Vibration Analysis of a Euler-Bernoulli Beam on a Winkler Foundation Subjected to Two Moving Distributed Loads

This study investigates the dynamic vibration of a simply supported Euler-Bernoulli beam resting on a Winkler foundation and subjected to two simultaneous moving distributed loads of constant or harmonic magnitude. Closed-form modal solutions to the governing fourth-order partial differential equation are derived using the finite Fourier sine transform, Laplace transform, and convolution theorem. Equal and unequal load magnitudes are considered, and the effects of foundation stiffness, axial prestress, and the two load magnitudes are quantified. An independent 80-element Euler-Bernoulli finite-element solution integrated by the average-acceleration Newmark method validates the analytical response: the peak midspan-deflection difference is 0.17% and the normalized root-mean-square difference is 0.26% for the stated benchmark. The results show that foundation stiffness and tensile prestress reduce dynamic deflection, whereas either load magnitude increases it linearly away from resonance. For harmonic loading, three resonance pathways arise from the interaction between the load oscillation and translational modal frequencies. The formulation therefore provides a verified analytical basis for assessing beams and beam-like transport structures subjected to concurrent moving actions.

Distributed Loads, Fourier Sine Transformation, Bernoulli-Euler Beam, Winkler Foundation, Moving Loads, Prestress Forces

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