Application of generalised functions and Fourier’s integral transformation in modelling the effect of a moving load on a beam resting on an elastic foundation

Abstract

The paper is dedicated to modelling the behaviour of an infinitely long beam resting on a solid elastic foundation under the concentrated moving load effect. The modelling is carried out in the context of determining the high-speed vehicle stock impact on the track structure. When developing the method of problem solving, the known elasticity theory factors, wave-propagation theory, complex variable theory, as well as integral transformations of generalised functions were used. The mobile concentrated force is represented by the Dirac delta function. The Fourier’s integral transformation is used to solve the equation of motion and determine the beam defect. Contour integration using Cauchy residue theorem is applied to calculate the defect values in the original space. The general dependence form the beam defect on the load velocity is determined. It is shown that the defect values in the beam increase with moving load increasing speed. It is shown how the moving load velocity affects the character of the beam defect time dependence. Solutions are given for Euler-Bernoulli, Rayleigh and Timoshenko models of beams on an elastic base. The considered mathematical models behaviour is compared. It is shown that the character of beam interaction with the moving load, the defect values, as well as the reaching critical velocities character do not differ significantly for the considered beam models. The character of reaching the maximum defect in the application area of the mobile load is analysed. It is shown that the maximum defect values of beams on an elastic base occur after the moving load. The presented methodology can be used to evaluate the impact of rolling stock on the track structure, including the design of high-speed lines.