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University of London Imperial College of Science, Technology and Medicine

ANALYSIS O F ELASTIC THIN SHELLS BY TH E TW O-SURFACE TRUSS MODEL

A thesis submitted for the degree of Doctor of Philosophy of the University of London by FRANCIS CHUKWUEMEKA MBAKOGU

Department of Civil Engineering September 1989

ii

Ill

TO MY MOTHER and TO THE MEMORY OF MY FATHER

IV

V

Of all things, but proverbially so in mechanics, the supreme excellence is simplicity

James Watt

VI

Vll

ABSTRACT

The work described in this thesis concerns research into the theory and numerical analysis of elastic thin shells. The thesis begins with a survey of the theoretical foundations of elastic thin shell theory, as well as a review of some of the available analytical and numerical methods for the calculation of shell structures of practical importance, such as, for example, shell roofs.

In particular, it is concluded that, among the available

numerical schemes of computation, the two-surface truss model possesses specific advantages that help clarify the

structural response of thin shells; and that,

therefore, extensions of the method, which had originally been devised for the analysis of shallow, transversely-loaded convex shells with rectangular plan and isotropic material properties, would be desirable. The rest of the thesis deals, in the main, with extensions of the truss-model scheme so as to encompass general surface tractions, arbitrary geometric shapes and anisotropic material properties as well as the implementation of the (extended) numerical scheme for automatic calculation.

With the general aim of facilitating

these extensions (which are validated on

the basis of comparisons with both

theoretical and finite-element solutions) and simplifying the computational process, generalizations and modifications to the original truss-model scheme are presented : these include some of the computational procedures associated with the numerical technique, such as the assignment of nodal areas of influence, the imposition of boundary conditions, and the exploitation of symmetry.

Also reported are certain

extensions of the two-surface approach to the formulation of the equations of shell theory, and it is shown that the use of this analytical technique in the derivation of the shell equations is not only elegant but is also helpful in understanding the computational logic inherent in the truss-model scheme.

Vlll

IX

ACKNOWLEDGEMENTS

The work described in this thesis is the result of research carried out in the Department of Civil Engineering at Imperial College, London.

During the course

of this research the author was in receipt of a Federal Government of Nigeria Scholarship; this financial support is gratefully acknowledged. The author is greatly indebted to Dr. M.N. Pavlovic", an inspiring scholar and teacher who suggested and supervised the present work, for his constant guidance, help and encouragement.

It was Dr. Pavlovic/ who introduced the author to the

study of shell structures, and, indeed, his own work on the two-surface truss model forms the basis for the research reported here.

In addition, Dr. Pavlovichs lucid

writings and his emphasis upon an understanding of structural behaviour have strongly influenced the author, and will, surely, remain a continued source of inspiration. The author owes much to Dr. M.D. Kotsovos (of the same Department), for his invaluable advice and encouragement throughout the course of this work. The author wishes to record his gratitude to Mr. T. Sippel-Dau (of the Imperial College Computer Centre),

for his advice on the implementation of

program TRUSS on the Imperial College CDC Cyber 960 machine, to Mr. D. Hitchings (of the Department of Aeronautics), for his guidance on the use of the finite-element analysis package FINEL, and to Dr. L.G. Booth (of the Department of Civil

Engineering),

for

his

advice

on

the

use

of

timber

in

shell-roof

is

gratefully

construction. The

excellent

typing

of

the

thesis

by Patricia

O'Connell

acknowledged, as is the help and advice of the author's friends, within and outside the College, who are too numerous to be mentioned individually. Finally, the author is deeply indebted to his wife, Chinwe, and the rest of his family, for their understanding, patience, encouragement and constant support.

X

XI

CONTENTS Page

ABSTRACT

vii

ACKNOWLEDGEMENTS

ix

CHAPTER 1.

ON SOME ASPECTS OF THE THEORY AND ANALYSIS OF ELASTIC THIN SHELLS

1

1.1

INTRODUCTION

1

1.2

THE THEORY OF SHELL STRUCTURES

2

1.3

THIN-SHELL THEORIES

4

1.3.1

The Bending Theory

4

1.3.2

The Membrane Theory

8

1.3.3

The Theory of Shallow Shells

14

1.3.4

The Two-surface Theory

19

1.4

THE STATIC-GEOMETRIC ANALOGY

21

1.5

RESPONSE OF SHELLS TO POINT LOADS

23

1.6

NUMERICAL SCHEMES OF COMPUTATION

24

1.6.1

The Finite Difference Method

25

1.6.2

The Framework Analogy Method

25

1.6.3

The Finite Element Method

27

XI 1

Page

1.6.4

The Two-surface Truss Model

30

1.7

SHELL ROOFS

31

1.8

STATEMENT OF RESEARCH AIMS

33

CHAPTER 2.

THE TRUSS MODEL FOR CONVEX SHELLS WITH RECTANGULAR PLAN

36

2.1

INTRODUCTION

36

2.2

THE TWO-SURFACE IDEALIZATION OF A SHELL

36

2.2.1

The Formulation of the Shell Equations

36

2.2.2

A Numerical Method Based on the Twosurface Model

2.3

43

DESCRIPTION OF THE TRUSS MODEL

44

2.3.1

Basis for the Truss Model

44

2.3.2

Assignment of Nodal Areas of Influence

46

2.3.3

Application of the Truss Model to the Membrane Analysis of Shells

2.3.4

Application of the Truss Model to the Bending Analysis of Shells

2.3.5

48

57

Calculation of Transverse Displacements for Non-membrane Shells

64

2.3.6

Boundary Conditions

66

2.3.7

Exploitation of Symmetry

67

Xlll

Page

2.4

2.5

CASE STUDY 1 : PARABOLOID OF REVOLUTION WITH SQUARE PLAN

71

DISCUSSION AND CONCLUSIONS

77

CHAPTER 3.

APPLICATION OF THE TRUSS MODEL TO SHELLS OF ARBITRARY FORM

3.1

INTRODUCTION

3.2

STRESS AND STRAIN MEASURES REFERRED TO

3.4

116

3.2.1

In-plane Stresses

117

3.2.2

In-plane Strains and S-surface Constitutive Relations

122

3.2.3

Ordinary Curvature Changes

125

3.2.4

Bending (and Twisting) Moments and 125

BOUNDARY CONDITIONS FOR STRESS AND STRAIN MEASURES ACTING IN ARBITRARY DIRECTIONS

128

3.3.1

S-surface Boundary Conditions

129

3.3.2

B-surface Boundary Conditions

145

CASE STUDY 2 : PARABOLOID OF REVOLUTION WITH EQUILATERAL TRIANGULAR PLAN

3.5

116

OBLIQUE CO-ORDINATES

B-surface Constitutive Relations 3.3

116

159

CASE STUDY 3 : PARABOLOID OF REVOLUTION WITH RIGHT-ANGLE ISOSCELES TRIANGULAR PLAN

162

XIV

Page

3.6

CASE STUDY 4 : PARABOLOID OF REVOLUTION WITH CIRCULAR PLAN

3.7

CASE STUDY 5 : CYLINDRICAL PARABOLOID WITH SQUARE PLAN

3.8

3.10

174

CASE STUDY 6 : SADDLE HYPERBOLIC PARABOLOID WITH SQUARE PLAN

3.9

166

178

CASE STUDY 7 : HYPERBOLIC PARABOLOID WITH STRAIGHT BOUNDARIES AND SQUARE PLAN

179

DISCUSSION AND CONCLUSIONS

181

CHAPTER 4.

APPLICATION OF THE TRUSS MODEL TO SHELLS SUBJECTED TO GENERAL SURFACE TRACTIONS, WITH SPECIAL EMPHASIS ON THE PROBLEM OF PRESTRESSING

4.1

INTRODUCTION

4.2

THE DIFFERENTIAL EQUATIONS OF SHALLOW

252

252

SHELLS SUBJECTED TO GENERAL SURFACE TRACTIONS 4.3

INCORPORATION OF GENERAL SURFACE TRACTIONS INTO THE TWO-SURFACE TRUSS MODEL

4.4

256

CALCULATION OF THE DISPLACEMENTS OF ARBITRARILY-LOADED NON-MEMBRANE SHELLS

4.5

252

258

CASE STUDY 8 : PRESTRESSED PARABOLOID OF REVOLUTION WITH SQUARE PLAN

259

XV

Page

4.6

CASE STUDY 9 : PRESTRESSED PARABOLOID OF REVOLUTION WITH EQUILATERAL TRIANGULAR

4.7

PLAN

264

DISCUSSION AND CONCLUSIONS

266

CHAPTER 5.

APPLICATION OF THE TRUSS MODEL TO ANISOTROPIC SHELLS, WITH PARTICULAR REFERENCE TO LAYERED SHELLS

5.1

INTRODUCTION

5.2

THE CONSTITUTIVE RELATIONS OF LAYERED

5.3

5.4

296

296

ANISOTROPIC SHELLS

297

5.2.1

Mid-surface Symmetric Layered Shells

307

5.2.2

Single-layered Shells

311

THE STATIC-GEOMETRIC ANALOGY IN THE EQUATIONS OF ANISOTROPIC SHELL THEORY

318

5.3.1

324

Single-layered Shells

THE FORMULATION OF THE GOVERNING DIFFERENTIAL EQUATIONS OF ANISOTROPIC SHELL THEORY

5.5

335

ANALYSIS OF ANISOTROPIC SHELLS BY THE TRUSS-MODEL SCHEME 5.5.1

5.5.2

339

S - and B-surface Constitutive Relations Referred to Oblique Co-ordinates

341

Boundary Conditions

351

XVI

Page

5.6

CASE STUDY 10 : LAYERED PARABOLOID OF REVOLUTION WITH SQUARE PLAN

5.7

379

CASE STUDY 11 : LAYERED PARABOLOID OF REVOLUTION WITH EQUILATERAL TRIANGULAR

5.8

PLAN

383

DISCUSSION AND CONCLUSIONS

387

CHAPTER 6.

NUMERICAL IMPLEMENTATION OF THE TRUSS-MODEL SCHEME, AND PROGRAM •TRUSS’

6.1

INTRODUCTION

6.2

SOME GENERAL ASPECTS OF THE

416

416

NUMERICAL IMPLEMENTATION OF THE TRUSS-MODEL SCHEME

417

6.2.1

Mesh Specification

417

6.2.2

Support, Material and Load Specifications

418

6.2.3

Matrix Assembly and Computational Strategies for Bending and Membrane Analyses

6.3

421

'TRUSS' : A COMPUTER PROGRAM FOR THE CALCULATION OF THIN SHELLS BY THE

6.4

TRUSS-MODEL SCHEME

424

6.3.1

Automatic Mesh Generation in TRUSS

425

6.3.2

Input-Data Requirements for TRUSS

429

DISCUSSION AND CONCLUSIONS

432

XVII

Page

CHAPTER 7.

CONCLUSIONS

APPENDIX A.

A SIMPLIFIED DERIVATION O F THE

449

EQUATIONS O F SHALLOW-SHELL THEORY IN CURVILINEAR CO-ORDINATES APPENDIX B.

THE ASSIGNMENT OF NODAL AREAS OF INFLUENCE BY TH E 'MEDIAN' METHOD

APPENDIX C.

457

489

A THEORETICAL MEMBRANE SOLUTION TO THE PARABOLOID OF REVOLUTION WITH EQUILATERAL TRIANGULAR PLAN

APPENDIX D.

492

THEORETICAL BENDING SOLUTIONS TO PARABOLOIDS O F REVOLUTION WITH RIGHT-ANGLE ISOSCELES TRIANGULAR AND EQUILATERAL TRIANGULAR PLAN

APPENDIX E.

496

A THEORETICAL BENDING SOLUTION TO TH E SHALLOW PARABOLOID OF REVOLUTION WITH CIRCULAR PLAN

APPENDIX F.

510

A THEORETICAL BENDING SOLUTION TO A RECTANGULAR TRANSLATIONAL SHELL COMPOSED O F AN ODD NUMBER OF ORTHOTROPIC LAYERS SYMMETRICALLY ARRANGED WITH RESPECT TO THE

APPENDIX G.

REFERENCES

MIDDLE SURFACE

524

INPUT DATA INSTRUCTIONS FOR 'TRUSS'

546

571

xviii

1

CHAPTER 1 ON SOME ASPECTS OF THE THEORY AND ANALYSIS OF ELASTIC THIN SHELLS

1.1.

INTRODUCTION Thin shells possess

the remarkable property of combining high strength with

light weight and, not surprisingly, lend themselves to a wide range of practical applications in diverse branches of technology, such as, to mention but a few, aircraft structures, building construction and chemical engineering*. Owing to their complex mode of

action, the utilization of shells poses a

serious problem to engineers, who are responsible for securing their (structural) integrity

under

a

variety of

operating

conditions.

As might

be

expected,

researchers have responded to the challenge presented by this structural form; and, indeed, the

theoretical foundations of the subject of shells have existed as a

well-defined branch of structural mechanics for about a hundred years.

Formerly

the preserve of a few specialists, the subject of shells has undergone a tremendous increase in development so that the currently available literature on shell structures is enormous, and still rapidly growing.

This high level of activity in the area of

shell structures can be explained, firstly, by the massive research effort required for the realization of the ever-widening practical applications of shells and, secondly, by the advent of thedigital computer which has

acted as a stimulus to the

development of extremely powerful numerical, as opposed to analytical, calculation schemes for predicting the structural response of shells. Although, by definition, the thickness of a thin shell is small in comparison with its other dimensions, it is, nevertheless, a three-dimensional body, just as a thin plate (or, for that matter, a long beam) is, in reality, a three-dimensional continuum. Thus, the mechanics of thin shells is, in principle, of the same kind as that of thin plates, in the sense that the same methods of continuum mechanics (such as the rigorous three-dimensional theory of elasticity) are equally applicable to both plate and shell problems.

*

A

However, on account of its curvature, a shell

com prehensive c hron icle o f th e

shell as a structural fo rm

m a n n e r in

w hich

the

in tro d uctio n

of a

th in

has s ig n ific a n tly c ontributed to the d ev elo pm en t o f several

d iffe re n t branches o f e ngineering can be found in the w o rk o f Sechler [2 2 8 ].

2

differs fundamentally in its load-carrying mechanism from that of a plate, for it is this intrinsic property of a shell that makes it, as a rule, a much more efficient structural form than a plate.

On the other hand, the very curvature of shells,

which is responsible for their outstanding structural efficiency, greatly complicates the formulation of shell problems and, therefore, makes the subject of shells, on the whole, much more difficult than that of plates.

1.2.

THE THEORY OF SHELL STRUCTURES In view of the considerable

mathematical difficulties encountered

in

the

solution of shell problems within the framework of the three-dimensional theory of elasticity, engineers have sought to simplify the theory by recourse to various approximations

that

behaviour of shells.

are

deemed

to

provide

a

reasonable

description

of the

The special theory founded on these approximations is the

so-called theory of shell structures. The theory of shells, like the simpler and more familiar theories of beams and plates, is set firmly in the field of structural mechanics, which works within the framework of the so-called scientific method and is concerned with the rational explanation

of phenomena

by means

of suitable

conceptual

models.

It

is

noteworthy that the use of conceptual models is suited not only to the study of the phenomena of man-made systems, with which engineers are generally concerned, but also to the study of natural phenomena.

As Naghdi [171] remarks : "The

notion of a model for an idealized body, a system or even a universe permeates the structure of classical physics; and is, in fact, the cornerstone of all field theories". Since the behaviour of a physical system may, in principle, be studied by means of quite different conceptual models (with concomitantly varying degrees of complexity), the analyst must exercise judgement in the choice of a model, the general aim being the selection of the simplest model that is adequate for the task in hand.

As Calladine [43] points out : "The advantage of a conceptual model

devised for a particular situation is that it leads to relatively simple mathematical manipulations and clear conclusions which can be put to the test.

Less specific

models, which have not been shorn of unnecessary features, tend to produce lengthy calculations which all too often are not brought to a conclusion, and consequently are of relatively little practical use". The classical conceptual model for a shell is the so-called two-dimensional

3

surface model, the essence of which is the idealization of the actual shell as a kind of 'physical surface', that is, as a surface of vanishing thickness which is endowed with mechanical properties in order to mimic the actual shell.

(Clearly,

this idealization of a shell is similar to the familiar idealization of a beam as a 'physical line'.)

Implicit in this two-dimensional

idealization of a shell is

the

fundamental assumption that the usual consideration of stresses in three-dimensional elasticity can be replaced, with sufficient accuracy, by consideration of statically equivalent

stress resultants

and

stress

couples,

independent surface curvilinear co-ordinates.

which

are functions

of

two

Thus, the surface idealization implies

that the complicated three-dimensional equations describing the behaviour of a shell can be reduced

to a

simpler, two-dimensional

form; and indeed Green

and

Zerna [88] point out that "the aim of shell theory is to reduce these equations to two-dimensional form and obtain equations for stress resultants and couples instead of actual stresses". It is important to note that the classical surface idealization of shells, on which the theory of shells is based, is, by necessity, of an approximate nature; and this clearly implies that the theory of shells cannot be exact.

As Koiter [123]

states: "An exact two-dimensional theory of shells cannot exist, because the actual body we have to deal with, thin as it may be, is always three-dimensional". Nevertheless, the theory is quite satisfactory for dealing with a wide range of practical problems. plates,

However, there are, in analogy with the theories of beams and

some aspects of the behaviour of shells -

such as, for

example, the

complicated three-dimensional states of stress that exist in regions of application of concentrated loads idealization.

which do not belong to the realm of a simple surface

In such cases one may have to resort either to more sophisticated

conceptual models which do not idealize a shell as a surface or to direct experimentation.

It is worth noting, however, that the above shortcoming of the

surface idealization is tempered not only by the generally localized nature of the regions in which it is inadequate, but also by the fact that the local details of the response of a shell are, in many practical cases, either unimportant or else may be considered more or less in isolation [43]. It is worth noting that, although it is now common practice for engineers to make use of versatile computer packages for the solution of shell problems, the theory of shell structures retains its importance, computational work.

and is,

There are several reasons for this.

indeed,

relevant to

First, the theory of shell

structures is useful in providing a basic understanding of the behaviour of shells; clearly, such an understanding should be a prerequisite of any computational work, for it is immensely helpful in the intelligent application of computer packages and

4

in the interpretation of the ensuing results.

In this connection, Calladine [44],

with particular reference to the widely-used finite element numerical scheme of calculation, states: " ... shell theorists are often in a strong position to interpret the output from a computer study : for example, they can point out regions in the solution in which 'membrane' theory appears to be satisfactory, and others where 'bending' effects are stronger.

Moreover, at an early stage in a computation they

can give advice about suitable element sizes in different regions, being in a position to anticipate the likely regions of more-or-less rapidly varying stress".

Secondly,

the equations of shell theory lend themselves to numerical solution, and are also useful in the formulation of certain numerical schemes of computation.

Finally,

the closed-form analytical solution of the equations of shell theory not only helps in the validation of numerical schemes of computation, but also permits the qualitative study of the effects of changes in problem parameters.

1.3.

THIN-SHELL THEORIES

1.3.1. The Bending Theory In the so-called bending (or general) theory of shells the stress resultants and stress couples, which, as noted earlier, are associated with the two-dimensional idealization of a shell, are regarded as being of comparable importance.

In other

words, the material of which a shell is made is, within the framework. of the bending theory, deemed to be endowed with mechanical properties in the form of elastic resistance to both extensional and flexural deformations. Since its origins in the last century, the formulation of the equations of the bending theory has been repeatedly examined in the literature [100, 122, 167, 170, 171].

In fact, a wide variety of equations for the bending theory have been

proposed by many investigators, not only because of the different approximations made by these authors, but also on account of variations in their elaboration of the approximations.

It is hardly surprising, therefore, that the bending theory of

shells has been a subject of much controversy. The equations of the linear bending theory of thin shells were originally formulated by Love [139, 140] in 1888.

This formulation, which is commonly

referred to as Love's first approximation [100, 208], occupies a prominent position in the subject of shells. In addition to the usual postulate (inherent in the classical linear theory of

5

elasticity) regarding the smallness of displacements, Love's first approximation to the theory of shells is based on two assumptions : first, it is assumed that the maximum value of the ratio of thickness t to the radius of curvature of the middle surface R is negligibly small in comparison with unity; secondly, it is assumed that the normals to the undeformed middle surface are deformed without extension into the normals to the deformed middle surface and that the components of stress normal to the middle surface are small in comparison with other direct-stress components. The first assumption, which is sometimes called the thinness assumption, defines what is meant by the term 'thin shell'.

Although no precise definition of

thinness is available, it is customary to adopt, for practical purposes, useful rules of thumb, such as that suggested by Vlasov [257], namely, that a shell may be regarded as thin if it satisfies the following inequality:

max

ft l z R

( 1 . 1)

30

It is evident from the above inequality (1.1), which is by no means the most restrictive criterion for thinness to be found in the literature [96, 125, 174], that, as mentioned earlier, the range of practical application of thin shells is very wide. The second of the above assumptions is easily recognizable as an extension to shells of intuitive notions of the kind employed in the well-known Kirchhoff method for plates [117]; and it is for this reason that the assumption is sometimes referred to as the Kirchhoff-Love hypothesis [180].

