A small strain, large rotation theory and finite element formulation of thin curved beams

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Citation: Mohamed, A.E. (1983). A small strain, large rotation theory and finite element formulation of thin curved beams. (Unpublished Doctoral thesis, City University London)

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A SMALL STRAIN LARGE ROTATION

THEORY

AND

FINITE ELEMENT

FORMULATION

OF

THIN CURVED

BEAMS

A THESIS SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF

CIVIL ENGINEERING AT THE CITY UNIVERSITY

BY

ABDELRAHMAN

ELZUBEIR MOHAMED

BSc, MSc

THE CITY UNIVERSITY

_{APRIL 1983}

To my PaAents

ABSTRACT

In this thesis _{a theory for the geometrically nonlinear analysis}

of thin curved beam-type structures is proposed and an associated

displacement _{finite element formulation developed.}

An exact two-dimensional _{large rotation} theory, _{which is based on}

an intrinsic coordinate _{system, has been developed.} Four

alternative Lagrangian formulations _{of the theory} have been

presented for comparison.

A family _{of two-dimensional thin curved beam elements has been}

developed by using the constraint technique to include the

convective _{coordinate system.} The elements are relatively _simple

and the minimum number of degrees of freedom necessary has been used.

The Total Lagrangian _{formulation has been shown to be numerically}

more effective than the Updated Lagrangian formulation. A new

Total Lagrangian _{formulation that includes the effect of curvature}

change on axial force in the incremental _{equilibrium equations} has

been developed. _{The formulation is based on the geometric strains}

and has the capability _{of predicting} true _axial force values in

large _{rotation and curvature problems. This approach can be used}

in the general continuum _mechanics largz; d3formation formulations.

A large _{rotation theory for three-dimensional beams and Total}

Lagrangian _{formulations of the theory, which are based on the}

Green strains _{and the geometric strains,} have been developed.

The theory _{correctly describes the large rotation elastic response}

of a thin _{eccentric curved beam of rectangular cross-section.}

Material _{nonlinearity, which is based an the von-Mises}

yield

function _{and the Prandtl-Reuss flow rule and in which isotropic}

hardening _{is assumed, has been included in the formulation.}

A family _{of three-diinensional beam elements, that can accurately}

accommodate the theory, has been developed _{by the constraint}

technique. _{The elements are suitable for use as stiffeners in}

the analysis _{of stiffened shell structures.}

The elements, _which have been developed, have been implemented

in the LUSAS finite _{element system. The accuracy of the results}

obtained has been demonstrated by comparison _{with*published}

results.

ACKNOWLEDGEMENTS

The research described herein _{was undertaken} in _the

Department _{of Civil Engineering of The City University.}

During _{the course of the research the author received}

valuable _{advice and helpful suggestions} from _{many persons.}

In particular the _{author wishes to express} his _gratitude

to the following:

To Professor _{J. E. Gibson, Head of the. Department, for the}

helpful _{way in which the project was accommodated.}

To Dr.

_{L. F. Boswell,}

_{who supervised}

_this

_work,

_for

_his

constant

_{encouragement,}

his valuable

_advice

_{and his}

guidance

throughout

the duration

_{of the project,}

_without

which

the effective

fulfilment

_{of this}

_{work would not}

have been possible.

To Dr. _{L. P. R. Lyons for his kind help, his constant}

supervision _{and his} helpful _{suggestions which} directly

contributed to the formulation _{of the ideas in this study.}

His _{help and advice regarding the use}

of the LUSAS finite

element _system is _{greatly appreciated.}

To Dr. _{S. H. Zhang, Dr. D. Irving, Mr. A. M. Kaream}

and

Mr. _{N. El-Kaddah for their}

generous help in many ways.

To the staff

_{of The City}

University

_Computer

_Centre,

whose effort

is appreciated.

To Mrs Calda _{Petrie who typed the manuscript.}

To The Sudan Government _{for providing fi-nancial support.}

Last, _{but not least, to my family and friends for their}

love _{and encouragement.}

NOTATION

Only the _{main symbols are} listed here. Other symbols

are defined as they first appear in the text:

A Cross-sectional Area

Ae First _{moment of area about} the _reference

line

A Matrix _of displacement _gradients

obtained from the nonlinear strains

a t Wanted element variables up to the i th

iteration _referred to the _{configuration}

at time t(t, = 0 or t)

ai Wanted element variables from the

c

beginning _{of an increment up to} the

ith iteration

B Strain _{matrix relating} the _variation

in _{the continuum mechanics strains} to

the _variation in displacements

R _{Strain matrix relating to the variation}

in _{the strain resultants} to the

variation in the nodal variables

B 0, Do

tt Infinitesimal strain matrix with

reference to the configuration at

time t (t--= 0 or t)

