of a fraglty surface developed for the loss of vertcal carryng capacty n a slab-column connecton. We used fraglty surfaces developed for dfferent structural components together wth response smulaton results of the testbed buldng, Fgure 2, to estmate the probablty of collapse n the LVCC mode, usng Eqs. (10) and (11). Fgure 10 presents the probablty of collapse computed for the testbed buldng as a functon of the ground moton ntensty level. As shown n the fgure, probablty of collapse at the system level for ntenstes smaller than 7.5 cm s 0. Incorporatng the results of Fgures 8, 9 and 10 n Eq. (8) we can estmate the condtonal expected loss of the system at dfferent levels of ntensty for the case study buldng. he result s presented n Fgure 11. he fgure also shows the contrbuton of collapse and non-collapse loss to the total loss of the system. It can be seen that at small levels of ntensty, less than 20 cm, the non-collapse losses s a major contrbutor to the total loss. Integratng the results from Fgure 11 wth the sesmc hazard at the ste, Eq. (7) we can estmate the expected annual loss of the system. For the testbed structure the EAL s $ 146,000. It should be noted that ths value s estmated based on two man assumptons; the most probable collapse mode of the system s LVCC and the components fraglty functons are lognormally dstrbuted. he senstvty of the EAL to these assumptons s under nvestgaton. Specally, for the case of second assumpton, prelmnary results show that the EAL s qute senstve to the ntaton of the loss n each ndvdual component. E [ L IM ] $ 10 M $ 8 M $ 6 M E [ L IM,NC ] $ 10 M $ 8 M $ 6 M DM EDP & EDP IM EDP IM DM EDP $ 4 M $ 2 M Collapse Non-collapse $ 4 M $ 2 M $ 0 M IM [ Sd (n) ] Fg. 11. Varatons of the system expected loss wth the level of ntensty. $ 0 M Fg. 12. Effects of uncertanty at DM EDP and EDP IM levels on the non-collapse expected loss of the system for a gven IM.

11 Senstvty study on the expected annual loss of the system he effects of two sources of uncertanty on the EAL of the system are nvestgated; the uncertanty n estmatng the damage as a functon of EDP, DM EDP level uncertanty, and the uncertanty n the estmaton of sesmc response as a functon of the level of ntensty, EDP IM level uncertanty. It can be understood from Eq. (8) that these two sources of uncertanty affect both the [ L IM,NC] P ( C IM ). E and Fgure 12 presents the senstvty of the non-collapse average loss of the system condtoned on IM to uncertanty at EDP IM and DM EDP levels. As can be seen n the fgure the non-collapse expected loss s not very senstve to the uncertantes at these two levels. hs observaton s very mportant snce t allows for sgnfcant smplfcatons n the estmaton of E [ L IM,NC], wth relatvely small amounts of error. he effects of uncertanty at EDP IM and DM EDP levels on the probablty of system collapse n the LVCC mode s shown n Fgure 13. he labels on the fgure shows whch sources of uncertanty are consdered n the estmaton of the probablty of collapse. For example, the curve wth EDP IM label presents the probablty of collapse when the uncertanty at EDP IM s consdered and not the one of the DM EDP. As can be seen n the fgure the uncertanty at the DM EDP level, plays a more sgnfcant role n the assessment of P ( C IM ) compared to the effects of EDP IM uncertanty. he effects of the two sources of uncertanty at the response level and the damage level on the expected loss of the system are nvestgated, usng Eq. (8). Fgure 14 presents the results of ths nvestgaton. Our observatons show that the uncertanty at the DM EDP s very mportant snce t can change the ntaton of the total loss at the system level, whch has a sgnfcant effect on the expected annual loss. For example, for the case of the testbed structure when certanty s assumed at the damage level the expected loss starts at 10 cm elastc spectral dsplacement, whle when the DM EDP uncertanty s accounted for, the ntaton of loss starts at around 2 cm. Our nvestgatons show that not consderng the uncertanty at the damage level can ntroduce 17% dfference n the EAL of the system, whle. assumng certanty at the EDP IM level ntroduces a 10% error n the system EAL. P ( C IM ) E [ L IM ] $ 10 M $ 8 M $ 6 M EDP IM & DM EDP EDP IM DM EDP IM [ S d (cm) ] Fg. 13. Effects of uncertanty at DM EDP and EDP IM levels on the probablty of collapse of the system n LVCC mode. $ 4 M $ 2 M $ 0 M DM EDP & EDP IM EDP IM DM EDP Fg. 14. Effects of uncertanty at DM EDP and EDP IM levels on the expected loss of the system condtoned on IM.