In essence, the adoption of

the Kirchhoff-Love hypothesis (or 'equivalent' assumptions [80, 122, 174, 229]), which implies the neglect of the deformations caused by transverse shearing and normal strains, enables the deformed state of a shell to be completely determined by reference to the deformed configuration of its middle surface. In the so-called higher-order approximations, one or all of Love's postulates, with the exception of the small-deflection assumption, may be suspended [125]. Thus, for example, Flugge [66], Lur'e [144] and Byrne [38] have independently developed equations, collectively called the Flugge-Lur'e-Byrne approximation, in which the thinness assumption is relaxed while the Kirchhoff-Love hypothesis is retained.

The modified thinness assumption of the above formulation results, in

general, in the appearance of several additional terms beyond those of Love's approximation.

Although the order of the shell equations (and the related

treatment of boundary conditions) in the Flugge-Lur'e-Byrne approximation is

6

precisely the same as in Love’s approximation (by reason of the neglect, in both formulations, of transverse shear deformations), the former does not exhibit any inconsistency

in

regard of

rigid-body

motions

approximation does result in such an inconsistency).

[120,

125]

(while

Love’s

However, several writers [122,

180, 209] have pointed out that the corrections to Love's first approximation obtained by the corrections

presentprocedure are of the same order of magnitude as

obtained

by

suspending

the Kirchhoff-Love

hypothesis,

so

the that

meaningful corrections to Love's formulation can only be achieved by simultaneously suspending both the thinness and the Kirchhoff-Love assumptions; as Koiter [122] remarks, "a refinement of Love's approximation is indeed meaningless, in general, unless the effects of transverse shear and normal stresses are taken into account at the same time". By abandoning the Kirchhoff-Love hypothesis, several writers [7, 87, 100, 169, 170, 213] have broadened the scope of the theory so as to include the effects of both transverse shear deformation and transverse normal stresses.

As might be

expected, the various formulations which admit of these effects differ from one another both in the assumptions adopted and in the rigour exercized in the ensuing developments.

Thus,

for example,

although

the

higher-order

approximations

developed, on the one hand, by Naghdi [169], and, on the other hand, by Green and Zema [87] include the effects of both transverse shear deformation and normal stresses, the latter approximation is based additionally on the thinness assumption of Love's first approximation, while the former depends, instead, on a relaxed thinness assumption of the kind employed in the Flugge-Lur'e-Byrne approximation (which it includes as a special case).

It is interesting to note that formulations of this

kind, in which the Kirchhoff-Love hypothesis is abandoned, are characterized by an increase in the order of the equations from eight (appropriate to such formulations as Love's first approximation and the Flugge-Lur'e-Byrne approximation) to ten. (Clearly, associated with this outcome is a corresponding increase in the number of boundary conditions that must be satisfied along an edge from four to five.)

To

elaborate, Kirchhoff's 'effective' shearing stress relations play no role in these formulations,

by virtue

of

the

inclusion

of

the

effects

of

transverse

shear

deformation, so that boundary conditions corresponding to the three shearing actions (namely, in-plane shear, transverse shear and twisting moment) may be prescribed separately along an edge of a shell.

It will be recalled that this increase (both in

the order of the equations and in the number of boundary conditions to be satisfied) has a direct parallel in plate theory, wherein the incorporation of the effects of transverse shear deformation results in an increase in the order of the (plate) equations from four to six and permits the satisfaction of all 'natural' boundary conditions along an edge [86, 152, 210].

7

In order to put the usefulness of the foregoing higher-order approximations into perspective, it

is worth noting that, with the exception of regions in

the

immediate vicinity of highly concentrated loads, the effect of transverse normal stresses is generally

negligible [125].

Similarly, the influence of transverse shear

deformation is generally insignificant for homogeneous thin shells [19]. with regard to the

Moreover,

order of accuracy obtainable from these formulations, it has

been shown [118, 125], by comparison with exact solutions based on the theory of elasticity, that, despite their sophistication, numerical results obtained with

the

higher-order approximations do not necessarily agree more closely with the exact results than do the results obtained on the basis of Love's first approximation. Thus, it is not surprising that, except for special cases (such as problems of sandwich

construction

and

stress

concentration

[92,

196,

212]),

Love's

first

approximation is generally used for the analytical treatment of shell problems in preference to the higher-order approximations.

Thus, with regard to Love's

first-approximation, Hildebrand et al. [100] remark : "It is generally believed that this formulation of the problem contains all the essential facts necessary for the treatment of thin shells, as long as special conditions do not require inclusion of the effect of transverse shear and normal stresses". Love's first approximation for thin shells (which, as is evident from the foregoing, is founded on a set of sound postulates) is not entirely consistent and, in fact, can be inadequate for the solution of certain shell problems involving nearly inextensional deformations [48, 122, 216].

With a view to removing the

inconsistencies in Love's first approximation, several writers have derived equations for thin shells, collectively called the improved first-approximation equations, in which all of Love's original postulates are retained.

Notable among formulations of

this kind is the so-called Sanders-Koiter set of equations, named after Sanders, who removed the inconsistency associated with rigid-body motions in Love's first approximation [35, 223], and Koiter, who is credited with the clarification of the accuracy of first approximation formulations [122].

However, it is worth noting

that, but for the rare cases involving nearly inextensional deformations, there is little difference,

from the standpoint

of accuracy,

equations based on Love's original postulates [122]. considerations,

the

equations

of

Love's

first

between

the

various shell

In view of the foregoing

approximation,

or

variants

and

extensions thereof, are generally regarded as constituting a reasonable description of the bending theory of shells, and this practice will be adhered to in what follows. Clearly, the solution of the governing equations of shell theory is dependent upon the boundary conditions relevant to the problem in question, i.e. a certain number of relations between the displacements, forces or functions of these actions

8

at the supporting edge or edges of the shell must be specified.

These relations

may be imposed either exclusively in terms of geometric quantities or exclusively in terms of static parameters or in terms of combinations of these parameters, depending on the manner in which the equations are formulated. As noted earlier, the number of boundary constraints required for the integration of the shell equations is four, as in the analogous (i.e. Kirchhoff) theory of plates.

However,

since the plate problem comprises two independent groups (namely, the stretching problem and the bending problem), each of which is characterized by a system of equations of the fourth order, the number of boundary conditions required for the solution of plate problems is accordingly reduced since the stretching and bending problems are uncoupled.

Hence, a greater variety of edge conditions may arise in

the analysis of shells than would be encountered in the solution of plate problems. Despite

the

approximations

to

the

rigorous

three-dimensional

theory

of

elasticity inherent in the bending theory of shells, the equations of the latter are, by reason of their complexity and high order, not always amenable to analytical treatment.

Thus, with the aim of simplifying the calculation of shells, a number

of additional approximations (beyond those on which the theory of shells is based) are often introduced into the theory.

Such approximations are usually justified by

the specific geometries of certain shell-types, by order-of-magnitude considerations, or both.

Thus, in certain cases, mathematical approximations that permit more

convenient solutions have been employed [33, 170, 176].

In this connection,

mention may be made of the so-called Donnell-Mushtari-Vlasov simplifications that are employed in the context of (but by no means limited to) 'shallow-shell' problems [174, 238], of the specialized treatment of specific geometries [231, 232, 259], and of the 'membrane' analysis of shells [54, 97].

1.3.2.

The Membrane Theory

The essence of the so-called membrane theory is the neglect of stress couples (and transverse shear forces) in the study of the equilibrium of a shell.

For

brevity, shells are sometimes classified as 'membrane' or 'non-membrane' according as they are analysed in keeping with the membrane theory or the formal bending theory [80].

The notion of a membrane or momentless state of stress, which is

deemed to exist in a membrane shell, results in a spectacular simplification of the equations of shell theory, thereby enabling many important practical problems to be tackled by means of relatively simple calculations. Consistent with the neglect of stress couples in the study of momentless shells

9

are two alternative approaches to the description of such shells (and hence of membrane theory).

First, it may be argued that the bending rigidity of a

membrane shell is negligibly small in comparison to its extensional rigidity [23, 125, 174, 236]; a simple interpretation of this, based on order-of-magnitude considerations, is that the shell is so thin that the third power of its thickness can be neglected in comparison with the thickness itself* [23, 125, 236].

Secondly, it

could be said that a membrane shell is one in which the changes of curvature and twist during deformation are negligible [125, 174, 229]. Before proceeding further, it should be noted that the classical terminology 'membrane theory', which has been used thus far, is, in fact, misleading : as Calladine [43] points out, the 'membrane' approach to the treatment of shell problems is, in essence, a hypothesis, in the sense that one assumes a p r io r i that bending action can be neglected; thus, one would expect an eventual check on the validity of the hypothesis to be made.

As noted by Calladine [43] (see also

Csonka [54]), there is a closely analogous state of affairs in the pin-jointed hypothesis for the analysis of triangulated trusses.

In the latter, the introduction of

frictionless joints at which the external loads are assumed to be applied (and the consequent neglect of flexural effects at joints) simplifies the analysis considerably; furthermore, by subsequently calculating the deformation pattern of the idealized structure, a check may be made on the validity of the initial hypothesis [184]. In the present context of thin shells, the various computational steps necessary for the complete treatment of shell problems in accordance with the membrane hypothesis are shown in flow-chart form in Fig. 1.1. problem is usually statically determinate,

Since the membrane

one may begin by determining the

in-plane stress components by integration of the equations of equilibrium (i.e. the so-called static equations of the membrane theory [80]), and this in turn leads to the associated in-plane strains through the constitutive relations of Hooke's law for elastic thin shells.

(In cases where the structure as a whole, as opposed to a

differential element thereof, is statically indeterminate, it would, of course, be necessary to combine the above steps.) considerations of geometry.

The next two steps are based on

Thus, in the third stage, the displacement components

are determined by integration of the strain-displacement relations (i.e. the so-called

It may be worth noting that the same conclusion may be reached, albeit more formally, by comparing the governing differential equations of equilibrium (in terms of displacements) of membrane shells with the corresponding equations of non-membrane shells (see, for example, Goldenveizer [80]).

10

geometric equations of the membrane theory [80]), while the fourth stage consists in

the

evaluation

of

curvature

curvature-displacement relations.

changes

from

the

associated

change

of

Finally, the bending and twisting moments may

be calculated by means of the appropriate version of the constitutive relations of Hooke's law. It is worth noting that it is not always essential to determine the bending moments before one can ascertain the validity of the membrane hypothesis.

Thus,

it is possible to detect the breakdown of the latter by examining the deformation pattern or even the stress field itself.

For example, a sharp discontinuity in the

stress field will, in general, lead to strain incompatibility, which can only be eliminated by local bending.

Furthermore, the validity of the membrane hypothesis

may be tested, rather crudely, by comparing the peak bending strain (stemming from the deformed pattern in-plane

strain component

predicted by membrane calculations) to the peak [43]; thus, it

may be

argued that the membrane

approximation is untenable unless the peak bending strain component is much less than its in-plane counterpart, that is, unless the deformation pattern satisfies the following inequality [43]:

“ m i x -1 ------------

< < 1

( 1. 2)

^ 6max

in which Kmax and

denote the peak change of curvature and in-plane strain

components respectively. The foregoing computational procedure for problems is not usually adopted in practice.

thetreatment of membrane-shell

To a large extent, this is due to the

fact that the scheme, although conceptually simple, generally requires fairly involved calculations and hence results in a loss of much of the simplicity supposedly implied by the membrane approximation.

This explains why most authors attempt,

instead, to provide broad guidelines, in the form of 'conditions', for the validity of the membrane hypothesis [9, 174, 202].

Clearly, there is some merit in such an

approach, for it engenders in practitioners an intuitive 'feel' for identifying certain circumstances, some of which are discussed below, under which the membrane approximation

may

be

applied with

confidence.

Nevertheless,

it must

be

emphasized that these guidelines should not be regarded as 'rules' of general validity,

and that the

speaking, only be

justification

of the

membrane

hypothesis can,

established a p o ste rio ri bymeans of the formal

summarized in Fig. 1.1.

strictly

procedure

11

The nature of the constraints at the edges of a shell has a marked influence on the validity of the membrane hypothesis.

To elaborate, it may readily be

shown that the static and the geometric equations of membrane shells, taken separately, represent systems of differential equations of the second order, which may be combined to obtain a fourth-order system of equations in terms of displacements [80, 174].

This, of course, implies that, for a membrane shell, only

two (edge) conditions, as opposed to four in the case of a non-membrane shell, may be specified along an edge (of the shell).

Clearly, this state of affairs is

connected with the omission, in membrane analysis, of flexural effects at all points of the middle surface, and hence also at the boundary of the shell.

Thus, it

follows that static (boundary) conditions at an edge of a membrane shell may only be specified in terms of forces acting in the (edge) tangent plane to the middle surface; and, correspondingly, one may only specify geometric boundary conditions in the form of (edge) tangential displacement components of the middle surface, since transverse displacement components and middle surface rotations are ruled out.

In this connection, it may be worth noting that, in his lucid treatment of

boundary conditions for membrane shells, Goldenveizer [80] refers to the set of admissible constraints relating to force and displacement quantities acting in the plane tangent to the middle surface as ’membrane boundary conditions', and to the set of inadmissible out-of-plane static and geometric constraints as 'non-membrane boundary constraints'. The admissible tangential boundary conditions required at the edge of a membrane shell cannot be specified in an entirely arbitrary manner.

It will be

recalled that the static and geometric equations of membrane shells are both of second order and that the two separate systems of equations may be combined into a single fourth-order governing equation in terms of displacements.

An obvious

consequence of this is that whereas the two required tangential constraints may be specified exclusively in terms of displacements, only one edge condition can be given in terms of forces; in other words, at least one tangential displacement constraint must be specified at an edge point of a membrane shell.

The physical

explanation for this mathematical requirement is rather simple : the 'compulsory' geometric constraint is necessary for securing the rigidity of the middle surface of a membrane shell against inextensional deformations, for the lack of a constraint at an edge point of a shell implies that its deformation pattern, under load, will be of an essentially flexural kind and that its state of stress will not correspond to a momentless state [80, 174].

(The requirement that a membrane shell must not be

deformed inextensionally is sometimes referred to as the 'Novodvorskii Condition', after Novodvorskii who, apparently, was the first to propose it [9, 80].)

12

Evidently, when the membrane boundary conditions at every edge point are of the 'mixed' type (that is, when one condition is geometric while the other is static), the shell as a whole, like its differential element, is rendered statically determinate, in the sense that the stresses may be obtained without reference to the displacements.

On the other hand, when both membrane boundary conditions

are of the geometric type, so that two displacement constraints are imposed at an edge point, the problem becomes statically indeterminate (in the sense that the stress distribution cannot be obtained unless the displacements are taken into account).

This is explained by the absence of a static (edge) condition with which

to carry out the integration of the static equations of membrane shells, so that the use of the fourth-order governing equations in terms of displacements (which may be integrated with the available geometric constraints) becomes compulsory. Following Goldenveizer [80], it is usually argued that the membrane solution gives an approximately true picture of the state of stress and deformation of a shell only at distances sufficiently removed from the so-called 'lines of distortion', to which belong : the edges of a shell, lines along

whichthe middle surfaceof a

shell has a break or the curvature of the middle surface changes abruptly,

lines

along which the rigidity of a shell undergoes sudden changes and lines along which discontinuities in the external surface loads occur. characterized

by more

or

less

significant,

These 'lines of distortion' are

but as

arule

quickly

diffusing,

disturbances of the basic state of stress of a shell, which only the formal bending theory can account for [80].

For example, a uniformly-loaded, simply-supported

cylindrical shell with closely-spaced reinforcing ribs cannot be expected to conform to the membrane approximation, by reason of the sheer density of the network of 'lines of distortion' (formed by the reinforcing ribs); on the other hand, the removal of the ribs would, ceteris p a r ib u s , result in a predominantly membrane state of stress, except, of course, in the vicinity of the edges. The sign of the Gaussian curvature of a shell also has a marked influence on the validity of the membrane hypothesis [71].

For shells of positive Gaussian

curvature differential equations of elliptic type result and disturbances of the membrane stresses are, in general, of a local nature, so that the membrane approximation may be profitably applied, in shells.

very many

practical cases, to such

On the other hand, shells of negative Gaussian curvature are particularly

prone to discontinuities in the membrane stresses, and these discontinuities are, as a rule, transmitted throughout the shells [68, 268]; as Flugge [68] points out, this is due to the hyperbolic nature of their governing equations, and indicates that, for such shells, the membrane approximation should be applied with caution.

13

Nevertheless,

for

many

practical

thin-shell

problems

the

membrane

approximation is adequate except in relatively narrow edge zones wherein, on account of support conditions that disturb the membrane state of stress, flexural actions predominate [99, 252].

Indeed, the bending solution to a problem of this

kind may be obtained by a simple method of calculation in which the membrane solution is a first approximation to which an edge disturbance is superimposed in order to satisfy the flexural boundary conditions.

Such an approach may be

explained from a mathematical standpoint; as Green and Zerna [88] point out, the membrane solution may be regarded as an approximate particular integral of the equations of the bending theory to which must be added the homogeneous solution (involving flexural actions, and in which all loads are taken to be zero) in order to obtain the overall solution.

(Although approximate, such a procedure usually leads

to accurate answers as the shell thickness tends to zero, and is sometimes 'exact* even when the thickness is finite [125].) For the treatment of arbitrary shells it is convenient to use the method of Pucher [198, 199], in which the difficult problem of determining the membrane stress field from the set of equilibrium equations of membrane shells is reduced to the integration of a single, second-order partial differential equation in terms of an Airy-type stress function.

A notable consequence of such an approach is the

revelation of the existence of an analogy between the problem of membrane-stress analysis and the analogy'

torsional problem of prismatic bars.

This so-called 'torsional

[53] opens up the possibility of adopting certain well-known. results

pertaining to the torsional problem of prismatic bars for the solution of analogous membrane-shell problems [53, 174, 250], It is worth

emphasizing that,

in virtue of its efficient utilization of the

material of a shell, the membrane state of stress is an ideal condition that should, as far as practical considerations permit, be aimed at.

Indeed, several writers have

devised an 'inverse' design method which abandons the conventional design practice, whereby stress analysis is performed on a shell having an assumed shape, in favour of one that is based on the selection of a shape which ensures the existence of a desired momentless state of stress [93, 200, 201, 235]. exists between the torsional problem

Interestingly, an analogy

of prismatic bars and

the

problem

of

shape-finding, so that, as in membrane stress analysis, available results from the torsional problem may be used to shells [200, 201].

advantage in obtaining 'membrane shapes' for

14

1.3.3.

The Theory of Shallow Shells

A shell is said to be 'shallow* when its middle surface deviates but little from a plane.

In mathematical terms, this is usually expressed by the requirement that,

for a shell to be considered shallow, its middle surface must satisfy the inequality

dz

2

>

dx

8z

0 .3 )

dy.

in which z - z(x,y)

(1.4)

represents the explicit form of the equation of the middle surface within the framework of the familiar orthogonal cartesian triad (x,y,z).

According to Reissner

[214] the above criterion for shallowness will, as a general rule, be met provided that

0z

z

dz

dx ' dy

1_

(1.5)

8

while the less restrictive requirement dz

dz 9x * d y

y

1

(

2

1 . 6)

will usually suffice for practical purposes. Alternatively, the shallowness criterion may be expressed in terms of the angle

7

between the normal to the middle surface of a shell and the normal to an

arbitrary plane of reference.

In this manner, a shell may be regarded as shallow

if the following approximations can reasonably be made [43, 84] : sin

7

*

7

, cos

7

* 1

(1.7)

It is worth noting, also, that a somewhat different criterion, due to Vlasov [257] and

widely used in the

literature

[9,

32,

85,

177],

consists in

the

requirement that, for a shell to be considered shallow, its rise (or fall) should not exceed one-fifth of its minimum plan dimension.

15

In many applications of thin shells (such as, for example, roof structures) the above requirements for shallowness are satisfied.

As it turns out, certain additional

simplifications connected with the smallness of the curvature of these shallow shells can, to a good approximation, be introduced into the equations of the general theory of shells.

The special theory of thin shells thus obtained, the so-called

theory of shallow shells, is of paramount importance; for, while adhering closely to the spirit of the bending theory (in the sense that it allows for the action of both stress resultants and stress couples), it offers considerable mathematical advantages over the general theory, thereby enabling many practical problems to be worked out with relative ease. The development of the theory of shallow shells is credited to Marguerre [149] and Vlasov [256].

Since their early works, the theory of shallow shells has

been elaborated further by several writers, including Reissner [211], Nowacki [177], Munro [162] and Ambartsumyan [9].

In addition to the fundamental assumptions

inherent in the general theory of shells, the theory of shallow shells is based on three, not unconnected, simplifying assumptions, which are considered below. First,

it

is assumed

that,

in

keeping with

the

foregoing mathematical

requirement for shallow shells (1.3), one may neglect the squares and products of surface derivatives (9z/3x, 3z/3y) in comparison with unity [157, 174, 218].

This

assumption, which is sometimes referred to as the geometric-shallowness assumption [218], implies that the intrinsic geometry of the middle surface of a shallow shell may be taken to correspond to the ordinary Euclidean geometry for a plane.

