Uncertanty Specfcaton and Propagaton for Loss Estmaton Usng FOSM Methods J.W. Baer and C.A. Corne Dept. of Cv and Envronmenta Engneerng, Stanford Unversty, Stanford, CA 94305-400 Keywords: Sesmc, oss estmaton, correaton, Frst-Order Second-Moment ABSTRACT: Probabstc predcton of structura and nonstructura damage costs due to future earthquaes s one component of oss estmaton currenty beng deveoped for use n performance-based earthquae engneerng. Sources of uncertanty n ths predcton ncude epstemc and aeatory uncertanty n the ste ground moton hazard, the budng response, the damage measures of each of the many budng eements, and repar cost of each of the eements. These are nter- and cross-correated random varabes. Two desred resuts are the tota uncertanty n annua osses, and the contrbuton of each uncertanty source to the tota uncertanty. Monte Caro smuaton s a smpe souton, but t can be computatonay expensve. Ths study proposes an aternatve approach usng Frst-Order Second-Moment (FOSM) methods for a but the (domnant) ground moton ntensty varabe. Suggestons for characterzaton of correatons are presented. A procedure for appyng FOSM methods n the cacuaton of tota uncertanty s outned. The proposed technque s very effcent, and easy used for senstvty studes. 1 INTRODUCTION Estmaton of annua osses n a budng due to earthquae damage s a quantty of nterest to decson maers, and s a current topc of study n performance-based earthquae engneerng. Among the quanttes to be determned are the uncertanty n the resut, and the contrbuton of each source of uncertanty to the tota uncertanty. The Pacfc Earthquae Engneerng Research Center (PEER) has proposed the foowng framng equaton for ths anayss: TC ( TC DVE DVE DM DM EDP EDP u v y x z ) f ( z, u) f ( u, v) f ( v, y) f ( y, x) d ( x) (1) wth terms defned n Secton 1.1. Ths equaton aows for moduar consderaton of the ground moton hazard, budng response, damage to budng eements, eement repar costs, and tota repar cost (Corne and Krawner, 000, Krawner, 00). One opton for cacuatng uncertanty n the resut s through Monte Caro smuaton (Porter, 00). Athough straghtforward, t can be very expensve computatonay, especay when mutpe runs are requred to cacuate senstvtes. The obectve of ths study s to propose an aternatve method of cacuatng uncertanty usng the Frst- Order Second-Moment (FOSM) method (e.g. Mechers, 1999). We sha use ths approxmate method to coapse out severa of the ntermedate condtona random varabes, eavng a mean and varance of Tota Cost (TC) condtoned on the Intensty Measure () of the ground moton. Ths nformaton can then be combned wth the ground moton hazard, d (x), to obtan the expected annua tota cost, varance n annua tota cost, and the annua rate of exceedng a gven tota cost. 1.1 Expanaton of the Framng Equaton The varabes n Equaton 1 are defned as foows: TC (z) s the annua rate of exceedng a tota repar cost of z, where tota repar cost, TC s the decson varabe under study. f TC DVE (z,u) s the PDF of TC, condtoned on the vector of damage vaues of each eement (DVE s the damage vaue of eement ). The assumpton n the framewor descrbed beow s that the tota cost of repar s the sum of a eement repar costs, but ths can be easy generazed. f DVE DM (u,v) s the PDF of the vector of damage vaues of each eement, gven the vector of damage states of each eement (DM s the damage state of
