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 arbitrary open cross sections. The proposed method takes the flexural–torsional coupling effect of tapered thin-walled beams with arbitrary open cross sections into account. The total potential energy is derived for an elastic behavior from the strain energy, the kinetic energy and work of the loads applied on the cross section contour. Free vibration is considered in the 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. As in the presence of tapering, stiffness quantities are not constant; therefore, 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, 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 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 the results of finite element analysis using Ansys software and those of other available numerical and analytical ones. It can be seen that the results of present study are in a good agreement with other available theoretical and analytical methods.