Corotational Formulation of a Piezoelectric Superconvergent Beam Element

Abstract

A corotational finite-element formulation is developed for piezoelectric plane frames subject to small strains but large deformations. In the corotating (local) system, the element is linear and superconvergent, with no electrical degrees of freedom. The formulation is derived from a piezoelectric beam theory that incorporates the Timoshenko displacement assumption and assumes a quadratic variation of the electric potential through the thickness of each piezoelectric patch. All linearizations required by the Newton–Raphson solution procedure, including those related to certain induced follower loadings, are explicitly and consistently derived. The proposed formulation (a) avoids membrane, transverse shear, and piezoelectric lockings; (b) yields assembled stiffness matrices that are not affected by the high condition numbers, typically found in piezoelectric finite-element models; (c) requires no special corotational algorithm; and (d) effectively handles large deformations, as demonstrated through numerical examples, including shape control applications.