Michael R. Horne

The behaviour of an assembly of rotund, rigid, cohesionless particles. III*

The mechanism of deformation implied in Howe’s stress-dilatancy theory for an irregular assembly of rigid cohesionless particles is examined. Limiting relations are established between the number of fixed and sliding contacts and the numbers of particles and sliding groups of particles. The ratio of energy transmitted through the assembly to energy supplied is expressed in terms of the orientations of the sliding contact planes and directions of sliding between pairs of particles. It is shown that these orientations and directions of sliding can be obtained by maximizing the energy transmission ratio, or by an equivalent procedure, namely the minimization of the ratio of energy absorbed in internal friction to energy supplied.

Elastic-Plastic Failure Loads of Plane Frames

When deformations become finite, the load factor required to produce rigid-plastic deformation of a plane frame under proportional loading differs from the load factor at collapse as given by simple plastic theory. The effect of all work terms involving the squares of member rotations is investigated and a simple formula derived. The effect of finite deformations on the load capacities of elastic-plastic frames is then studied. It is shown theoretically that, if certain approximations are made, the Rankin© load, based on the rigid-plastic failure load and the lowest critical load, give a close estimate of the actual failure load of such a structure. The nature of the approximations made to arrive at this result shows under what circumstances the Rankine load cannot be expected to provide a close estimate of failure. Experimental and theoretical results illustrating the degree of correlation are presented and discussed.

Minimum weight design

This chapter discusses the minimum weight design. The process of design is one in which the load factor is required to have a given minimum value, and the plastic moments of the various members of the structure are required. When a design is required for a single set of loads, any bending moment distribution satisfying the equilibrium and yield conditions constitutes a possible basis for design. The chapter reviews minimum weight design using prismatic members, and the use of members of continuously varying cross section is discussed. If the weight line is tangential to the permissible region over a finite range, a range of minimum weight designs is possible. A design gives the minimum weight if it satisfies four conditions: (1) equilibrium condition, (2) yield condition, (3) mechanism condition, and (4) plastic hinge condition. The first three conditions are identical with those for the plastic collapse of any structure, and it is the fourth condition that imposes minimum weight.

Chapter 1 – Plastic failure

Publisher Summary This chapter reviews the behavior beyond the elastic limit, with particular emphasis on the failure loads, of structures in which resistance to bending action is the primary means by which the loads are supported. The theorems of plasticity apply strictly only to rigid-plastic structures, and logical treatment is best achieved by regarding practical situations as approximations to the ideal situations solved by using the theorems. Consideration of several examples in the chapter shows that a bending moment distribution that represents the collapse conditions must satisfy various requirements: (1) equilibrium condition—the bending moments must represent a state of equilibrium between the internal and external loads, (2) mechanism condition—the plastic moment of resistance must be reached at a sufficient number of sections, and in the necessary senses, for a collapse mechanism to form, and (3) yield condition—the plastic moment of resistance, determined by the value of the yield stress, must nowhere be exceeded.

Variable repeated loading

This chapter discusses variable repeated loading. If a structure is subjected to a succession of differing load distributions varying between prescribed limits, it is not so easy to discover whether a state can be reached such that all subsequent load applications produce only elastic changes of stress. The order in which loads are applied has no effect on whether a structure can shake down, although again the order of loading may influence the rapidity with which a shakedown state is reached. An upper bound on the shakedown load can be obtained by considering either alternating yield or incremental collapse. A method of systematic elimination of unknowns from a set of linear inequalities, similar to that used for estimating static collapse loads, is more suitable if a digital computer is to be used. The importance of alternating yield is effectively a problem in low cycle fatigue. The fatigue life of structures of ductile material subjected to cycles that involve a stress range of the order of twice the static yield stress is certainly measured in hundreds, thousands, or even tens of thousands of cycles, except to the extent that joints can lead to severe stress-raiser problems.

