# Safety Evaluation of a Water-Immersed Bridge Against Multiple Hazards via Machine Learning
## 6 Fragility Curve Analysis  
A fragility curve is used to indicate the failure probability of the structure under different levels of hazard, and the fragility curve is mainly affected by the demand and the capacity of the bridge. The demand means the displacement ductility (\(R_{D}\)) analyzed by simulating a scoured bridge under specific earthquake magnitude. As for the capacity (\(R_{C}\)), it is calculated according to the definition proposed by Federal Emergency Management Agency (FEMA, HAZUS-MH/MR3, [23]), where \(1<R_{c}<2\) means minor damage, \(2<R_{c}<4\) means medium damage, \(4<R_{c}<7\) means major damage, and \(R_{c}>7\) can be considered that the structure completely collapsed. When the demand of displacement ductility is bigger than the capacity, it can be assumed that the bridge is damaged. Here, the lower bounds of each performance level (e.g., 1, 2, 4, and 7) were used as the threshold values. The displacement ductility demand (\(R_{D}\)) of the bridge is the ratio between the ultimate displacement (\(\Delta_{u}\)) and yielding displacement (\(\Delta_{y}\)), that is, \(R_{D}=\frac{\Delta_{u}}{\Delta_{y}}\). Assuming the pier of the bridge serves as the primary damage mechanism, in this case, the aforementioned term of \"displacement\" refers to the relative displacement between pier top and pier bottom. In view that it would be impossible to acquire an accurate ultimate displacement of the bridge by a nonlinear static analysis method, such as a pushover analysis, the ultimate displacement was therefore calculated by a nonlinear time history analysis, where the maximum value of the bridge displacement duration excitation is the ultimate displacement. Figure 12: Exceedance probability distribution of the bridge considered having a 50 year of lifespan. Figure 13: Occurrence probability distribution of the bridge considered having a 50 year of lifespan. The yielding displacement (\(\Delta_{y}\)) is another key in determining the displacement ductility demand (\(R_{D}\)), where \(\Delta_{y}\) is acquired from a nonlinear pushover analysis. Because the pushover curve is irregular in nature, the method of acquiring \(\Delta_{y}\) has always been a subject discussed by all sectors. In this study, the equal energy method proposed by [PERSON] et al. [24] was adopted to calculate the yielding displacement, and it will be explained in the following three steps. First, acquire area (A\({}_{1}\)) under the pushover curve. Then, draft the tangent lines on the origin and the ending points of the curve, where the corresponding slopes are considered as the initial stiffness (\(K_{1}\)) and yielding stiffness (\(K_{2}\)). Accordingly, the intersection point (\(P_{1}\)) is also obtained, as per Figure 14a below. Next, rotate the initial stiffness (\(K_{1}\)) line from its original point clockwise until area (A\({}_{2}\)), which is enclosed by the rotating and \(K_{2}\) lines, becomes equal to area (A\({}_{1}\)). Then, we have intersection point (\(P_{2}\)), as per Figure 14b below. Likewise, rotate the yielding stiffness (\(K_{2}\)) line from its ending point counter clockwise until area (A\({}_{3}\)), which is enclosed by the rotating and \(K_{1}\) lines, becomes equal to area (A\({}_{1}\)). Then, we have intersection point (\(P_{3}\)), as per Figure 14c below. Finally, a line which is orthogonal to the line of \(\overline{P_{2}P_{3}}\) and passes through the \(P_{3}\) point can be drafted in which the intersection point is the yielding point (\(P_{y}\)), and the corresponding displacement is the yielding displacement (\(\Delta_{y}\)), as per Figure 14d below. Indicated in Figure 15 is an example of the yielding displacement (\(\Delta_{y}\)) calculated from a pushover curve with 0 m of water depth and scouring depth. Figure 14: Schematic illustration of acquiring the yielding point in this study (\(V\) is the base shear and \(\Delta\) is the relative displacement between pier top and bottom). (**a**) Initial and yielding stiffness. (**b**) Illustration of P\({}_{2}\) point. (**c**) Illustration of P\({}_{3}\) point.