Fragility Based Rehabilitation Decision Analysis

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1 .171. Fraglty Based Rehabltaton Decson Analyss Cagdas Kafal Graduate Student, School of Cvl and Envronmental Engneerng, Cornell Unversty Research Supervsor: rcea Grgoru, Professor Summary A method s presented for assessng the sesmc performance of structural/nonstructural systems and developng ratonal strateges for ncreasng the sesmc reslence of these systems. The sesmc performance s measured by fraglty surfaces, that s, the probablty of system falure as a functon of moment magntude and ste-to-source dstance, consequences of system damage and falure, and system recovery tme followng sesmc events. The nput to the analyss conssts of () sesmc hazard, () structural/nonstructural systems propertes, () performance crtera, (v) rehabltaton strateges, and (v) a reference tme. Estmates of losses and recovery tmes can be derved usng fraglty nformaton, fnancal models, and avalable resources. A structural/nonstructural system located n New York Cty s used to demonstrate the methodology. Fragltes are obtaned for structural/nonstructural components and systems for several lmt states. Also, statstcs are obtaned for lfe tme losses and recovery tmes correspondng to dfferent rehabltaton alternatves. Introducton Captal allocaton decsons for a health care faclty nclude, for example, openng a new unt, extendng or closng some exstng unts, buyng new equpment, and relocatng the hosptal buldng. These decsons are based on lfe cycle capacty, vewed as the level of performance defned for a servce, and cost estmates. Exstng geotechncal, structural/nonstructural systems can be left as they are or can be retroftted usng one of the avalable rehabltaton alternatves. Leavng a system as t s seems to be reasonable for short-term decsons but retrofttng the system, despte ts ntal costs, mght be benefcal n the long run. A probablstc methodology s requred to make a rehabltaton decson snce sesmc hazard and system performance are uncertan. ost of the exstng earthquake loss estmaton methodologes usually calculate losses ncludng drect and ndrect economc and socal losses for a gven regon, based on the maxmum credble earthquake. The ATC-13 (ATC, 1985) methodology provdes damage and loss estmates, based on expertopnon, for ndustral, commercal, resdental, utlty and transportaton facltes. HAZUS (FEA, 1999) estmates potental losses on a regonal bass and these estmates are essental to decsonmakng at all levels of government, provdng a bass for developng mtgaton polcy, and response and recovery plannng. Both methods were developed to estmate losses for a large number of facltes n a specfed regon usng the maxmum credble earthquake and should not be appled to an ndvdual faclty. Losses estmated by usng the maxmum credble earthquake may not be accurate (Kafal and Grgoru, 24a). The man objectve of ths paper s the development of a methodology for evaluatng the sesmc performance and development of optmal rehabltaton strateges of ndvdual health care facltes Fraglty Based Rehabltaton Decson Analyss 47

2 durng a specfed tme nterval. The sesmc performance s measured by fraglty surfaces, that s, the probablty of system falure as a functon of moment magntude and ste-to-source dstance, consequences of system damage and falure, and system recovery tme followng sesmc events. Estmates of losses and recovery tmes, referred to as lfe cycle losses and recovery tmes, can be derved usng fraglty nformaton, fnancal models, and avalable resources. A health care faclty located n New York Cty s used to demonstrate the methodology. Fragltes and statstcs for lfetme losses are obtaned for ths structural system and some f ts nonstructural components. Proposed Loss Estmaton ethod The proposed loss estmaton method s based on () sesmc hazard analyss, () fraglty analyss and () capacty/cost estmaton. Fgure 1 shows a chart summarzng the loss estmaton methodology. Structural/nonstructural system defnton water tank ppng Sesmc hazard USGS sesmc actvty matrx mean annual Sesmc events 7 5 T (,R ).... tme Lfe cycle capacty/cost estmates Fraglty surfaces (for specfed lmt states) P f Damage D.... T Capacty 1% T Cost.. G.. C.... T tme tme tme Fgure 1. Loss estmaton Sesmc Hazard Analyss The nput to the sesmc hazard model conssts of (1) sesmc actvty matrx at the ste, (2) the projected lfe of a system, and (3) sol propertes at the ste. The sesmc actvty matrx s calculated usng the deaggregated matrces avalable at USGS webste ( Deaggregaton matrces at a ste gve the percent contrbuton of earthquakes wth dfferent moment 48

