One-dimensional equations for planar piezoelectric curved bars

A set of one-dimensional equations for coupled extension, flexure, and shear of a planar piezoelectric curved bar is obtained from the three-dimensional equations of piezoelectricity by expanding the mechanical displacement vector and the electric potential into power series in the cross-sectional coordinates and retaining lower-order terms. The equations are useful in the analysis and design of one-dimensional piezoelectric devices.