Finite Element Formulation for Linear Stability Analysis of Axially Functionally Graded Non-prismatic Timoshenko Beam

Abstract

An improved approach based on the power series expansions is proposed to exactly evaluate the static and buckling stiffness matrices for the linear stability analysis of axially functionally graded Timoshenko beams with variable cross-section and fixed-free boundary condition. Based on the Timoshenko beam theory, the equilibrium equations are derived in the context of small displacements, considering the coupling between the transverse deflection and angle of rotation. The system of stability equations is then converted into a single homogeneous differential equation in terms of bending rotation for the cantilever, which is solved numerically with the help of the power series approximation. All the mechanical properties and displacement components are thus expanded in terms of the power series of a known degree. Afterwards, the shape functions are gained by altering the deformation shape of the axially functionally graded non-prismatic Timoshenko beam in a power series form. At the end, the elastic and buckling stiffness matrices are exactly determined by the weak form of the governing equation. The precision and competency of the present procedure in stability analysis are assessed through several numerical examples of axially non-homogeneous and homogeneous Timoshenko beams with clamped-free ends. Comparison is also made with results obtained using ANSYS and other solutions available, which indicates the correctness of the present method.