B

L(ut)"'L (at) Linear strain matrix with reference

to the _{configuration at time} t

(t _{=0 or t)}

b Breadth _of the _{cross-section}

b Unwanted _{element variables}

1cs]i _{Matrix of direction cosines at point i}

D, De Elastic _{modulus matrices}

13, j5e _{Elastic rigidity matrices}

D(c, k), D(s, k) Elasto-plastic _{modulus matrices}

D(s, k) Elasto-Plastic _{rigidity matrix}

d _{Depth of the cross-section}

a _Displacement

vector of a general point

E _{Young's modulus of elasticity}

e Green _{strain measure of} the _reference

line

e Strains _referred to _{spatial coordinates}

in _the deformed _geometry

eyve, Eccentricities along the local y, z

axes respectively

F, f _{Yield functions}

F

yz Shear stress resultant

Vector _{of uniformly distributed loads}

acting in _{a fixed} direction

Gt _{Matrix relating the variation of the}

vector of the generalised strain

resultants to the _variation in the

nodal variables with reference to the

configuration at time t (t =0 or t)

G _{Shear modulus}

9 Natural base _vector in two dimensions

gx1g _Vgz Displacement _{gradient vectors} y

9y P9 z Vectors normal to the convected x

surface in the deformed _{configuration}

H _{Matrix relating the Green strains or}

the _{geometric strains to generalised}

strain resultants

H _{Matrix relating the variation in the}

geometric _strains to the _variation in

the Green _strains

I _{Second moment of area about the reference}

line

Unit _matrix

I

yy, I _zz Second moments of area about the

centroidal _{axes of} the _{cross-section}

iYj

,k Unit vectors in the global X, Y, Z

directions _respectively

Torsional _rigidity

KO, K

L(at) Infinitesimal strain and initial

displacement _{stiffness matrices}

respectively

KT _{Tangent stiffness}

matrix

K _{Initial stress stiffness matrix}

K* _{Additional initial stress stiffness}

matrix

K _{Physical curvature in two dimensions}

Kxy.

%K _xz Measures of bending curvatures about

the local _{y and z axes respectively}

k Strain hardening _parameter

kl, k

2 Factors included in the shear rigidity

terms to describe the torsion _{of a}

solid rectangular cross-section

ks Factor included in the _{shear modulus}

terms to improve the _shear displcement

approximation

L Length

M, M* _{Constraint matrix} in _{terms of} the

initial _{total element variables}

MM

A' _B Partitions of the constraint matrix

with reference to the _wanted and

unwanted variables respectively

M _{Bending moment}

Mt _{Bending moment with reference to the}

configuration at time t (t =0 or t)

M

xy'M xz -Bending moments about the local y, z

axes _respectively

Vector _{normal to the natural base}

vector in the deformed _geometry

Ni _{Heirarchical shape functions}

Ni _{Parabolic interpolation functions}

Cubic interpolation functions

n Unit _{vector normal} to the _natural

base _vector in the deformed _geometry

n ,n Unit vectors normal to the convected

y z

x surface in the deformed _{configuration}

P

ti The initial stress matrix with

reference to the configuration at

time t (t _{=0 or t)}

ti Additional initial stress matrix with

reference to the configuration at

time t (t _{=0 or t)}

P _{Applied load}

Pt _{Axial force with reference to the}

configuration at time t (t =0 or t)

P Vector _of follower _pressure loads

q Vector _of total _uniformly distributed

load _{per unit undeformed} length

R _{Vector of''conservative equivalent}

nodal forces

Rd _{Vector of deformation dependent nodal}

forces

Position _{vector of a general point}

before deformation

Position _{vector of a general point}

after deformation

r Position _{vector of a point on the} beam

reference line before deformation

Position _{vector of a point on the} beam

reference line after deformation

S, St _{2nd piola-Kirchoff stresses with}

reference to the configuration _at

time t(t _{=0 or t)}

Stost

Vector

_{of stress}

_resultants

_with

reference to configuration at time, t,

(t _{=0 or t),} for _the i th iteration

Vector _{of modified} initial _stress

resultants obtained from the geometric

strains

T Transformation _matrix

T Total torsional _moment

TyvT

z Torsional moments due to the torsions

of the y and the z normals respectively

U, V, W _{Displacement components in the global}

X, Y and Z directions _respectively

U, V, W Displacement _components in the local

x, y and z directions respectively

Ut Vector _{of continuum mechanics}

displacements _relative to the

configuration _at time t (t _{=0 or t)}

Displacement _{vector of a point on the}

beam reference line

Vt _{Volume in the configuration at time t}

(t _{=0 or t)}

v Volume in the deformed _{configuration}

W _{Constrained shape function array}

WN, WN* Unconstrained _shape function _array in

terms _of the initial _{total element}

variables

W

A' WB Partitions of the shap e function array

with reference to the _{wanted and the}

unwanted variables _respectively

X, Y, Z Global _coordinates

XIYIZ Local material coordinates

xIyPz Unit vectors in the local x, y and z

directions _respectively

ai Generalised _variables defining the

displacement _variation

a, a, e Measures _{of rotation about} the _global

X,. Y and Z axes respectively

Y Variables defined by the _shape function

array

Yxypyxzoyyz

Shear strains

6, s*

-Initial

total

element

variables

6W

_External

_virtual

_work

6W

int

Internal

virtual

work

Ct'Ci t

Vector

of strain

measures

w hich

are

explicit

functions

_{of the displacement}

gradients

_with

_reference

to the

configuration

_{at time}

t (t =0 or t),

for

_{the ith}

_iteration

Ct

Green-Lagrange

_strains

_{or geometric}

strains

with

reference

to the configuration

at time

t (t =0 or t)

