Report - PEER - University of California, Berkeley

deviations σ and correlation coefficientsDρ . The i-th demand can then beiD i D jexpressed, similarly to the corresponding capacity, as:Di( µ ) εiµ ⋅=D i x D(6)where the mean value µ of the demand is evaluated at the mean value of the basicDivariables µx, ε can be assumed to be Lognormal with unit mean, standardDideviation equal to the i-th demand coefficient of variation δD= σi D i/ µD iandcorrelation coefficient with ε equal toDρ = ρ .jij D i D jApart from the dependence of the demands on the basic variables x , all theelements are in place to evaluate the probability of failure of a completely generalsystem (not necessarily serial):Pf⎪⎧= Pr⎨⎪⎩nCUIj= 1 i∈ICjCi⎪⎫( , ε ) ≤ D ( ε ) ⎬ ⎪⎭x (7)CiDwithnCcut-setsC ( I denoting the set of indices of the failure modes belongingjC jto the j-th cut set).The system reliability problem in Eq.(7) can be evaluated either by FORM, firstsolving each component/failure mode and then using the multi-normal approximationfor general systems, or by Monte Carlo simulation, which is simpler and in this caseis comparatively inexpensive since it does not require any structural analysis.As a final step, it remains to account for the dependence of the demands on x .One possible approximate way of doing it is to consider this dependence as lineararound the mean value of x . This involves the first order partial derivatives of thedemands with respect to x evaluated in the mean µxof the basic variables. Thelatter, often called sensitivities with respect to the system parameters, can becomputed either numerically by a finite difference scheme, i.e., repeating the analysisfor perturbed values of the parameters, or, more efficiently, by the DirectDifferentiation Method (Franchin 2004, Kleiber 1997). In practice, the sensitivities∂ D / ∂x i jare computed as the mean values of the derivatives conditional on sampleaccelerogram:∂Di∂xj∂=∂xj⎛ 1⎜⎝ nn∑⎞ 1D ⎟ =⎠n∑∂Dxikikk = 1 n k = 1 ∂j(8)224