Critical velocity of a uniformly moving load on a beam supported by a finite depth foundation

In this paper, a new formula for the critical velocity of a uniformly moving load is derived. It is assumed that the load is traversing a beam supported by a foundation of a finite depth. Simplified plane models of the foundation are presented for the analysis of finite and infinite beams, respectively. Regarding the model for finite beams, only the vertical dynamic equilibrium is considered. Then the critical velocity obeys the classical formula with an augmented mass that adds 50% of the foundation mass to the beam mass. In the model for infinite beams, the effect of shear is added in a simplified form and then the critical velocity is dependent on the mass ratio defined as the square root of the fraction of the foundation mass to the beam mass. For a low mass ratio, the critical velocity approaches the classical formula and for a higher mass ratio, it approaches the velocity of propagation of shear waves in the foundation. The formula can also account for the effect of the normal force acting on the beam. Deflection shapes of the beam are obtained semi-analytically and the influence of different types of damping is discussed.