# Safety Evaluation of a Water-Immersed Bridge Against Multiple Hazards via Machine Learning
## 6 Fragility Curve Analysis
(**d**) Illustration of yielding point. Figure 15: Example of yielding displacement for a pushover curve. After acquiring the displacement ductility demand (\(R_{D}\)), the failure probability can be analyzed in order to obtain the fragility curve. When analyzing the failure probability, a limit state needs to be defined, in which \(PE=\frac{R_{C}}{R_{D}}-1\) was used in this study, indicating that when \(PE>0\), the displacement ductility capacity (\(R_{C}\)) of a bridge is bigger than displacement ductility demand (\(R_{D}\)), and the considered bridge is under safe conditions. Otherwise, it means that the bridge will be subject to damage. As such, the failure probability, when a bridge is subject to seismic force PGA = \(x\), can be expressed by the following formula:  
\[P(PE\leq 0|\mathrm{PGA}=x)=P(\frac{R_{c}}{R_{D}}\leq 1|\mathrm{PGA}=x)=P(R \leq 1|\mathrm{PGA}=x), \tag{16}\]  
in which \(R\) is \(\frac{R_{C}}{R_{D}}\). Assuming \(R_{C}\) and \(D_{C}\) are log-normal distributions, then the formula used for calculating the failure probability can be revised as:  
\[1-\Phi(\frac{\mu_{\mathrm{In}(R_{C})}-\mu_{\mathrm{In}(R_{D})}}{\sqrt{\sigma_ {Int(R_{C})}^{2}+\sigma_{Int(R_{D})}^{2}}}), \tag{17}\]  
where \(\mu_{\mathrm{In}(R_{C})}\) refers to the mean capacity and \(\sigma_{Int(R_{C})}\) refers to standard deviation of the capacity, and \(\mu_{\mathrm{In}(R_{D})}\) refers to the mean demand and \(\sigma_{Int(R_{D})}\) refers to standard deviation of the demand. Based on the research report [25] proposed by National Center for Research on Earthquake Engineering (NCREE) of Taiwan, the standard deviations of capacity of the respective performance level (e.g., slight, moderate, extensive, and collapse) are 0.5, 0.45, 0.4, and 0.4 respectively. \(\Phi(\cdot)\) represents the standard normal cumulative distribution function. The required mean value and standard deviation of demand were obtained from the regression of power law. To do so, 3 sets of data were used: the displacement ductility demands (\(R_{D}\)) for the 30 year return, 475 year return, and 2500 year return periods, respectively. Each set contained 7 structural responses from time history analyses using 7 earthquake excitations. Listed below are the formulas used to calculate the PGA, mean, and standard deviation displacement ductility demand. \[\mu_{R_{D}} =a(PGA)^{b}, \tag{18}\] \[\sigma_{R_{D}} =c(PGA)^{f}, \tag{19}\]  
where \(a\), \(b\), \(c\), and \(d\) represent the regression parameters. Note that at least 7 ground motions should be used if a time history analysis is used to measure the structural performance (LRFD seismic bridge design, AASTHO 2007). In addition, the \"response-spectrum-compatible\" time histories were used here. A response-spectrum-compatible time history refers to the response spectrum of the selected earthquakes falling between 0.2 and 1.5 T (T is the fundamental period), which may not be less than 90% of the corresponding design spectral acceleration for a damping ratio of 5%. In addition, the average value of the response spectrum within the designated period range may not be less than the average value of the corresponding design spectral accelerations. The 7 ground motions selected here were therefore converted into response-spectrum-compatible data for return periods of 30, 475, and 2500 years.