## 2 Computational Model
* In the wagon with the fixing blocks, it is assumed that the rails are immovably fixed, i.e., in the fixing block, it is assumed that we have a fixed beam (Figure 5a);
* At the locations of the supports \(A_{\rm i}\), it is assumed that unknown forces caused by the deformation of the beam act on the lateral direction (\(y\) axis) as shown in Figure 5b. The forces are directed perpendicular to the beam axis (\(x\) axis);
* Because of the large dimensions and deformations of the system and considering that the task is statically indeterminate, it is solved with a linear analysis method for the approximate determination of the forces in the supports. The obtained results contain some inaccuracies, which can be estimated by the studies presented in [25; 27];
* When moving in a curve with radius \(R=150\) m, the speed of the train composition is small; therefore, the inertial forces are neglected. The classical form of the differential equation of the bending line in the given coordinate system and for each section of the beam is given in Equation (2) [32]:  
\[\frac{d^{2}}{dx^{2}}\left(EI_{2}\frac{d^{2}u_{y}}{dx^{2}}\right)=q_{y} \tag{2}\]  
For the analytical solution, it is necessary to integrate Equation (2), where a significant number of integration constants appear, depending on the number of sections, which must subsequently be determined by the boundary conditions. For this purpose, the finite element method was applied [32; 33]. The large dimensions of the beams make it difficult to use precise models like those shown in [2; 34; 35]. The beam is divided into \(n\)-number of finite elements, and each of them is located between two adjacent supports \(A_{\rm i}\) (Figure 5). The type of finite element is shown in Figure 6 [32]. Figure 4: Cross section of one rail group. Figure 5: Calculation model of the fixed beam and lateral forces acting on it: (**a**) fixed beam in fixing block; (**b**) unknown forces acting on the lateral direction. The position of each element is described by four local coordinates--two displacements and two rotations \((u_{1y}^{(i)},u_{1z}^{(i)},u_{2y}^{(i)},q_{2z}^{(i)})\) of each of the nodes [32]. The stiffness matrix [32; 33] of the element \(i\) is given in Equation (3):  
\[\left[K^{(e)}\right]=\frac{2 EI_{z}}{{l_{i}}^{3}} \begin{bmatrix}6&3l_{i}&-6&3l_{i }\\ 3l_{i}&2{l_{i}}^{2}&-3l_{i}&{l_{i}}^{2}\\ -6&-3l_{i}&6&-3l_{i}\\ 3l_{i}&{l_{i}}^{2}&-3l_{i}&2{l_{i}}^{2}\end{bmatrix}  \tag{3}\]  
where \(l_{i}\) is the length of the element and \(E\) is the modulus of elasticity. The discretized model has \(n\) number of elements (\(n=12\) in Figure 7) with different lengths \(l_{i}\) and correspondingly different stiffness matrices. Its position is described by \(2n+2\) global coordinates (Figure 7a). The forces act on the nodes, and the moment acts in node \(A_{0}\) (Figure 7b). For the beam in Figure 7, Equation (4) is valid:  
\[\left[K_{0}\right]\{U_{0}\}=\{F_{0}\} \tag{4}\]  
where \(\{U_{0}\}=\left\{u_{0y},\varphi_{0z},u_{1y},\varphi_{12}......,u_{iy},\varphi_ {i2}......u_{ny},\varphi_{nz}\right\}^{T}\) is a vector of global coordinates with dimension \((2n+2)\times 1\) and \(\{F_{0}\}=\{F_{A0},M_{A0},F_{A1},0,F_{A2},0....F_{Ai},0..F_{An},0\}^{T}\) is a vector of node loads with dimension \((2n+2)\times 1\). \(\{F_{0}\}\) contains forces and moments applied at the nodes. Except for A0, there are no applied moments in all other nodes. \([K_{0}]\)  
Figure 6: Finite beam element [28]. Figure 7: Discretized model of the beam: (**a**) with defined nodes coordinates; (**b**) with defined loads in nodes. is the global stiffness matrix, with dimension (2n + 2) \(\times\) (2n + 2).