Abstract
A unified analytic solution approach to both static bending and free vibration problems of rectangular thin plates is demonstrated in this paper, with focus on the application to corner-supported plates. The solution procedure is based on a novel symplectic superposition method, which transforms the problems into the Hamiltonian system and yields accurate enough results via step-by-step rigorous derivation. The main advantage of the developed approach is its wide applicability since no trial solutions are needed in the analysis, which is completely different from the other methods. Numerical examples for both static bending and free vibration plates are presented to validate the developed analytic solutions and to offer new numerical results. The approach is expected to serve as a benchmark analytic approach due to its effectiveness and accuracy.
Highlights
To provide new benchmark solutions, we focus on the rectangular thin plates with four corners point-supported, which could rest on an elastic foundation
It should be noted that proper manipulation of the above simultaneous algebraic equations will lead to analytic solutions of more static bending and free vibration problems of point-supported plates with supported edges
A unified analytic approach is developed in this paper to solve static bending and free vibration problems of rectangular thin plates
Summary
Χ equals Kw2/2 − qw for static bending problems and K*w2/2 for free vibration, where K is the Winkler-type foundation modulus; q is the distributed transverse load; K* = K − ρhω[2], in which ρ is the plate mass density, h is the plate thickness, and ω is the circular frequency. Equations (3) and (4) are the Hamiltonian system-based governing matrix equations for static bending problems and free vibration of a thin plate, respectively. As will be shown in the following, the solution approaches to these two problems are similar, only different in solving the final simultaneous algebraic equations because one group is homogeneous while the other one is inhomogeneous. We will start with the solution of the inhomogeneous equation (3) and reduce to the homogeneous case based on the unified analytic approach
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