Saint-Venant’s Problem for Homogeneous Piezoelectric Beams

Abstract

Abstract This paper is devoted to the linear analysis of a slender homogeneous piezoelectric beam that undergoes tip loading. The solution of the Saint-Venant’s problem presented in this paper generalizes the known solution for a homogeneous elastic beam. The analytical approach in this study is based on the Saint-Venant’s semi-inverse method generalized to electroelasticity, where the stress, strain, and (electrical) displacement components are presented as a set of initially assumed expressions that contain tip parameters, six unknown coefficients, and three pairs of auxiliary (torsion/bending) functions in two variables. These pairs of functions satisfy the so-called coupled Neumann problem (CNP) in the cross-sectional domain. In the limit “elastic” case the CNP transforms to the Neumann problem, for a beam made of a poled piezoceramics the CNP is decomposed into two Neumann problems. The paper develops concepts of the torsion/bending functions, the torsional rigidity and shear center, the tip coupling matrix for a piezoelectric beam. Examples of exact and numerical solutions for elliptical and rectangular beams are presented.