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.

The problem

Chapter 4 was concerned with the steady combustion of the gases produced by vaporization of a linear condensate at its surface through pyrolysis or evaporation. The results were characterized by response curves of burning rate versus pressure (represented by the Damköhler number). If the applied pressure varies in time, then so also must the burning rate; the nature of the dependence is examined in this chapter.

The effect of variations in pressure on solid pyrolysis has received considerable attention because of its relevance to the stability of solid-propellant rocket motors. Acoustic waves bouncing around the combustion chamber will impinge on the propellant surface and thereby generate fluctuations in the burning rate. These fluctuations will affect the reflected wave which, it is argued, might have a larger amplitude than the incident wave. If so, the transfer of energy (provided it is greater than losses through dissipation and other mechanisms) implies instability.

Our discussion will focus on the response of a burning condensate (solid or liquid) to an impinging acoustic wave. Mathematically we must deal with the disturbance of a steady field containing large gradients; consequently, a frontal attack on the governing equations is not feasible. Six regions can be distinguished: condensate, preheat zone, flame, burnt gas, entropy zone, and far field; without rational approximation, the discussion soon degenerates into either a numerical or ad hoc analysis (or both).

Approach

The development of the equations governing combustion involves derivation of the equations of motion of a chemically reacting gaseous mixture and judicious simplification to render them tractable while retaining their essential characteristics. A rigorous derivation requires a long apprenticeship in either kinetic theory or continuum mechanics. (Indeed, the general continuum theory of reacting mixtures is only now being perfected.) We choose instead a plausible, but potentially rigorous, derivation based on the continuum theory of a mixture of fluids, guided by experience with a single fluid. Ad hoc arguments, in particular the inconsistent assumption that the mixture itself is a fluid for the purpose of introducing certain constitutive relations, will not be used.

Treating the flow of a reacting mixture as an essentially isobaric process, the so-called combustion approximation, is a safe simplification under a wide range of circumstances if detonations are excluded. But the remaining simplifications, designed as they are solely to make the equations tractable, should be accepted tentatively. They are always revocable should faulty predictions result; for that reason they are explained carefully. Nevertheless, whosoever is primarily interested in solving nontrivial combustion problems, as we are, can have the same confidence in the final equations as is normally placed in the equations of a non-Newtonian fluid, for example.

Prologue

The propagation speed of a plane deflagration wave is extraordinarily sensitive to changes in the flame temperature. The result (2.22) shows that, for fixed D and Js, an O(1) change in Tb produces an exponentially large change in the burning rate. More modestly, an O(θ–1) change in Tb produces an O(1) change in flame speed; it is perturbations of such a magnitude that concern us in this chapter.

Such a change may be engineered for the unbounded flame by an O(θ−1) change in Tf, an elementary example that is not of great importance either mathematically or physically. A much more interesting example is cooling by heat loss through the walls of a uniform duct along which the flame is traveling. It is well known that flames cannot propagate through very narrow passages (a key safety principle where explosive atmospheres are involved), and this can be adequately explained by such a heat-loss mechanism, as we shall see.

Perturbations of the same magnitude can also be produced by changes in the size of the duct that occur over distances O(θ). Now there will be slow variations in the combustion field developing on a time scale O(θ). Such slow variations can even be self-induced by residual perturbations of the initial conditions (on that time scale) in the absence of boundary perturbations. In all such cases an obvious conjecture is that the flame velocity is not close to the unperturbed value.

Diffusion flames

Earlier chapters have been concerned with flames for which the reactants are supplied already mixed. When two reactants are initially separate and diffuse into each other to form a combustible mixture, the term diffusion flame applies. A Bunsen burner with its air hole closed supports a diffusion flame between the gas supplied through the tube and the surrounding oxygen-rich atmosphere. A candle supports a vapor diffusion flame, so called because the fuel is produced by liquefaction and subsequent evaporation of the wax caused by the heat of the flame.

Premixed and diffusion flames share certain features, but there are important differences. For example, there is no unlimited plane flame with fuel supplied far upstream, oxidant far downstream. The only bounded solutions of the chemistry-free equations behind the flame sheet are constants, so that the oxidant fraction would be constant there, with no mechanism to generate the necessary flux towards the reaction zone. Supplying the fuel at a finite point upstream does not change this picture. Moreover, since cylindrical flames are geometrically attenuated versions of plane flames (cf. section 7.2), there is no cylindrical diffusion flame either.

To gain insight into the nature of diffusion flames we seek a simple onedimensional configuration that has some physical reality. One possibility is to introduce the oxidant at a finite point in the plane flame; details of that choice have been worked out by Lu (1981).