Keywords: elastic stability, simply-supported rectangular plates, Ritz energy technique, arbitrary loads, buckling coefficients, exact stress distributions, double Fourier series.

The stability of rectangular plates under in-plane forces (such as the flanges and webs in steel plate girders) has been the subject of numerous investigations over the last hundred years. Solutions to these problems are usually obtained on the assumption that the stress distribution throughout the plate is uniform. Moreover, for the relatively small number of cases solved by allowing for non-uniform stress fields (such as, for example, compressive patch loading), the buckling coefficients are not exact. The present article shows how the stability of simply-supported rectangular plates under arbitrary stress distributions can be tackled in an accurate -- indeed, practically 'exact' -- manner. The proposed method uses an analytical approach which, for convenience, is combined with Gaussian-quadrature evaluation of certain complex integrals. Several examples are provided which illustrate the generality of the approach, while the accuracy of the scheme is established by comparison with both earlier analytical attempts available in the literature and finite-element results.

In this study, the Ritz energy technique is used to derive the eigenvalue problem which leads to the buckling coefficient for a plate under non-uniform in-plane stresses. According to this energy method, two factors are crucial to the accuracy of the results: the deflection series and the stress distribution. The former is catered for by using the double Fourier series which, in principle, can represent any possible buckled profiles of the plate. Regarding the latter, the solutions to the 2-D elastic problem with rectangular boundaries under arbitrary external forces [1,2] are employed to derive the exact stress distributions within the plate.

Based on these exact stress distributions and the assumed buckled profile of the plate, the strain energy due to bending and the work done by the external loads can be derived. The former is solved analytically as in previous studies [3] while the latter is computed using numerical Gaussian quadrature to avoid the cumbersome mathematical manipulations since the expressions for the exact stresses are very complicated [2]. Then, the total potential energy is obtained by combining the strain energy due to bending and the work done by the external loads. Next, by minimizing the total potential energy, an eigenvalue problem emerges from which the buckling coefficient of the system under consideration is finally computed.

The use of double Fourier series and exact stress distributions guarantees the accurate determination of both the strain energy due to bending and the work done by the external forces, thus ensuring the accuracy of the buckling coefficient of the system. Theoretically, the results can be computed to any degree of accuracy provided enough terms are used in the Fourier series and in the series for the stress distribution (since the latter is also based on series [2]). In addition, the finite-element method is also utilized to compute the buckling coefficients for the buckling of plates under the same load conditions, thus providing the necessary data for comparative studies.

Since the exact two-dimensional elasticity solutions for stresses in a rectangular plate subjected to arbitrary stresses along its boundaries are combined with a general series for the buckled shape of such plates (simply-supported along its edges), 'exact' values for the buckling coefficients of these systems are obtained. The good working of the scheme is illustrated by means of several examples encompassing plates under direct, shear or 'bending' stresses (or combinations of these). In fact, three cases are considered: the first case revisits the buckling of plates under patch compression which also encompass the case of plates under concentrated forces; the second case deals with linearly-varying direct stresses along one axis of the plate; the last case considers a more complicated loading condition including direct and shear stresses.

The ensuing results are 'exact' (to the required number of decimal points). This is also evident through comparisons with previous investigations of the problems presently studied, as well as finite-element results. While, currently, the analytical Ritz technique is combined with a Gaussian-quadrature evaluation of certain complex potential-energy integrals, further work is under way to obtain closed-form results for these integrals so as to make the scheme fully analytical (and computationally more economical).

1

E. Mathieu, "Théorie de l'élasticité des corps solides", Second partie. Gauthier-Villars, Paris. 1890.

2

G. Baker, M. N. Pavlovic, N. Tahan, "An Exact Solution to the two-dimensional elasticity problem with rectangular boundaries under arbitrary edge forces", Philosophical Transactions: Physical Sciences and Engineering, the Royal Society of London, 343, 307-336, 1993.

3

P. S. Bulson, "The stability of flat plates", Chatto and Windus, London. 1970.

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