## 2 Description of the Assumed Semi-Analytical Model  
The applied model is an expansion of [PERSON]'s work [6]. [PERSON]'s work was related to vertical planes, and this method was adapted to the horizontal plane [5]. In the presented calculation model, a coordinate system (x,y) was assumed, which determines symmetry according to the y axis. The length of the deformation wave is 2 L, and the maximum displacement \(\delta\) will take place in the middle part of the system. At the ends of a buckled section, there are reactions: from lateral forces V, horizontal forces H, which are different from the values of longitudinal forces in beam N, and bending moment M\({}_{0}\)[5]. The assumed model is presented in Figure 1.  
* In the analyses, the following assumption was introduced: ballast resistance in lateral direction r\({}_{0}\) (kN/m) has constant value, independent from x and the lateral displacement y;
* Longitudinal resistance of beam has elastic character, described with the constant k (kN/m\({}^{2}\)), in the beam deformation zone this resistance was skipped--it is present only in sections adjacent to the deformation zone;
* Rails create a substitute beam with bending stiffness B (kN/m\({}^{2}\));  
Figure 1: Calculation model for constant resistance of the ballast r\({}_{0}\)[5]. * Area of cross-section of substitute beam A (m\({}^{2}\)) is equal to the sum of the areas of cross-sections of two rails;
* Thermal force and force other than thermal N (kN) are applied along the beam;
* Equilibrium for unstable and pre-buckling equilibrium path of CWR track is a symmetrical system in relation to point x = 0 of the assumed coordinate system. Basic equation of the equilibrium for system has a well-known simplified form: \[\text{B}\cdot\text{y}^{\prime\prime}=-\text{M}(\text{x})\] (3) where the bending moment describes the formula (comp. Figure 1): \[\text{M}(\text{x})=\text{M}_{0}+0.5\cdot\text{r}_{0}\cdot(\text{L}-\text{x})^{ 2}-\text{r}_{0}\cdot\text{L}\cdot(\text{L}-\text{x})+\text{H}\cdot\text{y}\] (4) After simple transformations, one can obtain: \[\text{M}(\text{x})=\text{M}_{0}+0.5\cdot\text{r}_{0}\cdot\text{x}^{2}-0.5 \cdot\text{r}_{0}\cdot\text{L}^{2}+\text{H}\cdot\text{y}\] (5) Substituting formula (5) to Equation (3), after rearrangement and knowing that M\({}_{0}\) = 0 (comp. [6]), differential equation of the system has the form: \[\text{y}^{\prime\prime}+\omega^{2}\cdot\text{y}=\text{\Psi}(\text{x})\] (6) where \[\omega^{2}=\frac{\text{H}}{\text{B}}\] (7) \[\text{\Psi}(\text{x})=\frac{1}{\text{B}}\cdot\left[-0.5\cdot\text{r}_{0}\cdot \text{x}^{2}+0.5\cdot\text{r}_{0}\cdot\text{L}^{2}\right]\] (8) H--horizontal reaction on the ends of the buckling zone (H \(\
eq\) N). M\({}_{0}\)--bending moment at the end of the deformed section, L--half of the length of the deformed section, \(\text{r}_{0}\)--lateral resistance In considerations based on the general integral of homogeneous equation \[\text{y}=\text{C}_{1}\cdot\cos(\omega\cdot\text{x})+\text{C}_{2}\sin(\omega \cdot\text{x})\] (9) Its interpretation, transformation into a particular integral of nonhomogeneous equation in the form of quadratic function, and reaching the solution are significant [5]: \[\text{y}=\text{C}_{1}\cdot\cos(\omega\cdot\text{x})+\text{C}_{2}\sin(\omega \cdot\text{x})-\frac{\text{r}_{0}}{2\text{H}}\cdot\text{x}^{2}+\frac{1}{ \text{H}}(\frac{\text{r}_{0}\cdot\text{L}^{2}}{2}+\frac{\text{r}_{0}\cdot \text{B}}{\text{H}})\] (10)  
To solve the above equation and obtain the aforementioned system symmetry, the following boundary conditions are assumed:  
\[ \begin{array}{l}\frac{\text{d}\text{y}}{\text{d}\text{x}}(\text{x}=0)=0\\ \text{y}(\text{x}=\text{L})=0\end{array}  \tag{11}\]  
As in Equation (6), there are two parameters the values of which we do not know, i.e., force at the end of the buckled section H, and the length of the buckled section L, we need to assume an additional boundary condition.