Finite Element Modelling for Buckling Analysis of Tapered Axially Functionally Graded Timoshenko Beam on Elastic Foundation

Abstract

In this study, an efficient finite element model with two degrees of freedom per node is developed for buckling analysis of axially functionally graded (AFG) tapered Timoshenko beams resting on Winkler elastic foundation. For this, the shape functions are exactly acquired through solving the system of equilibrium equations of Timoshenko beam employing the power series expansions of displacement components. The element stiffness matrix is then formulated by applying the developed shape functions to the total potential energy along the element axis. It is demonstrated that the resulting shape functions, in comparison with Hermitian cubic interpolation functions, are proportional to the mechanical features of beam element including the geometrical properties, material characteristics, as well as the critical axial load. An exhaustive numerical example is implemented to clarify the efficiency and simplicity of the proposed mathematical methodology. Furthermore, the effects of end conditions, material gradient, Winkler parameter, tapering ratio, and aspect ratio on the critical buckling load of AFG tapered Timoshenko beam are studied in detail. The numerical outcomes reveal that the elastic foundation enhances the stability characteristics of axially non-homogeneous and homogeneous beams with constant or variable cross-section. Moreover, the results show that the influence of non-uniformity in the cross-section and axially inhomogeneity in material characteristics play significant roles in linear stability behavior of Timoshenko beams subjected to different boundary conditions.