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.

Most of the references which I have cited are papers in journals, papers in proceedings of conferences, and sections of books. These journals, volumes of proceedings and books are all sources of further reading on the theory of shell structures. The following are some specific suggestions for further study.

The history of the subject is discussed from different viewpoints by Naghdi (1972) and Sechler (1974); and is also sketched by Flügge (1973, Bibliography).

The application of shell theory to practical problems in the aerospace field is described well by Babel, Christensen & Dixon (1974) and Bushnell (1981).

In his standard text on finite-element methods Zienkiewicz (1977) includes three chapters (13, 14, 16) on different types of finite-element calculation for shell structures.

A good example of the application of the membrane hypothesis to a shell of less simple form than those in chapter 4 is given by Martin & Scriven (1961).

Steele (1975) has written one of the few papers in the literature which uses change of Gaussian curvature as a variable for the description of distortion of surfaces (cf. chapter 6). His paper is concerned with the formation of a non-shallow shell (namely a cooking-pot) from a flat sheet by a process in which non-uniform surface stretching is imparted to the surface by beating.

For a discussion of non-symmetric behaviour of various non-cylindrical shells (cf. chapter 9) see Seide (1975).

Limit analysis (Chapter 18) was applied to the bending of curved pipes by Calladine (1974b).

The theory of shell structures is a large subject. It has existed as a well-defined branch of structural mechanics for about a hundred years, and the literature is not only extensive but also rapidly growing. In these circumstances it is not easy to write a textbook. The character of any book depends, of course, mainly on the author's conception of its subject matter. Thus it may help the reader if I set out my basic views on the theory of shell structures at the outset.

Most authors of books and papers on the theory of shell structures would agree that the subject exists for the benefit of engineers who are responsible for the design and manufacture of shell structures. But among workers who share this same basic aim, a wide variety of attitudes may be found. Thus, some will claim that they can give the best service to engineers by concentrating mainly on the form and structure of the governing equations of the subject, expressed with due rigour and in general curvilinear coordinates: for once the foundations have been laid properly (they say), the solution of all problems becomes merely a mathematical or computational exercise of solving the equations to a desired degree of accuracy; and indeed unless the foundations have been laid properly (they say), any resulting solutions are of questionable validity. Another group will argue, on the contrary, that they can serve engineers best by providing a set or ‘suite’ of computer programmes, which are designed to solve a range of relevant problems for structures (including shells) having arbitrary geometrical configuration; and indeed that the provision of such programmes renders obsolete, at a stroke, what was formerly called the theory of shell structures.

Introduction

The geometry of curved surfaces plays an important part in the theory of shell structures. Many practical shell structures are made in the form of simple surfaces such as the sphere, the cylinder and the cone, whose geometry has been well understood for centuries. It has, therefore, been argued by some workers that sophisticated geometrical ideas are not needed for the analysis and design of a wide range of practical shell structures.

Gauss (1828) made a breakthrough in the study of general surfaces. He showed that there were two completely different ways of thinking about a curved surface, either as a three-dimensional or a two-dimensional object, respectively; and that the two different views had a very simple mathematical connection involving a quantity which is now known as Gaussian curvature. The ideas which Gauss described in his paper are of great importance for an understanding of the behaviour of all shell structures, however simple their geometrical form happens to be. The main object of the present chapter is to explain Gauss's work in relation to the geometry of curved surfaces. In chapter 6 we shall proceed further along these lines, with an investigation of the geometry of distortion of curved surfaces.

The basic geometrical ideas which Gauss discovered are not difficult to grasp by those who have at their disposal relatively modest mathematical tools. However, Gauss gave a very thorough treatment of the problem in his paper, and in particular he developed the use of general curvilinear coordinates for the description of surfaces.

Introduction

In most of the chapters of this book we have assumed that the material from which a shell is constructed behaves under stress in a linear-elastic manner. The materials which are used in structural engineering generally have a linear-elastic range, but behave inelastically when a certain level of stress is exceeded. Moreover at sufficiently high temperatures irreversible creep may be the most significant phenomenon.

It is obvious that there are some circumstances in which it is necessary for the designer to understand the behaviour of shells in the inelastic range. This subject is a large one, and in this chapter we shall give an introduction to part of it.

The aim of the present chapter is to give a glimpse, mainly through a few specific examples, of the ways in which the structural analyst may tackle problems connected with inelastic behaviour of shells. In general our plan will be to set up the simplest problems which illustrate various important points. But first it is necessary to discuss some general questions in connection with the scope of plastic theory, and the circumstances in which it is valid.

Plastic theory of structures

Engineering problems involving shell structures in which plasticity of the material plays an important part may be divided roughly into three categories, as follows.

1. (i) In many shell-manufacturing processes large-scale plastic deformation over the surface enables flat plates to be deformed into panels of spherical shells, complete torispherical pressure-vessel heads, or highly convoluted expansion bellows. In all of these and similar cases the material undergoes strains well into the plastic range, and there are also large overall changes in geometry during the process of deformation.

2. […]

General remarks on shell structures

This book is about thin shell structures. The word shell is an old one and is commonly used to describe the hard coverings of eggs, Crustacea, tortoises, etc. The dictionary says that the word shell is derived from scale, as in fish-scale; but to us now there is a clear difference between the tough but flexible scaly covering of a fish and the tough but rigid shell of, say, a turtle.

In this book we shall be concerned with man-made shell structures as used in various branches of engineering. There are many interesting aspects of the use of shells in engineering, but one alone stands out as being of paramount importance: it is the structural aspect, and it will form the subject of this book.

Now the theory of structures tends to deal with a class of idealised or rarified structures, stripped of many of the features which make them recognisable as useful objects in engineering. Thus a beam is often represented as a line endowed with certain mechanical properties, irrespective of whether it is a large bridge, an aircraft wing or a flat spring inside a weighing machine. In a similar way, the theory of shell structures deals, for example, with ‘the cylindrical shell’ as a single entity: it is a cylindrical surface endowed with certain mechanical properties. This treatment is the same whether the actual structure under consideration is a gas-transmission pipeline, a grain-storage silo or a steam-raising boiler.

This appendix gives a brief sketch of various theorems in structural mechanics which are used in several parts of the book. These theorems apply to small deflections of elastic structures in the absence of buckling or other ‘geometrychange’ effects. In this appendix, for the sake of brevity and simplicity, they are described with reference to a simple plane pin-jointed truss which can be discussed in terms of a few, discrete, variables; but they can readily be translated into more general forms relevant to continuous structures (see appendix 2 on the idea of corresponding forces and displacements, etc.). The following description is restricted to frameworks whose members are made from weightless linear-elastic material, and which are stress-free in the initial configuration; but there is no difficulty in extending the scope of the theorems to include nonlinear elasticity and problems involving initial stress.

The description begins with the principle of virtual work, which makes a connection between the two distinct sets of conditions describing statical equilibrium and geometric compatibility of the various parts of the structure, respectively. This principle holds irrespective of the mechanical properties (or ‘constitutive law’) of the material from which the structure is made; and all of the various elastic theorems are derived directly from it by incorporation of the elastic material properties. (The theorems of the plastic theory of structures, which are used in chapter 18, are also derived directly from the principle of virtual work; but they are not proved here (see Calladine, 1969a).)

Features of a tube flame

Both inside and outside the laboratory it is commonly observed that a premixed flame can be stabilized at the mouth of a tube through which the mixture passes. Such a flame, usually conical in shape though not necessarily so, can be conveniently divided into three parts: the tip, the base (near the rim of the tube), and the bulk of the flame in between.

Elementary considerations of the flame speed and the nature of the flow explain the conical shape (see Figure 8.2 and the accompanying discussion). Simple hydrodynamic arguments provide salient features of the associated flow field, as we shall see in section 2; additional details are outlined in section 6.

An understanding of the nature of the combustion field in the vicinity of the rim is crucial in questions of existence and stability of the flame. Gas speeds near the tube wall are small because of viscous effects, so that if the flame could penetrate there it would be able to propagate against the flow, traveling down the tube in a phenomenon known as flashback. In actuality, the flame is quenched at some distance from the wall through heat loss by conduction to the tube (for a stationary flame); this prevents it from reaching the low-speed region. Such quenching enables unburnt gas to escape between the flame and the wall through the so-called dead space, a phenomenon that is described mathematically in section 5.

Existing combustion books are primarily phenomenological in the sense that explanation, where provided, is usually set in an intuitive framework; when mathematical modeling is employed it is often obscured by ad hoc irrational approximation, the emphasis being on the explanation of existing experimental results. It is hardly necessary to add that the philosophy underlying such texts is scientifically legitimate and that they will undoubtedly stay in the mainstream of combustion science for many years to come. Nevertheless, we are of the opinion that there is need for texts that treat combustion as a mathematical science and the present work is an attempt to meet that need in part.

In this monograph we describe, within a mathematical framework, certain basic areas of combustion science, including many topics rightly covered by introductory graduate courses in the subject. Our treatment eschews sterile rigor inappropriate for a subject in which the emphasis has been physical, but we are deeply concerned with maintaining clear links between the mathematical modeling and the analytical results; irrational approximation is carefully avoided. All but the most fastidious of readers will be satisfied that the mathematical conclusions are correct, except for slips of the pen.

Although the material covered inevitably reflects our special interests and personal perspectives, the entire discussion is connected by a singular perturbation procedure known as activation-energy asumptotics. The description of reacting systems characterized by Arrhenius kinetics can be simplified when the activation energy is large, corresponding to an extreme sensitivity to temperature.

Nonuniformities

An understanding of the response of a premixed flame to nonuniformities in the gas flow is important in many technological situations. To sustain a flame in a high-velocity stream the turbine engineer must provide anchors, and these generate strong shear. The designer of an internal combustion engine is concerned with the burning rate in the swirling flow of the mixture above the piston. Turbulence is ubiquitous; then the flame is subject to highly unsteady shear and strain. These situations are extremely complicated and it is unlikely that mathematical analysis will ever provide detailed descriptions; those must be left to empirical studies augmented by extensive numerical computations. Nevertheless, analysis of the response of a flame to a simple shear, for example, can provide useful insight into the interaction mechanism in more complex situations.

Moreover, there are simple circumstances in which such an analysis has direct significance. A burner flame is subject to shear in the neighborhood of the rim, and its quenching depends on the local character of that shear. A flame immersed in a laminar boundary layer experiences both shear (due to velocity variations across the layer) and strain (due to streamwise variations) and its quenching will depend on their local values.