Exact closed-form solutions for Lamb's problem (II): a moving point load

TL;DR

This paper derives an exact closed-form solution for the displacement caused by a moving point load on an elastic half-space, extending Lamb's problem to include moving loads, with potential applications in high-speed rail analysis.

Contribution

It provides the first explicit closed-form solution for a moving point load in Lamb's problem, expanding the classical static case to dynamic moving loads.

Findings

Closed-form solutions match numerical results, confirming accuracy.

Solution expressed with elementary functions and elliptic integrals.

Foundation for analyzing real-world moving loads like high-speed trains.

Abstract

In this article, we report on an exact closed-form solution for the displacement in an elastic homogeneous half-space elicited by a downward vertical point source moving with constant velocity over the surface of the medium. The problem considered here is an extension to Lamb's problem. Starting with the integral solutions of Bakker \textit{et al.}, we followed the method developed in Feng and Zhang, which focuses on the displacement triggered by a fixed point source observed on the free surface, to obtain the final solution in terms of elementary algebraic functions as well as elliptic integrals of the first, second and third kind. Our closed-form results agree perfectly with the numerical results of Bakker \textit{et al.}, which confirms the correctness of our formulas. The solution obtained in this article may lay a solid foundation for further consideration of the response of an actual physical moving load, such as a high-speed rail train.