A New Finite Element Formulation for Buckling and Free Vibration Analysis of Timoshenko Beams on Variable Elastic Foundation

In this study, the buckling and free vibration of Timoshenko beams resting on variable elastic foundation analyzed by means of a new finite element formulation. The Winkler model has been applied for elastic foundation. A two-node element with four degrees of freedom is suggested for finite element formulation. Displacement and rotational fields are approximated by cubic and quadratic polynomial interpolation functions, respectively. The length of the element is assumed to be so small, so that linear variation could be considered for elastic foundation through the length of the element. By these assumptions and using energy method, stiffness matrix, mass matrix and geometric stiffness matrix of the proposed beam element are obtained and applied to buckling and free vibration analysis. Accuracy of obtained formulation is approved by comparison with the special cases of present problem in other studies. Present formulation shows faster convergence in comparison with conventional finite element formulation. The effects of different parameters on the stability and free vibration of  Timoshenko beams investigated and results are completely new.