Laplace transform method for the deformation of an Euler–Bernoulli beam on a piecewise homogeneous foundation with moving point load

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The enhancement of railway infrastructure is increasingly important in meeting demand for heavier loads and higher-speed trains. Transition zones (e.g. bridges and culverts), where foundation properties change, are known to degrade more rapidly than standard track and so a comprehensive understanding of their mechanics is important. Available numerical methods capture complex geometries and behaviours however they are computationally intensive. It is of interest, therefore, to develop the availability of semi-analytical solutions for use in prototyping and comparison. The time-dependent response of an infinite Euler‒Bernoulli beam on a piecewise-homogeneous viscoelastic foundation with a moving point load is currently without a published solution. We present a novel solution using the Laplace transform and the method of undetermined coefficients. The Laplace-domain solution is given analytically in terms of a linear system and is inverted numerically. We compare three representative inversion algorithms (direct quadrature, Weeks, and Gaver–Wynn rho). Weeks' method performs satisfactorily at low velocities and is advantageous if the solution is required at many times; however, it fails to converge for realistic velocities. The direct quadrature method gives reliable convergence at higher velocities despite Gibbs phenomena at small times. Gaver‒Wynn rho consistently gives orders-of-magnitude better convergence with similar computation times.

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