# Collapse Potential of an Existing Reinforced Concrete Bridge Structure
## 2 Materials and Methods
### Analytic Model
The friction coefficient has been assumed to be 0.4 [28]. Figure 8 shows the considered load-deformation relationship for the bearings; k is the shear stiffness of the bearings. Shear keys significantly influence a bridge's response during an earthquake. They facilitate the transfer of seismic shear forces to the abutments and piers by restricting the transverse movement of the deck. This effect is especially critical for abutment bearings, where the superstructure may become more independent from the abutment, resulting in large displacements during seismic events. Shear keys continue to restrict the transverse movement of the deck due to friction resistance, even after reaching their maximum shear force capacity, according to the results of the study conducted by [PERSON] et al. [29]. Figure 9a illustrates the force-displacement relationship of an internal shear key. Point G represents the distance between the shear key and the adjacent structural element, while point B corresponds to 95% of this distance. From point B onward, the load increases at a constant slope until the shear force capacity of the block is reached. When the shear force capacity, denoted by V\({}_{1}\), is attained, shear failure occurs. This type of failure is brittle. Thus, a sudden decrease in load-bearing capacity is observed. If the block is subjected to a monolithic load, the load-displacement curve decreases linearly until it reaches the maximum displacement (D\({}_{\text{max}}\)). In the case of cyclic loading, the shear force capacity drops abruptly from V\({}_{1}\) to V\({}_{2}\), after which the behavior resembles that observed under monotonic loading. Figure 8: Load–deformation relationship of elastomeric bearing. Figure 5:0 Analytic model of a cap beam, elastomeric bearings, and shear keys. The following equations are the formulations used for the calculation of the shear force capacity (\(V_{max}\)) of the shear keys [29]. Figure 5:1  
Figure 5:2  
Figure 5:3  
Figure 5:4  
where \(V_{2}\),\(f_{c}\), \(b\), \(d\), \(K_{1}\), and \(\alpha\) are the cyclic capacity of the shear key, concrete strength, width, length, shear stiffness, and aspect ratio, respectively. The force-displacement relationship, shown in Figure 9b, is used in this study for external shear keys that are based on the research carried out by [PERSON] et al. [29], which involved a series of external shear key experiments on bridge abutments. Their findings revealed that shear keys can displace up to 8.9 cm before completely losing their load-bearing capacity, and then, the capacity of the seismic blocks essentially drops to zero. The shear key response is represented using zero-length elements that accurately capture this nonlinear force-deformation behavior. According to [29] the shear force capacity of an external shear key is calculated with 2 different approaches, namely shear capacity based on friction and shear capacity based on bar analogy. The lower one is taken as \(V_{max}=V_{1}\) in Figure 5:5 The difference between \(\mathrm{D}_{\mathrm{max}}\) and \(\mathrm{D}_{\mathrm{gap}}\) was found to be 8.89 cm by [PERSON] et al. [29]. The model proposed by [PERSON] and [PERSON] [30] was used to develop the analytical model of bridge abutment, as shown in Figure 5:6 It considers the transverse, vertical, and longitudinal behaviors of the abutment using spring elements. The abutment mass is calculated by taking into account both the mass of the abutment itself and a portion of the backfill soil behind the wall, according to [PERSON] and [PERSON] [31]. The longitudinal response of the bridge deck due to impact with the abutment occurs when the gap between the deck and the abutment closes, leading to a collision. Several factors influence this response, including the properties of the elastomeric bearings, the gap length between the deck and the abutment back wall, the geometric characteristics of the back wall, and the properties of the backfill material between the abutment and the soil.