Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beam. Cubic displacement polynomial and quadratic rotational fields are selected for this element. Moreover, it is assumed that shear strain of the element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energy and its stationing to shear strain. By exploiting these interpolation functions, beam elements' stiffness matrix is also examined. Furthermore, geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation on stability and beam free vibration. At last, accuracy and efficiency of proposed element are 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. 1. Introduction Two versions of theories have been developed for analysis of beams. In Euler-Bernoulli theory, the displacement of beam is considered without shear effects. This method gives appropriate and acceptable response in thin beam in which shear effect is insignificant. However, in this approach, by increasing the thickness of beam and shear effect deformation, the error of response is increasing [1]. Correspondingly, the effect of shear transformation is formulated in Timoshenko theory. Therefore, this method has a better result, especially in deep beams in which shear effect is impressive. Although the rotational inertia of thick beams was investigated by Rayleigh for the first time, Timoshenko has developed this theory and formulated shear effect. Due to the complexity of the governing equations of the free vibration and stability of beams in general, numerical methods such as finite element have been developed profoundly. Up to now, many elements have been presented based on Timoshenko theory. These elements are classified into two groups which are simple and higher order elements. Some researchers used simple two-node elements with four degrees of freedom [2–4]. Thomas et al. have examined the elements proposed by other researchers [3]. The first high-order element was proposed by Kapur with eight degrees of freedom [5]. Lees and Thomas formulated a complex element by applying independent polynomial series for displacement and rotation fields [6, 7]. Also, this method has been used by Webster [8]. Rao and Gupta have examined free vibration of rotating beams [9]. In some methods, like isoparametric formulation, displacement and rotation fields are