Analytic bending solutions of free rectangular thin plates resting on elastic foundations by a new symplectic superposition method

Abstract

Analytic bending solutions of free rectangular thin plates resting on elastic foundations, based on the Winkler model, are obtained by a new symplectic superposition method. The proposed method offers a rational elegant approach to solve the problem analytically, which was believed to be difficult to attain. By way of a rigorous but simple derivation, the governing differential equations for rectangular thin plates on elastic foundations are transferred into Hamilton canonical equations. The symplectic geometry method is then introduced to obtain analytic solutions of the plates with all edges slidingly supported, followed by the application of superposition, which yields the resultant solutions of the plates with all edges free on elastic foundations. The proposed method is capable of solving plates on elastic foundations with any other combinations of boundary conditions. Comprehensive numerical results validate the solutions by comparison with those obtained by the finite element method.

A New Simple Design Method for the Plate Foundation of a Transmission Tower in Subsidence Area

“…Z. Qu [12] have adopted the method of Fourier double transformation and have obtained bending analytic solution of a rectangular plate, with free boundaries at its four corners, which rested on an elastic foundation under the condition of a vertical load. R. Li [13] has presented a new simple method of supervision to solve the problem analytically by invoking Hamilton canonical equations.…”

Analytical Solutions for Functionally Graded Beams under Arbitrary Distributed Loads via the Symplectic Approach

“…The approach based on Hamiltonian system has shown great superiority in revealing the structure of solutions and their physical essence as well as predicting the accurate local behavior which is usually covered up by the Saint-Venant principle in the traditional elasticity analysis [16]. Furthermore, the symplectic approach as a rational and a unified manner has been widely applied to various problems in many various branches of applied mechanics [17][18][19][20][21][22][23]. In the numerical calculation, Zhao and Chen [24] reconstructed the symplectic expansion formula in the state space formalism to avoid matrix singularity and achieved the stability of numerical calculation.…”

Vibration of plate on foundation with four edges free by finite cosine integral transform method

“…This method has often been utilized to analyze some structural engineering problems, like in Sneddon (1981). However, based on the author's knowledge, there are no reports on using the finite integral transform to analyze the rectangular plate on elastic foundation, like in Zhong et al (2009, Li et al (2011) and Li et al (2013).…”