eement ). Mean repar costs can be estmated from sources such as R.S. Means Co. s pubshed materas on constructon cost estmatng (R.S. Means Co. 00). Addtona quantfcaton of repar costs s a topc of current research. f DM EDP (v,y) s the PDF of the vector of damage states, gven the vector of engneerng demand parameters. In current research, these damage states are typcay dscrete, and each state s descrbed by a fragty functon, whch returns the probabty of an eement exceedng the damage state at a gven EDP eve. See Asan and Mranda (00) for exampes. f EDP (y,x) s the PDF of the vector of engneerng demand parameters, gven the ntensty measure. For aeatory uncertanty, ths dstrbuton can be determned usng, for exampe, Incrementa Dynamc Anayss (Vamvatsos and Corne 00). d (x) s the absoute vaue of the dervatve of the annua rate of exceedng a gven vaue of the ntensty measure (the sesmc hazard curve). The absoute vaue s needed because the dervatve s negatve. See, for exampe, Kramer (1995) for bacground on hazard curves. ASSUMPTIONS A Marovan dependence s assumed for a dstrbutons n the framewor. For exampe, t s assumed that the dstrbuton of the DM vector can be condtoned soey on the EDP vector, and that nowedge of the provdes no addtona nformaton. In ths way, prevous condtonng nformaton does not need to be carred forward through a future dstrbutons, reducng compexty. A condtonng varabe that contans a necessary condtona nformaton s deemed a suffcent descrptor (Luco 00). A damage s assumed to occur on an eement eve. The tota cost of damage to the structure s then the sum of the damage cost of each eement n the structure. The excepton to ths assumpton s when coapse occurs, and repar costs w be a functon of the coapse, rather than ndvdua eement responses. The treatment of ths excepton s expaned beow. A reatons n the framewor are assumed to be scaar functons. For exampe, the condtona dstrbuton of the Damage Measure of eement s a functon of ony the th Engneerng Demand Parameter. Or aternatvey, f DM EDP ( v, y) f DM EDP ( v, y ) () Note aso that the functon s not condtoned on varabes from any prevous steps, because of the Marovan process assumpton descrbed earer. To cacuate tota uncertanty n our decson varabe, t w be necessary to account for both epstemc and aeatory uncertanty. The framewor outned here s approprate for ether source of uncertanty. These two sources of uncertanty are uncorreated, aowng ther contrbutons to be cacuated separatey for smpcty, and towards the end of the procedure. Ths s further dscussed n Secton 3.6. These assumptons are beeved to be consstent wth the most advanced current sesmc oss estmaton efforts. Most can be reaxed wthout forma dffcuty. 3 PROCEDURE The procedure outned maes use of FOSM approxmatons to cacuate the mean and varance of TC gven. Ths nformaton can then be combned wth the ground moton hazard, d (x), to obtan the expected annua tota cost, varance n annua tota cost, and (together wth a dstrbuton type assumpton), the mean annua rate of exceedng a gven tota cost. For ths fna combnaton wth the ground moton hazard, FOSM approxmatons are not used. The FOSM approxmatons are ustfed by the assumpton that the uncertanty n the hazard curve s the most sgnfcant contrbutor to varance of the tota oss. Therefore, we are retanng the fu dstrbuton for tsef, but usng the FOSM approxmatons for a moments condtoned on. In addton, we ey do not have nformaton about the fu dstrbutons of some varabes (for exampe, repar costs), and so negectng hgher moments of these dstrbutons does not resut n a sgnfcant oss of avaabe nformaton. Note that we are worng wth natura ogarthms of the varabes descrbed prevousy. Ths aows us to wor wth sums of terms, rather than products. We revert to a non-og form for the fna resut. The procedure s outned n the foowng sectons. 3.1 Specfy n EDP The proposed mode n ths study s EDP =H () (), where H () s the (determnstc) mean vaue of EDP gven, and () s a random varabe wth mean of one, and condtona varance adusted to mode the varance n EDP. (We ntroduce the random varabe notaton X Y, to denote that the mode of X s condtoned on Y.) Then when we use the og form of EDP, we have a random varabe of the form n(edp )=n(h ())+n( ()). Note that the expected vaue of n(edp ) s n(h ()), and the varance of n(edp ) s equa to the varance of n( ()). Both n(h ()) and n( ())], as we as the correatons between nedp s, can be determned from Incrementa Dynamc Anayss.
We w need the foowng nformaton for our cacuatons: n EDP ], denoted h () for a EDP (3) Fgure 1: Exampe Eement Fragty Functons n EDP ], denoted h* () for a EDP (4) ρ(n EDP, n EDP ), denoted ĥ () for a {EDP, EDP } (5) These functons w be used n Secton 3.3 beow. 3. Specfy DM n EDP and n DVE DM, and coapse to n DVE n EDP The dscrete states of the Damage Measure varabe found n current oss estmaton (Asan and Mranda, 00, Porter, 00) are not compatbe wth the FOSM approach, whch requres contnuous functons for the moments. To dea wth the dscrete states, we tae advantage of the fact that we can aways coapse the two dstrbutons DM n EDP and n DVE DM nto one contnuous dstrbuton n DVE n EDP by ntegratng over the approprate varabe: f DVE EDP ( u, y) fdve DM ( u, v) fdm EDP ( v, y) (6) v For a gven eement wth n possbe damage states, we use a set of eement fragty functons F 1, F F n, such that F (y)=p(dm>d EDP=y) (see Fgure 1). We aso defne F 0 1 (the probabty that each eement has at east zero damage s one). These functons w have a correspondng set of dstrbutons c 1, c c n of eement repar costs such that c (v) s a probabty dstrbuton of DVE, gven that the damage state equas d (see Fgure ). Wth ths nformaton, we can determne the frst two moments of the coapsed dstrbutons. For exampe, Asan and Mranda (00) document the deveopment of one set of these functons. Fgure : Eement Repar Costs From the tota probabty theorem, we now that n ths case, Equaton 6 can be wrtten n scaar form for each DVE as: f (7) f P DVE EDP DVE DM d DM d EDP DamageStates (reca our assumpton that each DVE s dependent on a snge EDP). For our FOSM purposes, furthermore, t s suffcent to fnd smpy the condtona means, varances, and covarances of the DVE s gven the EDP s. Thus, tang the mean of ths PDF, we have the resut: [ DVE EDP] F ( EDP) F 1 Damage States E ( EDP) (8) Appyng the same thnng to E [ DVE EDP], and recognzng that X]=X ]- X, we have the foowng resut: DVE EDP] DVE Damage States ( Damage States (( F ( EDP) F ) 1 ( F ( EDP) F ( EDP)) 1 ( EDP)) (9)
Fgure 3: Coapsed dstrbuton DVE EDP Fgure 3 shows an exampe of the mean and mean pus or mnus one sgma, as generated from the exampe dstrbutons shown n Fgure 1 and Fgure. 3..1 Quantfyng Correatons We now need to determne correatons among the DVE s of a eements n the structure. Note that e the mean and varance, these correatons are condtoned on the EDP s. Whe these cacuatons are straghtforward, estmaton of the necessary correaton nputs s a dffcut tas due to a ac of data. In the absence of addtona nformaton, t may be hepfu to use the foowng characterzaton scheme. Let us assume for ths purpose a mode of the form: n DVE n EDP g (n EDP ) n n n (10) m E where represents uncertanty common to the entre structure, represents uncertanty m common ony to eements of cass m (e.g. drywa parttons, moment connectons, etc.), and E represents uncertanty unque to eement. A of these s are assumed to be mutuay uncorreated. We then defne: n n ] (11) EDP n n ] for a m (1) m EDP E n n ] for a (13) E EDP Then the varance of ndve nedp s the sum of these varances. For ths speca case, a smpe cosed form souton exsts for the correaton coeffcent between two eement DVE s. If the two eements are of the same eement cass, then: (n DVE,n DVE n EDP,n EDP ) (14) If the eements are of dfferent eement casses, then ther correaton coeffcent s gven by: (n DVE,n DVE n EDP,n EDP ) E (15) Note that ths formuaton requres to be equa for a eement casses, and E to be equa for a eements. If ths s excessvey mtng, a cosed form souton aso exsts that aows to vary by cass, and E to vary by eement (Baer and Corne, 00). The prncpes used for ths souton are deveoped n Dtevsen (1981). The use of more than two uncertan terms, and the use of terms that vary by cass or eement are both generazatons of the basc equcorreated mode. Thus, we w refer to a mode ncorporatng any of these generazatons as a generazed equcorreated mode. The correaton matrx for a generazed equcorreated mode w have off-dagona terms that vary from term to term, as opposed to the strct equcorreated mode, where a off-dagona terms are dentca. We have now concuded the coapse of the dstrbuton ndve nedp. We have the condtona mean and varance functons of ndve nedp, obtaned by coapsng the dstrbutons provded (see Equatons 8 and 9), and correaton coeffcents determned usng the generazed equcorreated mode (see Equatons 14 and 15). We choose for future notatona carty to denote these resuts as: n DVE n EDP ], denoted g (n EDP ) for a DVE (16) n DVE n EDP ], denoted g* (n EDP ) for a DVE (17) ρ(n DVE, n DVE n EDP, n EDP ), denoted ĝ (n EDP, n EDP ) for a {DVE, DVE } (18) Wth ths nformaton quantfed, we can now use t aong wth the resuts from Secton 3.1 to cacuate n DVE. 3.3 Cacuate n DVE Usng nformaton from above, we can cacuate the frst and second moments of n DVE. Ths nvoves coapsng out the dependence on EDP, as suggested n Equaton 19 beow. f DVE E ( u, x) fdve EDP ( u, y) fedp ( y, x) (19) y To mantan tractabty, we sha use an FOSM approxmaton here. To remove dependence on EDP, we tae the expectaton of n DVE of wth respect to n EDP (gven ). We wrte ths as ndve ]=E EDP [ndve nedp ]], where E EDP [] denotes ths partcuar condtona expec
taton operator. Substtutng our notaton from Equatons 16 and 17, we have: n DVE ] g h ( ) (0) Usng a smar approach to condtona moments, and usng the resut from probabty theory: [ X Z] Var X ] X ] E Var Z (1) we can aso derve the varance and covarances of n DVE wth the usua FOSM approxmatons: g n DVE ] g * ( h ( )) n EDP h ( ) h * ( ) () Cov[n DVE ˆ, DVE ] g( h ( ), h ( )) g * ( h ( )) g * ( h ( )) (3) g g hˆ ( ) n EDP n EDP h ( ) h ( ) h * ( ) h * ( ) 3.4 Swtch to the non-og form DVE To swtch to the non-og form of DVE, we can use the frst-order approxmaton e X ] e X]. Then we have the foowng resuts: g ( h ( )) DVE ] e (4) DVE Cov[ DVE, DVE g ( h ( )) ] e n DVE ] (5) g ( h ( )) g ( h ( )) ] e Cov[n DVE,n DVE ] (6) These eement resuts can now be used to compute the moments of TC. 3.5 Compute moments of TC Under the assumpton that Tota Cost s the sum of eement costs, we can now aggregate the resuts from a ndvdua eements to compute an expectaton and varance for the tota cost of damage to the entre budng. The expected tota cost s the sum of expected eement costs: # eements TC ] DVE ] (7) 1 We denote the expected vaue computed n Equaton 7 as q(). The varance of tota cost can s the sum of eement varances, ncudng covarances between eement costs: TC ] # eements 1 DVE # eements # eements 1 1 ] Cov[ DVE, DVE ] (8) We denote the expected vaue computed n Equaton 8 as q*(). 3.6 Repeat Procedure to Cacuate Epstemc Uncertanty We assume a mode of the form TC =q(), R U where q() s the best estmate of the condtona mean as cacuated n Equaton 7, and R and U are uncorreated random varabes representng aeatory and epstemc uncertanty, respectvey. Then ntc n q( ) n n (9) Because R and U are uncorreated, we may dea wth them n separate steps. The above procedure usng aeatory uncertanty aone aowed us to fnd the varance due to R. We must now repeat the procedure to cacuate the varance due to U. The tota uncertanty can then be cacuated by combnng the two uncertantes as foows: ntc ] n ] n ] (30) We denote ths TC. Note that we have swtched to ogs agan to aow use of sums rather than products. The change can be made usng the foowng reatonshp: Var R[ TC ] n ] n 1 R (31) ER[ TC ] We denote n R ] as R. From ths vaue, we can then cacuate Var [ntc ] usng Equaton 30. Once we have cacuated Var [ntc ], accountng for both aeatory and epstemc uncertanty, we can denote ths vaue as q*(), and use t n the equatons to foow. When repeatng the procedure to fnd the varance n TC gven due to epstemc uncertanty, epstemc uncertantes for each condtona random varabe w need to be estmated, and characterzaton of correaton s agan a chaengng tas. It s suggested that the generazed equcorreated mode deveoped n Secton 3..1 may be used effectvey for ths probem. 4 ACCOUNTING FOR COLLAPSE CASES At hgh eves, the potenta exsts for a structure to experence coapse (defned here as extreme defectons at one or more story eves). In ths budng state, repar costs are more ey a functon of the coapse, rather than ndvdua eement damage. In fact, the structure s ey not to be repared at a. Thus, our predcted oss may not be accurate n these cases. In addton, the arge defectons predcted n a few cases w sew our expected vaues of some EDP s such as nterstory drfts, athough coapse s ony occurrng n a fracton of cases. To account for the possbty of coapse, we woud e to use the technque outned above for no-coapse R R U U
cases, and aow for an aternate oss estmate when coapse occurs. The foowng modfcaton s suggested. Note, n the foowng cacuatons, we are condtonng on a coapse ndcator varabe. To communcate ths, we have denoted the coapse and no coapse condton as C and NC respectvey. At each eve, compute the probabty of no coapse. Ths probabty, p ( NC ), s smpy the fracton of anayss runs where no coapse occurs Cacuate resuts usng the FOSM anayss as before, but usng ony the runs that resuted n no coapse. We now denote these resuts E [ TC, NC] and Var [ TC, NC]. Defne an expected vaue and varance of tota cost, gven that coapse has occurred, denoted E [ TC, C] and Var [ TC, C]. These vaues w ey not be functons of, but the condtonng on s st noted for consstency. The expected vaue of TC for a gven eve s now the average of the coapse and no coapse TC, weghted by ther respectve probabtes of occurrng: TC ] p( NC ) TC, NC] 1 p( NC ) TC, C] (3) The varance can be computed usng the property from Equaton 1: p( NC ) TC, NC] Var [ TC ] 1 p( NC ) TC, C] (33) p( NC ) TC ] TC, NC] 1 p( NC ) TC ] TC, C] The procedure can now be mpemented as before, usng these moments. Ths coapse-case modfcaton s probaby necessary for any mpementaton of the mode, as anayss of shang () eves suffcent to cause arge fnanca oss are ey aso to cause coapse n some representatve ground moton records. 5 INCORPORATE THE SITE HAZARD The expected vaue and varance of TC gven can now be ncorporated wth the ste hazard to compute the expected annua oss, and the rate of exceedng a gven Tota Cost. 5.1 Annua Loss Usng the functons q() and q*(), and the dervatve of the hazard curve, d ( ), the mean and varance of TC per annum can be cacuated by numerca ntegraton: E [ TC] q( ) d ( ) (34) TC] TC ] TC ]] q *( ) d( ) q ( ) d( ) TC] (35) Note that the frst term of Equaton 35 s the contrbuton from uncertanty n the cost functon gven, and the second two terms are the contrbuton from uncertanty n the. 5. Rate of Exceedance of a Gven TC The frst and second moment nformaton for TC can be combned wth a ste hazard to compute TC (z), the annua frequency of exceedng a gven Tota Cost z. For ths cacuaton, t s necessary to assume a probabty dstrbuton for TC that has condtona mean and varance equa to the vaues cacuated prevousy. The rate of exceedance of a gven TC s then gven by: TC ( z) F ( z, x) d ( x) (36) TC 5.3 Anaytc Souton Generay, the ntegra above w requre a numerca ntegraton. However, f the foowng smpfyng assumptons are made, an anaytc souton s avaabe: The dstrbuton of TC s ognorma TC s approxmated as constant for a ; we ca ths constant vaue * TC TC ] s approxmated by a functon of the form a b, where a and b are constants. Note that ths s consstent wth fttng the medan of TC wth a b, where 1 * TC a a' e (37) An approxmate hazard curve of the form (x)= 0 x - s ft to the true ste hazard curve Under these condtons, the annua rate of exceedng a gven Tota Cost s gven by: b z 1 TC ( z) 0 exp 1 * TC (38) a' b b We note that f the a from Equaton 37 s substtuted nto Equaton 38, then the resut becomes b z 1 TC ( z) 0 exp * TC (39) a b Ths equaton s usefu as an effcent estmate of TC (z), but t s aso very nformatve as a measure of the reatve mportance of uncertanty n the cacuaton. The term: b z 0 (40) a'
n the Equaton 38 woud be the resut f * TC were to equa zero that s f we made a cacuatons ony usng expected vaues and negected cost uncertanty gven. The term: 1 exp b 1 * TC (41) b n Equaton 38 s an ampfcaton factor that vares wth the uncertanty n TC present n the probem. Thus for ths speca case, t s smpe to cacuate the effect of uncertanty on the rate of exceedng a gven Tota Cost. As we sha show beow, t may not be unreasonabe for ths factor to ncrease TC (z) by a factor of 10, so the effect of uncertanty may very we be sgnfcant. However, even for arge vaues of * TC, the annua rate of exceedance s st domnated by the term from Equaton 40. It s for ths reason that t has been proposed here that the FOSM approxmatons of * TC performed above are suffcent to provde an accurate resut. For ustraton, et us assume that the expected TC as a functon of has been estmated usng the above technque as: TC ] 1 e (4) Ths functon coud be approxmated by the functon: 1.8 TC ] 1.4 (43) A pot of these two functons s shown n Fgure 4. Note that the anaytc functon s a good ft over the range 0<TC<0.5. Tota Cost 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 Expected TC Gven Exampe vaue of TC ] Functon ft for Anaytca Souton 0.00 0 0. 0.4 0.6 0.8 1 1. 1.4 Fgure 4: Expected TC, for 1.4 1.8 ft TC ] 1 e and 5.4 Comparson of Resuts from Numerca Integraton and Anaytc Souton Both the numerca souton and the anaytca souton outned above n Sectons 5. and 5.3, respectvey, can be evauated for severa vaues of TC. The resuts can then be potted to generate a oss curve, as shown beow n Fgure 5. Ths fgure was generated usng the expected TC curves shown n Fgure 4 above. For both soutons, we have assumed a hazard curve of the form: ( x) 0 x (44) where 0 and are constants equa to 0.00 and 3, respectvey. We have aso assumed TC equa to 0.6 for both soutons. Mean Annua Frequency of Exceedance 1E-01 1E-0 1E-03 1E-04 Annua Frequency of Exceedance of a Gven Tota Cost Anaytca Mode Numerca Integraton 1E-05 0 0. 0.4 0.6 0.8 1 1. 1.4 Tota Cost Fgure 5: λ TC (z): Comparson of numerca ntegraton and anaytc souton Fgure 5 aso aows us to compare the anaytca souton to the numerca one. For the functons gven n Equatons 4 and 43, the anaytc souton s a good approxmaton of the numerc souton over the TC range where Equaton 43 cosey ft Equaton 4 (0<TC<0.5). As we move to hgher TC eves, where the anaytca souton was not a good ft, the λ TC (z) resuts aso dverge. 