1993 dynamic investigation of semi-rigid tubular t- joints

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1993

Dynamic investigation of semi-rigid tubular T-jointsIraj HoshyariUniversity of Wollongong

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Recommended CitationHoshyari, Iraj, Dynamic investigation of semi-rigid tubular T-joints, Doctor of Philosophy thesis, Department of Civil and MiningEngineering, University of Wollongong, 1993. http://ro.uow.edu.au/theses/1240

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UNIVERSITY OF WOLLONGONG

DYNAMIC INVESTIGATION OF SEMI-RIGID

TUBULAR T-JOINTS

A thesis submitted in fulfilment of the

requirement for the award of the degree

Doctor of Philosophy

by

Iraj Hoshyari

Department of Civil and Mining Engineering

August 1993

To my family

Declaration

This is to certify that the work presented in this thesis was carried out by the author in the

Department of Civil and Mining Engineering, University of Wollongong, and has not been

submitted to any other university or institute for a degree except where specifically

indicated.

Iraj Hoshyari

i

Abstract

Offshore structures are fabricated from tubes or circular hollow sections. Tubular joints

which are created at the intersection of the circular members experience complicated

structural problems such as stress concentration, ultimate strength, joint stiffness and fatigue

life. S o m e aspects of tubular T-joints regarding joint stiffness and stress concentration are

investigated in this thesis. The methods employed in these investigations are based on

dynamic theoretical and experimental modelling techniques.

Semi-rigidity of tubular joints leads to different results when compared with the c o m m o n

assumption of rigid joint analysis. However, due to the lack of an efficient modelling

method, tubular joints are still assumed to be rigid in most analyses. A method based on the

model of a beam with end springs is used in this study to derive the bending and axial

stiffness of joints. This method can be distinguished from other techniques, as it employs the

natural frequencies rather than static measurements.

A n extensive Finite Element analysis is carried out to establish a series of stiffness equations

for inplane bending, out of plane bending and axial deformation of brace in tubular T-joints.

The proposed parametric equations include diameter to thickness ratio which is not

presently considered in any other formulae. The analysis results show that including

diameter to thickness ratio makes a significant difference in the stiffness of tubular T-joints.

The validity of the Finite Element analyses is investigated by experimentally testing eleven

steel T-joints.

The effect of dynamic loading on strain or stress concentration factors in tubular joints are

also investigated in this thesis. Stress concentration factors are used to calculate the

maximum stress values at the tubular joints. The results show that strain or stress

concentration factor is frequency dependent, when strains at tubular joints are subjected to

dynamic forces. This parameter has not been considered in the other studies of stress

concentration of tubular joints.

Another aspect of tubular joints investigated herein is the effect of joint stiffness

consideration on the fatigue life estimation of offshore towers. Results of the analyses of a

100m tower in this study show an average difference of 3 0 % in fatigue life due to the

flexibility consideration of tubular joints. Further detailed investigation though is required

into the joint stiffness effect on fatigue life estimate of tubular structures.

ii

Acknowledgments

The generosity of the Iranian Ministry of Culture and Higher Education in providing the

author with a scholarship is gratefully acknowledged.

This work could not have been completed without the support of a number of people to

w h o m I am deeply grateful:

Dr Richard Kohoutek, m y supervisor, whose guidance was constructive in maintaining a

proper direction towards the completion of this thesis.

M r Ian Bridge, a professional technician, who fabricated the test specimens and related

hardware.

Mr. Charles Mitchell and Tino Ferrero for their assistance in carrying out the experiments.

Other members of technical staff in the Department of Civil and Mining Engineering

especially M s . Lyn Middleton for proof reading of m y thesis, and Mr. Richard W e b b for his

general assistance in the laboratory.

Mrs. J. Kohoutek for drawing the sketches of the test specimens.

I wish to thank m y wife, Mahbobeh, for her unending patience and understanding through

all the difficulties w e encountered when this work was being carried out.

Iraj Hoshyari,

August 1993

iii

Table of Contents

1 INTRODUCTION

1.1 Preamble 2

1.2 Brief history of offshore platforms 2

1.3 Terminologies used in steel template platforms and tubular joints 4

1.4 W h y joint modelling is important? 8

1.5 Platform tower global analysis 9

1.6 Scope of this thesis 11

1.7 Summary 12

2 LITERATURE SURVEY

2.1 Introduction 14

2.2 General 14

2.3 Tubular joints 15

2.3.1 Analytical methods 15 2.3.2 Experimental and semi-experimental methods 18

2.3.3 Numerical methods 21

2.4 Effects of tubular joint flexibility on tower analysis 25

2.5 Model of joint in stiffness method 37

2.5.1 Bending rigidity factor 39

2.6 Summary 40

3 THEORETICAL MODEL

3.1 Introduction 42

3.1.1 Beam theory 42

3.2 Matrix formulation of beams 43

3.2.1 Beam with semi-rigid ends (bending deformations only) 45

3.2.2 Effect of shear deformations on bending stiffness 48 3.2.3 Stiffness determination of a joint 48

3.3 Bending rigidity factor 50

3.4 Stiffness of a joint due to brace axial loading 52

3.4.1 Axial rigidity factor 54

3.5 Summary.. 55

E X P E R I M E N T A L DETERMINATION O F JOINT STIFFNESS

4.1 Introduction 57

4.2 Test specimens 57

4.2.1 Survey of non-dimensional parameters of T-joints 58 4.2.2 Material properties and fabrication of test specimens 59

4.3 Test set-up 60

4.3.1 Test procedure 62

4.3.2 Test results and discussion 65

4.4 Calculation of rigidity factor and joint stiffness 67

4.5 Effect of joint support conditions 69 4.5.1 Provisions of end supports for test specimens 70

4.6 Summary 72

DETERMINATION OF JOINT RIGD3ITY USING FINITE ELEMENT METHOD

5.1 Introduction 74

5.2 A review of the Finite Element method 74 5.2.1 Galerkin Method (a finite element formulation) 75

5.2.2 Shell Finite Element 77

5.3 Description of the Finite Element Package: A L G O R 78 5.3.1 Shell Element in A L G O R 78 5.3.2 A Bench Mark on A L G O R 79

5.3.3 Comparison of shell and three dimensional element performances 80

5.4 Mesh generation procedure used to model T-joints 81

5.5 Boundary conditions for the FE models 82

5.5.1 Effect of joint support conditions 86

5.6 Finite Element analysis of the tested T-joints 88 5.6.1 Comparison of the Finite Element analysis and test results 89

5.7 Summary 91

RIGDDITY AS FUNCTION OF JOINT GEOMETRICAL PARAMETERS

6.1 Introduction 94

6.2 Finite Element analysis results 94 6.2.1 Inplane bending mode 96

6.2.2 Out of plane bending mode 102

6.2.3 Axial deformation of brace 106

6.3 Derivation of stiffness equations 111

6.4 Comparison with other stiffness equations 114

6.4.1 D N V (1977) 115

6.4.2 UEG(1985) 115

6.4.3 Efthymiou(1985) 116

6.4.4 Fessler(1986) 116

6.4.5 Uedaetal.(1990) 117

6.4.6 Chen B.etal. (1990) 117

V

6.4.7 Kohoutek and Hoshyari (1992) 117 6.4.8 Graphical Comparison of different stiffness formulae 117

6.5 Summary 123

7 DYNAMIC STRESS CONCENTRATION FACTOR

7.1 Introduction 126

7.2 Application of stress concentration factor in fatigue analysis 128

7.3 Methods of SCF determination 131

7.3.1 Experimental methods 132 7.3.2 Interpretation of stress concentration factor from measurements 133

7.3.3 The Finite Element Method (FEM) 134

7.4 Reliability approach for stress concentration factor 135

7.5 Dynamic strain reading 136 7.5.1 Test set-up 138 7.5.2 Test results 139

7.6 Dynamic strain calculation using stiffness method 145

7.7 Dynamic strain reading and SCF/SNCF reliability 147

7.8 Summary 148

8 SEMI-RIGIDITY EFFECTS ON BEHAVIOUR OF OFFSHORE TOWERS

8.1 Introduction 150

8.2 Tower geometry and joint specifications 150

8.3 Stiffness determination of the joints of the tower 152

8.5 Loading 152

8.6 Results 155 8.6.1 Deflections 155 8.6.2 Axial forces 157

8.6.3 Bending moments 158 8.6.4 Dynamic characteristics 160

8.7 Fatigue life estimation 167

8.8 Summary 172

9 SUMMARY AND CONCLUSIONS

9.1 Summary 175

9.2 Conclusions 181

9.3 Future research work 183

vi

REFERENCES 184

APPENDICES 192

Appendix A

Stiffness matrix of beam with semi-rigid ends 193

Appendix B

Stiffness matrix of truss member with semi-rigid ends 198

Appendix C

Mode shapes 1,2, and 3 for the tested T-joint specimens 201

Appendix D

List of sub-programs in A L G O R (used in the analysis of joints) 204

Appendix E

Description of TBC3 and GCS8 finite elements 205

Appendix F

Effect of chord length 207

Appendix G

Parametric SCF Formulae for inplane bending of T-joints 209

vii

List of Figures

1.1 A concrete gravity platform in the North Sea 3

1.2 A tension leg platform 3

1.3 Bullwinkle offshore oil drilling platform 4

1.4 Principal elements of an eight-leg steel template platform 5

1.5 Simple tubular T-joint and the common non-dimensional parameters 6

1.6 Some examples of complex joints 7

1.7 Concrete grouted joint 7

1.8 Example of a joint substructure constructed from shell finite elements 8

2.1 Kellogg's models of a tubular joint 16

2.2 Model of cylindrical vessel used by Bijlaard 16

2.3 Shell element used by Holmas

2.4 Extra D O F to express local joint behaviour used by Holmas 17

2.5 Test rig used by Fessler (1981) 19

2.6 Stress distribution assumed in Punching Shear Model 20

2.7 Model of joint substructure used by Bouwkamp (1980) 22

2.8 Rotations measured by Efthymiou for calculation of joint flexibility (1985) 22

2.9 Joint model proposed by Ueda (1986) 23

2.10 Joint super-element used by Souissi (1990) 24

2.11 The results of Kawashima's analyses on the frames with rigid and semi-rigid joints.. 26

2.12 K-braced frame analysed by Ueda and its load cases 27

2.12 Frame models analysed in U R 2 2 report by U E G (1984) 28

2.13 Nodal points considered in U R 2 2 Study to represent a joint 28

2.14 Predicted buckling load by Jong (1987) 32

2.15 Tower analysed by T. Chen (1990) 33

2.16 The frame analysed by Souissi (1990) 34

2.17 The structures analysed by Recho (1990) 35

2.18 Moment-rotation relationship for a semi-rigid joint 37

2.19 Joint model developed by Kawashima (1984) 38

3.1 DOFs for two dimensional beam element 43

3.2 Stiffness coefficients of beam with continuous mass due to the rotation of left end 44

3.3 Beam with semi-rigid ends (after Kohoutek 1991b) 45

3.4 Dynamic methods to determine rigidity of a joint 49

viii

3.5 B e a m model for definition of rigidity factor (unit length) 50

3.6 Relationship between rotational stiffness and beam end flexibility 51

3.7 Axially loaded member with semi-rigid ends 52

3.8 Relationship between axial stiffness and axial end flexibility 54

4.1 Distribution of f$ parameter for T-joints. The survey was carried out by UEG (1985) on six offshore structures 58

4.2 Distribution of y parameter for T-joints. The survey was carried out by U E G (1985) on six offshore structures 58

4.3 Distribution of y parameter for T-joints. The survey was carried out on the database in U E G (1985) 59

4.4 Distribution of P parameter for T-joints. The survey was carried out on the database in U E G (1985) 59

4.5 Distribution of T parameter for T-joints. The survey was carried out on the database in U E G (1985) 59

4.6 A typical weld cross section of a T-joint specimen 60

4.7 Diagrammatic test set-up for frequency measurement of T- joints (IPB) 61

4.9 Typical frequency spectrum of a tubular T-joint 62

4.10 Lumped mass model used to establish the mode shapes of joints 63

4.11 First two mode shapes of T-joint T10 from Dynamic Modal Analysis 65

4.12 Relationship between natural frequencies of specimens and diameter ratio 66

4.13 Stiffness calculation of a T-joint using measured natural frequency 67

4.14 Analytical model used for rigidity calculation of tested T-joints (TPB) 68

4.15 Support arrangement for inplane bending test 70

4.16 Support arrangement for out of plane bending test 71

5.1 Thin plate and shell element in ALGOR 78

5.2 L C C T 9 element assembly 79

5.3 Geometry and meshes used for cylinder under pressure 79

5.4 Typical mesh patterns used for the analysis of T-joints 81

5.5 Data generation procedure for tubular T-joints 82

5.6 Boundary conditions applied to F E models of the T-joints 83

5.7 T-joint T2, first three mode shapes in IPB mode 88

5.8 Weld cut-back at the saddle on joints with P = 1 90

5.9 Correlation between natural frequencies obtained from F E analysis and experiment. 91

6.1 Geometry of Finite Element models used for flexibility analysis of T-joints 95

6.2 Relationship between P and JPB stiffness 98

6.3 Relationship between y and IPB stiffness 99

ix

6.4 Relationship between x and IPB stiffness 100

6.5 Relationship between P and O P B stiffness 103

6.6 Relationship between y and O P B stiffness 104

6.7 Relationship between x and O P B stiffness 105

6.8 Relationship between P and Axial stiffness 108

6.9 Relationship between y and axial stiffness 109

6.10 Relationship between x and axial stiffness 110

6.11 Predicted values against data used for curve fitting 112

6.12 Model adopted for IPB joint stiffness with the absolute errors of curve fitting 113

6.13 Model adopted for O P B joint stiffness with the absolute errors of curve fitting 113

6.14 Model adopted for joint axial stiffness with the absolute errors of curve fitting 114

6.15 Comparison of different equations for axial joint stiffness 120

6.16 Comparison of different equations for IPB joint stiffness 121

6.17 Comparison of different equations for O P B joint stiffness 123

7.1 Stress distribution at the intersection of a simple T-joint 126

7.2 Stress distribution near weld toe 128

7.3 The geometry and forces of the T-joint considered in Example 1 128

7.4 Hot-spot definition for T and DT-joints recommended by E C S C 134

7.5a Variations of S C F for direction of axial loading 136

7.5b Variations of SCF for nominally identical joints 136

7.6 A typical strain gauge location diagram of a test specimen 137

7.7 Diagrammatic test set-up for the dynamic strain measurement 138

7.8 Dynamic strain readings for joint T10 139

7.9 Relationships between strain ratio and load frequency for joints with different p

ratios (IPB) 141

7.10 Relationships between strain ratio and load frequency for joints with different p ratios (OPB) 144

7.11 Beam model to study dynamic effects on strain ratio 146

7.12 Strain ratios resulted from beam models and test models for joints with a) P=l .0,

b) P=0.52, c) P=0.28 146

8.1 Geometry of the tower analysed 151

8.2 W a v e definition for load cases 2,3 and 4 153

8.3 W a v e forces calculated for load cases 2, 3, and 4 155

8.4 Deflected shape of CP1 and M P 2 frames for load case 1 156

8.5 Axial forces for load cases 2, 3 and 4 and their variations between flexible joint analysis and conventional analysis (MP2-CP1) 157

X

8.6

a) Bending moment diagram for frame M P 2 (Load Case 4) 158 b) Bending moment difference between M P 2 and CP1 (MP2-CP1) 158

8.7 Relationship between stiffness and frequency for a) CP1 and b) M P 2 towers 161

8.8 Mode shape 1 of towers CP1 and M P 2 163









8.17 Typical relationship between wave period and stress per wave height 168

8.18 Stress range per wave height of different wave periods for joints a) 5 and 6 and b) 7 and 8 169

xi

List of Tables

2.1a Kawashima's results for L-type frame 26

2.1b Kawashima's results for portal frame 26

2.2 Joint specification in Ueda's analyses 27

2.2 Joint parameters used in U R 2 2 Study 29

2.3 Summary of changes from U E G report on joint flexibility 30

2.4 Effect of flexibility consideration in the analysis, Chen T. (1990) 33

2.5 Effect of joint flexibility on internal forces by Recho (1990) 35

2.6 Fatigue life difference (NR/NF) when joint flexibility is considered 35

2.7 M a x i m u m response change for models with rigid joints obtained in various research works when joint flexibility was considered 36

4.1 Geometrical dimensions and non-dimensional parameters of tested joints 57

4.2 Metallurgical properties of the pipe sections used for test specimens 60

4.3 Natural frequencies of the T-joints using lumped mass theory (TPB) 64

4.4 Natural frequencies of the T-joints using Dynamic Deformation Method (IPB) 64

4.5 Natural frequency of the T-joints due to cantilever mode of vibration 66

4.6 Measured natural frequencies (co), rigidity factor (vb), and stiffness (k) of

the tested T-joints 68

4.7 Joint stiffness calculated for hinged and fixed analytical models using test results of specimens that have hinged supports 69

5.1 Results of different types of shell elements - cylinder under pressure 80

5.2 Comparison of shell and three dimensional solid elements 81

5.3 Application of internal boundary conditions to the F E model of T-joint T10 (IPB).. 84

5.4 Calculated results for the F E models of the T-joints with different support conditions (IPB) 87

5.5 Results of F E analyses and experiments 89

6.1 Non-dimensional parameters used for FE models of T-joints 95

6.2 Identification names and individual joint parameters of F E models 96

6.3a Calculated natural frequencies and stiffness of T-joint F E models, IPB (t = 0.2) 97

6.3b Calculated natural frequencies and stiffness of T-joint F E models, IPB (t = 0.5) 97

6.3c Calculated natural frequencies and stiffness of T-joint F E models, IPB (T = 1.0) 97

6.4a Calculated natural frequencies and stiffness of the T-joint F E models, OPB (T = 0.2) 102

6.4b Calculated natural frequencies and stiffness of the T-joint F E models, OPB (t = 0.5) 102

6.4c Calculated natural frequencies and stiffness of the T-joint F E models, O P B ( T =1.0). 102

xii

6.5a Calculated natural frequencies and stiffness of the T-joint F E models, axial

deformation (t = 0.2) 106

6.5b Calculated natural frequencies and stiffness of the T-joint F E models, axial

deformation (t = 0.5) 107

6.5c Calculated natural frequencies and stiffness of the T-joint F E models, axial deformation (t = 1.0) 107

6.6 T-joint stiffness formulae and the corresponding correlation factors Ill

6.7a Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 calculated using the formulae by others 118

6.7b Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 measured by others 118

6.7c Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 calculated by the formulae developed in this study 119

6.8 Comparison of the results of proposed equation with Kohoutek's equation (IPB).. 122

7.1 Hot-spot Stress Concentration Factors for the joint of Example 1 129

7.2 No. of occurrences for different wave heights used in the fatigue example 130

7.3 Nominal stress ranges in brace member for different deformation modes 130

7.4 Hot spot stress ranges for the T-joint in fatigue example 130

7.2 Dynamic strain readings and SNCFs for IPB mode 140

7.3 Dynamic strain readings and SNCFs for O P B mode 143

8.1 Geometrical properties of joints used in the analysed towers 151

8.2 Natural frequency and stiffness of the T-joint in M P 2 tower 152

8.3 Details of the load cases in the analysis of towers 153

8.4 Linear (Airy) Wave Theory equations 154

8.5 Deflection changes when joint flexibility is incorporated in analysis 156

8.6 Axial forces, their changes and axial stress changes for load case 4 158

8.7 Bending moment amplitude changes between M P 2 and CP1 (MP2-CP1) 159

8.8 Natural frequencies of towers CP1 and M P 2 (Hz) 161

8.9 Individual wave occurrences 'n' in a sea state of Hs=3m, T^= 7 sec, normalised to 1 year 168

8.10 Fatigue damage of joint 5 per year for each wave height and wave period 170




8.14 Fatigue life predictions and their changes when joint stiffness is considered in analysis 172

Nomenclature

fli coefficient of displacement vector

A sectional area

As shear area

C vector of integration constants

C\, C2, C3, C4.... integration constants

d brace diameter

D chord diameter

d' distance between weld toes at the saddle of a joint

Ds damage summation ratio

E modulus of elasticity

F. degree of fixity

g acceleration due to gravity

fa stress due to axial loading

fiPB stress due to inplane bending

fopB stress due to out of plane bending

h longitudinal displacement h(x,t)

I. moment of inertia

K global stiffness matrix

k local stiffness matrix of an element

ky member of a stiffness matrix

k'iQ-) stiffness of an axial spring at end /(/)

£,(,) stiffness of a rotational spring at end /(/)

L chord length

L differential operator

/ brace length, length, direction cosine

M bending moment

Mam added mass

m mass per unit length, direction cosine

M c continuous mass matrix

M L lumped mass matrix

n direction cosine, number of cycles

Af number of cycles before fatigue failure, shape function

Nt interpolation functions

NF the number of cycles to failure when connection is considered flexible

NR the number of cycles to failure when connection is considered rigid

xiv

P axial force

P force vector

Q matrix that converts end forces P to integration constants C

R discrepancy ratio

p water density

t brace wall thickness, time

T. chord wall thickness

U water particle velocity

U water particle velocity

u longitudinal displacement u(x), general displacement function

u approximated displacement function

Ub displacement function at the boundaries

V shear force

v transverse displacement v(x,f)

X body force per unit volume in x direction

x displacement vector

X surface force per unit area in x direction

Y. body force per unit volume in y direction

y transverse displacement y(x)

ys displacement due to the shear deformation

Y surface force per unit area in y direction

Z body force per unit volume in z direction

Z1 surface force per unit area in z direction

a non dimensional parameter 2L/D

ay) bending flexibility of beam end i(j) (a, = El/kil)

P non dimensional parameter dID

P' non dimensional parameter d'lD

P,(,) axial flexibility of beam end i(j) (P, = El/k'f)

A displacement vector

8 transverse displacement

<I> matrix that converts end displacements A to integration constants C

T rigidity factor, partial factor to give the appropriate level of confidence

y non dimensional parameter DI2T

r) longitudinal displacement

va axial rigidity factor

Vb bending rigidity factor

8 rotation, angle of diagonal brace in a Y-joint

p mass per unit volume

XV

Ox,y,z normal stress

x \JT

ixy,yz,zx shear stress

u) frequency

CA ith natural frequency

A / stress range

List of Abbreviations

AD Analog to Digital

DDM Dynamic Deformation Method

Det Determinant

DOF Degree Of Freedom

DSCF Design Stress Concentration Factor

ECSC European Community of Steel and Coal

FE Finite Element

FEM Finite Element Method

FS Factor of Safety

HS Hot Spot

IPB InPlane Bending

NF Natural Frequency

OPB Out of Plane Bending

PC Personal Computer

S-N Stress-Number of cycle curve for fatigue estimation

SCF Stress Concentration Factor

SNCF Strain Concentration Factor

U K DEn United Kingdom Department of Energy

UKOSRP United Kingdom Offshore Steels Research Program

Chapter 1

INTRODUCTION

Chapter 1: Introduction £

1.1 Preamble

The first oil drilling platforms in sea were built of wood, with much smaller dimensions than

their descendants which are almost the biggest structures on the earth. U p to the 1990s,

three major categories of fixed offshore platforms were developed: concrete gravity, tension

leg and steel template platforms. Approximately 98 percent of the existing platforms are of

the template type with heights varying from a few meters to more than 400 meters.

Concrete gravity platforms gain their resistance through their very large weight; they have a

heavy base, holding them in place against lateral loads. This type of platform is suitable for

regions with soft soil seabeds. Figure 1.1 shows a concrete gravity platform installed in the

North Sea. A tension leg platform is also shown in Figure 1.2. Tension leg platforms, which

are placed in the compliant category of offshore platforms have a flexible response to the

lateral loads. They adjust themselves horizontally with the water motion, and their

movements are controlled by mooring systems, which typically consist of chains, cables,

ropes and anchors (Demirbilek, 1989).

Template platforms are explained in Section 1.3.

1.2 Brief history of offshore platforms

In the 1890s, man first exploited oil from an offshore field on the coast of California, U S A .

Access to the oil well was through piers extending from shore. By 1900 eleven piers were

constructed in that area and drilling was being carried out in water 150m from the shoreline.

Offshore drilling continued through wooden platforms within sight of land in shallow

waters. In the 1920s timber platforms were erected in an oilfield off Venezuela.

After the Second World W a r with the development of technology, deeper water was

searched for oil and taller platforms were fabricated. The first steel platforms were

constructed in the Gulf of Mexico during 1947 and there were two platforms constructed in

that year which became the design standard for many years (Graff, 1981). One was located

29km offshore in 6 m of water. The total platform size was 53m by 33m. It had 268 steel

piles driven through the jacket legs. The other steel platform had a smaller deck area of

250m 2 and was placed in 5.5m of water offshore. Structural members of the tower had pipe

sections which showed less resistance to the forces generated by water. Pile type

foundations were used for transferring the loads to the supports. The pipes were driven

Chapter 1: Introduction 3

Figure 1.1. A concrete gravity platform in the North Sea Figure 1.2. A tension leg platform

through the main legs to a sufficient depth in the seabed and connected to the deck at the

top. A s a result, fixed steel offshore platforms are also called template platforms.

With the rapid growth of industry and its vital need to the oil products, fabrication of fixed

platforms continued and deeper waters were targeted. In September 1988, Shell Offshore

Incorporation installed Bullwinkle in the Green Canyon area, Gulf of Mexico. This platform,

which is 490 meter high, has the tallest tower in the world. Its design, fabrication and

installation took five years. Seventy thousand tonnes of steel were used to construct the

Bullwinkle platform (Sterling, 1989). Figure 1.3 shows the Bullwinkle oil drilling platform

tower.


<J> (sXD

® <a> ® ,<p? ?• £ I 140'' 2 I WL45- 9"-J>

®

LAN RT +15' \§) PLRN RT -370'

30'-

s rr

Q> T I ® 4 19' 3 "

140' 140' 140'

1 1 -30'

/-SZ\ \ /

ROW "RJ.B PLRN RT -1350'

Figure 1.3. Bullwinkle offshore oil drilling platform (after Digre, 1989)

1.3 Terminologies used in steel template platforms and tubular joints

A steel template platform consists of three main sections: 1) superstructure, 2) jacket, and

3) piles. The different components of an eight leg template platform is shown in Figure 1.4.

The jacket is a tubular steel structure sitting on the seabed and supporting the deck. It

surrounds the piles and holds them in position from the mudline to the deck; supports

conductors, pumps, sumps, risers, etc. and hence is called a jacket. The superstructure or

deck which is placed on top of the jacket provides required operational space. The deck sits

on girders and trusses, and eventually on the piles which extend from beneath the

superstructure into the seabed through the jacket legs.

The structural members that secure the jacket legs horizontally are called braces or

branches. Where one or more braces meet a leg, a joint is created. Figure 1.5 shows a

typical simple T-joint with commonly used notations and non-dimensional parameters.


skid beams

barge bumper

horizontal brace

vertical diagonal

jacket leg can

bracing stub

upper or drilling deck

wind girders

longittudinal truss

skirt pilre sleeve

launch truss

Figure 1.4. Principal elements of an eight-leg steel template platform (after Graff, 1981)

Chapter 1: Introduction

Brace or Branch

l r

Saddle -H D M -

P = 75 = _D

Y 27 /

a = 21

D

Figure 1.5. Simple tubular T-joint and the common non-dimensional parameters

Tubular joints are classified into four categories as follows (UEG, 1985):

1) simple welded joints,

2) complex welded joints, 3) cast steel joints, and

4) composite joints.

This classification is not strict, but is widely accepted. Simple joints are those without any

stiffener, gusset plate, diaphragm or grout. In multi braced simple joints the braces do not overlap. To strengthen a simple joint, the chord section is usually thickened in the connection zone. This section with higher strength is called joint can (see Figure 1.4).

The term complex is assigned to the following joints:

1) joints with uniplanar or multiplanar overlapping brace members,

2) joints with internal stiffeners or diaphragms, and

3) joints with external stiffeners.


Complex joints have higher strength and stiffness than simple joints. Some examples of

complex joints are shown in Figure 1.6.

ViewA-A

Figure 1.6. Some examples of complex joints (UEG, 1985)

Cast joints are made by a casting process in which the brace and chord are cast together.

Therefore, there is no welded connection between the brace and chord, creating better

shapes of fillet. Cast joints are potentially stronger than the welded joints (Edwards and

Fessler, 1985).

Composite joints are those filled fully, or partially between the leg and the pile passing

through the leg, with concrete. A double-skin grout reinforced joint which is placed in the

latter group of grouted joints is shown in Figure 1.7.


Pile centre m a y also be grout filled

Figure 1.7. Concrete grouted joint

1.4 W h y joint modelling is important?

A s explained in Section 1.2 the most suitable structural sections for jacket structures are

circular tubes. The best way to connect the tubes is direct welding creating tubular joints,

one of the most critical subjects in the analysis and design of an offshore platform. The

diameter to thickness ratios of the tubes at a joint zone (DIT) are generally between 25 and

60, which pronounces the shell characteristics in the structural behaviour of the joint.

To determine the stress resultants at a joint, nominal stress values are calculated through a

structural analysis, where the structural model of a jacket platform is usually a space frame,

or in simple cases a two-dimensional model. In this regard, it is c o m m o n in practice that

the connection of tubular members be considered as rigid. This means that the angle

between members does not change after the structure is gone under loading. However, a

better estimation to the internal forces in a jacket structure can be obtained by

incorporating the flexibility of joints in the analysis. The significance of stress distribution

at tubular joints is in connection with fatigue life. Because the internal forces are generated

under wind, waves, currents and vortices, stress fluctuation and ultimately fatigue failure

m a y be developed in the structural members of offshore platforms. B y considering the joint

stiffness, a designer can calculate the stress resultants at the joints more realistically and

consequently make a better estimate of the fatigue life. In addition to stress evaluation at

the joints, the deflected shape of a structure is calculated more accurately by employing the

flexibility of the joints. This is especially important for tall platforms. More accurate

estimation of the dynamic behaviour of tower frames is another feature of the joint

flexibility consideration. Furthermore, applying the joint flexibility provides more realistic

data for buckling design of compression members.


1.5 Platform tower global analysis

Presently, structural analyses are performed on computers and stiffness method based

programs are used. Regarding joint modelling in computer structural analysis, three basic

methods have been identified (Tebbett, 1982). They are:

1) addition of simple springs in line with brace,

2) entering of an effective length for the brace, and

3) use of a substructure of three dimensional finite elements for the tubular joints.

Tebbett labels the first alternative as the easiest and the third one as the most accurate.

Using the second alternative requires more attention to the length of the members, not to

invalidate wave loading and member stability checks. Figure 1.8 shows an example of the

third alternative, which uses a F E substructuring scheme.

Figure 1.8. Example of a joint substructure constructed from shell finite elements

The same methods as Tebbett identified are still used in studies where in essence most use

static Finite Element analysis as base. There has been a number of studies on the stiffness

or flexibility of tubular joints since the late 1970's. The c o m m o n limitation of these studies

is the lack of experimental backup supporting the results, which is indeed due to the

difficulties involved with the measurements of static deformations. In this regard, Fessler et

al. (1981) has carried out the only conclusive experimental study into the stiffness of

tubular joints and is described in Chapter 2.

Tebbett and Lalani (1986) has considered the local joint flexibility as an important

parameter in structural analysis, especially in the reassessment of offshore structures.

However, they have suggested that a consistent method should be used for flexibility

modelling of all joints in a structure.


Lack of efficient methods of structural modelling has caused the consideration of rigid

attachment between the beam members in tower global analyses (Barltrop, 1991). This

assumption introduces two important features in regard to the structural analysis

procedures such as the stiffness method. One feature is the simplicity of force-displacement

relationship used for the beam members when the joints are assumed rigid. Using the

c o m m o n matrix equation of kx = f for uniform beams, which includes a series of simple

relationships between moment of inertia, Modulus of Elasticity and beam geometry, saves

a lot of computation time on computer. More complicated relationships are required when

joint stiffness is implemented in the model of a structure. Another feature for using rigid

joints is that the required data is minimal for structural analysis. Extra data is needed when

joint stiffness is considered in the analysis.

Because of the difficulties due to the joint modelling in structural analysis, standard codes

for design and analysis of offshore structures have not addressed specific recommendations

for including the joint stiffness in the global analysis. Design codes nearly all have made

general comments on using accurate structural analysis methods.

The American Design Code API-RP2A (19th edition, 1991) states that: "Brace axial loads

and bending moments essential to the integrity of the structure should be included in the

calculation of acting punching shear."

The British Code of Standard B S 6325 (1982) gives no guidance for global analysis (UEG,

1985). Both API and B S codes allow the maximum eccentricity of D/4 for multi-braced

joints.

The Norwegian Code of Standard, D N V , requires that "Any local flexibility in the

connection of a member to a joint, which is of importance for the force distribution in the

structure should be accounted for in the stiffness or flexibility matrix of the total structure."

D N V requires the consideration of eccentricity in the analysis and recommends two

formulae, stated in Section 6.4.1, to determine the spring stiffness of T-joints for inplane

and out of plane bending, respectively.

The significance and necessity of conducting research on finding accurate, applicable and

simple methods to incorporate the stiffness of joints in the analysis of structures, especially

offshore platforms, can be observed in the practice. In this thesis a new method (Kohoutek,

1985a) which benefits from the dynamic behaviour of structures is examined to model the

joint stiffness of tubular T-joints. The study therefore has focused on the simplified models

of joints to avoid uncertainties due to complicated modelling.


1.6 Scope of this thesis

This study attempts to investigate two structural aspects of tubular joints using dynamic

methods of analysis in theory and experiment. These aspects are joint stiffness (flexibility)

and stress concentration. Joint stiffness is calculated based on a dynamic method using the

measured natural frequencies. According to the author's knowledge, this method is

examined for the first time on the tubular joints in this thesis. Using the same method, a

parametric study into the stiffness of tubular joints is also conducted based on the results of

Finite Element analysis. Finally the effects of joint stiffness on the structural behaviour and

especially fatigue life of an offshore tower is investigated.

Only tubular T-joints are studied here because of the time limit, but the method can be used

for other types of joints. The joints are assumed to have linear material properties, since

many fixed offshore structures respond linearly to the important fatiguing waves (Barltrop

and Adams, 1991).

Another feature of tubular joints, which is the focus of this study is the dynamic strain

measurements of tubular joints, as a means to determine stress concentration factor, and the

errors that can be made by performing static tests instead of dynamic tests. Determination

of SCFs are usually carried out through static tests, where the measurements in this study

show that there are some changes when dynamic loading is considered. It is shown here

that those changes are as significant, if not more, as those created by other parameters that

have been studied before.

The Objectives of this study are:

1) applying a new dynamic method to determine the stiffness of tubular joints

experimentally and analytically,

2) establishing a set of parametric formulae for stiffness of tubular T-joints,

3) investigation of dynamic loading effects on measurements of stress or strain

concentration factors in tubular joints, and

4) investigation of joint stiffness consideration on the structural behaviour and

fatigue life of tubular joints in offshore towers.

This thesis examines the structural performance of tubular joints as a crucial component in

the whole platform structure and is not concerned with the load calculation on Offshore

platforms. It assumes that only loading varies with time and therefore causes stress fluctu

ation in the structural members in time.


1.7 Summary

In this chapter a general introduction was given to the different types of offshore platforms

and the terminologies used to identify tubular joints. Tubular joints are the most critical

structural element in offshore structures and modelled to be rigid in today structural

analysis practice. The main reason for this assumption is the lack of an efficient method of

analysis for considering the joint stiffness. D u e to the same reason, the codes of standards

for offshore structures do not generally address any method to include the flexibility of

joints in the analysis.

The main reason that tubular joints impose special attention in the design and analysis of

offshore structures is fatigue problem. This thesis attempt to improve the estimation

techniques for fatigue life of tubular joints through establishing more realistic methods of

analysis.

Chapter 2: Literature Survey 14

2.1 Introduction

Extensive work has been carried out on tubular joints with the trend towards understanding

the fatigue behaviour and providing analysis methods to estimate the fatigue life of joints.

