PROBABILISTIC DECISION ANALYSIS FOR SEISMIC REHABILITATION OF A REGIONAL BUILDING SYSTEM

Transcription

1 3 th Worl Conference on Earthquake Engneerng Vancouver, B.C., Canaa August -6, 4 Paper No. 54 PROBABILISTIC DECISION ANALYSIS FOR SEISMIC REHABILITATION OF A REGIONAL BILDING SYSTEM Joonam PARK, Barry GOODNO, Ann BOSTROM 3 an James CRAIG 4 SMMARY Sesmc vulnerablty of bulng structures can be reuce wth approprate rehabltaton schemes. However, ecsons on rehabltaton of structures can epen on multple conflctng crtera such as cost, lfe loss, functonalty, etc. In ths stuy, a framework s evelope to support ecsons on sesmc structural rehabltaton. Three mult-crtera ecson moels are consere: an equvalent cost moel (ECM), mult-attrbute utlty theory (MAT) an Jont Probablty Decson Makng (JPDM). The ecson moels are apple to hosptal systems locate n Memphs, Tennessee, an the preferre rehabltaton optons are entfe base on the two ecson moels. INTRODCTION Sesmc falure or amage to bult systems n regons wth moerate to hgh sesmcty can exact a hgh toll, n lves lost, cost of amage, an other consequences of nterest an concern to stakeholers. However, the sesmc vulnerablty of such systems can be reuce wth approprate rehabltaton schemes (Abrams []). Structural rehabltaton ecsons can epen on multple crtera, such as structural performance, cost, aesthetcs, an functonalty. These crtera often conflct wth one another. There have been efforts on the ecson analyses for sesmc rehabltaton of bulng structures (e.g., Benthen [], Thel [3]). However, an effort to evelop a ecson support framework that takes nto account multple crtera nclung loss of lfe an bulng functon loss, whle utlzng comprehensve probablstc sesmc loss estmaton methos for structures (ether nvual structure or regonal systems) s sparse. In ths paper, a mult-crtera ecson support framework s propose to help ecson makers evaluate rehabltaton schemes, an to support regonal sesmc rehabltaton ecsons. Ths approach employs three mult-crtera ecson moels an equvalent cost moel (ECM), mult-attrbute Ph.D. Canate, School of Cvl an Envronmental Engneerng, Georga Insttute of Technology, Atlanta, GA , SA, Emal: [email protected] Professor, School of Cvl an Envronmental Engneerng, Georga Insttute of Technology, Atlanta, GA , SA, Emal: [email protected] 3 Assocate Professor, School of Publc Polcy, Georga Insttute of Technology, Atlanta, GA , SA, Emal: [email protected] 4 Professor, School of Aerospace Engneerng, Georga Insttute of Technology, Atlanta, GA 333-5, SA, Emal: [email protected]

