Report - PEER - University of California, Berkeley

2.4 Option C: Sufficient IMs: Estimation of P[C|IM] and λ CIn this section we introduce the notion of a “sufficient” IM and demonstrate theadvantages it brings to PBSA. An IM is sufficient if P [ C | IM , X ] = P[C | IM ], that isif the probability of the event given IM and X does not in fact depend on X at all. Inthis case λ = ∫∫ P[ C | IM , ] f ( IM | ) | d ( ) | = ∫ P[C | IM ] | d ( IM ) |CX X λ Xλ inIMwhich λ IM is simply the “hazard curve” of the IM, i.e., λ IM (u) is the mean annualfrequency that the IM exceeds a specific value u. FormallyλIM= ∫ G ( IM | X ) | d λ ( X ) | . This can be obtained by conventional hazard analysisand can be left to solely to the seismologist, provided the engineer has specifiedwhich IM he believes is appropriate for his particular structure. This may be as simpleas specifying that he wants the IM to be the spectral acceleration at a period in thegeneral vicinity of the (low strain) natural period of his structure. Estimation of λ Creduces to selecting a set of accelerograms, scaling them to each of a set of IM levels,estimating as above the probability P[C|IM] and then summing:λC≈ ∑ P [ C | IM ] ∆ λIM( IM ) . Assuming that the dispersion of forexample MIDR given a value of IM is about 0.3 to 0.4, each level will take order 10samples and there need to be 4 to 6 levels then the total number of runs is only about50. As discussed above this number can be reduced in special cases and by tools suchas regression. Incremental dynamic analysis is one scheme that may be employed inone or more ways (Jalayer, 2003), especially when one wants to use the same runs foranalyses of different C’s for which different IM’s may be most effective, e.g., S a forMIDR and PGA for peak floor accelerations. Applications of Option C are nownumerous; it has been widely used in the PEER Center, where S a at a period near thatof the natural period has been the IM of choice. The author’s students haveaccumulated many cases; see www.stanford.edu/groups/RMS for theses and papermanuscripts. The introduction of the sufficient IM raises the question of how toestablish sufficiency of a candidate IM and how to select the best from a collection ofsufficient IM’s. These questions will be addressed below. In Luco et al. (2002) allthree options A, B, and C are used and compared.2.5 Record Selection for PBEAWhenever assessment through nonlinear dynamic analysis is anticipated the questionof appropriate record selection always arises. The source of records for considerationranges from empirical recordings, through artificially “spectrum-matched”accelerograms derived from recordings, to various forms of synthetics, includingcolored Gaussian noise and geophysically based rupture simulations. Consider firstthe choice from among a catalog of recorded accelerograms. The question of recordselection is not unrelated to the discussions above. Under Options A and B above therecords must be selected appropriate to each of the several X bins (e.g., by magnitude,distance levels). In contrast, under Option C, because sufficiency (independence) with43