The major contribution has been made by oil companies in U S A and Europe to improve

the design, construction and maintenance procedures applied to offshore structures. With

regard to the stiffness behaviour of tubular joints, however, little published material is

available. Although the study on rigidity of open section joints goes back to the 1910s, the

topic of tubular joints was only attended to in the 1970s.

This chapter begins with the study of semi-rigidity of joints, in general. The rest of the

chapter is specifically about tubular joints, where results of other studies are reported and

discussed.

2.2 General

The importance of the end restraint provided by semi-rigid joints was realised over seventy

years ago. Wilson and Moore (1917) first investigated the flexibility of riveted structural

joints in 1917. Research workers in Britain (SSRC, 1931, 1934,1936), Canada (Young and

Jackson, 1934) and the United States (Rathbun, 1936), in three separate investigations

during the 1930s, measured the relations between end moment and relative angle changes

at beam to column connections in an attempt to provide data for semi-rigid joint design.

Since then numerous tests on riveted, bolted and welded joints have been reported.

Batho and Rowan (SSRC, 1931) proposed a graphical method for predicting the end

restraint provided by a connection for which the experimentally obtained moment-rotation

relationship was known. Young and Jackson (1934) investigated 1) the rotational capacity

and end restraint provided by joints to reduce the beam moment due to gravity loads and

also 2) the ability of joints in frames to resist horizontal deflections due to lateral or wind

loads. Methods of incorporating semi-rigid end restraint into slope-deflection and moment-

distribution methods were proposed by both Baker (SSRC, 1931) and Rathbun (1936)

independently.

From the early investigations into joint behaviour some possible economies were realised.

According to British investigations (SSRC, 1934), savings of as much as 20 per cent could

be achieved on the design of beams in frames by taking advantage of end restraint.

Chapter 2: Literature Survey _£

2.3 Tubular joints

Because this thesis is directly related to tubular joints, the literature survey has mostly

concentrated on this type of joints. It should be noted here that only static methods were

used in most of the reviewed articles and references. A m o n g them only the work by

Springfield and Brunair (1989) is based on a dynamic method of analyses.

Investigations on tubular joints are more concerned with stress and strain concentration

than flexibility analysis, nevertheless the same theory and models apply to both cases. The

methods used to study the behaviour of tubular joints can be categorised as:

1) analytical methods,

2) experimental and semi-experimental methods, and

3) numerical methods.

Each of the above methods has some advantages and deficiencies, however experimental

techniques can produce the most accurate results provided the test set-up is made according

to the assumptions adopted for the tests. Experimental accuracy and realisation of actual

conditions are very important for interpreting the experimental results. Analytical methods

based on plate and shell theory become very complicated when dealing with tubular joints.

They can, however, produce fast and relatively accurate results where applicable.

Numerical methods are those procedures which attempt to reach the solution of a problem

by somehow discretizing the domain of the function being studied. It is tried here to

differentiate between an analytical and a numerical method. Analytical procedures are

based on the theories of continuum mechanics and aim at the exact solution. However,

numerical methods approximate the exact solution. The Finite Element method is one of

the most powerful numerical methods for studying the behaviour of structures. However,

the complicated behaviour of tubular joints creates some inaccuracy and difficulty when

the Finite Element method is applied to the joints.

2.3.1 Analytical methods

Because of the complicated geometry of tubular joints, realistic analytical solutions to the

deformation or stress problems at the intersection of the tubes have not become available.

Investigators have tried to solve the problem by applying simplifying assumptions for

various loading cases.

Kellogg (1956) replaced the brace load with an equivalent distributed load shown in Figure

2.1. Based on the theory of beam on an elastic foundation, Kellogg derived the maximum

stress under the equivalent load. This method considers only axial load and/or inplane

bending moment on the brace. It gives approximate stress values for the chord and does not

have any reference to the brace (UEG, 1985).


Figure 2.1. Kellogg's models of a tubular joint

Another example of this type of analysis is Bijlaard's method (1955) which used a double

Fourier series to show the displacement field of a cylinder subjected to a rectangular

distributed load. Although the moment and deflections were computed for point O in the

model shown in Figure 2.2, equations were introduced for obtaining the moments at the

edges of the loaded area. The method needs to take into account a large number of terms in

the Fourier series to give a relatively accurate result. For example Rodabaugh (1980) used

21 terms in the hoop direction and 81 terms in the axial direction to determine the behaviour

ofK-joints (UEG, 1985).

r

Uniformly distributed radial loading

Q_3 © . < ,

Circumferencial moment longitudinal moment

Figure 2.2. Model of cylindrical vessel used by Bijlaard

Chapter 2: Literature Survey - 1_J_

Despite the agreement between experimental data and Bijlaard's results, the method is too

simplified for tubular joints. It may, however, be applied to the joints with small P ratio for

preliminary design purposes.

Dundrova (1965) has presented one of the most complete theoretical studies. She has

analysed a T-joint under axial load based on the classical theory of cylindrical shells. Her

solution finds the distribution of the forces acting on the chord wall by imposing

compatibility condition between brace axial displacement and chord wall deformation.

However, brace bending stiffness is not considered in Dundrova's solution. She was the

first one who considered the brace explicitly in the analysis.

Tubular joint flexibility was studied in a report by Holmas et al. (1985) using the classical

shell theory. The range of p considered by Holmas is between 0.1 to 0.5. The Donnell form

was used to express the forces and moments on a shell element as shown in Figure 2.3.

Bending moment and axial force in the brace were replaced by the equivalent forces,

shown in Figure 2.4.

Figure 2.3. Shell element used by Holmas Figure 2.4. Extra DOF to express local joint

behaviour used by Holmas

It appears that the model by Holmas is similar to the Dundrova's model. Holmas's model

suggest three extra degrees of freedom for every brace attachment, which are one

translational, and two inplane and out of plane bending degrees of freedom. The report

presents the variation of the axial and IPB stiffness of a T-joint for various D/T ratios and p

values. A model is suggested by Holmas for considering the high axial stiffness of the

brace based on the collocation method, but the bending stiffness of the brace is not taken

into account in this model.

B. Chen et al. (1990) have investigated the local joint flexibility of simple T, Y and

symmetrical K-joints for axial and inplane bending loads. They have used the classical


theory of thin shells and the Finite Element method to analyse tubular joints with the chord

and braces treated as substructures of thin shells while the intersection curve between any

two substructures is discretized into finite elements. Chen et al. have recommended a

formula for the stiffness matrix of a symmetrical simple K-joint. They have reached a good

agreement with other formulae by D N V (1977), Fessler (1986) and Ueda (1990) and some

experimental results by Tebbett (1982).

T. Chen et al. (1990) have introduced a similar analytical method to the method by B. Chen

(1990), using the two models by Holmas (1985) and Ueda (1986) for definition of joint

flexibility. These two models are based on solutions of shell equations and Finite Element

analysis, respectively. The model by T. Chen has the features of simple computations and

low C P U time. T. Chen has studied axial and inplane bending flexibility of T, Y and T Y -

joints.

2.3.2 Experimental and semi-experimental methods

There are some parametric formulae that are referred to in this section. The reader may see

Chapter 6 for those formulae and their ranges of validity.

Experimental methods

The theories of structures and continuous media are not used in these procedures. A

physical model which can range from small to full scale in size is tested under the

conditions similar to the real structure. The model can represent the whole structure or a

component of it.

In the study of tubular joints, test specimens selected earlier were from steel. Synthetic

materials such as acrylic and epoxy resin were used later as substitutes for steel since they

are cheaper, easier to handle and more flexible. Experiments are usually carried out by

loading the joints through static forces and measuring the desired quantity, which can be a

strain in any direction or displacement of a location with respect to a datum. Test

specimens from synthetic materials are on a small scale, whereas those from steel could be

the same scale as the prototype. Numerical methods are usually employed for curve fitting

the test results, where generally an equation or formula is established to be used for

analysis and design. Parametric formulae for stress concentration factors is a popular

example of the application of experimental methods to tubular joints.

The photoelasticity method is also an experimental technique involved in the experimental

stress analysis of tubular joints, where three dimensional stress distribution can be

deterrnined. The method is restricted to stress analysis and unless a relationship between

flexibility and stress is employed, it can not be used for the study of joint flexibility.


Fessler et al. (1981) developed a procedure to define and measure the flexibility of tubular

joints. Three loading modes were considered: 1) axial tension, 2) inplane bending moment,

and 3) out of plane bending moment. Fessler only considered T and non-overlapping Y-

joints by testing 25 joints made of precision-cast epoxy resin tubes. Methods, based on

experimental results, were proposed to determine the joint flexibilities of the different

deformation modes. A n equivalent brace length was proposed to consider the flexibility of

typical joints when the customary line model is used. A line model is constructed of one

dimensional beam elements being connected at the joints.

Fessler concluded that further work should include an analysis of simple frames of typical

structures. It appears that the experimental method proposed by Fessler includes a

relatively time consuming procedure and can be costly in terms of test equipment. Figure

2.5 shows the rig used for loading the test specimens. Deflections were directly measured

at various locations.

=£ Load ring fixed-7

to brace Channel section support Tape

transducer

(H

'-M

m TL

Out of plane bending

___ Support stand reacts load

Strain gauges

______

XI %4 Load frame-

Axial tension

Figure 2.5. Test rig used by Fessler (1981)

In another work, Fessler et al. (1986a) developed a set of parametric formulae for IPB,

O P B and axial deformation of brace in single brace tubular joints, using the same method

as in the 1981 paper. There were 27 tests on araldite models covering the c o m m o n range of

parameters in offshore structures. In comparison with experimental results Fessler's

formulae overestimate the bending stiffness of T and Y-joints.

In a companion paper, Fessler et al. (1986b) presented a set of equations for the cross-

flexibility between any two braces which may be in any orthogonal planes at a joint. This

work was also based on the same experimental procedures and actually on the same test


specimens as the other paper (1986a) by the same authors. The measurements on the end of

fictitious unloaded braces were determined from the measurements of the single brace joint

models. In both papers, the effect of the variations in brace wall thickness on joint

flexibility has been ignored. For non-overlapping joints, the proposed parametric equations

may overestimate the flexibility up to 7 0 % compared to the measured data when the

flexibility is significant.

Semi-experimental methods

W h e n compared with the experimental methods, semi-experimental procedures also benefit

from the analytical methods of structural analysis. In these procedures, a mathematical

model is employed and tuned using test results. Punching shear model, shown in Figure

2.6, is an example of this method. The punching shear stress, vp, is assumed to be

uniformly distributed. So it can be written as:

N v_ = p ndt

(2.1)

in which N, d and t are the axial force, diameter and thickness of the brace, respectively.

The axial force in terms of shear stress would be:

N=\pndt (2.2)

Design codes give the allowable values of the punching shear stress for different

geometrical parameters. The values have been derived from experiments on various test

models and then stated in the analytical form of punching shear stress formula.

Figure 2.6. Stress distribution assumed in Punching Shear Model

Chapter 2: Literature Survey £7

The method used in this study is a semi-experimental technique in which an unknown

analytical parameter is determined by experiment. A work on the support flexibility of pre-

tensioned cables has been carried out by Springfield and Brunair (1989) that implements an

approach similar to the method adopted in this thesis. The end fixity, as the main objective,

and the bending stiffness (EL) of an electrical transmission line were determined by

measurements of the displacements at certain locations of the line when it was vibrating

under a certain natural frequency. The theoretical model used by Springfield and Brunair is

an axially loaded, transversely vibrating beam supported at the ends through rotational

springs. Springfield and Brunair have concluded that consideration of end fixity leads to a

conductor bending stress of approximately one-lhird the value given by assuming a rigid

end. It was found in this study that only Springfield and Brunair have used a dynamic

method of analysis.

2.3.3 Numerical methods

Numerical and analytical methods were facilitated with the advent of computer, resulting in

the development of the analysis procedures in the theory of structures. The Finite Element

method, the dynamic deformation method and the flexibility method are some examples.

The dynamic deformation method is an improved version of the slope deflection method

where the inertia forces are also considered (Kolousek, 1939,1973).

The Finite Element substructuring is the most effective and non-expensive methods which

can be effectively used for parametric studies. However, there is always this argument that

the method does not include any deficiency unless purposely assumed in the model.

Furthermore, because the F E M is a numerical procedure its predictions can not be

absolutely reliable without any experimental verification.

Bouwkamp et al. (1980) developed a new procedure involving a modified three

dimensional Finite Element formulation for the modelling of a tubular joint substructure

and its subsequent insertion into a complete offshore platform computer model. The

substructuring technique used by Bouwkamp allows fast modelling of the tower frame

without having to do finite element modelling of each joint, when the super-element is

available. The technique is based on the results of the Finite Element analysis. There are

also some simplifications to easily model the super-element. Figure 2.7 shows a typical

substructure of a joint used by Bouwkamp.


T-brace.

Primary mesh

Embedded masrri

(-agonal brace

main chord

T-brace intersection

X __

Diagonal intersection

-meshes constrained' _r

Figure 2.7. Model of joint substructure used by Bouwkamp (1980)

Efthymiou (1985) has reported a Finite Element study on the local stiffness of unstiffened

tubular T, Y and K-joints subjected to inplane and out of plane bending. H e has defined the

local joint stiffness as the applied moment at the brace divided by the local joint rotation.

The rotation of the brace end due to the joint flexibility was calculated by deducting the

beam type rotation from the total rotation of the brace end. H e measured the rotation at the

end of the brace as shown in Figure 2.8. This is different from the other methods in which

the measurements are usually made on the chord wall.

M

QL_4| 'ML.,,

Brace—' ii • r - " « _ _ • "

r Chord

M

ib

Brace bending stiffness

Crown

0 Chord bending stiffness 12EIe/le

a) Physical model b) Theoretical model

Figure 2.8. Rotations measured by Efthymiou for calculation of joint flexibility (1985)


The F E program used by Efthymiou was P M B S H E L L , which had a thin shell element

implemented. T o verify the performance of P M B S H E L L , Efthymiou re-analysed one

geometry using another program called S A T E , which had a combination of plate and

membrane elements. The results of these two analyses showed a very good agreement. H e

established a set of parametric formulae, based on 24 F E analyses, for T, Y, and K-joints.

The Efthymiou's equations for T and Y-joints predict local stiffness to within 1 5 % of the

stiffness values used for curve fitting. The equations for K-joints are somewhat less

accurate. Their predictions are expected to be within 3 0 % of the measured stiffness. The

parameters considered by Efthymiou were |$ and y. His equations are inclusive of common

joint types used in offshore structures, but the database that he has used to establish the

equations does not seem to have adequate data. Furthermore, Efthymiou's study is based

only on F E analysis results and does not have any comparison with experimental findings.

Ueda et al. (1986) have developed a model for tubular joints. The model takes account of

joint flexibility in elastic as well as elastic-plastic ranges based on elastic fully-plastic load-

displacement relationship. It is stated by Ueda that the geometry of tubular joints makes it

difficult to obtain closed form analytical solutions to evaluate load displacement

relationships. In this respect, the method which is chosen in this thesis has a highly

theoretical base. It determines the natural frequency of a tubular joint from the measurement

and then employs it in the analytical model to produce a relationship between load and

displacement. The reader is referred to Chapter 4 for further explanation.

Ueda has proposed line elements for modelling the joint local behaviour, as shown in Figure

2.9. Elements *c' represent the local behaviour of the chord wall in Ueda's model. The

stiffness matrices for the elements are taken from another reference by the same authors.

The method is used for both elastic as well as plastic zones.

a

a

- - i

c,

- .

Chord

b * i

Branch stub

- -<

a

a - -<

CJ

T-joint Y-joint

Branch stub

«Chord

Figure 2.9. Joint model proposed by Ueda (1986)

The proposed method by Ueda for considering local joint behaviour is computationally

simple and does not need a great deal of computer memory. However, it is still based on

preliminary analysis by the Finite Element method to obtain the stiffness of the joints. The


method actually implements the stiffness results of a finite element analysis into a simpler

line model. Besides, the computational nature of the method allows no modification in the

joint model due to imperfections and other complicating factors involved in manufacturing

and fabrication. Such an ability could remove the approximations introduced by the Finite

Element and generally correct the F E model using experimental data.

In another paper, Ueda et al. (1990) developed a set of formulae for stiffness of T and Y-

joints. The database used for establishing the formulae was taken from F E analysis and

included 11 samples for IPB mode and 7 samples for axial deformation of brace.

The relationships between the stiffness of T and Y-joints used by Ueda et al., especially

axial stiffness, do not appear to be consistent with the results of others. For example Fessler

(1986) shows that:

fcY(«'a') = fcrsin"2-1^, and kY(tPB) = &Tsin"1,220 (2.3)

in which ky and kj are the stiffness of Y and T-joints, respectively. 0 is the brace angle in a

Y-joint. Efthymiou has obtained the following relation:

J.YaPB) = Min-(P+a4)e (2.4)

Whereas, Ueda used the same axial stiffness for T and Y-joints. To determine the IPB

stiffness of a Y-joint, Ueda used the stiffness value of a T-joint divided by sinG, where 9 is

the angle of diagonal brace.

Souissi (1990) has carried out a study on the flexibility of tubular T-joints using the Finite

Element Method. H e established a super-element to model a joint and attributed its property

to the fictitious centre nodes that were at the end of each tube, chord and brace (Figure

2.10).

Y 3 i • i

Figure 2.10. Joint super-element used by Souissi (1990)


Souissi considered inplane bending, out of plane bending and axial loading and performed

18 analyses for each case. His results showed good agreement with Efthymiou's (1985)

results. H e has recommended that corrective factors can be applied for each loading case to

consider the effects of x on joint flexibility.

2.4 Effects of tubular joint flexibility on tower analysis

This subject has been researched since the late 1970s. F e w references are available about the

effects of joint flexibility on the behaviour of tower structures. The Underwater Engineering

Group (UEG), which is a British non-profit-making organisation produced a comprehensive

report called U R 2 2 on the joint flexibility effects. This section provides a summary of this

report as well as other studies.

B o u w k a m p et al. (1980) summarised the results of a limited study into the effects of tubular

joint flexibility on the structural behaviour of deep water fixed offshore towers. Bouwkamp

produced a model, using Finite Element analysis results, to incorporate the joint flexibility

into the structural analysis. In order to illustrate the procedures used to assess the effect of

flexible joints, a two dimensional 330m high tower frame was analysed under dead and

wave loads, using the developed joint model as well as the so called line model. In the latter

model, joint effects are neglected. It was concluded that the effects of joint flexibility on the

structural behaviour of offshore towers can be significant The nature and magnitude of

these effects are dependent not only on the tower height, but also on its geometrical and

structural configuration. The effects were noted in the higher modes of vibration and in the

deflected shape of the tower under static loads. It was observed that joint flexibility effects

are more pronounced when stiffness of the member intersecting at a joint is relatively high.

The effect of joint flexibility on the deflected shape was seen to be very small for the nodes

at the top of the tower. However, larger displacements were observed for the joint model

below -170m of water level, with a maximum increase of 5 0 % over the line model at -300m.

Regarding member forces and moments, B o u w k a m p has shown: 1) a slight increase in

calculated leg axial forces (up to 2 % higher) and a considerable reduction in calculated

brace axial forces (up to 2 0 % ) ; 2) a modified distribution of pile loads with load transferred

towards piles through main legs; 3) an increase of up to five fold in legs moments.

Joint flexibility consideration in dynamic analysis was shown to lead to lengthening of the

fundamental periods particularly for higher modes, where changes in order of the mode

shapes were also observed.

Tebbett (1982) showed the effectiveness of grouting the legs of fixed jacket offshore

platforms which has become important with regard to the reappraisal of steel jacket

structures. T o do so, he has placed emphasis on considering the flexibility of tubular joints

in the analysis of the jacket structures. Tebbett has concluded that the effects of local joint


flexibility can be significant and should, if possible, be included in the structural analysis

during the reappraisal of jacket structures. Furthermore, if grouting is being considered, the

reduced local joint flexibility should be accounted for in the analysis.

Kawashima and Fujimoto (1984) checked their model by testing an L-frame and a portal

frame. Kawashima only studied the flexibility effects on the mode shapes and natural

frequencies by conducting dynamic analysis. The test models and the results obtained by

Kawashima and Fujimoto are shown in Figure 2.11. They obtained a good agreement

between the analytical results of the joint model and experimental results, especially for the

lower natural frequencies. The effect of flexibility consideration on the natural frequencies

showed a variation from - 2 5 % to 0 % between the calculated results. The - 2 5 % change

occurred for the first natural frequency of the portal frame. Joint stiffness introduced a

maxi m u m change of 1 0 % to the calculated natural frequencies of the L-frame.

600 h»-

900

Table 2.1a. Kawashima's results for L-type frame

Mode

1 2 3 4

Rigid

Cal. 23.3 51.4 74.5 142.8

Semi-rigid k-\ 200kg. cm/rad C=2.5kg.cm.sec

L Cal. 21.6 47.8 67.2 142.9

Exp. 21 46 65 138

a) L-type frame and its natural frequencies obtained from experiment and analytical models

900 («•-

600

Table 2.,

Mode

1 2 3 4 5 6 7 8

b. Kawashima's results for portal frame Rigid |

|

Cal. |i 9.4 26.6 65.8 I 74.9 100.4 | 173.1 | 220.2 1 232.7 J

Semi-rigid fc=1200kg.cm/rad C=2.5kc

Cal. 7.1 19.6 62.6 65.3 73.9 163.1 200.4 204.9

l.cm.sec

Exp. 6.9 20 59 65 73 144 180 217

b) Portal frame and its natural frequencies obtained from experiment and analytical models

Figure 2.11. The results of Kawashima's analyses on the frames with rigid and semi-rigid joints


Matsui et al. (1984) have studied the behaviour of truss beam columns composed of tubular

sections. Matsui considered the effect of joint flexibility on the buckling behaviour of the

web members with large diameter-thickness ratio. The flexibility analysis of Matsui is

based on a spring model from Sakamoto and Minoshima (1979). The results of Matsui's

analysis indicate a maximum of 1 0 % difference in buckling strength of a truss when only

bending moment is applied to the chords.

Ueda et al. (1986) carried out a parametric study on five K-braced, five-storeyed two

dimensional tubular frames as shown in Figure 2.12. Three horizontal point loads were

considered in Ueda's analyses.

4000

4000

4000

4000

4000

Figure 2.12. K-braced frame analysed by Ueda and its load cases

Table 2.2. Joint specification in Ueda's analyses

Model Model D T Initial

No. type load (kgf)

R15 R20 R30 R40 R

R50 R59 R67

F15 F20 F30 F40 F

F50 F30D

F40D

R: Rigid joints

F: Flexible joints

Chord Horizontal brace: Diagonal brace:

Yield stress:

1000

1000

64 50 33 25 20 17 15

64 50 33 25 20 33 25

DxT (Table 2.2) 400x25 400x25

mm mm

70 kgtfmm2

1410000

1100000 730000

550000 440000

374000 330000

1410000

1100000 730000 550000

440000 730000 550000

Ueda investigated the effects of the joint flexibility and strength on the structural behaviour

and collapse loads of the K-braced frames. It was found that joint flexibility may have only

a little effect on buckling of braces, but joint strength may have a great influence upon

collapse modes and strength. The lateral stiffness of the K-frame with D/T = 50 decreased

by up to 4 6 % when joint stiffness was considered in the analysis. There was, however, less

reduction of lateral stiffness when lower D/T ratios were assumed in the analysis.

The significance of the Ueda's study is investigating the effects of tubular joints on the

ultimate strength of tubular frames. The 3 point loads considered by Ueda in his study do

not simulate the loading from waves, current, etc. which exist in sea environment. The


results of Ueda's study could be more applicable to the offshore structures if different

loading were used.

UEG Report, Node Flexibility and its Effects on jacket Structures (1984, UR22)

This report presents an investigation into the effects of chord wall flexibility at brace

connections on the behaviour of oil production jacket structures. It has considered the

effects of joint flexibility on the inplane deflections, axial forces, bending moments, brace

buckling and natural frequencies of three different 100m tall vertical plane frames. The

overall geometry of the frames are shown in Figure 2.12. They have been modelled using

two dimensional beam elements with three inplane degrees of freedom at each end, two

translations and one rotation.

36.24m 63.094m 36.240m H H H H

structure 1 structure 2 structure 3

Figure 2.12. Frame models analysed in UR22 report by UEG (1984)

A simple representation of the joints was selected in the study. One nodal point was

provided on the chord and one on each brace at the brace to chord wall intersection as

shown in Figure 2.13. The nodal points 2, 3 or 4 were then all connected by a stiffness

matrix derived from the flexibility matrices provided by Fessler (1981).

Figure 2.13. Nodal points considered in UR22 Study to represent a joint


T w o types of analysis were carried out, one incorporated flexibility of the joints based on

the Fessler's model, the other did not consider flexibility which was called conventional

analysis as the braces were extended to the chord centre lines.

The various joints used were identified by three characters T I N (Type Intersection

Number). Type may be C-Conventional or M-matrix, T describes the intersection of the

braces and chords which is P-intersect at Point or E as Eccentric. 'N' is the joint Number

corresponding to the geometrical ratios characterising the joint geometry. The joint numbers

of the different geometrical parameters are shown in Table 2.2.

Table 2.2. Joint parameters used in UR22 Study

joint no. 1 2 3 4

D/T 25.3 50.6 25.3 25.3

d/D 0.53 0.53 0.33 0.75

For example M P 3 refers to the analysis, using matrix formulation for the joints, where the

braces are intersecting at a point having D/T=25.3 and _W)=0.33. Four load cases were

applied to each structure. The first load case was a point load applied at the top of the

frame. The other three were distributed wave load cases derived from a representative 100-

year storm wave with different phase angles: 0°, 90°, and 45°. The following results were

obtained from the analyses.

1) Global Deflections

The introduction of the joint flexibility into the analysis, made differences of up to

1 3 % to the overall sway of the structures analysed. A comparison of the

deflections for the structures with different joint types is given in Table 2.3.

2) Effect of Flexibility on Axial Forces

This effect was found to be negligible. The biggest change between the

conventional and flexible analysis was 1.5%. The maximum axial stress change was

less than 1 N/ram2.

3) Effect of Flexibility on Bending Moments

The largest change in brace end moment found was in structure 1 with joints M P 2 ,

where a horizontal brace moment increased to about three times the conventional

rigid frame analysis value implying a 2 0 0 % change. The largest variation of

bending stress for structure 2 was 6 0 % . Structure 3 had the largest change of

about 5 0 % . These changes are corresponding to a combination of the analysis


results of load case 2 and load case 3. The bending stress changes for all the

various structures and joints, under the wave load with a 45° phase angle, are

shown in Table 2.3. The largest stress changes in the structures under the same

loading were:

Structure 1: 30N/mm2,

Structure 2: 29 N/mm 2,

Structure 3: 4 N/mm2.

Table 2.3. Summary ofc

Change from

conventional analysis

Deflection change%

Chord

Axial stress 45 brace

change(NmnT2) 90 brace

Chord

Bending stress 45 brace

change% 90 brace

Buckling load 45 brace

change% 90 brace

1

2

3

4

5 Natural frequency 6

changes% 7

(rigid/semi-rigid) 8

9 10 11 12 13

hangesfrom UEG re port on joint flexibility

Structure 1 MPl

0

0 0 0

4 15 44

4

—

—

—

—

—

—

—

—

—

—

—

MP2 MP3

5 2

0 0 0.5 0 0 0

7 7 20 15 94 50

-9 —

2 — 3 — 1 — N — 3 — 3 — 1 — 5 — 15 — 0 — N —

MP4

-1

0 0 0

4 15 22

—

—

Structure 2 MPl

3

0 0 0

7 11 8

—

—

MP2 MP3

13 5

0 — 0 — 0 —

14 14 -25 -29 14 14

-10 — -13 —

6 — 9 — 3 — 26 — 3 — 4 — 3 — N — 17 — 15 — 26 —

MP4

-1

—

—

14 13 6

—

—

Structure 3 MPl

1

0 0

5 13 —

—

—

MP2

5

0 0

5 6 —

-12

2 1 3 82 13 8 1 3 2 0

MP3 MP4

1 -1

— — — —

5 9 6 6 — —

— —

— —

4) Effect of Joint Flexibility on Brace Buckling

The effect of joint flexibility on buckling load of the braces was determined in the

study. The results are shown in Table 2.3. The buckling load was reduced by about

1 0 % between the conventional CP1 and the most flexible M P 2 analysis. This was

caused by the flexible joints increasing the effective length of the brace.


5) Effect of Joint Flexibility on the Vibration Characteristics of Jacket Structures

The first few natural frequencies and their corresponding m o d e shapes were

calculated for each structure with the conventional C P 1 and the most flexible M P 2

joints. The natural frequencies of similar m o d e shapes were compared. Table 2.3

summarises the natural frequencies and reports the proportional changes. The

changes in the natural frequencies of corresponding modes were on average 4 % ,

1 2 % and 1 1 % for structures 1, 2 and 3, shown in Figure 2.12, respectively. The

greatest change in natural frequency of similar modes was 8 2 % and occurred for mode

shape 5 in structure 3.

The study showed that the increase in bending stress caused by incorporating joint

eccentricity of DIA in the conventional analysis was similar to that caused by joint flexibility.

It was concluded that the effects of joint eccentricity coupled with those of joint flexibility

could therefore be significant

The report U R 2 2 indicates the significance of incorporating joint flexibility of tubular joints

into the analysis of offshore towers. This report considers only one joint modelling

technique that is using joint stiffness matrices provided by Fessler. Other simulation

techniques could produce different results. Furthermore, it focuses only on the different

aspects of joint flexibility in the structural analysis, whereas an analysis of fatigue life seems

to show the significance of joint flexibility consideration more clearly.

Chapter 8 of this thesis reports the results of a similar analysis to U R 2 2 using the theoretical

model suggested in Chapter 3. The joint flexibility effect on the fatigue life is also

investigated in Chapter 8.

The effect of joint rigidity on the buckling behaviour of tubular members in trusses and

frames has been investigated by Jong and Wardenier (1987), using Finite Element method.

The moment rotation relationship of tubular joints have been employed in the derivation of

buckling load curve suggested by Jong. A formula is presented to determine the critical joint

stiffness of axially compressed members with semi-rigid joints. B y definition, there is not a

notable gain in the buckling load of a member when the joint stiffness is higher than the

critical value. The critical joint stiffness was found to be:

(X-20)2£/>ifX<2()thenX = 2() (25) 1 1400 L

C2=3Q (2.6)


The relationship suggested by Jong and Wardenier for buckling load of tubular members

with semi-rigid joints is:

/ V > ( 1 — — ) N m 1 10000

N2 = 0.98AL

(2.7)

(2.8)

The terms in Equations (2.5) to (2.8) are shown in Figure 2.14, below:

N0 the buckling load for a pin ended member

N-\ the buckling load belonging to C-\

Afc the buckling load belonging to C^

No. the buckling load for a member with rigid joints

X the slendemess ratio of the pin ended member in compression

100

Joint rigidity, C

Figure 2.14. Predicted buckling load by Jong (1987)

Joint rigidities were assumed to be the same at both ends of a member in compression. The

parametric formula for the joint stiffness allows the adjustment of the joint geometry to

achieve the desired strength for a compressive member intersecting at the joint

The proposed method by Jong and Wardenier is to some extent complicated for design

purposes. However, it is one of the few works on the buckling of tubular members

considering semi-rigid joints and the accuracy of the method also needs to be verified by

experimental analysis.

T. Chen et al. (1990) analysed a 5-storeyed tower, shown in Figure 2.15, considering

flexible tubular joints based on the data by Holmas and Ueda. The results of Chen's analyses

are reported in Table 2.4.


P1 = P2 = P3 =10000 kg

D=100cm d=40cm

7=23 cm t= 1.0 cm

E=2.1x10 kg/cm v=0.3

Figure 2.15. Tower analysed by T. Chen (1990)

Table 2.4. Effect of flexibility consideration in the analysis, Chen T. (1990)

Umoaicm) Vmax(cm)

Qmaxfrad)

Nmaxfkg)

Qmax(kg) Mmax(kg-cm)

computed results based on

rigid joints

7.198

0.365 -1.09E-3

3.45E5 1.26E5 1.99E6


Holmas'joints

8.784

0.386 -1.20E-3

4.27E5 1.54E5 1.71E6


Ueda's joints

8.722 0.379

-1.21E-3

4.23E5 1.53E5 1.92E6

Chen's results show a good agreement between the two methods by Holmas and Ueda. The

biggest difference was between the maximum bending moments calculated by the two

methods. Holmas's model produced a 1 4 % change in bending moment whereas Ueda's

model caused only a 4 % change.

According to Chen's results there was a maximum change of about 2 0 % in the horizontal

displacements, a 2 3 % change in axial forces and a 1 0 % change in bending moments when

semi-rigid joints were employed in the analysis.

Chen's results are due to a loading composed of three point loads as shown in Figure 2.15.

This type of loading does not occur as frequent as wave loading in the sea environment

Therefore, the results are not very applicable to the offshore structures. However, Chen's

results generally show the effects of joint stiffness on the behaviour of structures.

Souissi (1990) also compared the analysis results of two frames (Figure 2.16), one with

flexible and the other with rigid joints (conventional analysis).

x,u


H _,g _____'__. 8" 77 . 29'

senv-rigid ^M v_ 71 11/

>! -J£-_JL_____2J 2a 6<> te < _^

4' 15

3> 10 \V 18 24 -9 ' »

M 23

10m

25

22

8m

8m D=800mm d=240mm P=°-3 T=16mm "f=25

t=8mm ^ 5

L=4800mm " ^

8m

8m

Figure 2.16. The frame analysed by Souissi (1990)

His results are:

Loading N o 1 : H * 0 , V = 0

1) 9% to 11% underestimation of displacements for the conventional analysis, and

2) overestimation of bending moments up to 3 5 % at joints 10 to 12 for the conventional

analysis.

Loading N o 2: H = 0 , V * 0

overestimation of axial force up 37% for joint 10 and bending moment up to 23% at joints

10 to 12 for the conventional analysis.

Souissi has concluded the need of a simple method to assess the flexibility of joints from

analytical or numerical models.

Recho et al. (1990) have investigated the influence of flexibility on the fatigue design of

tubular T-joints. The joint stiffness has been determined by using Finite Element method

with static condensation technique. This method of stiffness calculation is the same as what

Souissi (1990) carried out in his study. Three series of curves, based on the F E analyses,

were established for the three load cases in T-joints (IPB, O P B and axial loading).

Recho analysed two different structures, shown in Figure 2.17, and calculated the fatigue

life change when the joint flexibility was applied in the analysis.


1000H "

5m

6D »

a

„

m

3 • 2\ Brace

l

3fc

«

Chord

_

h frtn

Typel

Same T-joint characteristics u sed in the two frames

".

4m 1000N—

i Am i

i

8m

3

*2 r

Chord

'JMVA

I

D = 80cm, T= 1.6 cm d=. .4 cm, d=0.8 cm

Brace

Brace

— 1 0 m

Type 2

3

2 i

Chord

1000N

1 »J

Figure 2.17. The structures analysed by Recho (1990)

The results obtained for the forces at joint 2 in the two frames are:

Table 2.5. Effect of joint flexibility on internal forces by Recho (1990)

Rigid joint Flexible joint Difference %

Typel Fa™, = 1739 N Faxial=1762N Faxial = +1.3%

A_n>B = 37 N m Mn»B = 45 N m Mn»B = +21.6%

Type 2 ^axi_l=763N

MIPB = 4.4 N m

F»i_!=724N

A-TPR = 4.8 N m

F„i_i=-5.1%

MIPB = +9.1%

Recho et al. then calculated the fatigue life of joint 2 using the French Standards ( A R S E M ,

1985) and compared the results of the rigid and semi-rigid analyses. The details of the

fatigue calculation is not given by Recho, however, the influence of flexibility on the fatigue

life of the two frames are reported. Table 2.6 shows this influence as the ratio of 7v*R (the

number of cycles to failure when connection is considered rigid) to 7v> (the number of cycles

to failure when connection is considered flexible).

