1. Project Goals/Objectives:

Develop and test the use of a vector-valued Intensity Measure in the Van Nuys Testbed context. By working jointly the investigators will address both the structural and seismological aspects of the vector-valued IM approach.

2. Role of this project in supporting PEER’s mission (vision):

PEER is pioneering the study of alternative Intensity Measures (IMs), which are the first (right-most) variable in the Framework equation. Several researchers and two PEER workshops on IMs have suggested that this IM should in the future be in vector format. Intensity measures form the link between seismology and engineering. This is the first project in which formal PEER-SCEC collaboration is involved.

3. Methodology Employed:

Within the PEER PBEE paradigm the purpose of vector-valued probabilistic seismic hazard analysis (VPSHA) coupled with a probabilistic EDP|IM characterization (where the bold font denotes a vector) is to provide the two inputs into a more efficient and accurate estimation the EDP annual frequency distribution, (EDP), e.g., the annual exceedance of a given maximum interstory drift level. VPSHA provides the annual frequency of the joint occurrence of two or more ground motion intensity measures. The EDP|IM assessment predicts the EDP as a function of two or more IMs. This vector-valued approach should prove superior when the response of a structure to ground shaking is strongly dependent on more than one ground motion intensity measure. For example, higher modes contribute significantly to the response of tall structures; for them, specifying the joint frequency of occurrence of spectral accelerations having specified levels at the fundamental mode and at the first higher mode will provide a more accurate and smaller variance representation of the response of the structure. In the near-fault region, the response of a structure may depend on both the period and amplitude of the forward rupture directivity pulse (Somerville et al., 2000; Somerville, 2001), requiring a vector-valued representation of the hazard. Other examples include the response of structures with significant cyclic degrading behavior, the amplification functions of soft soil deposits, the prediction of onset of liquefaction in saturated sandy deposits, and the assessment of stability of earth slopes. In all of these cases, the response may be driven strongly by multiple, joint ground motion intensity measures, including perhaps the duration of ground shaking.

The two basic parts of the project are developing the vector-valued hazard curves and developing procedures to determine the conditional distributions of EDPs (Engineering Demand Parameters) given the values of the IM vector. For the first part, the project has modified/developed existing PSHA software, constructed (joint) attenuation laws, and produced sample hazard curves and disaggregation results for the Van Nuys site and a site near the Hayward fault. For the second part we have conducted nonlinear analyses of existing models of the Van Nuys testbed building, prepared regressions (and “conditional regressions”) of interesting EDP’s vs. various vector IM candidates, studied their efficiency and sufficiency (relative to conventional scalar candidates) and coupled the hazard curve with these results to produce EDP hazard curves. This exercise is being repeated for a sample of generic building models using tabulated nonlinear response results from previous PEER projects under Prof. Krawinkler’s direction, and is progressing into the issues of near-source cases.

4. Brief Description of past year’s accomplishments (Year 6) & more detail on expected Year 7 accomplishments:

The project has focused initially on vector IMs that are made up of first-mode spectral acceleration (Sa1) plus either (1) additional variables such as magnitude, distance or “epsilon” (a measure of how far the specific record’s first-mode spectral acceleration deviates from the expected value) or (2) ratios of the spectral acceleration (to Sa1) at various periods (at and either-side of the first natural period). We have attenuation laws and vector hazard code for all these cases, as prepared by our collaborator, Dr. Somerville. Using the existing (Pincheira-DRAIN 2D) model results and

tabulated results for generic buildings we have evaluated various methods for obtaining the necessary conditional EDP|IM means and variances. The most effective is to scale the records to a given first-mode spectral acceleration, make several runs, and regress these results on the other IM (or IMs if the vector > 2 in length). This captures well how this conditional dependence changes with Sa1 level. In addition we have incorporated the effect of “collapses” (non-converging or “very large interstory drifts) by capturing the probability of collapse as a function of the vector IM. These results are then integrated with the vector hazard to obtain the EDP (drift) hazard curve.

The work has studied the benefits of a second Sa value (i.e., the ratio of Sa(T) to Sa1) as a function of T. Not unexpectedly the optimal (minimum variance value) is at T = second mode period for low ground motion levels, while at stronger motion levels the optimal T becomes greater than the first mode value and increases with this level (i.e., with Sa1 level). The latter trend is consistent with various researchers’ predictions for the “equivalent” period of a softening nonlinear system. The use of this optimal value can reduce the dispersion in the predicted drift by up to a factor of 2. A third variable gives much smaller additional gain in efficiency. (This work is reported in a WCEE 13 paper.)

More surprisingly we have discovered (and reported in an Annual Meeting poster) that epsilon proves to be virtually as effective as a second Sa – if not even better. The benefits of both contenders can be explained by their providing information about spectral shape. But epsilon can be obtained from current scalar PSHA codes by simply manipulating the “disaggregation” output; this makes it more practical. Equally importantly, study of the use of epsilon in the vector IM has found that a bias can by introduced in drift hazard estimates produced by the traditional scalar approach, especially at large drift levels and/or low probabilities. PSHA disaggregation has always shown that the rare, low probability ground motions are associated with large values of epsilon (1 to 2 or more). Because of the identified dependence of drift on epsilon, if this systematic effect is ignored, bias will result. Scalar procedures must select records carefully to have increasingly higher epsilon values at higher Sa levels; this is not considered in current record selection practice. (As shown in the figure, disaggregation results should be used to identify the conditional distribution of epsilon given Sa1.) Proper use of vector PSHA and EDP|IM will avoid or minimize this potential bias.

In the remainder of year 7 we shall extend our investigation into near-source conditions. Preliminary evidence suggests that spectral acceleration at two or more frequencies will capture the effects on nonlinear behavior of pulse like records.

5. Other Similar Work Being Conducted Within and Outside PEER and How This Project Differs:

Profs. Bray and Kramer are studying scalar IMs for geotechnical problems. Prof. Miranda is studying scalar IMs for EDPs for non-structural damage such as peak floor acceleration. Prof. Joel Conte (UCSD) has studied the efficiency of many vector intensity measures on simple SDOF oscillators for PEER. This project is studying a real (MDOF) structure and a suite of generic building models plus all of the seismological portions as well. We are unaware of any vector-valued work elsewhere.

6. Plans for Year 8 if project is expected to be continued:

N/A

7. Describe any actual instances where you are aware your results have been used in industry:

This is premature.

8. Expected Milestones & Deliverables: