Finite Element Method for Stability and Free Vibration Analyses of Non-prismatic Thin-walled Beams

Abstract

In this paper, a numerical method is presented for the free vibration and stability analyses of tapered thin-walled beams with non-symmetric cross-section. The proposed model takes into account for the flexural–torsional coupling and tapering of thin-walled beams with arbitrary open cross sections. The total potential energy is derived for an elastic behavior from the strain energy, the kinetic energy and work of the applied loads. Free vibration is considered in presence of harmonic excitations. The effects of the initial stresses and load eccentricities are also considered in stability analysis. The governing equilibrium equations, motion equations and the associated boundary conditions are derived from the stationary condition. In presence of tapering, stiffness quantities are not constant. The power series approximation is used to solve the fourth-order differential equations. Displacement components and cross-section properties are expanded in terms of power series of a known degree. Then, the shape functions are obtained by deriving the deformation shape of tapered thin-walled member as power series form. Finally, the static stiffness and mass matrices are carried out by means of the principle of virtual work along the member’s axis. In order to measure the accuracy and to check the validity of this method, the natural frequencies and buckling loads of non-prismatic thin-walled beams with web and flange tapering and various boundary conditions are obtained and compared to finite element analysis result using Ansys software and other available numerical and analytical results . It can be seen the results of present study are in a good agreement with other available theoretical and analytical methods.