Report - PEER - University of California, Berkeley

2.3 Option B: Estimation of P[C|IM, X] and λ CWith the objective of reducing the number of nonlinear analyses it can be helpful tointroduce the notion of an “intensity measure”, or IM. Familiar scalar examplesinclude PGA and spectral acceleration, S a . We shall restrict our attention here toscalar IM’s. An IM is scalar property of an accelerograms that can be found simplyand cheaply (at most be integration of the equation of motion of a simple oscillator.)With the introduction of this variable and the total probability theorem we may writeλC = ∫∫ P [ C | IM , X]f ( IM | X)| dλ(X)| in which f(IM|X) is the conditionalprobability density function of the IM given X, which is customarily available as an“attenuation law” in engineering seismology. The estimation of P[C|IM,X] wouldproceed as above except that the records selected in each X “bin” (e.g., each {M,R}pair) should also have a specified IM level (e.g., a given PGA value) usually obtainedby simply scaling the record to that level. For each of several levels of IM the set ofrecords is analyzed and the probability for that IM level and X bin, P[C|IM,X] isestimated as above as the ratio r/n’. Upon repetition over the set of Xbins, λC≈ ∑∑P[ C | IMj, Xi] ∆f( IMj| Xi) ∆λ( Xi) . The advantage ofintroducing the IM as that the dispersion (defined here as the standard deviation of thenatural log (or approximately the COV) of say the MIDR given IM and X is onlyabout 0.3 to 0.4 for a nonlinear MDOF frame at large ductility levels, implying, usingthe rule above, that only some (0.35/0.1) 2 or order 10 records are necessary in foreach first factor in the summation. However, assuming some 4 to 6 IM levels 3 and 10-20 X pairs the total required sample size is still in the range of 500. Again cleversampling or response-surface/regression modeling can potentially reduce this numbersubstantially. Indeed one result of performing regressions such as MIDR on IM andX is that observation that one or more of the variables in X, such as R the distance tothe fault, are statistically insignificant once IM is included in the equation. Thisimplies one can reduce by a factor root (10-20) or 4, say, the number of cases andhence the sample size necessary. In fact if the IM is well chosen experience showsthat all the variables in X may be found to be statistically insignificant, or at leastpractically so, i.e., the response (given IM) is no longer importantly sensitive to, say,M and R. This is not unexpected; in the limiting case of IM equal S a at period T, themaximum response of a simple linear SDOF oscillator with natural period T is totallyinsensitive to X once the IM is known. And it is common practice to assume both thatthe equal displacement rule holds for moderate-period, moderate-ductility nonlinearoscillators and that the maximum roof drift of a low to moderate-height frame isproportional to the response of a nonlinear oscillator. This insensitivity is exploited inthe next section.3 In fact if only a single event C is targeted, e.g., MIDR > 7%, this number of levels may be reduced to asfew as 1 or 2, Jalayer (2003).42