Elastic foundation beam analysis with horizontal friction and double shear effects

Considering the shear capacity and horizontal friction of elastic foundation and the shear deformation effect of beam, the equilibrium equation for Timoshenko beam resting on a two-parameter foundation was derived. The initial parameter solution and transfer matrix method were deduced. Using the initial parameter, a finite element formulation and the equivalent nodal forces of the distributed load, concentrated load and concentrated moment were presented. When the horizontal friction rigidity was zero, the foundation model degenerated into the classical two-parameter foundation, when the shear rigidity was zero, Winkler foundation was obtained. If the shear rigidity of beam approached infinity, Timoshenko beam can be converted into Euler beam. The present model was a general model. Using transfer matrix method and finite element method, the influences of different foundation modes on Timoshenko beam with free-free ends under concentrated moment and horizontal friction on stepped Timoshenko beam with hinged-free ends under concentrated load, moment and distributing load were analyzed. Example results show that the transfer matrix method is consist with the finite element method, the accuracy of finite element method does not depend on the mesh density, the horizontal friction may significantly affect elastic foundation beam.