Stress Gradient Effect on the Buckling of Thin Plates

This paper presents an analytical method to calculate the buckling stress of a rectangular thin plate under nonuniform applied axial stresses. Two cases are considered, buckling of a plate simply supported on all four sides and buckling of a plate simply supported on three sides with one unloaded edge free and the opposite unloaded edge rotationally restrained. These two cases illustrate the influence of stress (moment) gradient on stiffened and unstiffened elements, respectively. The axial stress gradient is equilibrated by shear forces along the supported edges. A Rayleigh-Ritz solution with an assumed deflection function as a combination of a polynomial and trigonometric series is employed. Finite element analysis using ABAQUS validates the analytical model derived herein. The results help establish a better understanding of the stress gradient effect on typical thin plates and are intended to lead to the development of design provisions to account for the influence of moment gradient on local and distortional buckling of thin-walled beams. Introduction The design of thin-walled beams traditionally involves the consideration of both plate stability (local buckling) and member stability (lateral-torsional buckling). Plate stability is considered by examining the slenderness of the individual elements that make up the member and the potential for local buckling of those elements. Member stability is considered by examining the slenderness of the cross-section and the potential for lateral-torsional buckling. Member stability modes, such as lateral-torsional buckling, occur over the unbraced length of the beam, which is typically much greater than the depth of the member (L/d >> 1). Classical stability equations for lateraltorsional buckling are derived for a constant moment demand over the unbraced length. For beams with unequal end moments or transverse loads, the moment is not constant and the moment gradient on the beam must be accounted for. In design, this influence is typically captured in the form of an empirical moment gradient factor (Cb) which is multiplied times the lateral-torsional buckling moment under a constant demand. The moment gradient, which so greatly influences the member as a whole, also creates a stress gradient on the plates which make up the member. In this paper, we investigate the influence of stress gradients on plate stability, which for unstiffened elements and potentially for distortional buckling of edge stiffened elements, may represent an important effect for properly capturing the actual stability behavior. In particular, for distortional buckling, ignoring the influence of moment gradient potentially ignores a source of significant reserve. In practice, one of the most common cases with a danger for distortional buckling includes high moment gradients: the negative bending region near the supports (columns) of a continuous beam. Since the compression flange behavior generally characterizes the distortional buckling of sections, the stress gradient effect on the flange is of practical interest to moment gradient influence on the beams. For plate stability, or local buckling, the influence of the stress gradient is typically ignored in design. Figure 1 provides a variety of classical plate buckling solutions that are intended to help indicate why moment gradient has been traditionally ignored for local buckling. The results are presented in terms of the plate buckling coefficient k as a function of the plate aspect ratio β=a/b for different numbers of longitudinal half sine waves, m. where: ) /( 2 2 t b D k cr π σ = and the plate of length a, and width b, is simply supported at the loaded edges and either simply supported (ss), fixed (fix) or free at the unloaded edges.