3 magntude ranges and rngs R j to the sesmc hazard at the ste. USGS provdes several deaggregated sesmc hazard matrces for any locaton n the Unted States at hazard levels of 1%, 2%, 5% and 1% probablty of exceedance n 5 years, where a hazard level s defned as the probablty that a ground moton parameter (e.g. peak ground acceleraton) exceeds a reference value durng a gven perod of tme. The mean annual rate ν j of earthquakes from bn (,R j ) can be calculated from deaggregaton matrx (Kafal and Grgoru, 24a). A onte Carlo algorthm can be developed for generatng () random samples of the sesmc hazard at the ste durng a gven perod of tme usng the sesmc actvty matrx, and () sesmc ground acceleraton samples for these sesmc hazard samples. Each sesmc hazard sample s defned by the number of earthquakes durng the tme, temporal dstrbuton, and magntude and source-to-ste dstance of each of them. Total number of earthquakes N() s assumed to follow a Posson dstrbuton wth mean annual rate ν=,j ν j, and the probablty that an earthquake havng a magntude n the range and comng from a source n the rng R j can be obtaned from P[, R R j ] =ν j /ν. The ground acceleraton A(t) s modeled by a non-statonary stochastc process A(t)=w(t)A s (t), where t s the tme, w(t) s a determnstc envelope functon and A s (t) s a statonary Gaussan process whose spectral densty functon s gven by the specfc barrer model. Input parameters of ths model are the moment magntude, source-to-ste dstance of the earthquake and the sol condton at the ste. The descrpton of specfc barrer model and how to generate samples of ground acceleraton tme hstores can be found elsewhere (Papageorgou and Ak, 1983 a and b; Kafal and Grgoru, 23a). Fgure 2 shows () the deaggregaton matrx for 1% probablty of exceedance n 5 years, () the sesmc actvty matrx, and () a sample of sesmc hazard scenaro over a lfe tme of 5 years, for New York Cty (NYC) area. Deaggregaton matrx of NYC Sesmc actvty matrx of NYC A sample of sesmc hazard contrbuton to hazard mean annual rate Fgure 2. Sesmc hazard Fraglty Analyss The probablty that a system response exceeds a lmt state vewed as a functon of and R s called system fraglty surface. onte Carlo smulaton and crossng theory of stochastc processes can be used to calculate fraglty surfaces of lnear/nonlnear systems and ther components for dfferent lmt states (Kafal and Grgoru, 23b, 24b). Fraglty s used to characterze the damage n the structural/nonstructural systems. Let D be a dscrete random varable characterzng the damage state of a nonstructural system after sesmc event characterzed by (,R ), =1,,N(t), where N(t) s the number of sesmc events n [,t]. Assume that the nonstructural system s n damage state d k, wth probablty p k, for k=1,,n, where n s the number of damage states. The probabltes p k, are obtaned from the fraglty nformaton of the nonstructural system and are functons of the lmt Fraglty Based Rehabltaton Decson Analyss 49

4 state defnng the damage state d k and (,R ). Smlarly, we can defne random varables characterzng the damage n structural system and components of the selected nonstructural system. Capacty and Cost Estmaton Capacty, for example patent per day capacty n a servce, and total cost are estmated for the case of no rehabltaton and for dfferent retrofttng technques. Usng these estmates effcent solutons can be determned. We assume that loss of capacty s caused solely by damage of nonstructural systems. The capacty at tme t s 1 % p % O(t) G exp(-γ (t-t )) S (p) t O( t ) = 1 N ( t ) = 1 G exp( Γ ( t T )), where T s the arrval tme of event, G and Γ are the loss n the capacty and the rate of recovery, after event, respectvely (Fg.3). Note that T Fgure 3. Capacty model S p ( t ) = N( t ) = 1 S ( p ), represents the total tme the system spends at or below p%-level capacty n [,t]. The cost relates to () structural falure, () retrofttng, () repar, (v) loss of capacty n servces, and (v) loss of lfe. Rehabltaton s only consdered for the nonstructural system. Costs due to (), (), (v) and (v) are random. The total cost n dollars at tme t n net present value s N( t ) TC( t ) = c + C /( 1+ dr ) = 1 where c s ntal cost related to the rehabltaton, dr s the dscount rate, T s the tme of arrval of event, and C s the cost related to event. Assume that C =CS, f structure fals and C =CR +CC +CL, otherwse. CS s the cost related to structural falure, CR s the repar cost of the nonstructural system, CC s the cost due to the loss n capacty, and CL s the cost of lfe losses. It s expected that wth an ncreasng ntal cost c, the cost C due to event wll decrease and for some rehabltaton alternatve we wll have the optmum soluton. Numercal Example An CEER Demonstraton Hosptal Project located n NYC s used to demonstrate the proposed methodology. Three dfferent levels of rehabltaton, namely, () no rehabltaton (rehab.1), () lfe safety (rehab.2) and () lmted downtme (rehab.3) are consdered. It s assumed that the structure s lnear elastc and cascade analyss apples, that s, the nonstructural system does not affect the dynamcs of the supportng structure. The nonstructural system consdered conssts of a water tank (comp.1) and a power generator (comp.2) located at the roof and at the frst floor, respectvely. It s assumed that () the components are not nteractng, () water tank s drft senstve, () power generator s acceleraton senstve, and (v) both components are lnear sngle degree of freedom oscllators. An llustraton of the hosptal model wth two components attached to t, the requred T, 5