C*

_t

Generalised

_strain

_measures

_{of the}

reference

line

obtained

from the

geometric

strains

_with

_reference

to

the configuration

at time

t (t =0 or t)

C9

_t

Infinitesimal

_strains

_with

_reference

to configuration

_{at time}

_{t (t =0 or t)}

CL

_t

Nonlinear

strains

with

reference

to

configuration at time t (t =0 or t)

E:

xx C yy Direct strain components

0t

Vector

_{of displacement}

_gradients

_with

reference

to the configuration

at

time

_{t (t =0 or t)}

exle

y '0 z

Measures

of rotation

about

the local

x, y and z axes respectively

X _{plastic strain rate multiplier}

V Poisson's _ratio

E Natural _coordinate

V Tiue Cauchy _{stresses or nominal} stresses

a _{effective stress}

a

_ya

Stress

_{components in the x and y}

yy

xx

directions

_respectively

a

Uniaxial

_yield

_stress

y

TXY#TXZ, T

yz

Shear stresses

T

IT

Torsions

about the local

x axis of the

Y _z

local

_{y and z normals respectively}

ýj

_Rotation

_{of node point}

_j

y

Angle between the local

y axes in the

deformed and undeformed configurations

ýz

_{Angle between the local}

_{z axes in the}

deformed and undeformed configurations

Nonlinear

_equilibrium

_equations

Rotation

_vector

(for

_{small displacement}

displacements)

CONTENTS

Abstract

Acknowledgments

Notation

Chapter 1 General Introduction

1.1 _{Introductory Remarks}

1.2 Scope _{and Outline of} Thesis

1.3 Large Deformation _Theory

1.3.1 Geometric Nonlinearity

1.3.2 The Eulerian Formulation

1.3.3 The Lagrangian Formulation

1.3.4 _Large deformation in Beam

and Shell Type Problems

1.3.5 _{The Geometric or}

Conventional Strains

1.3.6 _{Material Nonlinearity}

1.4 _{The Incremental Equilibrium}

Equations

1.4.1 _{Solution Procedures}

1.4.2 _{The Tangent Stiffness}

Matrix

1.5 _Element Geometry _{and Degrees of}

Freedom

1.5.1 Non-conforming Elements

1.5.2 _{Convergence and the Patch}

Page

1.6

Chapter ₂

2.1

2.2

2.3

2.4

1.5.3 _{Displacement Approximation}

within

Elements

39

Summary of Lagrangian

Formulation

42

1.6.1

_{The Total}

_Lagrangian

Formulation

45

1.6.2

The Updated

Lagrangian

Formulation

46

1.6.3

_{Stress-Strain}

_Relations

₄₇

Two-Dimensional

_{Large Deformation}

Curved Beam Theory

54

Introduction

54

Two-Dimensional

_{Large Rotation}

Theory 58

2.2.1 _{Internal Virtual Work in}

Terms _{of The Green Strain (31}

2.2.2 _{Internal Virtual Work in}

Terms _{of The Conventional}

Strain ₆₆

Stress-Strain _{Relations 70}

The Incremental _Equilibrium

Equations ₇₁

2.4.1 _{Total Lagrangian Formulation}

Based _{on Green} Strain

Measure (TLG) ₇₂

2.4.2 Updated Lagrangian

Formulation _{Based on Green}

Strain _{Measure (ULG) 77}

Page

2.4.3 Combined Updated _{and Total}

Lagrangian Formulation

Based on Green Strain

Measure (UTLG) 81

2.4.4 Total Lagrangian Formulation

Based _{on Conventional} Strain

Measure (TLC) 83

2.4.5 The External Virtual Work

and Equivalent Nodal forces 89

2.5 _Conclusions 94

Chapter 3 Formulation _{of Thin} Curved Two-

Dimensional Beam Finite Elemtnts 96

3.1 Introduction 96

3.2 _{The Requirements} for _the Elements 97

3.3 _{The Constraint Technique}

3.3.1 Non-conforming _{and Conform-}

ing Displacement Variations

within Elements 100

3.3.2 Outline _of the Constraint

Technique 104

3.4 ISOBEM 1 Element 110

3.5 ISOBEM 2 Element 113

3. '6 SUBBEAM Element 116

3.7 Computational Implementation 119

3.8 Conclusions 123

Page

Chapter 4 Two-Dimensional Curved Beam

Applications 130

4.1 Introduction 130

4.2 _Linear Solutions 131

4.2.1 Straight Cantilever Beam 131

4.2.2 Curved Cantilever Beam _-132

4.2.3 Shallow Clamped Arch 134

4.2.4 Deep Clamped Arch 135

4.3 Geometrically Non-linear Solutions 137

4.3.1 Cantilever _under Vertical

Load _{at Free} End 137

4.32 Symmetrical Buckling _of

Two-hinged Deep Arch 147

4.3.3 Cantilever _under Pure

Moment 149

4.3.4 Numerical Effectiveness _of

Formulations 151

4.3.5 _{Cantilever under Uniformly}

Distributed _{Load and}

Follower _Pressure Loading 153

4.3.6 The Post-bifurcation _{of a}

Simply Supported Beam 154

4.4 Conclusions 156

Chapter _{5 Three-Dimensional Lagre} Rotation

Elasto-Plastic Theory for _Thin

Curved Eccentric Beam Elements 224

5.1 _Introduction 224

I

Page

5.2 _{Geometric Definition of Element}

Reference Line 226

5.3 _{Deformation Geometry} 229

5.4 _{The Internal Virtual Work}

Expression _{and Nonlinear} Strain-

Displacement Relations 235

5.4.1 _{Conventional Strains at}

a General Point 235

5.4.2. _{Green-Lagrange Strains at}

a General Point 239

5.4.3 _{Strain Resultants in Terms}

of Displacement Gradients 241

5.5 _{Stress-Strain Relations 245}

5.5.1 _{Elastic Material 245}

5.5.2 _{Elasto-Plastic Material 250}

5.6 _{The Incremental Equilibrium}

Equations 253

5.6.1 _{Total Lagrangian Formula-}

tion Based _{on Green} Strains

(TLG) 254

5.6.2 _{Total Lagrangian Formula-}

tion Based _{on Conventional}

Strains (TLC) 258

5.6.3 _{Follower Pressure Loads}

and Applied Concentrated

Moments 263

Page

5.7 Conclusions 268

Chapter 6 Three-Dimensional Eccentric

Curved Thin Beam Elements-

Formulation _{and Applications} 274

6.1 Introduction 274

6.2 Formulation _{of Elements} 296

6.2.1 _{ISOBEM 3 Element} 281

6.2.2 _{ISOBEH 4 Element} 285

6.2.3 _{The SEMILOOF Beam. Element} 290

6.3 Linear Elastic Solutions 296

6.3.1 Straight Cantilever Beam 296

6.3.2 _L-Type Cantilever Frame 297

6.3.3 _{Semi-Circular} Beam 298

6.4 Geometrically Non-linear Solutions 298

Solutions 298

6.4.1 _Straight Cantilever _under

Point _{Load at Free} End 299

6.4.3 _{Three-Dimensional}

45-Degree _{Bend -ý} 300

6.5 _{Combined Geometric and Material}

Non-linearity _Solutions 303

6.5.1 Fully-Encastre _Beam

under Central Point Load 304

6.5.2 Simply Supported

Rectangular Strut 305

6.5.3 Continuous _{T-Beam. Strut 305}

6.6 _{Conclusions 306}

Page

Chapter 7 General Conclusions 34-2

7.1 Summary of Work 342

7.2 Concluding Remarks 343

References _{and Bibliography} 348

Appendix I Explicit Form of the Constrained

Displacement Field for the

2-D Straight Beam Element_ 366

I. 1 Incompatible Displacement Field 367

I. 1.1 Generalised Variables 367

1.1.2 Independent Interpolation 370

1.2 Conforming Displacement Field 373

Appendix. II Total Lagrangian Modified

Incremental Equilibrium Equations

for _{a Two-Dimensional} State _of

Stress 377

II. 1 Stress-Strain Relations 378

11.2 Strain-Displacement Relations 378

11.3 Finite Element Formulation 383

CHAPTER1

GENERAL

INTRODUCTION

1.1 INTRODUCTORY REMARKS

Modern developments in design _{and construction} have lead

to the increasing _{use of slender structural members.}

Typical _{examples of these are} the _{stiffened plated} box

girders which are used in bridge structures and the

stiffened shell constructions used in offshore structures.

The evolution of such complex structures*has resulted in

the _{establishment of extensive research and development}

programmes to improve the knowledge of the behaviour of

these _{structures when they are subjected} to load.

A well _{established method} for the _{study of structural}

behaviour _{is structural testing. The size of the}

structural _{members may be so large,} however, that full

scale testing becomes impractical. Thus, there _are two

alternative _{approaches available} for _structural testing.

The first _{is model testing (1), which has limitations}

due to the difficulty _{of accurately pealing} down the

prototype including all factors affecting performance,

such as initial imperfections and initial stresses. The

second approach requires the testing of panels with the

same geometry and support conditions as that of the

complete member (2). This approach needs a knowledge, of

the _{panel width required} and boundary, conditions to be

used for the panels. There are obvious advantage! if

the _{results obtained} from _structural testing could be

compared with those obtained from theoretical analysis.

