5.2 Damage Scenario Tree
5.2.3 Quantifying damage using EDP
EDP are used to quantify the probability of being in a DS at the system level, subsys- tem level, and component level. The EDP corresponding to the system level, subsystem
level, and component level damage assessments are denoted by EDPs, EDPss, and EDPc, respectively. EDPs, EDPss, and EDPcare random variables. Three primary sources of EDP uncertainty are the ground motion RTR variability of the structural response, the variability of the building system parameters, and the uncertainty regarding the accuracy of the model of the building used in the nonlinear structural response analyses.
For a specific subsystem, among all P subsystems considered for damage analysis (shown in Figure 5.2), the corresponding EDP is denoted by EDPss,p, where the subscript p indicates the pth subsystem. Similarly, for a specific component, among all Q components considered for damage analysis (shown in Figure 5.2), the corresponding EDP is denoted by EDPc,q, where the subscript q indicates the qth component.
EDP limit values (EDPDS,i) are used to distinguish between the (i − 1)thDS and ithDS for the system, for a subsystem, and for a component. The EDP limit values for the system level, subsystem level, and component level damage assessment are denoted by EDPs,DS,ns, EDPss,p,DS,nss, and EDPc,q,DS,n, respectively. Indices p and q indicate the pth subsystem and qth component, respectively. The indices ns = 0, 1, · · · , Ns; nss = 0, 1, · · · , Nss; and n= 0, 1, · · · , N specify the DS number for the system level, subsystem level, and component level damage assessments, respectively. As shown in Figure 5.2, one EDP limit value is required to separate the two DS of non-collapse and collapse at the system level (i.e., Ns= 1). The EDP limit value separating the non-collapse DS from the collapse DS is EDPs,DS,1= EDPs,C, where the subscript C stands for collapse. Similarly, one EDP limit value is required to separate the two DS of non-demolition and demolition at the subsystem level (i.e., Nss = 1). The EDP limit value separating the non-demolition DS from the demolition DS is EDPss,p,DS,1= EDPss,p,D, where the subscript D stands for demolition. At the component level, N EDP limit values are required to separate the N + 1 DS. The first component level DS, denoted by DSc,q,0 (see Figure 5.2), is the state of having no damage with no required repair action.
In the present analysis, the maximum (over all stories of the building) peak story drift ratio, θm, is used as the system level EDP (i.e., EDPs= θm). The θmlimit value, separating the non-collapse DS from the collapse DS, is EDPs,C= θm,C. The present study considers the SLFRS as the only subsystem in the damage analysis (i.e., P = 1). The maximum (over all stories of the building) residual story drift ratio, θr, is used as the subsystem level EDP (i.e., EDPss = θr) for the SLFRS [60]. The θr limit value, separating the non-demolition DS from the demolition DS, is EDPss,1,D= θr,D. Later in this chapter, where application of DSTA to a the 9SCBF archetype building is discussed, examples of EDPc,qand EDPc,q,DS,n are given.
Recognizing the epistemic uncertainty in the damage state criteria, the EDP limit val- ues (i.e., EDPs,C= θm,C, EDPss,1,D= θr,D, and EDPc,q,DS,n) are treated as random variables (rather than deterministic limit values). In other words, the EDP limit values are treated as random variables due to a lack of knowledge of the precise value of an EDP separating two DS. For example, the precise value of θr separating the non-demolition DS from the demo- lition DS is uncertain. Probability distributions for the EDP limit values can be estimated from analytical work, published test data, and post-earthquake reconnaissance reports [55]. The probability of a given EDP value (EDP = edp) exceeding an EDP limit value, EDPDS,i, is quantified by evaluating the cumulative density function (CDF) of EDPDS,iat edp as fol- lows:
P(EDP ≥ EDPDS,i|EDP = edp) = FEDPDS,i(EDP = edp) (5.1)
where edp is a value of the EDP, EDPDS,i is the ith EDP limit value separating the (i − 1)th DS from the ith DS, and FEDPDS,i is the CDF for EDPDS,i which is the i
th EDP limit value fragility function. For example, for the system level and the SLFRS subsystem level damage assessment, the collapse fragility function and the demolition fragility function are denoted by Fθm,C and Fθr,D, respectively. Fθm,C and Fθr,D represent the uncertainty in the θm,C and θr,Dlimit values, respectively, and are discussed in more detail later.
Figure 5.4 illustrates three damage scenarios from the damage scenario tree in Figure 5.2. The probability of occurrence of a damage scenario (such as those shown in Figure 5.4) is equal to the probability of the intersection of the DS at different levels, which form the damage scenario. The probability of each damage scenario shown in Figure 5.4 can be quantified using the EDP and EDP limit values as follows:
P(C|IM) = P(θm≥ θm,C|IM) (5.2) (a) (b) (c) GM | IM NC C ND D DSc,q,0 DSc,q,n DSc,q,N ͙ ͙ GM | IM NC C ND D DSc,q,0 DSc,q,n DSc,q,N ͙ ͙ GM | IM NC C ND D DSc,q,0 DSc,q,n DSc,q,N ͙ ͙
Figure 5.4: Damage scenarios; (a) collapse (C|IM): (b) non-collapse with demolition (NC ∩ D|IM); and (c) non-collapse, non-demolition with component damage (NC ∩ ND ∩ DSc,q,n|IM)
P(NC ∩ D|IM) = P(θm< θm,C∩ θr≥ θr,D|IM) (5.3)
P(NC ∩ ND ∩ DSc,q,n|IM) =
P(θm< θm,C∩ θr< θr,D∩ EDPc,q,DS,n≤ EDPc,q< EDPc,q,DS,n+1|IM) (5.4)
Equation (5.2) gives the probability of collapse of the building at a given IM value. Equation (5.3) gives the probability of non-collapse with demolition and reconstruction of the building at a given IM value based on the damage assessment of the SLFRS sub- system of the building. Equation (5.4) quantifies the probability of non-collapse and non- demolition with the qthcomponent being in the nth DS.