# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Computational Schemes for Rails--Wheels Contact Problems
In [14], contact stresses were calculated for six combinations of R50, R65, and UIC60 rails with wheels having standard conical [22] and nonlinear wear-resistant [24] profiles. The smallest contact stresses (\(\sigma_{eqp}\ \sim 600\) MPa) were observed for R50 and R65 rails with a standard wheel, the largest (\(\sigma_{eqp}\ \sim 1070\) MPa) for the combination of R65 rail with a wheel having a nonlinear profile. It is obvious that an increase in the contact stresses will lead to a rapid accumulation of fatigue defects in new rails, especially for metro tracks with increased stiffness and small axial loads. However, for worn rails on mainline tracks, this same combination, according to [51], is the most advantageous. In mountainous regions with small-radius curves, the standard profile is preferable [52]. The issue of choosing optimal profiles is very complex, if we take into account their gradual change during operation due to wear. Therefore, in most cases, recommendations for choosing \"optimal\" rail and wheel profiles will apply to homogeneous rolling stock and a certain track design. It should be noted that to assess the effectiveness of rail and wheel profiles, economic indicators related to the costs of grinding rails, turning wheel treads, and energy consumption for train traction should also be included [51, 52]. However, a detailed analysis of the known approaches is beyond the scope of this work. It is obvious that, in addition to the design parameters of the track, the level of contact stresses will also be influenced by the factors of the force interaction of rails and wheels. The analysis of theoretical and experimental methods for determining the forces of interaction of rolling stock with the railway track deserves a separate publication, since there are a lot  
\[q_{x}(x,y)=\mu_{x}p(x,y);\;q_{y}(x,y)=\mu_{y}p(x,y), \tag{9}\]1 Dependence of maximum normal stresses on the magnitude and direction of wheel lateral displacement relative to the rail. of works in this direction. Let us mention only the excellent monographs [53; 54; 55], which have already become classics, as well as some review articles [56; 57; 58]. In a simplified form, the force interaction between rails and wheels in the case of single-point contact is shown in the scheme (Figure 11), which shows the resultant forces and moments that arise as a result of slipping, and which are actually distributed over the contact area. Typically, the problem of the stress-strain state in the contact zone is solved in a quasi-static formulation, increasing the values of vertical, lateral, and longitudinal forces by dynamic coefficients, which are determined through modeling or experimentally using special vehicles with appropriate measuring equipment. In dynamic models, the vehicle and track, excluding the rails, are considered rigid multi-mass systems. Rails are modeled either as beams on a continuous elastic foundation (Timoshenko beams) [59] or as discrete supports (sleepers) with random viscoelastic characteristics [60]. Comparisons of modeling results with experimental data at different speeds on real track sections show that vertical forces are modeled with good accuracy (the relative error, depending on speed, ranges from 1.2% to 23%) [61]. Such a large spread of relative error is due to the fact that such estimates were made for different operating conditions. In real conditions, spatial oscillations of rolling stock and track occur. These oscillatory processes are influenced by many factors that cannot be fully taken into account in discrete dynamic models. The worst modeling results were obtained for ordinary main railway lines, and the best for special ones (metro and test sites). The accuracy of lateral force estimation is significantly lower, which, according to [62], is primarily due to discrepancies between modeled and actual wheel and rail profiles.