# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 5 Results and Discussions  
As shown in Figure 6, the main locations of seismic fragility in the piping system are considered to be the T-joints of the 2-inch pipes (e.g., T114 and T135), where leakage was observed in the dynamic cyclic test of the overall piping system. Therefore, to begin with, the seismic response analysis wasperformed based on the numerical model of the building-piping coupled system, and the seismic fragility of the piping system was assessed by applying the EPRI SOV method to the seismic responses of T104. When applying the EPRI SOV method, it was assumed that \(A_{DBE}\), which is the seismic intensity of the design basis earthquake flowing into the foundation of the building, was set to be 0.2 g based on the seismic design regulations of the Korean building code. The seismic response analysis of the building-piping coupled numerical model subject to the ground motion corresponding to the \(A_{DBE}\) evaluated the maximum rotation angle response of T104 as 0.003 rad. If taking into account the calculated maximum seismic rotation angle response and 0.0135 rad, which is the limit rotation angle of the 2-inch-pipe T-joint, the safety factor \(F\) can be estimated as 4.5. Furthermore, if the uncertainty related to the piping system is assumed to be a log-standard deviation value of 0.3 using the existing piping seismic fragility research results [44], the final seismic fragility curve can be represented by the green dash-dotted line in Figure 7 (\(A_{m}\) = 0.9 g, \(\beta_{c}\) = 0.3). As described above, since the EPRI SOV method cannot capture the nonlinear dynamic seismic behavior of the piping system as the seismic intensity increases, it is difficult to derive an accurate seismic fragility curve. Accordingly, the proposed method was applied to the seismic fragility analysis of the example piping system. Per the main process of the proposed method (see Figure 3), the values of the main parameters of the seismic fragility curve derived from the EPRI SOV method were utilized as the prior probability distribution information for the final fragility calculation. Specifically, the obtained prior probability distribution information was as follows: \(A_{m}\)-LN (0.9, 0.3) and \(\beta_{c}\)-Uniform (0.85 \(\times\) 0.3, 1.15 \(\times\) 0.3). Here, LN (\(a\), \(b\)) means a log-normal probability distribution with median \(a\) and standard deviation \(b\), and Uniform (\(c\), \(d\)) represents a uniform probability distribution with constant values between \(c\) and \(d\). In addition, based on this prior distribution, additional dynamic nonlinear seismic response analyses were performed at seismic intensity 0.6 and 1.2 g for further improvement of the accuracy of the seismic fragility curve. Based on these results, conditional failure probabilities were calculated at the corresponding seismic intensities. Finally, based on the Bayesian inference concept [45], the prior probability distributions of the main parameters of the seismic fragility curve obtained from the EPRI SOV method were updated to the posterior probability distributions using newly acquired conditional failure probabilities. Figure 8 shows the posterior probability distribution  
Figure 7: Piping seismic fragility curve regarding a limit state of seismically vulnerable T-joint T104. of the main parameters of the seismic fragility curve of T104 in the example piping system by Bayesian inference compared with the prior probability distribution. Derivation of the main parameter values of the updated posterior probability distribution based on Bayesian inference was performed by using the Metropolis-Hastings algorithm among Markov Chain Monte Carlo (MCMC) sampling techniques [46]. The total number of samples used was 10,000, among which the initial samples (also called burn-in samples) that did not satisfy the posterior probability distribution were not reflected in the parameter value calculation for the final probability distribution.