An alternative approach to assessing the stability of continuous welded rail
The article presents a mathematical model for the loss of stability in a continuous welded rail system, based on the analysis of the deformed shape of an equivalent beam that simulates the rail-sleeper grid. The deformed area of an equivalent infinitely long beam divided into a buckling region characterized by transverse displacements and adjacent regions where only compression occurs. The length of the regions is unknown and determined during the solution. The distinctive feature of the proposed model is to take into account the release of compressive forces after loss of stability, as well as the non-linearity of deformations, but with the preservation of the Bernoulli hypothesis. The law of changing the compressive force depends on the accepted law of changing the resistance of the environment in which the deformation occurs. Model has been improved to account for the nonlinearity of resistance to longitudinal and transverse cross-section displacements of the equivalent beam. Resistance functions are obtained by the approximation of experimental points using the least squares method, non-linear in case of transverse resistance approximation. The system of non-linear differential equations, which is closed by boundary conditions and transversality conditions, is solved by the finite difference method. The difference equations use central difference formulas, which require a minimum length of the template. The solution is made by a method of successive approximations, ensuring a sufficient number of grid nodes. The results of track stability parameters calculations are presented. Including the critical value of the heating temperature of the rails above the fixing temperature, at which a stable deformed shape can exist. The results of the solution according to the refined model are compared with the results of the model with constant resistance and with the results of the model without the release of the compressive force. Refinement of the model by taking into account non-linearities of various natures is important for assessing the pre-critical state of a continuous welded track.