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.

Introduction

Reinforced concrete shells have been used in the construction of roofs for many large buildings such as airport terminals, exhibition halls and factories. From a structural point of view a shell is attractive for this purpose, since the continuity of surface which is required to keep out the weather is provided by the structural member itself. From an economic point of view, however, reinforced-concrete shell roofs cast in situ are less attractive, largely on account of the labour-intensive effort which is needed in the construction of the formwork.

According to chapter 5, it is easy to construct a surface having zero Gaussian curvature from rectangular plywood sheets, whereas the construction of other kinds of surface makes it necessary to cut the sheets individually into non-rectangular shapes. It is not surprising therefore that cylindrical shells have been popular for the roofing of relatively simple rectangular buildings according to the scheme shown in fig. 10.1 and extensions of it. Shells of this kind, simply supported at their ends, form the subject of the present chapter.

Several authors have written on the structural analysis of cylindrical shell roofs of this sort, and at least one conference has been devoted to this subject: see Timoshenko & Woinowsky-Krieger (1959, § 126), Flügge (1973, §5.4.4.2), Gibson & Cooper (1954) and Witt (1954).

Almost all of the work which has been reported, however, is devoted to the analysis of particular examples having specific dimensions, and it cannot be said that any clear design principles have yet emerged from these studies.

Introduction

The ideas and equations which we have developed in chapter 8 provide tools for the solution of a wide range of practical problems. The aim of the present chapter is to give examples of some of these applications. The order follows roughly that of chapter 8.

The first example is concerned with a cylindrical tube which is partly full of a heavy fluid and acts as a beam between supports at its ends. We shall be able to make direct use of results from chapter 8 in determining the stresses and displacements in the shell, and we shall find that the pattern of behaviour depends strongly on the value of a certain dimensionless group Ω (to be defined in (9.9)) which involves the length, radius and thickness of the shell.

The next group of examples involves the response of a cylindrical shell to forces applied at one end while the other end is supported in various different ways which are met in engineering applications. The applied forces all vary periodically in the circumferential direction. The solutions will involve mainly the ‘long-wave’ behaviour of the shell, and it is advantageous to begin by setting out some standard ‘beam-on-elastic-foundation’ results which will be useful in the subsequent work. The last example in this group concerns the response of a cylindrical shell to a radial point load.

The final problem to be considered in this chapter is the response of a spherical shell to a radial point load: this may be discussed in terms of a ‘nearly-cylindrical’ shell.

Introduction

When a drinking-straw is bent between the fingers into a uniformly curved arc of steadily increasing curvature, there comes a point when the tube suddenly collapses locally and forms a kink. If the experiment is repeated with a fresh straw, and the specimen is observed more carefully, it is found that the cross-section of the entire tube becomes progressively more oval as the curvature increases: and the kink or crease which suddenly forms involves a complete local flattening of the cross-section, which then offers virtually no resistance to bending.

These observations suggest that the buckling of a thin-walled cylindrical shell which is subjected to pure bending involves behaviour which is of a different kind from that which we have encountered in chapters 14 and 15; for we have previously not come across major changes in geometry, spread over the entire shell, before buckling occurs.

The first investigation of this effect was made by Brazier (1927). He showed that when an initially straight tube is bent uniformly, the longitudinal tension and compression which resist the applied bending moment also tend to flatten or ovalise the cross-section. This in turn reduces the flexural stiffness EI of the member as the curvature increases; and Brazier showed that under steadily increasing curvature the bending moment – being the product of curvature and EI – reaches a maximum value. Clearly the structure becomes unstable after the point of maximum bending moment has been passed; and it is therefore not surprising to find that in experiments the tubes ‘jump’ to a different kind of configuration, which includes a ‘kink’.

Introduction

The subject of this chapter is the behaviour of thin elastic circular cylindrical shells when they are loaded by forces which are symmetrical about the axis of the cylinder. Cylindrical shells have structural applications in many fields of engineering, and the loading is often symmetrical, especially in pressure-vessel applications. Some of the results of this chapter will be directly useful and applicable in design. The behaviour of cylindrical shells when they are loaded by non-symmetric forces will be discussed later, particularly in chapters 8 and 9.

