# Seismic Vulnerability Assessment of Hybrid Mold Transformer Based on Dynamic Analyses
## 4 Dynamic Analyses and Development of Seismic Fragility Curves
### Fragility Curves and Discussions  
The peak response results obtained from the analytical model of the mold transformer are illustrated in Figure 17 in terms of the different parameters represented for damage states: the peak response acceleration at the coils in the Y-direction (Figure 17a), maximum displacement at the top beam in the Y-direction (Figure 17b), and maximum displacement at the bottom beam in the Z-direction (Figure 17c). As the peak ground acceleration increases, the maximum response from the analytical model increases as well, indicating increased physical damage to the mold transformer. Moreover, the limit values corresponding to the damage states for each performance level are presented in Figure 17. In order to establish the fragility curves, these analytical results provide the fraction of the analyses that lead to the exceedance of limit states corresponding to specific performance levels. In total, 1120 cases were analyzed with consideration of the variety of earthquake ground motion characteristics and uncertainty in modeling parameters to establish the fragility curves of the hybrid mold transformers. Figure 15: Critical damage states observed during the shaking table tests. To further elucidate the effects of uncertain modeling parameters on the performance of the mold transformer in an earthquake, Figure 18 presents the characteristics of the Friuli_Italy-01 Earthquake used for dynamic analysis in terms of the acceleration time histories in the Y and Z-directions (Figure 18a,c) and corresponding FT results in the frequency domain (Figure 18b,d). In both directions, at least four resonance frequencies could be clearly observed in the frequency domain, from 1.67 Hz to 19.3 Hz in the Y-direction. Figure 19 shows the effect of the mass coil, M\({}_{c}\), on the main responses of the transformer: peak response acceleration at the coil in the Y-direction, maximum displacement at the top beam in the Y-direction, and maximum displacement at the bottom beam in the Z-direction. In general, the peak response acceleration at the coil and the maximum displacement at the top beam in the Y-direction showed a nonlinear increase when the PGA was greater than 1.0 g, while the maximum displacement at the bottom beam in the Z-direction showed a linear increasing trend. Moreover, when the coil mass varied in the range of 0.6 to 1.2 M\({}_{c}\), at the PGA of 2.0 g, the peak acceleration response in the Y-direction varied in the range of 17 %, the maximum displacement at the top beam in the Y-direction varied in the range of 15%, and the maximum displacement at the bottom beam in the Z-direction varied in the range of 30%. In Figure 20, the effect of the translational stiffness, K\({}_{t}\), on the \begin{table}
\begin{tabular}{c c c} \hline
**Damage States** & **Performance Levels** & **Parameter of Limit States** & **Value of Limit States** \\ \hline DS1 & OP & PRA of coils in Y-direction (g) & 0.22 \\  & PR & 0.69 \\ DS2 & OP & Maximum displacement at the top & 50 \\  & PR & beam in Y-direction (mm) & 75 \\  & OP & Maximum displacement at the & 2.25 \\ DS3 & PR & bottom beam in Z-direction (mm) & 6.6 \\  & LS & 12.45 \\ \hline \end{tabular}
\end{table} 
Table 6: Values of limit states of the mold transformer. Figure 16: Correlation between damage states and performance levels of the mold transformer. main parameters was also presented. As shown in Figure 19(a), the effective translational stiffness has a great influence on the peak acceleration response. For example, at the PGA of 2.0, the peak acceleration response in the Y-direction in the case of 1.0 \(K_{t}\) was about 50% lower than that of 0.8 \(K_{t}\) and about 30% greater than that of 1.2 \(K_{t}\). Meanwhile, the effective translational stiffness had a negligible influence on the displacement in the Y-direction and the Z-direction (Figure 19(b),c). Moreover, in all cases, the effect of the rotational stiffness in the range of 0.8 to 1.2 \(K_{\theta}\) on parameters was also negligible.