Functionally graded beams (FGBs) with an arbitrary gradation of the material properties along the thickness of the beams are analyzed. Such FGBs are of special interest in civil and mechanical engineering to improve both the thermal and the mechanical behaviour of the beams. In [1] and [3] free vibrations of functionally graded Timoshenko and Euler-Bernoulli beams have been considered. The obtained analytical solutions are based on the work of Li [2], where closed-form solutions of stress distributions, eigenfrequencies and eigenfunctions have been derived by means of a single differential equation of motion for the deflection. However, these previous works did not take into account the coupling between the longitudinal and the transverse displacements and its effects on the deformation and internal forces of the FGBs. This approach is appropriate only for a symmetrical material gradation but it may not be valid for general cases with an arbitrary material gradation.

In this paper, the coupling effects of the longitudinal and transverse displacements on the deformation and internal forces of FGBs are investigated for different beam support conditions. Analytical solutions of the corresponding boundary value problem are derived. A comparison is also made with the numerical results obtained by the finite element method (FEM). (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)