Exact solutions for the buckling of rectangular plates having linearly varying in-plane loading on two opposite simply supported edges

Abstract

An exact solution procedure is formulated for the buckling analysis of rectangular plates having two opposite edges (x = 0 and a) simply supported when these edges are subjected to linearly varying normal stresses σ_x = −N₀[1−α(y/b)]/h, where h is the plate thickness. The other two edges (y = 0 and b) may be clamped, simply supported or free, or they may be elastically supported. By assuming the transverse displacement (w) to vary as, $\sin (m\pi x/a)$, the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (i.e., the method of Frobenius). Applying the boundary conditions at y = 0 and b yields the eigenvalue problem of finding the roots of a fourth order characteristic determinant. Care must be exercised to retain sufficient terms in the power series in calculating accurate buckling loads, as is demonstrated by a convergence table for all nine possible combinations of unloaded clamped, simply supported or free edges at y = 0 and b. Buckling loads are presented for all nine possible edge combinations over the range of aspect ratios 0.5 ⩽ a/b ⩽ 3 for loading parameters α = 0, 0.5, 1, 1.5, 2, for which α = 2 is a pure in-plane bending moment. Some interesting contour plots of their mode shapes are presented for a variety of edge conditions and in-plane moment loadings. Because the nondimensional buckling parameters depend upon the Poisson’s ratio (ν) for five of the nine edge combinations, results are shown for them for the complete range, 0 ⩽ ν ⩽ 0.5 valid for isotropic materials. Comparisons are made with results available in the published literature.

Introduction

For more than a century researchers in structural mechanics throughout the world have endeavored to obtain accurate theoretical results for the critical buckling loads of plates, as well as their corresponding buckling mode shapes. Several thousands of research papers on these topics have appeared in the international scientific and technical journals and in conference proceedings, most of them dealing with rectangular plates. Much of the useful results has been summarized in many texts and handbooks (Timoshenko and Gere, 1963, Volmir, 1967, Bulson, 1970, Japan Column Research Council, 1971, Szilard, 1974, Brush and Almroth, 1975, Trahair and Bradford, 1998).

Rectangular plates subjected to uniform in-plane stresses have been extensively analyzed in the buckling literature because the governing differential equation of equilibrium has constant coefficients, yielding exact solutions for buckling loads straightforwardly when two opposite edges of the plates are simply supported.

Of course, a plate may be loaded at two opposite edges by non-uniform, in-plane, axial forces (N_x), the first variation from the uniform loading being one which varies linearly. A special case of this is a pure, in-plane bending moment. In the non-uniform loading case the analysis is more formidable, and exact solutions are much more difficult to achieve. One finds considerable approximate results for plate buckling loads for such non-uniform stress fields, typically obtained by energy methods. Recently, for the case of linearly varying in-plane loadings some researchers have presented approximate results (Smith et al., 1999a, Smith et al., 1999b, Bradford et al., 2000, Smith et al., 2000). Exact solutions for S-F-S-F (Kang and Leissa, 2001) and S-C-S-C plates (Leissa and Kang, 2002) have also been obtained, where two opposite edges are simply supported and the other two are either free (F) or clamped (C).

Some researchers have also analyzed both the buckling and vibration of rectangular plates subjected to in-plane stress field (Kang and Leissa, 2001, Leissa and Kang, 2002, Bassily and Dickinson, 1972, Bassily and Dickinson, 1978, Dickinson, 1978, Kielb and Han, 1980, Kaldas and Dickinson, 1981). Bifurcation buckling may be regarded as a special case of the vibration problem; that is, determining the in-plane stresses which cause vibration frequencies to reduce to zero.

The present work presents exact solutions for the buckling loads and mode shapes for rectangular plates having two opposite edges simply supported when these edges are subjected to linearly varying in-plane normal stresses. The procedure is applied to all possible combinations of clamped, simply supported or free edge conditions applied continuously along the other unloaded edges. For the case of opposite edges being simply supported, a variables separable solution exists, which reduces the partial differential equation to an ordinary one having variable coefficients. This is solved by the classical power series method of Frobenius, and the convergence of the series is established. Comparisons are also made with results available in the published literature.