# Time-Variant Seismic Fragility of Offshore Continuous Beam Bridges Based on Collapse Analysis
## 5 Time-Variant Fragility of Components and System
### Time-Variant Fragility Coefficient of the System  
To discuss the fragility of each component of the bridge system in the original state, the time-variant fragility curve and time-variant \(a\) value curve of each component and bridge system in the complete damage state of WC 1 in the whole life cycle as shown in Figure 12.  
Figure 12a shows that the exceeding probability of the bridge system under complete damage state is the highest in the whole working cycle. The area enclosed by the PGA axis and the curve of fragility,  
Figure 12: Fragility analysis in (working condition) WC1: (**a**) Time-variant fragility curve of each component and bridge system; (**b**) Time-variant fragility coefficient of components and bridge system. namely \(S\) increase with the extension of the service period. When the service period reaches 100 years, \(S\) is 125.9% of that in 0 year. In addition to the 1\({}^{\#}\) pier and 4\({}^{\#}\) pier, the exceeding probability of other components in complete damage state is less than 0.1. When PGA = 0.5 g, the exceeding probability of each representative component (2\({}^{\#}\), 5\({}^{\#}\) pier and 1\({}^{\#}\), 3\({}^{\#}\) bearing) in complete damage state is that of 1\({}^{\#}\) pier and 4\({}^{\#}\) pier 0.00%, 0.00%, 0.98%, and 1.47%; and 9.88%, 14.82%, 13.45%, 20.18%, respectively. Compared with the 4\({}^{\#}\) pier, the ratio of other components' exceeding probability to 1\({}^{\#}\) pier is smaller that is more prone to occur complete damage during the service period. 2\({}^{\#}\), 3\({}^{\#}\), and 5\({}^{\#}\) pier will rarely occur complete damage during the service period. Figure 12b shows that the fragility coefficient of components and the bridge system is positively correlated with service life. Moreover, the fragility coefficient of any component is lower than that of bridge system. \(a\) value (the PGA value corresponding to the average exceeding probability) of any component is larger than that of bridge system. Hence, the bridge system is more prone to occur complete damage than any other component. During the service period, \(a\) value of 1\({}^{\#}\) pier is less than that of 4\({}^{\#}\) pier. Taking the 100 years as an example, \(a\) value of 4\({}^{\#}\) pier is 3.1% more than that of 1\({}^{\#}\) pier. Therefore, the most fragile component of the bridge system in WC1 is 1\({}^{\#}\) pier. According to Table 4 and Equation (30), the time-variant fragility coefficient of components and bridge system under different working conditions in the whole life cycle are shown in Figures 13 and 14. Figure 13: Time-variant fragility coefficient of each working condition: (**a**) WC1; (**b**) WC2; (**c**) WC3; (**d**) Fragility coefficient of the system. Figure 13a-c show the coefficient of fragility of every working condition promotes with the service period extension in the whole life cycle, and 4\({}^{\#}\) and 1\({}^{\#}\) pier with bearing are nearest to the bridge system. Taking 100 years as an example, the coefficient of fragility of the system is 105% in 1\({}^{\#}\) pier and 132% of the bridge system in 0 year. Figure 13d shows that the fragility coefficient of WC2 and WC3 are greater than WC1 in the whole service period, and the fragility coefficient of WC2 is the largest. The fragility coefficient of WC2 in 100 years is 227.53% and 117.85% of WC1 and WC3, respectively. It is  
Figure 14: A value of each working condition: (**a**) Fragility analysis of WC2; (**b**) Fragility analysis of WC3; (**c**) Comparison of each working condition. because that compared with the WC3 which is formed after the complete damage of 4\({}^{\#}\) pier, the system stiffness and redundancy of WC2 which is formed after the complete damage of 1\({}^{\#}\) pier is less than WC3. The seismic ability of the remaining bridge system of WC3 is better than WC2. Additionally, the fragility coefficient of WC2 and WC3 are not significantly different in 0-30 years, but they were significantly different in 30-150 years.