Bifurcation Criterion and Plastic Buckling of Plates and Columns

If a bifurcation of equilibrium paths occurs as the loading is varied on a conservative system, the preferred path is usually obtained by use of a stability criterion. However, even if the definition of stability is modified so as no longer to depend on the existence of a potential energy function, a stability criterion may not be useful if the system is nonconservative. The choice of a different type of criterion is discussed for the particular case of plastic buckling of plates and columns, and detailed calculations are given for the simplified column model introduced by Shanley. To find the deflection of a real column with arbitrary cross section in the neighborhood of the tangent modulus load, numerical methods must be used. Two such methods are outlined: One consists of a sequence of approximations to the simultaneous solution of two nonlinear integral equations, and the other consists of a sequence of graphical integrations based on the stressstrain curve of the material. Under certain conditions, a portion of the solution may be found in analytic form. The problem of plastic buckling of plates under compressive edge thrust has been most rigorously treated by Handelman and Prager. If their calculations are modified by the assumption of initial loading (in analogy with the tangent modulus theory for columns), then their results may be brought into fair agreement with recent experimental work by Pride and Heimerl. The assumption of initial loading may be verified a posteriori. STABILITY OF NONCONSERVATIVE SYSTEMS FROM THE ENGINEERING POINT OF VIEW, a s tructure is stable if the disturbances to which it is subject are not sufficient to produce excessive displacements. However, to permit mathematical t rea tment of stabili ty, it is customary to idealize in two ways: The structure itself is mechanically idealized (e.g., friction in pin joints, fmiteness of some dimensions, etc., are often neglected); and only such disturbances are considered as are certain to be applied to any structure because of the vagaries of nature—i.e., infinitesimal forces. If the idealized system is conservative, then the statem e n t t h a t infinitesimal forces will produce no finite deformation is equivalent to the s ta tement t ha t the potent ial energy of the position in question is a relative minimum. This lat ter s ta tement is essentially the classical definition of stabili ty; if the idealized system Received October 25,1949. * This paper is a condensed version of a thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate Division of Applied Mathematics at Brown University; the work was carried out at the suggestion of, and under the guidance of, Prof. W. Prager, Chairman of the Division. f Fellow in Applied Mathematics. Now, Research Associate, ^Gordon McKay Aeronautics Laboratory, Harvard University. is nonconservative, the definition of stability must be given with respect to infinitesimal disturbances instead of a potential energy function. The following definition is valid for both conservative and nonconservative systems: "A mechanically-idealized system will be said to be unstable if there exists a small distance e such tha t for any 5 > 0 it is possible to find a force distribution of intensity less than 8 which will produce an eventual displacement of the system in which a t least one material point is distant e from its original position. A stable system is one t ha t is not unstable ." Consider now an arbi trary mechanical system, either conservative or nonconservative when idealized. If the loading is varied, there can be no ambiguity in the equilibrium pa th followed by the real system, bu t such ambiguities may occur in the analysis of the idealized system. In the conservative case, the preferred pa th may usually be found by considering the stabilities of the alternative paths, although, because the exact differences between the real and idealized systems are not known, it is not possible to prove rigorously t ha t the real system will follow this preferred path . Of course, the less the idealization, the greater the likelihood t h a t this will be the case. However, stability need not be the only criterion; it might even happen t ha t several of the possible paths are stable, and in this eventuali ty an additional criterion must be sought. For a nonconservative system, the same remarks will apply, provided, of course, t ha t the modified definition of stability is used. In the problem of interest here—viz., the buckling of columns and plates—there is no difficulty if the buckling load occurs within the elastic range of stress, for a straightforward calculation shows t ha t only the buckled state is stable. If the stress is beyond the elastic limit, not only is the stability criterion difficult to apply b u t it seems likely tha t both the buckled and the undeformed states are stable; a different criterion is therefore required. I t is well known tha t it is often useful to examine the deformed state of a plate or column; if an initial deflection is introduced into the analysis, it will be found tha t the equilibrium pa th is always unique and approaches the buckling pa th as a limit when the initial deflection approaches zero. In other words, the convergence is nonuniform; the straight-line equilibrium position is obtained only as a result of zero initial deflection and is not approached by any sequence of