2 utlty theory (MAT) an jont probablty ecson-makng (JPDM) wthn a flexble confguraton to facltate use by a varety of stakeholers. ECM s a ecson technque n whch non-monetary values are converte nto equvalent monetary values. MAT s a wely use ecson theory that proves nsght nto preferences over a set of alternatves takng nto account the ecson maker s rsk atttues. JPDM s a ecson moel that gves an nex of system performance base on the probablty of achevng a preefne level of consequences for each attrbute of the system. To llustrate the ecson support framework, structures wthn a regonal system are ve nto several classes base on ther confguraton, functon an structural esgn. A sesmc hazar curve s use to represent the uncertanty n the sesmc hazar. Ths hazar curve an structural fraglty curves for the system are combne to obtan the overall probablstc strbuton of structural amage wthn a partcular tme pero. For the ecson analyses, consequences are efne n terms of crtera selecte by the ecson maker. Monte Carlo smulaton s use to estmate probablstcally the antcpate sesmc structural amage an overall consequences to the system, both wthout nterventon an wth alternatve rehabltaton schemes. The ecson analyses prove summary measures of consequences, to entfy the best rehabltaton scheme(s). The framework supports several forms of senstvty analyss, nclung ynamc restructurng of ecson crtera an rehabltaton alternatves, to prove ecson makers wth atonal nsghts nto the consequences of sesmc rehabltaton ecsons. SYSTEM DEFINITION Descrpton of the Bulng Systems Methost Healthcare s a hosptal system base n Memphs, Tennessee, servng the communtes of Eastern Arkansas, West Tennessee, an North Msssspp, an conssts of a number of hosptals an rural health clncs (Methost [4]). Among them, sx hosptal bulngs are selecte an examne to emonstrate the ecson support framework. Table shows the locatons (by zp coe) an the structural types of the hosptals. The locaton nformaton s use to efne the sesmc hazar, an the structural types are use to efne sesmc vulnerablty. Note that the table also shows the structural types base on HAZS bulng classfcatons (HAZS [5]) as loss estmaton n ths stuy follows a HAZS approach. Table Bulng Descrpton Hosptal ZIP Structural Type HAZS Moel Type Methost nversty Hosptal 384 Concrete Shear Wall (M-Rse) CM Methost North Hosptal 388 Concrete Shear Wall (M-Rse) CM T Bowl Hosptal 383 Concrete Shear Wall (M-Rse) CM Methost South Hosptal 386 Concrete Shear Wall (M-Rse) CM Methost Fayette Hosptal 3868 RM (Low Rse) RML Le Bonheur Germantown Hosptal 3838 Concrete Shear Wall (Low-Rse) CL Hazar Curves Earthquakes representatve for the locaton of the system of concern must be efne for use n the amage analyss. Groun moton ntensty s often characterze n terms of spectral splacement (S ) or spectral acceleraton (S a ). However, snce earthquakes are ranom events, whch epen on locaton, t s also necessary to entfy the probablstc characterstcs of the earthquake ntensty as well. sually the lkelhoo of fferent earthquake levels s expresse n terms of probablty of exceeance wthn certan tme lmts, for example, % probablty of exceeance n 5 years. The relatonshp between the earthquake ntensty an ts lkelhoo can be represente by a hazar functon H (Cornell [6], Yun [7]). The annual probablty of exceeance for earthquake ntensty (generally s a or s ) at the ste can be

3 obtane from the hazar functon. Accorng to Cornell [6], the hazar functon can be approxmate to le lnearly on a log-log plot. That s, f the hazar functon s efne n terms of spectral splacement s, the hazar functon can be expresse by the form = P[ S s ] = k k H ( s ) s () Parameters k an k are locaton-specfc. The hazar curve s shown schematcally n Fgure. H Ln(H) k S Ln(S ) H ( s ) Fgure Hazar Curve = P[ S s ] = k PROBABILISTIC EVALATION OF DAMAGE AND LOSSES The antcpate amage state of a bulng or a system of bulngs can be use to estmate sesmc losses. Before conuctng the amage assessment, the sesmc performance objectve for a structure must be specfe. A performance objectve can be efne n terms of the structural performance level an corresponng probablty that the performance level wll be exceee wthn a certan tme lmt (Yun [7]). Accorng to SAC [8], for example, the objectve performance level of a new bulng s that the bulng shoul have less than % chance of amage exceeng Collapse Preventon (CP) n 5 years. In other wors, the sesmc performance level of a structure can be represente n terms of the sesmc amage probablty. A close form soluton s avalable (Cornell [6]) to escrbe structural amage probablstcally. Three major sources of uncertanty n sesmc amage assessment for structural systems are: ) groun moton ntensty; ) structural eman; an 3) structural capacty. There are a number of ways to measure structural eman an capacty, nclung maxmum nter-story rft or varous types of amage nces. The generc expresson for the annual probablty that the eman D excees a specfc value s H D ( ) = p[ D ] = = P[ D S all x P[ D S = x] H( x) = x ] P[ S = x ] s k () The amage probablty (annual probablty of exceeng certan amage level) can then be expresse as