An

outcome of this postulate is that, when different from zero, the Gaussian curvature of the middle surface of a shallow shell is so small that terms (in the shell equations) in which it is a factor may be neglected [174, 179, 257]. Secondly, it is assumed that tangential displacement components may neglected in theexpressions

be

for changes of curvature and twist, so that these

bending measures are determined solely by the normal displacement component [9, 157, 174]. reference

The appropriatenessof this simplification can easily be explained to

the

analogous problem

of flat

plates,

in

which

by

the tangential

displacement components and the changes of curvature are completely uncoupled. Finally, it is assumed that, for shallow shells, the effect of transverse shear forces may be neglected in the tangential equilibrium equations [9, 179, 218]. in

the

preceding

assumption [218],

cases,

the

justification

for

this

so-called

As

static-shallowness

in virtue of whichtangential equilibrium is deemed to be

achieved purely by membrane action, is connected with the near-flatness of shallow

16

shells [43, 68, 215]. As noted earlier, the above assumptions are not unconnected. be shown,

by virtual work considerations,

Thus, it may

that the assumption regarding the

predominance of the influence of the transverse displacement components over the influence of their in-plane counterparts in the flexural response of a shell is equivalent to the assumption that the effect of transverse shear forces in the tangential equilibrium equations is negligible [157, 172, 174, 218].

Moreover, it

turns out that these two consistent assumptions are intimately connected with the geometric-shallowness assumption, to the extent that a refinement of the former (pair of postulates) must, in order to lead to meaningful corrections to the theory, be

accompanied

by a

corresponding

refinement

of

the

geometric-shallowness

assumption [218]. On the basis of the foregoing postulates, it may be shown that the behaviour of shallow shells is governed by a pair of coupled fourth-order partial differential equations in terms of a stress function (from which the in-plane stress resultants may be determined) and a displacement function (on which the stress couples and transverse shear forces depend)*.

In essence, these equations, which may

combined into a single eight-order equation in one variable,** element of a shallow shell, the deformation. equations

be

express, for an

conditions of equilibrium and compatibility

of

The two equations may be regarded as generalizations of the familiar

of plate stretching and plate bending, to which, taken separately, they

reduce in the degenerate case of vanishing curvature (i.e. flat plates).

The

auxiliary terms in the shallow-shell equations (i.e. those terms that are absent in

* This is the case when the equations are formulated in accordance with the so-called mixed (or combined) method, in which the basic unknowns are a displacement component and a stress function. However, it should be noted that this approach is not unique, as the general equations may be reduced to other systems of governing equations, depending on the choice of basic unknowns. In particular, by selecting the three displacement components of a point as the field variables, one obtains, in what is commonly referred to as the deformation method, a system of three governing equations [9, 23, 256].

The suspension of the Kirchhoff-Love hypothesis, on which the classical theory of shallow shells is based, results in an increase in the order of the shallow-shell equations by two to ten [8, 141, 168], as was pointed out in Section 1.3.1 when discussing the general theory of shells.

17

the analogous plate equations) are, in fact, due to the coupling effect of the curvature of a shallow shell, by virtue of which the stretching and bending effects co-operate in sustaining an applied load.

Thus, the equilibrium equation of a

shallow shell, which may be regarded as a generalized plate-bending equation, contains auxiliary term(s) representing the resistance to the bending of the shell offered by membrane forces; and, correspondingly, the compatibility equation of a shallow shell contains auxiliary term(s) representing the resistance to the stretching of the shell offered by flexural deformations, and may, in turn, be regarded as a generalized plane-stress equation.

The crucial point here is that the coupling

effect of the small, but finite, curvature of a shallow shell accounts for the radical difference between its mode of action and that of a flat plate; and it is for this reason that, under otherwise equal conditions, a shallow shell is, as a rule, much more rigid than a flat plate. In virtue of their relative simplicity, the shallow-shell equations have been used to considerable advantage in the solution of numerous practical problems. Several middle surface geometries with variously-shaped plan projections have been investigated by means of these equations. solutions

based

on

the

shallow-shell

Notable among the available analytical equations

are

those

pertaining

to

the

complicated translational shells which find wide application in civil engineering; detailed accounts of some of these solutions can be found in the well-known books by Ambartsumyan [9], Beles and Soare [23] and Vlasov [257], as well as in many technical papers [12, 13, 32, 85, 197].

An interesting feature which emerges from

calculations of this kind is the possibility of separating the actual solution to a shell problem into a portion recognizable as that of an analogous flat plate under the given loading and a complementary portion associated with the small curvatures of the middle surface of the shell [6, 69, 180]; this not only enables the achievement of significant savings in computational effort, but also enhances the qualitative study of the effect of the curvatures on the response of the shell.

In addition, mention

may be made of a strong similarity between a shallow shell and a flat plate resting on an elastic foundation [257, 258]. cases,

On the basis of this analogy, it is, in certain

possible to simplify the calculation of shallow shells by making use of

existing well-known solutions for plates on elastic media [23, 84]. It is noteworthy that the equations of the theory of shallow shells may also be used for the solution of problems which, p r im a f a c ie , do not belong to the realm of shallow-shell theory.

Clearly, this implies that the set of postulates inherent in

shallow-shell theory, or the consequences thereof, must bear a more general character. shells,

Thus, for example, even before the 'birth' of the theory of shallow

it had been established that the

assumption

regarding the neglect of

18

tangential-displacement components in the expressions for bending deformations (and hence the associated simplifications of the tangential equilibrium equations of the general theory) would generally be reasonable when, as is usually the case, the bending stresses are comparable in magnitude to, or smaller than, their membrane counterparts [37, 60, 172, 174].

In this connection, Novozhilov [174] points out

that "even without a special study, it is obvious on purely intuitive grounds that changes in curvature and twist caused by displacement components tangential to the middle surface must, as a rule, be unimportant".

Thus, the simplified change of

curvature-displacement relations of shallow-shell theory, which correspond precisely to those given by Aron [16] in his faulty

exposition of shell

theory [139],are

admissible as deliberate simplifications for a wide range of practical problems. Against this background, therefore, it is hardly surprising that the shallow-shell equations are applicable to the solution of problems of arbitrary shells of vanishing Gaussian curvature, such as, for example, cylindrical shells [9, 60, 174, 257]. Furthermore, as Goldenveizer [80] has shown, the shallow-shell equations can be used for the solution of a wide class of problems characterized by states of stress with 'large indices of

variation'.

In essence, quantities describing such a

state of stress exhibit very rapid variation (i.e. such quantities increase substantially with differentiation).

In formulating the governing equations for this class of

problems, it turns out that the terms which are neglected on the grounds that they contain low-order derivatives are identical with those which, in the development of the

shallow-shell

equations,

are

discarded

geometric-shallowness assumption [80, 174].

on

the

basis

of

the

so-called

Moreover, in such problems, the

transverse displacement components are considerably greater than their tangential counterparts [80], so that, as

in shallow-shell theory, the Aron-type expressions for

curvature-change components

may be adopted without undue lossof accuracy.

For

example, a state of stress of this type exists in regions of localized loading so that the (local) response of arbitrary shells subjected to concentrated loads may be studied with sufficient accuracy by means of the shallow-shell equations [43, 80]. Interestingly, the same conclusion may be reached from a physical standpoint. Thus, for example, Reissner [211] and Flugge and Elling [70] have pointed out that in many problems of localized loading the region of interest is shallow even if the shell itself is not necessarily so.

In particular, Flugge and Elling state that :

"Since almost any shell has a small slope within a restricted region, the solutions for concentrated loads applied to shallow shells can also be considered to be the solutions, in a localized region, for a more general shell having the same geometry around the point of loading as does the shallow shell".

The crucial point here is

that, while the description of the present theory as a theory of shallow shells is correct, it by no means reflects fully its wide field of application.

19

1.3.4.

The Two-Surface Theory

As noted earlier, loads applied to shell structures are generally sustained by a combination of bending and stretching actions.

The complex interaction between

these two effects can be effectively studied by means of a two-surface theory of shells proposed by Calladine [40].

In this theory the interaction between bending

and stretching actions is brought out in physical terms by the conceptual splitting of the actual surface of a shell into

two distinct,

but coincident, surfaces,

designated B and S, each of which is endowed with a different part of the stiffness of an element of the shell. The S - (or stretching) surface is endowed with in-plane stiffness, and carries membrane (i.e. in-plane) forces but is physically incapable of transmitting bending (and twisting) moments and transverse shear forces.

On the other hand, the B -

(or bending) surface is endowed with flexural stiffness, and sustains bending (and twisting) moments and transverse shear forces but offers no resistance whatsoever to in-plane forces.

Clearly, the notion of separating the material of a shell into a

'two-phase continuum' is merely a useful mathematical model.

As Calladine [40]

points out, this approach is conceptually akin to Terzaghi's model for saturated soils [245], in which the actual continuum is viewed as consisting of a

'soil

skeleton' and 'pore water', each with its own mechanical properties but both occupying the same space simultaneously. As might be expected, the separation of a shell into two surfaces necessitates the specification of appropriate interaction conditions in order to ensure that the surfaces, being coincident in reality, do not behave independently of one another. In the two-suface model the balance in load-sharing between the two surfaces is regulated by the use of a 'force' variable in the form of an interface stress (or pressure),

while the coincidence of the surfaces is ensured

'displacement'

variable

in

the

form

of

the

scalar

by means of a

Gaussian curvature-change

parameter (c.f. the treatment of a 'cut* in classical structural analysis).

In this

connection, it is worth noting that since, by definition, the B-surface cannot offer resistance to in-plane forces, any externally applied in-plane loads must be resisted solely by the S-surface; interestingly, an analogous state of affairs exists in the aforementioned soil model of Terzaghi [245], since, unlike the soil skeleton, the pore-water can only accept isotropic states of stress. From the mechanical characteristics assigned to the imaginary surfaces used in modelling a shell, it follows that the constraints at the edges of the shell must be applied partly to the S-surface and partly to the B-surface.

Thus, in-plane static

20

and geometric constraints must be applied to the S-surface, whereas specified forces and displacements perpendicular to the surface of the shell as well actions about its edge must

as flexural

be dealt with by the B-surface (c.f. Goldenveizer's

classification of boundary conditions into 'membrane' and 'non-membrane' types [80]; see also section 1.3.2). The two-surface theory

of shells has several distinct advantages

conventional presentations of the subject.

over more

First, the two-surface model of a shell

provides a direct and effective way of studying the complex interaction of bending and stretching actions, a grasp of which is crucial for the understanding of the structural behaviour of shells.

Indeed, the clarification of the interaction of

bending and stretching effects is a natural outcome of the two-surface formulation; as Calladine [43] points out, the device of separating the action of a shell into two distinct parts affords the possibility of thinking separately about the two aspects of behaviour, and the interaction between them.

Thus, the load-sharing between the

two surfaces provides insight into the regime of behaviour into which a given problem falls; for instance, this concept has been used to advantage in the analytical treatment of shell problems [42, 43]. The

two-surface

formulation also enables the

Gaussian curvature

change

parameter to appear explicitly in the governing equations for shells, a feature not usually found in the classical formulation of the shell equations.

Thus, the

compatibility of deformations

readily be

of the two imaginary surfaces may most

assured by a matching of the changes of Gaussian curvature of the two surfaces; this stems from the fact that, in keeping with Gauss' dual view of surfaces [75], the Gaussian curvature change variable can be computed separately for each of the two surfaces, that is, in terms of the in-plane strains of the S-surface and the ordinary curvature changes of the B-surface.

An important advantage of using the

Gaussian curvature change as a prime kinematic variable is that it enables the compatibility condition to be expressed in the most economical way since, in sharp contrast to the usual vectorial approach which requires the equality of three displacement components, only a single scalar deformation quantity (namely, the Gaussian curvature change variable) need be matched. The third advantage stemming from the two-surface formulation relates to the formal correspondence between the static and geometric relations of shell theory known as the static-kinematic (or 'static-geometric') analogy.

(This analogy is

discussed in some detail in the next section and hence will not be elaborated upon herein.)

The two-surface formulation of the shell equations not only exposes the

correspondence between the static and geometric relations very clearly, but also

21

enables an extension of the analogy, which had previously been restricted to homogeneous problems, to encompass shells loaded by normal pressure.

This

generalization of the static-geometric analogy to include normal loadings is made possible by the use, in the two-surface model, of the Gaussian curvature change as the prime kinematic variable.

Clearly, this represents a radical departure from

conventional formulations of the shell equations [80, 174, 257], in which the key potential

role

of the

Gaussian

curvature

change

parameter

had

never

been

exploited. Another advantage of the two-surface model is that it is ideally suited to numerical schemes of computation, for it reduces the highly indeterminate general shell problem to two intrinsically determinate sub-problems : the S-surface problem is statically determinate while the B-surface problem is kinematically determinate. Furthermore, it is evident that the extended static-kinematic analogy could be used to advantage in reducing the effort required in the setting up of the systems of equations for the two distinct surface models since it enables the equations for any one of the surfaces to be obtained, by mere inspection, once the corresponding equations for the other surface have been derived.

Indeed, these ideas have been

most profitably employed in a recently-devised specialist numerical technique for the analysis of thin shells known as the two-surface truss model [186] (see section 1.6.4).

Here, the scheme of computation is associated with the discretization of

the continuous surface into a polyhedral model which lends itself naturally to the ready calculation of the Gaussian curvature parameter (and, also, of its change). Finally, the two-surface

theory

enables considerable simplifications to

be

achieved in the derivation of certain governing equations of shell theory; this is well-illustrated by Calladine's elegant derivation of the well-known shallow-shell equations.

The reason for this is that, unlike conventional treatments, the general

problem is separated into two distinct, and simpler, problems which can be recombined at a later stage.

In addition to its directness, such an approach

possesses the didactic virtue of elucidating the meanings of the various quantities in the field equations in simple, physical terms.

1.4.

THE STATIC-GEOMETRIC ANALOGY The equations of the theory of thin shells possess a characteristic symmetry

which has led to the establishment of an intrinsic analogy between the static and geometric quantities employed in the formulation of shell theory.

This so-called

static-kinematic (or 'static-geometric') analogy, which is peculiar to shell theories

22

founded on the Kirchhoff-Love hypothesis [171], seems to have been established simultaneously and independently by Goldenveizer [79] and Lur'e [144]. The static-kinematic analogy is usually described as the formal correspondence between the static and kinematic relations of thin-shell theory, that is, between the equations of equilibrium and those of compatibility; this means that one set of equations can

be obtained

from

the

other

by interchanging stress/force and

strain/displacement variables, while at the same time permuting their subscripts.

In

addition, the analogy is also applicable to the constitutive relations of elastic shell theory, since the two sets of relations, for the bending and stretching actions, can be obtained from each other by the same interchange of stress and strain quantities as well as certain permutations of material properties.

It is for this reason that

some authors refer to the correspondence as a 'material-static-geometric' analogy [191, 231]. Until quite recently, it was thought that the static-geometric analogy was valid only in the absence of surface loading [40, 145]. the analogy is still useful in several ways.

In this limited sense, though,

A noteworthy use of the analogy is in

the clarification of certain mathematical similarities between physically independent problems.

Thus, for example, in the special case when the middle surface of a

shell degenerates to a plane, it can readily be shown that the governing equations of shell theory reduce to two independent systems, one describing the bending of a plate,

while

the

other coincides with

the

plane

problem

of

elasticity;

the

static-geometric analogy hereby indicates the well-known fact that the Airy stress function for the plane problem and the transverse-displacement function describing the plate-bending problem satisfy the same (for example, biharmonic) equation [109, 241].Again, as shown by

Goldenveizer [80], the analogy

is helpful in

exposing the similarity between the membrane (or 'extensionaP) approximation (see section

1.3.2)

and

theso-called

introduced by Lord Rayleigh neglected by

hypothesis.

inextensional

[205, 206],

(or

'bending')

approximation,

in which middle-surface strains are

Among other uses of the static-geometric analogy,

mention may be made of its important role as a criterion for testing the consistency of shell-theory formulations

[35,

163] and its application

in

the

formulation of 'shell elements' for the finite-element numerical analysis technique [108, 159]. As noted earlier, Calladine's ingenious two-surface idealization of a shell, in conjunction with his use of the Gaussian curvature change as a prime kinematic variable, makes possible an extension of the static-kinematic analogy to encompass normal-loading conditions [40].

Furthermore, it turns out that the normal pressure

23

and change of Gaussian curvature variables correspond to each other (in the present context).

Evidently, the suppression of the role of the Gaussian curvature

change parameter, which, as pointed out earlier, is a feature of the conventional formulations of the equations of shell restriction

of the

analogy to

theory,

is responsible for the

homogeneous problems.

Obviously,

earlier

Calladine's

extension of the static-geometric analogy to non-homogeneous cases (i.e. where loading is present) is a signal contribution to the subject of shells for, as intimated in the preceding section, it broadens the range of application of the analogy at both the conceptual and the computational levels, while, at the same time, making the analogy of relevance to a large number of practical structural problems.

1.5.

RESPONSE OF SHELLS TO POINT LOADS For purposes of structural analysis, it is expedient to replace the real loads

acting on a structure by idealized loads which, clearly, are deemed to approximate the real

loads to a sufficient degree of accuracy.

In so doing, one

usually

distinguishes between a distributed load, which is spread over a substantial surface area of

the structure, and a localized load which, by contrast, acts

relatively small surface area of the structure.

over a

In the limiting case when the area

on which a localized load acts is extremely small, such as when it is of the order of the smallest dimension of the structure, the load can be considered as a single force applied at a point only, that is, as a concentrated (point) load. The

point-load idealization is, of course, a gross simplification of the actual

distribution of a localized load and, on the basis of such an idealization, one cannot realistically expect to obtain other than an extremely crude estimate of the state of stress within the loaded region.

On the other hand, at distances

sufficiently removed from this loaded zone, it is reasonable to expect the actual details of the load distribution to be of but minor importance, so that, at such distances,the real state of stress may be predicted,

without significant loss of

accuracy, even with the crude point-load idealization. In shell problems the state of stress at the point of application of a point load is of a singular, as opposed to regular, character, in the sense that, at such points, certain state variables assume infinite values [67, 68, 142, 143].

Evidently,

these singularities do not accord with physical reality, and are indicative of the fact that a point load is a limiting case of a more complicated, but certainly more authentic load, a knowledge of the actual (or, at least, approximate) distribution of which is a c o n d itio sin e qua non for the realistic simulation of the local response

24

of the shell on which it acts. The response of a thin shell to point loads is strongly influenced by the rigidity of the shell.

For shells endowed with rigidity in regard to both extension

and flexure, singularities occur in the form of infinite bending moments and shear forces, while the deflections, although remaining finite, exhibit sharp gradients in the vicinity of loaded points.

By contrast, singularities manifested through infinite

deflections occur in shells endowed with only extensional rigidity, an outcome which clearly bears testimony to the importance of bending rigidity in the locality of point loads [67, 142]. Several more specific remarks can be made on the nature of point-load singularities in non-membrane shells.

Thus, regardless of the geometry of the

middle surface of a thin shell, the dominant part of the stress distribution in the vicinity of a singular point is the same as that for the corresponding problem of a flat plate [47 , 224, 225].

However, the behaviour of a shell in a region

sufficiently far removed from a singular point is influenced by both the shell's curvature and its boundary conditions [63].

Furthermore, under otherwise equal

conditions, the extent of the region of predominantly bending action is larger in shells of zero and negative Gaussian curvature than in positive-Gaussian curvature shells [70, 72, 142].

1.6.

NUMERICAL SCHEMES OF COMPUTATION Available analytical methods for the solution of engineering problems cannot

cater for arbitrary shells subject to general loadings and/or boundary conditions; and it is for this reason that numerical schemes of computation are widely used for the treatment of shell problems.

Unlike analytical techniques, which employ

vanishingly small quantities (or differentials) and thereby result

in continuous

expressions for problem unknowns, numerical methods are based on the use of finite, but small, quantities, and, for a given problem, lead to the determination of the unknowns only at points of a network defined a b in itio .

By enabling the

ready solution of large systems of simultaneous equations, the digital computer has extended radically the field of application of numerical methods.

Although they all

entail

techniques

the

discretization

of

continua,

the

various

numerical

differ

fundamentally in formulation, as will become apparent from the following brief review of some of the principal schemes that may be used in the treatment of shell problems.

25

1 .6 .1 .

T h e F in ite D iffe re n c e M e th o d

The finite difference method is a general numerical technique for the solution of physical problems.

It is based on the well-known mathematical concept of

expressing the derivatives of a function in terms of the values of the function itself at a discrete number of points [240].

In essence, the method consists in the

replacement of the governing differential equations as well as the equations for the boundary conditions of a problem by finite difference approximations in terms of the unknown values of the field variables at a finite number of discrete points; the resulting system of linear algebraic equations may then be readily solved for the problem unknowns. Notable success has been

achieved

problems by the finite difference method.

in the

numerical

treatment of shell

Within the framework of the bending

theory, it has been applied, for example, to the analysis of shells of revolution [195, 237] and translational shells [2, 77, 153].

The method has also been used

to advantage in the analysis of membrane shells [17, 185, 222]. that the numerical technique is particularly suited to the

It is worth noting

analysis of shells with

simple geometries, for which the finite difference equations may be formulated with relative ease.

