# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology
### Definition of Damage Indexes for Bearings
The bearing is considered to be damaged if the relative displacement ductility ratio is greater than \(\mu_{1}\). The severity of the damage is determined by the amount of the relative displacement ductility ratio. Among them, \(\mu_{1}\) is the relative displacement ductility ratio when the shear strain of the bearing is equal to 100%. This method is the most accurate of the three methods. However, it is also the most complex and difficult to implement. This is because the relative displacement ductility  
Figure 2: Damage index of bearing by relative displacement. ratio is a measure of the ability of the bearing to deform without sustaining damage. The relative displacement ductility ratio can be difficult to measure, especially in large bearings. The above three methods are summarized in Table 3 and Figure 3. The table shows the average values of the four algorithms for each method. The figure shows the relationship between the damage index and the displacement, shear strain, and relative displacement ductility ratio. It can be seen from the table and figure that the three methods are all effective in quantifying the damage index of a bearing. The method that is most suitable for a particular application will depend on the specific requirements of the application. Among them, for method 1, there are three situations: (1) Of the reference values given by scholars based on experience, there are currently three types (named B1-B3) widely used by scholars of bridge engineering. Additionally, the average value of the above three methods is also given here for comparative analysis, named A1. (2) Based on the quantification method proposed in Figure 2 (named B4), the damage index of each state is solved. (3) For comparison, the average values of the above four algorithms (B1-B4) are given, which is named A2. Method 2 is named B5. As for method 3, which is similar to method 1, there are three situations: (1) the reference values based on the experience of scholars are named B6-B9. Additionally, the average value of the above four methods is also given here for comparative analysis, named A3. (2) According to the quantification method proposed in Equation (18) (named B10), the damage index for each state can be determined. (3) For comparison, the average values of the above five algorithms (B6-B10) are given, named A4. Among the existing methods for quantifying damage indexes of bearings, two main approaches exist:  
(1) Empirical methods: provide reference values for the damage index based on the experience of scholars. These reference values are based on the observed damage to bearings after earthquakes. (2) Theoretical methods: calculate the damage index based on the properties of the bearing and the earthquake loading. These methods are based on established theories and are therefore more accurate than empirical methods. A detailed comparison and applicability analysis of the two quantification methods was conducted during the service life of the bearing when subjected to earthquakes. As shown in Table 4 and Figure 3, based on the relative displacement, the damage indexes obtained by the empirical methods (B1-B3) have the same trend, and the values are relatively close. To save space, the damage index for the relative displacement based on empirical methods can be represented by the average value (A1) of the above three methods selected here. Similarly, the damage index using the relative displacement ductility ratio also shows this phenomenon, so A3 is selected to represent the relative displacement ductility ratio using the empirical method. Figure 3: Damage index of bearing under different types: (**a**) based on relative displacement; (**b**) based on relative displacement ductility ratio. Furthermore, the damage index using the empirical method is evidently smaller than that based on the theoretical method. Especially in the serious damage and complete destruction states, the \(\Delta x\) obtained by the theoretical method (B4) is approximately 3.8 and 5 times that obtained by the empirical method (A1), respectively.