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

Abstract

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.