# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 2 Current Methods
### EPRI SOV Method  
The seismic fragility curve of the log-normal distribution type described above can be expressed as a more detailed distribution function as follows by separately considering the randomness and the uncertainty of the input earthquake ground motion and structural model [21]. Figure 2: Process used for piping seismic fragility evaluation of incremental dynamic analysis (IDA)-based maximum likelihood estimation (MLE) method. \[P_{f}(a)=\Phi\Bigg{[}\frac{\ln(a/A_{m})-\beta_{U}\Phi^{-1}(Q)}{\beta_{R}}\Bigg{]} \tag{4}\]  
Here, \(\beta_{R}\) and \(\beta_{U}\) are the logarithmic standard deviations indicating the randomness and the uncertainty, respectively. The parameter \(Q\) can be expressed as \(P\) [\(f<\)\(f^{\prime}\)] a], which is a subjective probability (or reliability), meaning the probability that the conditional failure probability \(f\) is less than \(f^{\prime}\) at a particular seismic intensity \(a\). The log-standard deviations of the randomness and uncertainty presented in Equation (4) can be combined into a single variable, which can be expressed as Equation (5). Based on this combined log-standard deviation, Equation (4) can be rearranged and represented as Equation (6), which finally takes the same form as Equation (3). \[\beta_{C}=\sqrt{\beta_{R}^{2}+\beta_{U}^{2}} \tag{5}\]  
\[P_{f}(a)=\Phi\Bigg{[}\frac{\ln(a/A_{m})}{\beta_{C}}\Bigg{]} \tag{6}\]  
The parameter \(A_{m}\) refers to the median seismic performance value (i.e., the seismic intensity when the probability of failure is 50%) as described above, and EPRI defines it as follows by introducing the concept of the safety factor. \[A_{m}=F\cdot A_{DBE} \tag{7}\]  
The parameter \(A_{DBE}\) is the seismic intensity of the design basis earthquake. The parameter \(F\) implies the safety factor of the target equipment/components/structures/systems, and this indicates the degree of margin resisting above the design basis earthquake. With regard to the piping system, this safety factor \(F\) can be further subdivided as follows, if necessary. \[F=F_{C}\cdot F_{RP}\cdot F_{RS} \tag{8}\]  
\[\beta_{R}=\sqrt{\beta_{R,C}^{2}+\beta_{R,RP}^{2}+\beta_{R,RS}^{2}} \tag{9}\]  
\[\beta_{U}=\sqrt{\beta_{U,C}^{2}+\beta_{UR,RP}^{2}+\beta_{UR,RS}^{2}} \tag{10}\]  
Here, FC refers to the capacity factor of the piping system, which is a factor indicating how much margin is greater than the defined design limit state. Especially, \(F_{C}\) can be further defined as \(F_{S}\times F_{\mu}\). The \(F_{S}\) can express design margins due to design stress (strength), and \(F_{\mu}\) can take into account design margins due to the nonlinear and inelastic behavior of structural systems. The mathematical basis of this combination is based on the feature that the median of the products of independent random variables following log-normal distributions is equal to the product of the medians of random variables [18]. Therefore, the log-normal cumulative probability distribution function, which represents the final seismic fragility, can include the nonlinear and inelastic effect and the other effects of various variables. Such an approach is in line with the way seismic fragility analysis is handled in the other studies [29; 30]. Specifically, an IDA performs nonlinear and inelastic seismic response numerical analyses according to the increase in seismic intensity in consideration of the randomness and the uncertainty of the earthquake and the structural system, nonlinear and inelastic behavior of the structural system, and so on. Based on the results of these IDA analyses, the key parameters of the log-normal cumulative distribution such as \(A_{m}\) and \(\beta\) for representing the seismic fragility curve is estimated using the statistical approach. That is, the general seismic fragility analysis approach of the structural engineering treats the nonlinear and inelastic behavior effects of the structure including various other effects through a log-normal distribution function, similar to the EPRI SOV seismic fragility analysis method.