Introduction

Classical elastic contact stress theory concerns bodies whose temperature is uniform. Variation in temperature within the bodies may, of itself, give rise to thermal stresses but may also change the contact conditions through thermal distortion of their surface profiles. For example if two non-conforming bodies, in contact over a small area, are maintained at different temperatures, heat will flow from the hot body to the cold one through the ‘constriction’ presented by their contact area. The gap between their surfaces where they do not touch will act more or less as an insulator. The interface will develop an intermediate temperature which will lie above that of the cold body, so that thermal expansion will cause its profile to become more convex in the contact region. Conversely the interface temperature will lie below that of the hot body, so that thermal contraction will lead to a less convex or a concave profile. Only if the material of the two bodies is similar, both elastically and thermally, will the expansion of the one exactly match the contraction of the other; otherwise the thermal distortion will lead to a change in the contact area and contact pressure distribution. This problem will be examined in §4 below.

A somewhat different situation arises when heat is generated at or near to the interface of bodies in contact. An obvious example of practical importance is provided by frictional heating at sliding contacts.

Real and apparent contact

It has been tacitly assumed so far in this book that the surfaces of contacting bodies are topographically smooth; that the actual surfaces follow precisely the gently curving nominal profiles discussed in Chapters 1 and 4. In consequence contact between them is continuous within the nominal contact area and absent outside it. In reality such circumstances are extremely rare. Mica can be cleaved along atomic planes to give an atomically smooth surface and two such surfaces have been used to obtain perfect contact under laboratory conditions. The asperities on the surface of very compliant solids such as soft rubber, if sufficiently small, may be squashed flat elastically by the contact pressure, so that perfect contact is obtained throughout the nominal contact area. In general, however, contact between solid surfaces is discontinuous and the real area of contact is a small fraction of the nominal contact area. Nor is it easy to flatten initially rough surfaces by plastic deformation of the asperities. For example the serrations produced by a lathe tool in the nominally flat ends of a ductile compression specimen will be crushed plastically by the hard flat platens of the testing machine. They will behave like plastic wedges (§6.2(c)) and deform plastically at a contact pressure ≈3Y where Y is the yield strength of the material. The specimen as a whole will yield in bulk at a nominal pressure of Y.

Micro-slip and creep

In Chapter 1 rolling was defined as a relative angular motion between two bodies in contact about an axis parallel to their common tangent plane (see Fig. 1.1). In a frame of reference which moves with the point of contact the surfaces ‘flow’ through the contact zone with tangential velocities V1 and V2. The bodies may also have angular velocities ωZ1 and ωZ2 about their common normal. If v1 and v2 are unequal the rolling motion is accompanied by sliding and if ωz1 and ωz2 are unequal it is accompanied by spin. When rolling occurs without sliding or spin the motion is often referred to as ‘pure rolling’. This term is ambiguous, however, since absence of apparent sliding does not exclude the transmission of a tangential force Q, of magnitude less than limiting friction, as exemplified by the driving wheels of a vehicle. The terms free rolling and tractive rolling will be used therefore to describe motions in which the tangential force Q is zero and non-zero respectively.

We must now consider the influence of elastic deformation on rolling contact. First the normal load produces contact over a finite area determined by the Hertz theory. The specification of ‘sliding’ is now not so straightforward since some contacting points at the interface may ‘slip’ while others may ‘stick’. From the discussion of incipient sliding in §7.2, we might expect this state of affairs to occur if the interface is called upon to transmit tangential tractions.

The subject of contact mechanics may be said to have started in 1882 with the publication by Heinrich Hertz of his classic paper On the contact of elastic solids. At that time Hertz was only 24, and was working as a research assistant to Helmholtz in the University of Berlin. His interest in the problem was aroused by experiments on optical interference between glass lenses. The question arose whether elastic deformation of the lenses under the action of the force holding them in contact could have a significant influence on the pattern of interference fringes. It is easy to imagine how the hypothesis of an elliptical area of contact could have been prompted by observations of interference fringes such as those shown in Fig. 4.1 (p. 86). His knowledge of electrostatic potential theory then enabled him to show, by analogy, that an ellipsoidal – Hertzian – distribution of contact pressure would produce elastic displacements in the two bodies which were compatible with the proposed elliptical area of contact.

Hertz presented his theory to the Berlin Physical Society in January 1881 when members of the audience were quick to perceive its technological importance and persuaded him to publish a second paper in a technical journal. However, developments in the theory did not appear in the literature until the beginning of this century, stimulated by engineering developments on the railways, in marine reduction gears and in the rolling contact bearing industry.

Frame of reference

This book is concerned with the stresses and deformation which arise when the surfaces of two solid bodies are brought into contact. We distinguish between conforming and non-conforming contacts. A contact is said to be conforming if the surfaces of the two bodies ‘fit’ exactly or even closely together without deformation. Flat slider bearings and journal bearings are examples of conforming contact. Bodies which have dissimilar profiles are said to be non-conforming. When brought into contact without deformation they will touch first at a point – ‘point contact’ – or along a line – ‘line contact’ For example, in a ball-bearing the ball makes point contact with the races, whereas in a roller bearing the roller makes line contact. Line contact arises when the profiles of the bodies are conforming in one direction and nonconforming in the perpendicular direction. The contact area between nonconforming bodies is generally small compared with the dimensions of the bodies themselves; the stresses are highly concentrated in the region close to the contact zone and are not greatly influenced by the shape of the bodies at a distance from the contact area. These are the circumstances with which we shall be mainly concerned in this book.

