Equivalent stiffness of a half-space interacting with a beam. Critical velocities of a moving load along the beam

Expressions are derived for the equivalent stiffness of an elastic half-space interacting with a beam with finite width. It is shown that the equivalent stiffness depends on the frequency and the wave number of the waves in the beam. The dependency is severe when the phase velocity of the waves in the beam is close to the velocity of the Rayleigh waves in the half-space. Expressions for the equivalent stiffness for long waves relative to the beam width are derived from general results and are approximated in terms of elementary functions. Critical velocities of a constant load moving at constant speed along a Euler-Bernoulli beam are determined. It is shown that there are two critical velocities, the first one equal to the velocity of the Rayleigh waves in the half-space and the second one mostly slightly smaller.