# Collapse Potential of an Existing Reinforced Concrete Bridge Structure
## 2 Materials and Methods
### Analytic Model
Before the impact, vertical forces from the superstructure are transmitted through the elastomeric bearings to the abutment face wall, then to the foundation and the ground. In the event of a seismic-induced impact, the longitudinal response generates an interaction between the abutment face wall and the backfill soil, mobilizing the full passive earth pressure. A similar situation occurs in the transverse direction. Horizontal forces are  
Figure 5:7 Load–deformation relationship of shear keys: (**a**) internal shear key, (**b**) external shear key [29]. transmitted sequentially from the elastomeric bearings to the shear keys, then to the abutment, foundation, and finally, to the ground. Thus, the bridge's transverse response relies on the stiffness provided by those elements and the backfill material. In this case, an interaction also occurs between the wing walls and the backfill. If there is backfill soil behind the wing walls, it decreases the horizontal displacements. Springs are used to take girder impacts on the shear keys into consideration, assuming that no displacement occurs in the backfill behind the wing walls. Several studies in the literature have adopted this approach [32]. This method is also considered to be a conservative approach to checking the shear capacity control of the shear keys. The vertical behavior of the bridge is modeled by considering the vertical stiffness of the elastomeric bearings and the soil backfill [33]. Figure 11 shows the computer model of the abutment. In the longitudinal direction of the bridge, a structural element is defined to model nonlinear behavior, as shown in the red color in Figure 5:8 It consists of a rigid component with shear and moment degrees of freedom, gap elements (GAP) at both ends that restrict movement to longitudinal translation, and a zero-length element that captures displacement induced by the backfill. In the transverse direction of the bridge, a series system similar to the longitudinal model is employed to capture nonlinear behavior. This system consists of a rigid element with defined shear and moment degrees of freedom, a gap element (GAP) that acts only in compression in the transverse direction, and a zero-length element at the end representing the external shear key. When the longitudinal gap between the bridge superstructure and the abutment back wall closes, the passive earth pressure of the backfill soil is mobilized. In this case, the initial stiffness of the backfill material plays a critical role. Figure 5:9 Abutment’s analytical model. Figure 3:0 Abutment model [30]. According to the Caltrans Seismic Design Criteria [34] Section 7.8.1-1, the initial stiffness for backfill material meeting standard specifications is defined by Equation (9). Figure 3:1  
According to the Caltrans Seismic Design Criteria [34], the passive longitudinal response of the backfill soil is modeled using an elastic perfectly plastic force-displacement relationship. This relationship is illustrated in Figure 12 for the abutment-supported model, while the abutment stiffness and the passive pressure force resisting the abutment movement are represented in Equations (10) and (11), where \(\mathrm{A_{e}}\) is the area of the back wall. The gap between the bridge superstructure and the abutment back wall is denoted as \(\Delta_{\mathrm{gap}}\) in Figure 12, whereas \(\Delta_{\mathrm{eff}}\) represents the effective longitudinal abutment displacement at the idealized yield point. Figure 3:2  
Figure 3:3  
To accurately determine the vibration characteristics of the bridge, it is crucial to account for the mass of the bridge abutment and the backfill behind it. In this study, the method proposed by [PERSON] and [PERSON] [31] is employed for this purpose, as mentioned before. This approach considers a specific portion of the backfill behind the wall, known as the critical length, which is calculated using Equation (12) below. Figure 3:4  
where \(S\), \(B_{c}\), and \(H\) are the backfill slope, abutment wall width, and height, respectively.