4 P PL = P[ C D] = = all P[ C ] H D P[ C D D = ( ) ] P[ D = ] (3) where C s a generc expresson for structural capacty. The amage probablty (annual probablty of exceeng certan amage level) n Equaton (3) can be approxmate as P PL ( D S + ) Cˆ k H ( S ) exp β β (4) = C where C S ˆ s the spectral splacement corresponng to the mean capacty. Therefore, f the eman hazar curve s efne for the regon an, f the mean capacty can be obtane along wth the spersons of the capacty an the eman, the amage probablty strbuton of a structure locate n a partcular regon can be obtane. The probablstc strbuton of sesmc losses of the structure can then also be obtane from the amage strbuton. It shoul be note that ths close form soluton for the amage strbuton shoul be use for a sngle structure or a class of structures wth the same structural type that are locate relatvely close to each other wthn a regon n whch the sesmcty can be represente by a sngle hazar curve. For aggregaton of losses of fferent types of structures, the close form expresson for the amage strbuton s rarely avalable. To assess expecte losses wthn a partcular tme pero, losses are estmate for a sute of earthquake levels. The amage strbuton of a structure or type of structure from a partcular earthquake level can be obtane from the fraglty curve for the structure. In ths stuy, HAZS [5] s use to estmate structural amage an losses, whch requres that the bulngs be classfe (nto one of 36 categores) base on structural type an heght. Bulng fraglty an capacty curves are use to etermne bulng amage state probablty n HAZS. The fraglty curves for a partcular structural type can be obtane for fferent coe levels (pre, low, moerate, an hgh coe level) n force when the structures were bult (assumng coe complance). HAZS proves an extensve lst of parameters that are neee to generate fraglty curves (for both structural an nonstructural amage) for all 36 types of structures an for four fferent coe levels. Base on these, fraglty curves can be generate for four fferent amage states slght, moerate, extensve, an complete amage. For etale escrpton of the amage states, see HAZS [5]. The HAZS loss estmaton methoology s assumes that there are strong relatonshps between bulng amage an major socal an economc losses (HAZS [5]). Socal losses nclue eath an njury, loss of housng habtablty, short term shelter nees, etc; economc losses nclue structural repar costs, nonstructural repar costs, bulng contents loss, busness nventory loss, loss of bulng functon, ntal rehabltaton cost, etc. The ecson maker chooses whch losses to assess n the rehabltaton ecson analyss, by selectng them as attrbutes of the system. Table shows the losses consere n ths example, where hosptals comprse the system of nterest. These are the system attrbutes for the ecson analyss. Note that only rect losses are consere, an nrect losses are not taken nto account here. The expecte sesmc losses are estmate for four fferent earthquake levels: %, %, 5%, an % probablty of exceeance n 5 years. The loss hazar curve can then be plotte for each kn of loss. For example, Fgure shows the hazar curve for monetary loss for four CM type structures among the structures lste n Table. The expecte loss s then calculate from the area uner the loss hazar curve.

5 Table Losses Consere Category Loss Descrpton Economc Loss Socal Loss Intal Cost Structural Repar Cost Nonstructural Repar Cost Loss of Bulng Contents Relocaton Expenses Loss of Functonalty Death Cost for sesmc rehabltaton or rebulng a new bulng to mprove structural performance Cost for reparng amage to structural components such as beams, columns, jonts, etc. Cost for reparng amage to nonstructural components such as wall parttons, panels, veneers, floors, general mechancal systems, etc. Cost equvalent to the loss of bulng contents such as furnture, equpment (not connecte to the structure), computers, etc. Dsrupton cost an rental cost for usng temporary space n case the bulng must be shut own for repar. Loss of functon for a hosptal may result n atonal human lfe losses ue to lack of mecal actvty. Number of eaths. Injury Number of serously njure people Loss Hazar Curve (Monetary Cost) Loss (,$) Probablty of Exceeance Fgure Monetary Loss Hazar Curve for CM Type Structures Four generc alternatves (sesmc rehabltaton alternatve schemes) are consere for each structural type: ) no acton; ) rehabltaton to lfe safety level; 3) rehabltaton to mmeate occupancy level; an 4) bul a new bulng to comply wth the current coe level. The rehabltaton levels mentone above are, as efne n FEMA [9], the target performance levels of the rehabltaton aganst an earthquake wth % exceeance n 5 years. The cost of sesmc rehabltaton of bulng systems epens on many factors, such as bulng type, earthquake hazar level, esre performance level, occupancy or usage type, etc. The ntal rehabltaton cost for fferent optons are obtane from FEMA ocuments (FEMA [] an FEMA []), whch prove the typcal cost for rehabltaton of exstng structures takng nto account above-mentone factors. For amage assessment of the alternatve systems, a specfc coe level, whch s utlze n HAZS, s assgne to each level of rehabltaton so that the fraglty curves can be obtane for each sesmc alternatve. It s assume that the no acton opton, whch means retanng the exstng structures, correspons to the low coe level. Rehabltaton to lfe safety level opton s