Table 2.6. Fatigue life difference (NtfNF) when joint flexibility is considered

Rupture Type 1 Type 2

at the saddle point 1.04 0.85


According to the above results, the influence of flexibility consideration on the fatigue life

can be negative or positive. Its cause is related by Recho to the geometry and boundary

conditions of a structure.

The study does not include a realistic loading c o m m o n to the offshore structures since the

loading in sea environment is a distributed load and depends on the wave or current

characteristics, whereas Recho considered one or two point loads in his analysis examples.

Furthermore, when comparing the fatigue life of rigid and flexible joints in Table 2.6, the

location of fatigue rupture is not specified. Failure of a joint under fatigue is because of

rupture at either saddle or crown locations, not likely both. Therefore, only two

comparisons out of the four shown in Table 2.6, correspond to the fatigue life of the joints

analysed by Recho.

Table 2.7 summarises the results of different investigations surveyed in this study

regarding the effect of joint stiffness on the behaviour of tubular structures. It is noted that

wave loading which is the dominant force acting on offshore towers is not considered in

most studies. Furthermore, a consistent trend can not be found in the results. The most

changes are in bending moments. This would be expected since tubular joints possess more

flexibility under inplane bending than axial loading. A s offshore structures transfer external

loads in a truss like fashion, introducing rotationally flexible joints will not make

significant changes in lateral deflections and axial forces. This can be also seen in Table

2.7.

The variations observed in the results of different studies indicate that joint stiffness has

different effects on the analysis results of tubular structures. These effects vary with

structural geometry, configuration, height and loading of the structure.

Table 2.7. Maximum response change for models with rigid joints obtained in various research works when joint flexibility was considered

Bouwkamp (1981)

U E G (1984)

Ueda (1986)

Chen T. (1990)

Souissi (1990)

Loading

dead load + point

loads at top

point load + wave

load

point load

point load

point load + dead

load

Deflection

+1%

at top

+13%

+13%

(DIT) = 26

+22%

+10%

Axial force

chord: +2%

brace: -20%

+1.5%

—

+24%

—

Bending

moment

+500%

-90%

—

-14%

-35%

Natural frequencies

-10%

-45%

—

—

—

Recho (1990) point load — -5% +22%


2.5 Model of joint in stiffness method

A typical model which is used commonly to represent a joint in a framed structure is a

linear spring. The idea has taken originally from the moment rotation relationship of a

semi-rigid connection, shown in Figure 2.18. For multi brace joints the joint stiffness may

be adopted in the form of a stiffness matrix to include the interaction of the braces upon

each other.

The linearized form of this relationship was for the first time implemented in the slope

deflection method by Rathbun (1936). His results showed that changing the end

connections of a beam from rigid to semi-rigid is equivalent to lengthening the beam by the

amount of3EI/k. k is the spring stiffness of the beam end.

5

Angle of rotation, 8

Figure 2.18. Moment-rotation relationship for a semi-rigid joint

In 1963, Monforton and Wu expanded the semi-rigid formulation introduced by Rathbun to

a matrix form, suitable for using in computer programs. The formulations developed by

Rathbun or Monforton and W u included only the stiffness properties without any reference

to inertia forces.

The works by Grant (1968), Lionberger and Weaver (1969) involved dynamic analysis but

the joint effects were not considered in the mass matrix formulation. They also used the

spring model to represent the joint flexibility.

Kawashima et al. (1984) modelled a beam with two subsidiary elements at each end. In this

model after derivation of the beam stiffness matrix, lengths of the end beams are taken

Test behaviour of a semi-rigid joint

Design range


close to zero which virtually simulate two rotational springs at the ends. Dashpots parallel

to the end springs are considered for modelling the energy dissipation.

In the Kawashima's study, the force displacement relationship for a beam with rotary

springs and dashpots is considered in a matrix form. The dynamic stiffness matrix has been

derived through the transfer matrix in an exact form in which a continuous mass property is

assumed. The elements of this matrix then has been expanded in a series form in terms of

circular frequency. This has resulted in the mass, damping and stiffness matrices whose

elements include two types of parameters designated as fixity factors and damping factors.

Fixity factor varies between zero and one, where it is zero for a hinged and one for a fixed

end. The significance of Kawashima's work is inclusion of the joint flexibility in the mass

and damping matrices. The joint model developed by Kawashima is shown in Figure 2.19.

Q~" 9

vy U -U L -U J actual lenght is zero actual lenght is zero

Figure 2.19. Joint model developed by Kawashima (1984)

Kawashima concluded that the effects of semi-rigid connections depend on the mode

shape. H e also obtained a good agreement between the results of analysis and experiment.

Kohoutek (1985a) used a different formulation from the spring model for a beam with

semi-rigid joints. The boundary conditions assumed by Kohoutek are:

y(0) = 0 y(l) = 0 (2.9a)

-(l-r^AO) + rvy{0) = o (i-r,,)y'(/) + _>/(/) = o (2.9b)

in which T is the fixity factor and is zero for a hinged or one for a fixed support. Equations

(2.9b) are aiming at balancing the second and first derivatives of the beam axis

displacement curve at the supports such that their sum becomes zero. y" and / correspond

to the bending moment and the slope of the beam axis. The bending moment is zero for a

hinged support whereas the slope of the beam axis is zero for a fixed support.

Using natural frequencies as a means to determine the rigidity of joints was introduced by

Kohoutek (1985a) who applied the method to open section joints.


2.5.1 Bending rigidity factor

It is preferred in practice to use a factor between 0 and 1 (or 0 % and 100%) to describe the

stiffness of a joint. U K O S R P has used a similar factor, called degree of fixity (F), to

correct for the effect of chord end fixity and chord length on stress concentration (UEG,

1985). Kohoutek (1985) modelled the joint rigidity by imposing the boundary conditions

shown in Equations (2.9a) and (2.9b). He called the joint rigidity F, which varies from 0 to

1. Monforton and W u (1963) defined a fixity factor vb as:

v"( V i a t j o i n , 0 =TT3_77r (X10)

in which E is the modulus of elasticity, / is moment of inertia and kt is the stiffness of the

end i of the beam. This definition was also used later by Grant (1968) and Kawashima

(1984). vb varies from 0 to 1, corresponding to k from 0 to oo. The base of Equation (2.10)

can be found from the work by Rathbun (1936) which states:

li = l+3EJ/ki (2.11)

in which // is the equivalent beam length to be used in the slope deflection formulae to take

the joint stiffness into account. Dividing both sides of Equation (2.11) by / and inverting

the result one can obtain:

- = l- (2.12) /.. 1 + 3EI/L

which is actually the definition of vbi in the studies by Grant (1968) and Kawashima

(1984).


2.6 S u m m a r y

The report of the survey on the available literature related to the flexibility of tubular joints

was presented in this chapter. The different methods of investigation on tubular joints were

described with the major applications on the stiffness of joints. These are:




It was pointed that the early analytical works on tubular joints by Bijlaard, Dundrova and

Kellogg are too simplified for the real and practical examples of tubular joints. Nevertheless,

the theoretical studies are still used and are seen in the works by Holmas (1985), B. Chen

(1990) and T. Chen (1990). The study of Fessler, which based on the surveyed articles here

is the only comprehensive experimental work on tubular joints, was also described. Fessler

has used araldite models in his tests. There was only one study found where a dynamic

method has been used to calculate the support stiffness of cables. Static techniques have

been used in all other works. The literature survey showed that semi-experimental methods

which have a theoretical and experimental base are not used for flexibility formulation of

tubular joints.

Finite Element method is used extensively in the researches on the flexibility of tubular

joints by Bouwkamp, Efthymiou, Ueda, Souissi and Recho. There is no comparison with

any test results in these studies. The effect of joint flexibility on fatigue life has been partially

studied by Recho which was on two very simple frames.

The reviewed studies show that consideration of joint stiffness makes significant changes in

the results of a structural analysis. The major effect has been recognised to be on the

bending moments. The fatigue life has hardly been attended in the studies related to

flexibility of tubular joints. The literature survey shows that the experimental investigations

on the various aspects of joint flexibility have not been carried out on frame specimens. This

is an essential step to verify and substantiate the results and conclusions already established

in this area.

The joint modelling techniques in the stiffness method use the spring concept most

extensively. The main task in the studies is finding the appropriate stiffness of the joints. In

this regard the rigidity factor is used to state the degree of fixity of a joint as a number

between 0 and 1.

Chapter 3

THEORETICAL MODEL

Chapter 3: Theoretical Model 42

3.1 Introduction

The theoretical part of this study considers the dynamic behaviour of beams in order to

establish a model for flexibility of joints. The stiffness matrix of a beam element with two

rotational springs at the ends is derived by exact solution of the differential equation of

displacement (in contrast with approximate or numerical solution). The axial flexibility is

also considered by applying translational springs to a joint. The formulation of axial

stiffness presented in Section 3.4 is derived by the author. The first difference between the

theoretical method incorporated here and those in other studies is the use of inertia forces

as a continuous property in the beam model. This would make the model more

representative of the real structures. Second is the dynamic approach employed in this

study for the calculation of joint stiffness which covers most stiffness and mass properties

of a joint. This is especially important for fatigue life estimation where inertia forces are

also involved.

Only prismatic beams are considered. Damping is not considered herein as this study deals

with steel structures and damping forces are not generally significant in this class of

structures, especially for lower natural frequencies. The stress-strain relationship is also

considered to be linear. These simplifying assumptions are employed herein to avoid a

complicated model. This study is carried out for the first time on tubular joints by the

author and different aspects of it is not known yet. It is, however, necessary to include

damping and especially non-linear material behaviour in the future works to achieve a

more refined model.

3.1.1 Beam theory

The relationship between flexural deformations and bending moments of a beam, is based

on the following assumptions:

1) planes normal to the axis of the beam remain plane and normal, after bending,

2) transverse displacements are attributed only to the longitudinal axis of the beam,

and

3) deformations are small.

The basic differential equation for flexural members then can be derived as:

EIy"=M (3.1)


in which the curvature of the longitudinal axis of a beam, y", is related to the bending

moment M. In equation (3.1), _ is the modulus of elasticity and / is the moment of inertia

of the beam section about its neutral axis. In order to solve or analyse a structure that is

composed of beam members, called a framed structure, each individual member should,

within its domain, satisfy Equation (3.1). Also, continuity of displacements and rotations at

the boundaries of the beam members should be satisfied. This solution result in a system of

linear equations, where the unknowns are the integration constants.

Several techniques have been developed for construction and solution of such linear system

of equations as easily and practically as possible, such as: moment distribution method,

slope deflection method formed into matrix analysis procedures. Matrix analysis

formulation is most c o m m o n because of its suitability for computer programming.

Equation (3.1) states the most simplified version of the flexural behaviour of prismatic

beams. Effects of shear and/or axial deformations and rotary inertia on the bending

deformation can also be included which creates a more complicated equation. The matrix

method of analysis is similar for all cases with more elaborate algebraic manipulation

involved in establishing the beam stiffness matrix when other assumptions are introduced.

This study focuses on the beam formulation excluding axial deformation and rotary inertia

effects, as they are not usually considered in practice, yet.

3.2 Matrix formulation of beams

Rigid connections are commonly used in computer programs for analysis of framed struc

tures. In the stiffness matrix of a beam each nodal point has three degrees of freedom in

plane, that is two translations and one rotation, as shown in Figure 3.1.

__C _}__. k I

6, 5j

Figure 3.1. DOFsfor two dimensional beam element

The system of equilibrium equations in dynamic analysis of framed structures is usually

established by using two types of mass distribution models called discrete and continuous

models.


Discrete model

A discrete mass model assumes that mass and stiffness are decoupled properties in

the equilibrium equations, as shown in Equation (3.2).

K x - © 2 M x = P (3.2)

There are two types of discrete mass models known as lumped and consistent

models. Mass of the members are attached to the nodal points in the lumped mass

model and the beam members are assumed to be massless. However, a more accurate

representation of mass is obtained by expressing a uniformly distributed mass in

terms of the end deflections and slopes. This will result in a full matrix which is

called the consistent mass matrix. Equations (3.3) represent the lumped and

consistent mass matrices for a beam element.

c E, 1, m, I, co 0

a) Degrees of freedom for the mass matrices

M L =

mill 0 0 0 0 ml/2 0 0 0 0 0 0 0 0 0 0

™ mL M c 420

156 54 22/ -13/

54 156 13/ -22/

22/ 13/ 4/2 -3/2

-13/ -22/ -3/2 4/2

(3.3)

b) Lumped mass matrix b) Consistent mass matrix

Continuous model

The stiffness matrix of a beam with continuous mass has been originally developed

by Kolousek in 1939 (Kolousek, 1973). He developed and published the dynamic

stiffness matrix of a beam using Dynamic Deformation method. The mass and

stiffness properties in the continuous model are coupled, therefore the stiffness

matrix includes the mass parameters and stiffness parameters together. A coefficient

of the stiffness matrix, &ij, is the force developed at the beam degree of freedom i due

to the amplitude of vibration one at the degree of freedom j. For example, the

coefficients of the third column of the stiffness matrix correspond to the deformations

and forces shown in Figure 3.2.


k =

0 0 kl3 0 0 0 k23 0 0 0 k33 0 L 0 0 k43 0

Figure 3.2. Stiffness coefficients of beam with continuous mass due to the rotation of left end.

The common form of representing the stiffness matrix of a beam modelled with

continuous mass is shown by Equation (3.4).

FT FT FT FT

TF6(X) TF5(X) TF4(X) TF3(X)

k =

FT FT FT FT

TF5(X) jF6(X) TF3(k) TF<(X)

FT FT FT FT

TF,{X) jF3(X) TF2(X) -pF^X)

L TF^ TF<(X) fF*(X) Tm)

(3.4)

In this equation the coefficients Fx are called the Frequency Functions (Kolousek,

1973). A, is a variable which includes the stiffness properties and frequency of the

beam and is defined later in this section. E, /and / are Modulus of Elasticity, moment

of inertia and length of the beam.

3.2.1 Beam with semi-rigid ends (bending deformations only)

The model used in this study is a beam with two rotational springs h and k] at its ends, as shown in Figure 3.3, where <g) represents a semi-rigid support.

end spring (for this study)

Figure 3.3. Beam with semi-rigid ends (after Kohoutek 1991b)

The stiffness matrix of a beam with end springs was introduced by Monforton and W u

(1963). They used the conjugate beam method to derive a relationship between the forces


and displacements at the ends of a beam member with semi-rigid connections. The elastic

behaviour of the springs was assumed to be linear as given by Equation (3.5).

M = kQ (3.5)

There was no reference to the inertia forces in the work by Monforton and Wu. Grant

(1968) used a similar model to Monforton and Wu's for dynamic analysis of frames but he

assumed a lumped mass formulation. The mass stiffness matrix that Grant used did not

include the spring stiffness of the ends. The stiffness matrix of a beam member with semi

rigid ends, considering flexural deformations only and continuous mass, is derived in this

section. Equation (3.6) shows the differential equation that expresses the equilibrium

condition on an infinitesimal beam element under no external load.

d4v 52v EI^ + m^- = 0 (3.6)

obc4 dt2

v(x,t) is the transverse displacement of the beam axis, and m is the mass per unit length of

the beam. For a harmonic boundary displacement with the frequency of co, the solution

v(x,r) =y(x)sm(ot can be assumed that yields:

EI^4-m(o2y = 0 (3.7) dxA y

or

^-(=^ = 0 (3.8) dx EI

Assuming X = l(mG>2/Er)A the solution of Equation 3.6 will be (Kohoutek, 1985b):

y(x) = Cicos(Xx/l) + C2sm(Xx/l) + Cje'^ + Qe'^^7 (3.9)

in which / is the length of the beam and C/, C_, C3 and Q are the integration constants. For

the degrees of freedom shown in Figure 3.1, the boundary conditions of a beam with semi

rigid joints will be:

8i = y(0) Sj=X/)

e/ = _MO)+/(o) o,=A/(0 (3.10)


in which k\ and k$ are the stiffness of the ends i and; respectively. Equations (3.10) relate

the displacements and rotations of the ends / and j (see Figure 3.1) to the integration

constants in Equation (3.9) and may be shown in a matrix form as:

A = <DC (3.11)

in which A is the displacement vector. Matrix <b contains a number of terms related to the

beam geometry, stiffness and frequency, and matrix C includes the integration factors.

Expanded forms of the matrices used here are given in Appendix A. The end shear forces

and bending moments can be related to the displacement function as:

Vi = V(0) = _i_>""(0) Vj = -V(/) = -_.//'(/)

Mi = -M(0) = -_s_>"(0) Mj = M(r) = EIy'XT) (3.12)

when substituted from equation (3.9) yields:

P = _QC (3.13)

in which P is the force vector. Matrix Q contains a number of terms related to the beam

geometry, stiffness and frequency. Combining Equations (3.11) and (3.13), the stiffness

matrix of a beam with semi-rigid joints can be derived as:

P = __/Q<_r _ (3.14a)

.-. k = ii/Q<D-1 (3.14b)

&ij, a member of matrix k is a function of modulus of elasticity (E), moment of inertia (i),

length (/), mass per unit length (m), frequency (co) and stiffness of the end springs (fa and/or

fcj). For example, k33 is (see Figure 3.2):

£33 = — EI

I

-A[(l-e2X)(cos>.-2aj>,sinA<)+sinl(l+e2x)j

2ex-X(ai+aj)(cosA,+siiiA)+A.e2X(ai+aj)(cosA,-sinA)+cOsA,(l+e2X)+2aiajA,2sinX(l-2e2X) ^ ' '


in which cti = Ellfal and ctj = El/kjl. The complete stiffness matrix of a beam with semi

rigid ends is given in Appendix A.

The relative stiffness of the beam with respect to the stiffness of the end springs is

represented by cti or ctj. For a fixed or a hinged support, a equals to zero or infinity,

respectively. The ideal fixed or hinged conditions, i.e. a support which produces y' = 0 or M

= 0, do not exist in reality. Therefore, a in practice has a value greater than zero.

3.2.2 Effect of shear deformations on bending stiffness

When the effect of shear deformations on bending stiffness is included, the differential

equation of displacement of the beam axis will be:

dx4 Afi dx1 EI )y v

in which As is the shear area of the beam section and G is the shear modulus of elasticity.

The modified boundary conditions for the rotational degrees of freedom are:

8,=j<0) &j=y(l)

e, = -^p-+y'(0)-y's(0) o, =+^+/(0-^(0 (3.17)

in which y's refers to the tangent of the shear displacement curve only. y's can be obtained

from the shear force, V, as:

V T7 _,_,d3y men2 dy.

Rest of the calculations are similar to the case of bending deformation only, described

earlier.

3.2.3 Stiffness determination of a joint

Generally two methods can be used to determine the stiffness or rigidity of a joint via

dynamic measurements:

1) Forced vibration that is measurement of forces and displacements under a forced

vibration and then using equation (3.14a) to calculate the rigidity (Figure 3.4a).

2) Free vibration that is using the property of structures regarding natural

frequencies. The determinant of the stiffness matrix is zero at the resonant conditions.

Natural frequencies are the eigenvalues of the stiffness matrix of a structure and


include the interaction of members meeting at a joint. Therefore, by knowing natural

frequencies, rigidity of a joint can be determined using Equation 3.19 below:

K x = 0 or Det[K] = 0 (3.19)

The procedures for determining the rigidity of a T-joint through dynamic testings are


P=P0Sin cot 5=5 0 Sin cot

Known parameters: frequency, geometry and material properties

Measured parameters: 50 and P0

Unknown parameter to be calculated: rigidity by using: K x 0 = P 0

s\

(1) Forced vibration

^r

6= Sin © ^

Unknown parameters: geometry and material properties

Measured parameter: frequency

Unknown parameter to be calculated: rigidity by using: K x 0 = 0

_____

(2) Free vibration

Figure 3.4. Dynamic methods to determine rigidity of a joint

W h e n using method (1) above, the displacement vector (A) is required in order to

determine the joint stiffness. A n accurate measurement of A is a difficult task in terms of

instrumentation and laboratory equipment. Method (2), shown in Figure 3.4, was used in

this study to evaluate the rigidity of the tubular joints. The advantage of this method is

employing the resonant conditions instead of using the force-displacement relationship of

Equation (3.13 a).

Forced vibration method was used by Springfield and Brunair (1989) on the end flexibility

determination of pre-tensioned cables. The main reason by Springfield and Brunair for

using a dynamic approach was that the real loading on the cables in their study had a

dynamic fashion and caused fatigue problem.


There are only few researchers w h o have used dynamic approach in their analyses.

Kohoutek (1985a) introduced the method of free vibration and applied it to the joints made

of open sections. The present study is using the same experimental method as Kohoutek's

with a different theoretical model described in Section 3.2.

By the knowledge of this author, the dynamic experimental method and analytical model

are examined for the first time on the tubular in this study .

Other methods for flexibility analysis of joints employ the static approaches through using

the ratio of moment to rotation concept. A review of such investigations are reported in

Chapter T w o as a part of the literature survey.

3.3 Bending rigidity factor

The model developed in Section 3.2 for rigidity of a joint produces a rotational stiffness,

which theoretically ranges from zero to infinity. However, it is c o m m o n in practice to

express the degree of fixity of a joint by using a non-dimensional coefficient between 0 and

1. This coefficient is called rigidity factor and is 0 or 1 for a hinged or fixed joint,

respectively. The model of a joint in the present study is similar to the model used by

Monforton and W u (1963) in using a spring to express the joint stiffness, but the rigidity

factor defined here is slightly different.

To define a rigidity factor, this author has followed the pattern of moment rotation

relationship of a beam with end springs. Considering the beam member in Figure 3.5, 0j = 1

and rest of the D O F s are restrained.

' 33 5j=9j=5j=0

Figure 3.5. Beam model for definition of rigidity factor (unit length)

The relationship between the bending moment and the rotation of end / is:

Mi = k33Qi = k33(\) = k33 (3-2°)


k33 is the diagonal coefficient of the stiffness matrix of a beam with semi-rigid ends

corresponding to the rotational degree of freedom (see Appendix A ) , which is maximum

for a rigid and zero for a hinged end. The variation of k33 with respect to the end flexibility

cti is shown in Figure 3.6 for different X values. X equals to zero represent the static case.

X = / (m(o2/EI)1M

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111

0 2 4 6 8 10 end flexibility, a j

Figure 3.6. Relationship between rotational stiffness and beam end flexibility

It is seen that a variation of X from 0 to 3.5 (X = 3.5 indicates a high level of dynamic

behaviour) does not have that much effect on the shape of the curves that state the

relationship between k33 and ctj. A n equation for the rigidity factor vb is proposed here,

based on the relationships shown in Figure 3.6, as:

1 u EI

vk = , where a = -rz l + 4a kl (3.21)

Equation (3.21) is also shown in Figure 3.6, which is actually a normalised form of k33

when X = 0.


3.4 Stiffness of a joint due to brace axial loading

Rotational flexibility is generally most important in the analysis of framed structures.

However, due to the radial flexibility of pipes in tubular structures (diameter to thickness

ratio effect), axial deformation of brace is also influenced by the flexibility of chord wall.

To consider the axial flexibility of a joint, an axially loaded member with two translational

end springs, as shown in Figure 3.7, is considered.

AAMAr — ^ I

*t*— —*t*— —*"

i , J 0 / 0

Figure 3.7. Axially loaded member with semi-rigid ends

The differential equation of equilibrium for the axially loaded member of Figure 3.7 under

no external load is:

EA^-m^ = 0 (3.22) dx2 dt2

in which h(x,i) is the function that describes the axial deformation, E is the modulus of

elasticity and A is the cross sectional area. A solution as h(x,t) = u(x)sm((at) may be

assumed which is a harmonic boundary displacement with a frequency of co. Therefore,

Equation (3.22) can be written as:

EA^- + ma2u = 0 (3.23) dx2

or,

^ + ( _ ! _ _ _ 1 ) W = 0 (3.24) dx2 " EA

Assuming \j/ = l(m(o2/EA),/2 the solution of Equation (3.23) will be

u(x) = C5sin(\|/x//) + C.cos(\|/x//) (3.25)


For the degrees of freedom shown below, the boundary conditions of an axially loaded

member with semi-rigid ends are given by Equation (3.26).

Ii "Hj PJ

T l , = - ^ + «(0)

1 ^=+-^r+«(0 (3-26)

fc'i and £'j are the axial stiffness of the ends / and j respectively, P(x) is the internal axial

force of the member. Substituting from Equation (3.25) in (3.26), it can be written in a

matrix form similar to Equation (3.10) that:

A_ = OflCa (3.27)

in which A_ is the displacement vector. Matrix <E>_ contains a number of terms related to the

geometry, axial stiffness and frequency, and matrix Cfl includes the integration factors, Cs

and C_. Expanded forms of the matrices used here are given in Appendix B. The axial

forces at the end of an axially loaded member can be written as:

Pi = -P(0) = -EAu'(0) P] = P{l) = EAu\l) (3.28)

when substituted from equation (3.25) in (3.28):

Pfl = i_Q_C_ (3.29)

in which P_ is the force vector. Matrix Qfl contains a number of terms related to the

member geometry, stiffness and frequency. The stiffness matrix of an axially loaded

member with semi-rigid ends can be obtained by combining Equations (3.29) and (3.27),

resulting:

ka = EAQaOa-1 (3.30)

fcj, a member of matrix k_ is a function of modulus of elasticity (E), sectional area (A),

length (/), mass per unit length (m), frequency (co) and stiffness of the end springs (k\

and/or kj). For example ka is:


ka = EA \j/(cosi|/ - Pjsinu/)

/ i|/cos\j/(pi+pj) + sinvi/(l-pipjV|/2) (3.31)

in which pi = EAIk'i and PJ = EA/k)l. The complete stiffness matrix of an axially loaded

member with semi-rigid ends is given in Appendix B. p values represent the relative axial

stiffness of the member and the end springs. P = 0 and P = oo are limit values for rigid and

sliding supports, respectively.

Axial stiffness of a joint can be determined similarly as for the bending case (Figure 3.4)

by conducting forced or free vibration tests. In the axial tests the force P or, in case of free

vibration, the excitations are acting in the axial direction of the brace.

3.4.1 Axial rigidity factor

A rigidity factor can be defined to show the degree of fixity of a joint under axial

deformations. The factor maps the joint stiffness to a range from 0 to 1. In this study, the

relationship between axial force and displacement at the end of an axially loaded member

is used to establish the function that converts the joint stiffness to the rigidity factor. This is

in fact the diagonal coefficient of the stiffness matrix of an axially loaded member. Figure

3.8 shows the variation of &," with respect to pi (= EA/k[l) for a member fixed at nodal

pointy.

v f / = 0 ' ( V f l = l + p 7)

F! = k°(1) = k?i

\y=l(m®2/EA}/2

J K

axial end flexibility, p,

Figure 3.8. Relationship between axial stiffness and axial end flexibility


Accordingly, the following equation for the axial rigidity factor is suggested:

v.^. where p-ff (3.32)

3.5 Summary

The dynamic stiffness matrix of a beam with semi-rigid supports was derived in this

chapter considering continuous mass and stiffness (ET) properties. The beam deformations

which were assumed included: 1) only bending properties, and 2) bending plus shear

properties. A coefficient of the stiffness matrix is a function of modulus of elasticity (E),

moment of inertia (7), length (/), mass per unit length (w), frequency (to) and stiffness of

the end springs (fa and/or kj).

The theoretical model used in this thesis for determination of the semi-rigidity of a joint

employs the resonant condition and natural frequencies. This approach is examined for the

first time on the tubular joints in this thesis. A new definition for rigidity factor, which uses

the bending behaviour of a beam, was given to produce a measurement number for rigidity

between 0 and 1.

The dynamic stiffness matrix of an axially loaded member with semi-rigid ends was also

derived in this chapter. The stiffness matrix include both stiffness and continuous mass

properties. For the axial case, a rigidity factor which varies between 0 and 1 is also

suggested.

Chapter 4

EXPERIMENTAL DETERMINATION OF

JOINT STIFFNESS

Chapter 4: Experimental Determination of Joint Stiffness 57

4.1 Introduction

A dynamic method of testing is used in this study for evaluating the rigidity of tubular joints.

The word dynamic is used here because inertia forces are involved in the applied experi

mental procedures. The method has been developed and previously used by Kohoutek

(1985a) for inplane rigidity of open section joints. In this study, the inplane and out of plane

flexibility of tubular T-joints are investigated experimentally. The theoretical base used to

determine joint stiffness from the experimental results is described in Chapter 3.

The dynamic method of analysis employed in this study to determine the stiffness of tubular

T-joints, is not as cumbersome as the c o m m o n procedures where load and displacements

are measured. Furthermore, the measurements of natural frequencies on steel specimens are

easier and less complicated compared to the static measurements of displacements.

4.2 Test specimens

Eleven tubular T-joints were tested with the geometrical dimensions and non-dimensional

parameters reported in Table 4.1. The specimens were fabricated from steel pipes readily

available in Australia. Subsequently, from a limited choice of diameter and thickness

dimensions available selection was made for the specimens. A survey of the non-dimensional

parameters P (= dID), y (= D/2T) and x (= tIT) was made to ascertain the apphcability of the

test pieces to real structures. The results of this survey are shown in the next section.

Table 4.1. Geometrical dimensions and non-dimensional parameters of tested joints

Joint

Tl T2 T3 T4

T5 T6 T7 T8

T9 T10 Til

D(mm)

219.1

219.1

219.1

219.1

219.1

219.1

219.1

219.1

219.1

219.1

219.1

T(mm)

8.2 8.2 8.2 8.2

6.4 6.4 6.4 6.4

4.8 4.8 4.8

Um)

1.50

1.50

1.50

1.50

1.50

1.50

1.50

1.50

1.50

1.50

1.50

d(mm)

219.1

114.3

114.3

60.3

219.1

114.3

114.3

60.3

219.1

114.3

60.3

t(mm)

4.0 6.0 4.8 3.9

4.8 6.0 4.8 3.9

4.8 4.8 3.9

Km)

1.23

1.23

1.23

1.23

1.23

1.23

1.23

1.23

1.23

1.23

1.23

P 1.00

0.52

0.52

0.28

1.00

0.52

0.52

0.28

1.00

0.52

0.28

Y

13.36

13.36

13.36

13.36

17.12

17.12

17.12

17.12

22.82

22.82

22.82

I

0.49

0.73

0.59

0.48

0.75

0.94

0.75

0.61

1.00

1.00

0.81

a

13.69

13.69

13.69

13.69

13.69

13.69

13.69

13.69

13.69

13.69

13.69

See Figure 1.5 for the notations used in this table.


The selected values for P and y cover the limits of the respective parameters in the offshore

structures plus one value approximately in the mid-range. The chord length parameter, a,

was kept 13.69 in all specimens. Strictly speaking, the length of the chord should be

selected according to the condition of a member in a real structure. However, other studies

show that a has a minor influence on the flexibility of a T-joint, which is also indicated by

the flexibility and SCF parametric formulae, not including any a parameter. The effect of a

on joint flexibility is discussed in Section 6.2. The reader is referred to Chapter 6 and

Appendix G, for flexibility and SCF parametric formulae, respectively.

4.2.1 Survey of non-dimensional parameters of T-joints (P, y and x)

The test specimens in this study were restricted to T-joints and therefore only the survey of

those joints will be shown here. To choose the practical ranges for the non-dimensional

parameters of the joints, two surveys were examined.

First, the survey reported in U E G (1985) for six different structures in the North Sea, which

were believed to include a common ranges of variables. The total number of intersections

were about 2700 for p population, and 1602 for y parameter. In this set, there was

insufficient information available to identify x ratio. The number of the joints corresponding

to P and y ranges are shown in Figure 4.1 and 4.2.

Figure 4.1. Distribution of P parameter for T- Figure 4.2. Distribution ofy parameter for T-joints. joints. The survey was carried out by UEG (1985) The survey was carried out by UEG (1985) on six on six offshore structures offshore structures

Second, was the database also referenced in U E G (1985). The results for p, y and x are

shown in Figures 4.3, 4.4, and 4.5. The non-dimensional parameters of the tested joints in

this thesis vary as follows:

0.28 <P< 1.0

13.36 < Y < 22.82 (4-1)

0.49 < x < 1.0


150- 140

26 36 46

Y Figure 4.3. Distribution of y parameter for T-joints. The survey was carried out on the database in UEG (1985)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4.4. Distribution of $ parameter for T-joints. The survey was carried out on the database in UEG (1985)

25 -

20 ..

1 5. o

£ 15 4-

10 ..

5 ..

21

6 6

| K S _ | KSSSI t KSSS11

14

l ^ t ^ l ^ <0.15 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 >0.95

T

Figure 4.5. Distribution of % parameter for T-joints. The survey was carried out on the database in UEG (1985)

As can be seen, the most common values of y in offshore structures are in the range of 10 to

20, which indicates the range chosen for the specimens. A range of 0.15 to 1.0 was found

for P with most joints having P values between 0.4 and 0.6. The survey also shows the

following limits for the non-dimensional parameters of T-joints:

0.2 < p < 1.0

5 < Y ^ 35

0.2 < x < 1.0

(4.2)

4.2.2 Material properties and fabrication of test specimens

The steel pipes used for the fabrication of the test specimens were seamed and had the

properties shown in Table 4.2. The pipes were m a d e of mild steel and manufactured in

Australia by B H P Company.


Table 4.2. Metallurgical properties of the pipe sections used for test specimens

Element

C P Mn Si S Al CE

Average%

0.14

0.023

0.74

0.12

0.007

0.022

0.32

M i n i m u m %

0.13

0.014

0.65

0.10

0.003

0.014

0.29

Maximum%

0.16

0.029

0.80

0.13

0.012

0.040

0.35

Brace sections were welded to the chords using Complete Joint Penetration Groove Weld.

Details of the weld used for the fabrication of the specimens are shown in Figure 4.6.

1) Weld size was 6mm. There were 3

passes on each joint.

3rd and last pass

2) Preparation on adjoining brace was a 45° bevel, feathered edge with 1.6mm gap

(penetration gap) for full penetration root

run.

3) Weld passes 1 and 2 were ground out

before proceeding.

Ground out edges

Figure 4.6. A typical weld cross section of a T-joint specimen

The Modulus of Elasticity (E) was confirmed experimentally to be between 200 and 210

GPa. Therefore, the common value of E = 200 GPa was used through this study in

calculations.

4.3 Test set-up

The test set-up used to determine joint flexibility is shown diagrammatically in Figure 4.7.

The main equipment for testings was as follows:

-Fourier Analyser, 2630 Tektronix,

-Personal Computer,

-accelerometer, and

-charge amplifier.


Chord

fWRffJ

Accelerometer

Brace

_

Charge amplifier

\ Spectrum analyzer

__

Figure 4.7. Diagrammatic test set-up for frequency measurement of T-joints (IPB)

The vibration induced by impacting the cantilever, was picked up by an accelerometer

mounted on the desired spot (usually tip of the cantilever on a joint). The output signal from

the accelerometer was amplified and then processed by the Fourier Analyser.

The Fourier Analyser, Tektronix 2630, digitised and calculated the frequency components

of the input signal. It was geared to a Personal Computer for its software needs. In addition,

the P C was used for recording the measured data generated by the Analyser.

Figure 4.8a shows one of the T-joints supported in place for testing. The support assembly

is shown in Figure 4.8b.

Figure 4.8a) T5, one of the joint specimens on the support assembly


Figure 4.8b) Hinged support assembly

4.3.1 Test procedure

The natural frequency of a joint corresponding to the cantilever mode of vibration was

measured in the tests. The joints were excited by a simple impact load through a light

hammer or hand and the frequency spectra were calculated by the spectrum analyser and

recorded on disk. Since the desired frequencies were lower than 500Hz, using hand or

hammer with a rubber tip seemed to be adequate. A typical frequency spectrum of a T-joint

is shown in Figure 4.9 with the peaks as the natural frequencies. It is necessary to verify the

recorded peaks to make sure that they are related to the desired mode shapes. This is

important to avoid any noise and interference from other sources.

