A Unified Solution for an Anisotropic Functionally Graded Piezoelectric Beam Subject to Sinusoidal Transverse Loads

The behavior of anisotropic functionally graded piezoelectric beams subject to sinusoidal transverse loads is investigated based on the equations for a generalized plane stress problem. Both the stress function and electric displacement function are assumed to consist of two parts. One corresponds to a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z). The other is represented by a linear polynomial of x with unknown coefficients also depending on z. The equations governing these z-dependent functions are presented. The expressions for stresses, electric displacements, resultant forces, displacements, and electric potential are then deduced, in which the integral constants are determined from the boundary conditions. Analytical solution is derived in the case that material coefficients vary exponentially along the thickness of the beam. Semi-analytical solution is also suggested along with the sub-layer approximation when the material properties vary in an arbitrary fashion along the thickness. Various boundary conditions at the two ends of the beam can be taken into consideration. Numerical results and the associated discussions are finally presented.