Numerical dynamic analysis of beams on nonlinear elastic foundations under harmonic moving load

As a main background practical context of the present numerical investigation, the appropriate description of track vibrations induced by high-speed trains looks crucial in contemporary railway engineering. The present paper is concerned with the modelization of the transient dynamic response of a simply-supported Euler-Bernoulli beam resting on a homogeneous in space Winkler elastic foundation, under the action of a transverse concentrated load with harmonic-varying magnitude, moving at constant velocity along the beam. Two types of constitutive laws are considered for the foundation subgrade reaction: (a) a linear law and (b) a nonlinear, cubic law. The governing linear/non-linear partial differential equation of motion is first semi-discretized in space with a Finite Element Method approach, by using cubic Hermitian polynomials as interpolation functions for the unknown deflection. Then, the dynamic solution is obtained numerically by a direct integration method, with focus on determining several characteristic response features, such as the critical velocities of the moving load, leading to high transverse deflections. Extensive numerical analyses are finally performed, with the following two main goals: (1) to demonstrate the reliability, consistency and accuracy of the present implementation, specifically by the comparison of the obtained numerical critical velocities with previously-published analytical and numerical results; (2) to investigate how the frequency of the harmonic moving load as well as its velocity do influence the response of the whole beam-foundation system, with or without taking viscous damping into account. Results show that such goals have been consistently achieved and outline new interesting trends, like the appearance of two critical velocities also for the nonlinear foundation, the first of which gets close to zero as the frequency of the load approaches the first natural frequency of the beam. The present outcomes reveal potential implications in practical terms, especially in lowering the ranges of admissible train speeds, as for structural requirement or for preventing passenger discomfort.