On the problem of a Timoshenko beam bonded to an elastic half-plane

The contact problem of beams, rods, ribs and plates bonded to a half-plane has been widely investigated by many Authors. In particular, the problem of prismatic beams resting on a finite or semi-infinite elastic substrate deserves great interest because its practical applications in many engineering application. As an example, Shield and Kim (1992) investigated the problem of an Eulero-Bernoulli beam resting on an elastic half-plane under symmetric loading conditions, founding the interfacial stresses as well as the SIFs at the edges of the beam. The Authors also studied the effect induced by an elastic-perfectly plastic cohesive interface. Nonetheless, a complete analytical study of the contact problem of a Timoshenko beam bonded to a half-plane cannot be found in literature. The present study concerns the contact problem of a Timoshenko beam resting on an elastic half-plane under general edge loading. The problem is solved by imposing the strain compatibility condition between the beam and the half-plane leading to a. system of 2 integral equations, which is transformed to an algebraic system equations by using series expansions in Jacobi polynomials for the displacement field and Chebyshev polynomials for shear and peeling stresses along the interface.