## 3 Vehicle-Bridge Interaction Model  
### Bridge Model  
A 15 m span bridge is modeled as a simply supported Euler-Bernoulli beam discretized into 150 elements 0.1 m long each. Hence, there are 151 nodes in total with 2 DoFs per node, representing the vertical translation and in-plane rotation of each node. The elementary mass \([M_{e}]\) and stiffness \([K_{e}]\) matrices are expressed in Equations (7) and (8), respectively.  
\[[M_{e}]=\frac{mL_{e}}{420}\begin{bmatrix}156&22L_{e}&54&-13L_{e}\\ 22L_{e}&4L_{e}^{2}&13L_{e}&-3L_{e}^{2}\\ 54&13L_{e}&156&-22L_{e}\\ -13L_{e}&-3L_{e}^{2}&-22L_{e}&4L_{e}^{2}\end{bmatrix} \tag{7}\]  
\[[K_{e}]=\frac{EI}{L_{e}^{3}}\begin{bmatrix}12&6L_{e}&-12&6L_{e}\\ 6L_{e}&4L_{e}^{2}&-6L_{e}&2L_{e}^{2}\\ -12&-6L_{e}&12&-6L_{e}\\ 6L_{e}&2L_{e}^{2}&-6L_{e}&4L_{e}^{2}\end{bmatrix} \tag{8}\]  
where \(m\), \(L_{e}\), \(E\), and \(I\), represent the mass per unit length, length of each element, modulus of elasticity and moment of area, respectively. The vertical translation of the left end node and its associated rotation are represented by the 1 st and 2 nd rows of the matrices in Equations (7) and (8), and similarly for the right end node by the 3 rd and 4 th rows. By assembling all the elementary mass and stiffness matrices, it is possible to obtain the bridge mass and stiffness matrices used in Equation (1). The dimensions of these matrices are \(302\times 302\) in this paper.  
Table 1 summarizes the values of the bridge properties, based on a reinforced concrete solid slab with a constant cross-section [34; 35].  
The first three frequencies associated with the longitudinal modes of vibration of the bridge are 5.655 Hz, 22.622 Hz, and 50.900 Hz. These values are consistent with the literature [36; 37], in terms of the relationship between the bridge fundamental frequency and its span length, i.e., approximately \(85/L_{b}\) for the first frequency of a 15 m concrete bridge. Damping is usually low in bridges, and it is neglected in this paper unless otherwise specified. For instance, [PERSON] et al. [38] report that nearly 60% of the damping ratios of vehicular and pedestrian bridges fall within the 1-5% range, with the majority of values concentrated around a value of 3% for vertical frequencies.  
[PERSON] et al. [39] propose a linear stiffness reduction as an approximation of the exponential function developed by [PERSON] and [PERSON] [40] for modeling the decay in stiffness due to a crack in steel beams. A different non-linear function is available to model the stiffness loss due to a crack in reinforced concrete beams [41], and it has been used by other authors in the context of moving vehicular loads [42]. For simplification purposes, the linear approximation by [PERSON] et al. is applied here to represent the stiffness loss in a reinforced concrete bridge. The damage severity of the crack, typically labeled as \(\lambda\), is defined by the ratio of the crack depth to the total beam depth. The crack is assumed to extend across the width of the entire section. If the total beam depth is \(h\), stiffness increases linearly from a minimum at the crack location to a maximum (healthy condition, i.e., 0.5273 m\({}^{4}\)) at a distance of 1.5\(h\) (Figure 1). If \(h=0.75\) m and \(\lambda=15\%\), the crack has a depth of 11.25 cm and the relative stiffness loss in percentage at the cracked section is approximately \((1-(1-0.15)^{3})\times 100=38.59\%\) based on a rectangular cross-section. For \(\lambda=5\%\), the crack depth is 3.75 cm and the relative stiffness loss is 14.26%.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Length, \(L_{b}\) & 15 & m \\ Width, \(w\) & 15 & m \\ Depth, \(h\) & 0.75 & m \\ Mass per unit length, \(m\) & 28,125 & kg/m \\ Modulus of elasticity, \(E\) & \(35\times 10^{9}\) & Pa \\ Second moment of area, \(l\) & 0.5273 & m\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters of the bridge model.  
Figure 1: Vehicle-bridge interaction model.