Lateral vibrations of an axially compressed beam on an elastic half-space due to a moving lateral load

The steady-state response of an axially compressed Euler-Bernoulli beam on an elastic half-space due to a uniformly moving lateral load has been investigated. It is assumed that the beam has a finite width and that the half-space and beam deflections are equal along the center line of the beam. To analyze the problem, firstly the equivalent lateral stiffness of the half-space is derived as a function of the phase velocity of waves in the beam. Then using the expressions for the equivalent stiffness, a dispersion relation is obtained for the lateral waves in the beam. Analyzing this equation, it is shown that lateral waves can propagate in the beam only when the axial force in the beam is larger than a 'cut-off compressional force'. The critical (resonance) velocities of a uniformly moving constant and harmonically varying load are determined as functions of the axial compressional force in the beam. It is shown that the critical velocity of the harmonically varying load is always smaller than that of the constant load. A comparison is made between the critical velocity of a vertical and lateral constant load showing that the lateral constant load is smaller.