Report - PEER - University of California, Berkeley

where e -λτ is the discounted factor of the loss over time t, λ is the discount rate peryear, E [ L T | IM ] is the expected loss in the building corresponding to a groundmotion intensity, IM, ν(IM) is the mean annual rate of exceeding a ground motionintensity IM. In (1) the time period t can correspond to the design life of the structure,the remaining life of an existing structure or another reference time period. Forpurposes of setting design actions in building codes or for setting insurance premiumslong t are usually assumed (Rosenblueth, 1976) and the effect of the finite life span ofthe facility becomes negligible.Since collapse (C) and non-collapse (NC) are mutually exclusive damage states,the expected loss in a building conditioned on ground motion intensity IM, can becomputed using the total probability theorem as followsE[ L | IM ] = E[ L | NC,IM ] ⋅[ 1−P(C | IM )] + E[ L | C] ⋅ P(C | IM )TTwhere E[L T | NC,IM] is the expected loss in the building provided that collapse doesnot occur for ground motions with an intensity level IM=im, E[L T | C] is expected lossin the building when collapse occurs in the building and P(C|IM) is the probabilitythat the structure will collapse conditioned on ground motion intensity.The expected total loss in the building provided that collapse does not occur at aground motion intensity IM=im, E[L T | NC,IM], is computed as the sum of the lossesin individual components of the building asNN⎡⎤E[ L ] = ⎢∑ ( ⋅ ) ⎥ = ∑ ⋅ [ ]T| NC,IM E aiLi| NC,IM aiE Li|NC,IM(3)⎣ i=1⎦ i=1where E[ L i |NC,IM ] is the expected normalized loss in the ith component given thatglobal collapse has not occurred at the intensity level im, a i is the replacement cost ofcomponent i and L i is the normalized loss in the ith component defined as the cost ofrepair or replacement in the component normalized by a i . Details on the computationof E[ L T | NC,IM ] and E[ L T | IM ] are given in Aslani and Miranda (2004b).The mean annual frequency of exceedance of a certain level of economic loss l Tis computed asIM[ L ] [ ]TlTP LTlTIM dIM∫ ∞ dν( )ν > = > | ⋅(4)IM0 dwhere P[L T >l T | IM], is the probability of exceeding a certain level of loss for a givenIM. For values smaller than 0.01 the mean annual frequency of exceedance of a loss l Tis approximately equal to the mean annual probability of exceedance.In Eq. (4), P( LT > lT| IM ) can be assumed lognormally distributed (Aslani andMiranda 2004b). On the basis of this assumption only the first two moments of theprobability distribution are required to evaluate this conditional probability. The firstmoment, the expected value, is given by equation (2) while the variance of the loss, iscomputed as follows222LT lT| IM = LT| NC,IM ⋅ 1 − P(C | IM)+ LT| C ⋅ P C | IMσ ( > ) σ ( ) [ ] σ ( ) ( )22{ E[ L | NC,IM] − E[ L | IM]} ⋅[ 1−P(C | IM)] + { E[ L | C] − E[ L | IM]} ⋅P( C IM)+ |T TTT(5)T(2)151