6 EFFECT OF UNCERTAINTY ON LOSSES The varance n annua osses s the resut that most expcty shows the effects of uncertanty. However, uncertanty aso has an effect on the annua rate of exceedng a gven TC. Ths s most ceary seen n the anaytca souton of Equaton 38, where * TC appears n the equaton for TC (z). Ths s dscussed n Secton 5.3. The antcpated effect s for uncertanty to ncrease the rate of occurrence of a gven TC. However, dependng on the sopes of the hazard curve and mean of TC as a functon of, (defned by the parameters and b), ncreasng * TC can potentay decrease TC (z), or have no effect at a. Usng the more genera numerca ntegraton of Equaton 36, we fnd smar resuts, as shown n Fgure 6. In ths exampe, usng the functons assumed prevousy, we see that the shft n resuts s mnor for * TC <1, but for * TC =, the expected annua frequency of occurrence of arge costs has ncreased by approxmatey an order of magntude. Because our expected repar cost functon does not ever produce a oss greater than 1, the ncuson of
uncertanty s crtca for estmatng occurrence of tota costs greater than 1, as seen n the fgure. Mean Annua Frequency of Exceedance 1E-01 1E-0 1E-03 1E-04 Annua Frequency of Exceedance of a Gven Tota Cost β = β = 1 β = 0 1E-05 0 0. 0.4 0.6 0.8 1 1. 1.4 Tota Cost Fgure 6: Effect of Uncertanty on Frequency of Exceedance of Tota Cost The mportant concuson to be drawn from ths resut s that usng expected vaues aone and gnorng uncertantes, athough temptng because of ts ease, can potentay ead to naccurate resuts. CONCLUSION A procedure for estmaton of uncertanty n repar costs due to earthquae damage has been proposed. Ths procedure wors wthn the framewor proposed by PEER for performance-based earthquae engneerng. Tota cost defned s a functon of repar costs for ndvdua budng eements, except n the coapse case, where a separate cost estmaton s used. Identfed aeatory and epstemc uncertanty n ground moton hazard, budng response, damage to budng eements and eement repar costs s combned to produce an uncertanty n tota repar cost. The proposed procedure uses the Frst-Order Second-Moment (FOSM) method to coapse severa condtona random varabes nto a snge random varabe. Numerca ntegraton s then used to ncorporate the ground moton hazard, where the uncertanty s most sgnfcant. The resutng nformaton s expected annua oss, varance n annua oss, and the annua rate of exceedng a gven cost. Baer, J.W. and Corne, C.A. (00). Technca Report on Uncertanty Specfcaton and Loss Estmaton Usng FOSM Methods. Report n Preparaton. Corne, C. A. and Krawner, H. (000). Progress and Chaenges n Sesmc Performance Assessment. Peer Center News, 3(). Dtevsen, Ove (1981). Uncertanty Modeng wth Appcatons to Mutdmensona Cv Engneerng Systems. McGraw-H Internatona, New Yor. Kramer, S.L. (1995). Geotechnca Earthquae Engneerng. Prentce Ha, New Jersey. Krawner, H. (00). A Genera Approach to Sesmc Performance Assessment. Proceedngs, Internatona Conference on Advances and New Chaenges n Earthquae Engneerng Research, ICANCEER 00, Hong Kong, August 19-0, 00. Luco, Ncoas and Corne, C. An (00). ture-specfc Scaar Intensty Measures for Near-Source and Ordnary Earthquae Ground Motons. Earthquae Spectra, (submtted). Mechers, Robert E (1999). tura Reabty Anayss and Predcton. John Wey and Sons, Chchester. Porter, K.A, Bec, J.L., and Shahutdnov, R.V. (00). Investgaton of Senstvty of Budng Loss Estmates to Maor Uncertan Varabes for the Van Nuys Testbed. Report to Pacfc Earthquae Engneerng Research Center, Bereey. 41pp. RS Means Co., Inc. (00). Means Assembes Cost Data. Kngston, MA. Vamvatsos, D. and Corne, C.A. (00). Incrementa Dynamc Anayss. Earthquae Engneerng and tura Dynamcs, 31(3): 491-514. ACKNOWLEDGEMENTS Fnanca support by PEER (proect number 305100) and the Shah Famy Fund feowshp s greaty apprecated. REFERENCES Asan, H. and Mranda, E. (00). Fragty of Sab-Coumn Connectons n Renforced Concrete Budngs. ASCE Journa of tura Engneerng, (submtted).