-90

dB Volts 5Q/Div

-10

10 /Div

- - - - ) - - - r - - - ( - - - r - - - l - - - T - - - | - - - T

- - -i- - J A - - -i- * - *• - M -i- - - * Jl - -1 - - - •*

A — -_^ _ _ k - _ — •-.-_-. _— • _\_ _ _•* . A — • - - - J

- Hertz- 500 Ch=1/ ASPECa f=95Hz

Figure 4.9. Typical frequency spectrum of a tubular T-joint


Three methods were examined to confirm the correspondence between natural frequency of

a joint and the related mode shape. The methods are:

1) Eigenvalue analysis by Finite Element method using one dimensional beam elements.

This would clarify the mode shapes which should be expected from the test samples and

give some general idea about the dynamic behaviour of the joints. Of course, using more

complicated finite elements such as plate and shell or three dimensional element will

produce more realistic results than using one dimensional beam elements. However, for

mode shape identification of a joint, simple methods of analysis still can be sufficient The

application of the F E M to the flexibility analysis of tubular joints will be discussed further in

Chapter 5.

A series of analyses were carried out to identify the mode shapes of the joints using ALGOR

computer program. This program assumes a lumped mass formulation for the inertia forces

in dynamic analysis. A typical joint model is shown in Figure 4.10, where the springs, at the

supports, are a result of the provisions used to simulate hinged conditions and will be

explained in Section 4.5.1.

lumped mass f

|-VWv\_-^-^-A-^^svWv^| * S77777 /t///9 ^

Figure 4.10. Lumped mass model used to establish the mode shapes of joints

The first two mode shapes due to the inplane bending mode of the specimens are calculated

by using ALGOR and shown in Table 4.3. The mode shapes of the joints are also reported in

Appendix C. Obviously, the connection of the chord and brace in the ALGOR analyses were

assumed to be rigid.

2) Eigenvalue Analysis using Dynamic Deformation Method. This method of analysis is

similar to method 1 above. The difference is only due to the mass formulation of the beam

elements. Method 2 assumes a distributed mass properties which is an improvement on the

assumption of lumped mass used in ALGOR program in Figure 4.10. First two natural

frequencies of the T-joints calculated by using method 2 for in plane bending mode are


shown in Table 4.4. T o make the analyses consistent with method 1, the connection of

chord and brace in this series of analyses were also assumed to be rigid.

Table 4.3. Natural frequencies of the T-joints using lumped mass theory (IPB)

Joint

T l

T2 T3 T4 T5 T6 T7 T8 T9 TIO Til

hinged support

1st mode 7.74

8.03

8.21

8.73

8.13

8.77

9.01

9.70

8.82

9.98

10.94

2nd mode 140.88

74.96

75.03

37.93

140.20

75.89

75.86

38.23

139.75

76.99

38.67

fixed support

1st mode 7.74

8.03

8.21

8.73

8.13

8.77

9.01

9.70

8.82

9.98

10.94

2nd mode 143.98

75.31

75.32

37.95

144.41

66.33

76.22

38.25

144.86

77.47

38.70

Table 4.4. Natural frequencies of the T-joints using Dynamic Deformation Method (IPB)

Joint

T l

T2 T3 T4 T5 T6 T7 T8 T9 T10 Til

hinged support

1st mode 7.74

8.0

8.22

8.73

8.13

8.76

9.01

9.70

8.82

9.98

10.93

2nd mode 143.09

76.27

76.37

38.62

142.49

77.22

77.20

38.92

141.96

78.35

39.37

fixed support

1st mode 7.74

8.03

8.22

8.73

8.13

8.76

9.01

9.70

8.82

9.98

10.93

2nd mode 146.28

76.63

76.67

38.65

146.82

77.67

77.58

38.94

147.22

78.83

39.40

3) Modal Analysis. This is an experimental procedure used to characterise the dynamic

properties of an elastic structure in terms of its modes of vibration (SMS, 1990). A mode of

vibration is a global property of a structure and is defined by a specific natural frequency

and a mode shape. The measurements for modal analysis are obtained by exciting the

structure and measuring its responses at various points across its surface. The

measurements in this study were acquired by using a Fourier Analyser. The different natural

frequencies and mode shapes were then processed using a program called STAR. Figure

4.11 shows the first two mode shapes of joint T10 calculated through modal analysis.


Frequency: 11.59Hz Frequency: 46.71 Hz

Mode#: 1 Mode#: 2

Figure 4.11. First two mode shapes of T-joint TIO from Dynamic Modal Analysis

Modal analysis can reveal any gross imperfection which may be present in the specimen

assembly and its supporting system. It is more realistic than methods 1 and 2, as it includes

direct measurements from a joint specimen. However the cost of modal analysis is higher

than the other two methods and in some cases it becomes very tedious such as in tubular

structures or generally in shell structures. Hence, method 3 may be used for identification of

the dynamic behaviour of a structure wherever a high accuracy is required or when the

computer modelling techniques do not work accurately. Obviously, only methods 1 and 2

can be used in the design stage before a structure is fabricated.

After the relevant natural frequency to the cantilever mode of bending is determined, rigidity

or flexibility of the joint can be calculated using the theoretical model. Similar procedures

were used for out of plane bending modes.

4.3.2 Test results and discussion

The measured natural frequency of the T-joints due to the cantilever mode of vibration are

reported in Table 4.5. This table also shows the calculated results by Dynamic Deformation

method ( D D M ) and ALGOR analyses. It is seen that there is a distinct difference between

the measured natural frequencies of the joints and those frequencies resulted from the

stiffness method analysis, using one dimensional beam element and rigid joints. It is also

seen that the natural frequencies resulted from lumped mass model (ALGOR) are less than

the values from Dynamic Deformation Method and closer to the measured results. This is

because the resultant inertia forces from lumped masses produce bigger deformations than

the distributed inertia forces in D D M model. This situation is similar to the different

maximum static moments under distributed and concentrated loads in the framed structures.


Table 4.5. Natural frequency of the T-joints due to cantilever mode of vibration

Joint

Tl T2 T3 T4 T5 T6 T7 T8 T9 TIO Til

Inplane bending

Measured 95 55 56.3

33.8

88.8

50 51.3

31.3

82.5

46.3

28.8

DDMt 143.09

76.27

76.37

38.62

142.49

77.22

77.20

38.92

141.96

78.35

39.37

ALGOR* 140.88

74.96

75.03

37.93

140.20

75.89

75.86

38.23

139.75

76.99

38.67

Out

Measured

46.1

30.8

33.4

27.3

36.8

24.4

26.5

23.1

33.3

20.1

17.9

of plane bending

DDMt 104.95

66.52

68.06

37.07

94.61

65.41

67.09

37.00

87.33

65.70

36.90

ALGOR* 103.72

65.72

67.23

36.60

93.54

64.63

66.28

36.54

86.51

64.92

36.44

fDDM: Dynamic Deformation Method (bending only).

XALGOR: Finite Element Package (lumped mass model).

Figure 4.12 shows the relationship between natural frequency of the joint specimen obtained

from measurements and Dynamic Deformation analyses. The difference seen between the

measured and calculated natural frequencies is the basis for determination of the rigidity of a

joint. In theory, a joint may be assumed rigid when its calculated natural frequency, using

Dynamic Deformation method, is equal to its measured natural frequency. Obviously, due to

the flexibility of tube walls there is no such joint in reality.

150-

140-

£ 130-

_ - „ -

g 110-S loos' 90-^ 80-• 70-3 60-36 50-

40-30-20

measurement • - - - calculation, Dynamic

Deformation method -D- -^13.36 -_- r="12 -0— fll.il a f=13M it f\l.\l * 0- 1*22.82

0

i 1 1 1 1 — | 1 1 1 1 1 | i | i |

0_> 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Diameter ratio, P

Figure 4.12. Relationship between natural frequencies of specimens and diameter ratio

It is noted in Figure 4.12 that the joints which have large diameter ratios (fJ) or small chord

to thickness ratios (y) show the highest frequencies. This indicates the higher stiffness of

these types of joints which will be also confirmed in Chapter 6 when the stiffness of the

joints are calculated.

http://fll.il


4.4 Calculation of rigidity factor and joint stiffness

Determination of the rigidity factor (v_) for a joint is, in principle, carried out using the

following equation:

K x = 0 (4.3)

in which K is the stiffness matrix of the joint and x is the displacement vector. Equation

(4.3) represents an eigenvalue problem, which can be used to detennine the natural frequen

cies of a structure. The non-trivial solution for the displacement vector x, is obtained when

the determinant of the stiffness matrix K becomes zero. The natural frequencies are the

roots of the resultant equation.

The method explained in Chapter 3 is used here for the calculation of rigidity factor (v*,) of

a joint. This method uses a continuous mass property to calculate the dynamic stiffness

matrix.

Following the concept of zero determinant for the stiffness matrix, rigidity factor can be

determined when the natural frequency is known. Elements of the stiffness matrix K in

Equation (4.3) are functions of E (modulus of elasticity), / (moment of inertia), / Qength),

vft (rigidity factor), m (mass per unit length) and co (frequency). After measuring the natural

frequency, Equation (4.3) can be solved for vb. Furthermore, the spring stiffness of the

brace connection to the chord in a T-joint can be calculated using Equation (4.4) shown

below:

1 , , 42*/ v, v„ = T-FT, a n d * = v" , AEI

1+ kl

L l-v„ (4.4)

The flow chart for the calculation of rigidity factor and finally stiffness of a T-joint is shown

in Figure 4.13.

Natural frequency measurement

(Fourier Analyser)

Kx = 0

*: = 4EI vb

/ l-v„

Figure 4.13. Stiffness calculation of a T-joint using measured natural frequency


The analytical model which suggests the flow chart of Figure 4.13 is a joint constructed

with one dimensional beam elements (line elements) as shown in Figure 4.14. The

connection between brace and chord is assumed to be semi-rigid with the spring stiffness k.

The supports were designed to simulate the ideal hinge condition for inplane bending as

closely as possible, but in the analytical model a semi-rigid connection can also be assumed,

provided the rigidities are known. However, as it will be shown in the next section, chord

supports can be safely assumed as hinged for inplane bending mode.

A similar model was used for out of plane bending mode. The supports, though, were

assumed to be rotationally fixed.

_ —77 y r^ /7777

Figure 4.14. Analytical model used for rigidity calculation of tested T-joints (IPB)

Calculated rigidity factors (v_), stiffness and related natural frequencies of the tested joints

for inplane and out of plane bendings are reported in Table 4.6. Rigidity and joint stiffness

have been calculated according to the flow chart in Figure 4.13.

Table 4.6. Measured na

Joint Tl T2 T3 T4 T5 T6 T7 T8 T9 TIO Til

tural frequencies (<£>), rigidity factor (v^), and stiffhe

IPB

CO (Hz)

95 55 56.3

33.8

88.8

50 51.3

31.3

82.5

46.3

28.8

vb

0.389

0.504

0.530

0.763

0.318

0.399

0.426

0.642

0.256

0.329

0.526

*(kNm) 6454

1984

1825

576 5620

1298

1197

323 4163

789 200

ss (k) of the tested T-joints

OPB

co(Hz)

46.13

30.75

33.38

27.25

36.75

24.38

26.50

23.13

33.25

20.13

17.88

vb

0.117

0.191

0.226

0.540

0.069

0.117

0.141

0.390

0.061

0.077

0.235

*(kNm)

1346

462 470 211 895 258 266 115 787 134 55


4.5 Effect of joint support conditions

Inplane bending

The effect of chord support conditions on the natural frequency was examined for inplane

bending m o d e using the following two approaches:

1) T w o models of the joints with the limit states for their boundaries, i.e. hinged and fixed,

were analysed for the first two natural frequencies using the Dynamic Deformation method.

In these analyses the joints were assumed to be rigid. The variation of the natural

frequencies of a joint when its support conditions vary from hinged to fixed will indicate

h o w much the structure is sensitive to the boundary conditions. The results of the analyses

are reported in Tables 4.3 and 4.4. It is noted that the effect of support fixity of a chord can

be as high as 3.7% on the IPB natural frequency of the joint when compared to a hinged

support.

2) Using results of the tests on the joints that have the hinged supports described in Section

4.5.1, the rigidity factors and stiffness of the joints were calculated based on two analytical

models described above (one supported on hinged and the other on fixed supports). The

results reported in Table 4.7, show that the maximum modelling error can be 4 % . This

indicates that the change of the support conditions of the analytical model from one extreme

to the other, has effectively no influence on the calculated stiffness of the joint

Table 4.7. Joint stiffness calculated for hinged and fixed analytical models using test results of specimens that have hinged supports

Joint


v„

(supports

0.389

0.504

0.530

0.763

0.318

0.470

0.399

0.642

0.256

0.329

0.526

fc(kNm)

are hinged)

6454

1984

1825

576 5620

1298

1197

323 4163

789 200

v_ (supports

0.381

0.502

0.528

0.763

0.310

0.398

0.424

0.642

0.250

0.327

0.525

fc(kNm)

are fixed)

6248

1968

1808

576 5428

1288

1186

323 4029

783 199

Khinged/Kfixed

1.03

1.01

1.01

1.00

1.04

1.01

1.01

1.00

1.03

1.01

1.00

According to the results of the analyses presented in this section, chord support conditions

have a negligible effect on the calculated IPB rigidity. This will be also shown in Section

5.5.1 when Finite Element analysis of the tested joints are described.


Out of Plane bending

In out of plane bending mode the chord of a joint is subjected to torsional moments.

Therefore, a fixed conditions at the supports were required for out of plane bending mode,

otherwise instability problems would have occurred.

4.5.1 Provisions of end supports for test specimens

Inplane Bending

Although the boundary conditions of a chord have a very small influence on the calculated

rigidity and stiffness of a T-joint for inplane bending mode, the best arrangement was made

to simulate a hinged condition for the supports of the test specimens. For inplane bending,

the support arrangement provided for the joints is illustrated in Figure 4.15.

The support conditions shown in Figure 4.15 were used for both ends of a chord. The

reeds, which were made from thin spring steel, could induce effectively no bending

resistance to the joint specimen. The reeds were glued in place to reduce the introduction of

any vibrational noise. The stirrups around the chord held it firmly along the perimeter

causing a uniform radial pressure on the chord wall. Because all the components in the

support assembly were held in contact with each other, vibrational noise was minimal and a

clear frequency spectrum could be obtained.

Figure 4.15. Support arrangement for inplane bending test


Out of plane bending

A fixed support condition was tried to be made in out of plane bending tests. The

arrangement for the supports is shown in Figure 4.16. There is a solid stirrup at each end of

the chord holding it in place. The reaction forces at the supports are torsional moments and

are provided by the frictional forces between the stirrups and the chord wall. If there was

any looseness between the stirrups and chord walls, it would have produced instability in the

test specimen and could be easily detected. Furthermore, the tests were conducted with a

low amplitude of vibration which did not produce large forces at the supports, so the

support conditions could be assumed to remain fixed when a joint specimen was vibrating.

The whole support assembly was bolted to the floor using high tensile strength bolts.

It will be seen in Chapter 5, though, that the integration which bolting has created

apparently has not been sufficient to produce vibrational fixity for the joint specimens with

large P ratios. This problem requires further investigation that can be pursued as the future

research work of the thesis.

Figure 4.16. Support arrangement for out of plane bending test


4.6 S u m m a r y

The experimental part of this study includes testing of eleven T-joints. The geometrical

specifications of the specimens are selected to cover the range of parameters commonly

used in offshore structures. To determine the range of parameters, a survey was carried out

on the non-dimensional parameters of a number of offshore structures. The database for the

survey was taken from U E G (1985). The survey indicated the highest population of joint

parameters: P, y, and x to be 0.5, 15, and 0.5 respectively.

The pipes used for the fabrication of the test specimens were seamed and made of mild

steel. The primary measurements in the tests were natural frequencies of the specimens

which was determined by a Fourier Analyser. A free vibration procedure as described in

Section 3.2.3 was employed in the experiments. The excitations were applied by hand or

through a hammer with a rubber tip. T w o modes of deformation i.e. inplane bending and

out of plane bending were considered. Computer modelling and modal analysis were used

to ascertain the correspondence between a measured natural frequency and the related

mode shape. A clear difference was seen between the measured natural frequencies and

those obtained from numerical modelling assuming a rigid joint. For example joint T9 had

a free vibration with a frequency of 83Hz for in plane bending whereas numerical analysis

showed 140Hz. This difference between measured and computed frequencies was the basis

for the calculation of joint stiffness.

The stiffness values of the tested joints were calculated based on the method presented in

Sections 3.3 and 4.4. It was noted that the joints with large diameter ratios or small chord

to thickness ratios possess higher stiffness values compared to other ranges of parameters,

both for inplane and out of plane bending modes. This is also discussed in Chapter 6 when

the stiffness of T-joints are calculated using F E Method.

Effect of support conditions on the natural frequencies of a joint was investigated for

inplane bending mode. The results show a maximum difference of 4 % in the calculated

stiffness of a T-joint when the assumed conditions for its supports are changed from hinged

to fixed which shows the insignificance of support fixity for IPB mode. Despite the small

effect of support conditions on the inplane bending behaviour of T-joints, a special support

arrangement was made to simulate the true hinged conditions.

Out of plane bending requires fixed supports as the chord of a joint is under torsional

loading. However, it is indeed difficult to build rotationally fixed support conditions,

especially when the structure experiences dynamic forces. This is clearly observed in the

difference between the F E analysis and the experimental results of the joint specimens in

IPB and O P B modes and presented in Section 5.6.1.

Chapter 5

DETERMINATION OF JOINT RIGIDITY

USING FINITE ELEMENT METHOD

Chapter 5: Determination of Joint Rigidity Using Finite Element Method 74

5.1 Introduction

The Finite Element Method (FEM) is the only analytical procedure that is widely used to

analyse the structural behaviour of tubular joints. The ability of the F E M to model arbitrary boundary conditions is a significant advantage over other methods used to solve different types of structural problems. However, it is still an approximate solution and requires to be verified by other methods that are mainly experimental.

This chapter gives a brief introduction on the F E method. Further, the results of the FE analyses of the test specimens of Chapter 4 is reported and discussed.

5.2 A review of the Finite Element method

The behaviour of most structural systems may be formulated by using the differential equation that expresses the interaction between the system components. The basic differential equation governing the behaviour of a three dimensional solid structure made of an elastic material can be determined by imposing the equihbrium conditions on a cube with

the sides dx, dy, dz, yielding the following set of equations:

_____.+__!_L+__!__.+p__lu

dx By dz dt

«*-_» Cri„ o «_ ,, ~

d<Tv dx„ dxvx d2uv _, n ,_ ..

*r+lf+lf+p#+r=0 (51)

^ _ + _ I _ + _^_.+p|^+Z = 0 dz dx dy dt2

o„, oy,..., %xz are the stress components due to the body and surface loads. The body forces

per unit volume are denoted by X, Y, Z in Equation (5.1). If the surface forces per unit area are assumed as X\ Y and Z', the equilibrium conditions at the surface can be represented by

Equations (5.2) below:

X' = Gxl + Xxym+ Xxtfl

Y' = Gym + Tyzn+ Xxy/ (5.2)

Z' = o_n + X_+ Tyz/n


I, m and n are the direction cosines of the external normal vectors at the point under

consideration. The six components of strain at each point can be calculated by using the

three displacement functions u, v and w. Hence, the components of strain can not be taken

arbitrarily as functions of x, y and z and they are interrelated. These interrelations are known

as the equations of compatibility or Saint-Venant's compatibility equations. The stress

solution of an elastic system must satisfy all conditions of internal and external equilibrium

plus the compatibility conditions.

The Finite Element Method employs the above conditions in terms of displacement

components, aiming to obtain the best approximation to the solution of u, v and w. This is

achieved by dividing the elastic body into smaller segments (finite elements) and

approximating the displacement fields in each element with an arbitrary function. The finite

elements are interconnected via nodal points. There are many numerical methods to

approximate a function, but the c o m m o n practice is assuming polynomial functions, called

shape functions or interpolation functions, with unknown coefficients. The equilibrium

conditions described by Equations (5.1) are imposed on each element, allowing the

evaluation of the unknown coefficients introduced by the shape functions. Generally, the

method is formulated so that the unknown coefficients are displacements or rotations at the

nodal points for a given load. After determination of the displacement functions for each

element, strains are calculated and then stress components can be obtained using Hook's

law.

The F E M is more sophisticated than as explained above with various numerical problems

ranging from computational stability of finite mathematics to the convergence of the

solutions. However, the method is generally very applicable to the stress analysis of

structural systems.

Another method for stress analysis of structures is the Finite Difference Method. This

method is not as popular as the F E M but is still used in practice. The difference between

Finite Element and Finite Difference method in the solution of a differential equation is that

the latter approximates the desired function as a whole, while the former attempts to satisfy

the differential equation for each individual element, while mamtaining the continuity

conditions.

5.2.1 Galerkin Method (a finite element formulation)

In this section a version of the Finite Element formulation, which is called the Galerkin

method, is described. Assuming a differential equation as:

L(M)+P = 0 (5-3)

with L ( M ) as a differential operator, a solution u can be assumed such as:


M

" (5.4)

where TV, are linearly independent trial functions or interpolation functions and a, are M

multipliers to be determined in the solution. ub satisfies the boundary conditions and X «.A. »=i

is zero at the boundaries. The unknowns a, can be determined by using Nj as weighting

functions, so that

jf/y,<L(«)+p)_/i_) = 0 (5.5)

where ii is the domain of u. Equation (5.5) provides adequate number of equations to

calculate a, values. It results in to the following equations:

r M

J Nj{L(ub) + X ««L(M) + p]dQ = 0, and (5.6) a »=i

fNJL(ub)dQ + X OifNUNdda + fp NjdQ = 0 (5.7) a t=i •__ 'a

Equation (5.7) can be restated in a matrix form to be used for computer programming

which suggests:

K a = f (5.8)

where,

kji = - / NjLiNddQ, and f} = - / NJUub)dQ - J NjPdCl (5.9)

in which kji, at and fi are the elements of k, a and f, called the stiffness matrix, the

displacement vector and the load vector, respectively.

Three main criteria should be observed by the interpolation functions in order to obtain

convergence upon the exact solution. The criteria are: 1) rigid body displacement, 2) a

constant strain condition, and 3) continuity. The rigid body condition ensures that no strain

will be developed when a constant displacement is imposed on an element The second

criterion, i.e. constant strain, is required upon convergence to the exact solution when the

finite elements are very small which demands effectively constant stress distribution in each

element The third condition is the continuity of displacements or certain derivatives of

displacements. The mathematical proof of these criteria is not in the scope of this thesis and

is available in the mathematics references.


5.2.2 Shell Finite Element

The F E M becomes complicated for certain types of structures such as plates and shells. A

review of the shell finite element literature reveals that shell and plate modelling have not

been solved as completely as two or three dimensional solid elements. Continuity and rigid

body modes are the properties that contribute most to the difficulty of a shell structure

modelling. There are basically three categories of finite elements that can be employed to

model a shell structure. These are:

1) plate elements, which are also called facet elements,

2) curved shell elements based on a shell theory, and

3) degenerated isoparametric elements.

Initially flat plate elements employing Kirchhoff theory were used for the finite element

analysis of shells. In this formulation shear deformations are neglected which makes it

difficult to satisfy interelement continuity on displacements and edge rotations. The

problems with using plate elements are the spurious bending moment they generate at the

intersection of the elements, and also uncoupled membrane forces and bending moments.

Huang (1989) states that in spite of certain shortcomings in such an approach, facet

elements are very efficient for the approximate analysis of many shell elements.

Despite the popularity of using shell elements which are formulated based on shell theories,

there are still some limitations concerning the true behaviour of a shell element. One of the

difficulties is with finding appropriate deformation idealisations which allow truly strain free

rigid body movements.

The third group of shell element formulation is based on Mindlin theory that includes the

effects of shear deformations. With this theory, the displacements and rotations of the mid-

surface normals are independent and the interelement continuity conditions on these

quantities can be satisfied as in the analysis of continua. This type of shell element which is

also called a degenerated shell element has become very popular. A well known type of a

degenerated shell element which is very efficient and simple, is the Semi Loof element a

family of A h m a d elements.

Finite Element analyses in this study were carried out using a computer package called

ALGOR. The shell element implemented in ALGOR is a flat plate element and will be further

explained in the next section. Although the results of F E analyses were up to 7 % different

from the experimental results for inplane bending, a comparison is also made in Section

5.3.3 between the performance of the flat plate element in ALGOR and other types of plate

and shell elements.


5.3 Description of the Finite Element Package: A L G O R

ALGOR is a Finite Element Package developed by a company with the same name, Algor.

The linear analysis processor of this program, which was used in this study, is similar to the

program SAPIV; developed at the University of California, Berkeley (Bathe et al., 1974).

The major differences between SAPIV and ALGOR, in addition to the fact that ALGOR runs

on PC, are due to the powerful and useful pre- and post-processors accompany the latter

program. ALGOR consists of different programs or modules performing different tasks. A

list of the modules which were used in different steps of a T-joint analysis are given in

Appendix D.

5.3.1 Shell Element in ALGOR

The plate and shell element in ALGOR is a general quadrilateral element formed as an

assemblage of four triangular elements called L C C T 9 , developed by Clough and Felippa

(1968). This element has been counted as one of the most efficient mesh units for both plate

and thin shell applications by Clough and Felippa. Figure 5.1 shows the quadrilateral

element used in ALGOR, which is composed of four constant curvature triangular L C C T 9

elements.

Figure 5.1. Thin plate and shell element in ALGOR

Each LCCT9 element is in turn a condensed form of three triangular elements as shown in

Figure 5.2. The use of three triangular elements is to provide compatibility in the L C C T 9

element The cubic interpolation functions used in L C C T 9 can not produce fully compatible

triangular plate elements, therefore L C C T 9 has been divided into three sub-elements. The

central nodal point of the quadrilateral element is located at the average of the coordinates

of the four corner nodes. Generally six degrees of freedom can be associated with each

nodal point which gives a total of 24 degrees of freedom for a quadrilateral element in a

three dimensional position. In the case of a flat plate or a shallow shell, the rotation about

the normal to the plate should be eliminated. Further information on the plate and shell

element of ALGOR is available in the paper by Clough and Felippa (1968).


Figure 5.2. LCCT9 element assembly

5.3.2 A Bench Mark on ALGOR

To evaluate the accuracy of the thin shell element in ALGOR, a simple example of a free

cylinder under internal pressure, shown in Figure 5.3, is analysed and its results are

compared with the results of other two types of plate and shell elements. This example was

originally reported in Knowles et al. (1976) to examine a flat plate and a curved shell

element, called T B 3 and G C S 8 respectively.

Overall geometry D

Mesh for GCS8 andQ12 D

Coarse mesh for TBC3 Fine mesh for TBC3

Figure 5.3. Geometry and meshes used for cylinder underpressure


W e call the shell element in ALGOR, Q12. The performance of the different shell elements,

TB3, G C S 8 and Q12, are shown in Table 5.1, indicating that Q 1 2 is suitable for

20 <_9/r<100, which is the applicable range in the offshore structures. It is also seen that

Q 1 2 performs reasonably satisfactory compared to G C S 8 element with a lower number of

nodal points. The plate element Q 1 2 does not produce a good result for D/T = 3.33, but this

is well outside the ranges of the tubular joint parameters in offshore towers. G C S 8 is a

semi-loof element and T B C 3 is a triangular element with six degrees of freedom at each

node. Descriptions of the finite elements, G C S 8 and T B C 3 , are reported in Appendix E.

Table 5.1. Results of different types of shell elements - cylinder under pressure

D/T

3.33

10

20

100

TB3

Coarse mesh Fine mesh

(NDOF* = 240)

(a) Displacements, Error of -7.5% to + 5 %

(b)Membrane strains, Correct

(a) Displacements, Error of 3 3 %

(b) Membrane stresses, Correct

(b) Displacements, Garbage

(b) Membrane stresses, Correct

(a) Displacements, Garbage

(b) Membrane stresses Correct

(NDOF* = 480)

(a) Error of ± 1 %

(b) Correct

(a) Error of ± 8 %

(b) Correct

(a) Error of -27% to +30%

(b) Correct

(a) Garbage

(b) Correct

GCS8

GCS8 mesh

(NDOF*=155)

(a) Correct

(b) Correct

(a) Correct

(b) Correct

(a) Correct

(b) Correct

Q12

GCS8 mesh

(NDOF* = 90)

(a) Error of-38%

(b) Error of - 8 %

(a) Error of+6%

(b) Error of -3%

(a) Error of-2%

(b) Error of-2%

(a) Error of-2%

(b) Error of - 2 %

* Number of Degrees of Freedom

5.3.3 Comparison of shell and three dimensional element performances

In order to verify the performance of the shell elements for F E modelling of the joints, two

analyses in IPB and O P B modes were carried out on joint T5, using shell element and three

dimensional solid element The results of the analyses are compared in Table 5.2. The shell

element has performed slightly better than the 3 D element for IPB mode, and its

performance in O P B mode is clearly better than 3 D element Both F E models had the same

mesh patterns as shown in Figure 5.4.


Table 5.2. Comparison of shell and three dimensional solid elements

Shell finite element 3D finite element Measurement

Natural frequency (Hz)

IPB 94.1

98.01

88.8

OPB 58.3

63.00

36.8

„ „ Finite Element Error% =

Measuremnet IPB O P B +6 +58

+10 +71

5.4 M e s h generation procedure used to model T-joints

A typical mesh used to model the T-joints is shown in Figure 5.4. The modes of

deformation considered were inplane, out of plane bending and axial deformation of brace.

These modes are resulted in to one plane of symmetry/anti-symmetry along the longitudinal

axis of the chord, and one plane of symmetry/anti-symmetry across the branch section.

Therefore a model of one quarter of a T-joint with the appropriate boundary condition was

sufficient for analysis. The general layout for the data generation of the T-joints is shown in

Figure 5.5. Mesh generation of the T-joints were carried out using Superdraw (SD2), the

computer aided design module in ALGOR. Intersection coordinates of a chord and its

branch was produced by using a separate Fortran program written by the author and applied

as input to Superdraw. The chord and branch meshes were generated separately and then

connected to each other by the program Substruc (one of the modules in ALGOR).

Boundary conditions were applied to the F E models in Superdraw.

a) Three dimensional solid element b) Plate and shell element

Figure 5.4. Typical mesh patterns used for the analysis of T-joints


Calculation of the nodal coordinates on intersection

brace I Construction of the brace

mesh in a planar coordinate system (ALGOR)

I Applying boundary

conditions and material properties (ALGOR)

I Decoding the graphic model into ASCII form (ALGOR)

I Folding the planar mesh

into a 3D co-ordinate system

I j.

chord

Construction of the chord mesh in a planar co

ordinate system (ALGOR)

I Applying boundary conditions and material properties (ALGOR)

I Decoding the graphic model into ASCII form (ALGOR)

I Folding the planar mesh into a 3D co-ordinate system

I Joining chord mesh to brace mesh (ALGOR)

Figure 5.5. Data generation procedure for tubular T-joints

5.5 Boundary conditions for the F E models

There are two categories of boundary conditions which can be applied to a T-joint or any

other structure, called here the internal and external boundary conditions. Internal boundary

conditions are those associated with the planes of symmetry or anti-symmetry; and external

boundary conditions are those assigned to the interface of a structure and its supports.

Applying the internal boundary conditions to an F E model will reduce the size of the model,

therefore less computer memory is required and also a higher accuracy for the solution can

be attained. However, it is essential that the internal boundary conditions produce exactly

the same deformations at the respective boundaries as when the structure was continuous at

those boundaries.

The T-joints had the conditions of symmetry and anti-symmetry, therefore only one quarter

of a joint was modelled. The boundary conditions applied to the T-joints are shown in

Figure 5.6.


U 0

* \

i I

i I

61 60 f 1000N

1B 17

\

ir 15 16

Loading and node numbering on T10 wfth reference to Table 5.2

© ©

©- -Q ©

©

-©

IPB

OPB

Plane

1-1, symmetry

2-2, anti-symmetry

chord ends

1-1, anti-symmetry

2-2, symmetry

chord ends

Sx

free

fixed

fixed

fixed

free

fixed

Boundary Conditions

6y

fixed

fixed

fixed

free

free

fixed

8Z

free

free

free

fixed

fixed

fixed

e* fixed

free

free

free

fixed

free

e-free

free

free

fixed

fixed

free

ez fixed

fixed

free

free

free

free

1-1, symmetry

axial 2-2, symmetry

chord ends

free

free

fixed

fixed

free

fixed

free

fixed

free

fixed

fixed

free

free

fixed

free

fixed

free

free

Figure 5.6. Boundary conditions applied to FE models of the T-joints

The chord walls at the supports were restrained from translation in x and y directions to

maintain the roundness of the chord section. This is because the joint is defined as a part of

the structure where the beam theory is no longer valid and must be replaced by the shell

theory. Therefore, the end cross sections of a joint behave as rigid planes with no

ovalization. The internal boundary conditions, assumed in the analyses of a quarter of a

joint, were examined by analysing two different F E models of the same joint, TIO, under a

static point load at the top of the cantilever. One was a full model of TIO and the other was

one quarter of it. Results of the analyses are reported in Table 5.3, indicating that the

assumed internal boundary conditions for the models, at symmetry and anti-symmetry zones

are acceptable.


Table 5.3. Application of internal boundary conditions to the FE model of T-joint TIP (IPB) (cont.)

