Actuator placement in static bending of smart beams utilizing Mohr’s analogy

An extension of Mohr’s analogy to bending of shear-deformable beams with eigenstrain-type actuation, such as a piezoelectric actuation, is presented first. Various refined shear-deformable beam theories are included by means of a theory-dependent parameter. The one-dimensional version of Reissner’s sixth-order plate theory is exemplarily addressed. The Bernoulli–Euler theory of beams rigid in shear, as well as the shear-deformable theory of Timoshenko, are included as special cases. Afterwards, the extended Mohr analogy is applied to the bending of smart beams with piezoelectric patch actuators. The following special problem of static shape control is solved: Seek a placement of single patch actuators, such that the displacement and the cross-sectional rotation vanish at some pre-selected locations of the beam, despite the beam is loaded by external forces. Using Mohr’s analogy, it is shown that the auxiliary loading of the adjoint beam must form a self-equilibrated system of loading in order to achieve the latter goal. The high potential of the proposed actuator placement is demonstrated for the case of a cantilever beam with a single force acting at the tip. Placements of single actuators are represented such that the tip displacement and the tip cross-sectional rotation vanish. The outcomes of shear-deformable theories are compared to the Bernoulli–Euler theory and to a Finite Element computation using piezoelectrically coupled elements.