BKiHtaiHu SCC'ETitS LIBM/ JUN 241986 P. G.Hodge, Jr. Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455 K.-J. Bathe Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 E. N. Dvorkin Universidad de Buenos Aires, Facultad de Ingeniera, Buenos Aires, Argentina Causes and Consequences of Nonuniqueness in an Elastic/Perfectly-Plastic Truss A complete solution to collapse is given for a three-bar symmetric truss made of an elastic/perfectly-plastic material, using linear statics and kinematics, and the solu- tion is found to be partially nonunique in the range of contained plastic deforma- tion. The introduction of a first-order deviation from symmetry and/or the inclu- sion of first-order nonlinear terms in the equilibrium equations is found to restore uniqueness. The significance of these effects is analyzed and discussed from mathematical, physical, modelling, computational, and engineering points of view. 1 Introduction In the linear theory of elasticity the solution to a "well- defined boundary-value problem" is known to exist and be unique. However, for an elastic/perfectly-plastic material neither of these facts is obvious. Indeed, if the load is equal to the so-called "yield-point load" the solution ceases to be unique; beyond this load no solution exists. The mathematical description of a boundary-value problem for an elastic/perfectly-plastic material involves partial dif- ferential equations and nonlinear constitutive relations. Further, if actual answers are required, it is necessary to ap- proximate the structure with a numerical model. Thus the dif- ficulty in combining mathematical rigor and physical intuition in the discussion of such questions as uniqueness is com- pounded by the necessity of distinguishing between the true nature of the continuum model and aspects introduced by the numerical model. However, many features of the general continuum are pre- sent in much simpler structures where they can be more clearly discussed. As an illustration of this approach, the present paper is concerned with a particular simple three-bar plane truss. We begin by defining a "well-defined equilibrium problem" for trusses as one in which the applied forces and displacements at the joints are such that: (a) overall equilibrium is not violated; (b) overall displacement constraints prescribe a unique allowable rigid-body motion (usually zero); (c) at each joint in each of two independent directions either the displacement or the applied load is prescribed. It is not difficult to prove that the solution to any well-defined Contributed by the Applied Mechanics Division for presentation at the Winter Annual Meeting, Anaheim, CA, December 7-12, 1986, of the American Society of Mechanical Engineers. Discussion on this paper should be addressed to the Editorial Department, ASME, United Engineering Center, 345 East 47th Street, New York, N.Y. 10017, and will be accepted until two months after final publication of the paper itself in the JOURNAL OF APPLIED MECHANICS. Manuscript received by ASME Applied Mechanics Division, November 27, 1984; final revision August 1, 1985. Paper No. 86-WA/APM-8. equilibrium elasticity problem is unique. Indeed, if two solu- tions exist and are denoted by primes and double primes, then it follows from the principle of virtual work that W"/' -F, ")(e,-' - e, ") = E Joint ,(P;' -P,")-(u,-' -u,") (1) where F, and e, are the bar force and elongation, respectively, and Vj and u y are the respective force and displacement vec- tors at joint./'. If each solution satisfies condition (c) above and we write the scalar product in terms of components in the in- dependent prescription directions, then either P'?=P" or u'=u" in each term on the right. Further, if each bar is elastic, then F, = k,e, (2) where the stiffness k, is positive. Thus equation (1) becomes Ebars^(e/'-e,") 2 = 0 (3) which clearly requires that each e,' = e," and hence each Fj'=Fj". Equilibrium equations then show each P y '=Pj", and truss kinematics plus condition (b) above lead to each u / -Uj•" =0. Therefore, the solution is unique. Let us generalize the material behavior of equations (2) to an "elastic/perfectly-plastic" truss where each bar behaves elastically under small bar forces but can elongate indefinitely when Fj reaches a certain limiting value Y f with similar behavior in compression. Thus during any sufficiently small time interval each bar is EITHER Elastic OR Plastic (F, 2 < F, 2 AND AF, = £,Ae,) (4a) (F,2=r, 2 F,Ae,>0) (4b) Consider first any well-posed equilibrium boundary-value problem where any prescribed displacements are zero and the loads Py are such that every bar is elastic. Clearly the same unique solution will hold as in the ideally elastic truss. Now consider this same truss under the set of loads XPy where X is slowly increased from X = 1. So long as F,- 2 < K,- 2 for each bar, the truss will remain fully elastic and the unique solution will increase in proportion to X. However, for some critical value X„ one or more bars will have their force Journal of Applied Mechanics JUNE 1986, Vol. 53/235 Copyright © 1986 by ASME Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 07/07/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use