Node No.*

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

Complete model

6>_) 6„<rad)

-1.7163E-12 -2.5008E-08

1.2821E-11 2.1821 E-03 6.4962E-12

7.1041 E-03 -8.4091 E-12

8.6872E-03

2.4173E-12

3.6357E-02

1.6454E-11 6.281 OE-03

-1.7312E-11 -5.3693E-02 -5.7971 E-11 -6.4790E-02

-3.1568E-11 -6.4209E-02 -7.4752E-12 -4.5478E-02

0.0000E+00 -2.1811E-02

0.0000E+00

-2.5357E-03

0.0000E+00

5.5674E-03

-1.1231E-12 3.3087E-03 -2.6760E-12 4.3068E-05 -1.0867E-12

7.0691 E-09

2.5844E-11 6.8927E-03

2.9054E-11 -1.7660E-04

2.9122E-11 2.2591 E-03

2.6045E-11 4.2533E-03

2.5998E-11 4.7094E-03

2.5995E-11 3.2768E-03

2.5833E-11 1.1407E-03

2.5418E-11 -3.8399E-04

2.5050E-11 -1.2925 E-03

2.4773E-11 -1.7899E-03

5y(m)

6y(rad)

-8.3464E-12

-7.8849E-02 -9.0210E-12

-8.0667E-02 -9.6672E-12

-8.6299E-02

-1.4899E-11 -9.2282E-02

-1.4501 E-11 -8.0755E-02

-1.2650E-11

-7.6181E-02

-3.4945E-11 -7.6540E-02

-6.5757E-11 -2.1633E-02

-3.6042E-11 1.4029E-02 0.0000E+00 2.9298E-02

1.3603E-11 1.8154E-02 7.4700E-12

-1.0221 E-03 3.3415E-12

-3.2668E-03 5.2698E-12 7.9826E-04

6.4871 E-12 2.2450E-03 5.9530E-12 1.9156E-03

-1.9544E-11 -1.1519E-01

-2.6657E-11 -8.7149E-02

-3.5038E-11

-9.2651 E-02

-3.7981 E-11 -9.8018E-02

-4.6567E-11 -9.9044E-02

-6.7108E-11 -1.0624E-01

-8.7552E-11 -1.0928E-01

-1.0748E-10 -1.1293E-01

-1.2651 E-10

-1.1650E-01

-1.4469E-10 -1.1995E-01

8z(m)

ez(rad) 1.3214E-05

-1.0487E-07 1.3574E-05

-2.8538E-08 1.1962E-05

8.1788E-08

9.6660E-06 3.0550E-08

5.1620E-06

-1.1088E-07

1.1240E-06

5.7145E-08 -6.3856E-07

3.761 OE-07 -3.4163E-06 3.3281 E-08 -6.4916 E-06 -1.7717E-07 -7.4030 E-06 -7.8433E-08 -6.9535 E-06

-2.3341 E-10 -5.4822E-06

1.4185E-08

-3.8531 E-06 1.6397E-09 -2.5970 E-06 -4.8369E-09

-1.6044E-06 6.4709E-10 -1.2053E-06 8.5135E-10

2.3400E-05

-2.3321 E-08 5.1465E-05 -3.3055 E-08

7.9433E-05

-1.5242E-08

1.0824E-04

-1.2445 E-08

1.3787E-04

-3.8042 E-08

1.9031 E-04 -4.1206E-08 2.4471 E-04 -4.0137E-08

3.0098E-04

-3.8842 E-08

3.5904E-04

-3.6682E-08

4.1885E-04

-3.5298 E-08

One quarter model

8x(m)

8„(rad)

0 0 0

2.1827E-03 0

7.1067E-03 0

8.6901 E-03 0

3.6381 E-02

0 6.2655E-03

0 -5.3699E-02

0 -6.4787E-02

0 -6.4209E-02

0 -4.5481 E-02

0 -2.1813E-02

0 -2.5360E-03

0 5.5681 E-03

0 3.3091 E-03

0 4.2995E-05

0 0 0

6.8920E-03

0 -1.8862E-04

0 2.2592E-03

0 4.2564E-03

0 4.7036E-03

0 3.2767E-03

0 1.1422E-03

0 -3.8608E-04

0 -1.2920E-03

0 -1.7903E-03

5y(m)

0y(rad)

0 -7.8851 E-02

0 -8.0670E-02

0 -8.6302E-02

0 -9.2286 E-02

0 -8.0752E-02

0 -7.6190E-02

0 -7.6542E-02

0 -2.1629E-02

0 1.4030E-02

0 2.9298E-02

0 1.8155E-02

0 -1.0215E-03

0 -3.2672 E-03

0 7.9813E-04

0 2.2451 E-03

0 1.9156E-03

0 -1.1531 E-01

0 -8.6915E-02

0 -9.2708 E-02

0 -9.8007E-02

0 -9.9008 E-02

0 -1.0629E-01

0 -1.0929E-01

0 -1.1294E-01

0 -1.1652E-01

0 -1.1997E-01

5z(m)

0z(rad)

1.3216E-05

0 1.3575E-05

0 1.1963E-05

0 9.6674E-06

0 5.1632E-06

0 1.1253E-06

0 -6.3742 E-07

0 -3.4155E-06

0 -6.4912E-06

0 -7.4031 E-06

0 -6.9538E-06

0 -5.4824E-06

0 -3.8532E-06

0 -2.5971 E-06

0 -1.6044E-06

0 -1.2054E-06

0 2.3402E-05

0 5.1468E-05

0 7.9436E-05

0 1.0824E-04

0 1.3788E-04

0 1.9032E-04

0 2.4472E-04

0 3.0099E-04

0 3.5905E-04

0 4.1887E-04

0


Table 5.3. Application of internal boundary conditions to the FE model of T-joint TIP, IPB (cont.)

Node No.*

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

Complete model

8x(m)

O^rad)

2.4592E-11 -1.9413E-03

2.4509E-11 -1.9517E-03

2.4473E-11 -1.8840E-03

2.4489E-11 -1.7925E-03

2.4527E-11 -1.7367E-03

2.4553E-11 -1.6482 E-03

2.4594E-11 -1.5825 E-03

2.4616E-11 -1.5399E-03

2.4638E-11 -1.4625E-03

2.4645E-11 -1.4153E-03

2.4632E-11 -1.3509E-03

2.4630E-11 -1.2947E-03

2.4611 E-11 -1.2532 E-03

2.4591 E-11 -1.1799E-03

2.4566E-11 -1.1256 E-03

2.4535E-11 -1.0774E-03

2.4515E-11 -1.0142E-03

2.4483E-11 -9.6329E-04

2.4458E-11 -8.9900E-04

2.4432E-11 -8.4961 E-04

2.4404E-11 -7.9028 E-04

2.4388E-11 -7.3456E-04

2.4361 E-11 -6.8374E-04

2.4345E-11 -6.1142E-04

2.4331 E-11 -5.6129E-04

2.4318E-11 -5.0563E-04

8y(m)

O^rad)

-1.6231 E-10

-1.2326E-01

-1.7972E-10

-1.2667E-01

-1.9760E-10

-1.2996E-01

-2.1497E-10 -1.3311 E-01

-2.3251 E-10

-1.3625E-01 -2.5013E-10 -1.3904E-01

-2.6792E-10 -1.4206E-01 -2.8578E-10

-1.4469E-01 -3.0374E-10 -1.4738E-01

-3.2176E-10 -1.4997E-01

-3.3977E-10 -1.5233E-01 -3.5782 E-10 -1.5477E-01

-3.7587E-10 -1.5697E-01 -3.9390E-10 -1.5910E-01

-4.1191 E-10 -1.6122E-01 -4.2988 E-10

-1.6309E-01

-4.4782E-10

-1.6500E-01

-4.6573E-10

-1.6672E-01

-4.8360E-10

-1.6837E-01 -5.0143E-10

-1.6994E-01 -5.1922E-10

-1.7136E-01 -5.3701 E-10

-1.7275E-01

-5.5536E-10

-1.7401E-01 -5.7307E-10

-1.7517E-01

-5.9077E-10

-1.7623E-01

-6.0844E-10

-1.7716E-01

8z(m) 6,(rad)

4.8044E-04 -3.4530 E-08

5.4369E-04

-3.4222E-08

6.1090E-04

-3.4163E-08 6.7750E-04

-3.4501 E-08 7.4567E-04 -3.4726E-08

8.1537E-04 -3.4974E-08 8.8652E-04

-3.5248 E-08 9.5909E-04 -3.5369E-08 1.0330E-03

-3.5581 E-08 1.1083E-03 -3.5605E-08 1.1848E-03

-3.5623E-08 1.2625E-03 -3.5680 E-08 1.3414E-03 -3.5647E-08 1.4214E-03 -3.5620E-08

1.5025E-03 -3.5542E-08 1.5846E-03 -3.5469E-08

1.6676E-03

-3.5434E-08

1.7516E-03

-3.5335E-08

1.8364E-03

-3.5285 E-08 1.9220E-03 -3.5192E-08

2.0084E-03 -3.5142E-08

2.0955E-03 -3.5109E-08

2.1863E-03 -3.5014E-08

2.2747E-03

-3.4982E-08

2.3636E-03

-3.4935E-08

2.4530E-03

-3.4896 E-08

One quarter model

8x(m)

6_<rad)

0 -1.9421 E-03

0 -1.9519E-03

0 -1.8847E-03

0 -1.7927E-03

0 -1.7370E-03

0 -1.6487E-03

0 -1.5829E-03

0 -1.5401 E-03

0 -1.4630 E-03

0 -1.4155E-03

0 -1.3514E-03

0 -1.2949E-03

0 -1.2537E-03

0 -1.1802E-03

0 -1.1259E-03

0 -1.0777E-03

0 -1.0145 E-03

0 -9.6360E-04

0 -8.9924E-04

0 -8.4980E-04

0 -7.9053E-04

0 -7.3475 E-04

0 -6.8394E-04

0 -6.1160E-04

0 -5.6142E-04

0 -5.0579E-04

8y(m)

0y(rad)

0 -1.2328E-01

0 -1.2669E-01

0 -1.2998E-01

0 -1.3313E-01

0 -1.3627E-01

0 -1.3906E-01

0 -1.4209E-01

0 -1.4471 E-01

0 -1.4740E-01

0 -1.4998E-01

0 -1.5235E-01

0 -1.5479E-01

0 -1.5698E-01

0 -1.5911 E-01

0 -1.6123E-01

0 -1.6311 E-01

0 -1.6502E-01

0 -1.6673E-01

0 -1.6839E-01

0 -1.6995E-01

0 -1.7138 E-01

0 -1.7277E-01

0 -1.7402E-01

0 -1.7518E-01

0 -1.7624E-01

0 -1.7718E-01

8_(m)

6_(rad)

4.8045E-04

0 5.4371 E-04

0 6.1092E-04

0 6.7753E-04

0 7.4570E-04

0 8.1540E-04

0 8.8656E-04

0 9.5914E-04

0 1.0331 E-03

0 1.1083E-03

0 1.1849E-03

0 1.2626E-03

0 1.3415E-03

0 1.4215E-03

0 1.5026E-03

0 1.5847E-03

0 1.6677E-03

0 1.7517E-03

0 1.8365E-03

0 1.9221 E-03

0 2.0085E-03

0 2.0956E-03

0 2.1864E-03

0 2.2748E-03

0 2.3637E-03

0 2.4532E-03

0


Table 5.3. Application of internal boundary conditions to the FE model of T-joint TIP, IPB (cont.)

Node No.*

53

54

55

56

57

58

59

60

61

Natural freque

Complete model

8x(m)

0x(rad)

2.4315E-11

-4.4219E-04 2.4316E-11

-3.8249E-04 2.4339E-11

-3.2054E-04

2.4392E-11

-2.6894E-04 2.4508E-11

-2.2389E-04 2.4693E-11

-2.5251 E-04 2.4990E-11

-2.5280E-04. 2.5289E-11

-1.6593E-04 2.5449E-11

-1.6276E-03

8><m)

0>(rad)

-6.2609E-10

-1.7802E-01

-6.4373E-10

-1.7876E-01

-6.6139E-10

-1.7941 E-01

-6.7914E-10

-1.7995E-01 -6.9715E-10

-1.8041 E-01

-7.1571 E-10

-1.8091 E-01

-7.3523E-10 -1.8061 E-01

-7.5620E-10

-1.8113E-01

-7.7948E-10 -1.8209E-01

ncies: co i *= 40.60 Hz a>2*=

5_(m)

0_(rad)

2.5429E-03

-3.4866E-08

2.6332E-03 -3.4862E-08

2.7239E-03

-3.4963E-08

2.8148E-03

-3.5283E-08

2.9060E-03 -3.6084E-08

2.9975E-03 -3.7457E-08

3.0890E-03

-4.0049E-08 3.1807E-03

-4.3600E-08

3.2725E-03 -4.9592E-08

319.27 Hz

One quarter model

8x(m)

0x(rad)

0 -4.4232E-04

0 -3.8261 E-04

0 -3.2063E-04

0 -2.6901 E-04

0 -2.2401 E-04

0 -2.5249E-04

0 -2.5280E-04

0 -1.6606E-04

0 -1.6281 E-03

a>i*= 40.69 Hz

dy(m)

O^rad)

0 -1.7803E-01

0 -1.7877E-01

0 -1.7943E-01

0 -1.7997E-01

0 -1.8042E-01

0 -1.8093E-01

0 -1.8062E-01

0 -1.8115E-01

0 -1.8211 E-01

* ©2 =

8_(m)

0_(rad)

2.5431 E-03 0

2.6334E-03 0

2.7241 E-03

0 2.8150E-03

0 2.9062E-03

0 2.9976E-03

0 3.0892E-03

0 3.1809E-03

0 3.2727E-03

0

319.63 Hz

*Nodal point locations are shown in Figure 5.6.

*Natural frequencies are only due to inplane bending mode of brace.

It is seen in Table 5.3 that the deformation values, including displacements and rotations,

have improved by adopting the correct internal boundary conditions. This is due to the less

mathematical operations carried out by the computer for the smaller model of the structure.

Therefore, a less round off error will occur.

In the example above, however, the improvement in the results have not been significant.

The values of 8„, o"y, and 0Z that should be theoretically zero due to the symmetry are in the

orders of 10"11, 10"11, and 10"8 respectively and have reduced to 0 in the one quarter model.

The maximum variations between the two models for 8Z, Qx and 0,, are 0.18%, 6 % and

0.27% at nodal points 7, 18, and 18, respectively. Furthermore, the computing time for the

full model was 29.9 minutes, whereas it was 1.6 minutes for the one quarter model

indicating a 18.7 times difference. Using a quarter of a joint mainly saves computational

time and the modelling effort. It should be stated that three analyses are required on a small

model in order to obtain the similar results of the analysis on a full model.

5.5.1 Effect of joint support conditions

The influence of chord support fixity on dynamic behaviour of a T-joint should be evaluated

in order to obtain a reliable estimate of the joint flexibility. T w o different support conditions

i.e. fixed and hinged were examined on the shell Finite Element models of the joint

specimens; similar to that discussed in Section 4.5 for the models constructed from one


dimensional beam elements. The results of the analyses, reported in Table 5.4, confirm the

conclusion made in Section 4.5, that the fixity of the end supports has very little influence

on the natural frequencies associated with the inplane bending. The results of the ID beam

element analyses are also reported in Table 5.4. It should be noted that these results are due

to the joints supported on hinges, with no transverse springs (the model shown in Figure

5.6). Therefore, there is a difference between the results presented in Table 5.4 and those in

Table 5.5 or Table 4.6.

Considering the higher accuracy of the Finite Element method, it can be concluded that

there is a very little influence from the chord support conditions on the dynamic behaviour

of a T-joint vibrating in inplane bending mode. It is seen in Table 5.4 that the variation of

boundary conditions from hinged to fixed has its biggest effect on the joints with large

diameter ratios (Tl, T 5 and T9). Hence, the joints with |_ = 1 or generally the joints with

large diameter ratios may be more sensitive to the boundary conditions. This property of

stiff joints will be also shown in Section 5.6.1 when comparing the results of F E analyses

and experiments.

Table 5.4. Calculated results for the FE models of the T-joints with different support conditions (IPB)

Joint

Tl (shell element)

(ID beam)

Tl

T3

T4

T5

T6

T7

T8

T9

TIO

Til

Natural free

Fixed supports

G)f(Hz) Q)f(Hz)

92.05 547.66

129.98 772.80

48.47 331.11

69.62 421.53

51.22 339.91

70.56 427.48

31.77 202.02

36.81 223.74

81.45 530.78

125.44 744.16

43.64 321.73

69.37 419.58

46.55 330.42

70.35 425.83

29.74 194.77

36.80 223.65

73.71 514.65

121.84 723.29

40.85 320.57

70.01 423.06

25.80 185.67

36.78 223.49

uency (IPB)

Hinged supports

0)f(Hz) 0)f(Hz)

91.01 539.54

127.08 743.72

48.34 330.08

69.28 419.07

51.09 338.93

70.27 425.36

31.76 201.93

36.80 223.64

80.35 522.36

121.61 715.48

43.51 320.82

68.95 416.64

46.43 329.56

69.99 423.28

29.73 194.68

36.78 223.51

72.46 501.20

117.28 695.85

40.60 319.27

69.56 419.89

25.78 185.58

36.75 223.32

Frequency ratio

cof/co? cof/cof

1.01 1.02

1.02 1.04

1.00 1.00

1.00 1.01

1.00 1.00

1.00 1.00

1.00 1.00

1.00 1.00

1.01 1.02

1.03 1.04

1.00 1.00

1.01 1.01

1.00 1.00 1.01 1.01 1.00 1.00 1.00 1.00

1.02 1.03 1.04 1.04 1.01 1.00 1.01 1.01 1.00 1.00 1.00 1.00

Chapter 5: Determination of Joint Rigidity Using Finite Element Method 8£

5.6 Finite Element analysis of the tested T-joints

Chapter 6 of this thesis reports the results of a parametric Finite Element study carried out

to establish a relationship between non-dimensional parameters and stiffness of the T-joints.

There are 270 analyses reported in Chapter 6 and it was required to experimentally verify

the F E results. T o do so, a series of Finite Element analyses were performed to obtain the

natural frequencies of the tested joints. One quarter of a joint with hinged and fixed

boundary conditions was modelled for inplane bending and out of plane bending modes,

respectively. The first three mode shapes, calculated for the F E model of the joint T 2 are

shown in Figure 5.7, as examples. The second natural frequencies of the joint specimens

were measured experimentally and compared with those obtained from F E analysis.

Mode 1 Mode 2 Mode 3

Figure 5.7. T-joint T2, first three mode shapes in IPB mode

Chapter 5: Determination of Joint Rigidity Using Finite Element Method 89_

5.6.1 Comparison of the Finite Element analysis and test results

The Finite Element analysis and experimental results for the IPB and O P B natural

frequencies of the T-joint specimens are reported in Table 5.5. The column labelled error

gives the relative difference between the results of the Finite Element analyses and those

from experiments.

Table 5.5. Results ofFE analyses and experiments

Natural frequency (Hz)

Joint


Finite Element

IPB OPB 101.7 67.6

53.0 33.5

55.1 36.0

32.9 27.1

94.1 58.3

48.8 27.5

51.1 29.7

31.2 23.9

88.1 53.2

46.1 23.7

27.8 18.6

Measurement

IPB O P B

95.0 46.1

55.0 30.8

56.3 33.4

33.8 27.3

88.8 36.8

50.0 24.4

51.3 26.5

31.3 23.1

82.5 33.3

46.3 20.1

28.8 17.9

„ _, Finite Element Error% =

Measurement

IPB O P B

+7 +47

-4 +9

-2 +8

-3 0

+6 +58

-2 +13

0 +12

0 +3

+7 +60

0 +17

-3 +4

It is seen in Table 5.5 that the results of the F E analyses are close to those obtained from

experiments, especially for inplane bending mode.

For the IPB natural frequencies the bigger discrepancies are seen for joints Tl, T5, and T9.

The diameter ratio (p) is noted to be one for all these three joints. Apparently this type of

tubular joints (p = 1) has always been problematic. Nearly all parametric formulae for the

estimation of stress concentration factors and also the D N V formulae for the stiffness

determination of T-joints do not cover the joints with p = 1. One reason for this, also stated

by Smedley and Fisher (1991), is the separation of the weld toes at the saddle points which

is smaller than the nominal diameter size of the brace, as shown in Figure 5.8. A formula is

recommended by Lloyd's Register as p' = l-(-sin065 \|0 to be used instead of P for the joints Y

with p = 1. \|f is the degree of weld cut-back as shown in Figure 5.8. Roesset and Pan

(1988) studied the behaviour of X-joints under axial loading and extensively used F E

method. They have observed that the elastic stiffness obtained experimentally for large

diameter tubes are always smaller than those obtained by computer modelling. This was also


observed on the joints with p = 1 tested in this study. However, the difference between

calculated and experimental results was caused by the reasons explained below.

10 30

7

ideal standard extreme

Figure 5.8. Weld cut-back at the saddle on joints with p = 1 (Smedley and Fisher, 1991)

Out of plane bending mode had the worst agreement with the experimental results. The

calculated natural frequencies of Tl, T 5 and T 9 were 4 7 % , 5 8 % , and 6 0 % higher than the

measured values respectively. This may be attributed to the inadequate fixity of the joint

support assembly to the ground. The support assembly, as shown in Figures 4.8 and 4.16,

was clamped to the floor with high strength bolts. It is seen in Table 5.5 that the most

flexible joints such as T4, T8, and Tl 1 show the best agreement between the calculation and

experiment either in IPB or O P B mode. The remaining joints which have a P equal to 0.52

and therefore possess a medium flexibility compared to the joints with P = 0.28 and P = 1

show a moderate agreement which is not as good as the joints with P = 0.28 but better than

the joints with P = 1. This variation of agreement between calculation and experiment

suggests that a different attachment system should be tried for connection of the support

assembly to the floor.

The correlation between the results of the shell F E analyses and experiments is shown in

Figure 5.9. These results indicate that, for engineering purposes, experiment can be replaced

by Finite Element analysis to predict the dynamic behaviour of a single T-joint. However,

this can be only used if the structure is not deficient such as the situation right after the

fabrication of a specimen which was the case for the T-joints in this study.


100-

N 80 I

3 S 60 L.

LJ

40-

20-

0-^

1 40 2. * * * * * Inplane bending

D a • a a Out of plane bending

T1

T5

T9 a

/ x»T3

T3 ' T2 °/ °*T4

T10 /*T6 /

/

or • T11

s

T8 T4

T6DrtK_,w

/

/

/

i_. 100

T1 /

T5

T9

/

* /

* /

* / /

-100

-80

-60

-40

— T 1 1 1 1 1 r -

20 40 60 80

Measured Frequency (Hz)

—r-

100

-20

Figure 5.9. Correlation between natural frequencies obtained from FE analysis and experiment

5.7 Summary

In this chapter a brief look at the Finite Element Method was made and results of the Finite

Element analyses of the tested T-joints were reported and discussed..

It was pointed out that the Finite Element method becomes complicated for shell and plate

structures due to satisfying the conditions of continuity and rigid body modes. The different

types of shell formulation were also explained to be 1) plate elements, 2) curved shell

elements, and 3) degenerated shell elements. The Finite Element computer program,

ALGOR, and its plate element used in this study were described. This type of element has

been proved to be efficient for the analysis of many shell structures. This was also shown by

comparing the results of a bench mark study on the plate element in ALGOR with those

from a curved thin shell element and a triangular flat plate element The results of such a

comparison indicate that a good performance can be obtained by using the plate element of

ALGOR in the analysis of a cylindrical structure. A comparison of the results of two

analyses on a T-joint using plate elements and three dimensional solid elements with


experimental results showed that plate elements can produce a more accurate result than 3 D

elements.

The mesh generation of the tubular joints and the boundary conditions adopted for the joint

models were described. It was shown that modelling a quarter of a joint decreases the

computing time by at least 6 fold with respect to a full model analysis time. Based on the FE

analyses, it was shown that the effect of boundary conditions on the inplane mode of

vibration is very little with the maximum of 4 % for the joints with P = 1 when the boundary

condition vary from hinged to fixed.

The F E analysis results had a good relationship with the test results, indicating that for a T-

joint with no noticeable deficiency Finite Element analysis works to an acceptable

engineering accuracy. For IPB mode, the mean error of the absolute differences between

Finite Element and experimental results were 3.1% with the maximum difference being for

the joints with P = 1. Similar error for O P B was 2 1 % when all eleven joint specimens were

considered and 8.3% when the joints with P=l were excluded.

Chapter 6

RIGIDITY AS FUNCTION OF JOINT

GEOMETRICAL PARAMETERS

Chapter 6: Rigidity as Function of Joint Geometrical Parameters 94

6.1 Introduction

Formulation of rigidity in terms of joint non-dimensional parameters is possible and

necessary in order to apply stiffness of the joints as a routine to the structural analysis

computer programs. Such formulae as discussed in Chapter 2 have been proposed by others

such as D N V (1977), Efthymiou (1985), Fessler (1986) and Ueda (1990) based on static

analyses. A set of formulae for different deflection modes of a T-joint (i.e. inplane bending,

out of plane bending and brace axial deformation) is proposed in this chapter, based on a

total of 270 dynamic Finite Element analyses. A beam member with rotational and axial end

springs, as described in Chapter 3, is employed as the theoretical model.

6.2 Finite Element analysis results

The flexibility of tubular T-joints are determined using resonant frequencies. A series of

Finite Element analyses were carried out to obtain the natural frequencies of the T-joints.

The deformation modes of brace include inplane bending, out of plane bending, and axial

deflection of brace. The following ranges for non-dimensional parameters of the joints are

considered:

0.28 <P< 1.0

13.36 < v < 28.83 (6.1)

0.2 < x < 1.0

in which p is the diameter ratio of the brace to the chord (dID), y is the chord diameter to

thickness ratio (DI2T) and x is the thickness ratio of the brace to the chord wall (tIT). When

these ranges are compared with those of the formulae proposed by D N V (1977), Efthymiou

(1985) and Fessler (1986), they cover a wider range of p values. Table 6.1 reports the

different non-dimensional parameters used in the Finite Element analysis of the T-joints in

this study. It is seen that there are totally 90 analyses required to cover the range of

parameters stated in Table 6.1. Since only one quarter of a joint was modelled and each

model was suitable only for one mode of deformation, therefore three analyses were carried

out for each parameter. This made the total number of analyses to be 270.


Table 6.1. Non-dimensional parameters used for FE models of T-joints

parameter value

V(d/D) y(DI2T) x(tlT) a (2UD)

0.28

13.36

0.20

13.69

0.41

15.22

0.50

0.52

17.12

1.0

0.64

22.82

0.75

28.83

1.0

The geometry of the Finite Element models is shown in Figure 6.1. The dimensions adopted

are similar to the test specimens described in Chapter 4. A chord length of 1500mm and a

diameter size of 219.1mm were assumed for all models which resulted in an a value of

13.69.

t variable

*J-

variable

All dimensions in millimeters M

1=1230

— z

*AL=1500

A-A

Figure 6.1. Geometry of Finite Element models used for flexibility analysis of T-joints

Few findings exist in the literature regarding the effects of a ratio on the measurements

carried on a tubular joint. This parameter represents the chord length significance. In

practice more than 6 0 % of tubular joints have a values in excess of 20 and, for 3 5 % of

joints a exceeds 40 (Smedley and Fisher 1991). Efthymiou (1985) has shown that for IPB

and O P B modes, a chord length of minimum 6D yields stiffness coefficients that are within

1 % to 4 % of the true stiffness. A n idealised analytical solution for decay length has been

reported by Efthymiou (1985) which agrees with the minimum length of 6Z>. This solution is

presented in Appendix F. Roesset and Pan (1988) conducted a study on the effects of chord

length and boundary conditions on the behaviour of tubular X-joints. They analysed six joint

geometries under tensile and compressive loads, using non-linear Finite Element method. It

is concluded by Roesset and Pan that for a chord length of two chord diameters or less, the

end conditions substantially affect the analysis results in relation to ultimate load and


stiffness of the joints. They added that for a chord length of four diameters or more the

effect of end conditions is relatively small.

In this study a = 13.69 (L/D = 6.85) was selected for the F E models as a representative for

the purpose of practical design.

The natural frequencies and mode shapes of a structure can be determined by calculating the

eigenvalues and eigenvectors of its stiffness matrix. This is called dynamic modal analysis in

ALGOR computer package and carried out by the program module SSAP1 in the package.

The identification names and the specification of the individual F E models are given in Table

6.2. There are 90 joints in Table 6.2, which are basically three batches of 30 joints. Each

batch has a different x value. The joint names with suffixes 'a', 'b' and 'c' have x values

equal to 0.2,0.5 and 1.0, respectively. T 2 and T6 have the same geometry as T3 and T7.

Table 6.2. Identification names and individual joint parameters of FE models

joint Tla,b,cT T2a,b,c T3a,b,c T4a,b,c T5a,b,c T6a,b,c T7a,b,c T8a,b,c T9a,b,c T10a,b,c Tlla,b,c

P 1.0 0.52 0.52 0.28 1.0 0.52 0.52 0.28 1.0 0.52 0.28

Y 13.36 13.36 13.36 13.36 17.12 17.12 17.12 17.12 22.82 22.82 22.82

joint T12a,b,c T13a,b,c T14a,b,c T15a,b,c T16a,b,c T17a,b,c T18a,b,c T19a,b,c T20a,b,c T21a,b,c T22a,b,c

P 0.41 0.64 0.75 0.41 0.64 0.75 0.41 0.64 0.75 1.0 0.75

Y 13.36 13.36 13.36 17.12 17.12 17.12 22.82 22.82 22.82 15.22 15.22

joint T23a,b,c T24a,b,c T25a,b,c T26a,b,c T27a,b,c T28a,b,c T29a,b,c T30a,b,c T31a,b,c T32a,b,c

P 0.64 0.52 0.41 0.28 1.0 0.75 0.64 0.52 0.41 0.28

Y 15.22 15.22 15.22 15.22 28.83 28.83 28.83 28.83 28.83 28.83

t Joint names with suffix 'a' have T values = 0.2, joint names with suffix 'b' have i values = 0.5, and joint names with suffix 'c' have x values = 1.0. Chord length parameter (a) is 13.69 for all models.

The theoretical model in this study employs the first natural frequency relevant to the

desired deformation mode. However, it is generally possible to include the higher natural

frequencies in the derivation of joint stiffness. The natural frequencies and stiffness of the T-

joints, analysed by the Finite Element method, are reported and discussed in Sections 6.2.1

to 6.2.3 for different modes of deformation.

6.2.1 Inplane bending mode

Tables 6.3 report the results of the analyses for inplane bending mode. Also, Figures 6.2 to

6.4 illustrate these results as relationships between p, y, x and joint stiffness. The

relationships suggest that an equation can be established to cover the calculated stiffness

values.


Table 6.3a. Calculated natural frequencies and stiffness of T-joint FE models, IPB (t =

joint Tla T2a T3a T4a T5a T6a T7a T8a T9a TlOa Tlla

NF*(Hz) *(Nm)

106.87 .645E7

60.57 .138E7 34.61 .508E6 105.25 .463E7

59.17 ~m€E6~' 34.77 .265E6 103.31 .308E7 57.64 591E6 32.60 .150E6

joint T12a T13a T14a T15a T16a T17a T18a i T19a T20a T21a T22a

N F ( H z ) 48.34 73.17 85.02 48.02 72.38 84.07 46.98 70.77 82.32 106.21 84.72

*(Nm) .794E6 .231E7 .349E7 .547E6 .160E7 .243E7 .339E6 .106E7 161E7 .547E7 .284E7

joint T23a T24a T25a T26a T27a T28a T29a T30a T31a T32a

NF*(Hz)

72.96 60.88 48.39 35.19 98.57 80.81 69.32 57.87 45.92 32.48

P.2)

Jt(Nm)

.188E7

.119E7

.644E6

.348E6

.195E7

.117E7

.740E6

.439E6 235E6 .971E5

*NF: Natural Frequency

Table 6.3b. Calculated natural frequencies and stiffness of T-joint i

joint Tib T2b T3b T4b T5b T6b T7b T8b T9b flOb Tllb

NF*(Hz) 101.74

52.65 31.56 87.30 —

50.64 30.50 86.30 48.32 28.29

Jfc(Nm) 8.58E6

1.62E6 4.74E5 6.10E5

1.07E6 2.98E5 4.18E6 6.81E5 1.53E5

joint

T12b T13b T14b T15b T16b T17b T18b T19b T20b T21b T22b

NF*(Hz)

42.76 62.82 72.49 41.62 61.13 70.65 39.81 58.67 68.05 89.54 71.68

*(Nm)

9.47E6 2.77E6 4.28E6 6.30E5 1.91E6 2.93E6 3.79E5 1.24E6 1.90E6 7.18E6

" 3.55E6

VEmode

joint T23b T24b T25b T26b T27b T28b T29b T30b T31b T32b

Is, IPB (x =

NF*(Hz)

62.10 52.31 42.34 30.93 84.03 65.94 56.63 47.54 38.18 27.74

P.5)

Jt(Nra)

2.31E6 1.35E6 7.62E5 3.59E5 3.05E6 1.35E6 8.45E5 4.97E5 2.60E5 1.24E5


Table 6.3c. Calculated natural frequencies and stiffness of T-joint i

joint Tic T2c T3c T4c T5c T6c T7c T8c T9c TlOc Tile

NF*(Hz) 76.44 45.19 —

28.86 74.51 42.83 —

28.43 72.46 40.69 24.72

*(Nm) .106E8 .185E7 —

.556E6

.763E7

.123E7 —

.414E6

.509E7

.773E6

.180E6

joint T12c T13c T14c T15c T16c T17c T18c T19c T20c T21c T22c

NF*(Hz) 37.97 53.83 60.63 36.22 51.66 58.42 33.96 48.89 56.53 75.69 59.48

*(Nm) .115E7 .316E7 .478E7 .746E6 .217E7

y- -327E7 .450E6 .140E7 .223E7 .898E7 .394E7

rE models, IPB (i =

joint NF*(Hz) T23c 52.82 T24c 44.98 T25c • 37.18 T26c | 28.75 T27c 70.75 T28c 54.33 T29c 46.72 T30c 39.41 T31c 32.05 T32c 24.10

1.0)

*(Nm) .263E7 .160E7 .897E6 .474E6 .379E7 .156E7 .982E6 .557E6 .294E6 .129E6


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12E6-

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2.E6

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-— " " ,

' 0. 0.2 6.4 6.6

X d.8 1.0

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8.E6- P=0.75

7.E6-

-.6.E6-

Z — 5.E6-I m 04.E6

*3.E6

2.E6

1.E6

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— i 1 —

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0.8 — i

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10E6-

8.E6

m n.

6.E6-

--4.E6

2.E6

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— i —

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(b)

6.E6-, P=10 1 P=0-75

5.E6

4.E6

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1.E6-

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— i —

0.8 -1

1.0

0. — I —

0.2 0.6

(d)

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1.0

4.E6

3.E6-

m

2.E6

1.E6-

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P=1.0 -P=0.75-P=0-64-p=0-52" P=0-41-P=0-28-

Y=28.83

0. ~02 04 06 0?8 LO

(e)

Figure 6.4. Relationship between . and IPB stiffness. a)y= 13.36, b)y= 15.22, c)y= 17.12,. d)y = 22.82,

e)y= 28.83


It is seen in Figures 6.2 and 6.3 that inplane bending stiffness of T-joints increases as P or y

ratios increase, or the joints with large diameter braces possess larger stiffness than those

with small diameter braces. It is also noted that thicker chords result in stiffer joints. These

effects should not be mistaken with the overall effect of a stiff chord on the moment

distribution in a structure. A thick chord causes high stiffness as regard to the global

behaviour of a structure, however it includes a high local stiffness which enhance the

stiffness of the joint.

Furthermore, the curves showing the relationships between &IPB and p for different y ratios,

in Figure 6.2, are almost equally spaced, whereas they correspond to the values of y =

13.36, 15.22, 17.12, 22.82, and 28.83 (Figure 6.2c). This indicates that IPB stiffness is

most sensitive to y ratio in the lower range of this parameter. In other words, the stiffness

of the T-joints that have thick chord walls has a large gradient with respect to y ratio.

The highest variation of inplane bending stiffness with respect to p ratio also occurs for the

joints with large diameter braces. This is seen in Figure 6.3c where the distance between

the stiffness curves, suddenly increases at p = 0.75. However, comparing Figure 6.2c and

6.3c, it is noted that kim is more sensitive to the variation of y ratio than p.

B y the knowledge of this author, the effect of x ratio on the stiffness of tubular joints has

not been considered until now. The diagrams in Figure 6.4 show that there is a rather

significant effect from x ratio on the inplane bending stiffness of T-joints. It is not,

however, as much as the effects of p and y ratios. The average sum of the ratios of the

stiffness values for the joints with x=l and x=0.2 based on the results reported in Tables

6.3c and Table 6.3a is 1.40. This indicates an average difference of 4 0 % in IPB stiffness of

a T-joint when x ratio is varied from 0.2 to 1.


6.2.2 Out of plane bending mode

Tables 6.4a to 6.4c report the results of the Finite Element analyses for out of plane bending

mode. Figures 6.5 to 6.7 also illustrate the same results as relationships between a joint non-

dimensional parameter and its stiffness.