6 assume to be a moerate coe level, an rehabltaton to mmeate occupancy level opton s assume to be a hgh coe level. For the rebul opton, a specal hgh coe s assume because hosptals are classfe as essental facltes. The alternatves an ther coe levels are shown n Table 3 along wth the total floor area of each type of structure. Note that the fraglty curves for CL are use for amage assessment of the sesmc alternatves of a RML type structure, as they are not avalable n HAZS. Str. Type Alternatves CM (4, ft ) CL (4, ft ) RML (4, ft ) No Acton Table 3 HAZS Coe Levels for Alternatve Systems Rehabltaton to Lfe Safety Level Rehabltaton to Immeate Occupancy Level Rebul Low Coe Moerate Coe Hgh Coe Specal Hgh Coe Low Coe Moerate Coe Hgh Coe Specal Hgh Coe Low Coe Moerate Coe (usng CL) Hgh Coe (usng CL) Specal Hgh Coe (usng CL) EQIVALENT COST ANALYSIS For equvalent cost analyses, consequences measure n fferent unts are converte nto a sngle composte measure usually a monetary measure by ntroucng converson factors. For example, one ay of constructon elay can be consere equvalent to three mllon ollars. Ths cost-beneft analyss approach (Keeney []) s calle an equvalent cost analyss n ths stuy because n ecson problems regarng sesmc events, that the only beneft apparent s the mnmzaton of losses (or costs). However, there are several known problems wth ths metho (Keeney []). In orer to use a smple atve metho for estmatng the prce out consequences, several assumptons must be verfe. These assumptons are: ) the monetary value of an attrbute can be etermne wthout conserng other attrbutes; ) the monetary value of an attrbute oes not epen on the overall monetary value level. Even when these assumptons are consere val, many mportant attrbutes such as the value of a lfe are very har (an sometmes consere mpossble or mmoral) to prce out. Moreover, attrbutes may be gnore or not nclue n the analyss when t s har to convert them nto monetary values usng market mechansms (e.g., aesthetcs). Nevertheless, the equvalent cost moel s stll wely use because of ts smplcty n use an straghtforwarness. Among the non-monetary attrbutes, the value of human lfe s very ffcult to etermne an has been hghly ebate. Moreover, the value of human lfe wll have a we range of values epenng on the ecson context. Accorng to FEMA [], the typcal value of a statstcal lfe ranges from $. mllon to $8 mllon per lfe (other authors have foun fferent ranges). In ths stuy, the ecson analyss wll not be performe wth fxe values for non-monetary attrbutes, but nstea wth a range of values ($.m to $8m for the value of a statstcal human lfe), to nvestgate the effects of the equvalent monetary values on the ecson. The equvalent cost for the loss of functon s expresse n terms of the functon recovery tme (ays) per, square feet. For example, f one ay of loss of functon of a hosptal wth the total floor area of, square feet s estmate to cost $,, the equvalent cost for 5 ays of loss of functon of 5, square feet hosptal woul be $,5,. Obvously, ths rough approach for etermnaton of equvalent cost for loss of functon nees future refnement. As escrbe n Table 3, the value of loss of functon shoul be taken nto account that the loss of functon may result n atonal loss of lfe. In ths stuy, senstvty analyss wll be performe for the value of loss of functon rangng from $ to $5, for one ay of loss of functon of a hosptal per, square feet. Table 4 shows the baselne values for the non-monetary attrbutes for the ecson analyss. Note that the value of njury s estmate (cruely) at 3% of the value of a statstcal lfe loss.

