# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 5 Results and Discussions
Figure 9 details the MCMC sampling process that satisfies this posterior probability distribution (red big dot: an initial sample, small black points: post-samples that come after the initial sample, dotted line: the trace of the extracted samples). We represented the final updated piping seismic fragility curve through the proposed method as a solid red line in Figure 7.  
To verify the piping seismic fragility analysis results of the proposed method, the IDA-based MLE method was applied to the same example piping system. For this purpose, dynamic nonlinear seismic response analyses were performed at 0.2 g intervals from 0.2 to 2.2 g seismic intensity, and the conditional failure probability information according to the defined seismic intensities was calculated based on these numerical results. The obtained conditional failure probabilities are shown discontinuously as dots in Figure 7. Based on the conditional probabilities of failure, the MLE technique was used to estimate \(A_{m}\) and \(\beta\), which were the main variables that define the seismic fragility curve of Equation (3), and the calculated seismic fragility curve based on the estimated variable values is represented in Figure 7 as a blue dashed line. As can be seen from Figure 7, it is confirmed that the IDA-based MLE method effectively captures the discrete failure probability data obtained from the fully nonlinear seismic response analysis approach. In addition, the results of Figure 7 show that the piping seismic fragility curves obtained through the proposed method and the IDA-based MLE method are quite similar to each other. The quantitative indicators related to the accuracy comparison of the seismic fragility curves and the total number of nonlinear seismic response analyses required for the two methods are summarized and presented in Table 1. The quantitative comparison of the accuracy of seismic fragility curve results was carried out by comparing the R2 and root mean squared error (RMSE) values. It should be noted that the IDA-based MLE method performed 231 (=\(11\times 21\)) nonlinear seismic response analyses regarding the 21 earthquake ground motions in the total 11 seismic intensities to calculate the final piping seismic fragility curve. On the other hand, the proposed method conducted a single seismic response analysis for the EPRI SOV method and additionally performed nonlinear seismic response analyses in the ground motion inputs of 0.6 g and 1.2 g seismic intensity, respectively; thus, a total of 43 (=\(1+2\times 21\)) numerical simulations were made. To be more specific, in the numerical example used in this study, the IDA-based MLE method, evenly divided the seismic intensities into 0.2 g intervals from 0.2 to 2.2 g and considered the uncertainty of the earthquake loadings to a total of 21 recorded ground motions; thus, this method finally made use of a total of 231 numerical analyses. On the other hand, the method proposed in this study used the EPRI SOV method to estimate the prior seismic fragility distribution in the first place. Here, the EPRI SOV method used only the results of a single numerical analysis regarding the design seismic load that satisfied the design criteria in order for estimating such seismic fragility distribution. The proposed method secured seismic fragility data only in the 0.6 g and 1.2 g seismic intensity where the probability of failure is sensitive, based on this prior seismic fragility distribution. Finally, this method calculated the final seismic fragility curve using the additional data and the Bayesian updating technique. Here, because the proposed method further considered each of the 21 ground motions at the two seismic intensities (i.e., 0.6 g and 1.2 g), the total number of final numerical analyses conducted was 43. In conclusion, the proposed method showed similar accuracy compared with the existing IDA-based MLE method in terms of seismic fragility results, but it reduced the numerical simulation cost of nonlinear seismic response analyses considerably.