# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 2 Methodology
### Definition of Damage Indexes for Bearings  
A crucial aspect in assessing the fragility of a structure is determining its damage state under an earthquake based on a performance level and quantifying the damage index under different damage states. [PERSON], [PERSON] (2018) found that the failure process of a structure can be described by five distinct damage states, namely: no damage, minor damage, medium damage, serious damage, and complete destruction. At present, there is no systematic theoretical standard for calculating a damage index for bearings. However, through the exploration of scholars in the past 20 years, the following three quantitative criteria can be obtained: (1) the relative displacement (\(\Delta x\)), i.e., the relative displacement between the bearing and pier top under earthquake action, (2) the shear strain (\(\gamma\)), i.e., the shear deformation of the bearing under earthquake action, and (3) the relative displacement ductility ratio (\(\Delta\mu\)), i.e., the measure of the ability of the bearing to deform under earthquake action without sustaining damage. It is calculated as the ratio of the relative displacement to the shear strain. Equations (14)-(16) provide more details on how to calculate these three criteria. \[\Delta x=\left|x_{b}-x_{p}\right| \tag{14}\]  
Figure 1: Time-dependent deterioration of longitudinal reinforcement in C40 concrete: (**a**) yield strength and corrosion rate; (**b**) rising rate of bond-slip and corrosion rate. \[\gamma = \frac{\Delta x_{max}}{t} \tag{15}\] \[\Delta\mu = \frac{\Delta x_{max}}{\Delta x_{\gamma 100\%}} \tag{16}\]  
where \(x_{b}\) is the bearing displacement (mm), \(x_{p}\) is the pier top displacement (mm), \(\Delta x_{max}\) is the maximum relative displacement of the bearing (mm), \(t\) is the bearing thickness (mm), and \(\Delta x_{\gamma 100\%}\) is the relative displacement (mm) at 100% shear strain of the bearing. Taking the movable spherical steel bearing of the example bridge as an example, the definition of the damage state for the example bearing is described below, referring to the restoring force model of the PTFE slide bearing. (1) Method 1: This method is based on the maximum allowable displacement of the bearing [20; 21]. The bearing is considered to be undamaged if the displacement is less than or equal to half of the maximum allowable displacement. The bearing is considered to be damaged if the displacement is greater than half of the maximum allowable displacement. The severity of the damage is determined by the amount of displacement. As shown in Figure 2, \(\Delta x_{1}\) is 1/2 of the maximum allowable displacement of bearing, \(\Delta x_{2}\) is the maximum allowable relative displacement of bearing, which is the distance from the edge of the upper seat plate to the edge of the upper rotating surface, and \(\Delta x_{3}\) is the distance between the outermost edge of the upper seat plate and the outer edge of the middle seat plate on the other side. This method is the most straightforward and easy to implement. However, it is also the least accurate. This is because the maximum allowable displacement is a conservative value that is based on the worst-case scenario. In most cases, bearings will not experience the maximum allowable displacement. (2) Method 2: This method is based on the shear strain of the bearing [4]. The bearing is considered to be undamaged if the shear strain is less than or equal to 100%. The bearing is considered to be damaged if the shear strain is greater than 100%. The severity of the damage is determined by the amount of shear strain. This method is more accurate than Method 1. However, it is also more complex and difficult to implement. This is because the shear strain to implement. This is because the shear strain can be difficult to measure, especially in large bearings. (3) Method 3: This method is based on the relative displacement ductility ratio of the bearing [22]. The bearing is considered to be undamaged if the relative displacement ductility ratio is less than or equal to \(\mu_{1}\).