This project is designed to continue and bring to a level of completion the main issues of the Year 5 work statement: uncertainty propagation, IM sufficiency and efficiency investigations, and record selection/processing. We find it novel to view the last two subjects as being strongly inter-related.

Uncertainty propagation:
Year 5 research on uncertainty propagation will be completed and documented. The Baker work on propagation of all variables from IM to DV will be documented. This procedure captures and carries through both aleatory and epistemic uncertainty in all the EDP, DM and DV vectors including their correlations.

IM efficiency and sufficiency:
This portion of the work consists of two parts. The first is taking part in the ongoing process of selecting and evaluating (to the degree possible with the testbed structures) the efficiency of the various IMs proposed by the IM working group. This focuses on EDP|IM because the subsequent steps (such as DM|EDP) are (or should be) by design (conditionally) independent of IM. It is expected that this work will be completed in Year 6. The second part consists of new research on sufficiency of IMs, with an emphasis on the role of near-fault ground motions on this subject. This work is to be integrated with the other two issues, particularly with respect to the record selection at sites affected by near-fault forward directivity.

Record selection/processing:
The subject of “sufficiency” of an IM addresses the issue of obtaining an accurate estimate of the EDP hazard curve at a particular site. (Recall the EDP hazard curve is an integrated product of the IM and EDP|IM distributions.) This in turn is related to the question of what is a proper selection of records (for time-history-based assessment of EDP|IM) and/or what “additional” processing of the output of the EDP|IM studies (such as unequal weighting of the samples) should be conducted to insure that the EDP hazard curve is an accurate statistical representation of the seismic threat at a particular site. From this perspective, information that is relevant includes: the disaggregation of the site’s IM hazard (with respect to magnitude for example), the sensitivity of the nonlinear EDP to spectral shape, the degree to which a specific IM captures that sensitivity, how these issues change with ground motion level, etc. Work in Year 5 on this subject will be completed and documented. It has used a strength-degrading model of the Van Nuys Holiday Inn as a testbed. It is limited, however, to “ordinary” records, which are the threat at this site. For Van Nuys and such cases specific criteria for selection of records and/or processing (e.g., of current records, if necessary) will be presented. These criteria may suggest that a second set of records be selected for this testbed for certain subsequent uses. In Year 6 this effort will be extended, like the IM studies above, to the near-source problem.

Introduction and development of probabilistic methods in seismic PBEE.

Analytical development of uncertainty propagation by FOSM; design of record sets and oscillator/structural analyses and statistical analysis to study IM and record selection.

1. In year 5 and 6 we have developed and documented a comparatively simple first-order, second-moment scheme for conducting the (aleatory and epistemic) uncertainty propagation through the several, long correlated random vectors of EDPs and DMs to the DVs in the PEER framing equation. The results are the first and second moments of the DVs given IM = x. These can be used with the IM hazard curve to produce mean and variance of each DV, and, with an appropriate distribution assignment, the DV “hazard curve”, i.e., the annual probability of exceeding DV level y vs. y. It is presumed that this scheme will be adequately accurate for treatment of these intermediate variables given the comparative uncertainty dominance of the IM variable. The report includes a discussion and examples of estimating the second moments in the case of EDP|IM

2. As described last year in Year 5 we developed and documented further methods for estimating the conditional probability distribution of EDP given IM = x as a function of x for nonlinear structures based on suites of nonlinear dynamic analyses.

3. In Year 6 we have worked further on the question of the “sufficiency” of Sa (at the first-mode period) as this issue is closely related to the problem of record selection. We demonstrated that even the two extreme cases (a very HF oscillator) and a long-period, multi-mode-dominated 20-story structure no statistically significant dependence on magnitude, distance or epsilon remains once one conditions on Sa (the definition of “sufficiency”) implying that the records used in the nonlinear dynamic analyses need not be selected with great care with respect to these features of the threatening seismicity. For a model of the Van Nuys Testbed structure we have demonstrated using a very large suite of records designed to accentuate the problem, we have demonstrated that Sa is a sufficient IM with respect to M, R and “epsilon” for this structure. The figure here shows this (with respect to the most difficult parameter, magnitude) by the mild and statistically insignificant slope of the residuals of max interstory drift (regressed on Sa) versus those of magnitude.

4. More directly, we have selected records at random from a catalog covering a broad magnitude distance range and showed that scaled to the same Sa level, they produced the same (statistically) speaking displacements as a set taken from a set representing the “strongest” records in the range (i.e., those from a large magnitude, close distance range). Further studies along these lines are continuing with the focus shifting to consider near-source records as well.