Finite Element Formulation for Free Vibration and Stability of Timoshenko beam

In this paper، a two-node element has been suggested for analyzing stability and free vibration of Timoshenko beam. A cubic displacement polynomial and a quadratic rotational field are selected for this element. Moreover، it is assumed that shear strain of element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by writing total beam energy and its stationing to shear strain. By exploiting these interpolation functions، beam stiffness matrix is also examined. Then، geometric stiffness matrix and mass matrix of the proposed element are also calculated by writing governing equation on stability and beam free vibration. At last، accuracy and efficiency of proposed element is evaluated through numerical tests. These tests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis.