Automated Ritz Method for Large Deflection of Plates with Mixed Boundary Conditions
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Automated Ritz Method for Large Deflection of Plates with Mixed Boundary Conditions Madyan A. Al‑Shugaa1 · Husain J. Al‑Gahtani1 · Abubakr E. S. Musa1 Received: 11 September 2019 / Accepted: 14 May 2020 © King Fahd University of Petroleum & Minerals 2020
Abstract In this paper, an automated Ritz method is developed for the analysis of thin rectangular plates undergoing large deflection. The trial functions approximating the plate lateral and in-plane displacements are represented by simple polynomials. The nonlinear algebraic equations resulting from the application of the concept of minimum potential energy of the plate are cast in a matrix form. The developed matrix form equations are then implemented in a Mathematica code that allows the automation of the solution for an arbitrary number of the trial polynomials. The developed code is tested through several numerical examples involving rectangular plates with different aspect ratios and boundary conditions. The results of all examples demonstrate the efficiency and accuracy of the proposed method. Keywords Ritz method · Large deflection · Plate bending · Mixed boundary conditions
1 Introduction In many applications involving thin elastic plates, the lateral deflection may increase beyond a certain limit producing midplane stresses, called the membrane stresses, that cannot be neglected as in the case of small deflection theory. For large deflection, an extended plate theory must be employed, accounting for the effect of these membrane stresses. The widely used formulation is the one due to von Karman [1] which is represented by a couple of two nonlinear partial differential equations in terms of the lateral displacement and the stress function (w–F formulation). The details of the w–F formulation are given in most of the standard textbooks on plates and shells, e.g., [2–5]. More general formulation in terms of the lateral displacement, w, and the in-plane displacements, u and v (u–v–w-formulation), can be derived using the same fundamental assumptions used in the w–F formulation [6].
* Madyan A. Al‑Shugaa [email protected] Husain J. Al‑Gahtani [email protected] Abubakr E. S. Musa [email protected] 1
KFUPM, Dhahran 31261, Saudi Arabia
Analytical solutions are available only for very few cases involving simple geometries and boundary conditions. It is hard to cite all previously published analytical solutions in this limited space, and therefore, only a representative sample of them are mentioned here. The first analytical solution was due to Levy [7] who approximated the lateral deflection of the plate by a double series of sine-shaped harmonics to solve the case of simply supported rectangular plates. Yamaki [8] obtained nonlinear analysis of simply and clamped supported square plates. He approximated the solution by representing the deflection as trigonometric expressions and the stress function by Fourier series. Iyengar and Naqvi [9] also presented an approximate solution for clamped and simply support
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