Concentrations of Pressure Between an Elastically Supported Beam and a Moving Timoshenko-Beam

The present paper is concerned with the motion of an elastically supported beam that carries an elastic beam moving at constant speed. This problem provides a limiting case to the assumptions usually considered in the study of trains moving on rail tracks. In the literature, the train is commonly treated as a moving line-load with space-wise constant intensity, or as a system of moving rigid bodies supported by single springs and dampers. In extension, we study an elastically supported infinite beam, which is mounted by an elastic beam moving at a constant speed. Both beams are considered to have distributed stiffness and mass. The moving beam represents the train, while the elastically supported infinite beam models the railway track. The two beams are connected by an interface modeled as an additional continuous elastic foundation. Here, we follow a strategy by Stephen P. Timoshenko, who showed that a beam on discrete elastic supports could be modeled as a beam on a continuous elastic Winkler (one-parameter) foundation without suffering a substantial loss in accuracy. The celebrated Timoshenko theory of shear deformable beams with rotatory inertia is used to formulate the equations of motion of the two beams under consideration. The resulting system of ordinary differential equations and boundary conditions is solved by means of the powerful methods of symbolic computation. We present a nondimensional study on the influence of the train stiffness and the interface stiffness upon the pressure distribution between train and railway track. Considerable pressure concentrations are found to take place at the ends of the moving train.