## 2 Theoretical Background  
A finite element software was developed in MATLAB [34] computational environment to facilitate the automation of the significant number of numerical analysis conducted for the parametric study. Finite element models are well-suited for the complex vehicle-bridge-interaction (VBI) problem as they can reproduce the geometry of the vehicle and the bridge while considering other parameters such as bearing stiffness and their impact on the VBI. In this study, the bridge is modeled using a series of Euler-Bernoulli beams discretized at several points while the train is modeled as a series of masses connected to a spring and dashpot, which simulate the stiffness and the damping characteristics of the suspension systems of the train, respectively. Once the stiffness, mass and damping matrices of the bridge and the vehicle are defined, the equations of motion for each subsystem, i.e., the bridge and the vehicle, can be written as:  
\[[M_{b}]\{\ddot{u}_{b}\}+[C_{b}]\{\dot{u}_{b}\}+[K_{b}]\{u_{b}\}=\{F_{b}\} \tag{1}\]  
\[[M_{v}]\{\ddot{u}_{v}\}+[C_{v}]\{\dot{u}_{v}\}+[K_{v}]\{u_{v}\}=\{F_{v}\} \tag{2}\]  
where \([M_{j}]\), \([C_{j}]\), and \([K_{j}]\) are the mass, damping, and stiffness matrices, respectively; \(F_{j}\) are the time variant forces, and \(\{\dot{u}_{i}\}\), \(\{\dot{u}_{j}\}\), \(\{u_{j}\}\) are the acceleration, velocity, and displacement, respectively. Sub-indices \(j=\dot{b}\) and \(j=v\) refer to the bridge and vehicle, respectively. The equations of motion for the bridge and vehicle given in Equations (1) and (2), respectively, can be solved using either coupled or uncoupled approach. In the uncoupled approach, Equations (1) and (2) for each sub-system are solved separately. The compatibility of the forces and the displacements between the vehicle and the bridge is ensured through an iterative process at each time step. Although the uncoupled approach provides an attractive alternative as it avoids numerical instabilities due to singularities in the time-varying stiffness and damping matrices associated with the coupled approach, it implicitly assumes that the effect of the time-varying properties of the VBI system on the dynamic features of the response is negligible compared to that of the bridge, i.e., the frequencies and mode shapes of the bridge are invariant with time [22]. Although this assumption can be reasonable for vehicles with mass negligible compared to the mass of the bridge, it may lead to inaccuracies when the vehicle mass is significant enough to alter the vibration frequencies and the mode shapes of the bridge [22]. Considering the relatively high mass of trains and their impact on the vibration frequencies of railway bridges [35; 36; 37], it was decided to use the coupled approach to solve the system of equations. In the coupled approach, the system of equations of motions of the bridge (Equation (1)) and the vehicle (Equation (2)) are combined and solved using accurate and efficient numerical integration methods [38] that are suitable for second-order ordinary differential equations. The mass, damping and stiffness matrices of the bridge and the vehicle are combined to create global mass, damping and stiffness matrices, respectively. If the interaction forces due to track irregularities are ignored, the equation of motion for the entire system can be written as:  
\[ \begin{bmatrix}M_{b}&0\\ 0&M_{v}\end{bmatrix}\begin{Bmatrix}\dot{u}_{b}\\ \dot{u}_{v}\end{Bmatrix} + \begin{bmatrix}C_{b}&C_{bv}\\ C_{vb}&C_{v}\end{bmatrix}\begin{Bmatrix}\dot{u}_{b}\\ \dot{u}_{v}\end{Bmatrix} + \begin{bmatrix}K_{b}&K_{bv}\\ K_{vb}&K_{v}\end{bmatrix}\begin{Bmatrix}u_{b}\\ u_{v}\end{Bmatrix} = \begin{Bmatrix}0\\ -M_{v}g\end{Bmatrix}  \tag{3}\]  
where the mass matrix and the force vector are time-invariant while the stiffness and damping matrices are time-variant, and \(g\) is acceleration of gravity.