# Collapse Potential of an Existing Reinforced Concrete Bridge Structure
## 2 Materials and Methods
### Analytic Model  
The inelastic spine models, as shown in Figure 4, were used for the nonlinear time-history analyses by using SAP2000 v.10 [25] software. The superstructure, including the cap beams, was modeled as an elastic element, since they were expected to remain elastic. The nonlinear behavior of the other structural elements was taken into account with their proper nonlinear properties. The bridge columns are modeled as a fiber model. The used concrete model is Mander [1; 26], which defines the stress-strain relationship of concrete whether it is confined or unconfined. The reinforcement model, described by the Turkish Bridge Seismic Design Code, was used for a similar relationship of reinforcement. The used model incorporates strain hardening. Figure 5 shows the stress-strain relationship of the used material models. Figure 4: Analytic model of the bridge. Figure 5: Stress–strain relationship of materials: (**a**) unconfined and confined concrete; (**b**) reinforcement steel. Figure 3: Section view of structural elements: (**a**) prestressed girders; (**b**) column. The existing bridge columns have an elliptical cross-section geometry as mentioned above. To overcome the modeling difficulties, the cross-section has been idealized as a rectangular shape following the method used by [PERSON] [27], as illustrated in Figure 6. The cross-section conversion is based on the confined concrete area, as the core concrete can sustain loads even at high deformation levels. Since unconfined shell concrete reaches its compressive strength at significantly lower deformation levels under bending, it is neglected in the cross-section transformation. The plastic hinge hypothesis assumes that nonlinear deformations were uniformly distributed along a specific length of the structural element, which is called plastic hinge length. The selected locations of the plastic hinges are at both ends of the columns. The following is the equation that gives the plastic hinge length, as proposed by the Turkish Bridge Seismic Design Code 2020 [1]. \[L_{p}=0.08L_{k}+0.022f_{ye}d_{bl}\geq 0.044f_{ye}d_{bl} \tag{1}\]  
where \(L_{p}\), \(L_{k}\), \(f_{ye}\), and \(d_{bl}\) are plastic hinge length, distance between the critical section and moment zero point, reinforcement yield strength, and diameter of the longitudinal reinforcement, respectively. Elastomeric bearings transfer the horizontal forces of the bridge, acting on the superstructure, to other structural elements by exhibiting shear behavior. They completely isolate the superstructure from the substructure, making the superstructure more susceptible to deformations. In this type of elastomeric bearing, the only mechanism preventing sliding is friction forces. Depending on the applied forces, they can undergo horizontal, axial, and rotational deformations. Thus, the analytic model should include those mechanical properties. Equations (2)-(4) present the stiffness values considered for the axial, shear, and rotational deformations. \(E_{c}\), \(G_{\mathit{off}}\), \(H\), \(I\), and \(A\) show modulus of elasticity, effective shear modulus, moment of inertia, and area of bearing, respectively. \[k_{v}=\frac{E_{c}A}{H} \tag{2}\]  
\[k_{H}=\frac{G_{eff}A}{H} \tag{3}\]  
\[k_{\theta}=\frac{EI}{H} \tag{4}\]  
Figure 7 shows a schematic representation of the analytic model of a cap beam, elastomeric bearings, and shear keys, which are located on it. This part of the bridge is indicated in Figure 4 with a red circle. Elastomeric bearings can be damaged when their shear, rotational, or axial force capacities are exceeded. The shear force capacity of the bearing is governed by the friction force capacity between the two surfaces (the bottom rubber layer and the concrete surface). Figure 6: Column section transformation. The force-deformation relationship that represents this behavior of the elastomeric bearing is assumed to be elastoplastic [27]. The friction force capacity, F\({}_{y}\), is the product of the dynamic friction coefficient (\(\mu\)) and the axial force (N).