Exact stiffness-matrix of two nodes Timoshenko beam on elastic medium. An analogy with Eringen model of nonlocal Euler-Bernoulli nanobeams

The paper deals with a general method to obtain a closed-form analytical solution of the problem of bending of a shear deformable beam resting on an elastic medium. Within a well posed analytical framework, the basic equations governing the interaction problem, can be obtained in strong form by a differential formulation based on both the constitutive equation of the Timoshenko beam and the direct and inverse constitutive equation of the supporting local elastic medium. A general finite element is then derived for shear deformable beams with or without a continuous Winkler type elastic support. The obtained analytical results are discussed in the light of nonlocal elasticity of Eringen differential type, applied to an Euler-Bernoulli beam model. As a result the stiffness -matrix and equivalent nodal loads of an Euler-Bernoulli nonlocal elastic beam, can be defined in analogy to those of a first order shear deformable beam. This conclusion allows handling the elastostatic problem of nanobeams, modelled according to Eringen ' s nonlocal elasticity, by slight modifications of the existing computational tools for the solution of the elastostatic problem of a local shear deformable beam. (C) 2016 Elsevier Ltd. All rights reserved.