# Calculation Schemes for Determining Contact Stresses in Railway Rails
## 2 Materials and Methods
### Analytical Solution of the Contact Problem  
The problem of normal contact between two elastic massive bodies, where the equations of the contacting surfaces are quadratic, was first solved by [PERSON] in 1882. He determined that when the characteristic size of the contact area is significantly smaller than the characteristic radii of curvature of the contacting surfaces, the contact area is elliptical. Simple formulas for determining the dimensions of the contact area, contact pressure distribution, and stress-strain parameters remain relevant today--they are used in many normative documents to estimate stress levels in the contact of various components (gear teeth, railway rails and wheels, rolling bearings, etc.). Another important application of these formulas is the verification and calibration of numerical methods used to solve more complex contact problems. Furthermore, as will be shown later, [PERSON]'s solution is used to build simplified models of boundary conditions for force loading when solving contact problems using numerical methods; hence, we will examine it in more detail. According to [PERSON]'s theory, under normal contact conditions (friction forces are neglected, and shear stresses on the contact area are absent), the distribution of contact pressure (Figure 2) on the elliptical area can be described as follows [16]:  
\[p(x,y)=p_{max}\sqrt{1-\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}},\quad p_{max}= \frac{3P}{2\pi ab}, \tag{1}\]  
where \(a\) and \(b\)--are the major and minor semi-axes of the ellipse along the principal axes \(x\) and \(y\), respectively; \(P\)--is the resultant normal force. The lengths of the ellipse axes are calculated using the formulas:  
\[a=m\sqrt[3]{\frac{3\pi}{4}\frac{P(K_{1}+K_{2})}{A+B}},\quad b=n\sqrt[3]{\frac {3\pi}{4}\frac{P(K_{1}+K_{2})}{A+B}}. \tag{2}\]  
Positive constants \(A,\ B\) are determined from the system of equations:  
\[ \begin{array}{c}A+B=\frac{1}{2}\Big{(}\frac{1}{r_{11}}+\frac{1}{r_{12}}+ \frac{1}{r_{22}}+\frac{1}{r_{21}}\Big{)},\\ B-A=\frac{1}{2}\Big{[}\Big{(}\frac{1}{r_{11}}-\frac{1}{r_{12}}\Big{)}^{2}+ \Big{(}\frac{1}{r_{22}}-\frac{1}{r_{21}}\Big{)}^{2}+2\Big{(}\frac{1}{r_{11}}- \frac{1}{r_{12}}\Big{)}\Big{(}\frac{1}{r_{22}}-\frac{1}{r_{21}}\Big{)}cos2\psi \Big{]}^{1/2},\end{array}  \tag{3}\]  
where \(r_{11},\ r_{12},\ r_{22},\ r_{21}\)--are the principal relative radii of curvature of the contacting surfaces. In the case of contact between a new rail and wheel,  
\(r_{11}=R_{w}\)--the radius of curvature of the wheel at the contact point (nominal wheel radius at the rolling circle). \(r_{12}=r_{w}\)--the radius of curvature of the wheel profile (for a conical profile \(r_{12}\rightarrow\infty\));  
\(r_{22}=r_{r}\)--the radius of curvature of the rail head at the contact point, \(r_{21}\rightarrow\infty\) for a straight rail in the vertical plane. \(\psi\)--is the angle between the planes in which the principal radii of curvature \(r_{11},\ r_{12}\) (for standard railway wheels with a conical profile \(\psi=0\)). For a railway wheel according to the DSTU 4835:2008 standard [22] and a rail according to the DSTU 4344:2004 standard [23] under central contact conditions: \(R_{w}=478.5\ {\rm MM}\), \(r_{r}=500\ {\rm MM}\). For the so-called conformal wheel profile [24], where the contour of the ridge zone (starting from the nominal rolling circle and ending with the radial transition to the ridge) is a mirror reflection of the corresponding part of the head of a moderately worn rail, \(r_{w}=514\ {\rm MM}\). The dimensions of the contact area depend on the relationship between the elastic properties of the wheel and rail materials. Therefore, coefficients are included in Formula (2):  
\[K_{1}=\frac{1-v_{1}^{2}}{\pi E_{1}},\quad K_{2}=\frac{1-v_{2}^{2}}{\pi E_{2}}, \tag{4}\]  
Figure 2: Distribution of normal pressure on the contact area. where \(E_{1}\), \(E_{2}\) are the moduli of elasticity, and \(v_{1}\), \(v_{2}\) --are the Poisson's ratios of the wheel and rail materials, respectively.