We introduce a class of nonlinear transformations called "resizing rules" which associate with optimal shape design problems certain equivalent distributed control problems while preserving the state of the system. This puts into evidence the duality principle that the class of system states that can be achieved, under a prescribed force, via modifications of the structure (shape) of the system can be obtained as well via the modifications of the force action, under a prescribed structure. We apply such transformations to the optimization of beams and plates, and in the simply supported or cantilevered cases, the obtained control problems are even convex. In all cases, we establish existence theorems for optimal pairs by assuming only boundedness conditions. Moreover, in the simply supported case, we also prove the uniqueness of the global minimizer. A general algorithm that iterates between the original transformed problems is introduced and studied. The applications also include the case of variational inequalities.