# Seismic Vulnerability Assessment of Hybrid Mold Transformer Based on Dynamic Analyses
## 4 Dynamic Analyses and Development of Seismic Fragility Curves  
### Methodology and Process for Developing Seismic Fragility Curves  
The seismic fragility curve is a mathematical function that illustrates the relation between a given intensity measure (IM) for a seismic event and the probability that the response of the component exceeds the limit state where the IM occurs. In the present study, the seismic fragility curves of the mold transformer component were established using truncated incremental dynamic analysis (TIDA) [28]. This approach involves a set of accelerograms scaled to certain levels of seismic intensities until the component collapses. Then, time history analyses were performed for each intensity level, and the maximum response of each component was recorded. The main advantage of this method is that it reduces the number of structural analyses and computational efforts. The uncertainties in the modeling components are considered by using a set of corresponding parameters after [PERSON] et al. [29]. Figure 11 shows a flowchart of the process of developing the seismic fragility curves for the mold transformer.  
The fragility curves used for seismic vulnerable evaluation are natural logarithmic functions providing the probability of exceeding an identified response of a component for different intensity levels (IL) of the earthquake corresponding to a specified performance level. Accordingly, log-normal distribution is widely used to establish the fragility curves, not only for structural elements, according to the study of [PERSON] et al. [30], but also for nonstructural elements [16; 17]. For a specified damage state (DS), the relation between the probability of exceedance and the specific IL values can be presented in terms of a log-normal distribution function as follows [31; 32; 33; 28]  
\[P(DS|IL=x_{i})=\Phi\left(\frac{\ln(x_{i}/\theta)}{\beta}\right) \tag{6}\]  
where \(P\) is the probability for the cases that the component response will exceed a specified performance level at a given ground motion with \(IL=x_{i}\), \(\Phi\) is the standard normal cumulative distribution function, and \(\theta\) and \(\beta\) are the median and the standard deviation of the fragility function, respectively.  
In order to fit the fragility curves from the observed analytical data, \(\theta\) and \(\beta\) could be evaluated by using the maximum likelihood estimator (LE) method for the entire data set [34; 28] as follows  
\[LE=\prod_{i=1}^{m}\left(\frac{n_{i}}{z_{i}}\right)p_{i}^{z_{i}}(1-p_{i})^{n_{ i}-z_{i}} \tag{7}\]  
where \(m\) is the number of PGA levels, \(p_{i}\) is the probability that the component response will exceed a specified performance level at a given ground motion with \(IL=x_{i}\) based on analytical results, \(z_{i}\) is the number of exceedances out of \(n_{i}\) ground motion, and \(\Pi\) denotes a product over all PGA levels. Finally, parameters \(\theta\) and \(\beta\) were determined by maximizing the likelihood.  
Figure 10: Comparison of maximum displacement of the mold transformer between analytical and experimental results.