Mathematical Model for Dynamic Behavior of Elastic Euler-Bernoulli Beam under Moving-Induced Distributed Load

This work examines the transient deflection of a simply supported Euler–Bernoulli beam on a Winkler elastic foundation subjected to a finite-length distributed load moving at constant speed. The excitation has either a constant-amplitude or sinusoidally modulated form and includes a dimensionally specified power-law induced-load factor. To correct the ambiguity in an earlier moving-front representation, the applied loading is confined to a finite contact footprint with explicit entry and exit from the span. A Fourier-sine expansion reduces the governing fourth-order beam equation to modal oscillators, which are evaluated using Duhamel convolution and average-acceleration Newmark integration. Independent two-node cubic Hermite beam elements provide numerical verification. For a representative 50.192 m beam with a 3 m footprint, 12.5 m/s speed, and Winkler coefficient of 0.20 MN/m², the 32-mode constant-load solution gives a peak midspan deflection of 4.5964 mm, compared with 4.5964 mm from a 100-element finite-element model. Increasing the induced-rate parameter from 0 to 0.20 s⠻¹ increases the calculated constant-load peak from 4.5964 to 6.6440 mm. Under the specified undamped harmonic forcing at the first supported-beam natural frequency, the peak reaches 54.2900 mm, although the outcome is strongly frequency dependent. Modal convergence, point-force limiting behaviour, finite-element agreement, frequency estimates, and foundation-stiffness and velocity sweeps support the consistency of the revised calculations. The results establish parameter-dependent amplification for the stated linear moving-force model, not a universal increase under harmonic excitation. Limitations concerning moving-mass inertia, damping, support variability, and experimental calibration are identified.