Table 6.4a. Calculated natural frequencies and stiffness of the T-jo

joint

Tla T2a T3a T4a

• • • • • • • •

T5a T6a T7a T8a T9a TlOa Tlla

NP(Hz) 86.66

48.53 31.64 83.58

44.31 ] 30.34 f

81.60 41.10 27.11

*(Nm) .316E7

484E6 .200E6 .212E7

.274E6

.107E6 ".148E7 164E6 .516E5

joint NF*(Hz) T12a _.__

T14a T15a

40.60 56.61 65.30 38.05

T16a j 52.65 T17a T18a T19a T20a T21a T22a

61.33 34.32 48.02 56.73 85.14 63.34

*(Nm) .304E6 .740E6 .119E7 .174E6 .452E6 .753E6 .935E5 .256E6 .446E6 .259E7 .951E6

int FE models, OPB (x = P.2)

joint T23a 24a T25a T26a T27a T28a T29a T30a T31a T32a

NF*(Hz) 54.64 46.89 39.54 31.65 79.12 53.35 " 44.61 37.45 31.31 25.16

Jfc(Nm) .593E6 .362E6 .227E6 .146E6 .102E7 .298E6 .171E6

' .973E5 .563E5 .290E5


Table 6.4b. Calculated natural frequencies and stiffness of the T-joint FE models, OPB (x = 0.5)

joint Tib T2b T3b T4b T5b T6b T7b T8b T9b TlOb Tllb

NF*(Hz)

67.61 —

37.71 26.86 65.11

29.66 24.86 65.34 27.86 21.25

Jfc(Nm) 4.11E6 —

5.56E5 2.12E5 2.87E6

2.27E5 1.20E5 2.13E6 1.51E5 5.51E4

joint T12b T13b T14b T15b T16b T17b T18b T19b T20b T21b T22b

NF*(Hz) 32.47 43.42 50.13 29.35 39.45 46.16 25.62 35.18 41.83 67.63 48.11

*(Nm) 3.38E5 8.65E5 1.45E6 1.99E5 5.20E5 __ 8.67E5 ~~ 1.05E5 2.98E5 5.26E5 3.72E6 1.13E6

joint T23b T24b T25b T26b T27b T28b T29b T30b T31b T32b

NF*(Hz) 41.38

JKNm) 6.76E6

35.77 4.21E5 30.97 2.63E5 25.66 1.54E5 61.36 1.45E6

~ 38.76 " 3.47E5 32.17 1.95E5 27.02 22.84 19.09

1.14E5 6.37E4 3.26E4


Table 6.4c. Calculated natural frequencies and stiffness of the T-joint FE models, OPB(x

joint

Tic T2c T3c T4c T5c T6c T7c T8c T9c TlOc Tile

NF*(Hz)

56.47 30.53 —

23.30 53.98 26.93

22.58 53.16 23.65 17.54

*(Nm) .591E7 .634E6 —

.245E6

.398E7

.366E6

.175E6

.270E7

.197E6

.646E5

joint

T12c T13c T14c T15c T16c T17c T18c T19c T20c T21c T22c

NF*(Hz)

27.15 35.07 39.58 23.99 31.42 35.92 20.58 27.63 32.78 55.23 37.61

*(Nm)_ .422E6 .103E7 .162E7 .232E6 .640E6 .101E7 .122E6 .356E6 .592E6 .473E7 .119E7

joint NF*(Hz) T23c 33.16 T24c i 28.92

T25c J 25.56 T26c 22.83 T27c 50.08 T28c i 30.10 T29c ' 25.03 T30c 21.16 T31c 18.12 T32c 15.74

= 1.0)

JKNm)

.818E6 499E6 .319E6 .201E6 .184E7 .391E6 .212E6 .128E6 .718E5 .387E5


103

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(C) 2.5E6-, 0=1.0

2.E6-

m a. „1.5E6-

1.E6-

0.

J=0.75 - -J=0.64 — 5=0.52 — 5=0.41 -5=0.28 —

5.E6-1

4.E6-

E z ^ 3 . E 6 -

m o W2.E6-__

1.E6-I

0 -0

0=1.0 P=0-75 P = 0 . 6 4 — - — Y=15.22 P=0.52 — B=0.41 ^ ^ 0=0.28 - ^

L „ _ — •• — — ~ -

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m o. o

1.E6

0=0.64 v=22.82 0=0.52 0=0.41 0=0.28

0. 0.2 0 4 06 Oe? T.O

(d)

7=28.83

0. 0.2 ~0A — i —

0.6 0.8 — i

1.0

(e)

Figure 6.7. Relationship between x and OPB stiffness. a)y= 13.36, b)y= 15.22, c)y= 17.12,. d) y =

22.82, e)y = 28.83


The relationship between out of plane bending stiffness and non-dimensional parameters of

T-joints are similar to the IPB mode. The O P B stiffness has an ascending trend with respect

to p and x ratios and a descending trend with y ratio. The T-joints with large brace

diameters, or low chord diameter to thickness ratios have a relatively high stiffness.

It is noted in Figures 6.5 that out of plane bending stiffness becomes most sensitive to y

ratio in the lower range of this parameter. It is also seen in Figure 6.6 that O P B stiffness of

T-joints becomes significandy sensitive to P for the joints with high (J ratios. In comparison

with IPB mode, the O P B stiffness varies more drastically for large ps (comparing Figure

6.3c and 6.6d).

It can be said that the O P B stiffness of the T-joints with large brace diameters, or the

stiffness of those which have small chord diameter to thickness ratios should be determined

more carefully.

There is also a variation observed for O P B stiffness with respect to x ratio. It is seen in

Figures 6.7 that the effect of x parameter should be necessarily considered on the O P B

stiffness. From the results shown in Tables 6.4a and 6.4c, the average sum of the ratios of

the stiffness values for the joints with x=l and x=0.2 is 1.44. This indicates an average

difference of 4 4 % in O P B stiffness of a T-joint when x ratio is varied from 0.2 to 1.

6.2.3 Axial deformation of brace

The results of Finite Element analyses due to the axial deformation of brace are reported in

Tables 6.5a to 6.5c. The graphical presentation of the same results are given in Figures 6.8

to 6.10.

Table 6.5a. Calculated natural frequencies and stiffness of the T-joint FE models, axial deformation ft = P.2)

joint Tl T2 T3 T4 T5 T6

. _T7 , T8 T9 TIO Til

NF*(Hz) 213.84

225.18 235.08 213.99

226.48 236.17 215.67 227.26 237.33

fc(N/m)

3.31E8

8.19E7 3.87E7

__ 2.38E8

5.71E7 3.08E7 1.80E8 4.19E7 1.62E7

joint T12 T13 T14 T15 T16 T17

JL18

T19 T20 T21 T22

NF*(Hz) 229.37 221.50 217.95 230.58 222.74 219.21 231.55 223.62 220.46 214.30 218.62

*(N/m) 5.31E7 1.13E8 1.51E8 3.74E7 7.91E7 1.06E8 2.60E7 5.29E7 7.67E7 2.79E8 1.29E8

joint T23 T24 " T25 T26 T27 T28 T29 T30 T31 T32 T31

NF*(Hz) 221.53 225.59 230.00 236.17 215.58 220.93

" 224.29 227.79 231.91 237.91

*(N/m) 9.02E7 6.95E7 4.54E7 3.28E7 1.03E8

" 5.62E7 4.04E7 2.79E7 1.90E7 1.33E7



Table 6.5b. Calculated natural frequencies and stiffness of the T-joint FE models, axial deformation

<x-0.5)

joint Tl T2 T3 T4 T5 T6 T7 T8 T9 TIO Til

NF(Hz) 179.67

201.91 220.24 180.07 —

202.14 220.15 180.27 202.49 220.63

*(N/m) 1.22E9

2.05E8 1.05E8 6.82E8

1.264E8 5.41E7 4.78E8 8.41E7 3.56E7

joint T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22

NF*(Hz) 209.82 195.79 190.02 209.86 196.15 190.59 209.36 196.12 190.90 180.22 190.29

*(N/m) 1.41E8 3.06E8 4.35E8 8.96E7 1.90E8

1 2.65E8 5.46E7 1.12E8 1.63E8 9.40E8 3.35E8

joint T23 T24

T26 T27 T28 T29 T30 T31 T32

NF*(Hz) 195.96 202.34 209.80 220.83 181.93 190.76 195.64 201.31 208.25 219.70

*(N/m) 2.40E8 1.67E8 1.14E8 8.88E7 .307E9 110E9 .752E8 .513E8 .345E8 .255E8


Table 6.5c. Calculated natural frequencies and stiffness of the T-joint FE models, axial deformation

(x = 1.0)

joint Tl T2 T3 T4 T5 T6 T7 T8 T9 T10 Til

NF*(Hz) 150.56 176.14

198.46 150.27 174.26

197.69 150.21 172.08 195.89

*(N/m) 1.99E9 3.42E8 —

1 27E8 1.09E9 1.52E8

| 7.12E7 6.62E8 8.99E7 4.34E7

joint T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 T22

NF*(Hz) 184.71 167.15 161.24 183.26 166.65 161.11 180.82 165.59 159.95 150.67 161.22

fc(N/m) 2.07E8 4.62E8 6.57E8 1.20E8 2.62E8 3.76E8 6.70E7 1.44E8 2.15E8 1.46E9 4.85E8

joint T23 T24 T25 T26 T27 T28 T29 T30 T31

T32 T31

NF*(Hz) 166.86 174.46 183.90 198.43 150.83 159.26 164.24 170.16 177-85 192.00

J.(N/m) 3.34E8 2.29E8 1.54E8 1.03E8 4.03E8 1.42E8 8.92E7 6.15E7 4.22E7 2.83E7


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— i 1 1 r-

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— i

1.0

(e)

Figure 6.1P. Relationship between x and axial stiffness, a)y= 13.36, b)y= 15.22, c) y = 17.12,. d)y-

22.82, e)y= 28.83

http://f6.ES


The main observation from the results shown above in Figures 6.8 to 6.10, is the effect of x

ratio on the axial stiffness of a T-joint. The average sum of the ratios of the stiffness values

shown in Tables 6.5a and 6.5c for the joints with x=l and x=0.2 is 3.32, which indicates a

mean change of 2 3 2 % in the axial stiffness of a joint when the x ratio varies from 0.2 to 1. It

is also seen in Figures 6.9 and 6.10 that axial stiffness of T-joints becomes more sensitive to

P in the higher range of this parameter.

Further, it is noted that a large brace diameter or a small chord diameter to thickness ratio

result in a high axial stiffness. Figure 6.8c shows that for the differences between y ratios

equal to 1.86, 1.9, 5.7, and 6.01, the curves of relationships between axial stiffness and P

are almost equally spaced. This indicates that axial stiffness becomes significantly sensitive

to y in the lower range of this parameter.

As a result, x ratio should be definitely considered when the axial stiffness of a T-joint is

calculated. Furthermore, more attention should be paid to determine the axial stiffness of

the joints which have large P or small y ratios .

6.3 Derivation of stiffness equations

The joint stiffness values which were calculated by using the results of the Finite Element

analyses, indicate a clear relationship with non-dimensional geometrical parameters of the

joints. Therefore, a formula may be established for each m o d e of deformation. A power

regression method is used in this study to derive the stiffness formulae. The results of such

an analysis yields Equations (6.2), (6.3), and (6.4) for IPB, O P B , and axial brace

deformation respectively, as reported in Table 6.6. This table shows also the correlation

factor of each equation (R2). R2 values are very close to one indicating that there is a very

good correlation between the data values used to establish the formulae.

Table 6.6. T-joint stiffness formulae and the corresponding correlation factors

k = o 169i_D3B(Yt/122+155)Y(p/2I7"182)x025 /?2 = 0.996 (6.2)

kOPB = 0.088__D3p(ir/86 4+0 31)Y(187P'3 27)x°212 R2 = °"997

k = 2 712__D3("Y/2 7 3 + 0 - 6 t )Y(1 3 8 M )x09 1 /?2 = 0.985 (6.4)

The proposed formulae in this study are valid in the following ranges of non-dimensional

parameters:

0.3<P<0.9, 13<Y<30, and 0.2<x<1.0 (6.5)


The predicted stiffness values by the developed equations are shown in Figure 6.11 against

the data from F E results to check the performance of the curve fittings. T o save space only

the comparison is made for the joints that have a x ratio equal to 0.5. It is seen that the

developed equations produce a good estimation for the stiffness values, especially for IPB

and O P B modes. The maximum curve fitting errors for IPB, O P B , and axial modes are

24.9%, 18.3% and 28.6% respectively. These are the maximum difference between the

predicted and calculated stiffness results for each deformation mode.

9.E6-,

8.E6

p7.E6,

Z6.E6 CO

__ 5.E6

^ 4 . E 6 -

3.E6-

2.E6

1.E6

0.

T = 0.5

measured -y==15.22 «-••••-•••• predicted

•y=13.36

7=28.83

I r I 1 I I I I TT T T I I I I I I | 1 TT I T T T T I ' I I I I T T I T I I |

0.2 0.4 0.6 0.8 1.0

P

a) Inplane bending mode

5..E6n

4.E6

3.E6-

m a. o

2.E6

1.E6-

b) Out of plane bending mode

1.4E91

1.2E9

. 1.E9

O 8.E8-o x O 6.E8

4.E8

2.E8

0. 0.2 0.4 0.6

P 0.8

7=28.83 - i

1.0

c) Axial deformation of brace

Figure 6.11. Predicted values against data used for curve fitting

To have a better picture of the relationships between joint stiffness and non-dimensional

parameters, the three dimensional plots of the stiffness equations adopted herein are made

and shown in Figures 6.12 to 6.14. These plots include only the results of the joints with x =

0.5. The absolute errors of the curve fittings are also reported in these diagrams. It is seen


that the stiffness of the T-joints under IPB, OPB and axial loading have increasing trends

with P ratio, but they decrease w h e n Y ratio is going up.

Figure 6.12. Model adopted for IPB joint stiffness (eq (6.2)} with the absolute errors of curve fitting

Figure 6.13. Model adopted for OPB joint stiffness (eq (6.3)) with the absolute errors of curve fitting


Figure 6.14. Model adopted for joint axial stiffness {eq (6.4)} with the absolute errors of curve fitting

6.4 Comparison with other stiffness equations

The main objective of establishing the parametric formulae is to be able to predict the

stiffness of a joint without having to carry out detailed analysis, either model testing or F E

analysis. It is usually preferred in practice to use simplified analysis and design methods. A

parametric formula can be employed in a structural analysis computer program to

incorporate the joint stiffness automatically in the analysis. Nevertheless, it is required to

conduct detailed studies in some special cases. The stiffness consideration of joints is

especially important in the reappraisal of existing offshore structures.

To examine the accuracy of the equations developed herein, a comparison is made between

the predicted results of the equations in this study and those reported by other researchers.

It will be shown that the method used in this study can produce compatible results with

other investigations. However, this study can be distinguished among other works on the

stiffness of tubular joints, as the dynamic approach was employed here.


The literature survey carried out in this study shows seven sets of formulae for the

calculation of joint stiffness or flexibility using non-dimensional geometrical parameters.

The formulae are by:

DNV (1977),

U E G (1985),

Efthymiou (1985),

Fessler (1986),

Ueda etal. (1990),

Chen B.etal. (1990), and

Kohoutek and Hoshyari (1992).

Each set of formulae is described in the following sections.

6.4.1 DNV (1977)

D N V (Norwegian Rules for the Design, Construction and Inspection of Offshore Structures,

1977), has recommended two formulae for the stiffness of T-joints, as follows:

km = 0.43__3(i-.01)(235-,5fJ) (6.6)

Y

kOPB = 0.0016_i_?3(215-P)(--.02)(245-,6p) (6.7)

y

The range of validity for Equations (6.4) and (6.5) are given as:

0.33<p<0.8, and 10<y<30. (6.8)

DNV Formulae are probably the first set of equations proposed for the stiffness of tubular

joints.

6.4.2 UEG (1985)

The Underwater Engineering Group (UEG, 1985) has surveyed the experimental data to

the date of publishing and recommended a set of formulae which has been taken from

Professor Fessler at Nottingham University. There have been only two sets of formulae

available to U E G in 1985, DNV's and Fessler's. The recommended formulae by U E G are:


k^M = E D y 2'3 exp(3.3p) sin"2G/2.3 (6.9)

kjpB = E D 3 y -h65 exp(4.6p) sin^O/H 1 (6.10)

kOPB = E D 3 y 2'5 exp(3.7P) sin"2e/48.1 (6.11)

No validity range is given for the UEG formulae.

6.4.3 Efthymiou (1985)

A set of equations were proposed by Efthymiou (1985) for inplane and axial stiffness of T

and Y-joints. The work is based on Finite Element analysis and then calculation of joint

stiffness using equations (6.12).

M P kiPB=~jj- md ^ « _ = y (6-12)

in which M and P are the bending moment and the axial force of the brace, respectively. G

and 8 are the local rotation and axial deformation of the joint. The recommended equations

by Efthymiou (1985) are:

kIPB = 0.163 ED3 ylM p("5+yi25) sm-<P+o.4)e (6.13)

koPB = 0.288 E D 3 Y(07(0.55-P)2-2.20) 02.12 -(0+1.3)0 (6 14)

with a range of validity imposed as:

0.3<p<0.8, and 10<Y<30. (6.15)

6.4.4 Fessler (1986)

Fessler has derived a set of formulae for the stiffness of T and Y-joints, besides the ones

published by U E G (1985), based on testing a number of araldite models. The formulae are:

kaxua = ED y215 (1-P)-13 sin-219e/1.95 (6.16)

kWB = ED3 Y'1'73 exp(4.52P) sin1229/134 (6.17)

kOPB = ED3 Y"2'20 exp(3.85P) sin-2169/85.5 (6.18)

A range of validity is imposed on the formulae by Fessler as:

0.3<P<0.8, and 10<Y<20. (6.19)


6.4.5 Ueda et al. (1990)

The paper by Ueda et al. was discussed earlier in Chapter 2. The equations developed by

Ueda using Finite Element analysis are:

k^ = 3.195 ED y2-3 P12 sin-29 (6.20)

kIPB = 0.237 ED3 j-1-7 P22sin^e (6.21)

No validity range is given for the Ueda's formulae.

6.4.6 Chen B. et al. (1990)

The formulae developed by Chen B. et al. are originally for a symmetrical K-joint. However,

it is recommended by Chen that the formulae can also be used for axial and inplane bending

of T and Y-joints. The Chen's formulae are very similar to those recommended by U E G .

The formulae by Chen are:

_ = ED Y'2 V'25" sin"202 6 / 4.71 (6.22)

k^B = ED3y-168 _4'58(isin"1-25e/169 (6.23)

The validity range of the above equations are given as:

0.3 < p < 0.8,7.5 < Y < 35, and 30 < 9 < 90 (6.24)

6.4.7 Kohoutek and Hoshyari (1992)

A formula has been suggested for IPB rigidity factor of tubular T-joints by Kohoutek

(1992). The formula is actually based on the experimental results of this study described in

Chapter 4. The proposed equation is:

T = 1.307 - 0.819 p - 0.0130 Y+ 0.488 P2 (6.25)

T varies between 0 and 1. A hinged or a fixed support is assumed to have a T = 0 or 1,

respectively.

6.4.8 Graphical Comparison of different stiffness formulae

Figures 6.15, 6.16 and 6.17 show the comparisons between the performance of the

equations of joint stiffness developed by other researchers and those developed herein for

axial, IPB and O P B modes. The significant difference between the formulae developed in

this study and other equations is the inclusion of thickness ratio or x parameter. Therefore,


using other equations gives the same results for different x values, while the developed

equations in this study predict different stiffness values with respect to x. This leads to three

samples on a line in Figures 6.15 to 6.17 corresponding to x =0.2, 0.5 and 1 which makes

the comparisons inevitably inaccurate to some extent

A discrepancy ratio, R, is defined as the ratio between the results of other equations and the

formulae of this study. The geometrical parameters, stiffness and R values of the joints

which are used in Figures 6.15 to 6.17 are reported in Tables 6.7a, 6.7b and 6.7c. The

following findings can be obtained from the comparisons of different formulae:

Table 6.7a. Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 calculated using the formulae by others

Equation P IPB tf OPB fit axial /?*

Fessler 0.33

0.8

0.33

0.8

13.36

13.36

20

20

.787E6

.659E7

.392E6

.328E7

1.18

1.36

1.18

1.22

.292E6

.179E7

.120E6

.735E6

1.06

1.01

1.17

0.82

f

.144e9

.692E9

.603E8

.290E9

135

1.43

1.21

1.22

DNV

Efthymiou

Ueda

0.33

0.8

0.33

0.8

0.33

0.8

0.33

0.8

0.33

0.8

0.33

0.8

13.36

13.36

20

20

13.36

13.36

20

20

13.36

13.36

20

20

.707E6

.486E7

.289E6

.279E7

.600E6

.483E7

.316E6

.267E7

.530E6

.372E7

.267E6

.187E7

1.06

1.00

0.87

1.04

0.90

1.00

0.95

0.99

0.79

0.77

0.81

0.70

.271E6

.151E7

.848E5

.744E6

.210E6

.141E7

.875E5

.590E6

—

—

—

—

0.98

0.85

0.82

0.84

0.76

0.79

0.85

0.67

—

—

—

—

—

—

—

—

—

—

—

—

.953E8

.276E9

.377E8

.109E9

—

—

—

—

—

—

—

—

.89

.57

.75

.46

R : discrepancy ratio = ifcothers/fahis study

Table 6.7b. Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 measured by others

Equation P Y IPB # O P B R* axial R*

Tebbett

McDermott

0.33

0.54

0.59

0.92

0.53

20

20

32

20

24

.395E6

.673E6

.491E6

—

—

1.19

0.68

0.85

—

—

* R: discrepancy ratio = fcxheis/tthis study

.081E6

.095E6

— .180E6

0.79

0.76

.487E8

.762E8

— .285E9

0.97

0.79

0.77


Table 6.7c. Geometrical parameters and stiffness of the joints of Figures 6.15 to 6.17 calculated by the formulae developed in this study

&

0.33

0.8

0.33

0.8

0.54

0.59

0.92

0.53

Y

13.36

13.36

20

20

20

32

20

24

X

.2

.5 .97 .2 .5 .97 .2 .5 .97 .2 .5 .97 .2 .5 .97 .2 .5 .97 .2 .5 .97 .2 .5 .97

IPB .551E6

.669E6

.745E6

.389E7

.485E7

.566E7

.278E6

.331E6

.359E6

.216E7

.269E7

.312E7

.810E6

.988E6

.111E7

.477E6

.576E6

.637E6

.318E7

.398E7

.467E7

.579E6

.702E6

.781E6

OPB .227E6

.276E6

.317E6

.147E7

.178E7

.205E7

.848E5

.103E6

.119E6

.731E6

.887E6

.102E7

.211E6

.256E6

.294E6

.103E6

.125E6

.144E6

.133E7

.161E7

.185E7

.137E6

.166E6

.191E6

axial .565E8

.107E9

.142E9

.219E9

.484E9

.831E9

.265E8

.500E8

.668E8

.108E9

.238E9

.408E9

.467E8

.962E8

.148E9

.245E8

.514E8

.809E9

.162E9

.368E9

.656E9

.329E8

.675E8

.106E9

Axial rigidity of brace

The predictions of the axial rigidity equation adopted here are within ± 1 0 0 % of the values

calculated by Fessler and Ueda's equations and also some experimental results by Tebbett

and McDermott. It should be noted that the 1 0 0 % difference in case of Ueda's equation can

not be counted very reliable because Ueda's equation is based on only 7 samples.

The following points are observed in Figure 6.15:

1) A good agreement is found when the axial stiffness is compared with the experimental

results of Tebbett (1982). There are only three samples from Tebbett, which their x ratios

are not quoted in Tebbett's paper. The average difference with Tebbett's equation obtained

for the joint parameters examined here is 16%.

2) The results of the equation of this study and Ueda's equation have a close relationship for

the joints with low P ratios.


3) Fessler's results show a good relationship with the results of the formula developed here,

except for the joint with P = 0.92; perhaps because Fessler's equation is not valid in this

range.

4) The variation of axial stiffness with respect to x has been obtained in this study to follow

the equation: pH'27-3*0-6*)^-91 - j ^ _ jmpijes m a t m e j^uits 0 f the other stiffness equations

which do not include x ratio are susceptible to a large variations as shown in Figure 6.15.

joint parameters for data points column p y

1 2 3 4 5 6

0.33 0.54 0.33 0.8 0.92 0.8

20 20 13.36 20 20 13.36

***** Fessler (1986) 00000 Ueda (1990)

Tebbett, experimantal (1982)

1C? 10" Kaxiai , developed equation

Figure 6.15. Comparison of different equations for axial joint stiffness, the arrows specify the range of

axial stiffness when X varies from P.2 to P.97

Inplane bending mode (IPB)

There is generally a good agreement between the results of all IPB formulae compared here,

including the developed equation. It is assumed that the IPB formula established here,

produces reasonably accurate stiffness values. The reason for this assumption is the good

correspondence obtained between the test and F E analysis results, reported in Section 5.6.1.

Furthermore, the curve fittings used in constructing the IPB formula showed small errors.

The following points can be deduced from Figure 6.16 about the performance of different

equations:

1) Fessler's equation is more appropriate for the joints with large x values. Fessler's equation

generally predicts high values for the stiffness of a T-joint

2) Ueda's equation is more appropriate for the joints with small x values. Ueda's equation

generally predicts low values for T-joint stiffness and shows the poorest consistency among

the equations compared here. It should be noted that Ueda et al. have considered only 11

joints to derive their equation for IPB stiffness of T-joints (Ueda, 1990).


3) The results of D N V and Efthymiou's equations are in a very good agreement with those

from the new equation of this study, especially for T-joints with x values around 0.5. DNV

and Efthymiou's equations produce higher stiffness results for small x ratios than the actual

values and lower results for large x values.

}P=0-8 y=20

} p=0-8 Y=13.36

***** Fessler (1986) +++++DNV (1986) 00000 Efthymiou (1986) DD-IDD Ueda (1987) O O 0 0 0 Tebbett, experimental (1982)-Table 6.7c

KIPB x10 , developed equation

Figure 6.16. Comparison of different equations for IPB joint stiffness

4) The variation of IPB stiffness with respect to x ratio is seen as p^^+i-ss^o.* _ j ^ ^

make a difference of approximately 40% in the joint stiffness of a joint which has a x = 1

compared to when x = 0.2.

5) The Tebbett's results shown in Figure 6.16 are taken from experiment and are also

reported in Table 6.7b. It is seen that the experimental results may be well predicted for the

practical design purposes. However, a realistic comparison is not possible since the x ratios

used in Tebbett's work are not reported.

6) The calculated stiffness values should be converted to rigidity factors in order to have a

comparison with Kohoutek's equation. The rigidity factors of the joints in Table 6.7a are

calculated using Equation (3.20) suggested in Chapter 3 and Kohoutek's equation. The

results are shown in Table 6.8.


Table 6.8. Comparison of the results of proposed equation with Kohoutek's equation (IPB)

p 0.33

0.8 0.33

0.8

Y 13.36

13.36

20 20

*IPB

.787E6

.659E7

.392E6

.328E7

vb 0.67

0.48

0.58

0.43

r 0.92

0.79

0.83

0.70

*t 1.37

1.65

1.43

1.63

R : discrepancy ratio = Vb/T

As the theoretical basis of Kohoutek's equation is different from the joint model adopted in

this thesis (stated in Chapter 2), the comparison of the two equations may not be very

realistic. The maximum difference of rigidity factors calculated by the two formulae is 6 5 %

for the batch of parameters selected here. However, the similar trend is observed between

the results of the two equations, indicating that a closer agreement can be obtained by

adopting a different conversion rule to convert the joint stiffness to rigidity factor.

Out of plane bending mode (OPB)

The comparison of the out of plane bending stiffness values does not include Ueda's

equations since they do not cover this mode of deformation.

The proposed O P B equation herein predicts higher stiffness values than the experiments of

this study for the joints with large p ratios. The poor agreement of F E and experimental

results, as discussed in Section 5.6.1, is due to the poor fixity at the support assembly of the

joints that have large brace diameters. The F E method, however, is believed to produce

acceptable results as it predicted the IPB stiffness with a high accuracy. The following

conclusions can be made by comparing the different formulae of O P B stiffness:

1) The correlation among different formulae of OPB stiffness compared here is generally

good. However, Fessler's equation is more in agreement with the new formula developed in

this study, compared to D N V and Efthymiou's. Efthymiou's equation shows the poorest

comparison. Fessler's equation is seen to be more appropriate for the joints with x ratios

around 0.5, whereas D N V and Efthymiou's equations are appropriate for the joints with

small x values.

2) It is seen that the predicted stiffness values for the joints with p = 0.8 are also in a close

relationship to the values resulted from other equations. This supports the accuracy of the

F E analyses of the joints with large p ratios, since the developed equations herein are based

on the F E analysis results.

3) The results of the established equations here are in a good agreement with the

experimental results of Tebbett (1982) and McDermott (1979). The maximum difference is

2 4 % as shown in Table 6.7b.


4) The variation of O P B stiffness with respect to x is up to 4 0 % ; the joints with higher x

values produce larger stiffness.

M

c o

? :

o ID

o x £ o __:

0.1-

parameters shown in Table 6.7c

iN.8 XY=20

} P=0-» 1*13.36

***** I l i t I-

00000

mnm ooooo

Fessler (1986) DNV (1986) Efthymiou (1986) Tebbett, experimental (1982) McDermott, experimental (1979)

T 1—I I I |

S • 7 • »

1 KQPB X 1 ° , developed equation

Figure 6.17. Comparison of different equations for OPB joint stiffness

6.5 Summary

The results of a parametric study carried out based on the FE analyses to establish a set of

equations for prediction of the stiffness of tubular T-joints were reported in this chapter.

There are 90 different geometrical parameters which were covered by the F E models. A

chord diameter to thickness ratio of 6.85 was adopted for all models. The non-dimensional

parameters of the F E models considered in the analyses were diameter ratio (P), chord

diameter to thickness ratio (v), and thickness ratio (x). Because only one quarter of a joint

was modelled, three analyses on each model with various boundary conditions were carried

out.

Finite Element results had generally a good correlation with the verification tests, reported

in Chapter 5. Therefore, it is believed that the parametric study based on the F E analyses

produces realistic results. The IPB, O P B and axial stiffnesses calculated for T-joints show

that the stiffer joints are most sensitive to the variation of non-dimensional geometrical

parameters. For the three modes of deformations considered here, the joints with high P and


low Y ratios are relatively stiff. Consequently, it is recommended that accurate methods to

be used to determine the rigidity of the stiff joints.

Three equations for IPB, O P B and axial stiffness of T-joints were established based on the

F E analysis results. A power regression method was used for curve fittings of the calculated

stiffness values. It was shown that the maximum prediction errors corresponding to IPB,

O P B and axial stiffness values were 24.9%, 18.3%, and 28.6 compared to the F E results,

respectively.

The developed formulae include thickness ratio (x), which to the knowledge of this author,

has not been covered by other formulae reported before. This parameter makes a difference

of up to approximately 4 0 % for O P B and IPB stiffness, and 2 3 0 % for axial deformation of

brace. The significance of x ratio in the stiffness of T-joints, especially for axial loading,

indicates that this parameter should be considered in the prediction of the stiffness of a

tubular joints.

The predicted results by the developed equations in this study were compared with other

equations for a number of joint parameters, and good agreements were generally obtained.

The average of the absolute differences between the results of other formulae and the

equations reported here are 15.4%, 15.2% and 27.4% for IPB, O P B and axial loading

modes, respectively. The variation of joint stiffness due to the change of thickness ratio, x,

was clearly observed in those comparisons. However, x ratio is not usually reported in the

research works which makes the comparison of the different stiffness formulae to some

extent inaccurate. It was shown that the stiffness formulae which do not include the

thickness ratio are consequendy suitable for certain joints with specific x values. Despite

that P=l is highly used in tubular joints of offshore structures, most parametric equations do

not cover the joints with large P ratios.

Chapter 7

DYNAMIC STRESS CONCENTRATION FACTOR

Chapter 7: Dynamic Stress Concentration Factor 126

7.1 Introduction

Fatigue design of tubular joints for offshore structures by the S-N method requires the

calculation of stress values to be as accurate as possible at the area surrounding the weld.

The complicated geometry of the tubular joints makes the stress distribution spatially

dependent and highly nonlinear, particularly near the weld toe, resulting in the need for an

extensive research into the solution of the stress problem at the intersection of tubes. A

typical stress distribution for a simple tubular T-joint, when the brace is axially loaded, is


Normal stress distribution away from itersection

Maldistribution of nominal stress

Saddle Chord

Figure 7.1. Stress distribution at the intersection of a simple T-joint (UEG, 1985)

Methods for investigating the local behaviour of tubular joints are time consuming and

expensive, as referred to in Section 2.3, whereas, in practice, the use of relatively simple

procedures are preferred for analysis and design of structural systems. Because of this

complexity a coefficient which is called S C F (Stress Concentration Factor) is employed for

the analysis of tubular joints. S C F is generally defined as the ratio between the stress at the

intersection area and the nominal stress in the brace, from which S C F may be interpreted as

the normalised stress with respect to the brace nominal stress.

Fatigue design of tubular joints requires the knowledge of the maximum stress range.

M a x i m u m stress occurs at a location called the hot-spot, and there are different definitions

for the hot-spot SCF. S o m e investigators have employed the maxi m u m principal stress at

the hot-spots in their works. Others have referred to the maxi m u m stress perpendicular to

the weld line as the hot-spot stress ( U E G , 1985), and there are other methods with different

Brace

Crown

\

t—x- -

III! Ill


definitions (Tebbett, 1984 and Lalani, 1986). The proximity of the hot spot to the weld toe

is a matter of concern too, i.e. h o w far from the weld toe the strain measurements should be

made.

All these factors have imposed an uncertainty on the SCFs determined by different methods.

A parameter which has not been taken into account (to the knowledge of this author) in

regard to the uncertainty of the calculated S C F is dynamic strain reading. As the results of

the experiment and computer analysis showed in this study, the way loads are applied to a

joint in order to detect the strains makes a substantial difference to the calculated values of

stress or strain concentration factors.

This chapter reports the results of a series of dynamic strain reading tests carried out on

tubular T-joints and also the analytical solutions compatible with the experiments.

The stress field at a tubular joint under service load can be described as follows (UEG,

1985):

1) Action of the joints under general loading on the structure causes nominal stresses.

These stresses are due to external loading and are calculated through a global analysis of the

structure. This analysis, which is generally carried out on computer using the stiffness

method, leads to the recovery of internal forces and moments in different structural

members. Nominal stresses can then be calculated using the internal forces and the formulae

from strength of materials. In this regard, the global modelling of a platform structure plays

an important role in the determination of nominal stresses.

2) The geometry of a joint at the intersection changes the internal forces in the members by

developing deformation stresses. The chord wall deforms locally to maintain the

consistency of deformation between the brace end and the chord body, thus introducing

deformation stresses. Variable flexural stiffness between the crown and the saddle results in

a large stress variation at a joint. The deformation stresses are two types, a membrane type

which is constant through the thickness of the wall and a bending type which is assumed to

vary linearly.

3) Notch stresses are developed because of the sudden thickness change introduced by weld

and consequently an additional stiffness that is created at the intersection. This effect comes

as a part of the deformation stress. For example, if the weld is not modelled in the F E

analysis, the result does not contain the notch stresses, and will be called gross deformation

stresses (UEG, 1985). The stress system due to the geometry of a joint affected by the weld

is called local deformation stresses. A n example of the latter type of stress is the

measurements obtained through strain readings by electrical strain gauges on test

specimens. Figure 7.2 shows a typical stress distribution close to the weld toe, in which the

linearity of the bending stress is disturbed (Abel, 1982).


strain concentration at weld toe

Chord wall

Figure 7.2. Stress distribution near the weld toe

1 1

4) Residual stresses which are c o m m o n in steel structures also exist in tubular joints. They

are generally created via two sources; one is welding and the other is a lack of fit in

members. The former sometimes causes stresses up to yield point. Residual stresses are

tensile at the position of the weld passes.

7.2 Application of stress concentration factor in fatigue analysis

The following example shows how the stress ranges in tubular joints are calculated using

SCFs. It is taken from U E G (1985) and summarised here. The same procedure is used in

Chapter 8 of this study to predict the fatigue life of tubular joints.

Example

A T-joint is considered for the evaluation of the fatigue damage ratio (nIN).

Figure 7.3 shows the joint geometry and the forces obtained from a computer

analysis. OPB

IPB

Axial

Chord 457x9.5mm L=5484mm

(6 far side) 8

Brace 457x6.3mm

2 (4 far side)

t 1 (5 far side)

Figure 7.3. The geometry and forces of the T-joint considered in Example 1 (UEG, 1985)


Using a set of S C F parametric formulae, the stress concentration factors for the

different deformation modes and the different locations around the intersection

of the brace and the chord can be calculated. The results of such calculations are

shown in Table 7.1, below.

