# Time-Variant Seismic Fragility of Offshore Continuous Beam Bridges Based on Collapse Analysis
## 5 Time-Variant Fragility of Components and System  
### Time-Variant Fragility Analysis of Pier  
Based on the above the article of [PERSON] [38], 150 ground motion and Equations (25)-(27), displacement ductility ratios of 150 seismic responses corresponding to each pier are selected for logarithmic linear regression analysis. Probabilistic seismic demand model (PSDM) as shown in Table 3 and Figure 6. Finally, the fitting functions are substituted into Equation (28) to obtain the regression coefficients of the 0-year probabilistic seismic demand model for each pier. R\({}^{2}\) refers to the degree of fitting in Table 3.  
Figure 6 shows that the seismic response of each pier increases linearly with the increase of PGA. The seismic response of 1\({}^{\#}\), 4\({}^{\#}\), 5\({}^{\#}\) pier are greater than that of 2\({}^{\#}\), 3\({}^{\#}\) pier because 1\({}^{\#}\), 4\({}^{\#}\), 5\({}^{\#}\) pier and 2\({}^{\#}\), 3\({}^{\#}\) pier are made of C40 concrete and C50 concrete respectively. In addition, 1\({}^{\#}\), 4\({}^{\#}\), 5\({}^{\#}\) pier have bearing, while 2\({}^{\#}\), 3\({}^{\#}\) pier are rigid piers. During the deformation process under the action of earthquake, both the bottom and top of rigid pier will form a plastic hinge zone that dissipates seismic energy and the pier with bearing forms a plastic hinge zone at the bottom of pier, so the seismic response of rigid pier is less than that of the pier with bearing. In this paper, 1\({}^{\#}\), 4\({}^{\#}\) pier have similar height and same concrete type. They also connect with the main beam and similar bending resistance during the service period, as do 2\({}^{\#}\), 3\({}^{\#}\) pier. Moreover, the fragility coefficient in the following text only needs the fragility curve in the state of complete damage. Hence, only 1\({}^{\#}\), 2\({}^{\#}\) and 5\({}^{\#}\) pier are represented to draw the time-variant fragility curve under the state of complete damage in the whole life cycle, as shown in Figure 7. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Pier Number** & **ln \(a\)** & \(b\)** & **R\({}^{2}\)** \\ \hline
1\({}^{\#}\) pier & 1.517 & 1.273 & 0.601 \\
2\({}^{\#}\) pier & \(-\)1.010 & 1.021 & 0.602 \\
3\({}^{\#}\) pier & \(-\)0.934 & 1.015 & 0.600 \\
4\({}^{\#}\) pier & 1.446 & 1.372 & 0.585 \\
5\({}^{\#}\) pier & 0.568 & 1.136 & 0.612 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The regression parameters and determination coefficients. Figure 6: Probabilistic seismic demand model (PSDM). Figure 7 indicates that the exceeding probability of pier in the whole life cycle is positively correlated with PGA and service period. In 0-30 years, the exceeding probability growth rate of each pier is the slowest because the cracking time of C40 concrete under chloride-induced corrosion is 27.1 years. The cracking time of C50 concrete under chloride-induced corrosion is 30.39 years, and it has almost no cracking in 0-30 years. When PGA = 1.0 g, the exceeding probability of 1\({}^{\#}\) pier and 5\({}^{\#}\) pier in 30, 50, 70, 100, and 150 years increased by 0.012, 0.035, 0.051, 0.055, 0.107 and 0.002, 0.006, 0.009, 0.01, 0.017, respectively, compared with 0 years. The exceeding probability increment of 1\({}^{\#}\) pier is greater than that of 5\({}^{\#}\) pier due to 5\({}^{\#}\) pier is higher and the bending resistance is better than that of 1\({}^{\#}\) pier. Figure 7b, c show that the exceeding probability in 150 years is \(1.03\times 10^{-6}\) and 0.049, respectively. Therefore, in the subsequent analysis, it can be assumed that 2\({}^{\#}\) and 5\({}^{\#}\) piers will not reach complete damage in the whole life cycle. In the same way, it is assumed that 3\({}^{\#}\) rigid pier has similar height and materials with 2\({}^{\#}\) pier will not reach complete damage in the whole life cycle.