# Comparison of Damage Indexes for Assessing Seismic Fragility of Bearings in an Offshore Bridge
## 3 Problem Definition
### Detailed Numerical Model of the Bridge Structure
pier with the bearing and the bottom of the girder is realized by the bearing simulated by a \"zero-length element\". See Figure 5 for details. Regarding the bearings, spherical movable steel bearings are utilized in this study. The structural composition and types of these bearings are illustrated in Figures 4-6, and detailed information can be found in Table 6. The QZ series spherical bearing is a special basin-type rubber bearing that consists of an upper bearing plate, lower bearing plate, spherical disc, polytetrafluoroethylene (PTFE) sliding plate, and rubber stopper. It is designed with a spherical PTFE plate instead of a rubber plate in traditional basin-type rubber bearings, which reduces the friction coefficient. The displacement is achieved by sliding between the upper bearing plate and the flat PTFE plate. It can be configured as a unidirectional movable bearing or a fixed bearing by incorporating guidance slots or rings on the upper bearing plate. The rotational movement of the bearing is facilitated by the sliding between the spherical disc and the spherical PTFE plate. In the numerical analysis, as the spherical bearing has nonlinear characteristics, an \"elastic-perfectly plastic material\" is used to define its stiffness in each direction, and a \"zero-length element\" is used to simulate the bearing. For piers with bearings, each pier is equipped with two bearings, which are multi-directional sliding bearings and longitudinal sliding bearings. Figure 5: Numerical model in OpenSees. Figure 6: Force–displacement curve of bearings. According to [PERSON], [PERSON] [22] and [PERSON], [PERSON] [1], the relative sliding between PTFE sliding plates and stainless steel plates in the movable pot bearing and movable spherical bearing allows for nearly complete relative displacement. Thus, a hysteresis model of PTFE sliding bearings is adopted to simplify the computational modeling. Test results from all types of PTFE slide bearings show that the dynamic hysteresis curve has a similar stress-strain relationship to that of an ideal elastic-perfectly plastic material. In summary, the hysteretic model shown in Figure 6 is adopted, and the calculation of the three-dimensional stiffness of the bearings is shown in Equations (21) and (22). \[K\cdot x_{y}=F \tag{21}\]  
\[\mu\cdot N=F \tag{22}\]  
where \(K\) is the horizontal shear stiffness (kN/m), \(x_{y}\) is the critical sliding displacement (0.002 m), \(F\) is the sliding friction force (N), \(\mu\) is the friction coefficient (0.02), and \(N\) is the load of the superstructure borne by the bearing (N). The horizontal stiffness of the multi-directional movable bearings and the longitudinal stiffness of the longitudinal movable bearings are calculated based on the actual vertical force borne by each bearing. The vertical stiffness of both types of bearings and the transverse stiffness of longitudinal movable bearings are calculated using the maximum bearing capacity of the bearings. Specifically, the maximum bearing capacity of the longitudinal movable bearings in the transverse direction is assumed to be 20% of its vertical bearing capacity (12,500 kN), as shown in Table 6. In summary, the nonlinear numerical model for the example bridge in this study was established by OpenSees, as shown in Figures 4-6. In the numerical model, it is assumed that the vertical stiffness of the bearing is completely rigid, and the vertical displacement of the bearing can be constrained by a large stiffness. The vertical stiffness (\(K_{3}\)) of the bearings is assumed to be 5 times that of \(K_{1}\), providing a strong constraint. intensity with regard to efficiency, practicality, sufficiency, and the ability to calculate hazard, and found that PGA is a highly sufficient _IM_ for conditioning fragility analysis. Meanwhile, a probability distribution for the PGA can be obtained using current seismic records [29; 30; 31].