Problem Types in the Theory of Perfectly Plastic Materials

Typical problems of the theory of perfectly plastic materials are discussed for the example given by the structure shown in Fig. 1. A simple graphical representation is developed which facilitates the study of the states of stress and residual stress produced by slowly varying the load. For monotonically increasing load, three domains of mechanical behavior are denned: elastic deformation, contained plastic deformation, and unrestricted plastic flow. In the domain of contained plastic deformation the instantaneous stresses in the structure depend only on the instantaneous load and can be found from the minimum principle of Haar and von Karman. However, if unloading is permitted, the stresses depend on the complete history of loading, and the principle of Haar and von Karman must be replaced by a minimum principle recently developed by H. J. Greenberg. Cases in which the precise loading program is known beforehand are not often encountered in engineering. As a rule, only the extremes are known between which the load will vary. Under certain conditions, the structure will then "shake down" to a state of residual stress such that all further variations of the load between the given extremes are supported in a purely elastic manner. This "state of residual stress" is independent of the precise program of loading. Conditions for the existence of such a state of residual stress are discussed, and a minimum principle is conjectured from which this state may be determined. Finally, structural stability in the plastic range is discussed. I t is pointed out that the customary formulation of the stability problem for conservative systems (elastic range) is not adequate for nonconservative systems (plastic range). ANY METHOD of theoretical or experimental stress analysis is based on a stress-strain law. Hooke's law is generally accepted as an adequate basis for stress analysis in the elastic range. In the plastic range, however, no stress-strain law has found equally general acceptance. The selection of a stress-strain law adequate for the treatment of a specific problem is therefore important. This selection is facilitated by a classification of the problems encountered in the theory of plasticity. In the following, such a classification of problems is attempted for plastic materials that do not exhibit strain-hardening (perfectly plastic materials). To simplify the mathematical work as much as possible, Received February 25, 1948. * This paper was presented at the Symposium on Plasticity held at Brown University on February 9-10, 1948, under the joint sponsorship of the Office of Naval Research and the Bureau of Ships (Contract N7onr-358). t The author is indebted to F. R. Shanley for valuable suggestions concerning the discussion of structural stability in the plastic range. t Professor of Applied Mechanics. the various problem types will be discussed for the example given by the system shown in Fig. 1. The two-force members OA, OB, OC join the point 0 to the fixed points A, B} C. The system is symmetric with respect to the vertical OB, and the load P is acting along this axis of symmetry. The subscripts 1, 2, 3 will be used to refer to the bars OA, OB, OC, respectively, and the forces acting in these bars will be denoted by Si, S2, S3. On account of the symmetry of the system, Si — S3. Consider, first, the mechanical behavior of the system in the elastic range. The elastic strain energy of the system is given by an expression of the form