12 On the bass of above observatons t can be concluded that for the case of the testbed structure when the expected annual loss s the target performance measure, a more accurate estmaton of fraglty functons, whch results n decreasng the uncertanty at DM EDP level, should be performed. he concluson cannot be generalzed, however, snce the effects of modelng uncertanty at dfferent levels, specfcally at the EDP IM level, has not taen nto account and s currently under nvestgaton. Estmaton of the mean annual frequency of the system loss he MAF of the system can be computed as ν [ L > l ] = P [ L > l IM ] dν ( IM ) 0 (12) where P [ L > l IM ] s the probablty of losng a certan level of loss, l, n a gven scenaro, IM. he probablty of exceedng a certan level of loss at a gven scenaro can be expanded as P [ L l IM ] = P [ L > l IM,NC][ P ( C IM )] + P [ L l C] P ( C IM ) > 1 (13) > where P [ L l IM,NC] structure does not collapse, P [ L l C] structure collapses. In ths study we assume P [ L l C] > s the probablty of losng money n the system condtoned on IM when the > s the probablty of losng money n the system when the > s lognormally dstrbuted. From central lmt theorem n the theory of probablty, we can assume that the sum of n random varables s normally dstrbuted for large enough n s. Hence, non-collapse probablty of losng money n the system at a gven scenaro when the structure does not collapse can be assumed normal l - E [ ] [ L IM, NC] P L > l IM,NC = 1 Φ (14) σ [ L IM,NC] E L IM, NC s where Φ s the standard cumulatve normal dstrbuton of the loss at the system-level, [ ] estmated from Eq. (9), σ [ IM,NC] s the standard devaton of the L,when the structure does not collapse for a gven scenaro and can be expanded as L n n [ L IM,NC] = σ a a ρ σ σ (15) = 1 j = 1 j L L j IM, NC L IM, NC L j IM,NC where ρ L L j IM, NC s the correlaton coeffcent between the losses n ndvdual components as a functon of the level of ntensty, σ NC, σ IM, NC s the standard devaton of the loss n a gven scenaro for IM, L L j the th and jth components and can be estmated as descrbed n the component-level loss estmaton secton. f the losses n ndvdual components are not correlated, Eq. (15) smplfes to [ L IM,NC] = = n 2 σ 2 L IM,NC σ a (16) 1

13 f the losses n ndvdual components are correlated, then the correlaton coeffcent can be wrtten as σ L L j IM, NC ρ L L j IM, NC = (17) σ σ L IM, NC L j IM,NC where σ s the covarance between the losses n the th and jth components for a gven scenaro, L L IM, NC j IM. Estmaton of σ L L j IM, NC requres the nowledge of correlaton between ndvdual components at three dfferent levels, EDP IM, DM EDP, and DV DM. Detaled formulaton on the evaluaton of correlaton between losses n ndvdual components s presented n Aslan and Mranda [5]. Estmaton of the mean annual frequency of the loss at the system-level for the testbed structure he scenaro-based probablty of losng money for collapse and non-collapse cases, P [ L > l IM,NC] and P [ L > l C] respectvely, are estmated for the testbed structure, usng formulatons and assumptons presented n prevous secton. he buldng loss curve, MAF of loss at the system level, s then computed usng Eqs. (12) and (13). he results are shown n Fgure 16 and are dscussed n the next secton. Senstvty study on the effects of correlaton on the mean annual frequency of the system loss o nvestgate the effects of correlaton on the MAF of the loss for the testbed structure, two sets of loss estmatons are accomplshed. In the frst set, we assume that the losses n ndvdual components are not correlated and estmate the standard devaton of the loss at the system level, L, usng Eq. (16). In the second set,we assume that the losses n ndvdual components are correlated. he correlaton s computed usng the methodology and data presented n Aslan and Mranda [5]. Fgure 15 compares the standard devaton of L at dfferent levels of ntensty for correlated and non-correlated cases when the structure does not collapse (left graph) and when t collapses (rght graph). As can be seen n the fgure the effects of correlaton on standard devaton of L s sgnfcant both for collapse and non- collapse cases. When the structure does not collapse the standard devaton of L almost trples for the case of correlated losses n ndvdual components compared to the non-correlated case. When the structure collapses the standard devaton of L ncreases by 75% from correlated to non-correlated case. σ [ L IM,NC ] σ [ L C ] $ 10 M $ 8 M Correlated losses Non-correlated losses $ 10 M $ 8 M Correlated losses Non-correlated losses $ 6 M $ 6 M $ 4 M $ 4 M $ 2 M $ 2 M $ 0 M $ 0 M Fg. 15. Effects of correlaton on the standard devaton of the loss at system-level at dfferent levels of ntensty, IM, when the structure does not collapse (left graph) and when t collapses (rght graph).

