Title: Exact Shape Functions for Timoshenko Beam Element
Abstract: This paper derives exact shape functions for both non-uniform (non-prismatic section) and inhomogeneous (functionally graded material) Timoshenko beam element formulation explicitly.In this paper, the shape functions formula embedded the explicit functions and its derivatives describing the non-uniformity and inhomogeneity of a beam element.The shape functions are made interdependent by requiring them to satisfy three homogeneous differential equations associated with the Timoshenko's beam theory.With the formulated axial, transverse and rotational displacement shape functions, the stiffness and mass matrices and consistent force vector for a two-node Timoshenko beam element are developed based on Hamilton's principle.Comparison studies with reference work on the accuracy and computational efficiency for non-uniform and inhomogeneous Timoshenko beam structures are highlighted.Static and vibrational analyses of the beams element by using the exact shape functions can predict the displacement, and natural frequencies of nonuniform and inhomogeneous Timoshenko beams by using only one/the least element accurately.