7 Table 4 Baselne Values for Non-monetary Attrbutes Attrbute Equvalent Cost Value of Death $5,, / person Value of Injury $,5, / person Value of Loss of Functon $, / ay to recover /, ft If a temporal trae-off s consere n performng a ecson analyss, future costs may be scounte to net present value, f the ecson maker consers them less panful. If we have a tme stream of costs (c, c,, c T ), the total net present value of the cost can be expresse as follows: c npv = c T t t t= ( + λ) (5) where λ s the effectve pero-to-pero scount rate. Accorng to FEMA [], several fferent approaches have been use to estmate the scount rate for publc nvestments, wth the resultng scount rates rangng between 3% an %. Determnaton of the tme pero T also epens on the ecson maker. In ths stuy, a 3-year tme pero an wth 6% scount rate are use as baselne values, wth senstvty analyses on tme peros rangng from years to 5 years, an scount rates from 3% to %. Note that the probablty of exceeance of fferent earthquake levels must be calbrate to be consstent wth the tme pero. Fgure 3 shows the loss hazar curves for each type of structure wth the expecte equvalent losses corresponng to fferent earthquake levels. Note that the losses shown n ths fgure are the equvalent cost, where non-monetary attrbutes are prce out. Table 5 shows the expecte earthquake losses for each rehabltaton scheme, whch are obtane from the loss hazar curves, along wth the ntal costs for the rehabltaton, followe by the total expecte losses (for 3 years of tme pero). Ths specfc expecte equvalent cost analyss ncates that none of the rehabltaton actons are justfe. 7 Loss Curve (Money Cost) 8 Loss Curve (Money Cost) 8 Loss Curve (Money Cost) Loss ($M) No Acton Rehab LS Rehab IO Rebul % 5% % 5% Prob. of Exceeance Loss ($M) 6 4 No Acton Rehab LS Rehab IO Rebul % 5% % 5% Prob. of Exceeance Loss ($M) No Acton Rehab LS Rehab IO Rebul % 5% % 5% Prob. of Exceeance (a) CM Structures (4 unts) (b) CL Structure ( unt) (c) RML Structure ( unt) Fgure 3 Loss Hazar Curves for CM type Structures (4 unts) Fgure 4 shows the senstvty plots for fferent values of functon loss (the senstvty plots for other varables are not shown n ths paper). Among the varables nclue n ths ecson analyss, the relatve fferences of the expecte equvalent costs of the alternatve systems are most senstve to the change of the value of functon loss. Note that the slopes (senstvty) of the optons are fferent an ecson reverses occur when the value of functon loss excees approxmately $,.

8 Table 5 Expecte Equvalent Costs ($Mllon) of Dfferent Rehabltaton Schemes Intal Cost (Rehab. Cost) Expecte Earthquake Loss (n 3 years) Total Expecte Cost (n 3 years) No Acton CM Rehab LS (4 unts) Rehab IO Rebul CL ( unt) RML ( unt) No Acton.9.9 Rehab LS Rehab IO Rebul No Acton Rehab LS Rehab IO Rebul Senstvty to Value of Functon (for CM) Senstvty to Value of Functon (for CL) Senstvty to Value of Functon (for RML) Equvalent Cost ($M) No Acton 4 Rehab LS Rehab IO New..4.6 Value of Functon Loss ($M/ay/,sq.ft) Equvalent Cost ($M) No Acton 6 Rehab LS 4 Rehab IO New..4.6 Value of Functon Loss ($M/ay/,sq.ft) Equvalent Cost ($M) No Acton 6 Rehab LS 4 Rehab IO New..4.6 Value of Functon Loss ($M/ay/,sq.ft) (a) CM Structures (b) CL Structure (c) RML Structure Fgure 4 Senstvty Plot for Value of Functon Loss MLTI-ATTRIBTE TILITY ANALYSIS Mult-attrbute utlty theory (MAT), whch has been wely use n the fel of ecson analyss, ncorporates ecson makers unque preferences for multple attrbutes, thus allowng ncorporaton of multple crtera nto a ecson. Preferences (or values) are measure n terms of utlty functons, whch can be lnear or nonlnear. For multple crtera ecson-makng problems, a mult-attrbute utlty functon s generate as a functon of a number of sngle utlty functons, conserng ther relatve mpact on the overall value as well as ther nteractons. The etals of the theory an the technques for utlty elctaton are well escrbe n the lterature (Keeney []). If uncertanty s nvolve n the problem, the expecte utlty s obtane for each alternatve an the alternatve wth hghest expecte utlty s the one wth hghest prorty. To examne the effect of nclung rsk atttues, a set of utlty functons s assume n ths stuy. From the fact that ecson makers ten to be rsk seekng (.e., the shape of the utlty functon s convex) for losses (Kahneman [3]), four rsk seekng utlty functons are assume as shown n Fgure 5. Note that loss of functon s measure as ays of loss of functon multple by the sze of the faclty (n terms of, ft ). In constructon of the mult-attrbute utlty functon, the utlty functons are assume to be atve for smplcty, an the scalng factors are efne as shown n Table 6. For the purpose of comparng these results wth the equvalent cost analyss, the scalng factors for the attrbutes

