# Approximate solutions for the problem of a load moving on the surface of a half-plane

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## Abstract

The classical problem of determining the dynamic response of an elastic half-plane to a load moving with constant speed on its surface is revisited. The problem is first solved analytically in an exact manner by a simple and efficient method that employs complex Fourier series involving the horizontal coordinate and the time to reduce the partial differential equations of motion into ordinary ones, which can be easily solved to provide the system response. Then, the problem is solved again by the same method under various simplifying assumptions that effectively reduce the system of two partial differential equations of the problem into a single equation. These assumptions are zero horizontal displacement, zero horizontal normal stress and zero horizontal normal stress plus zero derivative of the horizontal displacement with respect to the vertical coordinate. The resulting three approximate solutions are much easier to derive and simpler than the exact solution but do not satisfy the zero shear stress on the surface boundary condition and the equation of motion along the horizontal direction. Nevertheless, comparison of these approximate solutions against the exact solution by means of numerical parametric studies demonstrates that only one of them is practically acceptable in the range of sub-Rayleigh load speeds, which are of interest in road pavement dynamics.

## Keywords

Moving load Elastic half-plane Exact solution Approximate solutions Approximation errors## Preview

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