# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Computational Schemes for Rails--Wheels Contact Problems
The maximum equivalent contact stresses, calculated using the finite element method for a straight track section with the same rail type, were approximately 610 MPa at a depth of 3.12 mm [14]. Therefore, for tunnel track sections of metro systems, which exhibit high stiffness, the bending and torsion stresses can be neglected. Including these stresses may only affect the distribution of shear contact stresses \(\tau_{yz}\) and \(\tau_{zx}\), whose magnitudes are approximately 200 MPa [14]. For tracks on wooden sleepers, which have significantly lower stiffness in the vertical plane, greater rail deflections will result in a slight decrease in contact stresses due to an increased contact area. A significant factor affecting the level of contact stresses is the inclination angle of the rail base relative to the sub-rail foundation (rail inclination; see Figure 6), which is adjusted using special rubber pads. In Ukraine, the standard rail inclination value for railways with a gauge width of 1520 mm is 1:20. For European railways with a gauge width of 1435 mm, this angle can be 1:20, 1:30, or 1:40, depending on the country. The dependence of the maximum contact pressure on the rail inclination for two types of rails is shown in Figure 7. For UIC60 rails, the value of the contact pressure depends little on the rail inclination angle; only in the absence of a slope \(p_{max}\) does it increase significantly. Figure 6: Refined computational scheme for the interaction of rails with rolling stock wheels. Figure 7: Dependence of contact pressure on rail inclination. In [14], it is shown that in the case of a single-point contact of a railway wheel according to DSTU 4835:2008 [22] with rails R50, R65 according to DSTU 4344:2004 [23], the use of a 1:20 rail inclination provides minimal contact stress values \(\sim\) 600 MPa (Figure 8). This is due to the fact that there is no inclination of the conical part of the wheel rolling surface to the rail rolling surface and the condition \(\delta_{L}=\delta_{R}=0\) is fulfilled (see Figure 6). For nonlinear wheel profiles, the magnitude of contact stresses under these conditions increases significantly (up to 1020 MPa for a wheel with a conformal profile [24]), leading to the rapid wear of new rails. For UIC60 rails, the value of the contact stresses depends little on the rail inclination angle; only in the absence of a slope \(p_{max}\) does it increase significantly (up to 1092 MPa). The rail inclination can change due to the compression of pads, partial destruction of sleepers, fasteners, etc. If the rail inclinations in the track vary, the wheelset shifts toward the rail with a greater inclination angle. In this case, dynamic (impact) two-point contact occurs, with the contact patches located in the same transverse plane (see Figure 5). The absence of rail inclination significantly increases the level of contact stresses. A similarly significant impact on contact stress levels is exerted by the magnitude of the outer rail elevation in curved track sections. In such cases, the inclination angles \(\delta_{L}\) and \(\delta_{R}\) (see Figure 6) will be different in magnitude and will not be equal to zero. Consequently, contact stress levels will increase on both rails. However, in this situation, the contact patches on the outer rail will be located in different transverse planes, with the distance between them depending on the track curvature radius. When calculating the influence of rail elevation on contact stress levels, it is necessary to consider the possible redistribution of wheel loads on the rails caused by unbalanced centrifugal acceleration and the eccentric position of the vehicle's center of mass. This effect will be more pronounced for taller  
Figure 8: Dependence of contact stresses on rail inclination. vehicles. However, the magnitude of this influence on the level of contact stresses remains an open question, as there are few examples of such calculations in the literature [40].