Analytic Solution of Seismic Probabilistic Risk Assessment

Probabilistic seismic risk assessments (PSRA) are used to quantify the seismic damage probability of complex engineering structures due to seismic events. In such systems, the seismic capacities of individual relevant components are determined by structure mechanics. A significant digits computation method to quantify the mean seismic failure probability (fragility) for given component capacity and seismic intensity is proposed. The overall system failure probability is then calculated using standard fault tree / event trees models. Most of the tools available to quantify such logical models implement established techniques based on the rare event approximation. However, for high failure probabilities as seen in PSRAs, the rare event approximation produces conservative results. A method based on binary decision diagrams (BDD) is proposed to quantify such seismic model analytically. A simplified, representative binary decision diagram model is developed in order to evaluate the impact of the rare event approximation on such models. This paper focuses on PSRAs in the nuclear industry. Capacity factors are derived from several sources of information including plant-specific design reports, test reports, generic earthquake experience data, and generic analytical derivations of capacity based on governing codes and standards. Both structural and functional failure modes are considered in developing capacity factors for equipment. 1.4 Component seismic fragility Fragility is defined as the conditional probability of failure for a given ground motion level. Fragilities are developed on a component-specific basis or for a group of similar component s, considering component location. A typical fragility curve and its associated uncertainty range is shown in Figure 2. Figure 2. Example of seismic fragility curve for a given component capacity (am=0.92), including 5 th percentile; median (50 percentile); 95 percentile; mean. 1.5 Correlation between failure modes Many of the potential failure modes of safety-related equipment are not considered to be completely independent. For instance, the collapse of a structure is also expected to result in failure of the equipment located in that structure. Some degree of response correlation exists for all items and for all modes of failure since they are all excited by the same seismic event. 1.6 Accident sequence modeling The event tree delineates the possible accident scenarios for the seismic event. The event tree functional logic is developed to address seismic-specific aspects of the analysis. The event trees are coupled with the system models through fault tree linking to develop a comprehensive, integrated model. 1.7 Accident sequence quantification The possible combinations of randomly and seismically induced failures are modeled using logical fault tree and event tree structures. Accident sequences are thus expressed in terms of Boolean equations or minimal cut sets, or prime implicants. Seismic risk quantification is performed for several discrete ground motion levels. Those discrete seismic risks are then summed up to obtain the overall seismic risk. 2 SIGNIFICANT DIGIT COMPUTATION OF THE FRAGILITIES 2.1 Governing equation The fragility curves are expressed in terms of probability of failure as a function of the sustained ground motion level. The variability for the fragility curves is represented by two parameters. These parameters take into account the inherent randomness of the capacity of a particular type of component (ßr) and the uncertainty in the median capacity (ßu). Such a formulation is important in order to separate the effects of randomness and uncertainty in the seismic risk. The fragility F(a,Q) is represented by the following equation for a given component capacity (am, ßu, ßr) (American Nuclear Society and National Institute of Electrical and Electronic Engineers 1983) (1):