## 3 One-Dimensional Seismic Fragility Analysis
### Correlation Coefficients of Vulnerable Components  
The diagram of the 2D joint performance limit state curves is shown in Figure 10. The elliptic curves represent the projection contour of the joint probability density function surface of the two evaluation indexes on the X-Y plane, and there are four performance limit state curves: Slight damage (S\({}_{1}\)), Moderate damage (S\({}_{2}\)), Serious damage (S\({}_{3}\)), and Complete damage (S\({}_{4}\)). When \(N=1\), the seismic capability of two components is assumed to be completely correlated, and the performance limit state surface is projected as a straight line on the X-Y plane. When \(N=\infty\), the seismic capacity of two components is assumed to be mutually independent, and the performance limit state surface is projected as two vertical lines on the X-Y plane. According to the shape of the 2D joint performance limit state curves, \(N\) directly affects the relationship between the performance limit state surface, and further influences the seismic fragility curves of the entire bridge. Therefore, the correlation between the seismic demand and the respective seismic capacities of the plate rubber bearing and the pier must be thoroughly investigated in order to obtain the fragility curves in accordance with the internal relationship of the structural seismic response. The Pearson correlation analysis method was used to calculate the correlation coefficients on the peak responses of the plate rubber bearing and the pier under different ground motions, as shown in Table 7. The relevant value \(\lambda\) can be obtained based on a fitting analysis. The results show that \(\lambda\) decreases as PGA increases. The bridge mainly performs the elastic response under ground motions with small PGA, and the \(\lambda\) between different components is massive. The entire bridge suffers nonlinear damage to a certain extent as PGA increases, and the \(\lambda\) between different components becomes smaller. According to Table 7, when the entire bridge is subjected to severe ground motions \(\lambda\) is approximately 0.8, indicating that the plate rubber bearing and the pier have significant relevance under severe ground motions. This is because the plate rubber bearing and the pier are connected directly, meaning that the seismic responses of the two components interact with each other. Due to the opposite correlation between \(N\) and \(\lambda\) based on the correlation analysis, in this paper \(N\) is taken as the reciprocal value of \(\lambda\). Considering the short-changing of \(\lambda\) under various ground motions with different PGA, all values of \(N\) under different performance limit states are chosen to be 1.215. According to the seismic thresholds and \(N\) between the plate rubber bearing and the pier under different performance limit states, the 2D joint performance limit state equations are as shown below:  
\[\left(\frac{\Delta_{LS}}{0.050}\right)^{1.215}+\frac{\mu_{LS}}{1.000}-1=0 \tag{21}\]  
\[\left(\frac{\Delta_{LS}}{0.075}\right)^{1.215}+\frac{\mu_{LS}}{1.322}-1=0 \tag{22}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**PGA (g)** & \
\begin{tabular}{c} **Calculation** \\ **Coefficient** \\ \end{tabular} & \(\lambda\) & \(N\) & **PGA (g)** &
\begin{tabular}{c} **Calculation** \\ **Coefficient** \\ \end{tabular} & \(\lambda\) & \(N\) \\ \hline
0.05 & 0.902 & 0.846 & 1.182 & 0.55 & 0.830 & 0.821 & 1.217 \\
0.10 & 0.881 & 0.844 & 1.185 & 0.60 & 0.846 & 0.819 & 1.221 \\
0.15 & 0.862 & 0.841 & 1.189 & 0.65 & 0.887 & 0.816 & 1.225 \\
0.20 & 0.856 & 0.839 & 1.192 & 0.70 & 0.830 & 0.814 & 1.228 \\
0.25 & 0.780 & 0.836 & 1.196 & 0.75 & 0.795 & 0.812 & 1.232 \\
0.30 & 0.726 & 0.834 & 1.199 & 0.80 & 0.781 & 0.809 & 1.236 \\
0.35 & 0.813 & 0.831 & 1.203 & 0.85 & 0.794 & 0.807 & 1.240 \\
0.40 & 0.833 & 0.829 & 1.206 & 0.90 & 0.800 & 0.804 & 1.244 \\
0.45 & 0.801 & 0.826 & 1.210 & 0.95 & 0.808 & 0.802 & 1.248 \\
0.50 & 0.806 & 0.824 & 1.214 & 1.00 & 0.823 & 0.799 & 1.251 \\ \hline \hline \end{tabular}
\end{table}