Table 7.1. Hot-spot Stress Concentration Factors for the joint of Example 1

Stress Concentration Factor

Location

1 2 3 4 5 6 7 8

Axial

Chord 7.64

8.62

9.61

8.62

7.64

8.62

9.61

8.62

Brace 5.81

6.43

7.05

6.43

5.81

6.43

7.05

6.43

IPB

Chord 0.00

2.30

3.26

2.30

0.00

2.30

3.26

2.30

Brace 0.00

2.16

3.05

2.16

0.00

2.16

3.05

2.16

OPB

Chord 9.55

6.75

0.00

6.75

9.55

6.75

0.00

6.75

Brace 7.02

4.96

0.00

4.96

7.02

4.96

0.00

4.96

Points 2, 4, 6 and 8 are located on a 45 degree angle with respect to the crown

points, as shown in Figure 7.3. From Table 7.1 the hot-spot stress range can be

calculated for points one to eight using Equation (7.1):

A/«s = A/_ SCFa + AfwB SCFIPB + AfoPB SCFOPB (7.1)

in which A/_, AfipB and AfoPB are the nominal brace stress ranges due to axial,

inplane and out of plane loadings, respectively. A better estimation of the hot

spot stress under combined loadings on a joint can be obtained by using the

formulae or methods which consider the interaction between the loads.

The nominal stress ranges are usually obtained from a computer analysis of the

platform structure under wave loading, wind loading, operational loads, etc.. To

assess the fatigue damage to the joint, the stress ranges corresponding to a

specific load case is calculated using a linear damage summation as shown in

Equation (7.2). It is usually assumed that fatigue failure occurs when under total

effects of individual loads the damage summation I,(n/N) becomes equal or

greater than one.

_ ; ^ A ( = I . O ) (7.2)


Considering a wave distribution as shown in Table 7.2:

Table 7.2. No. of occurrences for different wave heights used in the fatigue example

Wave height

(m) 1.5 4.5 7.5 10.9

17.1

Occurrence

(Direction from south)

1026467

252347

13621

1211

601

The following nominal stress ranges are obtained from a structural analysis,

when the third wave with the height of 7.5m is considered:

Table 7.3. Nominal stress ranges in brace member for different deformation modes

Load type Nominal stress range

Axial 32.16 IPB 23.26 OPB 15;39

Using Equation (7.1) with the stress range values of from the above table and

SCFs from Table 7.1, hot-spot stress ranges for the eight points around the

intersection can be determined as shown in Table 7.4 below :

Table 7.4. Hot spot stress ranges for the T-joint in fatigue example

Location Hot-spot stress range (N/mm2)

1 2 3 4 5 6 7 8

Chord 393 435 385 435 395 435 385 435

Brace 295 333 298 333 294 333 298 334

It is seen in table 7.4 that the hot-spot locations are points 2, 4, 6 and 8 with the

m a x i m u m stress values of 435 N/mm 2 . Using a design S-N curve, N value is

assumed to be 29888, which is the number of cycles for fatigue life


corresponding to the constant stress range of 435 N/mm2. The cumulative

damage then would be:

i _ ! _ _ _ o . 4 5 6 (7.3) N 29888 V '

The value 0.456 shows the degree of fatigue damage to the joint, produced by

the 7.5m high waves. The same calculations should be carried out for the other

wave heights and loadings, and the damage summation then compared with

unity. W h e n a factor of safety (FS) is assumed the maximum permissible

damage ratio will be reduced to 1/FS.

This simple example shows how crucial S C F is in the fatigue design of tubular joints. It is

the key parameter for converting a nominal stress, calculated in a structural analysis, to the

hot spot stress used for fatigue calculation. This study is trying to make improvements on

the calculation of both nominal stress values and stress concentration factors. The stiffness

consideration of the joints in the structural analysis increases the accuracy of the calculated

nominal stresses. It is shown in this chapter that the dynamic aspect of the loading on

tubular joints should also be considered in order to have better predictions of the SCFs.

The SCFs, in the above example, were calculated by using parametric formulae which is the

method commonly exercised in practice. However more comprehensive methods, which will

be explained in the next section, are required to establish the S C F parametric formulae.

7.3 Methods of SCF determination

The methods used to study the local behaviour of tubular joints are also applied to the

investigation of SCFs. Referring to Section 2.3, S C F may be evaluated via the following

methods:


2) experimental methods, and

3) Finite Element method.

The major study on tubular joints involves stress distribution and stress concentration at the

joints. In this regard, experimental and Finite Element methods are most successful and

extensively used by the researchers. Using any method other than experiment is an attempt

to by-pass direct measurement and inevitably introduces some approximations. However,

when a non empirical model is confirmed by a systematic experimental approach, a formula

may be established from the results and used instead. The parametric formulae for S C F or


stiffness of tubular joints are the examples. In this section a brief introduction is given to

strain gauging for strain and stress recovery in tubular joints. There will be also a general

review on some Finite Element studies.

7.3.1 Experimental methods

A m o n g the experimental methods of stress and strain identification, applying strain gauges

is the most popular and perhaps the oldest procedure. This method can be applied to steel

models as well as the models made of synthetic materials such as araldite. The most popular

strain gauges are foil gauges that were made in 1932 for the first time in the U S A . The foil

gauges are produced by a printed circuit process from selected alloys, which have been

rolled to a thin foil. The foil is placed on a backing material and encapsulated in an

insulating matrix. The surface area of the foil conductor per its cross-sectional area is very

large which provides good heat transfer between the grid and the specimen. The significant

economic measure in strain gauging is the total cost of the complete installation and the cost

of strain gauge itself is not ordinarily a prime consideration. There are of course some

applications in the industry that require very expensive strain gauges, too.

Strain measurements on tubular joints are generally carried out under static loads. The

specimens used for strain gauge tests require elaborate test rigs in order to be supported

properly, especially for ultimate strength tests in which large loads are applied, resulting in

expensive procedures. Steel tubular joints must be fabricated according to the standard

offshore procedures. Furthermore, models should simulate the real joints as closely as

possible in terms of geometry, material, fabrication procedures, loading and restraint

conditions. Other testing precautions are also necessary such as longitudinal seam welds on

the pipes which may affect the readings and should not coincide with the expected hot

spots, or weld size that could also be a problem for scaled down specimens.

The first strain measurements on tubular joints were conducted in the 1960s. Toprac and

Beale (1967) established the first parametric formulae using a limited database from steel

joints. Since that time a large number of experiments have been carried out on strain

measurements of tubular joints. U K O S R P (United Kingdom Offshore Steels Research

Program) has established several projects on the comparison of different methods of

studying the stress distribution of tubular joints. Based mainly on U K O S R P results, U E G

(1985) recommends experimental methods for the cases where,

1) high accuracy is required, or

2) the problem is such that the numerical methods are either not cost effective

or not readily interpreted and/or accepted (because of accuracy).

Strain gauges, like any other measurement device, have some particular deficiencies which

preclude their absolute accuracy. These are:


1) gauge length that produces an average strain reading.

2) They are very difficult for determination of strain gradient, especially when

there is a high stress gradient, physical dimensions of strain gauges become a

problem.

3) Strain gauge implementation is impossible or sometimes very difficult on the

inside surface of tubes.

Apart from the above problems interpretation of strain readings for fatigue design purposes

requires a rational approach in order to produce consistent results from different tests. This

will be discussed in the following section.

7.3.2 Interpretation of stress concentration factor from measurements

There is debate on h o w strain measurements can be incorporated into S C F determination.

Several methods have been proposed, which generally try to avoid picking up notch

stresses, since their accurate measurement is not possible. This is due to the rapid three

dimensional variation of the notch stresses within a few millimetres of the weld toe and also

because the notch stresses depend on the weld geometry and vary from joint to joint

Therefore, the S-N curves established for fatigue life prediction of tubular joints are specific

to particular weld profiles. For the same reason a general S-N curve e.g. an as-welded S-N

curve becomes very conservative. The European Community of Steel and Coal (ECSC)

recommends a method for strain measurement to determine SCF. This method, which is

similar to that of the U K Department of Energy (DEn) Guidance Notes is shown in Figure

7.4.

There are two issues involved in the E C S C or similar methods. One is the location of strain

gauges relative to the weld toe and the other is the form of extrapolation on the strain

readings to obtain the hot spot strain or stress. Gurney (1979) recommends a distance of

approximately 0.35 times the wall thickness from the weld toe for applying strain gauges.

This has been obtained based on Finite Element analyses of simple fillet-welded joints in flat

plates. H e has not suggested any distance for the second strain gauge or guidance on how

to extrapolate to the toe position.

The following methods are generally applicable to the S C F evaluation of tubular joints,

when stress or strain extrapolation are required (Lalani et al., 1986):

1) linear extrapolation of maximum principal stresses,

2) curvi-linear extrapolation of maximum principal stresses,

3) curvi-linear extrapolation of maximum principal strains,

4) individual curvi-linear extrapolation of strains parallel, perpendicular and at

45° orientations to the weld toe and combining them to obtain the maximum

principal stress, and 5) curvi-linear extrapolation of strains perpendicular to the weld toe.


0.6S(rt)

a = 0.2(d) , but not smaller than 4 m m

Chord

Figure 7.4. Hot-spot definition for T and DT-joints recommended by ECSC

In a paper in 1986, Lalani et al. describe the third method above as being increasingly

favoured by investigators. In that article, however, it is recognised that the linear

extrapolation of maximum principal stresses is appropriate for vertical braces, and the curvi

linear extrapolation is for joints with inclined braces. It is emphasised by Lalani that the

resulting hot-spot stress should be compatible in terms of extrapolation technique with an

appropriate S-N curve.

7.3.3 The Finite Element Method ( F E M )

Numerical methods have performed extremely well in connection with structural analysis

since the advent of computers. The Finite Element method has become very popular as a

numerical tool to solve the deformation problem of continuum media, and most of the time

performs with sufficient accuracy. Greste (1970) analysed tubular K-joints using F E M . H e

used a mesh generating routine for definition of joint geometry which produced a well

designed mesh. Kuang et al. (1975) derived a set of S C F parametric formulae via Finite

Element analysis. Several other S C F studies have been conducted using F E M , e.g. Gibstein

(1978), Buitrago et al. (1984), Efthymiou et al. (1985) and Hellier et al. (1990), resulting to

the parametric S C F formulae. Some of the analyses included weld modelling. Similar to

strain measurement, the location of maximum stress is a controversial matter in the Finite

Element analysis of tubular joints. Most investigators have selected the brace mid-wall to


calculate the hot-spot SCF, which differs from that which is practiced on test specimens. It

has been recommended (UEG, 1985) that a specific distance from the weld toe similar to

the electrical strain gauges be used for stress determination around the tubular intersections.

Apart from the problems regarding shell element in the Finite Element method discussed

briefly in Chapter 5, there are other deficiencies associated with the method. For example

residual stresses can be hardly considered in F E M . The weld is not usually included in the

Finite Element model of a joint, producing false bending moments for the areas near the

intersection. Mesh generating is another task that requires sufficient attention. Progressively

reducing the size of elements provides convergence to the exact solution, but becomes

costly due to the small mesh size. The actual mesh chosen then is a compromise between

acceptable accuracy and cost of analysis.

With all descriptions given above, one has to be cautious when modelling a tubular joint by

the Finite Element method to recover strain or stress values. In order to verify the F E

results, convergence studies or comparisons with test results are essential.

7.4 Reliability approach for stress concentration factor

In 1984, Tebbett and Lalani introduced a reliability approach to the stress concentration

factor concept used in design of tubular joints. The subject of their paper is the probability

that the stress concentration factor (SCF) used in calculating the hot spot stress range for a

defined input loading, will underestimate the actual hot spot stress range. The paper

compares the test results of steel tubular joints with the S C F prediction of parametric

formulae, considering variability of the S C F estimation for similar conditions.

Variations in evaluated SCFs stem from the following factors:

1) general form of parametric equation,

2) the method used for generating the original SCFs (e.g. F E M or model testing),

3) out of roundness tolerances on tubes,

4) wall thickness tolerances on tubes,

5) accuracy of fabrication,

6) weld size and profile,

7) accuracy of load application and measurement,

8) accuracy of gauge position and position measurement, and

9) accuracy of stress/strain measurement or conversion.

In the follow-up paper, Lalani et al. (1986) have continued the reliability approach they

introduced in 1984 and showed the variations in SCFs when different parametric formulae

are used. Figure 7.5a shows the distribution of the ratios of the SCFs obtained from

compression tests to the SFCs of the same joints from tension tests. It is seen that for 42


specimens, approximately 37 S C F ratios (90%) are between 0.9 and 1.1. This indicates that

an S C F which is determined in a compression test can be ± 1 0 % different from when it is

calculated in a tension test. Further change in S C F is expected for the remaining 1 0 % of the

results. Also, nominally identical joints showed ± 1 0 % change in measured SCFs. This is

shown in Figure 7.5b where 29 SCFs out of 33 have a variation of ± 1 0 % with respect to the

average SCF.

Whether a test is performed dynamically or statically can influence the results as much as

those 1) to 9) above. Offshore rigs operate under dynamic loading that contributes to

fatigue in the critical areas such as joints. The different methods referred to, in Section 2.3

are generally performed statically and make no distinction when applied to a structure under

dynamic or static loads. Of course, dynamic loading will not entirely change the internal

forces of the structural members in comparison with static loading. It is important to

consider all parameters that influence the calculation of SCF, in order to establish a

reliability approach for SCFs. This will provide a higher accuracy in the structural analysis

Sample size = 33

JHL_ .9 1. 1.1 1.2 1.3 1.4 Measured SCF Average SCF

ariations of SCF for nominally s (after Lalani, 1986)

7.5 Dynamic strain reading

Electrical strain gauges can be used on the structures for dynamic strain measurement,

however more expensive and sophisticated instruments are required. A series of strain

measurements were carried out on eleven T-joints in this study for inplane and out of plane

bending modes. The geometry and non-dimensional parameters of the joints are reported in

Table 4.1. A typical diagram of the strain gauge locations is shown in Figure 7.6. Strain

gauges D and E are located at the proximity of the hot spots for inplane and out of plane

modes, respectively. Strain gauges C and B measure the nominal strain in the brace. The hot

and a higher safety in the structural design.

1.1 1.2 1.3 1.4 Compression SCF Tension SCF

Figure 7.5a. Variations of SCF for direction of axial loading (after Lalani, 1986)

Figure 7.5b. Vt identical joint.


spot strain gauges were located 5 m m , and the brace strain gauges were positioned at least

2d from the weld toes, where d is the brace diameter.

gauge length = 6 mm

Figure 7.6. A typical strain gauge location diagram of a test specimen

The purpose of the tests was to compare the SCFs of a joint under sinusoidal loading when

the load frequency was varying. It is realised that the real loading on the offshore structures

has a random distribution. However, the generation of inertia forces is the most important

difference between static and dynamic loads which is also simulated by sinusoidal loading.

Strain comparison serves the same purpose as S C F comparison because of the linear

relationship generally assumed between stress and strain in steel offshore structures.

The outcome of each test was a series of strain readings for two points on a joint, for

example £ D and Zc at points D and C in Figure 7.6. Hence, every two strain readings

correspond to a certain load frequency. The load was also measured to check the quality of

the input excitation to the joint specimen. In static tests the strain readings are single

numbers whereas for dynamic loads they are distributed quantities varying with time and

may be shown by a curve as in Figure 7.8. After the strain readings were recorded, the ratio

of the average readings for each frequency (£D(_v/ec,av.) was calculated and used for

comparison with the ratios from other frequencies.

The strain gauges used had the following specifications :

Gauge length = 6 m m ,

Electrical resistance = 120 Ohms,

Gauge factor = 2.04.


7.5.1 Test set-up

T w o series of tests were carried out to study the relationship between strain concentration

factor (SNCF) and load frequency for inplane and out of plane bending modes. A

diagrammatic test set-up for the dynamic strain readings of a tubular T-joint under inplane

bending is shown in Figure 7.7. A similar set-up was used for out of plane bending with the

load acting perpendicular to the specimen plane defined by the chord and brace. The load

was measured through the force transducer installed between the shaker and the joint. The

alternating analog signal (voltage) which was produced by the strain gauges under dynamic

deformation of the system, was converted to a digital signal. Diagrams of load and strain for

the same time base could be read directly on the monitor of the PC. Therefore, any noise or

interference from other sources would show on the screen and if necessary the test could be

repeated.

M

=n r

_____

v -v

1500mm

1- Shaker 4- Strain gauge amplifier

8- Force transducer

2- Shaker amplifier

5- AD converter

9-Strain gauge

3-Charge amplifier

6 and 7- PC

Figure 7.7. Diagrammatic test set-up for the dynamic strain measurement


Only sinusoidal loads were applied to the joints, with the frequencies from 0.5Hz to 55Hz

and 5Hz increments. N o provision was made for temperature compensation of strain gauges

since the rapid change of strain did not allow any temperature effects on the leadwires.

Typical strain readings for IPB mode, taken at the crown point and on the brace is shown in

Figure 7.8.

20m/Div inplane T10/3 200

Micro Strain

50 /Div

-200 0 - seconds - 200m

Figure 7.8. Dynamic strain readings for joint TIP

In the following section the results of inplane and out of plane strain measurements will be

presented and discussed.

7.5.2 Test results

Inplane bending mode

Table 7.2 presents the results of the measurements carried out for the inplane bending case.

There are two strain measurements reported for each frequency taken at points C and D

(see figure 7.8), on the brace and hot spot, respectively. The ratio of the two strains is also

reported.

Strain ratio and load frequency relationships are also shown in Figures 7.9a to 7.9c, where

an increasing variation with respect to load frequency is generally observed, however in

some cases the variation is not very pronounced.


Table 7.2. Dynamic strain readings and SNCFs for IPB mode (mm/mmxlP6) Jo

int

1

2

3

4

5

6

7

8

9

10

11

strain

location

EC

ep/ec

EC

E D

ED/EC

Ec E D

ED/EC

Ec

ED

ED/EC

EC

E D

ED/EC

EC E D

ED/EC

EC ED

ED/EC

Ec

E D

ED/EC

EC

E D

ED/EC

EC

E D

ED/EC

EC

E D

ED/EC

Frequency (Hz) 0.5 5 10 15 20 25 30 35 40 45 50 55 3.035 4.894 4.95 4.538 4227 4.442 4.608 4.794 4.92 5.12 5.471 5.945

3.586 5.445 5.512 5.08 4.752 5.045 5.338 5.625 5.895 6.284 6.893 7.718

1.182 1.113 1.114 1.119 1.124 1.136 1.158 1.173 1.198 1.227 1260 1298

6.632 11.86 10.49 10.43 10.47 12.48 14.03 17.17 23.85 44.31 63.87 23.79

12.33 21.07 18.68 18.73 18.99 22.99 26.04 33.01 46.88 89.59 133.5 51.56

1.859 1.777 1.781 1.796 1.814 1.842 1.856 1.923 1.966 2.022 2.09 2.167

7.595 13.16 12.74 12.51 12.51 14.9 16.09 19.31 25.61 41.64 74.42 37.1

1124 18.62 18.05 17.85 18 21.75 23.88 292 39.66 66.16 122 62.8

1.48 1.415 1.417 1.427 1.439 1.46 1.484 1.512 1.549 1.589 1.639 1.693

39.03 6728 69.48 72.61 70.51 113.3 1242 8321 50.91 31.96 22.34 16.54

24.94 41.89 43.65 46.44 462 76.59 87.42 61.66 39.86 26.68 20.07 16.18

.639 .623 .628 .640 .655 .676 .704 .741 .783 .835 .898 .978

1.697 2.851 2.601 2.448 2.416 2.62 2.645 2.801 2.961 3253 3.675 4.341

6.085 9.748 8.894 8.387 8249 8.957 9.043 9.486 1029 1128 12.78 15.09

3.586 3.419 3.419 3.426 3.414 3.419 3.419 3.387 3.475 3.468 3.478 3.476

8.869 13.94 12.87 12..9 13.64 15.9 19.67 27.42 49.09 33.67 24.67 13.57

17.17 28.06 25.95 26.32 2822 3321 4226 60.51 112.7 80.01 61.13 34.64

1.936 2.013 2.016 2.04 2.068 2.089 2.148 2207 2296 2.376 2.478 2.553

8279 14.07 13.63 1424 14.19 16.95 18.17 23.14 3725 68.12 35.86 18.15

16.58 26.5 25.74 27.14 27.44 32.92 36.5 47.88 79.16 149.8 81.83 43.34

2.003 1.883 1.888 1.906 1.934 1.942 2.009 2.069 2.125 2.199 2282 2.388

37.11 64.39 65.51 68.72 8925 1032 87.33 56.74 31.91 24.45 16.8 12

46.7 78.34 80.6 86.31 115.4 138.8 122.7 84.85 51.08 42.38 31.92 25,45

1258 1217 123 1256 1293 1.345 1.405 1.495 1.601 1.733 1.9 2.121

2.453 4257 4.008 3.896 3.534 4.02 4.143 4.347 4.618 5.116 5.91 7.603

5.69 9.563 9.052 8.88 8.091 9248 9.785 10.54 11.58 13.32 16.07 21.85

2.32 2246 2258 2.279 2289 2.3 2.362 2.425 2.508 2.604 2.719 2.874

8.379 14.35 14.13 14.35 15.39 18.47 2425 36.64 54.51 28.58 14.72 8.949

29.76 47.89 47.41 48.77 53.4 65.34 88.71 139.1 216.6 119.5 65.31 42.62

3.552 3.337 3.355 3.399 3.47 3.538 3.658 3.796 3.974 4.181 4.437 4.763

43.3 66.75 68.37 71.31 92.44 84.83 61.95 4023 26.1 17.12 11.95 8.431

77.18 112.8 1172 125.5 168.9 162.9 126.5 88.42 62.74 45.86 36.45 30.19

1.782 1.69 1.714 1.76 1.827 1.92 2.042 2.198 2.404 2.679 3.05 3.581

E D is measured at hot spot (crown). _c is measured on brace.

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The diagrams in Figure 7.9 have a clear indication that the strain ratios calculated from the

readings at the foot of the welds and brace members have a variation with respect to the

load frequency. Apart from joint T5 in Figure 7.9a, it is seen that the strain ratios of the

joints with large y ratios have the biggest and the steepest variations with respect to the load

frequency. The maximum changes of the strain ratios calculated in a frequency span of

0.5Hz to 55Hz are 2 3 % , 3 4 % , and 1 0 0 % for T9, TIO, and Til respectively. It is also noted

that the variations of strain ratios with respect to load frequency increase for small P ratios.

This means that there is a more possibility of S C F variation due to dynamic loading in the

joints with thin chord walls or small brace diameters. According to the findings in Chapter

5, these types of joints possess a low stiffness.

The real loading on offshore structures has a frequency of less than 1Hz which means a

period of greater than 1 second. Although the frequencies of the loadings used in this study

span well beyond 1Hz but the results show that the strain ratio or S C F is frequency

dependent. The range of frequency used for loadings in the tests herein would only magnify

the observed phenomenon.

Out of plane bending mode

Out of plane bending measurements produced similar results to the inplane bending case.

Measurements of the hot spot (saddle point) strains and the nominal brace strains are

reported in Table 7.3. Points E and B correspond to the hot spot and brace, respectively.

Figures 7.10a to 7.10c shows the relationships of strain ratio with respect to load frequency.

Although the strains in O P B tests were measured up to 55Hz, but due to the very large

ratios obtained for T9, TIO, and Tl 1 after 40 Hz, only the ratios related to 0.5Hz to 40Hz

were plotted in Figure 7.10.

It is seen in Figure 7.10 that the change of strain ratios with respect to load frequency in

O P B mode is bigger for the joints with large y ratios compared to other joints. For this

range of y, the strain ratio shows also the steepest variation especially at high frequencies.

Each diagram in Figure 7.10 is for a different P ratio. The maximum variations of the strain

ratios with respect to 0.5Hz frequency are 3%, 7 5 % , and 1 2 0 % for T9, T10, and Til,

respectively. It is seen that, similar to IPB mode, the joints with low p values show the

greatest variations in strain ratio. The findings here indicate that out of plane bending SCF

has the largest variations with respect to load frequency in the joints with small P or large y

ratios.

Comparing the results of IPB and O P B modes in Figures 7.9 and 7.10, it is seen that higher

strain ratios have been resulted for O P B mode. The increase in strain ratios is noted to

occur drastically for high load frequencies in the O P B tests.


Table 7.3. Dynamic strain readings and SNCFs for OPB mode Jo

int

1

2

3

4

5

6

7

8

9

10

11

strain

location

EB EE

EE/E B

EB

EE EE/EB

EB

EE

Efi/Efl

EB

EE

EE/E B

EB

EE

£_/eB EB

EE

EE/EB

EB

EE

E E / E B

EB

EE

EE/E B

EB

EE

EE/E B

EB

EE

_E/e_

EB

EE

EE/EB

Frequency

0.5 1.777

3.889

2.19

8.93

46.26 5.18

8.475 38.49 4.542

91.39 191.3 2.093

1.621

11.72 7.230

8.37

65.97

7.882

9.502

42.36 4.458

43.72 130.3 2.980

1.532

26.11

17.04

8.96

82.99 9.262

41.91 244.8

5.841

5 3.069

6.701

2.18

14.82

72.5 4.892

15.23 64.55 4.238

69.48

136.4 1.963

2.94 20.27

6.895

14.45

105.7 7.315

16.19 67.24 4.153

68.93 190.1 2.758

2.479 40.19

16.21

15.53

129.7

8.352

66.44

355.1 5.345

10 2.952 6.431

2.18

15.19 74.64

4.914

15.11 64.65

4.279

71.89 143.3 1.993

2.735

18.85

6.892

16.29 121.7

7.471

16.51

69.64 4.218

69.95 197 2.816

2.643

42.54 16.10

17.63

150.7 8.548

63.09 352 5.579

15 2.956

6.415

2.17

17 84.4

4.965

16.16 70.38

4.355

76.72 157.3 2.050

3.026

20.89 6.904

19.83

153.8

7.756

18.14

78.95

4.352

74.23 217.7

2.933

2.922 47.2

16.15

23.02

209.1

9.083

57.53

344.3

5.985

20 3.074

6.65

2.16

22.16

111.8

5.045

18.88

84.59 4.480

86.04

183.6 2.134

3.463 23.96

6.919

30.21 246.8

8.169

25.7

116.9 4.549

76.16 236.6 3.107

3.607

58.89 16.33

29.26

287.1

9.812

39.69 267 6.727

25 3.373 7.274

2.16

35.54 183.4 5.16

25.87 120.4 4.654

85.42 192.2 2.250

4.19 29.05

6.933

36.01 316 8.775

41.94

203.2 4.845

57.69 194.3 3.368

4.916

79.72 16.22

13.68

150.3

10.99

23.89 188.7

7.899

30 3.967

8.498

2.14

47.17

249.9 5.298

45.03 220.4

4.895

59.94

144.3 2.407

6.558

45.62 6.956

13.8

132.8

9.623

20.9 108.8

5.206

33.01 131.3 3.978

10.45 170 16.27

6.006

77.46

12.90

13.69 127.9 9.343

35 5.197

11.05 2.13

21.74 118 5.428

29.16 151 5.178

36.37 94.24

2.591

11.66

81.94 7.027

6.428

70.05 10.90

9.577 54.81 5.723

19.97 83.66

4.189

6.058

101.9 16.82

2.946

47.66 16.18

6.758

87.79

12.99

40 8.577

18.13

2.11

10.71

60.04 5.606

13.03 71.89 5.517

22.84 65.42 2.864

6.536 45.85

7.015

3.423

43.91 12.83

5.136

33.24 6.474

12.26 60.7

4.951

3.059 50.4

16.48

1.372

35.19 25.65

2.997 66.44

22.17

45 18.54

38.69

2.09

6.751 39.46

5.845

7.419 44.32 5.974

14.76 47.68

3.230

3.318

23.28

7.016

1.997

30.98 15.51

3.014 23.25

7.714

7.408 45.65 6.162

1.788

28.96 16.20

.6697

22.51 33.61

.5956

52.72

88.52

50 9.174

18.91 2.06

4.297 28.06 5.630

4.756 31.17

6.541

10.4 38.84 3.735

2.097

14.71

7.015

1.081 23.76

21.98

1.812 16.48 9.095

4.441 36.92 8.313

1.245

19.39 15.57

.2274

17.66 77.66

1.169 43.55

37.25

55 4.725 9.628

2.04

3.407

21.75 6.384

3.201

23.32 7.285

7.197

32.2 4.472

1.419 9.931

6.999

.495 18.08

36.52

1.048 13.06 12.46

2.366 31.06 13.13

.9432 14.11 14.96

.3658 13.98

38.22

2.37 36.57

15.43

E E is measured EB is measured

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7.6 Dynamic strain calculation using stiffness method

In this section the performance of the stiffness method in predicting the strain ratios at

tubular joints under dynamic loading is examined and the results are compared with

experiment In the model shown in Figure 7.11 a continuous mass system is assumed for

calculation of the inertia forces. W e call this model a beam model as distinct from a test

model (experiment). Since the stress or strain concentration can not be considered by the

stiffness method, the strains at half way from the intersection to the chord support are

calculated (point D in Figure 7.11) instead of a location close to the weld. Therefore the

experimental data which are used in conjunction with the results of the stiffness method is

different from those previously showed in this chapter. The difference is the location of the

strain gauges on the chords. Only inplane bending is considered in this section since out of

plane bending produced very small strains at point D and therefore a reliable strain

measurement was not possible. The loading was a sinusoidal force at the top of the

cantilever equal to 1000N. The strains were calculated at points D and C. The strain gauges

were applied to the test models at the same locations as on the beam models (points D and

C in Figure 7.11) to obtain comparable results.

The strain ratios obtained for the beam models and the test models are shown in Figure 7.12

for a number of joints. The joints with a diameter ratio of 1 are compared in Figure 7.12a,

which are three joints with different y ratios. It is seen that the results of the test and the

stiffness method are approximately 1 0 % different for Tl which can be counted as a good

agreement. However, for T 5 and T 9 the difference between test and theory becomes much

bigger than Tl. Furthermore, the variation of the strain ratios for the test models show a

bigger slope than the beam models, especially in high frequencies.

The same trend is observed for the test and analytical results in Figures 6.7b and 6.7c as in

Figure 6.7a. The stiffness method and experiment on the joints with low diameter to

thickness ratios (T2 and T4) show a good agreement. It is again seen that for the joints with

large y ratios, there is a bigger difference between the results of the test and stiffness

method. Although, the beam model has not predicted the strain ratios of tubular joints as

accurate as experiment, however it shows a dependence of strain ratios on the load

frequency, as observed on the calculated values.

The comparison of stiffness method and test results shows that:

1) Stiff T-joints are less prone to the variation of strain concentration factor due to

load frequency.

2) Stiffness method does no produce reliable results for the variation of strain

concentration factor under dynamic loading.


P = PeSln(ot

1230

h-375 -H z.

c-

D

All dim. in m m

T h

___.

1500

h=3S0 tor T2,14,16,18,110, and 111

h=420forl1,l5,andl9

T9,T= 22-82 T5,7=17.12 T1,T= 13.36

m i TI 1111 n 11 n r TI p i ii n n 11 I I H i n i tji 111111

20 30 40

Frequency (Hz)

60

Figure 7.11. Beam model to study dynamic

effects on strain ratio

a) Joints with $ = 1.0

0.35

0.05"

* * * * * Test Model o o o o o Beam Model T10

T6,T= 1712

T2,T= 13-36 0.0 111111111111 11111111111111111111111111111

0 10 20 30 40 Frequency (Hz)

50 60

0.096-,

0.080

J" 0.064-

h 0.048 d

M 0.032

0.016

0.0

__*-_ Test Model o o o o o Beam Model

IPB, P=0.28

T11.T = 22.82

T8,T= 1712 T4.7= 13.36

T11

11 m i n i | m i |iin |IIIIIIIII| iiii|illllinl|

0 10 20 30 40 50 60

Frequency (Hz)

b) Joints with p = 0.52 c) Joints with p =0.28

Figure 7.12. Strain ratios resulted from beam models and test models for joints with a)$ = LP, b)$ = P.52,

and $ = 0.28


7.7 Dynamic strain reading and S C F / S N C F reliability

The variation of strain concentration factor under dynamic loading is an indication of

different response behaviour of a structure to dynamic loads compared to static loads.

Stress or strain concentration factors are currently calculated based on static tests or static

Finite Element analyses. Therefore, the variation of stress concentration factor with respect

to load frequency is not considered in the analysis procedures. However, offshore structures

operate under dynamic loadings and therefore their behaviour should be studied and

predicted under such conditions. Dynamic loading on a structure produces inertia forces

acting all along the members, which is described well by the dynamic deformation

formulation used in this study. These distributed forces vary with load frequency and create

a variation of the nominal forces or stresses in the structural members. Therefore, S C F or

S N C F become variable with respect to load frequency.

Tebbett and Lalani (1984) suggest four primary parameters to be considered in the

calculation of the probability of failure by fatigue. These variables are:

1) loading spectrum,

2) accuracy of SCF,

3) scatter in fatigue life for known 'hot spot' stress range, and

4) relationship of stresses at the point of consideration to calculated SCF.

Dynamic strain reading may be placed in the second category of the variables suggested by

Tebbett. It can improve the accuracy of the measurements on physically tested models, and

therefore reflects one of the reasons for the reliability approach to the calculation of SCF or

SNCF.

The best solution to take the load frequency into account is to employ a frequency

dependent factor in the parametric S C F formulae. The determination of such a factor

requires further analytical and experimental studies.

Another solution to consider the variation of S C F with respect to load frequency is also

possible by using a reliability approach. Since the environmental loads impose typically low

frequencies on the offshore structures, it seems to be possible to introduce a constant factor

to give the appropriate level of confidence of SCFs. This approach comes with the concept

of design S C F to be used in the design equations. Design S C F is given by Equation (7.4)

below (Tebbett and Lalani, 1984):

SCF D S C F ________ (7.4)

r in which DSCF is the design stress concentration factor and T is the partial factor selected

to give the appropriate level of confidence. Depending on the approach taken for an


individual design, T could range between 0.6 and 1.5. The T values lower than one will give

increased confidence in design, Le. less chance of a higher actual SCF.

7.8 Summary

It has been shown in this chapter that the static methods of strain/stress determination do

not produce necessarily conservative strain/stress concentration factors when a structure

experiences dynamic loads. The variation of S C F due to the load frequency becomes

important in fatigue life estimation of tubular joints in offshore structures. The parametric

equations, which are used to determine the hot spot stress at a tubular joint, are all based on

the databases from static experiments. Therefore, they do not recognise the dynamic loads

which are applicable to the offshore platforms. As the experiment and analytical solutions

showed in this study, whether the loads applied to a joint are static or dynamic makes a

substantial difference to the calculated values of stress or strain concentration factors.

A reliability approach was introduced by Tebbett and Lalani (1984) to the S C F parametric

equations. The reason for introduction of such approach was the inconsistent results

obtained from different formulae or tests on the joints with the same specifications.

Dynamic load effects on S C F estimation can be considered the best by employing a

frequency dependent factor in those equations. Another solution can be employed based on

a reliability approach in the form of a constant factor, because the dominant loads on

offshore structures have low frequencies. However, in both solutions there is a need to

evaluate those factors realistically using analytical or numerical dynamic methods of analysis

supplemented by experimental investigations.

Chapter 8

SEMI-RIGIDITY EFFECTS ON BEHAVIOUR

OF OFFSHORE TOWERS

Chapter 8: Semi-Rigidity Effects on Behaviour of Offshore lowers 150

8.1 Introduction

In order to study the effects of joint rigidity on the behaviour of offshore structures, a tower

is analysed in this chapter for two states of rigid and flexible joints. To be able to compare

the results with another reference, the geometry of the tower is taken similar to the one

used in the 'Node flexibility and its effect on jacket structures,' report U R 2 2 by the

Underwater Engineering Group (UEG, 1984).

