Closed-form solution for the Timoshenko beam on elastic soil

The theoretical model of elastic beams on springs is of great interest especially for its usage in many technical applications. The closed-form solution is obtained analysing the behaviour of an elastic beam on uniformly distributed springs. Each spring generates a reaction proportional to displacements. These reactions are continuous as commonly considered in the foundation beam models (Winkler's theory), where soil is modeled by a set of infinitely close bilateral springs, elastic and independent (main assumption). To complete the formulation of interaction the beam model is associated to the soil model. The problem has been extensively investigated and many works in the scientific literature can be found on Euler-Bernoulli model. Much less investigated is the interaction that takes into account the shear deformation contribution of the structural elements ("Timoshenko beam model"). The present paper is focused on derivation of a closed-form solution for the interaction problem of a Timoshenko beam model. In a first step usual differential equations have been analysed, while in a second step the stiffness matrix for beams has been developed. The objective has been to provide explicit expressions of stiffness matrix elements. This formulation represents the exact solution for the Timoshenko beam model on elastic soil, to be implemented in numerical analyses, as a single unified formulation even if soil interaction is inactive.