Symplectic elasticity approach for exact bending solutions of rectangular thin plates

Abstract

This thesis presents a bridging analysis for combining the modeling methodology of quantum mechanics/relativity with that of elasticity. Using the symplectic method that is commonly applied in quantum mechanics and relativity, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some basic problems in solid mechanics and elasticity that have long been stumbling blocks in the history of elasticity. Specifically, the approach is applied to the bending problem of rectangular thin plates the exact solutions for which have been hitherto unavailable. The approach employs the Hamiltonian principle with Legendre’s transformation. Analytical bending solutions are obtained by eigenvalue analysis and the expansion of eigenfunctions. Here, bending analysis requires the solving of an eigenvalue equation, unlike the case of classical mechanics in which eigenvalue analysis is required only for vibration and buckling problems. Furthermore, unlike the semi-inverse approaches of classical plate analysis that are employed by Timoshenko and others in which a trial deflection function is predetermined, such as Navier’s solution, Levy’s solution, or the Rayleigh-Ritz method, this new symplectic plate analysis is completely rational and has no guess functions, yet it renders exact solutions beyond the scope of the semi-inverse approaches. In short, the symplectic plate analysis that is developed in this paper presents a breakthrough in analytical mechanics, and access into an area unaccountable by Timoshenko’s plate theory and other, similar theories. Here, examples for rectangular plates with 21 boundary conditions are solved, and the exact solutions are discussed. Specially, a chapter on benchmarks of uniformly loaded corner-supported rectangular plate is also presented. Comparison of the solutions with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only for other types of boundary conditions, but also for thick plates, vibration, buckling, wave propagation, and so forth. Remarks and directions for future work are given in the conclusion.