# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology
### Fragility Analysis Method for Bearing  
Fragility analysis involves examining the correlation between the probability of structural damage caused by seismic events and the intensity of ground motions during an earthquake. Common methods for structural seismic fragility analyses include: (1) Fragility analysis based on expert opinion: This method uses the judgment of experts to estimate the damage probability of a structure. This method has high levels of uncertainty. (2) Empirical fragility analysis method based on existing seismic damage analyses: This method uses data from past earthquakes to estimate the damage probability of a structure. This method is only applicable when using data from similar sources, and subjective differences in data collection can occur. Furthermore, many areas lack actual ground motion records. (3) Theoretical fragility analysis method based on numerical analysis: This method uses finite element analysis to simulate the behavior of a structure under earthquake loading. This method can comprehensively consider corresponding seismic records and accurately evaluate the seismic performance of structures under earthquakes. (4) Mixed fragility analysis methods: These methods combine two or more of the above three methods [1; 15]. Quantitative methods based on expert opinion have high levels of uncertainty. Empirical fragility analysis methods are only applicable when using data from similar sources, and subjective differences in data collection can occur. Furthermore, many areas lack actual ground motion records. In contrast, theoretical fragility analysis methods based on finite element analysis can comprehensively consider corresponding seismic records \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Type** & & **LS1** & **LS2** & **LS3** & **LS4** \\ \hline \multirow{8}{*}{Relative displacement (mm)} & B1 [22] & 90.00 & 150.00 & 200.00 & 250.00 \\  & B2 [3] & 37.40 & 104.20 & 136.10 & 186.60 \\  & B3 [20] & 50.00 & 80.00 & 100.00 & 112.50 \\  & A1 & 62.50 & 111.40 & 145.40 & 183.00 \\  & B4 [21] & 95.00 & 195.00 & 550.00 & 905.00 \\  & A2 & 70.60 & 132.30 & 246.50 & 363.50 \\  & B5 [4] & 100.00 & 150.00 & 200.00 & 250.00 \\  & B6 [22] & 1.00 & 1.50 & 2.00 & 2.50 \\  & B7 [22] & 0.52 & 0.79 & 1.05 & 1.31 \\  & B8 [22] & 0.20 & 0.55 & 0.71 & 0.98 \\  & B9 [20] & 0.26 & 0.42 & 0.52 & 0.59 \\  & A3 & 0.50 & 0.82 & 1.07 & 1.35 \\  & B10 [21] & 0.50 & 1.00 & 2.50 & 4.50 \\  & A4 & 0.50 & 0.85 & 1.36 & 1.98 \\ \hline \hline \end{tabular}  
* LS1–4 represent minor damage, medium damage, serious damage, and complete destruction, respectively.  
\end{table} 
Table 4: Damage indexes of bearing under different types. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Relative Displacement** & **Shear Strain** & **Relative Displacement** & **Ductility Ratio** \\ \hline Case 1 & Case 2 & Case 3 & Case 4 & Case 5 & Case 6 & Case 7 \\ A1 & B4 & A2 & B5 & A3 & B10 & A4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Case division of damage indexes for bearing. and accurately evaluate the seismic performance of structures subject to seismic actions. Meanwhile, [PERSON], [PERSON] [10] conducted analyses on the service performance of bridges using the theoretical fragility method, thereby providing further evidence of the accuracy and feasibility of this approach. Consequently, the fragility analysis method, which relies on nonlinear dynamic time-history analysis of structures, became a crucial tool in the realm of performance-based seismic engineering. It enables a comprehensive assessment of structural performance by considering the complex nonlinear behavior of the system under seismic forces [2], as shown in Equation (17). \[P_{f}=P[D\geq C|IM] \tag{17}\]  
where \(P_{f}\) is the conditional probability of damage. \(D\) is the seismic demand of the structure under random earthquake action. \(C\) is the structural response under earthquakes, corresponding to its bearing capacity. \(IM\) is the ground motion intensity.