## 4 Fragility Analysis Based on IDA Method
### Incremental Dynamic Analysis (IDA)
For the same seismic record, the slope of the IDA curve for the hybrid isolation structure is steeper than that for the base-isolated structure. This indicates that for the same seismic intensity S\({}_{\text{a}}\) (T,5%), the shear strain in the base-isolated structure is greater than that in the hybrid isolation structure, and the increment in TP value is also larger for the base-isolated structure. This suggests that the hybrid isolation structure provides better control over the horizontal displacement and tensile stress in the isolation bearings, reducing the likelihood of structural overturning during an earthquake and enhancing the overall seismic performance. A single IDA curve cannot fully capture the impact of different seismic records on the seismic performance of the two isolation structures. By summarizing the IDA curves from 10 different seismic records for each structure, we obtain IDA curve clusters, as shown in Figures 12 and 13. Due to the randomness and variability of seismic motions, the DM values are relatively small, and the differences between the curves are minimal when the seismic intensity S\({}_{\text{a}}\) (T,5%) is low, resulting in a more concentrated IDA curve cluster. As the seismic intensity S\({}_{\text{a}}\) (T,5%) increases, the variability among the IDA curves becomes more pronounced. To mitigate the influence of data variability on the final analysis, a quantile  
Figure 11: IDA curve comparison between isolation structure and hybrid isolation structure: (**a**) \(\gamma\); (**b**) LRB bilinear model; (**c**) TP. regression statistical method is applied to the IDA data, yielding three quantile curves at the 16%, 50%, and 84% levels. This approach facilitates the identification of statistical patterns in the data. To quantitatively compare the anti-overturning capacity of the base-isolated structure and the hybrid isolation structure, the spectral acceleration Sa (T,5%) corresponding to the 50% quantile curves under different limit states was recorded, as shown in Table 6. The results indicate that both shear strain and TP value increase synchronously with the rise in seismic intensity. Furthermore, the hybrid isolation structure equipped with viscous dampers requires higher seismic intensity to reach the same limit state under different seismic excitations compared with the base-isolated structure. When shear strain \(\gamma\) is used as the DM, the maximum increase is 27.1%, and when TP is used as the DM, the maximum increase is 30.4%. This is because, during seismic excitation, the dampers in the hybrid isolation structure dissipate a portion of the energy absorbed by the isolation layer, resulting in smaller horizontal displacements and tensile stresses in the isolation bearings. Consequently, the anti-overturning capacity of the structure is significantly enhanced. Figure 12: IDA curve clusters for shear strain \(\gamma\) of isolation bearings in base-isolated and hybrid isolation structures: (**a**) Base-isolated structure. (**b**) Hybrid isolation structure. Figure 13: IDA curve clusters for TP value of isolation bearings in base-isolated and hybrid isolation structures: (**a**) Base-isolated structure. (**b**) Hybrid isolation structure.