14 ν ( L > l ) 1.E+00 1.E-01 Correlated Uncorrelated 1.E-02 1.E-03 1.E-04 1.E-05 $ 0 M $ 4 M $ 8 M $ 12 M $ 16 M Loss ($) Fg. 16. Effects of correlaton between losses n ndvdual components on the loss curve of the testbed structure. Presented n Fgure 16 s the comparson between the loss curves of the testbed structure when the losses n ndvdual components are correlated and when they are assumed non-correlated. It can be seen that for small levels of loss whch can also be referred to as hgh-probablty events, losses smaller than $ 2 M, the dfference between the two curves s very small. For low-probablty events, losses larger than $ 10 M, however, the dfference between the correlated and non-correlated MAF s are sgnfcant. For example, the MAF of losng more than $ 12 M when the losses at the component-level are assumed to be non-correlated s almost half of the MAF computed for the correlated case. CONCLUSIONS Identfyng the mportant sources of uncertanty n estmatng economc losses at the component-level and system-level n buldngs s nvestgated. Four sources of uncertanty contrbute to the loss estmaton at the component-level; uncertanty n the sesmc hazard, IM, uncertanty n the mposed sesmc demand at a gven scenaro, EDP IM uncertanty, uncertanty caused by the damage n the component, DM EDP uncertanty, and the uncertanty n the ncurred loss n the component, DV DM uncertanty. he contrbuton of each of these sources of uncertanty on the expected annual loss (EAL) and the mean annual frequency (MAF) of exceedng a certan level of loss s modeled by extendng the PEER framng equaton. he extended methodology s then appled to a renforced concrete slab-column connecton. he results show that the most contrbutng source of uncertanty for the case of EAL s the uncertanty n the sesmc hazard curve. For the MAF of loss, however, the effects of the DV DM uncertanty are the most on the loss curve compared to other sources of uncertanty. At the system-level, the effects of EDP IM level uncertanty and DM EDP uncertanty are nvestgated on the EAL. he two sources of uncertanty effect both the scenaro-based expected loss when the structure does not collapse, E [ L IM, NC ], and the probablty of collapse at a gven scenaro, P ( C IM ). Our nvestgaton shows that the EAL s manly senstve to the DM EDP uncertanty.

15 A senstvty analyss s preformed on the effects of correlaton on the buldng loss curve, MAF. Frst, we developed a formulaton to account for the effects of correlaton between losses n ndvdual components. he formulaton estmates the correlaton between component losses as a functon of the correlaton at three levels of EDP IM, DM EDP and DV DM. he methodology s then appled to a renforced concrete testbed structure. Our observatons show that the correlaton between component losses sgnfcantly ncrease the dsperson of the loss at the system-level, ncrease of more than 300% for the non-collapse dsperson of the loss at dfferent scenaros. he effect of correlaton on the buldng loss curve s then nvestgated. It s concluded that assumng non-correlated component losses s not conservatve and for the case of low-probablty events, e.g. collapse, can lead to underestmatons of more than 50% n MAF n some cases. For the case of hgh probablty events, however, assumng that the losses are non-correlated at the component-level does not ntroduce sgnfcant approxmaton to the loss curve. he results s promsng snce t allows for great smplfcatons n the loss estmaton methodology and ncreasng ts practcalty for hgh probablty events. ACKNOWLEDGEMENS he authors gratefully acnowledge the fnancal support of ths study from the Pacfc Earthquae Engneerng Research (PEER) Center under the Earthquae Engneerng Research Centers Program of the Natonal Scence Foundaton (Award number EEC ). REFERENCES 1. Mranda E, Aslan H. Buldng-specfc loss estmaton methodology. Report PEER , Pacfc Earthquae Engneerng Research Center, Unversty of Calforna at Bereley, Bereley, Calforna, Aslan H, Mranda E. Probablstc damage assessment for buldng-specfc loss estmaton. PEER report under preparaton, Brownng J, L RY, Lynn A, Moehle JP. Performance assessment for a Renforced concrete frame buldng. Earthquae Spectra 2000; 16(3): Jalayer F. Drect probablstc sesmc analyss: mplementng non-lnear dynamc assessments. Ph.D Dssertaton, Stanford Unversty, Stanford, CA, Aslan H, Mranda E. Investgaton of the effects of correlaton for buldng-specfc loss estmaton. Report n preparaton, Pacfc Earthquae Engneerng, aghav S, Mranda E. Response assessment of nonstructural elements., PEER Report No. 2003, Pacfc Earthquae Engneerng Research Center, Rchmond, Calforna, 2003.