The development _{of powerful computers} has resulted in

the _{application of numerical methods} to solve structural

problems. These methods have been shown to be very

effective and to give extremely good results. To define

the _special features _required for the _analysis of

stiffened structures it is necessary to state the main

modes of failure of these structures. Due to their

slenderness the failure of these structures is pre-

dominantly _{by buckling C3 - 7).} Depending _{on the relative}

sizes _{of shells and stiffeners,} failure may occur by

local _{buckling of panels and stiffeners while} their

junctions _{remain straight and while the shell} is _still

el ast ic, by 'overall buckling in the form of significant

radial _{movements of stiffeners or as a complex} interaction

of the two modes. In some cases these _{may also} be

combined _with lateral torsional instability (tripping)

of the _stif femer&. It is usual not.. to include tripping

in _{an analysis and the present} design _{procedures adopt}

the _{criterion of} limiting the depth _{of the stiffener} in

order to preclude the tripping effect (6,81'. An

additional factor-which must be taken into account in

the _analysis is that the fabrication _{of such structures}

results in initial imperfections and introduces _residual

stresses. Hence, the behaviour of these structures is

such that large displacements and rotations can develop.

This, _{coupled with} the _{need to make use of} the full

load _{carrying capacity, may also} lead _{to plastic}

deformation. Thus the _{analysis procedure must} take

into _{account both geometric and material nonlinearity..}

Since these _{structures are normally constructed using}

steel, however, the strains can be assumed to be small.

This

_nonlinear

behaviour

is, best

_treated

incrementally

especially

because

of the incremental

nature

of the laws

governing

plastic

deformation.

Originally

the effect

of stiffeners

was taken

into

account

employing

the

concept

_{of smeared stiffeners,}

thus

ignoring

the

possibility

_{of local}

buckling

(3).

A study of the

behaviour

_{of stiffened}

_cylinders

_{under moderately}

large

rotations

has been carried

_{out by Bushnell}

(4,5,9)

using _{a double} loop iteration _{subincremental} technique

and the finite difference _{energy method} taking into

account the discrete form _{of the stiffeners. He concludes}

that _{agreement between test and theory is improved if}

the _{analytical model reflects the discrete behaviour}

of _stiffeners. 'For _{stiffened plates,} Webb and Dowling

(7) have _{presented solutions} for flat, _{angle and Tee}

section _stiffeners by applying dynamic _relaxation to the

finite _{difference equivalents of the governing equations.}

The significant features of the stiffened structures,

namely the interaction of the buckling modes, the

yielding and discrete behaviour of the stiffeners and

the influence _of initial imperfections _{and residual}

stresses, are more effectively considered using the

finite _{element method (10 - 14). Crisfield} (6) has used

the finite _{element method to obtain solutions} for

eccentrically stiffened steel plates for moderately

large displacements. Plasticity is _{considered by using}

approximate yield criteria that are direct functions of

the _{stress resultants and plastic curvatures, obtained}

from _{a modification of} Ilyshin's _{yield function} by

making _{some allowance} for _yield in the fibres before

full _{section yield. The stiffeners are modelled as two}

uniaxial line _{elements and a central rectangular} bar

element. Bathe _{and Bolourchi} (15) have _presented

compatible _{shell and bending elements} formulated-by

interpolating _{the element geometry using the mid-surface}

nodal _{point coordinates and mid-surface nodal point}

normals. Ferguson _{and Clark} (16) have developed two-

and three-dimensional _{super-parametric eccentric} beam

elements _suitable for _{use as stiffeners} based _{on relaxing}

the Kirclhoff _{hypothesis by assuming that plane sections}

remain plane, but not necessarily normal to the beam

axis. Both formulations, however, include _shear

deformation _{and the performance of such elements is}

known _{to deteriorate} for _{very thin beams especially for}

large _{rotation problems (11,17).}

Curved beam element formulations have _attracted

considerable interest due to their relative simplicity

while giving insight to the more complex problems of

shell elements. This accounts for the large number

and different types of beam elements presented in the

literature _{(16 - 32).} However, to the _author's

knowledge, _{a generally curved eccentric} beam element

formulation _{that can accommodate} large _rotations is _not

yet available.

1.2 _{SCOPE AND OUTLINE OF THESIS}

For large

deformation

_analysis

_{of beam and shell}

type

problems

two alternative

_approaches

are generally

adopted

(15).

In the classical

_approach,

_{a plate}

or

shell

theory,

_{which has been developed}

from the'three-

dimensional

_continuum

_methanics

_equations

_{by incorporating}

various

_assumptions

_appropriate

to the structural

behaviour,

is used as a starting

_point

_{of the element}

formulation

(10,30).

_{The alternative}

_approach

_{uses the continuum}

mechanics

_equations

_{and then introduces}

_{approximations}

in the element formulation

(10,15,35,36).

The latter

approach

_{amounts to using}

_{a general}

_shell

_{or plate}

theory.

In this

thesis

_{an approach}

_similar

_{to the classical}

_one

is adopted.

Using the Kirchhoff

hypothesis

_{the shape}

of-the

_element

is defined

before

_{and after}

deformation.