9 are etermne such that the ratos of the scalng factors are same as the ratos of the equvalent costs for the maxmum values. These scalng factors are presente as baselne values an are subjecte to senstvty analyss. The mult-attrbute utlty functon can then be formulate as u x, x, x, x ) = k u ( x ) + k u ( x ) + k u ( x ) + k u ( ) (6) ( x4 where u( x, x, x3, x4 ) s the mult-attrbute utlty functon, k s are the scalng factors an u ( x ) s are the margnal utlty functons of the attrbutes. Table 6 Scalng Factors of Attrbutes Attrbutes Mn. Value Max. Value Scalng Factor Monetary Cost ($M) k =. Functon Loss ( ays, ft ) 5, k =.6 Death 3 k 3 =.8 Injury 55 k 4 = y = e -.5x.6.4. y = e -.x Cost (Mllon $).8 (a) Monetary Cost Loss of Functon (ays*area/,) (b) Functon Loss.6.4. y = e -.5x.6.4. y = e -.x Death (c) Death Injury () Injury Fgure 5 tlty Functons (Rsk-Seekng) for Attrbutes tlty functons are measure over the range of total consequences, an shoul be performe on the system as a whole. Accorngly, the alternatve systems are efne n terms of the combnatons of the alternatves for each nvual structure or each type of structures. In ths example, eght combnatons of the alternatve systems are analyze, usng two alternatves for each type of structure (here the best two optons from the equvalent cost analyss). The expecte utlty of a rehabltaton scheme can be calculate as

10 E =... u( x, x, x, x ) f X, X, X, X ( x, x, x, x x x x x (7) x x ) where E s the expecte utlty of th scheme an f X, X, X 3, X 4 ( x, x, x3, x4 ) s the jont probablty ensty functon for the rehabltaton alternatves corresponng to the th scheme. Same as the equvalent cost analyss, the Monte-Carlo smulaton s performe to obtan the expecte utlty of each alternatve combnaton scheme. Table 7 shows the lst of combnatons of the alternatve systems; expecte utltes for these combnatons are gven as well. Note that the expecte utltes are obtane for two fferent values of the scalng factors for loss of functon, k. These two values of k are obtane such that they are consstent wth the case that the values of functon loss n the equvalent cost analyss are $, an $,, respectvely. The utlty hazar curves for selecte combnatons of the alternatve systems are shown n Fgure 6 showng the expecte utltes corresponng to fferent earthquake levels. Ths analyss suggests that none of the rehabltaton actons are justfe, as n the equvalent cost analyss, unless the relatve mportance of the functon loss s very hgh. Wth a scalng factor for functon loss (k ) of.75, scheme T s preferre. It shoul be note that although T4 omnates n both plots n Fgure 6,T4 s less preferre than ether T or T overall because T an T are preferre (because of low ntal costs) over T4 when there s no earthquake, whch s hghly lkely. Table 7 Expecte tltes of the Combnatons of the Sesmc Alternatve Schemes (wth rskseekng utlty functons) Scheme CM CL RML Expecte tlty k =.6 k =.75 T No Acton No Acton No Acton T No Acton No Acton Rehab LS T3 No Acton Rehab LS No Acton T4 No Acton Rehab LS Rehab LS T5 Rehab LS No Acton No Acton.9.96 T6 Rehab LS No Acton Rehab LS T7 Rehab LS Rehab LS No Acton T8 Rehab LS Rehab LS Rehab LS tlty Hazar Curve tlty Hazar Curve Expecte tlty T T. T3 T4.% 5.%.% 5.% Probablty of Exceeance Expecte tlty T T T3 T4.% 5.%.% 5.% Probablty of Exceeance (a) wth k =.6 (b) wth k =.75 Fgure 6 tlty Hazar Curves for Selecte Combnaton Schemes

