Beam on viscoelastic foundation: an extension of Winkler’s model

Models in the field of Applied Mechanics originate less from thought experiments but rather from technical problems. So does the so-called Winkler model: elastic beam on deformable foundation. It stems from the then (1870) High Technology the railway system. The question was: What is the stress state in a continuously bedded beam, the sleeper, loaded by singular rail loads? Winkler came up with a convincing closed form solution by linearising the behaviour of the subgrade which consists of a compressed layer of stones. The Winkler model is time-independent. The extension is to make the subgrade time dependent, in the simplest case to make it viscous. This also applies to the problem of a wheel set consisting of a beam ring, a rigid disc, and in between a pre-stressed rubber sheet. Such wheel sets are or have been in use in nowadays tram and railway systems. In this paper an analysis of such a rotating wheel set under a singular load is given. It is shown that the stress state in the ring beam depending on rotational speed decreases linearly with increasing rotational speed.