A General Procedure for the Dynamic Analysis of Finite and Infinite Beams on Piece-Wise Homogeneous Foundation under Moving Loads

Abstract Transversal vibrations induced by a load moving at a constant speed along a finite or an infinite beam resting on a piece-wise homogeneous visco-elastic foundation are studied. Special attention is paid to the amplification of the vibrations which arise as the point load traverses a foundation discontinuity. The governing equations of the problem are solved by the normal-mode analysis. The solution is expressed in the form of an infinite sum of orthogonal natural modes multiplied by the generalized displacements. The natural frequencies are obtained numerically exploiting the concept of the global dynamic stiffness matrix. This ensures that the frequencies obtained are accurate. The methodology is neither restricted by load velocity nor damping and is simple to use, though obtaining the numerical expression of the results is not straightforward. A general procedure for numerical implementation is presented and verified. There is no restriction for finite structures, however, for infinite structures, validity of the results is restricted to a “region of interest” of finite length. To illustrate the methodology, the probability of exceeding an admissible upward displacement is determined when the load travels at a certain velocity according to the normal distribution. In this problem, the given structure has an intermediate part of adaptable foundation stiffness, which is optimized in a parametric way, enabling to draw important conclusions about the optimum intermediate stiffness. The results obtained have direct application on the analysis of railway track vibrations induced by high-speed trains crossing regions with significantly different foundation stiffness.