11 To nvestgate the effect of fferent rsk atttues, the analyss s performe wth the rsk-averse utlty functons are shown n Fgure 7. Note that the same set of scalng factors s use for the analyss. The analyss s performe n the same manner as escrbe above. Overall expecte utltes of the fferent rehabltaton schemes are shown n Table 8. In contrast to the results wth rsk-seekng utlty functons, the analyss ncates that T5~T8 are preferre over T~T4 for both sets of scalng values. Conserng the fact that CM type structures consttute the majorty of the system of nterest (at least n terms of square footage), the analyss shows, as one woul expect, that rehabltaton actons are generally recommene when ecson makers are rsk averse y = -(/) 3 x y = -(/5) 3 x Cost (Mllon $)..8 (a) Monetary Cost Loss of Functon..8 (b) Loss of Functon.6.4. y = -(/3) 3 x y = -(/55) 3 x Death (c) Death Injury () Injury Fgure 7 tlty Functons (Rsk-Averse) for Attrbutes Table 8 Expecte utltes for alternatve rehabltaton schemes (wth rsk-averse utlty functons) Scheme CM CL RML Expecte tlty k =.6 k =.75 T No Acton No Acton No Acton T No Acton No Acton Rehab LS T3 No Acton Rehab LS No Acton T4 No Acton Rehab LS Rehab LS T5 Rehab LS No Acton No Acton T6 Rehab LS No Acton Rehab LS T7 Rehab LS Rehab LS No Acton T8 Rehab LS Rehab LS Rehab LS JOINT PROBABILITY DECISION MAKING Bante [4] evelope Jont Probablstc Decson Makng (JPDM) as a tool for mult-objectve optmzaton an prouct selecton problems n aerospace system esgn. In ths metho, a jont probablty strbuton functon for multple objectves can be obtane ether mathematcally or from

12 emprcal strbuton functons. sng jont probablty strbuton functons, a unque value calle Probablty of Success (POS), whch ncates the probablty of satsfyng specfc ecson makng objectves, can be calculate to prove a barometer wth whch the ecson can be mae. The POS can be mathematcally expresse as follows. POS = P{ ( z = zmax zmn mn... z z max ) ( z mn z z zn max f Z Z Z ( z, z,... z N ) z z N zn mn max z )... ( z N Nmn z N z Nmax ) } (8) where, z s the crteron value, f Z (,,... ) Z... Z z z z N N s the jont probablty functon of the crtera, an z mn an z max are the mnmum an maxmum range of the objectve crteron value, respectvely. In JPDM the alternatve wth maxmum probable postve consequences s preferre. Note that because JPDM requres specfyng specfc threshols for success (ecson crtera values), the preferre ecson n JPDM may not be same as that resultng from an expecte value (or expecte utlty) approach. Value nformaton (.e., success) n JPDM s expresse n terms of the crtera values. The consequental fference between ECA an MAT s the ncorporaton of rsk atttues. JPDM s a categorcal approach, n that the ecson maker specfes at the outset what values of each ecson attrbute of nterest (.e., crteron values) wll be consere success (or acceptable ). JPDM s esgne to help the ecson maker maxmze the jont probablty of attanng success (.e., acceptablty) on all attrbutes of nterest. Table 9 shows the crteron values assume for the JPDM analyss n ths example. These ncate the range of the consequences the ecson maker consers successful (.e., acceptable). The Monte Carlo Smulaton s use to calculate the probabltes of success (POS) of the alternatve schemes an fferent earthquake levels. Table shows the overall expecte values of POS for alternatve rehabltaton schemes. In ths analyss, JPDM gves hgh prorty to T6 an T8. Table 9 Crteron Values for JPDM Attrbutes Mnmum Crteron Value Maxmum Crteron Value Monetary Cost ($M) Functon Loss ( ays, ft ) Death Injury Table POSs of the Combnatons of the Sesmc Alternatve Schemes Scheme CM CL RML POS T No Acton No Acton No Acton.93 T No Acton No Acton Rehab LS.94 T3 No Acton Rehab LS No Acton.9334 T4 No Acton Rehab LS Rehab LS.944 T5 Rehab LS No Acton No Acton.954 T6 Rehab LS No Acton Rehab LS.9644 T7 Rehab LS Rehab LS No Acton.9548 T8 Rehab LS Rehab LS Rehab LS.9694

