# Seismic Vulnerability Assessment of Hybrid Mold Transformer Based on Dynamic Analyses
## 4 Dynamic Analyses and Development of Seismic Fragility Curves
### Fragility Curves and Discussions
Figure 17: Variation of critical dynamic responses according to PGA obtained from analytical results. Figure 18: Characteristics of Friuli_Italy-01 earthquake (Y and Z-directions). Figures 21-23 present the effects of uncertainty in the modeling parameters on the fragility curves for the coil mass, effective translational stiffness, and rotational stiffness of the mold transformer. As shown in Figure 21, the variability of coil mass ultimately has little effect on the fragility curves corresponding to DS1; meanwhile, the fragility curves corresponding to DS2 and DS3 become flatter as the mass coil decreases. This is mainly attributed to the fact that the decrease of the mass coil leads to an increase in the natural time period, which leads to the decrease of deformation and finally the probability of exceeding limit states. Figure 22 shows that the variation of translational stiffness in the range of 0.8 to 1.2 \(K_{t}\) has almost no effect on the fragility curves corresponding to DS1 and DS2, but does have a significant effect on DS3. The increased translational stiffness significantly reduced the deformation in the Z-direction with the increase of PGA, leading to the decreased probability of exceeding limit states and the fragility curves becoming flatter. Similarly, Figure 23 indicates that the rotational stiffness has no effect on response acceleration and deformation in the Y-direction, leading to no change in the fragility curves corresponding to DS1 and DS2. Nonetheless, the change in rotational stiffness leads to slight variation of vertical deformation of the mold transformer, triggering the variation in the fragility curve with respect to DS3; however, this variation is not significant overall. In order to estimate the complete fragility curves of the mold transformers, all analytical cases were combined to consider both variability in PGA and uncertain modeling parameters. Accordingly, the values of the median (\(\theta\)) and the standard deviation (\(\beta\)) were evaluated to establish fragility curves for different performance levels corresponding to damage states, and these are summarized in Table 7. By using Equations (6) and (7), the fragility curves of the mold transformer were derived for various damage states according to different performance levels, and these were presented in Figure 24. In Figure 24, the data points represented the fraction of exceedance of limit states for specific performance levels calculated from dynamic analyses. In addition, Figure 25 illustrates the differences in the probability of exceedance between analytical data and the fitted model according to PGA of different damage states. Figure 19: Effect of coil mass variation on dynamic responses according to PGA. Figure 20: Effect of variation of effective translational stiffness on dynamic responses according to PGA. The critical PGA values corresponding to a specific probability of exceedance, which may play an important role in the seismic vulnerable evaluation; seismic design purposes were also determined from the fragility curves. Previous studies by [PERSON] et al. [44], [PERSON] et al. [45], and [PERSON] et al. [46] considered 50% probability of exceedance as the acceptance criteria. In this study, in order to consider the safety and the abundance of electrical transformer components in buildings, the value of 40% probability of exceedance was proposed as the acceptance criteria. Figure 24 also presents the critical PGA values corresponding to 40% probability of exceedance of the mold transformer. In detail, for the DS1 (Figure 24a), the 40% probability of exceedance corresponding to the OP and PR performance levels were predicted at PGA levels of 0.15 g and 0.4 g, respectively. For the DS2 (Figure 24b), the 40% probability of exceedance corresponding to the OP and PR performance levels were predicted at PGA levels of 0.3 g and 0.45 g, respectively.