Bounds on Minimum Weight Design 271

A somewhat limited design procedure for elastic-perfectly plastic structures was developed previously []]**. It is extended here to provide upper and lower bounds on the minimum weight of three dimensional structures and is specialized to safe one and two dimensional structures in which either direct stresses or bending stresses are negligible. The generalization also includes the influence of body forces. In principle, therefore, such troublesome factors as the weight of the structure itself or centrifugal "forces" may be designed for in a direct manner. Radially symmetric plane stress and plate bending examples are solved to demonstrate direct design procedures. Introduction. Although design rather than analysis is the real problem in machines and structures, far more research effort is spent on analysis. The reason is primarily the specific nature of the problem posed and the greater possibility of obtaining an unambiguous solution. In what follows the material of construction is idealized as elastic-perfectly plastic. This first approximation to the behavior of real structural metals beyond the elastic range and the accompanying techniques of limit analysis are suitable for problems in which load carrying capacity is of primary importance. In addition to providing more realistic answers for ductile materials, the great advantage of limit analysis over elastic analysis is its relative simplicity. The assumption of perfect plasticity opens the possibility of direct design in the strict sense as opposed to preliminary guess and repeated analysis with the end point not necessarily the best that can be achieved. For problems of such difficulty that direct plastic design is not feasible, it is possible to obtain bounds on an optimum design. Such bounds provide a sound basis of comparison with any proposed structure and indicate the gains, if any, which can be achieved by further refinement. An economical structure ordinarily will be one of least weight within the restrictions of the procedures of fabrication. Framed steel structures, for example, often are best constructed of beams of constant cross-section between joints. Heyman [2], Foulkes f3], Prager [4] and Livesley [5] have studied minimum weight frames of this type. The present paper dealing with beams, plates, sheets, and space structures is in the spirit of Michell [6] where the minimum weight is sought without regard to problems and costs of manufacture and construction. Prior studies of plates in bending from this point of view have been made by Hopkins and Prager [7] [8], Freiberger and Tekinalp [9], A start on a broad theory was made [1] by the establishment of a criterion for absolute minimum weight design for structures which are subjected to direct or membrane stresses and for relative minimum weight in the case of beams or plates in transverse bending. These criteria which include that of Michell as a special case suffer from the same disadvantage. They cannot be satisfied in all, and probably not in most, problems which arise. As part of the extension of this earlier work, the matter of existence of