Furthermore, the method is most conveniently applied to shell

problems governed

by low-order equations since, in such

cases, fairly accurate

results can usually be obtained without the use of excessively dense networks. However, the finite difference method has shortcomings which severely limit its application to

arbitrary shells and

programmes (based on the

make

the development of general-purpose

numerical technique) relatively difficult [226].

In

addition to the difficulties associated with the formulation of finite difference equations for arbitrary networks [229], problems may also be encountered in the edge regions of shells [50, 226].

It should be noted, however, that excellent

accuracy can be obtained in a great number of cases by judicious application of the finite difference method, a fact which accounts for its use in the validation of 'elements' developed for the powerful finite element method [49, 116, 220, 265].

1.6.2.

The Framework Analogy Method

The framework (or lattice) analogy method is a numerical technique in which continua are modelled by 'equivalent' bar systems.

The basis of the method is the

notion of replacing finite continuum elements by analogous framework 'molecules' (or 'cells').

The motivation for such an approach was the prospect of utilizing the

26

w e ll-es tab lis h ed

techniques

fo r

th e

analysis

of

la ttic e

systems

in

th e

tre a tm e n t

of

th e m o re d iffic u lt problem s o f con tin u a [1 0 5 , 1 0 6 , 1 4 8 ].

In the lattice analogy method the continuous material of the elastic body under investigation is replaced by a framework composed of a number of cells, the constituent bars of which are endowed with elastic properties consistent with the nature of the problem in hand.

The cells of a framework are usually arranged in

a definite pattern and each cell is connected to its neighbours at its peripheral joints.

For a given type of continuum problem, the stiffness assigned to the

members of a lattice cell must be such as to ensure that it behaves like the actual continuum element which it idealizes; clearly these stiffnesses depend both on the material properties of the actual continuum and on the geometry of the cell (i.e. on the shape of the cell as well as on the layout of its constituent bars). The lattice analogy method can be used for the solution of a variety of problems, such as those of plate-stretching, plate-bending and three-dimensional elasticity.

In each case, the primary consideration is the derivation of a lattice cell

that is suited to the task in hand.

A lattice cell that is suitable for the modelling

of an element of a thin shell subject to the bending theory may be obtained by a combination

of

the

bar

plate-stretching lattice cells.

stiffnesses

of

its

corresponding

plate-bending

and

On the other hand, the modelling of a membrane

shell can be achieved by means of lattice cells, the members of which are endowed with only stretching stiffnesses. Prior to the advent of the digital computer, the principal shortcoming of the framework analogy method, which militated against both its adoption as a practical analysis tool and its further development, was the huge effort generally required for the analyses of the 'equivalent* framework models.

As might be expected, the

availability of suitable computing facilities more or less removed this shortcoming and gave new impetus to the development of the framework method.

In

particular, renewed interests in the numerical technique led to the establishment of its variational basis [1, 24, 221] and to the development of more sophisticated lattice cells [181, 182, 266, 267]; thus, for example, it became possible to tackle the problem posed by the dependence of certain lattice-cell arrangements on particular values of the ratio of Poisson, albeit at the expense of increasing the number

of degrees of freedom of the framework cells.

Nevertheless, and despite

its good predictive capability, the method has largely failed to achieve significant popularity as an approach to the calculation of shells, an outcome doubtless connected with the emergence of, and shift in emphasis towards, the more sophisticated, and less restrictive, finite element method.

27

1 .6 .3 .

T h e F in ite E le m e n t M e th o d

By virtue of its versatility, the finite element method has become the foremost numerical technique for the

analysis of structural

systems.

The

method

is

well-established, and full details

of its theoreticalfoundations, as

well as its

compact formulation by the use

of matrix algebra, can be found in standard

textbooks [20, 51, 269]. The

finite element procedure

conceptual model of a

is

based on the

notion of idealizing the

physical structure as an assemblage of

connected to one another at a discrete number of nodal peripheries.

distinct elements points along their

The elements may be of one - , two - or three-dimensional form;

also, they may have curved sides or faces and may contain internal nodal point(s) in addition to the peripheral nodes. The

finite element scheme may

be formulated in several different ways,

depending on the choice of primary nodal variables.

The most widely-used

formulation is the so-called displacement-based finite element method in which nodal displacements are chosen as the basic variables; once the displacement field for a given problem is

known, the other design

readily be obtained.

In essence,

quantities,

such as stresses, can

the displacement-based method

(on which

attention is primarily focussed here) consists in the obtention of nodal displacements through

the

solution of the

equilibrium

equations

relating nodal forces

and

displacements for the complete assemblage of elements subject, of course, specified

(displacement)

boundary

conditions.

In

formulating

the

to

equilibrium

equations for a given finite element model, the individual element stiffness and load matrices,

which must be assembled into the corresponding structure matrices, are

derived on the basis of aprioristically

postulated deformation patterns, and it is

mainly on these that the accuracy of the predictions of the computational model depend. The difficulties associated with the finite-element modelling of thin shells (which, arguably, finite-element

constitute the

analysis)

have

led

most to

difficult structural form the

development ofa

encountered

in

wide variety

of

approaches to the problem, and to a large number of element types.

There are

three broad avenues to the finite-element representation of thin shells [65, 74], and these are outlined in what follows. One approach, commonly referred to as the facet-shell formulation, consists in the idealization of the curved middle surface of a shell as a multi-faced polyhedral

28

surface formed by the assembly of flat elements (or facets), the vertices of which are situated on the middle surface of the shell.

The stiffness matrix of a

facet-shell element is obtained by superimposing the separate stiffness matrices of a flat plate subject in turn to bending and stretching modes.

The coupling between

bending and stretching actions which exists throughout a curved shell, but is absent within a facet-shell element, only occurs in the facet formulation when adjacent elements lying in different planes are joined procedure.

in

the

usual element-assembly

In general, it is prudent to use fairly low-order facet-shell elements,

as this is consistent with the need, in numerical computations based on this approach, to use rather large numbers of elements in order to minimize the physical idealization errors inherent in the facet-formulation.

Moreover, the use of

a large number of elements enhances the simulation of the continuous coupling between bending and stretching effects in shells.

A range of facet elements have

been developed by combining the properties of various available plate-bending and plate-stretching elements.

Unlike rectangular facet-shell elements, which are only

suited to the modelling of some specialized geometries [269], elements of triangular shape are very popular [14, 22, 128, 178] by reason of their amenability to the treatment of arbitrary contours.

Despite the non-conforming nature of facet-shell

elements and the availability of more sophisticated elements based on

other

formulations (which are discussed below), the facet-shell approach is still used in practice; this is explained, firstly, by its simplicity in regard to both formulation and usage and, secondly, by its ability to yield accurate results for engineering purposes at low cost [14, 22, 128, 178].

(Notwithstanding the scope of the present

work, it is well to note that the latter point is of crucial importance in non-linear analyses.) In a second approach use is made of curved elements formulated on the basis of thin-shell theory.

Such an approach is, in principle, superior to the facet-shell

formulation since, by avoiding the physical idealization errors inherent in the latter, it enables the proper simulation of the continuous coupling between bending and stretching effects that is characteristic of shells.

Nevertheless, the better physical

idealization of shells obtained by the use of shell-theory formulations will not necessarily result in superior computational models since, as noted earlier, the predictive capability of a (displacement-based) finite-element model is intimately tied to the nature of the assumed displacement field.

Indeed, despite the

conventional use of simplified shell equations founded on the

Kirchhoff-Love

hypothesis [159, 160], shell-theory finite-element formulations are rather complex, due to the difficulties associated with the fulfilment of compatibility requirements and the representation of rigid-body motion [74, 161].

(A closely analogous state

of affairs exists in finite-element plate formulations

based on the

Kirchhoff

29

method.) some

It may be worth noting that poor performance in certain applications of

shell-theory-based

elements

may

be

attributed

to

the

use,

in

such

formulations, of polynomial surface-displacement fields of low-order [55]; in this connection, Morley [158] has demonstrated that the use of quadratic expressions for tangential-displacement components in conjunction with cubic expressions for the associated normal component would result in accuracy consistent with that of first-approximation shell-theory formulations if the lateral dimensions of a (shell) finite element are of the order of the thickness of the shell.

Despite the

complexity of shell-theory formulations, however, highly sophisticated conforming elements have been developed for arbitrary geometries by the use of high-order polynomial displacement fields [15, 61]; in addition, simpler elements have been developed for specialized geometries, such as shallow [52, 204, 265] and cylindrical [29, 45, 137] surfaces.

Although they are quite capable of yielding accurate

results, shell-theory formulations are not widely employed for the treatment of arbitrary shells, on account of their complexity in use and their high computational cost [121, 178, 183]. In the third,

and probably most popular, approach

to the finite element

modelling of thin shells use is made of surface-type isoparametric elements [4, 21, 107, 165].

The formulation of these elements, which is essentially a generalization

of the very popular Mindlin formulation for flat plates [101], is based on the assumption that particles of a shell originally on a line that is normal to the undeformed middle surface of the shell

remain on a straight line which is not

necessarily normal to the deformed middle surface; thus, elements of this kind admit of transverse shear deformations

andtheir deformation patterns can

be

completely defined in terms of independent translational and rotational displacement components.

In

addition

to

beingnumerically

applications, Mindlin-type shell elements

more

stable

in

thin-shell

are more economical than corresponding

solid (i.e. three-dimensional) elements [165].

The popularity of the method as an

approach to the formulation of shell elements can be explained by its avoidance of both the physical idealization errors associated with the facet-shell approach and the complexities of shell-theory formulations.

In general, accurate results can be

obtained even with coarse meshes; this

outcome is partlyattributable to the ease

with which the geometry of a structure

can be modelled by these elements and

may be contrasted with the usual practice of employing relatively large numbers of facet-shell

elements

in

order

to

represent

the

geometries of

shells

closely.

Furthermore, the incorporation of the effects of transverse-shear deformations not only enhances the

fulfilment of

inter-element

compatibility requirements

shell-theory formulations), but also broadens their field of application.

(c.f.

30

1 .6 .4 .

T h e T w o -S u rfa c e Truss M o d e l

The two-surface truss model is a simple numerical technique developed by Pavlovic [186] specifically for the analysis of thin shells.

The key ideas on which

the model is based, and from which its simplicity stems, are three-fold [189]. The

first fundamental concept is the use of the Gaussian curvature

change

parameter as the prime kinematic variable : this enables the otherwise difficult treatment of shell geometry to be considerably simplified.

The second key notion

is the conceptual splitting of the actual middle surface of a shell into two distinct, but coincident, surfaces, S and B, which carry separately the

'stretching' and

'bending' stresses, respectively : this two-surface idealization of a shell possesses the computational virtue of reducing the highly indeterminate general shell problem to two simpler, and intrinsically determinate, subproblems (see section 1.3.4).

The

third fundamental idea consists in the idealization of the middle surface of a shell by a triangulated network of pin-jointed bars : this enables the response of the actual continuum to be worked out by reference to a comparatively simple, statically-determinate skeletal structure with which engineers are well acquainted. In the two-surface truss model the actual continuous shell surface (and hence, also, both the S - and B - imaginary surfaces) is replaced by a single-layered triangulated mesh of pin-jointed bars.

The response of each of the imaginary

surfaces can, in principle, be reproduced by means of a series of computations performed on the truss model which follow the logic of operations implicit in the differential formulation of these surfaces.

In accordance with the principles of the

two-surface theory, in-plane static and geometric constraints are applied to the S-surface, while flexural boundary conditions are dealt with exclusively by the B-surface; furthermore, the two imaginary surfaces are connected by means of simple interaction conditions of overall equilibrium and compatibility.

The extended

static-geometric analogy is of great value in the numerical scheme, as it enables the systems of equations for any one of the imaginary surfaces to be obtained by inspection from the corresponding equations of the other surface, thereby resulting in considerable savings in computational effort. A most useful feature of the truss-model computational scheme is its ability to clarify the structural response of a shell by reference to the portion of the total applied load sustained in 'stretching' action, on the one hand, and in 'bending' action, on the other.

This is, of course, a consequence of the utilization, in the

numerical scheme, of the two-surface concept, and, in fact, it is the interface pressure parameter, which generally varies throughout a shell and regulates the balance in load-sharing between the imaginary surfaces, which turns out to be the

31

prime (force) variable of the numerical scheme [186-188, 190]. Although the truss-model technique is aimed primarily at the calculation of shells in accordance with the formal bending theory, it may also be used for the analysis

of

hypothesis. in itio ,

shells

subject

to

the

simplifying

assumptions

of

the

membrane

The latter type of computation is achieved by simply transferring, ab

the entire applied load to the S-surface.

Moreover, the numerical method

not only permits the determination of the membrane-stress distribution in a shell, but also incorporates the necessary check on the validity of such a hypothesis. Evidently,

the

truss-model

scheme

is related

to

the

framework analogy

method, in the sense that both techniques are based on the modelling of continua by ’equivalent' space trusses (see section 1.6.2).

Nevertheless, the two numerical

methods differ radically from each other, both in formulation and in scope.

Thus,

for example, the framework analogy method has a variational basis and can be used for the solution of a wide range of two- and three-dimensional elasticity (including thick-shell) problems,

whereas the truss-model scheme is rooted in

strongly intuitive notions and is but a specialist technique for the analysis of thin shells.

Furthermore, in marked contrast to the lattice analogy method, the space

trusses for the truss-model technique are statically determinate, and their bar sizes need not be worked out a p r io r i.

Moreoever, the number of degrees of freedom

of an 'equivalent' truss is, in the truss-model scheme, unaffected by the material properties of the continuum which it idealizes. The truss model had originally been devised for the analysis of shallow, transversely-loaded, properties [186].

convex shells with

rectangular plan and

isotropic material

Within this limited framework, the model has been shown to be

quite capable, even with coarse networks, of yielding results in good agreement with well-known analytical solutions [186-188].

1.7.

SHELL ROOFS The widespread utilization of shells for roofing purposes stems from their

strong aesthetic appeal, and their singularly high strength-to-weight ratios which enable the achievement of considerable economies in the covering of large areas without intermediate supports.

The advantages that shell forms have over other

structural systems in the roofing of great spans has long been known to mankind, as is evidenced, for example, by the celebrated ancient (stone and brick) Roman and Byzantine domes.

As might be expected, these massive projects were executed

32

on the basis of empirical knowledge gained over the years and the intuitive application of the principles of structural mechanics.

With present-day technology

and materials of construction, the early shell roofs may, of course, be built with but a very small fraction of their thicknesses [23, 77, 97]. A wide

variety of shell

forms

have

been

used

for

purposes of

roof

construction [64, 112, 227]; and research into new, and more efficient, forms is continuing [25, 110].

Domical, cylindrical and paraboloidal translational surfaces

constitute the majority of the shapes used thus far in roof construction.

Domical

forms are suited to the covering of circular, or nearly circular, areas and have found considerable application in the roofing of isolated structures of unusual character,

such as,

planeteria.

to

mention

but

a

few,

churches,

theatre

auditoria

and

Unlike the early domes, which were generally (nearly) hemispherical,

most of the modem domes (which are sometimes referred to as 'skull-cap' domes [112]) consist of shallow surfaces, mostly of spherical or parabolic form, resting on vertical supports.

Cylindrical shells have been used to great advantage in the

covering of rectangular areas.

These shells are largely used in industrial-type

structures since, from an aesthetic standpoint, the cylindrical shape is generally considered to be less suitable for structures built for social and cultural purposes. Another shortcoming of cylindrical surfaces, which militates against their use in shell-roof construction, is the necessity, particularly in long-span construction, for expensive special measures, such as an increase in wall thickness, in order to take care of the large flexural effects that generally occur in the edge zones of these shells

[23].

Perhaps

the

elliptic

and

hyperbolic

paraboloids,

which

are

doubly-curved translational surfaces with positive and negative values of Gaussian curvature respectively, constitute the most extensively-used forms in recent years. These shapes lend themselves to the roofing of a wide variety of plan areas, either singly or in combination.

Unlike their elliptic counterparts, hyperbolic paraboloids

are ruled surfaces, so that they can be formed by straight-line generators; and it is partly on account of the constructional (and hence cost) advantages stemming from this property of ruled surfaces that hyperbolic paraboloids have achieved great popularity in shell-roof construction [30]. Of the

many

materials

that

lend

themselves

to

shell-roof construction,

reinforced concrete and timber, taken together, have found the broadest field of application.

Reinforced concrete is doubtless the more widely-used of the two,

largely on account of the ease with which it can be moulded to practically any desired form; and, indeed, it was this material which was responsible for the radical revolution in shell-roof construction at the end of the last century [112]. It is worth noting, also, that the scope of concrete shell construction has been

33

considerably widened by the development of prestressing techniques; the various benefits accruing from the prestressing of concrete discussed in the literature [3, 26, 112, 113, construction,

timber,

in

comparison

with

shells have been extensively

136].For purposes of shell-roof reinforced

concrete,

has

inferior

mouldability and fire-resistance characteristics; on the other hand, the superior strength-to-weight

ratio

of

cost-effective [131, 251].

timber

largely

makes

its

use,

as a

rule,

more

The resistance of timber to fire may be considerably

improved by the use of fire-retardant impregnation and fire-retardant paints, while the difficulties associated with moulding the material to arbitrary form suggest that it is particularly suitedto shallow shell roofs

and to non-shallow shell roofs with

developable and

Timber

ruled surfaces [131, 251].

shells are often constructed

from several layers of boards placed in different directions and nailed together; the boards need not be made of high-grade material since the stresses induced in them are usually of low-order [227].

It is useful to note, also, that laminated timber

shells are usually constructed with symmetry about their middle surfaces [31, 82]; as one would expect, this mid-plane symmetry results in considerable simplification in their structural analyses.

1.8.

STATEMENT OF RESEARCH AIMS The key to all engineering design is an understanding of the relevant physical

phenomena.

Such an understanding should be a prerequisite for any engineering

design work, for it not only enhances innovation

development and progress in

design, but also helps in the intelligent application of engineering-analysis computer packages and in the interpretation of the ensuing results.

Clearly, this point is of

paramount importance in the context of shell structures,

by reason of their

complexity and special nature. It is true that the finite element method enables the solution of a range of problems for shells having arbitrary geometrical configuration. that the

However, it is felt

'mechanical' use of available finite element computer packages is a

somewhat 'brutal' way of dealing with shell structures; thus, the usually voluminous information

gained

by the

application of these

constitute 'knowledge' ratherthan 'understanding'. desirability of any numerical

computer

packages tends to

Evidently, this indicates the

scheme of computation which permits not only the

solution of shell problems that do not yield to analytical methods, but also the clarification of the structural response of shells in a manner that engenders in engineers a level of understanding that is adequate for purposes of design.

34

Loads applied to shell structures are generally sustained by a combination of bending and stretching actions, and a grasp of the complex interaction between the two effects is crucial for the understanding of the structural behaviour of shells. The ingenious two-surface model for a shell provides a direct and effective way of studying this interactive behaviour at both the conceptual (analytical) and the computational levels. At the computational level, the application of the two-surface idealization in the recently-devised truss-model scheme enables the clarification of the structural response of a shell by reference to the portion of the total applied load sustained in stretching action, on the one hand, and in bending action, on the other.

This

numerical scheme, which had originally been devised for the analysis of shallow, transversely-loaded, convex shells with isotropic material properties and rectangular plan, embodies the essence of the equations of thin-shell theory as well as aspects which make shells different from other structural forms; among the latter are the 'membrane' philosophy, the 'bending' solution and the static-geometric analogy.

In

addition to its close adherence to the spirit of shell theory, the truss model is remarkably simple, for it enables the response of a shell structure to be worked out by reference to a comparatively simple, statically-determinate skeletal structure with which engineers are well acquainted. At the conceptual level, on the other hand, the two-surface idealization permits an elegant derivation of the equations of shell theory.

Such an approach,

which had hitherto been used within the limited framework of the cartesian co-ordinate

system

and in

the

context of

isotropic

material

properties

and

transversely-applied loads, enhances the qualitative study of the behaviour of shells: apart from its directness, it elucidates the meanings of the various quantities in the field equations in simple, physical terms. The primary object of the research described in this thesis is the extension of the truss-model scheme so as to encompass general surface tractions, arbitrary geometric shapes and homogeneous as well as heterogeneous anisotropic material properties.

Additionally, it is. the aim of the present work to develop a general

computer program for the analysis of shell structures by the truss model : this would greatly facilitate the application of the model to a wide range of practical problems, while, at the same time, enabling the validation of the afore-mentioned proposed extensions to be carried out.

Furthermore, it is the object of the

present work to generalize the elegant and instructive two-surface approach to the formulation of the shell equations so as to cater for anisotropic material properties, tangential loadings and curvilinear co-ordinate systems.

35

MEMBRANE HYPOTHESIS Assume flexural actions to be zero or negligible

Equilibrium

__________ l ________ i)

In-plane stresses

Hooke's law

V/___________ ii) In-plane strains

Geometry

iii) Displacements

Geometry

J. iv) Curvature changes

Hooke’s law

____________ 1 _________ v)

Fig. 1.1:

Flexural actions

Formal procedure for checking the validity of the membrane 'theory'

36

CHAPTER 2 THE TRUSS MODEL FOR CONVEX SHELLS WITH RECTANGULAR PLAN

2.1.