13 CONCLSIONS Ths paper outlnes a ecson framework that ncorporates state of the art earthquake engneerng nformaton an ecson maker preferences nto a flexble tool to support earthquake rsk mtgaton ecsons. Three ecson moels are use to prove nsght nto the value of system nterventons to reuce earthquake rsks: ) an equvalent cost moel, ) mult-attrbute utlty theory an 3) jont probablty ecson makng. To llustrate the kns of nsghts the framework can prove, t s apple to a set of hosptals n Memphs, Tennessee, to assess the relatve value of structural rehabltaton optons. Wth the assume baselne values of scount rate, value of functon loss, value of eath an njury, an tme pero, an the assume set of utlty functons, no rehabltaton acton s justfe n ether the equvalent cost analyss or the utlty analyss. However, senstvty analyss suggests that the RML structure n the hosptal system shoul be rehabltate to lfe safety level f the relatve mportance of hosptal functon loss s hgh. If the ecson maker s rsk-averse, the analyss ncates rehabltaton actons are generally justfe. Wth JPDM, two rehabltaton schemes are preferre. The results llustrate the kns of nsghts the system coul prove to ecson makers, recognzng that any such analyses requre sgnfcant assumptons, whch shoul be probe wth approprate techncal support. ACKNOWLEDGMENT Ths research was sponsore n part by the M-Amerca Earthquake Center through Natonal Scence Founaton Grant EEC However, all results, conclusons an fnngs are solely those of the authors an o not necessarly represent those of the sponsors. REFERENCES. Abrams, D. P., Elnasha, A. E., an Beavers, J. E. (), A New Engneerng Paragm: Consequence-Base Engneerng, M-Amerca Earthquake Center. Benthen, M. an von Wnterfelt, D. (), A Decson Analyss Framework for Improvng the Sesmc Safety of Apartment Bulngs wth Tuckuner Parkng Structures, workng paper, School of Polcy, Plannng an Development, nversty of Southern Calforna,.S.A. 3. Thel, C. C. an Hagen, S, H. (998), Economc Analyss of Earthquake Retroft Optons: an Applcaton to Wele Steel Moment Frames, the Structural Desgn of Tall Bulngs, v7, pp Methost Healthcare System (3), Methost Healthcare, < 5. HAZS (999), Techncal Manual, Feeral Emergency Management Agency, Washngton, D.C. 6. Cornell, C. A., Jalayer, J., Hamburger, R. O., an Foutch, D. A. (), Probablstc Bass for SAC Feeral Emergency Management Agency Steel Moment Frame Guelnes, Journal of Structural Engneerng, v8, n4, pp Yun, S.Y., Hamburger, O. O., Cornell, C. A., an Foutch, D. A. (), Sesmc Performance Evaluaton for Steel Moment Frames, Journal of Structural Engneerng, v8, n4, pp SAC Jont Venture (), Recommene Sesmc Desgn Crtera for New Steel Moment Frame Bulngs, Report No. FEMA-35, Feeral Emergency Management Agency, Washngton, D.C. 9. Feeral Emergency Management Agency FEMA (999), Example Applcatons of the NEHRP Guelnes for the Sesmc Rehabltaton of Bulngs, Report 76, Washngton, D.C.,.S.A.. Feeral Emergency Management Agency FEMA (99), A Beneft-Cost Moel for the Sesmc Rehabltaton of Bulngs, Report 7, Washngton, D.C.,.S.A.. Feeral Emergency Management Agency FEMA (995), Typcal Costs for Sesmc Rehabltaton of Exstng Bulngs, Vol. Summary, Report 56, Washngton, D.C.,.S.A.

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