# Efficient Seismic Fragility Analysis for Large-Scale Piping System Utilizing Bayesian Approach
## 5 Results and Discussions
These numerical example results are a single case study of the efficiency of the proposed method compared to the current MLE approach, and the specific values related to this example may vary depending on the degree of segmentation in the seismic intensities, the number of the earthquake ground motion records, and so on. However, unlike the MLE approach, the proposed method can minimize the number of seismic fragility data required by utilizing the prior probability distribution; thus, this can be expected to achieve the efficiency in most case studies relatively compared to the current approach. The overall process of the seismic fragility assessment applied to the limit state of T104 in the piping system described above was similarly applied to the remaining seismically vulnerable piping components (i.e., T114, T125, and T135). The obtained piping seismic fragility curve results for these components are presented in Figures 10-12, respectively. Moreover, the quantitative indicators related to the accuracy comparison of the piping seismic fragility curves for T114, T125, and T135, and the total number of nonlinear seismic response analyses required are summarized in Table 1 together with the results regarding T104. As can be seen from the figures and the table, the proposed method applied to T114, T125, and T135 also can be confirmed to be more efficient in that it obtains piping seismic fragility curves with similar accuracy while reducing the numerical cost of nonlinear seismic response analyses. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Fragile Location** & \multicolumn{2}{c}{**T104**} & \multicolumn{2}{c}{**T114**} & \multicolumn{2}{c}{**T125**} & \multicolumn{2}{c}{**T135**} \\
**Method** & **MLE** & **Proposed** & **MLE** & **Proposed** & **MLE** & **Proposed** & **MLE** & **Proposed** \\ \hline
**\# of nonlinear seismic analyses** & 231 (+11\(\times\) 2) & 43 (+1 + 2 \(\times\) 2) & 231 (+11\(\times\) 2) & 43 (+1 + 2 \(\times\) 2) & 231 (+11\(\times\) 2) & 43 (+1 + 2 \(\times\) 2) & 231 (+11\(\times\) 2) & 43 (+1 + 2 \(\times\) 2) \\
**R\({}^{2}\)** & 0.986 & 0.984 & 0.987 & 0.990 & 0.990 & 0.991 & 0.993 & 0.994 \\
**RMSE** & 0.045 & 0.049 & 0.039 & 0.033 & 0.036 & 0.036 & 0.030 & 0.036 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of the accuracy and the numerical cost for the piping seismic fragility curves according to two methods, i.e., incremental dynamic analysis (IDA)-based maximum likelihood estimation (MLE) method and the proposed method (Bayesian approach with the EPRI SOV method). Figure 8: Distribution of main parameters for piping seismic fragility curve (regarding the limit state of T104): (**a**) a prior probability distribution and (**b**) a posterior probability distribution. Figure 10: Piping seismic fragility curve regarding a limit state of seismically vulnerable T-joint T114. Figure 9: The posterior distribution of main parameters for piping seismic fragility curve (regarding the limit state of T104), and the related Markov Chain Monte Carlo (MCMC) sampling process.