Hierarchical FEM modelling of piezo-electric beam structures

A class of new one-dimensional finite elements for the static analysis of composite beams embedding piezo-electric layers is presented in this paper. Elements main unknowns consist in displacements and electric potential that are approximated above the beam cross-section via Lagrange’s polynomials in a layer-wise sense. Within the framework of the principle of virtual displacements, the stiffness matrix is derived in a nucleal form by means of a unified formulation. This nucleal form does not depend upon displacements and electric potential approximation order over the cross-section and the number of nodes along the axial direction. In a such a manner, higher-order displacements-based theories that account for non-classical effects are straightforwardly formulated. Beam embedding full piezo-electric layers as well as piezo-patches are investigated. Results are given in terms of displacements, electrical potential and stresses. Comparison with three-dimensional finite elements models are provided, showing that the proposed class of finite elements matches the reference results as long as the appropriate approximation over the cross-section is considered.