## 5 Seismic Risk Analysis of the Entire Bridge
### Seismic Risk Analysis of the Entire Bridge
#### 5.2.2 Calculation Process of Seismic Risk Analysis  
The 2D seismic fragility curves and the hazard curve at the bridge site have already been obtained in Sections 4 and 5.1, respectively. However, additional processing is required before beginning the seismic risk analysis due to the exceedance probability. According to the definition of seismic fragility, the structural failure probability of a specific level of  
Figure 12: The hazard curve at the bridge site. performance limit states under ground motions of varying intensities can be calculated by employing Formula (31), as illustrated in Figure 13:  
\[P\big{[}S=S_{j}|A=a_{i}\big{]}=\left\{ \begin{array}{ll}1-P\big{[}S\geq S_{j+1}|A =a_{i}\big{]}&j=0\\ P\big{[}S\geq S_{j}|A=a_{i}\big{]}-P\big{[}S\geq S_{j+1}|A=a_{i}\big{]}&1\leq j< m\\ P\big{[}S>S_{j}|A=a_{i}\big{]}&j=m\end{array} \right. \tag{31}\]  
where \(j=0\) refers to the structure's basically intact state, \(P\big{[}S\geq S_{j}|A=a_{i}\big{]}\) refers to the exceedance probability that the structure may be in the j-class performance limit state when encountering ground motion with PGA of \(a_{i}\), and \(m\) is the number of performance limit states, which is four in this paper. Similarly, according to the definition of the site hazard, the occurrence probability of ground motions of varying intensities during a 100-year design reference period can be approximately calculated by employing Formula (32), with the results shown in Table 8:  
\[P[A=a_{i}]=\left\{ \begin{array}{ll}P[A\geq a_{i}]-P[A\geq a_{i+1}]&1\leq i< 20\\ P[A=a_{i}]&i=20\end{array} \right. \tag{32}\]  
where \(P\)\([A=a_{i}]\) refers to the occurrence probability of ground motions that the structure may encounter with PGA of \(a_{i}\) during the design reference period and \(i\) is the number PGA values, which is twenty in this paper. In view of the difficulty and subjectivity of estimating the indirect economic loss and direct input cost of earthquake relief, this paper only quantitatively estimates the direct economic loss of the entire bridge using the loss ratio defined by the American earthquake  
Figure 13: Failure probabilities of the entire bridge under different performance limit states. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**PGA (g)** & **Occurrence** & **PGA (g)** & **Occurrence** & **PGA (g)** & **Occurrence** & **PGA (g)** & **Occurrence** & **PGA (g)** & **Occurrence** \\ \hline
0.05 & 0 & 0.30 & 0.00204 & 0.55 & \(4.091\times 10^{-6}\) & 0.80 & \(8.001\times 10^{-8}\) \\
0.10 & 0.14040 & 0.35 & 0.00043 & 0.60 & \(1.650\times 10^{-6}\) & 0.85 & \(4.211\times 10^{-8}\) \\
0.15 & 0.73821 & 0.40 & 0.00011 & 0.65 & \(7.132\times 10^{-7}\) & 0.90 & \(2.296\times 10^{-8}\) \\
0.20 & 0.10621 & 0.45 & \(3.275\times 10^{-5}\) & 0.70 & \(3.273\times 10^{-7}\) & 0.95 & \(3.247\times 10^{-8}\) \\
0.25 & 0.01254 & 0.50 & \(1.102\times 10^{-5}\) & 0.75 & \(1.582\times 10^{-7}\) & 1.0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Probability distribution of peak ground acceleration. risk assessment system HAZUS [50]. The so-called loss ratio refers to the ratio between the maintenance or reconstruction cost and the initial cost of the structure under different performance limit states. Using the value indicated in Table 9, the loss ratio of the complete damage state is \(2/n\), where \(n\) is the total number of bridge spans. Now the seismic hazard and seismic risk probability for the entire bridge during 100-year design reference period can be calculated by employing Formula (18), as shown in Table 10.  
It can be seen from Table 10 that the seismic risk probability for the entire bridge ranges from 0.00454 to 0.00608. The bridge is most likely to remain basically intact or slightly damaged during ground motion. Although the loss ratio is relatively high when the structure is seriously damaged, the seismic hazard is less due to the small possibility of large ground motions.