This

definition

_{of geometry}

_{is used to obtain}

_exact

strain

displacement

_relations.

Elements

that

_{can correctly}

accommodate

these relations

_{are then developed.}

Generally _{a small strain} large _rotation formulation

for _{beam and shell} type _problems is best treated by

using a frame indifferent measure of strains. Hence

the _{most accurate approach} is' to develop the theory

using an intrinsic coordinate system. The strains

obtained from such a description of motion are the

Green-Lagrange _{strains. In a displacement} finite

element model for generally curved thin beam and shell

type _{elements such a coordinate system can be defined}

accurately using the constraint technique. In the

constraint'technique _geometric relations are used to

correctly define the _shape of the element and the

variation _of the local _variables. In fact the technique

can be used to develop completely conforming elements

if _required.

A further _{point of importance is the definition of} the

stress-strain _{relations to be used in the virtual work}

expression in _{a Total} Lagrangian friLme _{of reference.}

This _{is necessary when attempting to obtain the same}

results for both _{stresses and displacements using either}

the Total _{Lagrangian or Updated Lagrangian} formulation.

Taking _{into account the added numerical effort required}

to calculate the _shape function _array for _elements based

on the _constraint technique, the Total Lagrangian

formulation _{will be numerically more effective than the}

Updated Lagrangian formulation. The displacements _of

a structure, _which have been obtained _{using an objective}

theory _{as a basis} for the formulation, _{will be exactly}

the _{same using either} formulation. For _{the calculation}

of stresses, however, for large rotation large curvature

problems _{a difference} in the _{response obtained using}

either formulation _{can be expected unless appropriate}

material constants are used. Hence, a transformation

of the _{modulus matrix will} be necessary _{when a Total}

Lagrangian formulation is _{used. To avoid this and at}

the _{same time reduce the computer time required} for

solution, _{a combined} Total _plus Updated Lagrangian

formulation _{can be used.}

Alternatively,

_{the conventional}

_{or geometric}

_measure

of strains

_{can be used since}

_{the stresses}

_calculated

using

these

_measures

_{of strain}

_{can be assumed to be}

equal

to the true

_stresses

if

_{the strains}

_{are small,.}

Thus the nominal

_stress

_{and the conventional}

_strain

_can

be used as work conjugate

_variables

_{in the virtual}

_work

equation.

Such an approach

_{is equivalent}

_{to assuming}

_a

strain

_energy

function

_{in terms of the Green's'deforma-}

tion

_tensor

_{and satisfies}

_{the requirement}

_{of frame}

indifference

_(37).

_{For beam and shell}

_type

_elements

this

_{approach. takes}

_into

_account

_{the effect}

_{of curvature}

on the axial

force

thus resulting

in true

_axial

force

values.

The approach

is general,

however,

_{and may be}

used in continuum

_mechanics

based formulations

(see

Appendix

_II).

The objective ofthis thesis is:

1. To develop _{a small strain} large _{rotation exact}

theory for _curved two-dimensional beam type

structures in Total, Updated and Combined Lagrangian

frames _{of reference based on the} Green _strain

measure.

2. To develop _{a general} incremental formulation for

geometric

nonlinearity

based on the conventional

definition

_{of strains}

_{as the change in length}

per unit

initial

length

and the change in the right

angle,

and to show the capability

of this

theory

to

evaluate

correctly

the stresses

using

the Total

Lagrangian

formulation.

3. _{To demonstrate the possibility of obtaining non-}

conforming and exactly conforming curved elements

using the constraint technique.

4. _{- To examine} the _{effectiveness of} these _elements

in _correctly dealing _{with the objective-theory}

developed _{with a special reference to the convergence}

requirements.

5. _{To present comparisons between the Lagrangian}

formulations _{developed and hence decide on the}

numerical _{effectiveness of either} formulation.

6. _{To examine} the _{possibility of extending} the _small

strain large rotation theory to three dimensions

for _{thin curved beam elements} including _material

nonlinearity and hence develop thin generally

durved _{eccentric three-dimensional beam elements}

using the constraint technique. Such elements

would be suitable for use as stiffeners.

Outline _{of Thesis:}

Sections 1.3 to 1.5 _{of this chapter give a brief general}

account of the conditions and difficulties encountered

in developing

_amllstrain

large

_rotation

finite

_element

formulations _{based on beam type elements and suitable}

for _{use in the analysis of stiffened shell construct_4ons.}

The remedies suggested in the literature to overcome

these difficulties _{are also outlined.} In Section 1.6

a summary of the mathematical formulation based on the

Lagrangian description _{of motion} is _given.

Chapter _{2 presents the exact two-dimensional large}

rotation theory for thin _curved beam elements. Incremental

equilibrium _{equations which are} based _{on four alternative}

descriptions _{of deformation are given and these are:}

i. _{Total Lagrangian formulation based on the Green}

strain _measure (TLG).

Updated Lagrangian formulation based _{on the} Green

strain measure (ULG).