INTRODUCTION This chapter describes in detail the two-surface truss-model scheme for the

analysis of shallow elastic thin shells.

The exposition relates to transversely-loaded

convex shells with rectangular plan and isotropic material properties, for which the model had originally been devised.

Certain extensions of, and modifications to, the

original truss-model scheme are presented; these include a simple technique for the calculation

of the

transverse

displacements

of

non-membrane

shells

and

the

introduction of a simpler algorithm for the assignment of nodal areas of influence. In addition, the two-surface approach to the formulation of the equations of shell theory is described and generalized so as to encompass general curvilinear co-ordinates corresponding to the lines of curvature of the middle surface of a shell.

Irrespective of the co-ordinate system used, this analytical approach to the

derivation of the shell equations is helpful in understanding the computational logic inherent in the truss-model scheme.

2.2.

THE TWO-SURFACE IDEALIZATION OF A SHELL

2.2.1. The Formulation of the Shell Equations In what follows, the equations describing the behaviour of a shallow shell within the framework of plane cartesian co-ordinates (x, y) are formulated on the basis of the two-surface theory of Calladine [40].

This serves as a prelude to the

proposed extensions of the method; moreover, a knowledge of this formulation is helpful in understanding the numerical treatment of shell problems in accordance with the two-surface idealization.

A generalization of the method to encompass

curvilinear co-ordinate systems is presented in Appendix A. Fig. 2.1 shows the various stress resultants and couples acting on a differential shell element, as well as the applied normal surface traction (or pressure) p.

The

diagram defines the positive senses of the in-plane direct and shearing stress

37

resultants Nx, Ny,

(= N ^), out-of-plane (or transverse) shear stress resultants

Qx, Qy and bending and twisting moments (or couples) Mx, My,

(= M ^).

The shell is assumed to be shallow, so that the x-y plane may be taken to be tangential to the element; this, of course, implies that the x and y directions may be regarded as the principal directions along the middle surface of the shell. Furthermore, the material of which the shell is made is assumed to be isotropic and to have uniform thickness. It will be recalled that

in

the

two-surface

model the shell surface

conceptually split into two distinct, but coincident, surfaces (see section 1.3.4). shown in Figs. 2.2a and 2.2b,

is As

the S-surface carries only the membrane (or

in-plane) stress resultants (Nx, Ny, N ^) while the B-surface sustains the bending actions (Mx, My, M ^, Qx, Qy) exclusively.

Thus, instead of formulating the

equilibrium, compatibility and constitutive relations for the actual shell e le m e n t , as in conventional treatments, the various equations describing the behaviour of each of the imaginary surfaces are derived separately in the present scheme [40].

The

equations may be collected together in two separate columns for the respective surfaces as follows:

B-surface

S-surface

N%

Ky

Ny

------ + --------Ps Ri R2

3N x — 0x

0Ny

(2-1S)

R,

B kv

3N x v +

—H -



-

0

( 2 .2 S )

3y

B kx

3N

--- + ------0 3y Bx

— 9x

(2.3S)

kx + ------------g B

(2.3B )

Bx

3 2w

(2.4B )

(2.4S ) 3y2

3 2$ N

y

3 2w

( 2 . 5B)

(2.5S )

3x 2

3x2

38

02$

3 2w

(2.6S)

NX y



Kx y

(2.6B)

-

3x3y

ex

-

i r

3x3y

(2.7S)

My - D(Ky +

(2.7B)



‘'Ny>

PKX )

(N *

(N y y

» ,N X )

(2.8S)

Mx = D ( k x +

vKy

)

(2.8B)

(2.9S)

2Mxy - 2D(1 -

1 £y “ — Et

2(1 + Y x y

v) N xy

Et

4-

3x2

3x3y

a 2 **

v

)

kx

(2.9B)

y

3 2Mx

3 2Mxy

3 2My

3x2

3x3y

3y2

3y2

The set of equations

(2.1 S ) -

under the normal loading ps.

(2 .1 0 B )

-P B

-gS (2.10S)

(2 .1 OS)

describes the behaviour of the

Expressions

(2 .1 S )

-

S -surface

(2.3S) represent the three

equilibrium equations of the S-surface; as is well-known, it is often convenient to deal with equations of this type by introducing the Airy stress function $ 2 6 0 ],

and this is defined by relations

(2 .4 S ) -

Equations

(2 .6 S ).

(2 .7 S ) -

[24 7 , (2 .9 S )

are the constitutive relations for the S-surface; the quantities ex, ey and

are

the direct and shearing strains corresponding to the in-plane stresses (N* Ny, N ^) while the parameters E,

and t denote the values of Young's modulus, Poisson's

v

ratio and the (uniform) thickness of the shell, respectively.

Expression

(2 .1 OS)

corresponds to the compatibility equation relating the S-surface Gaussian curvature change gs to the in-plane strains [39, 43]. to

zero

in the

above expression,

two-dimensional elasticity problems

the

It is worth noting that, on setting gs well-known

[24 7 , 2 6 0 ]

compatibility

equation

for

is obtained.

The behaviour of the B-surface is described by the set of equations (2.1 B) (2.1 OB).

Expressions (2.1 B) - (2.3B) are the compatibility equations relating the

B-surface components

Gaussian k x,

Ky

curvature

and

k^

.

A

change s

gB to

the

ordinary

curvature-change

in plate theory [248], the latter may be expressed

in terms of the normal (or transverse) displacement component w; the relations are listed as expressions (2.4B) - (2.6B).

The constitutive relations for the B-surface

are given by equations (2.7B) - (2.9B), in which the parameter D denotes the

39

flex u ra l (o r b ending) rig id ity , d e fin e d by:

E t3 D _ -----------------12(1 - v 2 )

(2.11)

Finally, equation (2.1 OB), relating the applied B-surface pressure pB to the bending and twisting moments acting on equation for the B-surface.

the shell element, represents the equilibrium

The expression (2.1 OB) is readily recognizable as the

equilibrium equation for an element of a flat plate which is locally tangential to the curved middle surface of the shell and is subjected to the (normal) loading pB [248].

This can easily be understood since, by definition, the B-surface cannot

sustain loads by in-plane action, and hence it is obliged to act in a plate-like manner. The formal similarity between the

corresponding expressions for the two

surfaces - as indicated by the common numbering - is evident, and is, in fact, a consequence of the so-called static-geometric analogy for thin shells (see section 1.4).

Thus, the set of equations describing the behaviour of the S-surface can be

transformed into the corresponding set of equations for the B-surface, and v ice versa.

Specifically, the equilibrium equations for one surface can be transformed

into the compatibility equations for the other, and vice versa , while the constitutive relations for the two surfaces correspond to each other.

It is easily seen that the

interchanges of analogous variables and material constants required for the complete transformation of one set of equations to the other are:

Ky

(2 .1 2 )

Ny

(2 .1 3 )

NXy

i -------

XX 1

(2 .1 4 )

ex

My

(2 .1 5 )

«-------

Mx

(2 .1 6 )

T xy

-2M*y

(2 .1 7 )

$

-w

Kx

>>

6y

T

Nx

(2 .1 8 )

40

Clearly,

the

^(S) -••(B)

(2 .1 9 )

1 Et

(2 .2 0 )

-----> D

Ps

sin2A) 1

. Ne + 2cosA . Ne 2

12

- 0

(3.94)

141

(c o s 2A

The

- p s in 2A) . Ne + Ne + 2cosA . Ne 1

solution of these

2

12

- 0

(3.95)

two equations can most conveniently be achieved

by

expressing the direct-stress components (N®, N®) in terms of their shearing-stress counterpart (N f2).

In this way one obtains:

Ne - Ne - A 1

2

S

. Ne

(3 .9 6 )

12

where

A

2cosA sin2A (1 + v) « ----------------------------------s (cos2A - i*sin2A ) 2 - 1

(3 .9 7 )

By modifying equation (3.70) in the light of relation (3.96), one obtains:

Ne

-c s . A

1

1 sinA

Ne

-c s . A 2

1

-C S

1 2

2

c s .A

s

1

2

c s .A

S

1

2

C S

1

1

c .A s

S

s

S

2

NC

s

X

A jj

NX + Ny +

k2 2

=

Njjy

(5.1 79 )

^22

A, “Y x y

n 2 6

A2 6 A6 6 — Ny + ----- N.xy A2 2 A2 2

^ 2 2

22

(5 .1 8 0 )

and

Mx

- D11

My

= D„

®12 ®18 Kx + ----- Ky + 2 ----- KXy Di i '1i

D, D ,,

D1 6 ^xy *“ ^1 i

D2 2

'2 6

D1 1

D1 1

(5 .1 8 1 )

kxy

D.2 6 D,6 6 K„ + ----K .. + 2 ----Kxy x y D 11 ^ ii

Evidently, the following interchanges, together with the correspondences

(5 .1 8 2 )

(5 .1 8 3 )

(2.12) -

(2.17) and (5.162), enable the transformation of any one of the two sets of constitutive relations (5.178) - (5.180) and (5.181) - (5.183) into the other:

Am

D2 2

A2 2

Dn

Al 2

»12

A2 2

Du

Ag 6 A2 2

A

(5 .1 8 4 )

(5 .1 8 5 )

Dg 6

(5 .1 8 6 )

Dm

Al 6

D26

A2 2

Dm

(5 .1 8 7 )

323

^2 6 Die ----- 4---- ----> -2 ----A2 2 D„

(5.1 88 )

In our final example, the two sets of constitutive relations (5.70) - (5.72) and (5.67 - (5.69) are recast, respectively, as

A 12

Am ex

» A n e6

Nx As 6

A 22

A se

(5 .1 8 9 )

Nxy As 6

-

A 26 Ny +

Nx +

“ ^66

T x y “ A 66

Ny 4A G6

A 12 cy

Ai G

4-

(5 .1 9 0 )

Nxy Ag 6

Ag 6

A1s A2 s ----- Nx + ----- y Ag g Ag 6

(5 .1 9 1 )

xy

and

M*

= 4D 6 6

j D,2 1 Die 1 D ,, /( + — ----- Kv + — y 2 4 D 4 Dg 6 ^6 6 6

My

- 4D 6 6

T

- 4D6 6

■■

1

1

2 D6 6

1 ®22 *x * T

°16

Kx +

D6 6

■■ ■■■

D6 6

1 D26 ^

l Ky

+ T

Ky

+

xy

^26 D66

Kxy

1

.

Kx y

(5 .1 9 2 )

(5 .1 9 3 )

(5 .1 9 4 )

D6 6

It can readily be seen that any one of the above two sets of constitutive relations (5.189) - (5.191) and (5.192) - (5.194) can be transformed into the other by invoking the interchanges (2.12) - (2.17), (5.164) and the following:

Am As 6

1 D22 4 D6 6

(5.195)

324

A2 2 ^66

Ai 2 l CO CO <

1 D ,, 4 6

(5 .1 96 )

1 »12 4 6

(5 .1 9 7 )

^1 6

1 2

^6 8 A2 6

6

(5 .1 9 8 )

^6 G

1 D,6 2 ^6 8

Ag 6

(5 .1 9 9 )

In each of the above three examples, a total of six interchanges of elastic constants, and their ratios, is required (in addition to the correspondences (2.12) (2.17)) for the complete transformation of one set of constitutive relations into the other, as expected. medium,

the

total

Evidently, in the particular case of a specially orthotropic number of

(non-trivial)

interchanges

required in each of the examples is reduced to four.

(of elastic

constants)

Furthermore, it is a

straightforward matter to verify that the correspondences (5.173) - (5.177), (5.184) - (5.188) and (5.195) - (5.199), between ratios of the material stiffnesses and compliances, are consistent with the interchanges (5.161) - (5.166), in the light of which they can, indeed, be directly established.

(Note that other interchanges

between ratios of the material compliances and stiffnesses may also be established.)

5.3.1

Single-layered Shells Let us now consider the special case when the shell is single-layered, and has

a uniform thickness t.

For such a body, the interchanges (5.161) -

(5.166),

corresponding to the case of planar-anisotropy, can be recast as follows: t3 B22 —

(5 .2 0 0 )

12

3 22 -----

(5.2 02 )

B, 12

t3

'ss

1G

-> B 6 6

(5 .2 0 3 )

-> -B 2 6

(5 .2 0 4 )

t3

•2 S

(5 .2 0 5 )

**B, 6 6

As one would expect, the above interchanges (5.200) - (5.205), together with the correspondences (2.12) - (2.17), are sufficient for the complete transformation of the (S-surface) constitutive relations (5.118) - (5.120) to the associated (B-surface) relations (5.114) - (5.116), and v ic e ve rsa.

Furthermore, the ;

(5.200) -■ (5.205) may be expressed in terms of the engineering way, the correspondences (5.200) - (5.205) can be written as:

1

t3

h i 1 - 7?xMx]

(5 .2 0 6 ) 12

Qf

1

t3

E x[l - ’lyj'y]

(5 .2 0 7 ) 12

Qf

V

vx

vy

Ext

v

>

+ Vx Vy }

t3

Qf

12

E*[ vy + TlyVx} Qf

1

[J

' ’’x-y]

rt

Qf

(5.208) 12

t3 (5.2 09 )

1

^xy

t3

3

326

Vx

Ey[^y

Px

Ex t

+

Gxy t

t3

■ •xftc]

6

Qf

Gx y [ ^ y

+

^ y ’l x ]

Ex [Mx

hr

Eyt

Gxy t

v y t *y

J

t3 6

Qf

Gx y [^ix

(5 .2 1 0 ) 6

Qf

Vy

t3

^x^l yj

t3 (5 .2 1 1 ) 6

Qf

where the quantity Qf is defined by expression (5.110). It can readily be verified that, when the shell is composed of a specially orthotropic material, the interchanges (5.210) and (5.211) become trivial, while the interchanges (5.206) - (5.209) reduce to the following: 1 (5 .2 1 2 ) Ext

12[l

1

-

rxV y\

Ex»2 12 [ l

V

-

(5 .2 1 3 )

r x vy ]

'x E y t3 Ex t

Ey t

12 [ l

-

vx vy

)

Ex t 3

(5 .2 1 4 )

1 2 [l - vx vy ]

1

t3 (5.2 15 )

is the ratio of Poisson characterizing the transverse compression in the y'

direction due to tension in the x' direction.

Each layer of the shell has a

thickness of 20mm, and the laminate is arranged such that the direction of the grain of the outer layers coincides with the co-ordinate direction y = constant, while the direction of the grain of the inner layer coincides with the co-ordinate direction x = constant (clearly, the material of the shell is cross-laminated, in the sense that the axes of orthotropy of contiguous laminae are rotated through 90*). Thus, the elastic constants of the shell layers can be defined in terms of the global (x,y) co-ordinate system as follows

380

e' X

-

e'" X

- E11 y

E1 y

pill ■ E

G1

■* rG11

x y

V

I X

V

I y

y

x y

V

111 y

X

-c"' xy

■= V I I I = X

-=

e"



V

V

II y

11

X

- 14.1285 x 106 KN/m2

(5 .4 8 7 )

- 1.0474 x 106 KN/m2

(5 .4 8 8 )

- 0.8419 x 10s KN/m2

(5 .4 8 9 )

= 0.37

(5 .4 9 0 )

* 0.02743

(5 .4 9 1 )

where the superscripts I, il and ill denote the layer numbers starting with I as the bottom layer (i.e. concave side). The structure is subjected to the following two loading conditions: Loading condition 1: Downward vertical uniformly distributed pressure of magnitude 0.3KN/m2, corresponding, roughly, to the self-weight of the shell (the density of the material of the shell is 480kg/m3 (0.48g/cm3) [27, 28]). Loading condition 2 : Downward concentrated load of magnitude 30KN applied at the apex, or crown, of the shell. One quadrant of the structure is analysed, on account of symmetry (c.f. case study 1, where only one-eighth of the shell could be analysed), by means of the uniform (6 x 6) truss-model idealization shown in Fig. 3.16.

From the diagram, it

can be seen that the point of application of the concentrated load (for loading condition 2) corresponds to node 49, the x,y and z co-ordinates of which are 0, 0 and 0,

respectively.

The theoretical bending solution to the present

problem,

which enables the verification of the truss-model scheme in the context of material anisotropy, is given in Appendix F.

It should be noted that only representative

results relating to some of the nodal points which lie within, and on the

boundary

of, the computational zone under consideration (i.e. the quadrant of

the shell

shown in Fig. 3.16) are presented below.

3 8 1

(a) Loading condition 1 - membrane solution, (in-plane'l stresses: the in-plane stresses corresponding to the membrane

solution

to

the

present

counterparts, appear in Table 5.1.

problem,

together

with

their

theoretical

It can be seen that the correlation between the

truss-model values and their theoretical counterparts is very good.

Also, the above

two sets of stress distributions for the membrane shell, being unaffected by the material properties of the latter (see Appendix F), exhibit symmetry with respect to the shell diagonal x = y; this, of course, implies that the membrane stress distribution for the (entire) structure admits four planes of symmetry, namely x = 0, y = 0, x = y and x = -y, as in the corresponding isotropic-shell problem (see sections 2.3.7 and 2.4).

In this connection, it should be noted that the present

sets of membrane stresses can be obtained directly from their counterparts, listed in Table 2.7, for case study 1 by multiplying the latter (sets of in-plane stresses) by a factor of 27/200 (this factor is

obtained by simply dividing the magnitude (i.e.

3/10 KN/m2) of the uniformly distributed load acting on the present structure by the magnitude (i.e. 20/9 KN/m2) of the corresponding load applied to the structure considered in case study 1). -

membrane solution, transverse displacements: the transverse displacements of

the membrane shell are listed in Table 5.2. numerical and theoretical values is very good.

Here also the correlation between the Unlike the corresponding membrane

stress distributions, the deformation fields (numerical and theoretical), being affected by the material properties of the (membrane) shell, do not exhibit symmetry with respect to the diagonal x = y and bear no particular relationship with their counterparts for case study 1. - S - and B - surface load distribution: the load distributions are given in Table 5.3.

The results show that a major portion of the shell (particularly the central

region) responds primarily by in-plane action, while the converse is true in the comer regions. - bending solution, in-plane stresses: the in-plane stresses are listed in Table 5.4. The correlation between the truss-model values and their theoretical counterparts is very good.

Unlike their counterparts for the membrane solution, the stress fields

(numerical and theoretical) neither exhibit symmetry with respect to the diagonal nor bear any special relationship with their counterparts for case study 1, as might be expected. the

Further, by comparison of Tables 5.1 and 5.4, it can be seen that

in-plane stress distributions corresponding to

the

membrane and

bending

solutions agree rather closely with each other, particularly in the interior zone.

382

This is not surprising, on account of the greater role generally played by stretching action in the shell. -

bending solution,

bending and twisting moments: the

moments are listed in Table 5.5.

bending and twisting

The agreement between the truss-model results

and their theoretical counterparts is very good.

It can be seen that the moments

in the non-membrane shell are more or less evenly distributed throughout the shell; the penetration of the bending disturbance into the interior of the shell may be attributed to the very small curvature of the shell (the rise-to-span ratio of the shell is only 3/100). -

bending solution, transverse displacements: the transverse displacements of the

structure are displayed in Table 5.6.

The correlation between the truss-model

results and their corresponding theoretical values is very good.

It can be seen that

the displacement values increase towards the interior of the shell, maximum deflection occurring at the crown.

with the

Furthermore, it should be noted that

the displacement values approach their membrane-analysis counterparts as one moves towards the interior zone of predominant stretching action. (b)

Loading condition 2

- membrane solution, fin-plane') stresses: the in-plane stresses appear in Table 5.7. The tabulated results show that the stresses increase substantially in magnitude as one moves towards the loaded point.

Further, the maximum direct stresses occur

at the (singular) point of application of the concentrated load; and these stresses can be expected to increase with mesh refinement (see section 3.5).

It is worth

noting that, like its counterpart for loading condition 1, the in-plane stress field exhibits symmetry with respect to the diagonal, as one would expect. - membrane solution, transverse displacements: the results are listed in Table 5.8. As expected, the deformation field does not exhibit symmetry with respect to the diagonal.

In marked contrast with the corresponding results for loading condition

1, wide variations occur in the computed displacements of the interior nodes of the shell; in addition, the displacements fluctuate between positive and negative values, indicating a somewhat undulating deformation pattern.

It will be recalled that a

similar pattern of behaviour was observed in the corresponding (isotropic) problem of case study 3.

As in the latter case, the displacements of the present structure

increase in magnitude towards the loaded point, at which the maximum deflection occurs. region

Clearly, the relatively very large displacements which occur in the loaded suggest

that

the

dominant

part

of

the

stress

distribution

in

the

383

neighbourhood of the loaded point is of a flexural kind, as expected. - S - and B-surface load distribution: from the results displayed in Table 5.9 it can be seen that, at the loaded point (i.e. at node 49), roughly 81% of the applied

load

concentrated

is load

taken

by

causes

the a

B-surface,

predominantly

indicating, flexural

as

expected,

response

in

its

that

the

vicinity.

Evidently, the ratio Pg/P at the apex can be expected to approach unity as the mesh of the truss model is refined. - bending solution, in-plane stresses: the in-plane stresses are displayed in Table 5.10.