Plastic moments under shear force and axial load

This chapter discusses the plastic moments under shear force and axial load. In the presence of axial load, either tensile or compressive, the neutral axis no longer divides a section into two equal areas. The plastic moments of doubly symmetrical sections, such as, rectangular sections and rolled I-sections are always reduced by the presence of an axial force, but the case of mono-symmetric sections bent in the plane of symmetry, such as, the T-section is not straightforward. The effects of axial thrust and tension are identical, but the case of axial thrust is usually of greater practical importance. Under axial thrust or tension, the neutral axis may lie in the web or in the flange, depending on the sense and value of the axial load and the sign—hogging or sagging—of the bending moment. The Bernoulli theory of elastic bending of a beam is only correct for a beam subjected to a uniform moment. Unless the moments are applied at the ends in the same way as the elastic stress distributions derived by the theory, that is, varying linearly with distance from the centroidal axis, the beam must be long to eliminate the effects of the precise end loading conditions.

Methods of plastic analysis

This chapter discusses methods of plastic analysis. The uniqueness theorem is being used by ensuring the simultaneous satisfaction of the three conditions of equilibrium, mechanism, and yield. In simple structures in which it is not clear where hinges will form, because of the presence of many loading points, the use of free and reactant bending moment diagrams is of considerable assistance. This method has been extensively applied to the analysis of single bay bents. A typical example of such an application is discussed in the chapter. The extension of the method to multi-bay bents is described. In analyzing structures with larger numbers of members and joints, it is found to be easier to invoke either the lower or the upper bound theorem. The kinematic or upper bound theorem provides the most convenient hand methods of deriving failure loads. As collapse mechanisms are easily visualized, difficulties over sign conventions are avoided, and intuitive ideas of structural behavior are readily introduced.

Chapter 6 – Stability

Publisher Summary This chapter discusses the general features of stability in relation to plastic collapse. Actual structures undergo elastic deformations before any plastic deformation occurs, followed by still larger deformations as the degree of plasticity increases. Whether or not these deformations are sufficient to affect significantly the failure load as compared with the rigid-plastic collapse load depends on the particular structure, and also on the loading pattern. Very slender structures may approach a condition of failure due to elastic instability well before the theoretical collapse load given by rigid-plastic theory is reached, but for a wide range of structures the failure loads are found to be very close to the rigid-plastic collapse values. Change of geometry may be classified according to whether their influence is primarily: (1) in the structure as a whole—“overall stability,” (2) within the length of a member relative to the points of support or attachment of the member—“member stability,” and (3) within the member, affecting its cross-sectional shape—“local stability.”

Rigid-Jointed Frames

This chapter focuses on rigid-jointed frames. In a triangulated frame, any set of loads, provided all loads act at joints, could be supported in equilibrium by a system of internal forces acting as axial loads in the members without any bending action. Non-triangulated frames will support certain load systems also in this way, but only if the load systems are suitable. The behavior of frames so loaded is similar to that of triangulated frames statically determinate in their primary stresses, buckling modes being theoretically possible at a series of critical loads. The essential difference from triangulated structures is the incidence of sway modes involving the translation of one end of a member relative to the other. This chapter describes the calculation of critical loads of this type of frame. A frame that sustains the applied loads entirely in axial compression or tension is structurally the most efficient, but it is not possible so to support any arbitrary combination of joint loads acting on a rigid frame. The chapter also explores multistorey single-bay portals.

Elastic—Plastic Behaviour

This chapter explores the elastic–plastic behavior of a structure. When considering elastic stability, the stiffness of each member of a structure affects the buckling load, and it is incorrect to speak of the buckling load of an individual member. The same is true in the elastic–plastic range, but in many cases, final failure may take place in one member only. Imperfections are always present in some degree, because of lack of straightness or lack of fit or both, and may be important in their effect on the failure loads of compression members in triangulated structures, or of columns in multistory frames restrained against sway. Ultimately, a mechanism forms with hinges at both ends and near the centre of length. If the surrounding members are weaker than the failing member, then hinges may form in these adjacent members. Axial deformations are important in triangulated frames, in which they induce joint translations with accompanying double curvature bending.