# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Computational Schemes for Rails--Wheels Contact Problems  
The success of solving contact interaction problems between rails and rolling stock wheels depends on the accuracy of the computational schemes, the choice of numerical method, software, and the computational power of the equipment. This explains why the results of different studies are often difficult to compare, not only quantitatively but sometimes even qualitatively. In this work, we will limit ourselves to considering the modeling of the stress state of rails using software implementing the finite element method. Today, computing technology and available software for solving spatial contact problems have limited capabilities. Therefore, researchers have to use simplified versions of all components of the calculation scheme in problems of contact interaction of rails and wheels. Here, it is appropriate to recall [PERSON]'s statement: \"Everything should be made as simple as possible, but not simpler.\"  
Notable examples are the first attempts at numerical modeling of such problems. In works [33; 34], a physical model of surface failure of bodies under friction conditions due to the initiation and growth of internal cracks was proposed. Given the computational limitations of that time, numerical modeling of this process was restricted to a plane problem (Figure 4a). For numerical modeling, the ADINA software package was used, and the finite element mesh is shown in Figure 4b. Figure 3: Graphs showing the variation in relative equivalent stresses at points along the major (1) and minor (2) semi-axes of the ellipse. These works continue to attract attention even today because they demonstrated that high-quality results can be obtained even with relatively simple computational schemes. However, it is clear that the accuracy of stress and deformation determination with such a sparse mesh is very low. Let us highlight some features of the computational scheme shown in Figure 4, which have been utilized in many subsequent works for calculating the stress-strain state of rails. The first feature concerns the method of load application. In some works [11; 27], a stationary load is defined, distributed over an elliptical contact area whose dimensions are determined by Formula (2), while the intensity of the distributed load is described by law (1). To improve calculation accuracy, the finite element mesh in the contact zones must be significantly refined (of course, taking into account the capabilities of computational equipment, as the number of calculations is estimated to grow exponentially with decreasing element size). To apply the load, it is necessary to calculate the values of the contact pressure for the known coordinates of each mesh node within the elliptical contact area and define them as normal surface nodal forces. In works [11; 27], a specialized program is used for this purpose. Figure 5 shows the distribution of contact pressures under conditions of a two-point contact between the wheel and rail on curved track sections [27]. One advantage of this method of load application is the ability to account for the influence of tangential forces that arise due to wheel slippage relative to the rails. Their nodal values can be calculated considering different coefficients of friction for rolling and sliding in longitudinal and transverse directions relative to the rail axis:  
\[q_{x}(x,y)=\mu_{x}p(x,y);\;q_{y}(x,y)=\mu_{y}p(x,y), \tag{9}\]  
Figure 4: Loading scheme (**a**) and finite element model (**b**) of the problem [34]. Figure 5: Distribution of normal pressure under two-point contact conditions. where \(q_{x}(x,y)\) and \(q_{y}(x,y)\)--are the magnitudes of tangential forces at the contact point with coordinates \((x,y)\); \(\mu_{x}\) and \(\mu_{y}\)--are the coefficients of friction in the longitudinal and transverse directions, which may vary for each contact zone in the case of two-point contact.