## 4 Analysis of the Relationship Between the Number of Input Ground Motions and Seismic Fragility
### Example Facility: Six-Story Commercial Building
When 10 ground motions were selected and the seismic fragility curve was derived (repeated 100,000 times), the median ranged from 0.14 g to 0.79 g, with an average of 0.31 g, representing a 2.06% difference from the reference value. Additionally, the reference value is 0.72 for the log-standard deviation, with a minimum of 0.15, a maximum of 1.30, and an average of 0.68, resulting in a 6.43% difference from the reference value. Figure 4: \(Cont.\)  
The left graph in Figure 4 presents the results of the seismic fragility analysis, with the solid line representing the median of the fragility curve calculated after 100,000 repetitions. The right graph displays the frequency distribution and probability density function for the mean (top right graph) and the log-standard deviation (bottom right graph) of the results; the red solid line represents the reference value, the blue solid line represents the mean, and the black solid line represents the results after 100,000 repetitions. Seismic fragility curves were calculated for the 16%, 50%, and 84% levels using the results from selecting 10 ground motions at random based on the values presented in Table 5 and summarized at Table 6.  
For Performance Level 1, the 16% fragility level ranged from 0.04 g to 0.68 g, revealing a variation of over 17 times (Table 5). At the 84% fragility level, it ranged from 0.17 g to 2.90 g, exhibiting a variation of over 17 times. These differences can cause overestimation or underestimation when using fragility curve values for further analysis, such as vulnerability assessments of target facilities, cost evaluations for earthquake damage, or seismic performance analyses of structures. Figure 4: Seismic fragility curves and histograms of the six-story building (N = 10). (**a**) Performance Level 1; (**b**) Performance Level 2; (**c**) Performance Level 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Performance Level**} & \multicolumn{4}{c}{**Median (g)**} & \multirow{2}{*}{**Difference**} \\ \cline{2-2} \cline{4-6}  & **Ref** & **Min** & **Max** & **Avg** & **(\%)** \\ \hline
1—IO & 0.3000 & 0.1442 & 0.7947 & 0.3062 & 2.0638 \\
2—LS & 0.9638 & 0.4782 & 2.4424 & 0.9850 & 2.2061 \\
3—CP & 1.0148 & 0.4944 & 2.5765 & 1.0371 & 2.1951 \\ \hline \multirow{2}{*}{Level} & \multicolumn{4}{c}{Log-standard deviation} & \multirow{2}{*}{Difference} \\ \cline{2-2} \cline{4-6}  & Ref & Min & Max & Avg & (\%) \\ \hline
1—IO & 0.7237 & 0.1542 & 1.3016 & 0.6771 & 6.4377 \\
2—LS & 0.7430 & 0.1387 & 1.4723 & 0.7059 & 4.9873 \\
3—CP & 0.7471 & 0.1421 & 1.4444 & 0.7107 & 4.8720 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Seismic fragility curve parameters range of the six-story building (N = 10). Figure 5: EDS spectrum analysis of 17-4 PH (**a**), 316L (**b**), and 304 (**c**). Figure 6: _Cont._  
As the number of considered ground motions gradually increases, the frequency distribution becomes a smoother bell-shaped curve (Figures 4-6), indicating that the distribution of the seismic fragility input variables tends to follow a normal distribution. Additionally, the range of minimum and maximum values for each performance level decreases, implying that the input variables of the fragility curve converge toward the reference \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Performance Level**} & \multicolumn{4}{c}{**Median (g)**} & \multirow{2}{*}{**Difference**} \\ \cline{2-2} \cline{4-6}  & **Ref** & **Min** & **Max** & **Avg** & **(\%)** \\ \hline
1—IO & 0.3000 & 0.1946 & 0.4747 & 0.3015 & 0.4959 \\
2—LS & 0.9638 & 0.6463 & 1.5458 & 0.9689 & 0.5290 \\
3—CP & 1.0148 & 0.6802 & 1.5781 & 1.0201 & 0.5158 \\ \hline \multirow{2}{*}{Performance Level} & \multicolumn{4}{c}{Log-standard deviation} & Difference \\ \cline{2-6}  & Ref & Min & Max & Avg & (\%) \\ \hline
1—IO & 0.7237 & 0.3770 & 1.0909 & 0.7112 & 1.7229 \\
2—LS & 0.7430 & 0.3989 & 1.1347 & 0.7338 & 1.2345 \\
3—CP & 0.7471 & 0.4493 & 1.1461 & 0.7381 & 1.2081 \\ \hline \hline \end{tabular}
\end{table}