Shape control of moderately thick piezoelectric beams

The present contribution focuses on shape control of thick beam-type structures. First the governing equations of a multi-layered beam are derived by taking advantage of the Timoshenko assumptions and the constitutive relations of piezoelectric materials. The deflection curves are explicitly given for a piezoelectric cantilever subjected to a polynomial distribution of the vertical load and the applied electric voltage. In order to find a solution for the optimal shape control voltage an objective function, which depends on the quadratic deflection curve over the beam length, is minimized. Finally several benchmark examples are given for thick beams and the outcome is compared to finite element results and previously derived shape control results from the scientific literature that hold for thin piezoelectric beams. The presented shape control method shows a better agreement with the numerical outcome than the analytical shape control results within the Bernoulli-Euler theory, but the desired voltage distribution only slightly differs from the outcome for thin beams. Furthermore it is found that for a given total thickness-to-length ratio piezoelectric bimorph structures may be more difficult to be perfectly controlled than three-layer beams with thin piezoelectric layers. This is due to higher order piezoelectric effects which are not considered by the present theory (e.g. the thickness deformation caused by the thickness piezoelectric coupling constant).