Behavior of Piezoelectric Beams under Axially Non-uniform Distributed Loading

The paper presents a generic solution methodology for a quasi-static homogeneous monoclinic piezoelectric beam under axially distributed electric and mechanical surface loads and body forces expressed as polynomials of degree K≥ 0 of the axis variable. (In the absence of any electrical loading, this problem is known as the Almansi–Michell problem). The stress and the electrical displacement components are presented as a set of polynomials of degree ≤K+2 of the axis variable (“solution hypothesis”) containing 4K unknown tip loading constants and 3K stress functions of two variables. The cases K=0,1 stand for uniform or linear distributed loads in the axis direction. The analysis is initiated by the Kth level and continues down to lower levels. The main result of this work generalizes the “elastic” solution given recently by O. Rand and the first author (2005). Examples of solutions for axially uniform distributed loads (K=0), and equilibrium in which the stress and the electrical displacement do not depend on the axis variable, are presented. The applications to constant body loads and a hydrostatic pressure are considered.