Keywords: Ritz method, plastic buckling, deformation theory of plasticity, incremental theory of plasticity, the Mindlin plate theory, arbitrary shape.

The Ritz method [1] is a commonly used method for plate analysis because of its simplicity. The Ritz method does not require mesh design as in the finite element method and therefore the preparation of input data for the Ritz method is relatively easy. However, in the Ritz method, it is necessary to have admissible displacement functions that satisfy the geometric boundary conditions in order to ensure convergence to the exact solution. It is difficult to obtain admissible displacement functions for a wide range of plate shapes and boundary conditions. This has been a major set back for the Ritz method until Liew and Wang [2] proposed the so-called p-Ritz method. In the p-Ritz method, the displacement functions are defined by the product of mathematically complete two-dimensional polynomial functions and boundary equations raised to appropriate powers that ensure the satisfaction of the geometric boundary conditions. With regard to plastic buckling of plates, various theories of plasticity have been proposed in the literature. The two most commonly used theories are the incremental (flow) theory of plasticity (IT) [3,4] and the deformation theory of plasticity (DT) [5,6]. So far, the authors could find only a few papers that used the Ritz method for the plastic buckling analysis of plates, despite many papers been published on elastic buckling analysis of plates using the Ritz method. Smith et al. [7] applied the Ritz method and IT for the plastic local buckling analysis of steel plates subjected to in-plane axial, bending and shear loading. Their study is, however, confined to the treatment of rectangular plates and is based on the classical thin plate theory. In this paper, we present the p-Ritz method for plastic buckling analysis of arbitrarily shaped plates with boundaries defined by polynomial functions. In addition, we also include the effect of transverse shear deformation which becomes significant when the plate is thick. We have adopted both IT and DT theories of plasticity and the Ramberg-Osgood stress strain relation for the plate material [8]. In order to demonstrate the validity, convergence and accuracy of the method, we analyzed the plastic buckling problems of rectangular and isosceles triangular plates under a uniform compressive stress.

Rectangular plates with two opposite sides simply supported while the other two sides take on any combination of boundary conditions are first considered because such plates admit analytical solutions and hence allow us to verify the convergence of the Ritz solutions [9]. Next, the simply supported equilateral triangular plates under uniform compression are considered and the results are checked against the analytical solutions [10]. It has been found that the p-Ritz method is able to furnish very accurate plastic buckling results. New plastic buckling stress parameters for the square, rectangular and isosceles triangular plates with other boundary conditions are determined. The buckling stress parameters predicted by DT theory are found to be consistently lower than their incremental theory counterparts. The difference is more pronounced when the plate boundary is more restrained and when the plate is thick. As expected, the plastic buckling stress parameters decrease with increasing plate thickness due to the effect of transverse shear deformation. The effect is more pronounced when the DT theory of plasticity is employed.

1

W. Ritz, "Über eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik", Journal für die Reine und Angewandte Mathematik, 135, 1- 61, 1909.

2

K.M. Liew, C.M. Wang, 1993. "pb-2 Rayleigh-Ritz method for general plate analysis", Engineering Structures, 15(1), 55-60, 1993. doi:10.1016/0141-0296(93)90017-X

3

G.H. Handelman, W. Prager, "Plastic buckling of rectangular plates under edge thrusts", NACA Technical Note No. 1530, 1948.

4

M.J. Sewell, "A general theory of elastic and inelastic plate failure, Part 1", Journal of Mechanics and Physical Solids, 11, 377. doi:10.1016/0022-5096(63)90016-4

5

A.A. Illyushin, "The elastic plastic stability of plates", NACA Technical Note No. 1188, 1947.

6

E.Z. Stowell, "A unified theory of plastic buckling of columns and plates", NACA Technical Note No. 1556, Washington, USA.

7

S.T. Smith, M.A. Bradford, D.J. Oehlers, "Inelastic buckling of rectangular steel plates using a Rayleigh-Ritz method", International Journal of Structural Stability and Dynamics, 3(4), 503-521, 2003. doi:10.1142/S0219455403001026

8

W. Ramberg, W. Osgood, "Description of stress-strain curves by three parameters", NACA Technical Note No. 902, 1943.

9

C.M. Wang, Y. Xiang, J. Chakrabarty, "Elastic/plastic buckling of thick plates", International Journal of Solids and Structures, 38, 8617-8640, 2001. doi:10.1016/S0020-7683(01)00144-5

10

C.M. Wang, "Plastic buckling of simply supported, polygonal Mindlin plates", Journal of Engineering Mechanics, 130(1), 117-122, 2004. doi:10.1061/(ASCE)0733-9399(2004)130:1(117)

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