5 modal propertes of the structure (dampng rato s 3% for all modes), and natural frequences and dampng ratos of the components for the dfferent rehabltaton alternatves are show n Fgure 4. comp.1 Structural system Nonstructural system 51 ft mode ω Γ (rad/sec) comp.1 ω (rad/sec) ζ comp.2 ω (rad/sec) rehab ζ comp rehab rehab Fgure 4. System propertes Fraglty surfaces are obtaned for the structural/nonstructural systems, comp.1 and comp.2 for dfferent lmt states assumng statonary ground acceleratons. Structural system s assumed to fal when the roof dsplacement exceeds 5. Lmt states are {.12,.25,.5 } and {1.g,1.5g} for comp.1 and 2, respectvely. The nonstructural system has three damage states () no damage, when both components have no damage; () extensve damage, when ether of ts components fals; and () moderate damage, otherwse. Fgure 5 shows fraglty surfaces of structural/nonstructural systems and comp.1 and comp.2 for dfferent rehabltaton alternatves and lmt states. Structural system Comp.1 rehab.3 ls.3 Comp.2 rehab.1 ls.1 Pf Pf Pf Fgure 5. Fraglty surfaces Estmates of the total tme the system spends at or below 8%-level capacty and the total cost TC, durng a projected lfe of =1 years, for the three rehabltaton alternatves are obtaned by onte Carlo smulaton. Followng nformaton s used to obtan these estmates. G and Γ are dscrete random varables takng values {,.5,.9} and {,.5,.3}, respectvely, wth probabltes obtaned from the nonstructural system fraglty. The dscount rate s 7% and the rehabltaton cost c takes values {,1,5}, for no rehabltaton, lfe safety and lmted downtme rehabltaton, respectvely. CR =C 1, +C 2,, where C 1, and C 2, are dscrete random varables takng values {,6,14,2} and {,5,1}, respectvely, wth probabltes obtaned from the correspondng component fraglty surfaces, and they represent the repar costs for comp.1 and 2, respectvely. CS =cs.q, where cs=47 s the cost related to the downtme and the constructon of a new faclty and q s the probablty of system falure obtaned from the structural Fraglty Based Rehabltaton Decson Analyss 51

6 system fraglty. CL =cl.x, where cl=22 s the cost of one person's lfe loss and X s a bnomal random varable wth parameters n x =1 and p x =.1, representng the number of people losng ther lves, respectvely. CC =cc.rt, where cc=23 s the cost due to the loss n capacty per day and RT s the tme to reach 1% capacty gven by RT = for G =Γ =, and RT =-ln(.1/g )/Γ, otherwse. Fgure 6 shows P(S p (t)/ >s/) and P(TC(t)>c). P(S p (t)/ >s/) P(TC(t)>c) s/ (=1 years) c (n 1) Fgure 6. Estmates of the capacty and total cost A possble measure for comparng the effectveness of dfferent rehabltaton alternatves s the probablty that the total cost exceeds a level c. Accordngly, the optmal soluton s the one wth the lowest P(TC(t)>c), and depends on the selected value of c (see Fg.6). For example, the optmal solutons are rehabltaton alternatves 1 and 2, for c=5 and c=4, respectvely. Concludng Remarks A method was developed to dentfy an optmal retrofttng technque for structural/nonstructural systems. The method () consders a realstc sesmc hazard model rather than usng the maxmum credble earthquake, () ncludes all components of costs, that s, the costs related to the structural falure and downtme, retrofttng, repar, loss of capacty n servces, and loss of lfe, and () s desgned for ndvdual facltes rather than a large populaton of them. The method s based on onte Carlo smulaton, probablstc sesmc hazard, fraglty surfaces and capacty/cost analyses. Acknowledgements Ths research was carred out under the supervson of Dr.. Grgoru, and supported by the ultdscplnary Center for Earthquake Engneerng Research under NSF award EEC References ATC-13 (1985): Earthquake damage evaluaton data for Calforna. Appled Technology Councl, Redwood Cty, CA. FEA (1999): HAZUS 99: Estmated annualzed earthquake losses for the Unted States, FEA, Washngton, D.C. 52

7 Kafal, C. and Grgoru,. (23a): Non-Gaussan model for spatally coherent sesmc ground motons. In proceedngs of the ICASP9, San Francsco, CA., pp Kafal, C. and Grgoru,. (23b): Fraglty analyss for nonstructural systems n crtcal facltes, In proceedngs of ATC-29-2 semnar, Newport Beach, CA, pp Kafal, C. and Grgoru. (24a): Rehabltaton decson analyss. In preparaton. Kafal, C. and Grgoru,. (24b): Sesmc fraglty analyss. In proceedngs PC24, Albuquerque, N.. Papageorgou, A. S. and Ak, A. K. (1983a): A specfc barrer model for the quanttatve descrpton of the nhomogeneous faultng and the predcton of strong ground moton. Part I, Bulletn of the Sesmologcal Socety of Amerca 73, Papageorgou, A. S. and Ak, A. K. (1983b): A specfc barrer model for the quanttatve descrpton of the nhomogeneous faultng and the predcton of strong ground moton. Part II.. Bulletn of the Sesmologcal Socety of Amerca 73, Fraglty Based Rehabltaton Decson Analyss 53

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