Partially Hinged Rectangular Plates: Theoretical Analysis And Applications

Partially Hinged Rectangular Plates: Theoretical Analysis And Applications The aim of the thesis is to present a rigorous theoretical analysis of the dynamics of partially hinged rectangular plates and their possible role in mathematical models of footbridges or suspension bridges expanding on the recent paper by Ferrero and Gazzola [1]. It is known that simplified models can prove useful since finer analytical methods can be applied to provide valuable insights into the physical properties of the structures, furthermore if the simple model exhibits a behavior that can be considered dangerous for the structure, we can expect that the more precise model will behave similarly. On the other hand, a reliable model of bridges --even a very simple one-- should have enough degrees of freedom to express the manifestation of torsional oscillations, which are considered by engineers a crucial factor in many collapse events. Motivated by these considerations, in [1] a simplified mathematical model of bridge is proposed where the deck is indeed seen as a long narrow thin plate hinged at short edges and free on the remaining two; this reflects the fact that the deck is supported by the ground at short edges. The dissertation is opened by a compendium of suitable theoretical background material on higher order Sobolev spaces and elliptic boundary value problems reported here for the convenience of the reader, furthermore proper references are given to the update literature. Then we proceed by recalling the bending elastic energy of a deflected plate originating from the classical Kirchhoff-Love theory of elasticity, then the fourth order partial differential equation satisfied by the equilibrium position of the plate is derived together with the associated boundary conditions, which are of Navier type on short edges and of Neumann type on the remaining edges. This is followed by a rigorous theoretical study of existence, uniqueness and regularity of solutions of the boundary value problem obtained. Additionally, the explicit form of the solution is written by applying a suitable technique of separation of variables, refined to reflect the presence of peculiar boundary conditions. We improve on the results given in [1] by relaxing the assumptions on the dependence of the load from the position, thus extending the result to arbitrary forcing terms. As shown in [1], the same separation of variables technique may be exploited to characterize completely the oscillating modes of the plate, i.e., the spectrum and the corresponding eigenfunctions of the biharmonic operator under partially hinged boundary conditions. References [1] Alberto Ferrero and Filippo Gazzola. A partially hinged rectangular plate as a model for suspension bridges. Discrete Contin. Dyn. Syst, 35(12):5879–5908, 2015.