## 2 Theoretical Background
More specifically, the stiffness matrix, \(K\), is time variant and, along with the time variant damping matrix, mathematically consider the interaction between the vehicle and the bridge. As an example, the stiffness matrix when the spring mass is located on the vertical degree of freedom \(i\) of the discretized bridge model can be written as:  
\[ \begin{bmatrix}K_{b}&K_{vb}\\ \hline K_{bv}&K_{v}\end{bmatrix} = \begin{bmatrix}k_{1,1}&k_{1,2}&\cdots&k_{1,j}& \cdots&k_{1,n-1}&k_{1,n}&0\\ k_{2,1}&k_{2,2}&\cdots&k_{2,i}&\cdots&k_{2,n-1}&k_{2,n}&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ k_{i-1,1}&k_{i-1,2}&\cdots&k_{i-1,j}&\cdots&k_{i-1,n-1}&k_{i-1,n}&0\\ k_{i+1}&k_{i,2}&\cdots&k_{ij}+k_{v}&\cdots&k_{i,n-1}&k_{i,n}&-k_{v}\\ k_{i+1,1}&k_{i+1,2}&\cdots&k_{i+1,j}&\cdots&k_{i+1,n-1}&k_{i+1,n}&0\\ \vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\ddots&\vdots\\ k_{n-1,1}&k_{n-1,2}&\cdots&k_{n-1,j}&\cdots&k_{n-1,n-1}&k_{n-1,n}&0\\ k_{n,1}&k_{n,2}&\cdots&k_{n,j}&\cdots&k_{n,n-1}&k_{n,n}&0\\ \hline 0&0&\cdots&-k_{v}&\cdots&0&0&k_{v}\end{bmatrix}  \tag{4}\]  
where \(k_{i,j}\) are the elements of the stiffness matrix of the bridge and \(k_{v}\) is the stiffness of the vehicle. When the vehicle is represented using a single-degree-of-freedom system, the submatrices \(K_{vb}\) and \(K_{bv}\) are \(1 xN\) and \(Nx1\) vectors, respectively, where \(N\) is the number of degrees of freedom on the bridge. When the vehicle is represented using a multi-degree of freedom system with \(N_{v}\) degrees of freedom, the submatrices \(K_{vb}\) and \(K_{bv}\) are \(N_{v}xN\) and \(NxN_{v}\) vectors, respectively. In Equation (4), the interaction between the vehicle and the bridge is achieved through the \(i^{th}\) elements of the \(K_{vb}\) and \(K_{bv}\) vectors as well as the diagonal term in \(K_{b}=k_{i,i}+k_{v}\). Using Rayleigh damping, a similar equation to Equation (4) can be written for the damping matrix as it is a combination of the time-invariant mass matrix and the time-variant stiffness matrix. As such, the vehicle-bridge interaction is achieved through time-variant stiffness and damping matrices. The developed FE software is based on the theoretical background summarized above and elsewhere [22]. As the next step, the developed FE software is validated by computing the response of an undamped simply-supported beam under a single sprung mass that was initially presented by [39] and used in several studies such as [8,16]. Figure 1 presents the modeling parameters of the sprung mass and the bridge used in the verification of the developed software. Figure 1: Verification case. The acceleration and displacement response of the bridge and the sprung mass was computed using the developed software, and the results are compared with the solution presented by [PERSON] et al. [8] as well as the analytical solution obtained considering the first mode response only. The results depicted in Figure 2 show that the developed software can capture the acceleration and displacement response of the bridge under moving sprung mass. Figure 2: Comparison of the acceleration and displacement response obtained from the developed software with the literature. The bridge stiffness matrix, \(K_{b}\), can be modified to include the stiffness of the bearings leading to the numerical model depicted in Figure 3. In this model, each moving axle consists of a mass, a spring and a dashpot, which simulate the inertia, stiffness and the damping characteristics of the vehicle, respectively. While the interaction between each axle and the bridge is considered, pitching and rolling of the train cannot be considered with the developed model.