# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Computational Schemes for Rails--Wheels Contact Problems
This approach simplifies the solution of contact problems (especially in three-dimensional formulations) significantly but has some disadvantages that affect the accuracy of stress and deformation calculations for real wheel-rail friction pairs. It should be noted that this method can only be used when Hertzian theory conditions are satisfied. Therefore, it is not applicable to areas where conformal contact conditions exist (i.e., the initial contact occurs not at a point but over a sufficiently large area). This applies particularly to worn rails as well as certain combinations of profiles of new rails and wheels. The main source of error in this method lies in its inability to account for the evolution of the contact patch as plastic deformations occur, which significantly impacts the calculated stress levels. Nonetheless, the simplicity of this scheme outweighs potential inaccuracies in contact stress calculations, which is why it is widely used for assessing the fatigue life of rails under conditions of rolling and sliding friction [11; 27; 28; 29; 32]. It should be noted, however, that such assessments, for the reasons mentioned above, are conservative in most cases. The second feature of the computational scheme in Figure 4 is the refinement of the finite element mesh in stress concentration zones. Theoretically, the accuracy of stress calculations increases exponentially as element size decreases, so in practice, the number of elements often needs to be balanced against the capabilities of computational equipment. Under these circumstances, mesh refinement is performed only in the contact zones between rails and wheels. Optimal element sizes in the contact zone can be estimated by comparing calculated stress values with theoretical ones using Formulas (\(6-8\)) [30; 35]. In work [35], it is noted that even for sufficiently refined meshes, errors may exceed 25%. Therefore, it is essential to ensure node consistency between finite element meshes on the contact surfaces. Often, the contact of two disks is considered as a model problem [11; 30; 35; 36; 37]. There is an exact analytical solution to this problem. This allows us to estimate the accuracy of the numerical solution and select the optimal parameters of the calculation scheme, for example, the degree of condensation of the finite element mesh. To achieve sufficient (<5%) accuracy in determining contact stresses, the average linear dimensions of elements, modeled as tetrahedrons, should not exceed 0.5 mm [30]. In this case, the calculated durability can be compared with experimental results, as this mechanical testing scheme is straightforward to implement and widely used. The accuracy of stress level assessment in the wheel-rail contact zone depends not only on the degree of finite element mesh refinement. It is also necessary to take into account factors related to the track design, such as  
* The stiffness of the sub-rail foundation, which affects deflections and angles of the rail twisting under wheel loads;
* The inclination of the rail base relative to the sub-rail foundation;
* The degree of outer rail elevation in curved track sections;
* Track gauge width;
* The profiles of the contact surfaces. Incorporating these factors significantly complicates the computational scheme shown in Figure 6. Therefore, we will examine which of these factors should be prioritized and which can be considered secondary. In most studies that attempted to calculate rail durability based on crack initiation criteria, the stresses from rail bending and torsion were not considered. Indeed, when comparing the levels of these stresses with contact stresses, the latter are significantly higher. In work [38], based on the simplified method of calculating rails proposed in paper [39], the maximum compressive normal stresses in the head of a type R50 rail from bending and compressed torsion were calculated for a curved section of metro track with a 400 m radius. Even considering dynamic loading, these stresses did not exceed 100 MPa.