## 4 Fragility Analysis Based on IDA Method
### Incremental Dynamic Analysis (IDA)  
In this study, IDA was conducted on both the base-isolated structure and the hybrid isolation structure by subjecting them to 10 selected ground motion records. Nonlinear time-history analysis was performed, and the resulting IDA curves were compared and analyzed. The IDA curves in this study are plotted with the DMs TP and \(\gamma\) as the horizontal axis and the IM S\({}_{\text{a}}\) (T.5%) as the vertical axis. Figure 11 presents a comparative analysis of IDA curves between the base-isolated structure and the hybrid isolation structure, using seismic record 10 as the excitation input for the computational model. The comparison is illustrated through two different DMs: shear strain \(\gamma\) and TP value. As depicted in the figure, the IDA curves for both structures exhibit similar trends, with both shear strain \(\gamma\) and TP value showing a positive correlation \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Destructive State** & **Description of Destruction State** & **Value of Destruction State** \\ \hline Basically Intact (DS1) & The load-bearing components are intact; minor damage to individual non-load-bearing components; varying degrees of damage to accessory components. & \(\leq\)LS1 \\  & Some load-bearing components have slight cracks, while & \\ Slight Damage (DS2) & some non-load-bearing components have obvious damage; & (LS1, LS2) \\  & the auxiliary components have varying degrees of damage. & \\  & Most load-bearing components have slight cracks, with & \\ Moderate Damage (DS3) & some showing obvious cracks; serious damage to & (LS2, LS3) \\  & individual non-load-bearing components. & \\ Severe Damage (DS4) & Most load-bearing components are severely damaged or & (LS3, LS4) \\  & partially collapsed. & \\ Destruction (DS5) & Most load-bearing components have collapsed. & \(\geq\)LS4 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Classification of damage levels. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Limit State** & **LS1** & **LS2** & **LS3** & **LS4** \\ \hline TP & 2.5 Mpa & 5 Mpa & 6 Mpa & 6.5 Mpa \\ \(\gamma\) & 50\% & 130\% & 300\% & 400\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Limit values for extreme states. with the spectral acceleration S\({}_{\text{a}}\) (T,5%) of the seismic record. However, the growth trends of the IDA curves differ based on the chosen DM. In Figure 11a, the gray-shaded area indicates that the slope of the curve changes with the increase in shear strain \(\gamma\). When the spectral acceleration S\({}_{\text{a}}\) (T,5%) is relatively low, resulting in smaller shear strains, the IDA curve exhibits a steeper slope, indicating that the rubber bearings are in the elastic phase with pre-yield stiffness. As S\({}_{\text{a}}\) (T,5%) increases, leading to larger shear strains, the slope of the IDA curve becomes gentler, reflecting the post-yield stiffness of the rubber bearings. This behavior aligns with the bilinear model of the LRB in the finite element model shown in Figure 11b, where the post-yield stiffness is less than the pre-yield stiffness. In Figure 11c, the gray-shaded area reveals an inflection point in the IDA curve around a TP value of 5 MPa, where the slope of the curve increases. This arises because, according to the definition of the TP value, when the TP value exceeds 5 MPa, the tensile stress T in the isolation bearing is positive, indicating that the bearing is in tension. Conversely, when the TP value is below 5 MPa, the tensile stress T is negative, indicating compression. The finite element model used in this study assigns different stiffness values to the isolation bearings under tension and compression, with the tensile stiffness being one-tenth of the compressive stiffness. Consequently, for the same increase in seismic intensity, the increment in TP value is greater when the structure is under compression compared with when it is under tension.