The correlation between

the

truss-model

values and

their theoretical

counterparts is good, even in the locality of the loaded point.

As one would

expect, the stress fields predicted by both theory and model do not exhibit symmetry with respect to the diagonal x = y. - bending solution, bending and twisting moments: the results appear in Table 5.11.

The good agreement between the truss-model results and their theoretical

counterparts is evident.

The only significant exception occurs at the loaded point,

which is, as noted earlier (refer to case studies 1, 3 and 5), a point of bending-moment singularity. -

bending solution, transverse displacements: the displacements appear in Table

5.12.

It can be seen that, in marked contrast with the results obtained from the

membrane solution, all the internal points of the shell deflect downwards, and the displacement values are much more uniformly distributed throughout the interior of the shell.

The good correlation

between the truss-model results and

their

theoretical counterparts, particularly at nodal points which are far removed from the loaded point, is evident.

The accuracy of the numerical results may, of

course, be improved with mesh refinement, especially in the loaded region, where the state of stress exhibits very rapid variation.

5.7 CASE STUDY 11 : LAYERED PARABOLOID OF REVOLUTION WITH EQUILATERAL TRIANGULAR PLAN In this section the

truss model

is applied to

a mid-surface

symmetric

three-layered orthotropic shell in the form of a paraboloid of revolution with equilateral triangular plan.

The middle surface of the shell is described by

equation (2.115), and its edges are assumed to be hinged.

The dimension a of

each side of the base (equilateral) triangle, the radius of curvature R of the middle

384

surface, and the material thickness t of the shell are given by equation (3.165). The shell is built up of three layers of Larch Western timber the elastic constants of which are defined, with reference to the principal axes of orthotropy x' and y', by equations (5.483)-(5.486) (refer to section 5.6).

Each layer of the shell has a

thickness of 20 mm, and the laminate is arranged such that the direction of the grain of the outer layers coincides with the co-ordinate direction y = constant, while the direction of the grain of the inner layer coincides with the co-ordinate direction

x

= constant;

thus,

in

analogy

with

the

paraboloid

of revolution

considered in the preceding case study, the elastic constants of the shell layers can be defined in terms of the global (x,y) co-ordinate system through equations (5.487)-(5.491). The structure is subjected to the following two loading conditions: Loading condition 1: Downward vertical uniformly distributed pressure of magnitude 0.3KN/m2, corresponding, roughly, to the self-weight of the shell (refer to the preceding section). Loading condition 2 : Downward concentrated load of magnitude 20KN applied at the apex, or crown, of the shell. The entire structure is analysed by means of the uniform 9 x 9 x 9 truss-model idealization shown in Fig. 3.7.

As in case study 2, we chose to

analyse the entire structure, rather than - by exploiting symmetry - a portion thereof, so as to be able to construct such a remarkably

'balanced' mesh.

However, the results to be presented below relate to one-half of the structure.

It

should be noted that the point of application of the concentrated load (for loading condition 2) corresponds to node 31, the x,y and z co-ordinates of which are 0, 0 and 0, respectively. (a) Loading condition 1 - membrane solution, (in-planel stresses: the in-plane stresses corresponding to the membrane solution to the present problem appear in Table 5.13. verified that the above stresses can be obtained directly

It can easily be

(disregarding minor

rounding errors) from their counterparts, listed in Table 3.1, for case study 2 by multiplying the latter (stresses) by a factor of 1/5 (this factor is obtained by simply dividing the magnitude (i.e. 3/10KN/m2) of the uniformly distributed load acting on the present structure by the magnitude (i.e. 3/2KN/m2) of the corresponding load applied to the structure considered in case study 2); obviously, this outcome stems

385

from the fact that the membrane stress fields corresponding to both case studies are unaffected by the material properties of the associated shells. - membrane solution, transverse displacements: the transverse displacements of the membrane shell are listed in Table 5.14.

It can be seen that the deformation

field, being affected by the material properties of the (membrane) shell, bears no particular relationship with its counterpart for case study 2. - S - and B-surface load distribution: the load distributions are given in Table 5.15.

The results show that the portion of the total load which is carried by the

B-surface increases towards the edge and corner region, and, correspondingly, that the percentage of the total load which is sustained by the S-surface increases as one moves towards the interior zone. - bending solution, in-olane stresses: the in-plane stresses are listed in Table 5.16. Unlike its counterpart for the membrane solution, the above stress field bears no special relationship with the corresponding stress field, listed in Table 3.4, for case study 2, as might be expected. -

bending solution,

bending and

twisting moments: the bending and twisting

moments are listed in Table 5.17.

It can be seen that the moments in the

non-membrane shell are more or less evenly distributed throughout the shell; the penetration of the bending disturbance into the interior of the shell may be attributed to its very small curvature (c.f. case study 2). - bending solution, transverse displacements: the transverse displacements of the structure are displayed in Table 5.18.

It can be seen that the displacement values

increase towards the interior of the shell, but that the displacement values for the interior nodes differ markedly from their counterparts, shown in Table 5.14, for the membrane solution. (b) Loading condition 2 - membrane solution, (in-olanel stresses: the in-plane stresses appear in Table 5.19.

The results show that the stresses increase substantially in magnitude as one

moves towards the loaded point.

Further, the maximum direct stresses occur at

the (singular) point of application of the concentrated load; and these stresses can be expected to increase with mesh refinement.

386

- membrane solution, transverse displacements: the results are listed in Table 5.20. In marked contrast with the results obtained for loading condition 1, wide variations occur in the computed displacements of the interior nodes of the shell; in addition, the displacements fluctuate between positive and negative values, somewhat undulating deformation pattern.

indicating a

It can be seen that the displacements

increase (in magnitude) towards the loaded point, at which the maximum deflection occurs.

Clearly, the very large displacements which occur in the loaded region

suggest that the dominant part of the stress distribution in the neighbourhood of the loaded point is of a flexural kind, as expected. - S - and B-surface load distribution: from the results displayed in Table 5.21, it can be seen that, at the applied

load

concentrated

is

taken

load

by

causes

loaded point (i.e. at node 31), roughly 90% the

B-surface,

a predominantly

indicating, flexural

as

expected,

response

in

its

of the

that

the

vicinity.

Evidently, the ratio Pg/P at the apex can be expected to approach unity as the structural idealization is refined. - bending solution, in-olane stresses: the in-plane stresses appear in Table 5.22. One can easily see that the in-plane stress field corresponding to the bending solution differs markedly from its counterpart for the membrane solution, an outcome which highlights the fact that the membrane solution cannot provide a realistic estimate of the actual response of the shell. - bending solution, bending and twisting moments: Table 5.23 shows the bending and twisting moments of the non-membrane shell.

It should be noted that the

truss model predicts finite values for the bending moments at the loaded point, which is, in theory, a point of bending-moment singularity; clearly, the truss-model results are meaningful only in so far as they provide an estimate of the average bending moments which occur in the 'influence' zone associated with the loaded point. - bending solution, transverse displacements: the transverse displacements appear in Table 5.24.

It can be seen that, in marked contrast with the results obtained

from the membrane solution, all the interior points of the shell deflect downwards, and the displacement values are much more uniformly distributed throughout the interior of the shell.

387

5.8

DISCUSSION AND CONCLUSIONS This chapter was begun

with

a

review of the

constitutive

relations of

anisotropic shells of layered construction; these relations are similar in form to their counterparts for non-layered anisotropic shells, so that, from the standpoint of their form, they may be taken to represent the constitutive relations of general anisotropic

shells.

It

arbitrarily-constructed

was

(layered)

shown

that

anisotropic

the

shells,

in

constitutive marked

relations

contrast

to

of the

corresponding relations of classical isotropic shell theory, generally exhibit coupling between bending and stretching actions, in the sense that these relations contain (additional) terms characterizing the reciprocal influence of bending and stretching effects. there

Thus, for example, in the expressions for in-plane stress components appear

(additional)

terms

containing curvature-change

components,

and,

reciprocally, in the expressions for bending moments there appear (additional) terms containing in-plane strain components.

This bending-stretching coupling not only

leads to a breakdown of the membrane hypothesis, but is also out of keeping with the very concept of separation (of stretching and bending effects) inherent in the two-surface idealization of a shell; in other words, for shells which exhibit the bending-stretching coupling phenomenon, both the membrane hypothesis and the two-surface theory are untenable.

We have also seen that the constitutive relations

of certain anisotropic shells (notably the mid-surface symmetric layered shells) do not exhibit the

bending-stretching coupling phenomenon; for such shells,

the

constitutive relations reduce to two separate groups (one linking stretching quantities exclusively, the other

connecting flexural

actions only),

corresponding relations of classical isotropic shell theory.

in analogy with

the

As one would expect,

both the membrane hypothesis and the two-surface theory may be applied to anisotropic shells which do not exhibit the coupling phenomenon. The static-geometric analogy has been extended, by reference to (mid-surface symmetric) layered anisotropic shells, so as to encompass the constitutive relations of general anisotropic shells which do not exhibit the bending-stretching coupling phenomenon.

This essentially constitutes a generalization of the earlier works of

Visarion and Stanescu

[242, 243, 254, 255], for single-layered (orthotropic and

anisotropic) shells, and Librescu

[132-134], for mid-surface symmetric layered

isotropic and specially orthotropic shells. The

governing

arbitrarily-loaded,

differential

shallow

equations

anisotropic

shells

describing which

do

the not

behaviour exhibit

of the

bending-stretching coupling phenomenon have been derived on the basis of the two-surface theory and with reference to a mid-surface symmetric layered shell.

388

The derivation is essentially a generalization of the earlier formulation, described in section 4.2, tractions.

for a single-layered

isotropic

shell subjected to

general

surface

In analogy with the simpler case of the single-layered isotropic shell,

the governing equations for the anisotropic shell can be obtained by substituting the appropriate expressions (in terms of the stress and deflection functions) for the four variables defining the S - and B-surface Gaussian curvature changes and normal loadings into the geometric

two interaction

compatibility.

conditions

Furthermore,

the

of normal-force

equilibrium

expressions defining

the

and

S-surface

Gaussian curvature change and the B-surface normal loading differ from their counterparts

for

the

case

of

the

single-layered

isotropic

shell,

while

the

corresponding expressions for the S-surface normal loading and the B-surface Gaussian curvature change are identical in both cases.

Thus, the present approach

explicitly reveals the invariant terms (associated with the S - and B-surfaces) in the field equations, an outcome which clearly stems from the concept of separation inherent in the method (c.f. section 4.7).

It is worth noting that, as in the case

of the single-layered shell, there is no symmetry between the expressions for the S-surface Gaussian curvature change and normal loading and their counterparts for the B-surface, by reason of the breakdown of the static-geometric analogy in the presence of tangential surface tractions.

On the other hand, when the applied

loading is purely normal to the shell mid-surface, the static-geometric analogy holds true, and may indeed be used to advantage in reducing the effort required for the derivation of the associated governing differential equations. It has been pointed out that, while the truss-model scheme may not be applicable

to

anisotropic

shells

which

exhibit

the

bending-stretching

coupling

phenomenon, the numerical technique can readily be applied to anisotropic shells for which the coupling phenomenon is absent.

The application of the numerical

scheme to the latter class of anisotropic shells has been investigated, by reference to

mid-surface

symmetric

laminated

shells.

It

is

seen

that,

apart

from

modifications associated with the constitutive relations for the (discrete) S - and B-surfaces, the computation of such shells by the truss-model scheme can proceed in the same manner as for single-layered isotropic shells; in particular,

the

static-geometric analogy is applicable in the assembling of the system matrices for the S - and B-surfaces as well as in the ensuing operations, in analogy with the case of the single-layered isotropic shell.

It has been shown how the constitutive

relations of Hooke's law (relating stretching as well as bending stress and strain quantities) for an anisotropic truss/shell may be expressed in terms of an oblique frame of reference.

It is worth noting, in particular, that, on the basis of these

constitutive relations, transformation formulae relating material compliances as well as stiffnesses referred to two distinct orthogonal sets of axes (each of which can be

389

obtained from the other by (axes) rotation) have been established; and it is seen that the (B-surface) stiffness-transformation formulae can be obtained, directly, from

the

(S-surface)

compliance-transformation

invoking the static-geometric analogy.

formulae,

and

vice

versa ,

by

Furthermore, the simple scheme for the

imposition of boundary conditions involving in-plane strains and stress couples has been generalized so as to encompass anisotropic shells : clearly, this generalization should facilitate the application of the truss-model scheme to anisotropic shells with arbitrary boundary contours.

Finally,

the truss model has been applied, for

purposes of illustration, to two layered orthotropic paraboloids of revolution (one with square plan, the other with equilateral triangular plan); and full details of the theoretical solution corresponding to the shell with square plan, which formed the basis for the presented.

verification

of the

model in the

present context,

have been

390

Table 5.1:

Stresses (KN/m) for the membrane solution of case study 10 (refer to Fig. 3.16) subject to loading condition 1. The first value in each column refers to the truss model while the number in brackets denotes the corresponding theoretical value.

Node Number

Nx

Ny

Nxy

1

0

(0)

0

(0)

-94 .4 (-00)

3

0

(0)

0

(0)

-26 .9 (- 3 3 .6 )

5

0

(0)

0

(0)

-9,.7 (- 1 3 .4 )

7

0

(0)

0

(0)

0

(0)

9

-22,.5 (-22 .5)

-22,.5 (-2 2 .5 )

-47,.4 (-4 3 .6 )

10

-31,.3 (-31 .7)

-13,.6 (- 1 3 .3 )

-31 .7 (-3 0 .5 )

11

-35,.4 (-35 .6)

-9..6 ( - 9 .4 )

-21..1 (- 2 0 .5 )

12

-37,.4 (-37 .6)

-7,.6 ( - 7 .4 )

-13..0 (- 1 2 .7 )

13

-38..4 (-3 8 .5)

-6 ..6 ( - 6 .5 )

-6..3 ( - 6 .1 )

14

-6 ..3 ( - 6 .2 )

15

-38..7 (-3 8 .8) 0 (0)

17

-22,.5 (-22 .5)

-22..5 (-2 2 .5 )

-24..6 (- 2 3 .8 )

18

-27..8 (-28 .0)

-17..2 (- 1 7 .0 )

-17..5 (- 1 7 .0 )

19

-14., 1 (-1 3 .9 )

-11.,2 (-1 0 .8 )

21

-30..9 (-31 .1) -33.,0 (-33 .2)

-12.,0 (- 1 1 .8 )

25

-22..5 (-22 .5)

-22..5 (- 2 2 .5 )

28

-28..3 (-28 .6)

-16.,9 (-1 6 .4 )

29

0

(0)

0

0

(0)

(0)

0

(0)

-26,.9 (- 3 3 .6 )

0

(0)

-13,.1 (- 1 2 .6 ) 0

(0)

-9..7 (- 1 3 .4 )

31

-14..1 (-13,.9)

-30. 9 (- 3 1 .1 )

-11.,2 (- 1 0 .8 )

32

-19..1 (-19,.0)

-25.,9 (-2 6 .0 )

-8 ..7 ( - 8 .2 )

33

-22,.5 (-22 .5)

-22..5 (-2 2 .5 )

-6.,1 ( - 5 .5 )

35

-24. 5 (-25,.2)

-20..5 (-2 0 .0 )

37

-6. 6 ( - 6 .5 )

-38..4 (-3 8 .5 )

-6..3 ( - 6 .1 )

39

-17,.3 (-17 .0)

-27..8 (-2 8 .0 )

-4 .,5 ( - 4 .1 )

41

-22. ,5 (-22,.5)

-22. 5 (-2 2 .5 )

-3 . 3 ( - 1 .4 )

42

-21.,1 (-23,.2)

-23. 9 (-2 1 .8 )

43

0

(0)

0

(0)

0

(0)

0

(0)

0

(0)

-33..0 (-3 3 .2 )

0

(0)

46

-12..0 (-11 .8) -16..9 (-16 .4)

-28..3 (-2 8 .6 )

0

(0)

47

-20..5 (-20. 0)

-24. 5 (-2 5 .2 )

0

(0)

49

-22..5 (-22,.5)

-22..5 (-2 2 .5 )

0

(0)

45

391

Table 5.2:

Transverse displacements (mm) for the membrane solution of case study 10 (refer to Fig. 3.16) subject to loading condition 1. The first value refers to the truss model while the number in brackets denotes the corresponding theoretical value.

Node Number

Di splacement

1

0

(0)

3

0

(0)

5

0

(0)

7

0

(0)

9

-4 7 .5 (-3 3 .0 )

10

-3 0 .4 (-2 9 .0 )

11

-2 5 .4 (-2 6 .0 )

12

-2 3 .4 (-2 4 .4 )

13

-2 1 .9 (-2 3 .7 )

14

-1 9 .4 (-2 3 .4 )

15

0

(0)

17

-3 8 .5 (-3 6 .2 )

18

-3 6 .6 (-3 5 .7 )

19

-3 4 .7 (-3 4 .8 )

21

-3 2 .1 (-3 3 .9 )

25

-4 1 .5 (-4 0 .7 )

28

-4 1 .2 (-4 1 .8 )

29

0

(0)

31

-3 8 .1 (-3 8 .3 )

32

-4 3 .4 (-4 3 .0 )

33

-4 6 .2 (-4 5 .6 )

35

-4 7 .4 (-4 7 .0 )

37

-2 8 .5 (-3 0 .2 )

39

-4 3 .6 (-4 4 .1 )

41

-5 1 .0 (-4 9 .4 )

42

-5 1 .8 (-4 9 .9 )

43

0

(0)

45

-3 6 .4 (-3 8 .3 )

46

-4 3 .6 (-4 4 .4 )

47

-4 8 .3 (-4 8 .2 )

49

-5 7 .1 (-5 0 .9 )

392

T a b le 5 .3 :

S-

and

B -s u rfa c e

load

d istribu tio n

(K N )

fo r

case study 10

(re fe r

to F ig . 3 .1 6 ) subject to lo ading c on d itio n 1 .

Node Number

Ps

Pb

P ( t o t a l load)

9

0.02

-0 .7 0

-0 .6 8

10

-0 .2 8

-0 .4 0

-0 .6 8

11

-0 .4 6

-0 .2 2

-0 .6 8

12

-0 .5 5

-0 .1 3

-0 .6 8

13

-0 .6 0

-0 .0 8

-0 .6 8

14

-0 .7 2

0.0 4

-0 .6 8

17

-0 .5 5

-0 .1 3

-0 .6 8

18

-0 .6 6

-0 .0 2

-0 .6 8

19

-0 .7 0

0.02

-0 .6 8

21

-0 .7 4

0.0 6

-0 .6 8

25

-0 .6 5

-0 .0 3

-0 .6 8

28

-0 .7 6

0.0 8

-0 .6 8

31

-0 .5 2

-0 .1 6

-0 .6 8

32

-0 .6 5

-0 .0 3

-0 .6 8

33

-0 .7 1

0.03

-0 .6 8

35

-0 .7 3

0.05

-0 .6 8

37

-0 .3 4

-0 .3 4

-0 .6 8

39

-0 .6 6

-0 .0 2

-0 .6 8

41

-0 .7 2

0 .0 4

-0 .6 8

42

-0 .7 2

0 .0 4

-0 .6 8

45

-0 .5 5

-0 .1 3

-0 .6 8

46

-0 .6 6

-0 .0 2

-0 .6 8

47

-0 .7 1

0.03

-0 .6 8

49

-0 .8 1

-0 .0 9

-0 .9 0

393

Table 5.4:

In-plane stresses (KN/m) for the bending solution of case study 10 (refer to Fig. 3.16) subject to loading condition 1. The first value in each column refers to the truss model while the number in brackets denotes the corresponding theoretical value.

