Infinite plate loaded with normal force moving along a complex path

Abstract

Introduction. A technique of solving the problem on an infinite plate lying on an elastic base and periodically loaded with a force that moves along an arbitrary closed trajectory and according to an arbitrary law; is considered. Materials and Methods. An original method for solving problems on the elasticity theory for plates loaded with a force moving arbitrarily along a closed trajectory of arbitrary shape is considered. The problem on an infinite plate lying on an elastic foundation is investigated. The plate is loaded with a normal force moving at a variable speed. The load is decomposed into a Fourier series on a time interval whose length is equal to the time of its passage along the trajectory. The solution to this problem is realized through a superposition of solutions to the problems corresponding to the load defined by the summands of the specified Fourier series. The final problem solution is presented in the form of a segment of the Fourier series, each summand of which corresponds to the solution to the problem on the action on an infinite plate of the load distributed along a closed trajectory of the force motion. The fundamental solution to the vibration equation of an infinite plate lying on an elastic foundation is used to construct these solutions. Results. A solution to the problem of an infinite plane, along which a concentrated force moves at a variable speed, is presented. A smooth closed curve consisting of arcs of circles was considered as a trajectory. The behavior of displacements and stresses near the moving force is investigated; and the process of the elastic wave energy propagation is also studied. For this purpose, a change in the Umov-Poynting vector is considered. Discussion and Conclusions. The results obtained can be used in calculations for road design. The study of the propagation of the energy of elastic waves from moving vehicles will provide the assessment of the impact of these waves on buildings located near the road. Analysis of the behavior of displacements and stresses near the moving force will allow assessing the wear of the road surface.