Finite Elements on Generalized Elastic Foundation in Timoshenko Beam Theory

Using the Vlasov foundation model, a modified approach of the continuous beam on elastic supports, leading to both a mechanical model and the proper foundation parameters of the generalized foundation is shown. Two formulations of the beam finite-element with shear deformation effect, resting on a two-parameter elastic foundation, characterized by distinct contributions of normal and rotary reactions are presented. The behavior of the second foundation parameter in the two formulations is governed by the bending cross section rotation of a beam. The first formulation, yielding a free-of-meshing stiffness matrix and equivalent nodal load vector, is based on the transcendental or “exact” solution of the governing differential equation of the beam resting on the elastic layer of constant thickness. Considering a linear variation of the layer thickness along the beam, the second formulation is based on the assumed polynomial displacement field. Numerical comparisons with the exact approach show that the cubic formulation leads to better results when the foundation parameters are variables. The practical utility of the analogy between a tensile axial force and the second foundation parameter is exemplified, too.