# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Computational Schemes for Rails--Wheels Contact Problems
The position of the wheelset relative to the two rails in the track also affects contact stress levels, even on straight track sections. During movement, the bogie of the rolling stock undergoes transverse oscillations, leading to different combinations of wheel-rail contact zones. This results in changes in the size and shape of contact patches and, accordingly, significant variations in stress distribution. In certain cases, two-point contact with impact interaction between the flanges of the rolling stock wheels and the rails may occur. Works [41; 42] present the results of mathematical modeling of the interaction between 60E2 rails and wheels with S 1002 profiles. Calculations of contact stresses using the finite element method in the elastic formulation were performed for straight track sections with a rail inclination angle of 1:40 for two levels of static wheel loads: 60 and 90 kN. The possibility of lateral displacements of the wheel relative to the rail was also considered. The term \"lateral displacement\" refers to the distance \(\Delta\eta\) between the symmetry axis of the rail and the plane of the wheel's rolling circle (Figure 9). Positive \(\Delta\eta\) values correspond to the rolling plane being to the right of the rail's symmetry axis, while negative values indicate the lateral side of the rail approaching the wheel flange. For a nominal track gauge width of 1435 mm and a distance of 1425 mm between the inner faces of the wheelset flanges, the calculated lateral displacement is +3 mm from the neutral position of the wheelset relative to the track's symmetry line. Lateral displacements of the wheelset relative to the track can range from +7.5 mm to \(-\)3 mm. At \(\Delta\eta=-3\) two-point contact occurs as shown in Figure 5.  
Figure 10 shows how the maximum normal stresses change depending on the magnitude of the wheelset's lateral displacement relative to the track for two levels of wheel load. The figure also illustrates the calculated shapes of contact patches for several values of \(\Delta\eta\) (the sizes of the images are kept constant for convenience). The data indicate that, as the rail's lateral side approaches the wheel flange, the contact patch gradually loses its elliptical shape, and its area decreases by nearly 20%, while maximum stresses localize in a relatively small zone. The difference between the maximum stress values for different contact zones reaches 300 MPa, indicating a significant influence of track gauge width on contact stress levels. Certain combinations of rail inclination, rail elevation in curves, and track gauge width can clearly be used to reduce contact stress levels. However, such analysis requires multivariate calculations for each pair of rail and wheel profiles. \[q_{x}(x,y)=\mu_{x}p(x,y);\;q_{y}(x,y)=\mu_{y}p(x,y), \tag{9}\]0 Scheme for measuring the lateral displacement of the wheelset. Compatibility assessments of different rail and wheel types are presented in many studies [14, 43, 44, 45, 46, 47, 48, 49]. The primary goal of such research is to select the optimal geometric dimensions and shapes of the working surfaces of rails and wheels. However, the set of optimization criteria is quite broad. In [44], profile compatibility is evaluated based on criteria such as  
* Contact should be single-point and conformal;
* The rolling surfaces should have equivalent conicity that maximizes the wavelength of the bogie's lateral oscillations and meets the requirements of the international standard UIC Code 519 [50]. In works [43, 45, 46, 47, 48, 49], the optimality criterion is taken as the minimum value of equivalent stresses in the contact zone, a decrease in the wear rate of the rail and wheel material, or a decrease in the accumulation rate of contact fatigue damage. The data presented in these works show that the influence of the wheel-rail friction pair profiles is significant, and this factor should be taken into account first.