The problem of infinite plate loaded with normal force following a complex trajectory*

Abstract

Introduction. A method for solving the problem of an infinite plate on an elastic foundation is proposed. The plate is affected by a periodic load in the form of a force following an arbitrary closed path. The work objective is to develop a numerical method for solving problems of the elasticity theory for bodies under a moving load. Materials andMethods. Given the periodicity of the load under consideration, it is decomposed in a Fourier series in a time interval whose length is equal to the load period. The solution to the original problem is constructed by superposition of the solutions to the problems corresponding to the load specified by the terms of the Fourier series described above. The final solution to the problem is presented as a segment of a series. In this case, each term corresponds to the solution of the problem of the impact on an infinite plate of a load distributed along a closed curve (the trajectory of the force motion). To find these solutions, the fundamental solution to the equation of vibration of an infinite plate lying on an elastic base is used.Research Results. A new method is proposed for solving problems on the elasticity theory for bodies with a load following a closed path of arbitrary shape. The problem of an infinite plane along which a concentrated force moves at a constant speed is solved. It is determined that the trajectory of motion is a smooth closed curve consisting of circular arcs. The behavior of displacements and stresses near a moving force is considered. The energy propagation of the elastic waves is studied. For this purpose, the coordinates of the Umov – Poynting vector are calculated. The effect of the force motion speed on the length of the Umov – Poynting vector is investigated.Discussion and Conclusions. The method is applicable when considering more complex objects (plates of complex shape, layered plates, viscoelastic plates). Its advantage is profitability since the known problem solutions are used to build the solution. The final decision is expressed in a convenient form – as the sum of curvilinear integrals. The results obtained can be used in the road design process. Studying the energy propagation of elastic waves from moving vehicles will enable to evaluate the impact of these waves on buildings near the road. The wear of the pavement is estimated considering data on the behavior of displacements and stresses

Keywords

• infinite plate
• moving force
• arbitrary closed path
• energy of elastic waves