William Prager

The Theory of Plasticity: A Survey of Recent Achievements

After a brief historical introduction, recent achievements in the theory of plasticity are surveyed with emphasis on applications in mechanical engineering.Kinematic models are presented that indicate the complexities of mechanical behaviour in the plastic range. The fundamental theorems of limit analysis are discussed, and their application to two- and three-dimensional problems is illustrated by examples. Shakedown analysis and limit design are defined. Problems involving large plastic deformations are discussed with special reference to metal forming processes. Applications of the theory of plasticity to impact testing and blast damage are reviewed. Recent changes in the theory of structural stability in the plastic range are mentioned. Throughout the lecture, impending developments of the theory of plasticity are indicated.

Problems of Optimal Structural Design

Optimal sandwich structure design for stiffness, natural frequency, buckling load and safety through plasticity

Optimal structural design for given deflection

ZusammenfassungEs wird ein Prinzip der stationären gegenseitigen potentiellen Energie aufgestellt für zwei Belastungssysteme eines elastischen Balkens veränderlicher Biegesteifigkeit. Aus diesem Prinzip wird eine hinreichende Bedingung für stationäres Gewicht eines Sandwichbalkens abgeleitet, wenn die von einer Belastung an einem bestimmten Querschnitt erzeugte Durchbiegung vorgeschrieben ist. Für statisch bestimmte Balken wird gezeigt, dass diese Bedingung ein globales Minimum des Gewichts sicherstellt. Anwendungsbeipiele und Erweiterungen werden besprochen.

The load carrying capacities of circular plates

Abstract This paper is concerned with the load carrying capacities of circular plates made of a perfectly plastic material that obeys the yield condition of Tresca and the associated flow rule. Various conditions of rotationally symmetric loading and support are discussed.

Optimal Layout of Grillages

ABSTRACT The paper deals with the minimum-weight design of elastic grillages in which the absolute value of the axial stresses nowhere exceeds a prescribed value. After discussing optimality conditions for this problem, a geometrical method for obtaining the optimal beam directions is presented. Considering quadrilateral grillages, the effect of translation and rotation of sides on the optimal layout is investigated. Finally, the concept of “beam weaves” is introduced and the optimal layout of beams near a free edge is discussed.

Strain Hardening Under Combined Stresses

Experimental investigations of the strain hardening of metals under combined stresses are usually conducted so that the directions of the principal stresses as well as the ratios of their magnitudes remain constant during any one test. The paper is concerned with incompressible isotropic materials which are stressed in this manner and deform in accordance with certain postulates. The most general stress‐strain relation which can arise under these circumstances is established, and some special cases of this relation are discussed.

OPTIMIZATION OF STRUCTURAL GEOMETRY

Publisher Summary This chapter focuses on the optimization of structural geometry. Much of the literature on structural optimization is concerned with the optimal choice of cross-sectional dimensions in a structure whose layout has already been determined by other considerations. When the layout and the cross-sectional dimensions are at the choice of the designer, structural optimization becomes a more challenging problem. This chapter reviews the state of the art in the optimization of the layout of trusses and grillages. The chapter discusses grillages and trusses. It also discusses the optimality condition for trusses and describes the truss-like continua.

On Michell trusses

Abstract A modification of Michells problem1 is formulated. This concerns the minimum-weight design of a truss T that transmits a given load to a given rigid foundation with the requirement that the axial stresses in the bars of the truss stay within an allowable range. It is then shown that the truss T cannot be less stiff in the elastic range or in stationary creep than any other truss that uses the same amount of material and respects the same allowable range of axial stress. Finally, it is shown that a truss of minimum weight that supports given point masses from the given rigid foundation and has a given fundamental natural frequency has the same layout as the truss T but, in general, different cross-sectional areas.

Limit analysis of arches

Abstract In the limit analysis of frames it is usually assumed that the limiting bending moment of a cross section is not significantly reduced by the presence of axial forces. In the limit analysis of flat arches this assumption is not permissible as a rule. In the present paper the basic concepts of limit analysis are extended to members in which axial forces as well as bending moments must be taken into account. Methods of determining the critical load intensity of arches are developed and illustrated by numerical examples.

Optimal design of multi-purpose structures*

Abstract Minimum-weight design of a sandwich member that has to serve as a tie in some circumstances and as a beam in others is used to illustrate a general method of optimal design of sandwich structures that have to meet several design requirements. The extension of the method to solid construction is discussed.