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.

A central notion in the concept of virtual work (appendix 1) is that both the external force and displacement quantities and the internal tension and elongation quantities are related to each other in the sense that the product of corresponding variables represents a quantity of work. If a single force P acts at a joint, the ‘corresponding’ measure of displacement of the joint is the component of the displacement in the (positive) direction of the line of action of the force. More generally, if the components of a force are specified, say U, V, W, in mutually perpendicular directions, the ‘corresponding’ displacements are the components of displacement u, v, w in the same directions; and the appropriate (scalar) work product is simply Uu + Vv + Ww.

We are not, however, limited to discussion of loads on structures in terms of force as such. A structure may be loaded by a couple, for which the corresponding displacement is an angle of rotation (measured in radians); or a pressure, for which the corresponding displacement is a ‘swept volume’; or a uniform line load, for which the corresponding displacement is a ‘swept area’.

In relation to internal variables we saw in appendix 1 that we must multiply the tension in a bar by the elongation in order to obtain the appropriate work quantity. For a uniform bar of length L and cross-sectional area A, precisely the same quantity would be obtained by evaluating σ∈V, where σ = T/A is the tensile stress, ∈ = e/L is the tensile strain and V = AL is the volume of the bar.

Introduction

The subject of this chapter is the behaviour of thin elastic shells of revolution which are subjected to loads applied symmetrically about the axis of revolution. In chapter 4 we studied the same problem, but there we worked under the simplifying conditions of the ‘membrane hypothesis’; and the analysis involved only the equations of statical equilibrium. In the present chapter we shall not exclude in this way the possible occurrence of normal shearstress and bending-stress resultants; and in consequence we shall need to consider not only the equations of equilibrium but also the conditions of geometric compatibility and the generalised Hooke's law. In the main we shall assume, for the sake of convenience, that the shell is stress-free in its initial, unloaded state. This is by no means always true in practice, and we shall discuss some important exceptions in section 11.6. However, it is always correct to regard our analysis as giving properly the change of stress resultants, displacements, etc. on account of a change of loading. Throughout the chapter we shall adopt the ‘classical’ assumption that displacements, strains and rotations are so small that the various equations may be set up in relation to the original, undeformed, configuration of the shell. Some remarks on the validity of this assumption are made in section 11.6.

On account of the symmetry of both the shell and its loading, the problem becomes one dimensional, in the sense that all of the relevant quantities are functions of a single variable which describes the position of a point on the meridian.

If the various shallow-shell equations from chapter 8 which govern the behaviour of the S-and B-surfaces are assembled together, some remarkable formal analogies between them become obvious. Analogies of this kind were first pointed out in the 1940s by Lur'e and Goldenveiser (see Lur'e, 1961; Goldenveiser, 1961, §30), and they are known collectively as the ‘static-geometric analogy’. They are peculiar to the theory of thin shells and have no counterpart in, e.g. the classical equations of three-dimensional elasticity.

These analogies emerge particularly clearly in the formulation of the equations of elastic shells in terms of the static and kinematic interaction of distinct ‘stretching’ and ‘bending’ surfaces. The following exposition follows closely that given by Calladine (1977b). Since it relates explicitly to shallowshell equations, for which in particular the coordinates are aligned with the directions of principal curvature, the discussion cannot be regarded as complete. In fact the analogy holds when the equations are set up in terms of the most general curvilinear coordinate system; but it is usually regarded as being restricted to shells with zero surface loading (Naghdi, 1972, p.613). As will be seen, the introduction of change of Gaussian curvature (g) as a kinematic variable makes possible the extension of the analogy to shells loaded by pressure (p); and indeed these two variables turn out to be analogous in the present context.

The engineering design of space-frames is often facilitated by the use of an idealisation in which the actual structure is replaced conceptually by an assembly of rods and frictionless ball joints, or (as Maxwell put it) a collection of lines and points. If the idealised assembly is rigid when all of the bars or lines are inextensional – as distinct from being a mechanism – then the actual physical structure under consideration can be expected to carry loads applied at its joints primarily by means of tension and compression in its members. The next stage of the engineering calculation for such a structure is to perform a statical analysis of the tensions in the members, to invoke Hooke's law and then to compute the displacements of the assembly. But for the purposes of this appendix, we are concerned only with the question of the rigidity (or otherwise) of idealised frameworks made up from inextensional bars or lines.

This problem is one which attracts the attention of pure mathematicians. (Consideration of elasticity etc. would make the problem ‘applied’.) These workers are inclined to think of the assembly of lines and points as their real structure, and any physical representation of the system by means of (e.g.) rubber connectors and wooden bars, or even structural steelwork, as conceptual idealisations of the reality under consideration. Here, then, we have a complete inversion of the engineer's view that the geometrical array of lines and points is a conceptual idealisation of the physical reality under consideration; the mathematician's is the platonic as opposed to the aristotelian view of nature.

Buckling of structures

Buckling is a word which is used to describe a wide range of phenomena in which structures under load cease to act in the primary fashion intended by their designers, but undergo instead an overall change in configuration. Thus a rod which was originally straight, but has bowed laterally under an end-to-end compressive load has buckled; and so has a cylindrical shell, which has crumpled up under the action of the loads applied to it.

The buckling of structures is an important branch of structural mechanics, because buckling often (but not always) leads to failure of structures. It is particularly important in shell structures because it often occurs without any obvious warning, and can have catastrophic effects.

The buckling of shells has been studied intensively for about four decades, and the information now available on the subject is enormous. The aim of this chapter and the two following is to give an introduction to the subject in the simplest possible terms.

The ‘classical’ theory to be described in this chapter is merely an extension into the field of shell structures of what is often described as the ‘Euler’ theory of buckling of simple struts which are initially straight. For some problems in shell buckling this kind of theory is adequate, and well attested by experiment; but for other problems it is inadequate and indeed can be positively misleading.

Introduction

We are now in a position to bring together the work of chapters 2, 4 and 6 in order to calculate the displacement of elastic shells which carry applied load, according to the membrane hypothesis, by direct-stress resultants only. In the case of a shell which is statically determinate according to the membrane hypothesis the procedure is straightforward, and consists of the same three steps which are used in the calculation of distortion of other kinds of statically determinate structure:

1. (i) Given the shell and its loading, and appropriate edge support conditions, use the equilibrium equations to find the direct-stress resultants, as in chapter 4.

2. (ii) Given the elastic properties of the material of which the shell is made (E, v) and the thickness of the shell, use Hooke's law (chapter 2) to determine the surface strains in the shell.

3. (iii) Solve the strain–displacement equations, as in chapter 6, together with the appropriate boundary conditions, to determine the displacement of the shell.

Most of the problems which we shall investigate in the following chapters will involve interaction between stretching and bending effects in shell structures. It may seem odd therefore to wish to perform the sequence of calculations listed above, since in practice the membrane hypothesis will rarely be valid. And indeed, most of the results which will be obtained in the present chapter will reappear later as special cases of more general analyses, incorporating bending effects, which will be performed in subsequent chapters.