The points of surface contact which are found in engineering practice frequently execute complex motions and are called upon to transmit both forces and moments.

Introduction

In chapter 3 we studied the way in which stretching and bending effects combine to carry axially symmetric loads applied to a uniform elastic cylindrical shell. In chapter 4 we investigated the way in which applied loads of a more general kind are carried by in-plane stress resultants alone, according to the membrane hypothesis. It became clear when we examined the resulting deflections of the shells in chapter 7 that this hypothesis is untenable in certain circumstances, and that indeed, in such cases the bending effects, explicitly neglected in the membrane hypothesis, might well sustain a major portion of the applied loading.

We have reached the point, therefore, where we must consider the more general problem of a shell which is capable of carrying the loads applied to it by a combination of bending and stretching effects. This is our task in the present chapter.

An important idea which we shall develop is that it is advantageous to regard the shell as consisting of two distinct surfaces which are so arranged to sustain the ‘stretching’ and ‘bending’ stress resultants, respectively; and indeed, the chapter as a whole explores various consequences which flow directly and indirectly from this idea.

In section 8.2 we introduce the ‘two-surface’ idea to the equilibrium equations, and show that we may treat the two surfaces separately provided we introduce appropriate force-interactions between them. The ‘stretching surface’ is identical to a shell analysed according to the membrane hypothesis, and we may therefore use directly the work of chapters 4–7 for this part.

Introduction

The main task of this chapter is to investigate some purely geometrical aspects of the distortion of curved surfaces. In general, if a given surface is distorted from its original configuration, every point on the surface will under-go a displacement; and at every point the surface will experience strain (‘stretching’) and change of curvature (‘bending’). Clearly the components of strain and change of curvature are, in general, functions not only of the components of displacement but also of the geometry of the surface in its original configuration.

In the present chapter (and indeed throughout the book) we shall consider only the limited class of distortions in which displacements, strains and changes of curvature are regarded as small, just as they are in the classical theory of simpler structures. In consequence of this simplification, the functional relationships between strain, change of curvature, and displacement will be relatively simple, and indeed linear.

In chapter 3 we have already established ad hoc expressions for change of curvature and hoop strain in terms of radial displacement for symmetrical deformations of a cylindrical shell surface. Our present task includes not only the investigation of more general types of distortion of cylindrical shells but also the consideration of distortion of other kinds of surface.

The arrangement of the chapter is as follows. First we investigate some aspects of the distortion of initially plane and cylindrical surfaces: these have the advantage that they may be described in terms of simple Cartesian coordinates.

Introduction

Piping systems are an indispensable feature of many industrial installations. In such systems straight tubes predominate; but problems of plant layout, etc., obviously make it necessary for pipes to turn corners. There are, broadly, four ways of getting the line of a pipe to turn a corner. First, fig. 13.1a shows a so-called long-radius bend in which the radius b of the centre-line of the curved portion is much larger than the radius a of the tube itself. A rightangle bend is illustrated, but it is obvious that the angle through which the line of the pipe turns is arbitrary, in general. On the domestic scale, bends of this sort may be made, ad hoc, in ductile metal pipes by the use of a pipe-bending machine; but the resulting cross-section of the curved portion is usually not circular: see later. Second, fig. 13.1b shows a so-called shortradius bend, in which the ratio b/a has a value of less than 4, say. The curved section is specially fabricated by casting, or welding together suitably curved panels; and the curved unit is connected to the straight pieces by bolted or welded joints. The types shown in fig. 13.1a and b are known as smooth bends. Third, fig. 13.1c shows a single-mitre bend, which is made by joining a pipe which has been ‘mitred’ by a plane oblique cut. A mitre joint may either be unreinforced (as shown) or reinforced by an elliptical ring or flange.

Introduction

Most of this book is concerned with the performance of shells under static loading. In contrast, the present chapter is concerned with an aspect of the response of shells to dynamic loading. The response of structures to dynamic loads is an important part of design in many branches of engineering: examples are the impact loading of vehicles, the aeroelastic flutter of aircraft, and wave-loading on large marine structures.

In this chapter we shall be concerned with the vibration of cylindrical shells, and in particular with the calculation of undamped natural frequencies. Calculations of this kind sometimes give the designer a clear indication that trouble lies ahead for a proposed structure; but if the design can be altered so that the natural frequencies of vibration of the structure are sufficiently different from the frequencies of the exciting agency, the occurrence of vibration can often be avoided.

For reasons of brevity, this chapter is restricted to cylindrical shells. The methods of the chapter may be adapted to the study of other sorts of shell, e.g. hyperboloidal shells used for large natural-draught water-cooling towers: see Calladine (1982).

Two very early papers on the subject of shell structures, by Rayleigh and Love, respectively, were on the subject of vibration, and the present chapters represent in fact only a relatively small advance on their work. Rayleigh (1881) was concerned with the estimation of the natural frequencies and modes of vibration of bells.