The paper deals with the geometrically nonlinear theory and finite element method for the simulation of large amplitude vibration control of straight and curved beams, or what is equivalent, cylindrical deformations of plates and shells, consisting of a master structure with integrated thin piezoelectric layers or patches. The theory is based on the Bernoulli or Kirchhoff-Love hypothesis, respectively, and constitutes an entirely Lagrangean displacement formulation for small strains with no restrictions imposed on the magnitude of rotations or displacements. Based on modified strain measures the strain-displacement relations are only quadratic in terms of displacements and their gradients. Work-conjugate stress resultants and stress couples are derived from the internal virtual work associated with the first approximation of the strain energy density. Based on the principle of virtual displacements the FE equations for the simulation of static and dynamic response control are derived.