## 4 Fragility Analysis Based on IDA Method
### Seismic Fragility Analysis  
Seismic fragility reflects the probability of structural damage exceeding a specified limit state under varying seismic intensity. It serves as a critical metric for evaluating the safety of structures at specific seismic intensity levels and their inherent capacity to withstand seismic actions. This analysis is of significant importance for developing reliability-based seismic design codes. The probability value can be calculated by using Formula (7):  
\[P_{\text{f}}=\text{P}(LS/IM)=\text{P}(\text{C}<\text{D}|\text{IM}) \tag{7}\]  
The seismic capacity parameter C and the seismic demand parameter D are assumed to follow a lognormal distribution. Under this assumption, the failure probability can be formulated as  
\[P_{\text{f}}=\phi\left(\frac{\ln(-\hat{C}/\hat{D})}{\sqrt{\beta_{\text{c}}^{2 }+\beta_{\text{d}}^{2}}}\right)=\phi\left(\frac{\ln(a*Sa(T,5\%)^{b}/\hat{C} )}{\sqrt{\beta_{\text{c}}^{2}+\beta_{\text{d}}^{2}}}\right) \tag{8}\]  
In this context, \(\hat{C}\) and \(\hat{D}\) correspond to the median values of the seismic capacity parameter and the seismic demand parameter, respectively. Similarly, \(\beta_{c}\) and \(\beta_{d}\) represent the logarithmic standard deviations associated with \(C\) and \(D\). According to HAZUS99, \begin{table}
\begin{tabular}{c c c c c c} \hline
**DM** & **Structure** & **LS1** & **LS2** & **LS3** & **LS4** \\ \hline \multirow{3}{*}{\(\gamma\)} & Isolated structure & 0.1117 g & 0.2367 g & 0.45 g & 0.5707 g \\  & Hybrid isolation structure & 0.1341 g & 0.307 g & 0.6098 g & 0.7827 g \\  & Growth rate & 16.7\% & 22.9\% & 26.2\% & 27.1\% \\ \hline \multirow{3}{*}{TP} & Isolated structure & 0.109 g & 0.199 g & 0.391 g & 0.458 g \\  & Hybrid isolation structure & 0.124 g & 0.286 g & 0.473 g & 0.552 g \\  & Growth rate & 12.0\% & 30.4\% & 17.3\% & 17.0\% \\ \hline \end{tabular}
\end{table} 
Table 6: Comparison of Sa (T,5%) values for different limit states of 50% quantile curves. \begin{table}
\begin{tabular}{c c} \hline
**Structure** & **Probabilistic Seismic Demand Model** \\ \hline Isolated structure & \(\ln(\gamma)=1.971+1.208\ln(Sa(T,5\%))\) \\ Hybrid isolation structure & \(\ln(\gamma)=1.758+1.306\ln(Sa(T,5\%))\) \\ Isolated structure & \(\ln(\gamma)=2.388+0.723\ln(Sa(T,5\%))\) \\ Hybrid isolation structure & \(\ln(\gamma)=2.210+0.617\ln(Sa(T,5\%))\) \\ \hline \end{tabular}
\end{table} 
Table 7: Probabilistic seismic demand model. when defining the IM as Sa, the parameter \(\sqrt{\beta_{c}^{2}+\beta_{\rm d}^{2}}\) is typically assigned a value of 0.4 [56]. Additionally, \(\phi(*)\) denotes the mathematical expression of the standard normal distribution function. \[\phi(x)=\frac{1}{\sqrt{2\pi}}{\int_{-\infty}^{x}\exp\biggl{(}-\frac{t^{2}}{2} \biggr{)}}dt \tag{9}\]  
The exceedance probabilities of structural damage measure reaching various limit states under different seismic intensities are determined by substituting the data from Table 7 into Formula (8). By connecting the data points corresponding to different limit states, the fragility curves are constructed. As demonstrated in Figure 14, this figure provides a comparative analysis of fragility curves for base-isolated structures and hybrid base-isolated structures, with shear strain and TP values acting as distinct DMs. Compared with seismic isolation structures, hybrid seismic isolation structures have smaller values on the vertical axis of the fragility curve under different ultimate states. This indicates that under identical seismic intensities, hybrid base-isolated structures have lower failure probabilities when their seismic isolator shear strain reaches the threshold, suggesting reduced fragility and a decreased likelihood of reaching critical failure states. Additionally, under the same limit state conditions, the disparity in failure probabilities between hybrid base-isolated structures and base-isolated structures widens as seismic intensity increases.