We formulate a contact-adjoint problem of lateral bending of the Kirhhoff plates with uniformly stressed reinforcement. We perform a qualitative analysis of the corresponding system of constitutive equations and boundary conditions. The iteration method proposed here allows us to simplify the problem solution significantly. The method can be applied for rational designing of rectangular and polygonal simply connected and doubly connected plates with rectilinear and curvilinear edges. We discuss the particular analytic and approximate solutions obtained for rectangular and square plates subjected to bending and for doubly connected plates with straightedges.