# Seismic Response Mitigation of Base-Isolated Buildings
## 5 Numerical Study
### Seismic Fragility Analysis  
Seismic fragility is defined as the probability of exceeding a defined damage state due to a given seismic intensity measure (IM). It happens when the seismic applied load (demand) is greater than the structural capacity. Seismic fragility can be expresses as,  
\[\text{Fragility}\ \approx\text{P}[\text{Demand}>\text{Capacity}] \tag{18}\]  
For the fragility analysis of the buildings, following two steps are considered: (1) A failure criteria is defined and (2) a set of seismic intensity measures are selected. Here, the failure criterion is considered as maximum isolator displacement, which ensures the safety of isolation system. In this study, the collapse state of damage is based on limit states proposed by [PERSON] et al. (2018). They proposed a limit state for isolator displacement for elastic, elastic-plastic, and plastic states. The maximum target displacement assuming the plastic state is considered 330 mm in this study. Once the target displacement is specified, the analysis of BI buildings equipped with different TMD schemes using the set of 40 earthquakes, each scaled 80 times is carried out. The peak absolute isolator displacement is recorded. The recorded peak response of the BI building with and without TMD schemes for the generated number of ground motions (N\({}_{\text{gen}}\)) are then compared with the corresponding seismic demands. Thereafter, the probability of failure \(P_{f}\) at any certain PGA level is defined as following  
\[P_{\text{f}}=\frac{N_{\text{fail}}}{N_{\text{gen}}} \tag{19}\]  
where N\({}_{\text{fail}}\) denotes the number of the cases which satisfies Equation (18). Then the process is reiterated for the range of the above-mentioned PGA levels, and the respective fragility curves are achieved assuming a lognormal distribution. The procedure of obtaining the fragility curves for the BI building equipped with the TMD schemes is schematically described in Figure 9. The bearing displacement fragility curves for 5-story BI building equipped with TMDs is depicted in Figure 10.  
Figure 8: Cumulative probability distribution functions of maximum bearing displacement, peak top floor acceleration, and normalized shear force for 10-story building.  
The TMD schemes reduce the probability of the considered damage state by about 10% across a range of PGA. The MTMDs and d-MTMDs are found to perform slightly better than the STMDs.  
Figure 10: Bearing displacement simplified fragility curves for 5-story base-isolated building equipped with TMDs.  
Figure 9: Procedure for seismic fragility analysis of base-isolated building equipped with TMD schemes.  
In addition, similar trend of response reduction was observed for the case of 10-story BI building equipped with TMDs and the results are shown in Figure 11. Since these fragility curves represent uncertainty in ground motion intensity and frequency content, while the structure is assumed to be deterministic, the resulting fragility curves are called as simplified fragility curves in the sense that they might be different from real fragilities of a structure with uncertain parameters. There is future scope for experimental verification of the concept like reference [45].