The problem of the assessment of the continuous welded rail track stability at its thermal elongation is considered in this paper. The method for determining the stress-strain state of various elastic objects developed by the specialists of JSC “VNIKTI” is analyzed on the example of a rail of a continuous-welded track section. The essence of the method is to determine the dependence of the natural frequencies of rail vibrations on the applied longitudinal force. As a means of obtaining such a dependence, a modal analysis of a track section model carried out using the finite element method was chosen. The methodology for calculating the natural frequencies of rail vibrations and the description of the created model of the track section and its properties are presented in the paper, the contact interactions of the model elements and boundary conditions are defined in it. Calculations for determining the natural frequencies and modes of rail vibrations are performed, the first four modes of rail vibrations obtained by calculation and experimental methods are shown. The assessment of the adequacy of the created model of the track section is carried out by comparing the natural frequencies and modes of rail vibrations obtained by calculation, by modal analysis methods and experimentally when measuring vibrations on a full-scale object - assembled rails and sleepers, having a similar design compared with the model. It was decided to use the distance between the vibration nodes as a criterion for comparing the obtained modes of natural rail vibrations. The first mode of vibration is selected for further studies. Calculations were carried out and the dependence of the natural frequency of the first vibration mode of the rail on the longitudinal force applied to it was obtained using the created model of the track section. The obtained dependence can be used in the method proposed by the specialists of JSC “VNIKTI” in order to determine longitudinal force, applied to the rails on the full-scale section of the continuous welded rail track.