Combined Total _{and Updated} Lagrangian formulation

based _{on the} Green-strain _measure (UTLO).

iv. Total Lagrangian formulation based _{on the}

conventional strain measure (TLC).

Chapter

3 deals

_with

the development

_{of curved}

two-

dimensional

_{beam elements}

_which

_{can represent}

the theory

presented

in Chapter

2.

The elements

are formulated

using

the constraint

technique.

Chapter _{4 examines} the _{convergence requirements,} gives

an assessment of the performance of the elements and

demonstrates the _{possibility of obtaining exactly} the

same results for both displacements and stresses, using

the _alternative descriptions _of deformation. The results

obtained are compared with published results.

Chapter _{5 presents} the three-dimensional large _rotation

theory _which is _{an extension of} the two-dimensional

theory. Neglecting distorsion, the theory _exactly

represents the deformation of a beam of eccentric solid

rectangular cross-section. Material nonlinearity based

on the von-Mises yield criterion and the Prandtl-Reuss

flow _{rule with isotropic hardening is also included in}

the formulation, For beams under torsion, however, the

material nonlinearity presented is _{only correct} for

concentric square cross-sections.

Chapter _{6 presents the} formulation _{and application of}

the-three-dimensional thin _{eccentric beam elements.}

The elements are developed using the constraint technique.

The applications presented are intended to show the

excellent capabilities of the theory _{and elements} which

have been developed.

Chapter _{7 presents general} discussions _{and conclusions}

with

some suggestions

for

possible

future

_work.

One of the advantages of the finite _{element method} is

that _{once a general purpose program comprising input.,}

assembly _{and solution} facilities is _written, it becomes

a relatively _simple task to add new elements. One such

very efficient _{general purpose} finite _{element program}

is _{LUSAS (38,39). The elements and theory developed}

were implemented

in the LUSAS system.

In this

_presenta-

tion, however, _{attention is focused on operations to be}

performed _{on a single element.}

1.3 LARGE DEFORMATION THEORY

The nonlinear equilibrium equations for almost any

structure can be obtained using a variational principle

and such an approach is basic to the finite element

formulation _C10,11). Mainly _{there are three} types _of

finite _{element formulations:}

a) The displacement _method: based _{on minimizing} the

total _{potential energy where the} displacements

are the unknowns.

b)

_{The equilibrium}

_method:

_{based on minimizing}

_the

complementary

_energy

_{where the stresses}

are the

unknowns.

c) Mixed finite _{element models:} based _{on the} Riessener

variational _{principle where} both displacements

and stresses constitute the unknowns.

The displacement _{finite element method is the most}

widely used because _of its _{simplicity and sound} theoretical

basis. _{The virtual work principle, which is suitable}

for both linear _{and nonlinear problems, is the basis of}

the finite _{element displacement formulation (10,11).}

1.3.1 Geometric Nonlinearity

The virtual work principle can either be formulated in

terms _{of variables} in _the deformed _{configuration or in}

terms _{of variables referred to the undeformed}

configuration. For the first _approach the _work conjugate,

variables are the true Cauchy _{stresses and the} strains

referred to a spatial _{coordinate system and measured}

in the deformed _{configuration. For the second approach}

the work conjugate variables are the second Piola-

Kirchhoff _{stresses and the Green-Lagrange strains}

referred to referential or material coordinates in the

undeformed configuration (37). These alternative

descriptions _{of deformation are termed Eulerian or}

Lagrangian _{respectively. The Kirchhoff stresses and}

the Cauchy _{stresses are related by the ratio of the}

volume in the. reference _state to the _{ratio of} the _volume

in _{the current state (12,40). Thus, for small strains}

these _{stress measures are equal.}

1.3.2

_{The Eulerian}

_Formulation

The virtual _{work principle} is _written in _{terms of the}

unknown deformed _{configuration} for the Eulerian

formulation. _{Thus a solution is only possible by}

assuming that the _{reference configuration} instantaneously

coincides _with the _{current state.} McMeeking _{and Rice}

(401 _{have presented an Eulerian formulation based on}

Hill's _{variational principle. The special feature of}

their formulation is _{the additional term in the} initial

stress stiffness matrix. The term is _{a function} of the

strain matrix _{and the} initial Cauchy _{stresses and results}

from _{relating the nominal stress to a spin-invariant}

stress rate. The authors _{also point out,} however, the

possibility _{of obtaining} the _{same results when work}

conjugate _{measures of stress and strain are used} in the

virtual work equation.

1.3.3 _{The Lagrangian Formulation}

The Lagrangian _{approach is more suitable for use in}

continuum _{mechanics since one can always assume} the

existence of _{an undeformed configuration} for _many

structures. An important _point to _note here, however,

is _{that the equilibrium equations while written in the}

undeformed _{configuration must be satisfied} in the deformed

configuration C371. This implies that the _{stresses are}

measured _relative to material _coordinates in the deformed

configuration. Also, their directions _{are always normal}

to the _{cross-sectional areas in the deformed configuration,}

to which the _{respective axes that the stresses are}

referred to _{are normal} in the _{undeformed configuration.}

Hence, _{if a stress-strain relationship is to be writteny}

either the _{stresses, which are measured} in _{the deformed}

configuration, _must be referred back _{to the undeformed}

configuration _{or the strains must be referred to the}

deformed _{configuration so that both tensors in the}

virtual work principle are referred to the same axes.

