A loaded Timoshenko beam bonded to an elastic half plane

Abstract

The problem of a Timoshenko beam of finite length loaded by concentrated forces and couples and perfectly bonded to a homogeneous elastic and isotropic half plane is considered in the present work. In particular, the effects induced by shear deformation of the beam on the contact stresses arising at the interface between the beam and the underlying half plane are investigated accurately. An asymptotic analysis of the stress field at the beam ends and in the neighborhood of the loaded section of the beam allows us to characterize the singular nature of the peeling and shear stresses. The problem is formulated by imposing the strain compatibility condition between the beam and the half plane, thus leading to a system of two singular integral equations with Cauchy kernel. The unknown interfacial stresses are expanded in series of Jacobi orthogonal polynomials displaying complex singularity. This approach allows us to handle the oscillatory singularity and to reduce the integral equations to a linear algebraic system of infinite equations for the unknown coefficients of the interfacial stresses, which is solved through a method of collocation. The interfacial peeling and shear stresses and, in turn, the displacement field along the contact region have been calculated under various loading conditions applied to the beam. The internal forces and bending moments along the beam have been calculated varying the shear and flexural stiffness of the beam. The complex stress intensity factors and the strength of the stress singularities have been assessed in detail.