Node Number

Nx

Ny

Nxy

1

0

(0)

0

(0)

-58.,5 (-4 7 .5 )

3

0

(0)

0

(0)

-33. 6 (-3 6 .8 )

5

0

(0)

0

(0)

-14.,3 (-1 7 .7 )

7

0

(0)

0

(0)

0

(0)

9

-0,.9 (-9.15)

2,.5 (- 5 .4 )

-35..3 (-3 9 .1 )

10

-10.,8 (-17 .7 )

-7 .,9 ( - 8 .4 )

-32. 6 (-3 2 .8 )

11

-20..8 (-23 .6 )

-10,.1 (- 9 .3 )

-25.,2 (-2 4 .6 )

12

-27.,5 (-27 • 5)

-9..2 ( - 9 .0 )

-16.,7 (-1 6 .1 )

13

-32..0 (-2 9 .7 )

-8..2 ( - 8 .5 )

-8,,2 (- 7 .9 )

14

-36.,0 (-3 0 • 3)

-12,,2 ( - 8 .3 )

15

0

(0)

0

(0)

0

(0)

-27..7 (-3 0 .7 )

-15,.8 (-1 4 .8 )

-24..5 (-2 4 .7 )

18

-20,.8 (-1 9 • 8) -27.,0 (-27 • 0)

-16..9 (-1 6 .6 )

-18.,3 (-1 8 .9 )

19

-30..8 (-31 • 9)

-16,.1 (-1 6 .5 )

-12..2 (-1 2 .6 )

21

-32..9 (-35 • 7)

-16..4 (-1 5 .4 )

25

-21..9 (-2 2 • 7)

-21,.4 (-2 1 .8 )

28

-29..6 (-3 0 .6 )

-20,.9 (-2 0 .7 )

17

29

0

(0)

0

(0)

0

(0)

-13..0 (-1 3 .1 ) 0

(0)

-10.,7 (-1 2 .9 )

31

-13..2 (-13 .5 )

-21,.7 (-2 1 .9 )

-10..5 (-1 0 .6 )

32

-18..5 (-1 8 .7 )

-24..8 (-2 5 .1 )

-8..4 (- 8 .2 )

33

-22,.4 (-2 2 .5 )

-25,.2 (-2 5 .4 )

-5,.6 (- 5 .6 )

35

-25..0 (-25 .5 )

-24,.0 (-2 4 .3 )

37

-6,.1 ( - 6 .:2)

-16,.5 (-1 4 .4 )

-6..3 (- 5 .9 )

39

-17,.3 (-1 6 .6 )

-26..8 (-2 7 .0 )

-4..1 (- 4 .0 )

41

-21..9 (-2 2 .0 )

-26..1 (-2 6 .7 )

-2..0 (- 1 .3 )

42

-21,.8 (-2 2 .7 )

-26,.4 (-2 6 .3 )

43

0

(0)

0

(0)

0

(0)

0

(0)

0

(0)

45

-13..1 (-11 .4 )

-23..4 (-2 3 .9 )

0

(0)

46

-16..8 (-15 .9 )

-27.,1 (-2 7 .6 )

0

(0)

47

-19..9 (-1 9 .2 )

-27,.5 (-2 8 .0 )

0

(0)

49

-18..6 (-21 .9 )

-21..7 (-2 7 .0 )

0

(0)

394

Table 5.5:

Bending and twisting moments (KNm/m) for the bending solution of case study 10 (refer to Fig. 3.16) subject to loading condition 1. The first value in each column refers to the truss model while the number in brackets denotes the corresponding theoretical value.

Node Number

M*

My

Mxy

1

0

(0)

0

(0)

0,.21 (0 .1 7 )

3

0

(0)

0

(0)

0 .05 (0 .0 8 )

5

0

(0)

0

(0)

0

(0 .0 2 )

7

0

(0)

0

(0)

0

(0)

9

-0,.41 (-0 .3 5 )

-0 .09 (- 0 .0 8 )

0 .10 (0 .0 9 )

10

-0 .43 (- 0 .3 6 )

-0 .13 (- 0 ,1 2 )

0 .07 (0 .0 6 )

11

-0..29 (-0 .2 6 )

-0..14 (-0 .1 3 )

0..04 (0 .0 3 )

12

-0.,15 (-0 .1 6 )

-0..12 (- 0 .1 4 )

0..02 (0 .0 2 )

13

-0,.04 (-0 .0 9 )

-0 .10 (-0 .1 3 )

0

(0 .0 1 )

14

0..03 (-0 .0 7 )

-0,.09 (- 0 .1 3 )

0

(0)

15

0

(0)

0

(0)

0..02 (0 .0 4 )

17

-0..50 (-0 .4 9 )

-0..07 (-0 .0 7 )

0,.04 (0 .0 3 )

18

-0..37 (-0 .3 8 )

-0..08 (-0 .0 8 )

0..03 (0 .0 2 )

19

-0..23 (-0 .2 4 )

-0,.08 (- 0 .0 9 )

0..02 (0 .0 1 )

21

-0 .,08 (-0 .1 1 )

-0..08 (-0 .0 8 )

0

25

-0..40 (-0 .4 1 )

-0..04 (- 0 .0 4 )

0..02 (0 .0 1 )

28

-0..14 (-0 .1 5 )

-0..05 (- 0 .0 4 )

0

(0)

0

(0 .0 1 )

29

0

(0)

0

(0)

(0)

31

-0..49 (-0 .5 2 )

-0 .,03 (- 0 .0 3 )

0..01 (0 .0 1 )

32

-0.,41 (-0 .4 3 )

-0 ..03 (- 0 .0 3 )

0.,01 (0 .0 1 )

33

-0 .,30 (-0 .3 0 )

-0 ..03 (-0 .0 3 )

0..01 (0 .0 1 )

35

-0. 19 (-0 .1 8 )

-0 . 03 (- 0 .0 3 )

0

(0)

37

-0..40 (-0 .4 5 )

-0 .,02 (-0 .0 2 )

0

(0)

39

-0..40 (-0 .4 3 )

-0.,02 (- 0 .0 2 )

0.,01 (0)

41

-0..25 (-0 .2 3 )

-0 .,03 (-0 .0 2 )

0.,01 (0)

42

-0 .,24 (-0 .2 0 )

-0 .,02 (-0 .0 2 )

0

(0)

0

(0)

43

0

(0)

0

(0)

45

-0 .,45 (-0 .5 2 )

-0 .,01 (-0 .0 2 )

0

(0)

46

-0 . 39 (-0 .4 3 )

-0 . 02 (- 0 .0 2 )

0

(0)

47

-0.,30 (-0 .3 2 )

-0 .,03 (-0 .0 2 )

0

(0)

49

- 0 . ,27 (-0 .2 1 )

-0 .,04 (-0 .0 2 )

0

(0)

395

Table 5.6:

Transverse displacements (mm) for the bending solution of case study 10 (refer to Fig. 3.16) subject to loading condition 1. The first value refers to the truss model while the number in brackets denotes the corresponding theoretical value.

Node Number

Displacement

1

0

(0)

3

0

(0)

5

0

(0)

7

0

(0)

9

-6 .5 (- 9 .8 )

10

-1 2 .4 (-1 6 .8 )

11

-1 6 .8 (-2 0 .8 )

12

-1 9 .6 (-2 2 .8 )

13

-2 1 .1 (-2 3 .5 )

14

-2 1 .5 (-2 3 .7 )

15

0

(0)

17

-2 0 .6 (-2 5 .6 )

18

-2 7 .9 (-3 2 .4 )

19

-3 2 .7 (-3 6 .0 )

21

-3 6 .3 (-3 8 .0 )

25

-3 4 .0 (-3 7 .9 )

28

-4 4 .7 (-4 5 .5 )

29

0

(0)

31

-2 7 .1 (-3 1 .3 )

32

-3 7 .1 (-4 0 .6 )

33

-4 3 .9 (-4 6 .0 )

35

-4 9 .1 (-4 9 .5 )

37

-1 4 .9 (-1 8 .0 )

39

-3 8 .6 (-4 1 .9 )

41

-5 0 .1 (-5 0 .7 )

42

-5 1 .5 (-5 1 .6 )

43

0

(0)

-2 8 .4 (-3 2 .5 )

46

-3 9 .0 (-4 2 .3 )

47

-4 6 .5 (-4 8 .3 )

49

(-5 2 .3 )

K l, have the same co-ordinate values, then node number K2 is replaced by Kl element topology definitions.

in the relevant

This, of course, means that node K2 no longer

exists, and it is then necessary to reduce all node numbers from node number (K2 + 1) to the highest node number by 1, so as to ensure that the nodal numbering is sequential.

This re-numbering procedure is carried out systematically, beginning

by comparing the co-ordinates of node 1 with those of all the other nodes and repeating the process for nodes 2, 3, etc. until all repeated nodal co-ordinate positions have been eliminated. The present quadratic isoparametric mapping scheme is generally adequate for practical purposes.

Clearly, errors are introduced in the generation of curves

which cannot be described exactly by parabolas (see, for example, Henshell [95], Singh [233] and Zienkiewicz and Phillips

[272]).

However, such errors are

generally very small, and may be reduced, or altogether eliminated, by various means; in the case of a mathematically described surface, for instance, Zienkiewicz and Phillips [272] suggest the use, if necessary, of a special adjustment program for the modification of incorrectly generated mesh co-ordinates.

Finally, it is

useful to note that the present scheme can easily be extended so as to enable the direct mapping of (curved) triangular-shaped regions as well as the

use of

429

higher-order

(say,

cubic)

interpolation

functions;

such

an

extension

should

considerably enhance the accuracy and flexibility of the scheme.

6.3.2

Input-Data Requirements for TRUSS The data-input format for TRUSS is presented in Appendix G.

It can be

seen that the data-input process is divided into a number of separate blocks, consistent with the program structure shown in Fig. 6.4.

Our discussion here will

be limited to a brief description of the input-data requirements for each of the blocks.

For purposes of illustration, sample input data for the bending solution of

case study 10 are presented in Fig. 6.5.

The run (or execution) time for the job,

on the CDC Cyber 960 machine, was found to be approximately 136 CP seconds (note that the two relevant load cases are solved for, and that, for each (load) case, all three vectorial displacement components at each node of the truss/shell are computed).

On the other hand, the corresponding execution time for the

(complete) solution of the same truss/shell model in accordance with the membrane hypothesis was found to be approximately 32 CP seconds, 4.25 times less than the run time for the corresponding bending solution (note that, for the solution based on the membrane hypothesis, the entry 4 on data line (or row) 2 of Fig. 6.5 must be replaced by the number 3, in keeping with the instructions for data input given in Appendix G). (a)

Problem title and control/ geometric data Each problem is identified with a title, or heading, which is specified as input

data.

Certain control information, used during subsequent data input as well as in

the ensuing analysis, must also be specified; such control information includes, for example, data defining the type of analysis to be carried out and the manner in which the mesh topology is to be defined. Geometric data consists essentially of information required for the definition of mesh topology and, if necessary, material thickness.

For mesh generation by the

isoparametric mapping concept, it is necessary to specify (co-ordinate) data defining the outlines of the zones into which the domain in question has been subdivided; in addition, the material identification number corresponding to each zone and the required fineness of subdivision within it need to be specified.

Furthermore, when

nodal thicknesses are to be generated by the mapping process, it is necessary, too, to specify the values of the shell thickness at the nodal points which define the zonal outlines.

On the other hand, for the alternative case wherein mesh data is

430

prepared manually, it is necessary to specify the co-ordinates of all the joints of the truss/shell under consideration as well as the nodal connectivities and material identification numbers of the associated triangular elements (or facets). (b)

'Fictitious'-bar data 'Fictitious'-Jbar data need to be specified in cases where symmetry is exploited.

It will be recalled (section 2.3.7) that errors generally occur in the computation, by direct application of the virtual-work equations, of stress (and, by virtue of the static-geometric analogy, curvature-change) fields corresponding to nodal points situated along planes of symmetry when, as is often the case, the mesh patterns at such nodes do not exhibit the symmetries which actually exist at the points : errors of this type can, of course, be eliminated by introducing 'fictitious* bars at these joints (thereby modifying the associated mesh patterns), with a view to simulating the relevant symmetries, prior to the application of the virtual-work equations; and, in applying the latter, it is necessary to ensure that the nodal areas of influence (corresponding to the modified mesh patterns) are assigned accordingly, and that the appropriate values of the tensions in the truss bars having end-points that lie on the axes of symmetry are used.

The present set of data provides the necessary

information for the correct formulation of the virtual-work equations required for the determination of stress and curvature-change fields corresponding to nodes situated along planes of symmetry. Each prescribed 'fictitious' bar is defined by specifying: (i) its bar number, which should be the same as the bar number of the corresponding (or 'real') bar within, or along the boundary of, the actual computational zone; (ii) the node number of the truss/shell joint, lying along the relevant plane of symmetry, to which the 'fictitious' bar is connected; and (iii) the (x,y) co-ordinates of a second (or 'imaginary') joint, lying outside the computational zone, to which the 'fictitious' bar must be connected.

Furthermore, it is necessary to specify whether or not the

'real' bar corresponding to each 'fictitious' bar lies along the relevant plane of symmetry.

Finally, one must also prescribe a suitable multiplication factor which

should enable the calculation of the 'influence' area corresponding to the modified (i.e. 'symmetry-consistent') truss-bar arrangement from its counterpart calculated purely on the basis of the bar arrangement associated with the actual structural zone modelled. (c)

Boundary-condition data This set of data provides the necessary information for the imposition of

431

boundary conditions pertaining to stress and strain measures acting in arbitrary directions on the middle surface of a shell (and hence, also, on the associated (x-y)

tangential

corresponding

plane

to

of reference).

vanishing

in-plane

Several stress

types of

and/or

strain

boundary conditions components

of

the

S-surface, on the one hand, and vanishing curvature changes and/or bending moments of the B-surface, on the other, are catered for; and a type number is assigned to each of these boundary conditions.

A given boundary condition can be

prescribed by specifying: (i) the (node) number of the joint at which it is imposed; (ii) its type number; and (iii) the direction angles defining the orientation of the imposed constraint(s) with respect to the global (x,y) co-ordinate system. (d)

Link data This data set furnishes information for the specification of the supporting (or

foundation) links attached to the edge nodes of a truss/shell model; and, when only a portion of a structure is analysed on account of symmetry, it also provides information for the definition of the reactive (or 'symmetry') links attached to nodal points situated along the relevant plane(s) of symmetry.

Each link is

completely described by specifying its nodal incidence (i.e. the (node) number of the nodal point to which it is connected) and the angles it makes with the global (x,y,z) directions. (e)

Material data Material data is required for other than pure membrane stress analyses.

Four

different options, associated with each of which is a type number, are available for the specification of this data set; and it is left to the user to choose the option that is most suited to the specification of the material data for a given problem. For the first method, the values of the Young's modulus, Poisson's ratio and wall thickness of each different material zone in a truss/shell model make up the input, while, for the second method, the required material data are the zonal stretching and bending stiffnesses referred to an orthogonal co-ordinate system (the orientation of which must also be specified with reference to the global (x,y) system of axes). The third method specifically applies to mid-surface symmetric layered shells (note that the single-layered shell also belongs to this family); and here the required material data are the layer thicknesses and coefficients of deformation referred to an orthogonal system of axes (the orientation of which must also be specified with reference

to

the

(x,y)

co-ordinate

corresponding to the fourth option

system).

Finally,

the

material

data

are the zonal coefficients of deformation

referred to an orthogonal co-ordinate system (the orientation of the latter with

432

respect to the global (x,y) frame of reference must also be specified) and, when necessary, the values of the wall thickness at all the nodal points of the truss/shell in question. (f)

Load data Two different load types may be prescribed, namely: (i) discrete (concentrated)

loads applied at nodal points and referred to the global (x,y,z) co-ordinate directions; and (ii) uniformly distributed surface loadings (or tractions) referred to the global (x,y,z) directions.

The form(s) of loading that are of relevance to a

particular load case, or problem, are specified with the aid of certain control parameters; to elaborate, the values assigned to these parameters indicate the load type(s) that consideration. (ii)

need

to

be

prescribed

for

the

load

case,

or

problem,

under

A concentrated joint load is prescribed by specifying: (i) its value;

the (node) number of the joint at which it is applied; and (iii) a number

signifying the particular (global) direction in which it acts.

Uniformly distributed

loadings, on the other hand, are specified with reference to the triangular facets (or elements) of a truss/shell model; this enables patch loadings to be handled in simple fashion, as these can be taken to be applied to an appropriate number of facets.

A uniformly distributed load is prescribed by providing data describing its

value, its direction, and the triangular elements on which it acts. (g)

Data defining required displacement components This final set of data need only be prescribed when only a portion of the

total number of the vectorial displacement components of a truss/shell is required. Each required displacement component is prescribed by specifying the

(node)

number of the joint at which it occurs as well as a number indicating its direction.

6.4. DISCUSSION AND CONCLUSIONS This chapter was begun with an examination of some general aspects of the numerical implementation of the truss-model scheme.

The basic input data

requirements for the numerical scheme have been discussed, and guidelines on the input formats for their specification presented.

In particular, it was pointed out

that, in practice, the manual preparation of mesh data for the numerical scheme is, generally speaking, tedious, time-consuming and error-prone; and that the solution of practical problems by the numerical technique can greatly be enhanced by the facility for the automatic generation of mesh data.

433

The necessarily sparse character of the matrices of coefficients of the system equations for a truss/shell model has been highlighted. that the

Further, it has been shown

assembly of these system matrices can be achievedby means of

very

simple operations which lend themselves readily to computer programming. addition,

computational strategies for the

membrane

and

In

bending analyses of

truss/shells have been presented : it is seen that the actual computational process for each case is very easy to follow, and that its step-by-step nature is well-suited to a modular programming approach. A special-purpose computer program, TRUSS, for the calculation of thin shells by the truss-model scheme has been briefly described. program,

which is written

implemented on the CDC presented.

The basic structure of the

in standard FORTRAN 77 language and

has been

Cyber 960 machine at Imperial

has been

College,

The program enables the analysis of shells in accordance with, both,

the formal bending theory and the membrane hypothesis, as described in detail in the preceding four chapters, and it includes facilities for the automatic as well as manual definition of mesh topology and (uniform or varying) material thickness. The automatic mesh (and thickness) generation scheme used in TRUSS has been described in detail, as has

the input data format for the

program.All the

numerical results corresponding to the case studies considered in the present work were obtained by means of TRUSS : for purposes of illustration, a sample set of input data used for the solution of one of these case studies has been displayed; this data set complies with the instructions for data preparation for TRUSS, which are presented in Appendix G.

434

435

436

437

438

*

Retrieve

*

Re-calculate PB = [G]

[G]

\/ STOP

Fig. 6.1:

^

Flow chart for a computational scheme for the bending analysis of a truss/shell model

439

440

Fig. 6.2:

Flow chart for a computational scheme for the stress/deformation analysis of a membrane truss/shell model

4 A1

442

Fig. 6.3:

Flow chart for a computational scheme for the (complete) analysis of a truss/shell model in accordance with the membrane hypothesis

4 43

START \ N/ Read problem t i t l e and c o n t r o l / g e o m e t r i c da ta s/

Generate to po lo g y

>/ Read 1 ink d a t a

444

!

I I I

C a l c u l a t e the r e q u i r e d shell actions

STOP F ig . 6 .4 :

Basic flo w c h a rt fo r T R U S S

445

EXAMPLE : LAYERED ELPAR WITH SQUARE PLAN (UNITS - KN, m) 4

0 1,1

1 0 8,1

1.9.0. 9.0,-0.54 2.4.5.9.0, -0.3375 3.0.

0.9.0,-0.27

4.0. 0.4.5,-0.0675 5.0. 0.0.0.0.0 6.4.5.0. 0,-0.0675 7.9.0.

0.0,-0.27

8.9.0. 4.5,-0.3375 1,1,1,2,3,4,5,6,7,8 1 ,6,6

1.0.

1 . 0 . 1 . 0 . 1 . 0 . 1 . 0 . 1.0

1.0,1.0,1.0,1.0,1.0,1.0 27 7,2,16,-1.5,9.0,1 14,2,18,-1.5,9.0,1 14,2,35,-1.5,7.5,1 21,2,37,-1.5,7.5,1 21,2,54,-1.5,6.0,1 28,2,56,-1.5,6.0,1 28,2,73,-1.5,4.5,1 35,2,75,-1.5,4.5,1 35,2,92,-1.5,3.0,1 42,2,94,-1.5,3.0,1 42,2,111,-1.5,1.5,1 43.2.97.9.0, -1.5,1 44.2.98.9.0, -1.5,1 44.2.100.7.5, -1.5,1 45.2.101.7.5, -1.5,1 45.2.103.6.0, -1.5,1 46.2.104.6.0, -1.5,1 46.2.106.4.5, -1.5,1 47.2.107.4.5, -1.5,1 47.2.109.3.0, -1.5,1

48.2.110.3.0, -1.5,1 48.2.112.1.5, -1.5,1 49,4,120,-1.5,0.0,2 49.4.114.0.

0,-1.5,2

49.4.113, -1.5,1.5,1 49.4.113.1.5, -1.5,1 49.4.113, -1.5,-1.5,1 26 1.2.0.

0.0.0

2,2,0.0,0.0 3.2.0. 0.0.0 4.2.0. 0.0.0 5.2.0. 0.0.0 6.2.0. 0.0.0 7.2.0. 0.0.0 8 . 2 . 0 . 0 . 0.0 15.2.0. 0.0.0 22.2.0. 0.0.0 29.2.0. 0.0.0 36.2.0. 0.0.0 43.2.0. 0.0.0 1.8.0.

0.0.0

2,8,0.0,0.0 3.8.0. 0.0.0 4.8.0. 0.0.0 5.8.0. 0.0.0 6.8.0. 0.0.0 7.8.0. 0.0.0 8.8.0. 0.0.0 15.8.0. 0.0.0 22.8.0. 0.0.0 29.8.0. 0.0.0 36.8.0. 0.0.0 43.8.0. 0.0.0 1.90.0. 90.0.180.0 2.90.0. 90.0.180.0 3.90.0. 90.0.180.0 4.90.0. 90.0.180.0 5.90.0. 90.0.180.0 6.90.0. 90.0.180.0

447

7.90.0. 90.0.180.0 8.90.0. 90.0.180.0 15.90.0. 90.0.180.0 22.90.0. 90.0.180.0 29.90.0. 90.0.180.0 36.90.0. 90.0.180.0 43.90.0. 90.0.180.0 7.180.0. 90.0.90.0 14.180.0. 90.0.90.0 21.180.0. 90.0.90.0 28.180.0. 90.0.90.0 35.180.0. 90.0.90.0 42.180.0. 90.0.90.0 49.180.0. 90.0.90.0 43.90.0.

