Analysis of Three-Dimensional Piezo-Electric Beams via a Unified Formulation

Abstract

A Unified Formulation for deriving several higher-order theories and related finite elements for beams is presented within this paper.Three-dimensional structures with piezo-electric layers are considered.Static and free vibration analyses are carried out.Models' main unknowns are the displacements and the electric potential.They are approximated above the beam cross-section via Lagrange's polynomials in a layer-wise sense.Finite elements stiffness and mass matrices are derived in a nucleal form using d'Alembert's Principle.This nucleal form is representative of the generic term in the approximating expansion of the displacements and electric potential over the cross-section.It is, therefore, invariant versus the theory expansion order and the element nodes' number.In such a manner, higher-order displacements-based theories that account for non-classical effectssuch as transverse shear deformations and cross-section in- and out-of-plane warping are straightforwardly formulated.Results are given in terms of displacements, electrical potential and stresses.Comparison with three-dimensional finite elements models are provided, showing thataccurate results can be obtained with reduced computational costs.