## 4 Results and Discussion
### The Seismic Fragility Analysis of Isolated Bridge Components
According to the reliability theory, when the seismic demand on components is completely related, the maximum seismic fragility of components is the system seismic fragility, which constitutes the lower bound of the first-order limit method; when the seismic demand on components is completely unrelated, any component failure will lead to the failure of the series system, which constitutes the upper bound of the system seismic fragility. Therefore, the first-order boundary of the seismic fragility of the bridge system is:  
\[\max_{i=1}^{m}[P(F_{i})]\leq P_{sys}\leq 1-\min_{i=1}^{m}[1-P(F_{i})] \tag{16}\]  
where, \(P(F_{i})\) indicates the probability of failure in the \(i\)-th damage state. \(P_{sys}\) is the failure probability of the bridge system in each damage state. \(m\) is the number of bridge system components. By substituting the seismic fragility function of piers and bearings into Equation (16), the upper and lower boundaries of the seismic fragility of the system could be obtained. For comparison, the seismic fragility boundary of the bridge system based on the copula function and that based on the boundary method of first-order reliability theory are shown in Figure 8.  
Figure 7: The seismic fragility curves of the girder collision (general bridge and isolated bridge). As seen in Figure 8, the seismic fragility curve of the overall structure based on the first-order reliability theory could be used to evaluate the failure probability range of the overall structure of the bridge under earthquake action. Under the rare earthquake action (PGA = 0.51 g), the failure probability of the overall structure of the bridge was between 21 and 28%, and the upper boundary seismic fragility curve of the failure probability of the overall structure of the bridge and the seismic fragility curve of the bearing failure almost coincided. However, the seismic fragility curve of the bridge system based on the copula function was between the upper and lower bounds of the first-order boundary in the whole range of seismic intensity, and it was closer to the lower bound. If the lower bound described the seismic fragility of the bridge system, it would underestimate the fragility of the structure, while if the upper bound described the seismic fragility of the bridge system, it would obviously overestimate the fragility of the structure. With the increase in ground motion intensity, the difference between the upper and lower bounds obtained by the first-order reliability theory limit method increased, and the difference between the upper and lower bounds obtained by the copula function method and the seismic fragility of bridge system also increased. Therefore, when the intensity of ground motion is large, the limit method of the first-order reliability theory will fail to estimate the fragility of the seismic system of the structure and cannot reasonably evaluate the safety and reliability of the structure. It can also be seen from Figure 8 that the seismic fragility curve of the bridge system based on the copula function method, under the rare earthquake action (PGA = 0.51 g), indicated that the failure probability of the whole bridge structure was 19%. However, no matter whether they were based on the first-order reliability theory or the copula function method for seismic fragility analysis, the pier and the bearing of the bridge structure system will not fail under frequent earthquake action (PGA = 0.11 g), and it is more likely to cause minor or medium damage. In the rare earthquake action (PGA = 0.51 g), the probability factors of complete failure of the pier and of the bearing were 8 and 21%, respectively. This embodied the structural requirements for multiple seismic designs. In conclusion, compared to the first-order reliability theory, the failure probability of the whole bridge structure based on copula function analysis was smaller and the probability interval was safer and more reliable. Figure 8: The seismic fragility curve of the overall bridge based on the copula function method.