180.0.90.0

44.90.0. 180.0.90.0 45.90.0. 180.0.90.0 46.90.0. 180.0.90.0 47.90.0. 180.0.90.0 48.90.0. 180.0.90.0 49.90.0. 180.0.90.0 2 1 1.592100.0. 327800.0.23500.0.50500.0.0.0.0.0.0.0 1.248.111.27.856.7.047.15.1542.0. 0.0.0.0.0 2 0,1

1 1.72.1.3, -0.30 1,0

1 49.3,

-7.50

Fig. 6.5: Input data for the bending solution, by means of TRUSS, of case study 10.

448

Fig. 6.6:

Basic (8-noded isoparametric) zone for mesh generation (keynodes are numbered 1 to 8)

4 49

CHAPTER 7 CONCLUSIONS

This final chapter summarizes the main results of the dissertation.

For a

more comprehensive discussion, reference should also be made to the last sections of the preceding chapters. The thesis was begun with a review of some general aspects of the theory and analysis of elastic thin shells. membrane

theories,

The discussion covered, in te r a lia , the bending and

the static-geometric

analogy,

the two-surface theory,

numerical schemes for the treatment of shell problems.

and

In particular, it was

argued that it would be desirable to aim for any numerical scheme of computation which permits not only the solution of shell problems that do not yield to analytical methods, but also the clarification of the structural response of shells in a manner that engenders in engineers a 'feel* for purposes of design.

In this

connection, it was noted that, although the powerful finite-element method enables the solution of a wide range of practical shell problems to be achieved, the ’mechanical' use of available finite-element computer packages is a somewhat inelegant - even perhaps brutal - way of dealing with shell structures, by reason of their complexity and special nature.

Furthermore, it was pointed out that a

grasp of the complex iteraction between bending and stretching effects (which generally combine to sustain loads applied to shell structures) is crucial for the understanding of the structural behaviour of shells, and that the two-surface model for a shell provides a direct and effective way of studying this interactive behaviour at both the conceptual (analytical) and the computational levels. computational

level,

the

application

of

the

two-surface

Thus, at the

idealization

in

the

recently-devised truss-model scheme for the calculation of shallow, open-type shells (of the kind commonly employed in civil engineering roofing structures) enables the clarification of the structural response of a shell by reference to the portion of the total applied load sustained in stretching action, on the one hand, and in bending action, on the other.

Similarly, at the conceptual level, the two-surface model

permits an elegant derivation of the equations of shell theory, and, also, enhances the qualitative study of shell behaviour. Against this background, it seemed worthwhile not only to broaden the range of application of the truss-model scheme, which had originally been devised for the analysis of shallow, transversely-loaded convex shells with rectangular plan and

450

isotropic material properties, but also to extend the two-surface approach to the formulation of the shell equations, which had hitherto been used within the limited framework of the Cartesian co-ordinate system and in the context of material isotropy and transversely-applied loads.

The rest of the thesis, the main results of

which are summarized in what follows, is, broadly speaking, devoted to

: (i)

extensions of the truss-model scheme so as to encompass general surface tractions, arbitrary geometric shapes and anisotropic material properties as well as the implementation of the (extended) numerical scheme for automatic calculation; and (ii) generalizations of the two-surface approach to the formulation of the shell equations so as to cater for anisotropic material properties, arbitrary loadings and curvilinear co-ordinate systems. In the first phase of the present research into the truss-model scheme, the original numerical technique was described.

As noted earlier, in its original form

this was restricted to the calculation of transversely-loaded convex shells with rectangular plan and isotropic material properties.

Within this limited framework,

certain extensions of, and modifications to, the numerical technique were presented. Among these were : (1)

The introduction of a simpler algorithm for the assignment of nodal

areas of influence.

It was found that this so-called 'median' algorithm leads to

results which are generally only slightly less accurate than - but sometimes superior to -

corresponding results obtained by means of the original,

bisector' method.

'perpendicular-

Moreover, for the sort of mesh sizes that one would normally

adopt in practice, there is little to choose between the two methods, from the standpoint of accuracy.

Since it was felt that its simplicity more than makes up

for any possible, generally only slight, loss in numerical accuracy, the 'median' approach was adopted in the present work. (2)

The

presentation

of a

simple

scheme

transverse displacements of non-membrane truss/shells. on the

well-known

'dummy'

unit-load

for the

calculation

of

the

The scheme is based partly

method and partly on

the

two-cycle

computational scheme for the bending analysis of truss/shells, and is, in essence, a generalization of the more straightforward approach to the calculation of the corresponding displacements of membrane truss/shells.

Interestingly, the scheme

was necessarily restricted to the calculation of transverse displacement components, on account of the limitation, at the time, of the two-cycle computational process to normal-loading conditions. (3)

The introduction of a simple scheme for the exploitation of symmetry in

451

the truss-model scheme, which lends itself rather well to computer implementation and enables the achievement of significant savings in computational effort as well as a reduction in computer-storage requirements. vanishing displacement

In this scheme, the condition of

(or translation) perpendicular to a

plane

of symmetry

(associated with the portion of a truss/shell being analysed) is simulated by means of rigid reactive links attached to nodal points situated along the plane; these 'symmetry' links, together with the relevant foundation links (attached to edge nodes), provide a proper constraint, in space, of the domain in question.

It was

pointed out that, in using the present technique, care should be taken to avoid the occurrence of spurious shell actions (such as, for example, shearing stresses) at nodes situated along planes of symmetry when, as is often the case, the bar arrangements corresponding to such nodal points do not simulate the relevant symmetries at the joints.

With the aim of overcoming this problem, an artifice

was proposed consisting of the

introduction,

conceptually, of

'fictitious'

truss

members, which are connected to nodal points lying on axes of symmetry and defined such as to simulate the appropriate symmetries, prior to the formulation of the relevant truss-model equations for such nodal points. The second phase of the present work on the truss model was concerned with an extension of the numerical scheme so as to encompass isotropic shells of arbitrary geometric form.

It was noted that the concept of deriving the truss

model for an open shell from that for a simply-closed shell is not subject to restrictions of shell contour, so that the numerical scheme may be applied to shells of arbitrary form, provided that the geometrical layouts of the trusses used in modelling the shells and/or their support conditions do not result in 'critical' truss forms.

With the aim of facilitating the calculation of shells with general boundary

shapes by means of the numerical technique, a simple scheme for the imposition of boundary conditions pertaining to stress and strain measures acting in arbitrary directions on

the middle

surface of a shell was presented

: the scheme

is

well-suited to computer implementation, and is essentially a generalization of the earlier treatment relating to stress and strain quantities acting along directions parallel

and

orthogonal

to

the edges

of

a

shell

with

rectangular

plan.

Furthermore, the truss model has been applied to various shells of positive, zero and

negative

Gaussiancurvature,

with the general

aim

of

evaluating

the

performance of the numerical scheme and of gaining useful qualitative insight into the 'mechanics' of these shells.

The main results of these studies are outlined

below. (1)

The predictive capability of the model is very good for shells of positive

and zero Gaussian curvature with variously-shaped plan projections : this conclusion

452

was reached on the basis of analytical results, the full details of which have also been presented. the

structural

As one would expect, the truss model enabled the clarification of response

of

such

shells

as

evidenced,

in

particular,

by

the

load-sharing between bending and stretching effects. (2)

The difficulty in constructing triangulated networks which fully reflect the

conditions of symmetry that exist at all joints of axi-symmetric truss/shells subject to axi-symmetric loads was highlighted, and it was shown that spurious shell actions will generally occur at nodal points with bar arrangements which do not conform exactly

to

the

symmetry

conditions.

Nevertheless,

it

was found

that,

for

truss-model idealizations which do not fully reflect the conditions of symmetry, the computed spurious shell actions generally tend to

be negligibly

small (at least in

comparison with the corresponding principal values) and that the actual shell actions will be predicted with sufficient accuracy provided, of course, that the truss-model idealization is reasonably well-conditioned; this is borne out by results obtained for a paraboloid of revolution with circular plan. (3)

Disappointing results were obtained in

Gaussian curvature.

the case of

shells of negative

Two such shells - a saddle hyperbolic paraboloid and a

hyperbolic paraboloid with straight boundaries - were considered; and in each case two different truss-model idealizations were investigated.

For the saddle hyperbolic

paraboloid, neither of the two truss/shells considered could be analysed, owing to the occurrence of mechanisms.

For the hyperbolic paraboloid

with straight

boundaries, on the other hand, only one of the two idealizations investigated could be analysed; even so, the results obtained from the truss-model solution (for the *non-critical* truss/shell) were unexpectedly poor, indicating that the rigidity of a truss-model idealization does not guarantee the good performance of the numerical scheme. shells;

It would seem that the model may not be applicable to non-con vex however,

further

research

is

obviously

needed

before

more

definite

conclusions can be reached. In the third phase of the work on the truss-model scheme, the numerical technique was extended to include general loading conditions.

It was seen that,

even in the presence of tangential surface tractions (which, by necessity, are sustained exclusively by the discrete S-surface), the overall interaction conditions of force equilibrium can effectively be imposed solely in terms of normal loading components.

Furthermore, it was shown that the breakdown in symmetry between

the discrete S - and B-surfaces resulting from the presence of tangential loadings is of no practical consequence in the context of the numerical scheme, so that the static-geometric analogy is applicable in the assembling of the system matrices and

453

their ensuing operations, even under arbitrary loading conditions.

The extension of

the numerical scheme to encompass general loading conditions (which was validated on the basis of comparisons with finite-element solutions) led to the following: (1)

The application of the 'dummy' unit-load method to the calculation of

the normal and tangential displacements of non-membrane truss/shells subjected to general surface loadings: thus, the earlier limitation of the above method to the computation

of

the

normal-displacement

components

of

transversely-loaded

non-membrane shells was removed, and the simple calculation scheme for the determination of the

normal and

tangential displacements of arbitrarily-loaded

membrane shells generalized. (2)

An investigation into the response of vertically-supported shells subjected

to self-weight loading and horizontal forces simulating edge post-tensioning, with reference to two paraboloids of revolution (one with square plan, the other with equilateral triangular plan).

The truss-model scheme was seen to elucidate the

effect of prestressing on the behaviour of these shells.

In particular, it was found

that the effectiveness of the prestressing forces, in reducing (edge) bending (owing to self-weight loading), can be assessed on the basis of the reduction in deflection (of the interior region of the shell in question) resulting thereof.

Furthermore, it

was seen that, for combinations of self-weight loading and prestressing forces which result in

(near-) vanishing deflections

throughout a shell,

the internal

stress

distributions corresponding to the membrane and bending solutions agree closely with one another : this outcome stems from the predominance of stretching action throughout the shell, as evidenced, in the bending solution, by the distributions of S - and B-surface loadings. The fourth phase of research into the truss-model scheme was concerned with the application of the numerical technique to the calculation of anisotropic shells, with particular reference to

shells of layered construction.

The

constitutive

relations of layered anisotropic shells (which are identical in structure to the corresponding relations of non-layered anisotropic shells) were reviewed.

It was

found that: (1)

The constitutive relations of arbitrarily-constructed (layered) anisotropic

shells, in marked contrast to the corresponding relations of classical isotropic shell theory, generally exhibit coupling between bending and stretching actions, in the sense that these relations contain (additional) terms characterizing the reciprocal influence of bending and stretching effects.

This bending-stretching coupling leads

to a breakdown of the membrane hypothesis, and is, also, at odds with the very

454

ra iso n d 'e tr e

(2)

of the two-surface model for a shell.

The

constitutive relations

of certain

anisotropic

shells

(notably the

mid-surface symmetric layered shells) do not exhibit the above bending-stretching coupling phenomenon; and for such shells the constitutive relations reduce to two separate groups (one linking stretching quantities exclusively, the other connecting flexural actions only), in analogy with the corresponding relations of classical isotropic shell theory.

As one would expect, both the membrane hypothesis and

the two-surface theory may be applied to anisotropic shells which do not exhibit the coupling phenomenon.

Furthermore, the static-geometric analogy has been

extended, by reference to (mid-surface symmetric) layered anisotropic shells, so as to encompass the constitutive relations (and hence, also, the classical theory) of general anisotropic shells which do

not exhibitthe bending-stretching coupling

phenomenon. In view of the preceding discussion, it is evident that the numerical technique can readily be used to analyse anisotropic shells for which there is no coupling phenomenon.

The application of the numerical scheme to the latter class of

anisotropic shells has been investigated.

It was seen that, apart from modifications

associated with the constitutive relations for the (discrete) S - and B-surfaces, the computation of such shell problems

by means

of the truss-model scheme can

proceed in the same manner as for single-layered isotropic shells.

With a view to

facilitating the application of the numerical scheme to anisotropic shells with arbitrary boundary contours, the simple scheme for the imposition of boundary conditions involving in-plane strains

and stress

couples (which depends on the

material properties of the shell in question, and had been limited to conditions of material isotropy) was generalized so as to cater for anisotropic material properties. The model was applied to two layered orthotropic shells, and the good working of the model in the context of material anisotropy was shown by reference to the relevant theoretical solutions. In the final phase of the present work on the truss model, the implementation of the numerical scheme for automatic calculation was outlined. data

requirements for the

input-data formats given.

model

were discussed,

and

The basic input

guidelines on

possible

It was shown that the system matrices for the scheme

can be assembled by means of very simple operations which lend themselves readily to computer programming. Also, computational strategies for the membrane and bending analyses of truss/shells were presented computational step-by-step

process for nature

each

case

is well-suited to

: it was seen that the actual

is very easy to

follow,

and

that

a modular programming approach.

its A

455

special-purpose computer program, TRUSS, for the membrane as well as bending analyses of thin shallow shells by means of the truss-model scheme was briefly described.

All the numerical results corresponding to the case studies considered in

the present work were obtained by means of TRUSS.

The basic structure of the

program, which was written in standard FORTRAN 77 language and implemented on the CDC Cyber 960 machine at Imperial College, was presented.

The program

has facilities for both automatic and manual definitions of mesh topology and (uniform or varying) material thickness.

The automatic mesh (and thickness) -

generation scheme used in TRUSS was described in detail : this scheme is based on the quadratic isoparametric mapping of (curve-sided) quadrilateral domains, and can

easily be extended

so as

to

include

higher-order

(for example,

cubic)

interpolations as well as the direct mapping of (curved) triangular domains; such an extension should considerably enhance the accuracy and flexibility of the scheme. In addition, the input-data requirements for TRUSS, and the associated instructions for data preparation, were presented. Finally, a brief summary of the work related to the formulation of the equations of shell theory is in order.

Research in this area was begun by

recapitulating the two-surface approach

to the derivation of the shallow-shell

equations, i.e., by describing the basic conceptual method (which, as noted earlier, had hitherto been used within the limited framework of the Cartesian co-ordinate system and in the context of material isotropy and transversely-applied loads) and then

generalizing

it

so

as

to

encompass

general

curvilinear

co-ordinates

corresponding to the (mutually orthogonal) lines of curvature of the middle surface of a shell.

Special cases of the above formulation corresponding to spherical,

cylindrical, polar and quasi-polar co-ordinates were also presented.

As in the case

of Cartesian co-ordinates, it was shown that this analytical approach to the formulation

of the shell equations is, in the spirit of the two-surface idealization

of a shell, more direct, simpler and, above all, more instructive than conventional presentations; and, in addition, that the static-geometric analogy can be used to advantage in reducing the effort required for the derivation of the shell equations. In the second phase of the analytical work on the two-surface approach to the formulation of the shell equations, an extension of the method was achieved within the framework of Cartesian co-ordinates and in the context of material isotropy, to include general surface tractions.

It was noted that, in keeping with

the two-surface idealization of a shell, tangential surface loads must be applied exclusively to the S-surface, since, by definition, the B-surface cannot resist loads lying in its plane; and that such a move is tantamount to a neglect of the (feeble) tangential-force interaction between the S - and B-surfaces, in accordance with

456

conventional shallow-shell theory.

In addition, it was pointed out that there is no

symmetry between the expressions of the S-surface and their counterparts for the B-surface, by reason of the breakdown of the static-geometric analogy in the presence of tangential surface tractions.

It was shown that the coupled equations

describing the behaviour of a shallow shell subjected to general surface tractions can be derived by substituting the appropriate expressions (in terms of the stress and deflection functions) for the four variables defining the S - and B-surface Gaussian curvature changes and normal loadings into the two interaction conditions of normal-force equilibrium and geometrical compatibility, in analogy with the simpler case of shells loaded purely by normal surface tractions.

Furthermore, it

was seen that a knowledge of this analytical approach to the formulation of the above equations is helpful in understanding the

treatment of general surface

tractions in the truss-model computational scheme. The final phase of the analytical work on the two-surface approach as regards derivation of shell equations was concerned with an extension of the method, within the

framework

of

Cartesian

co-ordinates

and

by

reference

to

(mid-surface

symmetric) layered shells, so as to encompass arbitrarily-loaded anisotropic shells which do not exhibit the bending-stretching coupling phenomenon.

The derivation

is, in essence, a generalization of the earlier formulation for single-layered isotropic shells subjected to general surface tractions.

It was noted that, as in the case of

the single-layered isotropic shell, there is no symmetry between the expressions of the

S-surface and their counterparts

for the

B-surface,

on

account of the

breakdown of the static-geometric analogy in the presence of tangential surface tractions.

In addition, it was seen that in the special case wherein the applied

loading is purely normal to the shell mid-surface, the static-geometric analogy holds true, and may indeed be used to advantage in reducing the effort required for the derivation of the associated governing differential equations.

Finally, it

seems clear that this formulation can readily be generalized so as to encompass curvilinear co-ordinates corresponding to the lines of curvature of the middle surface

of a

shell;

and,

indeed,

the

present

two-surface

approach

to

the

formulation of the shell equations may be extended to cater for general, oblique frames of reference.

457

APPENDIX A A SIMPLIFIED DERIVATION OF THE EQUATIONS OF SHALLOW-SHELL THEORY IN CURVILINEAR CO-ORDINATES

In what follows the coupled differential equations describing the behaviour of a shallow shell within the broad framework of curvilinear co-ordinates corresponding to the lines of curvature of the shell middle surface are derived on the basis of the

two-surface

shell

theory.

As

noted in

section 1.3.3,

shallow-shell equations, which were originallydeveloped by Vlasov

these

so-called

[256], may also

be used for the solution of some problems which do not belong to the realm of shallow-shell theory.

The present derivation of the equations is, in essence, a

generalization of Calladine's earlier formulation in terms of Cartesian co-ordinates (see section 2.2.1). equations referred

The present method is also used to formulate the shell to

spherical,

cylindrical, polar

and quasi-polar co-ordinate

systems : these equations, like their counterparts for Cartesian co-ordinates, may, of course, be derived from the aforementioned, more general, equations in terms of orthogonal curvilinear co-ordinates.

It will be seen that the present approach

to the formulation of the shell equations is, in the spirit of the two-surface idealization of shells, more direct, simpler and, above all, more instructive than the conventional presentations [174, 177, 211, 229, 238, 256, 257]. Fig. A.l shows the various stress resultants and couples acting on a differential element of a shell, as well as the surface tractions to which the element is subjected.

The orthogonal curvilinear co-ordinates q 1 and ot2 coincide with the

lines of curvature of the middle surface of the shell. of a normal pressure pN and a

pair of tangential surface tractions q 1 and q 2

aligned with the directions of principal curvature. senses of the in-plane direct and

The applied loading consists

The diagram defines the positive

shearing stress resultants N t l , N 22, N 12 (=

N 21), out-of-plane (or transverse) shear stress resultants Q ,, Q 2 and bending and twisting stress resultants (or couples) M ^ , M 22, M 12 (= M21) (c.f. Fig. 2.1). Let the principal curvatures of the middle surface of the shell be denoted by K 1 (= 1/R ,), K 2 (= 1/R2) whilethe associated Lame parameters be denoted A, = A -jfa,, a 2), A 2 = A ^ a , ,

a 2).

by

It will be recalled, from the theory of

surfaces, that these geometric parameters are related thus:

458

8

1

8a 2

8

8 a,

8a2

1

8a, -K,K2 A,A 2

8 a , A,

8

(Al)

A2 8 a2

8a 2

(K2 A 2) - K, --8a, 8a,



8 —

(K,A,) =

k

8a 2

(A2)

8a, 2 --8a 2

(A3)

The above relations are the well-known Gauss-Codazzi equations (the first is Gauss's equation, while the other two are those of Codazzi), the significance of which rests on the fact that the quantities A ,,

A 2, K,

and K 2 cannot be

expressed arbitrarily as functions of the co-ordinates (a ,, a 2) of a point on the surface [73, 261]. Now consider the equilibrium of the differential shell element.

Instead of

formulating the equilibrium equations for the actual shell element directly, as in conventional treatments, we shall, in keeping with the two-surface idealization, set up the corresponding equations for each of the distinct conceptual surfaces used in modelling the shell.

As shown in Fig. A.2(a) and Fig. A.2(b), the S-surface

element carries only the membrane stress resultants while the B-surface element sustains the flexural actions exclusively.

The diagrams also show that the B-surface

element is subjected to a normal pressure p j^ and two tangential surface tractions q ,B,

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