The main reason for the inclusion of the present topic early in the book is that it is uniquely instructive. Although it is obviously a particularly simple problem on account of the symmetry, it does nevertheless illustrate well some basic features of the behaviour of shell structures which reappear repeatedly, as we shall see, in much more complicated problems later in the book. In particular we shall be able to see clearly how the shell mobilises both stretching and bending effects in order to carry the applied loading. The problem will also illustrate how the choice of suitable dimensionless groups enables us to present useful results in the most economical way. Lastly we note that the symmetrically-loaded cylindrical shell provides a good introduction to the behaviour of symmetrically-loaded general shells of revolution, which will be discussed in chapter 11.

Our first problem (section 3.4) is the simplest of all: a semi-infinite shell which is loaded either by uniformly distributed radial shearing force or bending moment at its edge, as shown in fig. 3.1.

Introduction

We are now ready to establish the mechanical properties of a typical small element of a thin uniform elastic shell. We have already decided to replace the shell itself by a model consisting of a surface, and we must now furnish this surface with appropriate mechanical properties.

This task is equivalent to the well-known piece of work in the classical theory of beams in which the beam is shown to be equivalent to a ‘line’ endowed with a flexural stiffness EI, where E is Young's modulus of elasticity of the material and I is a geometrical property of the cross-section. But the present task is more complex than the corresponding one for the beam, in two distinct ways. First, an element of a shell is two dimensional, whereas an element of a beam is one dimensional. Second, an element of shell is in general curved rather than flat.

A basic idea, which was proposed in the early days of shell theory, is that in relation to the specification of the mechanical properties of an element of a shell it is legitimate to proceed as if the element were flat, and not curved. Legitimate, that is, as a ‘first approximation’. Much work has been done by many authors on the question of the degree of inaccuracy which is introduced by this idea: see, for example, Novozhilov (1964), Naghdi (1963). We shall not attempt to justify this idea formally.

Introduction

Triangulated structural trusses of the kind used for bridges, electric power transmission towers, etc. carry the loads applied to them mainly by tensile and compressive stresses acting along the prismatic members. But the applied loads are also carried to a minor extent by transverse shear forces in the members, which are related to bending moments transmitted between members at joints of the frame. It is usual to begin the analysis of structures of this type by imagining that the joints are all made with frictionless pins, and also that the loads are applied only at these joints. In direct consequence of this idealisation there are no bending moments and transverse shear forces, and the analysis is much simpler than it would be otherwise (e.g. Parkes, 1974).

The displacements of the simplified structure are relatively easy to compute. But they involve, in particular, relative rotations of the members at the joints; and thus it is possible to use these computed rotations in order to assess the order of magnitude of the bending moments which were dispensed with at the outset. If these turn out to be substantial, the initial hypothesis that bending moments are negligible is clearly not justified, and the whole calculation must be abandoned in favour of one which pays proper respect to the bending effects.

There is a closely analogous state of affairs in the action and analysis of thin-shell structures.

Introduction

The ‘classical’ analysis of the buckling of thin elastic cylindrical shells was presented in the preceding chapter. There it was pointed out that in some circumstances the results of the classical analysis are reliable, and can form the basis of rational design procedures; but also that there are other circumstances in which the results of the classical analysis can be grossly unsafe. The task of the present chapter is to consider this second class of problem. It was pointed out in sections 14.1 and 14.2.3 that the key to the situation is the use of nonlinear theories of elastic buckling. This is a very large subject indeed, which has attracted much attention from many workers since the early days of the use of thin-sheet metal for the construction of aircraft. The aim of this chapter is to investigate the particular problem of a thin-walled cylindrical shell under axial compression, and to attempt to describe, by the means at our disposal, some important features of the behaviour. In chapter 16 we shall tackle a different problem which also demands nonlinear analysis; and indeed we shall find some common aspects with chapter 15. At the end of the two chapters the reader should be in a position to appreciate the literature of the field (e.g. Hutchinson & Koiter, 1970; Brush & Almroth, 1975; Bushnell, 1981): this covers not only the general theory of elastic post-buckling behaviour, but also specific applications to cylindrical and non-cylindrical shells and panels, with and without reinforcing ribs, etc.