1. Project Goals/Objectives:

Extend new developments for the post-quake environment and develop a new paradigm for pre-quake design/assessment that explicitly recognizes the pre-quake importance to owners of (future) post-quake tagging (e.g., evacuation or not), downtime and their relationships to life safety. The motivation is utilize and integrate lessons learned and products of PEER, SCEC, and SAC activities. These include post-quake damage states and their global capacities via (dynamic and/or non-linear static) IDAs; importance of downtime costs and their dependence on the tagging state which may induce a loss of functionality; the degree of post-quake aftershock hazard (its dependence on site and mainshock magnitude) as well as its temporal evolution and role in re-occupancy decisions; and the role of epistemic uncertainty in risk calculations and its dependence on the degree of information available (e.g., degree of damage and estimated capacity of the damaged structural system as a function of level of information about the ground motion experienced, degree of inspection and level of analysis, etc.)

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

This project addresses the costs of degree and length of loss of functionality due limits on occupancy. It applies the PEER PBEE paradigm in a new quantitative focus on the role aftershocks should play in assessment and potentially design.

3. Methodology Employed:

The project is designed to consolidate and build on recent efforts in the PEER Lifelines Building Fragility project (two papers accepted in the 13th WCEE). These studies have provided new tagging decision guidelines and post-quake assessment procedures based on the severity of aftershock threat to life safety, which depends on the post-quake damage state (or what might be known about it), the time-varying aftershock hazard, and whether or not the building is occupied during repair (which also impacts downtime costs, the repair time and hence the duration of exposure to the aftershocks in the reduced-capacity building), These efforts have also explored in a preliminary way the decision-theoretic (decision/event tree) formulation of value of information questions (e.g., should more detailed, costly inspection to ascertain more precisely the true state of damage be conducted before making this evacuation decision, what is the role of more precise information about the level of ground motion experienced, what other epistemic-uncertainty-measures might be cost-effective). One initial portion of this project will be to consolidate and report on those efforts. The major project effort will be to address and ‘solve’ a new pre-quake PBEE assessment formulation that recognizes that post-quake there will not only some (random) damage state but also an aftershock environment that depends on the (random) size and location of the mainshock, and hence (following the previous discussion) a set of (random) post-quake decisions (in particular evacuated or not during repair) and outcomes (e.g., life loss due to aftershock-induced collapse of a non-evacuated building). This two-stage aspect of the problem appears to make it a so-called “dynamic programming” problem, for which there exists a body of literature and algorithms. It is our challenge to try to capture the essence of that view of the pre-quake assessment and decision PBEE problem in a conceptual form analogous to the PEER framing equation in order to communicate the essence of the problem and its “solution”. A second step will be then to seek closed form or compact analytical representations of this formulation under simplifying analytical assumptions about the inputs. A third step will be to analyze a specific structural problem using these tools. This will involve first determining (as now the likelihood of different earthquake events and their consequences (damage state). Then for each damage state one would seek, for example, the collapse capacity with respect to future aftershocks and the likelihood of the aftershock-induced collapse event as well as its consequences (which both depend on the post-quake decisions deemed optimal at that time). We shall enter this structural analysis phase with the experience gained in the Lifelines program about the effect of main-shock damage on aftershock capacity. That work (both earlier nonlinear static based, and recent nonlinear dynamic based being conducted by Luco and Bazzurro AIR for PEER (13 WCEE paper accepted for publication) and the Building fragilities project) remains in its infancy. We shall select a moderately sized structure (a testbed?) upon which to develop and apply the concepts and tools. A theme throughout will also be assessment of the “value of information”. For example, this PBEE formulation will permit one to directly address questions such as: should I invest in instrumentation near or even in my building to improve my state of knowledge (reduce my epistemic uncertainty) about the actual state of damage and capacity after the main shock. What is the level of future possible benefit of such information and is it worth it to invest in it today? The outcome of such an analysis may help convince building owners to invest in instrumentation voluntarily (as the city of San Francisco is trying to encourage them to do.)

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

The project has developed an analogue to the PEER Framing equation to describe the aftershock problem and its similarities and differences to the more familiar problem (Fig. 1).

Figure 1. Relative to the familiar PEER Framing Equation this New Aftershock Framing Equation for the T year Interval after the Mainshock contains a Conditioning on the Magnitude of the Mainshock, MS, a Conditioning on the State of Damage Due to the Mainshock, S_i and an Implicit Treatment of the Time-Varying Nature of the Aftershocks Via, _T_a a New Concept of “Equivalent Constant Rate” of Occurrence of Aftershocks

This was presented in an Annual Meeting poster. We developed two formal solution approaches to the expected cost analysis problem. One is based on an extension to the non-homogeneous (aftershock regime) of continuous time Markov chain decision analysis (a pioneered by Prof. Ronald Howard) and a second is a homogeneous, dynamic programming formulation applied step-wise to a large number of homogeneous finite time intervals over the period of significant aftershock activity. In both cases states represent levels of damage. This solution approach was suggested in consultation with Prof. Peter Vinott of Stanford’s MS&E Department. The former is a more direct formulation but the latter proves to be computational more efficient and open to richer generalizations of the model, e.g., to random duration of the repair and hence random downtimes. We have implemented and tested these procedures against each other and against much simpler known solutions (Esteva, Wen, etc.) for main shocks. The next efforts will be in documentation and demonstration.

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

While we aware of nothing similar to the general goals, there has been some work reported recently by several Italians using one or two records to look at dynamic response of SDOFs and one MDOF model damaged by a main shock. We are aware of efforts by Prof. Stojadinovic (UCB) to look at the static load-carrying capability of bridges after main shock damage as a measure for evaluating them for “tagging”. This is analogous to previous work by Prof. Deierlein (Stanford) studying post-quake building static-load stability for this purpose. In our case, we dynamically evaluate the structure for aftershock capacity. Ned Field (USGS Pasadena) is implementing through SCEC an aftershock PSHA in his Open-SHA software.

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

Year two will address procedures to implement this paradigm and applications. One example may be use of this formulation in the context of making decisions about benefits vs. costs of various levels of investment in instrumentation (in the building, near the building, or locally (e.g., ShakeMap).

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

The results of the predecessor work in the Lifeline Building Fragilities project are being applied by consultants to PG&E, the local utility, as described in a “nugget” of the PEER annual report.

8. Expected Milestones & Deliverables:

In year one (2003-4) the milestones are an ”updated” Framing Equation that reflects the pre-main shock assessments of the (random) potential future damage states (as now) and also the aftershock environments and decisions made then with (hopefully explicit) collapse probability equations for the combined pre/post assessment problem based on life-safety (collapse limit state) considerations and (2) a generalized pre-quake (collapse limit state and expected cost) analysis formulation that includes cost/benefit (e.g., downtime; inspection) in the post-quake decision phase.

Year 1:

Deliverable: A paper/report on the formulation of and illustrations of post-quake decision-making assessments (e.g., tagging, degree of inspection, etc.).

Year 2:

Deliverables: Papers and a report on this more general PBEE formulation including analytical developments, treatment of uncertainty and value of information, and worked illustrations.