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).)