Maximum load potential of hinged plates with non-homogeneous thickness

Abstract

In this paper, a PDE-constrained optimization problem corresponding to the bi-Laplacian operator is investigated. The optimization problem arises when we search for the maximum load potential in a uniform elastic plate hinged at its boundary with a given total amount of force. It is established that there is a bang–bang solution to the optimization problem, and an optimality condition is derived. Based on the optimality condition, we obtain some topological and geometrical properties of the maximizer. Although analytical solutions for PDE-constrained problems are very rare, an analytical solution is determined if the plate is circular and its thickness has a particular form. In those cases, it is proved that the solution is unique. For plates of general shape, an efficient numerical algorithm is developed to solve the PDE-constrained optimization problem, and its convergence is studied. According to the results, one can determine the profile of the regions in a plate where the largest and smallest external force should be exerted to have the maximum load potential. According to the research results, the thickness of the plate plays a crucial role in determining the shape of those regions. Therefore, disregarding the impact of plate thickness in our studies for the sake of simplicity would be impractical and might result in inaccurate outcomes. By utilizing our findings, engineers and designers can determine the optimal external force pattern that yields the maximum load potential for plates. This, in turn, enables them to create plates that are more efficient and effective, tailored to meet their specific needs and requirements.