Functionally graded piezoelectric beam under arbitrary loading

Analytical solutions for functionally graded beams under arbitrary loading are presented in this paper. The material coefficients vary exponentially along the thickness of the beams. The arbitrary loading is expanded in terms of sinusoidal series. For plane stress problems, the stress function and electric displacement function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The expressions for stresses, electric displacements, resultant forces and displacements and electric potential are then deduced, with integral constants determinable from the boundary conditions.