Stress and free vibration analysis of functionally graded beams using static Green's functions

In this paper static Green's functions for functionally graded Euler-Bernoulli and Timoshenko beams are presented. All material properties are arbitrary functions along the beam thickness direction. The closed-form solutions of static Green's functions are derived from a fourth-order partial differential equation presented in 2]. In combination with Betti's reciprocal theorem the Green's functions are applied to calculate internal forces and stress analysis of functionally graded beams (FGBs) under static loadings. For symmetrical material properties along the beam thickness direction and symmetric cross-sections, the resulting stress distributions are also symmetric. For unsymmetrical material properties the neutral axis and the center of gravity axis are located at different positions. Free vibrations of functionally graded Timoshenko beams are also analyzed 3]. Analytical solutions of eigenfunctions and eigenfrequencies in closed-forms are obtained based on reference 2]. Alternatively it is also possible to use static Green's functions and Fredholm's integral equations to obtain approximate eigenfunctions and eigenfrequencies by an iterative procedure as shown in 1]. Applying the Sensitivity Analysis with Green's Functions (SAGF) 1] to derive closed-form analytical solutions of functionally graded beams, it is possible to modify the derived static Green's functions and include terms taking cracks into account, which are modeled by translational or rotational springs. Furthermore the SAGF approach in combination with the superposition principle can be used to take stiffness jumps into account and to extend static Green's functions of simple beams to that of discontinuous beams by adding new supports. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)