Report - PEER - University of California, Berkeley

predictions and hence required sample sizes even further. This subject has been theobject of previous studies and recent PEER investigations.The candidates for improved IMs include both scalars and vectors. The scalarsare developed as functionals of several variables shown to carry information about theresponse of a particular structure, e.g., a function (1) of S a and magnitude, M, ifstudies show that S a is not sufficient with respect to magnitude for a particularstructure (e.g., a tall long period structure), or (2) of the two S a ’s at both the first andsecond mode periods, or (3) of the S a ’s of the first-mode S a and the S a at some longerperiod. Both of the latter examples are designed to capture spectral shape informationin the period ranges of interest to a specific structure, second mode in the first caseand that of an effective-period-lengthened nonlinear structure in the second. Luco(2002) proposes a scalar that is a SSRS-like combination (employing modalparticipation factors) of the inelastic displacement of an elasto-plastic oscillator (withyield displacement equal that derived from a static push-over analysis of the MDOFstructure) and the second-mode elastic spectral displacement. The vectors mayinclude similar such variables (e.g., Bazzurro (2002), Baker (2004b)).As discussed above the sufficiency is typically demonstrated by showing the lackof dependence of the response on certain X variables given the IM level. Forexample, Figure 1a shows that for this structure the S a is sufficient with respect tomagnitude as the residuals from a regression of MIDR on S a show no significantdependence on M. In contrast when seeking a better IM one looks for additionalvariables (beyond S a , say) that demonstrate additional explanatory power. Figure 1bshows that epsilon does this for this structure.Van Nuys 7 StoryResidual of the "original" regression on S a(FMF)10 010 1 Residual of the regression of magnitude on ln S a(FMF)a = 2.31e-015b = 0.1443σ ln residual|residual2= 0.2648σ b= 0.08303one-sided p-value = 0.04378θ max10 -210 -1-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 110 -1 Epsilon10 -3-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5Figure 1. (a) Shows MIDR residuals (given S a ) versus magnitude; (b) showsresiduals versus epsilon. Baker(2004b).Therefore one can ignore M (as S a is sufficient with respect to it for this structure),but epsilon needs further consideration, for example, as a member of X or by carefulrecord selection as discussed above. Further epsilon deserves to be considered as acandidate for inclusion in an improved scalar or vector IM, one that would eliminatethe insufficiency of S a with respect to epsilon and, given its clear explanatory power,should decrease the dispersion in predicting the MIDR (Baker (2004b)). Figure 245