## 3 Results from the Numerical Calculations
### Results from Calculation of Lateral Forces between Rails and Vertical Columns of the Wagon  
To determine the coordinates of the points \(A_{\rm i}\) (and the displacements \(u_{\rm ij}\), respectively) along the \(y\)-axis, it is assumed that the composition is in a curve with radius \(R\) (Figure 2a). It is assumed that the points \(A_{1}\), \(A_{2}\), \(A_{4}\), \(A_{5}\), \(A_{7}\), \(A_{8}\), \(A_{10}\) and \(A_{11}\) lie on the theoretical arc of a circle because that is where the central bearings of the wagons are positioned. The remaining points \(A_{3}\), \(A_{6}\), \(A_{9}\) and \(A_{12}\) are located on the chords between the above points (Figure 3). Their position is determined by the line connecting the central bearings of the wagon. Point \(A_{1}\) is located at the beginning of the curve. The theoretical curve of the bending line, described with the coordinates from Table 1, is shown in Figure 8.  
The coordinate system \(A0 xy\) is introduced so that the \(x\)-axis is in the direction of line A\({}_{0}\)-A\({}_{1}\), and its direction relative to the curve is determined by the two central bearings of the wagon with the fixing blocks (Figure 2a). The determination of the unknown forces \(F_{\rm Ai}\) and the moment \(M_{A0}\) is carried out in the MATLAB program environment using the displacements from Table 1 (third column). Table 1 (column 5) also contains the obtained values for force reactions in points A\({}_{\rm i}\). It  
Figure 8: Bending line with coordinates of points A\({}_{\rm i}\). should be noted that the values are obtained under the condition that there is no horizontal clearance between the rails and the support stanchions, i.e., \(\Delta h=0\) (Figure 2d). During the calculations, the presence of large forces was found (Table 1, last column), especially in the points that do not lie on the theoretical arc of a circle (A\({}_{3}\), A\({}_{6}\) and A\({}_{9}\)). This is logical and is dictated by the fact that the clearance \(\Delta h\) is not provided in the model, i.e., \(\Delta h=0\) (Figure 2d). To increase the adequacy of the model, a limited horizontal displacement of the points was allowed due to the available constructively provided clearances between the rails and the support frames. With successive iterations based on a possible displacement of the bending line in the support points, new results for the \(y\)-coordinates of points A\({}_{\text{i}}\) and values for the force reactions in points A\({}_{\text{i}}\) were obtained and are presented in Table 2.  
Some of them have been adjusted relative to their original position up to 90 mm (constructively provided clearance between the rails and the support frames) in the lateral direction. In this case, the group of rails does not necessarily rest against each of the side supports, but settles freely, hindered by the minimal frictional forces at the base of the rail. This fact was established during real measurements carried out on another specialized train composition for the transport of long rails. The clearance and possible settlement of the beam at a given point was obtained by iterations through a certain step with a numerical experiment until the actual measured values were obtained. The iterations of horizontal clearance between the rails and limiting pins \(\Delta h\) (Figure 2d) show that the reaction forces decrease because the bending line changes. With new coordinates from Table 2 (third column), new values of reaction forces are determined and are presented in the last column of Table 2.  
An analysis of the data from Table 2 shows that the maximum lateral forces are at points A\({}_{0}\) and A\({}_{1}\) of the wagon with fixing block and at points A\({}_{11}\) and A\({}_{12}\) of the front (closest to the locomotive) wagon. These lateral forces are transferred via central bearings to the bogie frame and then on the wheels, acting on wheel-rail contact and causing additional lateral force.