## 3 One-Dimensional Seismic Fragility Analysis
### The Seismic Fragility Curves of Single Components  
The IDA method was first proposed by [PERSON] in 1977 [40], and is based on a sufficient number of dynamic time-history analyses that can reflect the seismic performance of structures under different ground motions of varying intensities in order to provide \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Components**} & **Damage** & **Slight** & **Moderate** & **Serious** & **Complete** \\  & **Indexes** & **Damage** & **Damage** & **Damage** & **Damage** \\ \hline Plate rubber bearing (m) & \(\mu\) & 0.050 & 0.075 & 0.100 & 0.125 \\ Pier & \(\Delta\) & 1.000 & 1.322 & 2.133 & 3.000 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The seismic thresholds of the damage indexes under different performance limit states. comprehensive and realistic evaluation of structural seismic performance [12; 16; 19]. This method has become widely employed in performance-based seismic engineering thanks to improvements in computational capability and structural analysis software. Therefore, the IDA method was employed to simulate 300 groups of bridge-ground motion analysis sample pairs based on OpenSees. The peak displacement \(\Delta\) of the plate rubber bearing and the peak displacement ductility ratio \(\mu\) of the pier were calculated. The statistical information is shown in Table 6.  
The seismic demand \(\mu_{d}\) of the structure or components is assumed to conform to logarithmic normal distribution [35], and the relationship between \(\mu_{d}\) and IM should satisfy the following formula:  
\[u_{d}=x\;\text{IM}^{y} \tag{10}\]  
Taking the logarithm on both sides, we obtain  
\[\ln(u_{d})=\ln(x)+y\ln(\text{IM}) \tag{11}\]  
The fitting regression equations between \(\mu_{d}\) and PGA are established by employing statistical regression analysis, as shown in Figure 7. The mean standard deviation \(\beta_{d}\) of the fitting regression equations can be calculated according to Formula (12). A smaller \(\beta_{d}\) indicates a better linear relationship, which is more suitable for the structural fragility evaluation index:  
\[\beta_{d}=\sqrt{\frac{\sum_{i=1}^{n}[\ln(\text{DM})-\left(\ln x+y\ln(\text{IM} )\right)]^{2}}{n-2}} \tag{12}\]  
When the fitting regression equations of the plate rubber bearing and pier are substituted into Formula (8), the following formulas for calculating the exceedance probability of single component under different performance limit states are further obtained:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Plate Rubber Bearing**} & \multicolumn{3}{c}{**Pier**} \\ \cline{2-7}
**PGA (g)** & \begin{tabular}{c} **Mean of** \\ **Logarithm** \\ \end{tabular}  & \begin{tabular}{c} **Logarithmic** \\ **Standard** \\ **Deviation** \\ \end{tabular}  & \begin{tabular}{c} **Coefficient of** \\ **Variation** \\ \end{tabular}  & \begin{tabular}{c} **Mean of** \\ **Logarithm** \\ \end{tabular}  & \ \begin{tabular}{c} **Logarithmic** \\ **Standard** \\ **Deviation** \\ \end{tabular}  & \begin{tabular}{c} **Coefficient of** \\ **Variation** \\ \end{tabular}  \\ \hline
0.05 & \(-\)4.979 & 0.318 & \(-\)0.064 & \(-\)2.642 & 0.171 & \(-\)0.065 \\
0.10 & \(-\)4.189 & 0.234 & \(-\)0.056 & \(-\)1.920 & 0.