This fact is basic to _all the difficulties in the geometric

nonlinear formulations presented in the literature.

Thus, in the Lagrangian description _of deformation two

alternatives are possible in order to relate the work

conjugate stresses and strains. In the first, which

is known _{as the} Total Lagrangian _approach the _stresses

must be referred back to the undeformed configuration.

This _requires the'use _{of appropriate material constants.}

For finite _elasticity Oden (12,41) _{has used a strain}

energy function to define the stress-strain relations.

Hibbitt _{et al} (42) have _given the transformation _necessary

for _{the constitutive relations} for _{metal structures.} This

is _{obtained by adopting a linear relation} between true

stress and strain increments which is necessary because

the _{measure of stress and strain} increments _must be

independent _{of the current rate of rigid body motion}

when using the Prandtl-Reuss flow rule (defined in true

stress vs logarithmic strain) together with the von Mises

yield criterion (43). Hence, the transformation of the

modulus matrix is necessary in _{a Total} Lagrangian

formulation. _{In almost all the Total Lagrangian}

formulations _{presented in literature, however, this}

transformation is _{neglected (10,15,20,44,45).}

The second approach using the Lagrangian formulation

refers the strains to the deformed configuration by

considering that this configuration coincides

instantaneously _with the _{current state} C20,44 _{- 47).}

This _approach, termed the Updated Lagrangian, is

equivalent to the Eulerian formulation. Hence, for

the Updated Lagrangian formulation direct _{proportionality}

between _{the stresses and strains} in the _{virtual work}

principle can be assumed for large rotation problems

without introducing errors.

Thus, _provided that _{appropriate material constants are}

used the results obtained using either the Total

Lagrangian _{or the} Updated Lagrangian _should be identical.

The question, which formulation to be used, now arises.

An answer to this is that _{the. choice of method should}

only depend on the relative numerical effectiveness

and ease of implementation of either method. Clearly

the _computer time _required for _{an Updated} Lagrangian

solution _will be considerably _{more than} that required

for _{a Total Lagrangian solution. This} is basically due

to the _{necessity of recalculating the element shape}

functions _{each time the geometry is updated. Bathe}

and Bolourchi (20) have _concluded that for three-

dimensional _{beam elements the Updated Lagrangian}

formulation _{is numerically more effective than the}

Total Largangian formulation. This is because _their

Total _{Lagrangian formulation requires a transformation}

of the interpolation functions as a consequence of

the definition _of the _axes they _use.

Most _{researchers agree} that the difference between the

Total _{and Updated} Lagrangian _results, if _{any, will}

be due to the _{choice of material constants} (15,20,44,

45). It has been _{previously stated} that the _modulus

matrix transformation required by the Total Lagrangian

formulation is _{always neglected.} This _{approximation} is

valid for small strains and small rotations. For large

rotations and/or plastic deformations, however, such

an approximation is invalid and large errors will result

if _{the transformation} is _{neglected (42). The approxima-}

tion _amounts to _assuming direct _{proportionality} between

the 2nd Piola-Kirchhoff _{stresses and the} Green-Lagrange

strains in the virtual work equations. Thus, neglecting

the transformation, the Total Lagrangian formulation

while giving correct displacement estimations for large

rotation problems will result in the wrong stresses.

This is _{the only serious} disadvantage _of the _general

Total Lagrangian formulation. _{The remedy} to this

problem is _provided in this thesis (see Section 1.3.5).

1.3.4 Large Deformation _{in Beam and Shell} Type

Problems

The continuum mechanics nonlinear Green strain-

displacement _{relations are exact} (37). For beam and

shell type problems approximations, such as the

Kirchhoff hypothesis, _{are used} to describe the deformation

at a general point within an element in terms of the

deformation _{of the neutral surface. This introduces}

curvature terms which _{can be extremely} complex. In fact,

one. of the main difficulties in dealing with beam and

shell type problems is accounting for the effects of

curvature. In a finite element context this difficulty

is _Pronounced for _{elements which} include _{rotations as}

degrees _of freedom in the _{non-conservative nature of}

moments which are referred to fixed axes (48). This

is _{due to the fact that finite rotations referred to}

fixed _{axes are not vectors (37) and hence not commutative}

even for small strain formulations (see Figure 1.1).

Thus, _{as has been stated by Argyris et al (48), the}

strains _must be expressed _{as at least second order}

functions _{in the non-commutative nodal rotations.}

Frey _{and Cescotto C27) and Surana (491 have eliminated}

the _{restriction on rotation magnitude by defining the}

displacement _{field as a nonlinear function of the nodal}

rotations _{and basing} the derivation _of the _element

properties _{on this} displacement field (Figure _1.3).