# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 3 Results and Discussion
In order to take into account, the main design features of the upper track structure, namely the rail inclination angle, track width, and the elevation of the outer rail in curves, it is necessary to consider a sufficiently complete geometric model of the track, which contains the entire wheel pair, which rests on two rail threads (see Figure 6). Additionally, such a model can be used to assess the influence of the stiffness of the sub-rail base on the level of contact stresses. However, if the sub-rail base is considered as completely elastic, then a change in its stiffness has little effect on the level of maximum contact stresses [63]. A significant effect can be if there is a local change in stiffness (for example, due to the destruction of the sleeper). But for this, the sub-rail base should be considered unequally elastic [60], which significantly complicates the calculations. The indicated scheme (see Figure 6) will also be useful for assessing the influence of the bending stiffness of the rolling stock bogie axis on the stressed state of the rail. Significant bending of the bogie axis in the vertical plane can change the position of the contact points of the wheels with the rails, and this will affect the level of contact stresses. For quasi-static stress analysis of rails on straight track sections, external vertical \(P\) and lateral \(Q\) forces should be applied to the bogie axis (see Figure 6). In this case, there will be three or two contact zones (in the case of conformal profiles or when \(Q=0\)). Solving such a problem requires quite powerful computers, so it is often limited to considering the right half of the calculation scheme in Figure 6. However, in this case, the analysis of the influence of the above-mentioned design factors is significantly complicated and, in most cases, only approximate results can be obtained. The results of calculations by various authors, given in Section 2.2, show that the above-mentioned track design parameters must be taken into account first in the calculations of the stress state of the rails. Within the framework of one publication, it is impossible to consider other extremely important factors that affect the accuracy of the assessment of the level of contact stresses. Such factors include the initial distribution of technological stresses, mechanical properties of the wheel and rail materials, the magnitude and nature of the change in axial and lateral loads, consideration of friction, anisotropy of materials, temperature, etc. These issues are somehow related to the formulation of the physical model of the finite element method. In such cases, it is advisable to first consider the contact interaction using 2D and 3D fragments of the wheel and rail in order to obtain high-quality results. Such an approximate formulation of the contact problem is advisable to use for modeling surface plastic deformation under conditions of rolling with slipping, analyzing the influence of surface roughness on the stress-strain state of surface layers of parts, stress concentration on defects (inclusions and microcracks) of the material, etc. The authors plan to consider these aspects in subsequent publications.