On Plastic Buckling of a Compressed Strip

Abstract

A n ACpl -At P 2 = cos L vi + «i J ^ (i _ k) vm J-^v This is in agreement with the solution obtained by Evvard. 5 SUMMARY Three types of edges can border a region: (1) a leading edge, (2) a trailing edge, and (3) an edge bordering another region. For types (2) and (3) the method given in references 2 and 3 applies, without regard to the homogeneous solution to the integral equation. For type (1) an additional analysis similar to that given here must be used. restated the inconsistency of deformation theories and pointed out that a more complex incremental theory than the PrandtlReuss should be used if initial imperfections of the specimens did not prove to be responsible for the mentioned disagreement. Here, it is shown that, if a small initial deflection is taken into account, the incremental theory can afford an explanation of experimental facts in plastic buckling of compressed plates, even in the case where the discrepancy appears most marked.Let s be the thickness and b be the width of the strip. One of its long sides is free; the other is simply supported. Let <r be the compressive stress in the direction of the length L ^> b. The twist of the buckled strip produces shear stresses r and strains y. We suppose that the ratio r/y has the same value anywhere in the plate, depending only upon <r. According to this hypothesis, if, under the stress a, the angle of twist at station z passes from the initial value /3o to /So + & the maximum shear strain at the same station attains the value s(dft/dz). LetIf /So varies according to a sinusoidal halfwave, 0 also follows the same law; the condition of equilibrium of moments yieldsSupposing Poisson's ratio =0.5, the Prandtl-Reuss equations givewhere e is the longitudinal strain. Expressing the octahedral shear stress as a function of the corresponding strain yields r2 + (,2/3) = /(T2 + 3e2)The function /, a characteristic of the material, is deduced from simple stress-strain measurements. In particular, if from a shear test the tangent modulus of rigidity Gt = dr/dy is obtained, the derivative/' of/with respect to its argument iswhere r e and y e are the "equivalent" stress and strain re = Vr 2 + (<r 2 /3); 7e = VV + 3e 2 From Eqs.(2), (3), and (4), we getThe integration of this equation may be performed by step-bystep procedures. In the calculations carried out here, the accuracy was improved by successive approximations; at each step, for the new value of y, the quantities were computed according to the following cyclic order: da do-•--(guess); <r; r; r e ; y e ; f ; e; --ay ayIf deformation theory is used, Eq.(3) must be replaced by 3er = ycr (8)

Plastic Buckling Paradox: An Updated Review

“…Neale (1974) estimated the plastic buckling of cylinders subjected to torsion using the flow theory of plasticity in the presence of imperfections and found that the critical torsional stress was reduced by extremely small and unavoidable imperfections, but that it was not sensitive to the magnitude of imperfections. Cicala (1950) concluded in his approximate analysis of strips that small imperfections in the specimen would reduce the bifurcation load predicted by flow theory to the level of the deformation theory. Hutchinson (1972) investigated the imperfection sensitivity of spherical shells subjected to external pressure in the plastic range.…”