T w o aspects of joint stiffness are studied in this chapter. The first aspect is the effect of

joint stiffness on the results of structural analysis including bending moments, axial forces

and natural frequencies. This is generally similar to what was carried out in the report

U R 2 2 and is required to apply the results in fatigue analysis. However, the method of

flexibility implementation in the analyses herein, which uses a continuous mass model for

beam members, is different from U R 2 2 study and is described in Chapter 3. The second

aspect of study in this chapter is the effect of joint stiffness on the fatigue life of tubular

joints. The fatigue life is calculated for a wave condition taken from the book 'Dynamics of

fixed marine structures,' (Barltrop, 1991). To the knowledge of the author this part of the

study is not carried out before. The purpose of the analyses herein is to investigate the

effects of joint rigidity on the structural behaviour of a tower frame. This will indicate how

much variation a designer will achieve in the results of his analysis assuming flexible

joints. In this study a frame with rigid joints is called a conventional model.

All computer programs used in this chapter are written by the author. A continuous mass

formulation is used in the analysis of the towers. Therefore, there was no need to assume

extra nodal points on beam elements between the joints in dynamic analyses.

8.2 Tower geometry and joint specifications

U R 2 2 study, from which the geometry of the towers in this chapter is taken, has considered

three frames with different configurations. The frames in U R 2 2 have been intended to be

representative of the platforms in the North Sea. In order to show the semi-rigidity effects

on the structural behaviour, the most flexible tower configuration in U R 2 2 study was

selected and analysed here. This tower and the geometry of its members are shown in

Figure 8.1.

Chapter 8: Semi-Rigidity Effects on Behaviour of Offshore lowers 151

In order to be consistent with the U R 2 2 study the tower with rigid joints is called CP1 and

the one with flexible joints M P 2 (see Section 2.4 for the tower identification names). The

non-dimensional parameters of the joints in CP1 and M P 2 frames are reported in Table 8.1.

The chords of the joints in the tower are assumed to be stiffened with joint cans at the

intersections. Also brace stubs are considered for the horizontal braces. To strengthen a

joint, the chord section is usually thickened in the connection zone. This section with

higher strength is called joint can. Similar zone on brace is called brace stub.

10000

17.0x33.5 '

18045

22054

26955

tane =0.1 y = 45°

32946

36240 dim. in m m

Figure 8.1. Geometry of the tower analysed

Table 8.1. Geometrical properties of joints used in the analysed towers (units in millimeters)

frame with:

rigid joints

semi-rigid joints

name

CP1

MP2

d45

900

900

braces

t45 d90

25

25

750

—

t90

25

—

D

1700

joint stiffness based on

T d45 d90 D/2T

33.5 900 25.4

d/D

0.53

Stiffness of the joints was accommodated in the analyses by using the modified stiffness

matrix for the members, described in Chapter 3. Therefore, both frames with rigid and

semi-rigid joints were modelled using the centerlines of the members. Joint cans were not

modelled separately but the stiffness of the joints were calculated using the chord can

geometry.

Chapter 8: Semi-Rigidity Effects on Behaviour of Offshore Towers 152

Because the focus of this study is on single braced joints, the inclined braces have been

assumed to have hinged connections to the chords; i.e. they do not produce any eccentricity

or bending stiffness at the joints. Therefore, only T-joints were considered to be semi-rigid,

and the inclined braces were assumed as truss members.

8.3 Stiffness determination of the joints of the tower

The stiffness of the T-joints in M P 2 was determined by using F E Method in conjunction

with the theoretical model described in Chapter 3. A Finite Element analysis of a T-joint

using shell elements with the following details was carried out for its natural frequencies.

_>1700mm, 7=33.5 mm, 1=13.5 m

d=900 m m , r=25 m m , 1=5 m

The calculated results for the natural frequencies and stiffness of the T-joints in M P 2 tower

and also the values used by TJR22 Report for the same joints are shown in Table 8.2. In this

table, a model constructed from one dimensional beam elements is called the line model. It

is seen that the calculated stiffness of the T-joints using the method of this study are close to

those used in U R 2 2 study especially IPB results. Therefore, to maintain the consistency of

the input data with U R 2 2 study, the stiffness values of U R 2 2 were also used in the analyses

of this chapter.

Table 8.2. Natural frequency and stiffness of the T-joint in MP2 tower

Mode Joint type Natural frequency (Hz) Joint stiffness

Line model FE model This study UR22 study

IPB T 31.68 15.65 292 M N m 379 M N m

Axial T 21.15 21.51 375 M N / m 559 M N / m

8.5 Loading

Four load cases were selected which are the same as U R 2 2 study. A concentrated load of

10 M N was applied at the top of the tower to study the transverse deflection changes. To

investigate the effects of joint flexibility on the internal forces, three wave loads derived

from a representative 100 year storm wave passing through the towers with three phase

angles were considered (see Table 8.3 and Figure 8.2). Loading produced by sea currents

was not considered in the analyses.

The square root of the sum of the squares of the instantaneous responses to two waves 90°

phase angle apart was calculated to obtain the response amplitude of the bending moments

(Barltrop, 1991). This simple method is used to make an estimate to the response of a

structure under sinusoidally varying waves with various phase angles. Therefore, the


amplitude of bending moments was calculated using the results of load cases 2 and 3 from

Table 8.3. The response amplitude, x_ is generally defined as:

*_= i / ( * _ + * 3 2 ) C8-1)

where xa = response amplitude, x-i = response from load case 2, *3 = response from load case 3.

Response amplitudes of bending moments are compared to determine the overall effect of

allowing for joint flexibility. To compare the distribution diagrams of a response such as

axial force or bending moment, a further wave load with phase angle 45° (case 4) was

chosen. It applies both horizontal sway forces and local loading on individual vertical,

inclined and horizontal members and therefore includes all the important effects of a wave

loading (UEG, 1984).

Table 8.3. Details of the load cases in the analysis of towers

Load case number

Load type and direction Numerical

details

1 Concentrated load, P in x direction applied to

Node 10

Linearly varying distributed loads acting perpendicular to all members derived by using linear (Airy) Wave Theory and Morison's

Equation for member loading.

P=10MN

<|> = 0o

4) = -90°

4> = -45°

Mean water level, d = 85m Wave period, T = 16sec Wave height, H = 30m Wave phase angle,*

M W L

Direction of w a v e propagation

<|> = 0 • = -90 $ = 5

Figure 8.2. Wave definition for load cases 2,3 and 4 (UEG, 1984)

The wavelength (_.) was calculated to be 360.5m by using Equation 8.2 (Barltrop, 1991,

Equation 6.97):

2 UJ\

27t In

\^J (8.2)


in which g is the acceleration due to gravity and T is the wave period. Wave loading was

calculated using a Fortran program written by this author where linear (Airy) Wave Theory

and Morison's equations, given in Table 8.4 below (Barltrop, 1991), were implemented.

Table 8.4t Linear (Airy) Wave Theory equations (Barltrop, 1991)

Surface elevation = y cos 27i( j - j) J = 9f cos A H

2

Horizontal particle velocity: u

Vertical particle velocity: w

Horizontal particle acceleration: du/dt

Vertical particle acceleration: dw/dt

Force/unit length of a member: F

(perpendicular to member axis)

_ nH cosh(k(z + d)) 14 — ;—•——__ ~ COS A

w

Tsinh(kd)

nHsinh(k(z + d)) sin A

Tsmh(kd)

du 2n2Hcosh(k(z + d))

dt T2 sinh(JW)

dw -2n2 Hsmh(k(z + d))

sin A

COS A

in which

dt T2 sinh(kd)

F = 0.5CdpD\U\U+CmpAU

H = wave height,

d = water depth, z = distance above mean water level,

L = wavelength,

and k = 2-K/L.

T See Figure 8.2 for a better explanation of the notations used in this table.

Drag and inertia coefficients (Cd and C^) for cylindrical sections were assumed to be 0.7

and 1.8, respectively (Barltrop, 1991). The analyses were carried out under static loads

where dynamic magnification was not considered.

The wave forces calculated for load cases 2, 3 and 4 are shown in Figure 8.3. The

specifications of the waves are reported in Table 8.3 and Figure 8.2. The forces were

calculated for each member between two nodes and assigned to those nodes. As it is seen

in Figure 8.2, the wave crest in load case 2 is at the top of the tower. However, in load

cases 3 and 4 the water surface is lower than the frame top level.

It is seen in Figure 8.3 that the forces in load cases 2, 4, and 3 corresponding to phase

angles 0°, -45°, and -90° are decreasing in the order of magnitudes. However, the response

of a structure to wave loading will be a combination of the responses to the waves with

various phase angels, and should be calculated using an appropriate method.

Chapter 8: Semi-Rigidity Effects on Behaviour of Offshon? Towers 155

a) Load case 2, § = 0° b) Load case 3,§ = -90° c) Load case 4,§= -45°

Figure 8.3. Wave forces calculated for load cases 2, 3, and 4

As it is seen in Figure 8.3, the maximum force intensity occurs at the top of the tower in

each load case and the wave force decreases towards the seabed. The forces are calculated

perpendicular to the members. However, at a node where a horizontal brace meets the leg

the sum of the forces will not be perpendicular to the leg member, due to the force

component from the brace.

8.6 Results

A detailed report is given in this chapter from the analyses carried out on the frames M P 2

and CP1. Where numerical values are reported, the locations selected for the output forces

or deflections have been chosen to correspond with those in U R 2 2 report.

8.6.1 Deflections

Load case 1 was selected in U R 2 2 for studying the deflection changes. However the

deflections herein are calculated for all four load cases with the maximum values as shown

in Table 8.5. It is seen in Table 8.5 that the difference between the maximum deflections of

CP1 and M P 2 is about 10%, comparing 3 4 0 m m and 373mm, for the location under the

point load of 10MN.


The maximum deflections under wave loads indicate a negligible change when the joint

stiffness is employed in the analyses. These are the cases where the loads are distributed

over the members. The reason for the small differences of displacements is because the

lateral stiffness of the tower depends on its overall stiffness as a cantilever, which in turn

depends on the cross sectional area of the legs and their distance from each other.

Table 8.5. Deflection changes when joint flexibility is incorporated in analysis

Frame

CPl

MP2

Load Case 1

0.3401

0.3732

%•

9.7

max deflection (meter)

Load Case 2 %*

0.0679 —

0.0689 1.5

Load Case 2 %*

0.0259 —

0.0263 1.5

Load Case 3

0.0494

0.0500

%*

1.2 * %: Deflection change of M P 2 with respect to CPl.

The deformed shapes of CPl and MP2 under the point load of 10MN in load case 1 are

shown in Figure 8.4. It is noted that almost all the deflection change in the frame with semi

rigid joints is due to the axial flexibility of joints 9 and 10.

i-WW • k=559E9MN/m

k=623E9MN/m

A5=10 (~+ ^ - ) = 0.034 m = 34 mm

10MN

rigjd=dashed line (CP1) semi-ngid=sloid line (MP2)

0.5m

I 1 1 h

Figure 8.4. Deflected shape of CPl and MP2 frames for load case l(point load at the top corner)

To verify whether the deflection difference between CPl and M P 2 is related to the

flexibility of joints 9 and 10, beam element 13 which connects the two joints is considered.

The axial force in this element is 1 0 M N . It is seen in Figure 8.4 that the total axial

deflections due to the springs at the ends of beam 13 is 34mm. From Table 8.5 the

difference in the displacements of CPl and M P 2 is: 373 - 340 = 3 3 m m which is very close

to 34mm. This confirms that the flexibility of the joints under point loads introduce the most

changes of deflections between the two models. This is important when the effects of

impact loads are considered on a platform. The U E G study, U R 2 2 , reported a change of

1 6 m m or 5 % for M P 2 in load case 1, which is almost half the value calculated here. Tall

platforms, though, may show more variation in lateral deflections when joint flexibility is

considered as reported by B o u w k a m p (1981).


8.6.2 Axial forces

The diagrams of axial forces for load cases 2, 3 and 4 and the changes of the forces between

the conventional and flexible joint analyses are shown in Figure 8.5. There was generally a

very small difference between axial forces resulting from the analyses on CPl and M P 2 . The

small variation of axial forces may be assigned to the unique load path exist in the towers.

Further analyses are required on the towers with redundant members and different

geometrical configurations to study the effect of joint stiffness on axial force distribution.

0 1 0 M N I I I I I I

load case 2 (<|> = 0) load case 3 («|» =-90) load case 4 (<)> = -45 )

a) Axial force diagrams for different load cases

0 0.1 MN l l l l l l

$ = 0 ^ = -90 <|> = -45

b) Axial force changes

Figure 8.5. Axial forces for load cases 2, 3 and 4 and their variations between flexible joint analysis and conventional analysis (MP2-CP1)


The axial forces for a number of elements corresponding to load case 4 are reported in

Table 8.6. It is seen that the maximum change is due to element 11 and is 0.18 M P a or

4.3%. The maximum change in the response amplitude of axial force occurs in member 10

and is 4.9%.

Table 8.6*. Axial forces, their changes and axial stress changes for load case 4. Chord 45° Brace

Member No. 2 15 Analysis

t Element numbers referred to here, are shown in Figure 8.1.

16

Horizontal Brace 11

Axial force (MN)

Axial force change (%)

Axial stress change (MPa)

CPl MP2

CPl MP2

CPl MP2

3.452 3.453

—

0.03

—

0.01

0.6598 0.6623

—

0.38

—

0.02

-2.189 -2.194

—

0.23

—

0.07

1.704 1.701

—

-0.18

—

-0.04

0.2387 0.2284

—

-4.3

—

-0.18

8.6.3 Bending moments

Figure 8.6 shows the bending moment diagram of CPl for load case 4 (<|> = -45°) and the

difference between the results of bending moments in M P 2 and CPl.

0.3MNm

(a) (b)

Figure 8.6 a) Bending moment diagram for frame MP2 (Load Case 4)

b) Bending moment difference between MP2 and CPl (MP2-CP1)

Table 8.7 also reports the response amplitudes of the bending moments at joint 7 of CPl

and M P 2 and their differences. Joint 7 was selected because it showed the maximum

variation of bending moment between the too frames.

"S a

I -Cs

.8

1 as

1

Is e

o

eb

_S OH

I s 3

I §

I

u

§1 u

is S3 8

Amp

diffa U

2 s

2

2

OH

u

2

OH

u

2

I

Q

03

+

o O Ov en

o

U

o O Ov en

o

o CS

U

1 5 s u

.5 o

_.

a

u

u

03

oo "1 oq r-; •«*' en

—; q CO -H

33 ^_ __ oo «/-> r— oo ^ __ vo o S §

^ CS S3 S3

d o d o 9 9

en § -H

©9

CS OV en r~ en <—i

oo r-

d d

-H r-O vp

es o

d d

o 00

VO en

en

CS vo en ^H

5 vo

6b vo

d d

2 " r. r~» \o r-oo i en

r~ Ov

d

VO Ov

d

00

cs cs d

cs d

o oo -H r~ oo r- «-« cs ~+ 1-H oo en

d o d o

38 vo cs d d

cs cs

d d

8 3 f» vo

d d

S O v en

IT) -H

d d

r~ cs

o o + +

° cs

°-9

00 vo CS ON CS r+

d d + +

O VO Ij 00 oo •* q o

99 +9

o 9g

9'°-

ir> en VO ^ H

o d

9?

r~ cs 00 O «t -H

d d

ir> r» r» oo r~ ov

cs -H -t

/5fl

Chapter 8: Semi-Rigidity Effects on the Behaviour of a Frame 160

It is seen in Table 8.7 that the maximum amplitude change of bending moment is 26.7% at

the brace section and 24.8% at the chord section. There is no change of moment in diagonal

members because they were assumed to have hinged connections to the leg members. It is

noted in Figure 8.6 that there are bigger changes of bending moments from CPl to M P 2 in

the members towards the top of the tower compared to the members at the levels further

down. This is because the vertical water particle velocity is largest towards the top of the

structure. The water particle velocity becomes zero at the seabed. Therefore, horizontal

braces at the top bays will be under loadings of large magnitudes. This will, consequendy,

cause a bigger change of bending moment in the joints of top bays.

8.6.4 Dynamic characteristics

The first nine natural frequencies and mode shapes of CPl and M P 2 were calculated in

order to compare the dynamic characteristics of the two frames. Similar to U R 2 2 study, a

deck with 5000 tonnes mass was considered in the dynamic analyses. The members were

assumed unflooded but the added mass and the members' own mass were considered. The

added mass, M^, for a tubular section submerged in water and undergoing oscillation is

calculated from Equation 8.3 (Hallam, 1978).

Mam= pnd2 (8.3)

in which p is the water density and d is the tube diameter. The added mass in the axial

direction of a member is assumed to be zero.

The natural frequencies, reported in Table 8.8, indicate bigger variations for the frequencies

of the higher mode shapes compared to the lower modes. The maximum change is 21.7%

where the sixth natural frequency of CPl has decreased from 2.48Hz to 2.03Hz. The

second and third frequencies are due to the local modes of the inclined braces and have not

changed at all. Rest of the natural frequencies have changed less than 4%. It is, therefore,

necessary to consider joint stiffness when the load frequency might be close to a natural

frequency, in other words when the frequency margin is small to avoid coincidence of the

structural natural frequency and loading frequency. Consequently, dynamic response of the

structure will become critical and very sensitive to the loading and demands high accuracy

in analysis.

Another difference which is observed between the dynamic behaviours of CPl and M P 2 is

an extra natural frequency when the joint stiffness is modelled in the analysis. The seventh

natural frequency in M P 2 does not exist in CPl. This can be also counted as one of the

important reasons for considering the flexibility of joints in the structural analysis.


Mode

1 2 3 4 5

CPl

0.388 0.760 1.132 1.667 1.763

Table 8.8.

MP2

0.379 0.760 1.132 1.658 1.752

Natural frequencies of towers

Frequency ratio (CP1/MP2) 1.024 1 1 1.005 1.006

Mode

6 7 8 9

CPlandMP2(Hz)

CPl

2.475 —

2.539 3.024

MP2

2.034 2.469 2.565 2.923

Frequency ratio (CP1/MP2) 1.217 —

0.990 1.035

Plots of stiffness versus frequency for CPl and M P 2 are shown in Figure 8.7a and 8.7b,

where the locations at which the stiffness becomes zero are natural frequencies. The

correspondence between the natural frequencies and mode shapes of the two towers can be

constructed by investigating the mode shapes, shown in Figure 8.8 to 8.16. It is seen that

point A in Figure 8.7b, which corresponds to the 7th natural frequency, is missing from

Figure 8.7a. The mode shapes shown in Figure 8.13 correspond to the sixth natural

frequencies, co6, shown in Figure 8.7a and 8.7b. However, mode shape 7 of M P 2 in Figure

8.14 does not have any counterpart in CPl.

Stiffness ..--25 °

X10

"10oo 0.2 '6.4 6.6 "'"as i:6' i'._ V.4 i!- ii"""2.0 2.2 2.4 2 . 6 _ . B 3 . 0 3 . 2

Frequency (Hz)

a) CPl - model with rigid joints

Figure 8.7. Relationship between stiffness and frequency for a) CPl and b) MP2 towers (cont.)


Stiffness X10-29

- 1 0 T*TTTTTTTTTTTTTTT*"rn IIIIIIIM|IMIIIIII|MIIIII'M|IIIIIIIII|MMIIIII|MIM II1111 rT*T1TTT'| I I I V I I ril | TT1 *1 TTTT'I VVI 1111 f| *1 *T* TTII | • I M 1111 TF> IMIIril|l IMII lll|

0.0 0.2 0.4 0.6 0.B 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.B 3.0 3.2

Frequency (Hz)

b) MPl - model with semi-rigid joints

Figure 8.7. Relationship between stiffness and frequency for a) CPl and b) MP2 towers (cont.)

M o d e shapes 1 to 9 of CPl and M P 2 are shown in Figures 8.8 to 8.16. The changes in

mode shapes are most significant for higher mode shapes. It is seen that the bouncing action

in mode shape 5 is stronger in M P 2 than in CPl. Noting mode shape 9, the amplitudes of

displacements in M P 2 are smaller than those in CPl.

Furthermore, the axial joint flexibility in mode shape 6 has resulted in bigger transverse

displacements in the top bay of M P 2 compared to CPl. This becomes important, similar to

the effect of axial flexibility on transverse deflections, when impact loads from ships and

loading barges are applied to the top of a platform.


a) CPl - rigid joints b) MP2 - semi-rigid joints

Figure 8.8. Mode shape 1 of towers CPl andMP2





Figure 8.10. Mode shape 3 of towers CPl and MP2




This mode shape does not

exist for CPl








8.7 Fatigue life estimation

The results of a structural analysis may be direcUy used for fatigue life prediction of the

tubular joints in a structure. The maximum stress range for a particular joint is calculated

through multiplying the nominal stress range of the brace meeting at the joint by the

appropriate stress concentration factor. W h e n different types of loadings are involved (IPB,

O P B or axial) an interaction formula, or conservatively, Equation (8.4) m a y be used.

A/ks = 4/a S C F a + A/JPB SCFjpB + A/oPB SCFQPB (8.4)

in which A/a, A/n>B and A/ O P B are the nominal brace stress ranges due to axial, inplane and

out of plane loadings, respectively. A/H_ is the hot spot stress range. In order to evaluate the

effect of joint flexibility on fatigue life prediction of tubular joints a typical wave loading is

considered on towers C P l and M P 2 , and the fatigue lives of joints 5 to 8 (see Figure 8.1)

are calculated. The wave specifications are taken from Barltrop (1991) and are shown in

Table 8.9. The significant wave height, Hs, is the average height of the waves that observers

will typically report. The mean zero crossing period, T_, is obtained from the mean time

between the up-crossings of the mean water level.


Table 8.9. Individual wave occurrences '«' in a sea state ofHs=3m, 7_= 7 sec, normalised to 1 year Wave height (meters) Wave period range

Min Max 1-5 5-7 7-9 9-11 11-13 13-15 15-20 13 12 11 10 9 8 7 6 5 4 3 2 1

60 13 12 11 10 9 8 7 6 5 4 3 2 1

2 5079 96949

1 22 554 7666

58555 242227

496608 244095

0

3 69

1220

13449

91053 366080 809832 814392 194854

5 271 6707 66778 213342 99300

9 1922

42917 49993

74 11002

27451

5 5414 31317

102030 1049728 2290952 386403 94841 38527 36736

Number of waves/year 3999214

To determine the maximum stress range at a certain location on the towers, it is required to

calculate the stress range per unit wave height for each wave height. A n example of such a

relationship is shown in Figure 8.17, where the stress response has a very large increase at

the natural period.

E

25

20 -

15 -

10 -

S -

0

•

/

7 '—*—\—

il

ll | 1 . H (wave height) = 1 m

I \ / /— H = 5m I \ / / t H -10m

I \ / / / /—H = 1Sm

i i I 1 1 1 1 — 1 - H 1 1 1 1 1 i "

10 12 14 16 18 20

Wave period, T (sec)

Figure 8.17. Typical relationship between wave period and stress per wave height (after Barltrop, 1991)

Since the scope of this thesis is concerning the structural aspects more than the loading

details, the stress range per unit wave height was calculated only for 1 meter high waves.

The stress range per unit height of the higher waves will be bigger than the range of the 1

meter high waves, as it is seen in Figure 8.17. Therefore, the fatigue life predictions which


are made here are not very accurate. This will not, however, affect the comparisons

between the fatigue lives of the joints in CPl and M P 2 towers.

In order to calculate the nominal stress values, a series of dynamic analyses were carried

out. There was a different analysis performed for every wave period and wave height,

according to Table 8.9, to construct the curves of stress range per wave height of each

joint. A phase angle of -45° was assumed for the waves. A deck mass of 5000 tonnes and

the added mass effects were also included in the analyses.

The curves of the hot spot stress range per unit wave height at joints 5, 6, 7 and 8 are

shown in Figure 8.18. Comparing the results of rigid and semi-rigid analyses, it is clearly

seen that lower stress values have been obtained in the latter case.

CP1 (rigid joints) o e e e o M P 2 (semi-rigid joints)

8q

Joint 6

i i i i i i i i i i i i i i i i i i i i i i i i i i i i I I i

4 6 8 10 12 14 16 18 Wave period, T (sec)

(a)

CP1 (rigid joints) ooooo M P 2 (semi-rigid joints)

0 - i i i | i i i | i i i | i i i | i i i | i i i i i i i | i i i | i i i |

0 2 4 6 8 10 12 14 16 18 Wave period, T (sec)

(b)

Figure 8.18. Stress range per wave height of different wave periods for joints a) 5 and 6 and b) 7 and 8

The stress concentration factor equation by Efthymiou and Durkin (1985) and Equation

(8.4) were used to convert the brace nominal stress range to the hot spot stress range. The

values of S C F were calculated as: SCFA = 20.5 and SCF^ = 4.8.

The S-N curve T from 'Offshore Installations: Guidance on design, construction and

certification,' (DEn, 1990) with the equation _v* = 1.458xl012(AoT3 was used to evaluate

the fatigue damage corresponding to the number of wave occurrences. Table 8.10 to 8.13

report the damage per year of the joints, calculated for different wave heights and periods.


Table 8.10. Fatigue damage of joint 5 per year (xl06)for each wave

Wave height

(meters)

Min Max 1-5 CPl MP2

5-7 CPl MP2

height and 1

Wave period range (seconds)

7-9 9-11

CPl MP2 CPl MP2

11-13

CPl MP2

wave period

13-15

CPl MP2

15-20

CPl MP2

13 12 11 110 9 8 7 6 5 4 3 2 1

60 13 12 11 10 9 8 7 6 5 4 3 2 1

0 0 0

0 0 0

0 2 8 13 6 0

0 2 6 9 4 0

0 2 29 192 711 1346 1085 236 2

0 2 22 149 554 1048 845 183 2

0 2 23 84 58 1

0 2 20 74 51 1

0 2 10 0

0 2 8 0

0 2 0

0 2 0

0 1 0

0 1 0

0 0 23 21 3603 2805 168 148 12 10

Frame CPl (includes rigid joints):

Total damage: DjXlO6 = 3809

Fatigue life (years) =263

Model M P 2 (includes semi-rigid joints):

Total damage: DjXlO6 = 2987


Table 8.11. Fatigue damage of joint 6 per year (x!06)for each wave height and wave period

Wave height Wave period range (seconds)

(meters)

Min Max

13 12

11

110

9 8

7

6

5 4

3 2 1

60

13 12

11

10

9

8

7

6

5 4

3 2

1

1-5 5-7 7-9 9-11 11-13 13-15 15-20

CPl MP2 CPl MP2 CPl MP2 CPl MP2 CPl MP2 CPl MP. CPl MP2

0

0

0

0 0

0

0 1

5 7

3 0

0 1

3 4

2

0

0 0


Total damage: _V<106 = 5722

Fatipue life (years) =175

0

4

43 284

1054

1994

1607 349

3

0

3 32

216 800 1514

1220

265 2

0 4

47 171

118 2

0

3 37

133 91

2

0 4

17

1

0 3 14

1

0 3

0

0 2

0

0 1

0

16 10 5338 4052 342 266 22 18 3


Total damage: £>_xl06 = 4349


0 1

0


Table 8.12. Fatigue damage of joint 7 per year (xlP6)for each wave height and wave period

Wave height Wave period range (seconds) — — (meters) Min M a x 1-5 5-7 7-9 9-11 11-13

CPl MP2 CPl MP2 13-15 15-20

CPl MP2 CPl MP2 CPl MP2 CPl MP2 CPl MP2 13 12 11 110 9 8 7 6 5 4 3 2 1

60 13 12 11 10 9 8 7 6 5 4 3 2 1

0 1 1

0 1 0

0 1 9 67 242 365 162 3

0 1 9 67 242 365 162 3

0 4 47 315 1169 2211

1782

387 3

0 3 34 228 847 1602

1292 281 2

0 2 25 89 62 1

0 2 18 65 45 1

0 1 6 0

0 1 5 0

0 1 0

0 1 0

0 0 0

0 0 0

1 849 849 5918 4289 179 131 1 1 0 0

Frame CPl (includes rigid joints): Total damage: DgXlO6 = 6956 Fatigue life (years) = 144

Model M P 2 (includes semi-rigid joints): Total damage: Dp<106 = 5277 Fatigue life (years) = 190

Table 8.13. Fatigue damage of joint 8 per year (xlP6)for each wave

Wave height

(meters)

Min M a x 1-5 CPl MP2

5-7 CPl MP2

height and

Wave period range (seconds)

7-9 9-11

CPl MP2 CPl MP2

11-13

CPl MP2

wave period

13-15

CPl MP2

15-20

CPl MP2

13 12 11 :io 9 8 7 6 5 4 3 2 1

60 13 12 11 10 9 8 7 6 5 4 3 2 1

0 0 0 0

0 0 0 0

0 0 2 1 25 17 186 132 670 473 1010 713 447 316 8 6

2348 1658


Total damage:

Fatigi ie life (years)

DgXlO6 = 5609

= 178

0 2 25 170 629 1191 960 208 2

3187

0 2 20 131 485 917 739 161 1

2456

0 1 10 36 25 0 72

0 1 9 0 33 1 23 3 0 0 66 4

0 1 0 0 0 3 0 0 0 0 0 0 0 4 0 0 0


Total damage:

Fatigue life (years)

D_xl(r = 4184 = 239

0 0 0 0


The fatigue lives obtained for the joints 5, 6, 7, and 8 and their changes after the joint

stiffness was considered are shown in Table 8.14. The fatigue life estimates of all four joints

have increased about 3 0 % on average in M P 2 model where the joints were assumed to be

flexible. However, this result is just an example and shows the significance and possible

advantage of considering stiffness of joints in analysis. Further study is required to draw

general conclusions.

Table 8.14. Fatigue life predictions and their changes when joint stiffness is considered in analysis

Joint Fatigue life (years) Ratio

CPl (rigid joints) M P 2 (semi-rigid joints) (CP1/MP2)

5 263 335 1.27 6 175 230 1.31

7 144 190 1.32

8 178 239 1.34

8.8 S u m m a r y

The aspects of the structural behaviour investigated in this chapter, to recognise the effects

of joint flexibility on the analysis of a 100m fixed offshore tower, were:

1) global deflections,

2) axial forces,

3) bending moments, and

4) dynamic characteristics.

There was almost no change of lateral deflections when distributed loads were acting on the

towers. However, under a point load the flexibility of the joints in the vicinity of the load

can generate extra deformations which may be up to 1 0 % of the maximum deflection of the

tower.

The axial forces were almost the same when joint stiffness was considered in the analysis.

This equality exists mainly because there was a unique load path for axial loads. However,

further studies are required on the structures with redundant members and different

geometrical configurations.

W h e n the stiffness of the joints was included in the analysis, bending moments had

variations up to 2 7 % at the brace sections and 2 5 % at the chord sections . The overall

changes of bending moments were more significant in the leg members than in the brace

members.


The flexibility allowance in the dynamic analysis altered the higher natural frequencies the

most The sixth frequency decreased by up to 2 2 % . The first natural frequency decreased by

only 2.4%. Furthermore, there was an extra m o d e shape when the joint stiffness was

considered in the analysis.

Fatigue life of the tubular joints increased by up to 3 0 % on average when flexibility was

considered in the analysis. The results obtained herein indicate the significance of joint

flexibility effects on the fatigue life of offshore structures. Further, analytical and experimen

tal studies on the structures with different geometrical configurations are required to

produce more evidence and to draw conclusions on the effects of joint stiffness

consideration on the fatigue life prediction of tubular joints.

Chapter 9

SUMMARY AND CONCLUSIONS

Chapter 9: Summary and Conclusions 175

9.1 Summary

Offshore steel structures are benefited extensively from the advantageous properties of

tubular sections. However, thin walls of tubes create some problems in design and analysis

of tubular joints, such as: joint flexibility, ultimate strength, stress concentration, and

fatigue. Perhaps ultimate strength and fatigue impose the most crucial criteria on the

methods used for design of tubular joints. However, joint flexibility and stress concentration

are directly involved in the calculation of stress values, and consequendy influence the

estimation of the fatigue life of a tubular joint

Offshore structures and particularly tubular joints are introduced in Chapter One. The scope

of this thesis is explained and the views of different codes of standards about joint flexibility

and its consideration is quoted. Tubular joints are commonly assumed to be rigid in the

structural analysis, and the main reason for this assumption is the lack of an efficient method

of joint modelling (Barltrop, 1991). The highlights of the work carried out in this thesis are

as follows:

1) an investigation into the flexibility behaviour of tubular T-joints using natural

frequencies as an indication of joint rigidity. The stiffness matrix of a beam with semi

rigid ends and a continuous mass property is calculated for two cases of bending

deformations only and bending plus shear deformations. The dynamic approach is for

the first time used herein to calculate the stiffness of tubular joints.

2) Establishment of parametric formulae for stiffness of tubular T-joints based on the

natural frequencies obtained from Finite Element analysis. The modes considered are

inplane bending, out of plane bending, and axial deformation of brace.

3) A n investigation into the effect of dynamic strain reading on the stress or strain

concentration factor of tubular T-joints. This parameter is being presently ignored in

design practice and research.

4) A study into the effect of joint flexibility consideration on the fatigue life of tubular

joints in offshore structures. Tubular joints are presently assumed to be rigid when

designed for fatigue.

The work on the stiffness or flexibility of joints, in general, has been started in the 1910s.

However, it was completely related to the joints made of open sections. Tubular joints have

been attended, though, in the 1970s and the main reason has been their application in the


offshore structures. As, this thesis focuses on the tubular joints in particular, thus the

literature survey concentrates on these types of joints. The different investigation methods

of tubular joints is described in Chapter T w o . This chapter reports a review of the available

literature on the flexibility of tubular joints. The various methods of analysis used to study

the behaviour of tubular joints are described as:




Analytical procedures are still used by researchers in view of their numerical accuracy and

theoretical bases. The only known experimental work on the stiffness of tubular joints,

which is on araldite specimens by Fessler (1981), was described. Most studies have used the

Finite Element (FE) method to determine the joint stiffness and they hardly include any

comparison with experimental data. All parametric formulae for the stiffness of tubular

joints have been derived by curve fitting the measured data. A semi-experimental model may

establish a better mathematical form for the stiffness formula (refer to Section 2.3.2).

The studies on the effects of flexibility consideration on the behaviour of tubular structures

are also reviewed in Chapter T w o . Bending moments are influenced the most, when

flexibility of the joints is introduced in the analysis. However, the literature survey shows

various effects on the same response in the structures, implying that the effects of joint

flexibility depends on the type, geometry and configuration of the structure. There was only

one study into the joint stiffness effects on fatigue estimate which was carried out on two

simple one-storeyed frames (Recho et al., 1990).

Regarding stiffness determination of joints, the aim of most studies is to calculate an

equivalent spring stiffness for the joint model. There is also another approach to define a

non-dimensional rigidity factor between 0 and 1 to state the degree of fixity of a joint

The research on the stiffness aspects of tubular joints, on one hand, has concentrated mostiy

on the joints as a discrete element of a structure. O n the other hand the effects of joint

stiffness on the behaviour of structures have only been studied analytically. The high cost of

experiment perhaps has prevented testing of large scale specimens.

This study offers Dynamic Deformation method to derive the stiffness matrix of a beam with

semi-rigid ends. The formulations developed include bending deformations and/or shear

deformations. In these formulations, the inertia forces have been modelled as continuous

properties. The method of analysis in this thesis uses the resonant frequencies to calculate

the stiffness of a joint. A new definition is given for the rigidity factor of joints compatible

with the moment rotation relationship of beams with semi-rigid joints. The stiffness matrix


of an axially loaded member with flexible ends is also derived. A definition is introduced for

the rigidity factor of a joint loaded axially.

The definitions introduced for joint rigidity factor, a number between 0 and 1 to express the

stiffness of a joint as a percentage of full fixity are:

Bending stiffness: vb = —J=J> Axial stiffness: va = —l-=rr~

The experimental part of this study includes eleven tubular T-joints tested for inplane and

out of plane bending modes. The test specimens are selected to bound the c o m m o n ranges

of parameters used in offshore structures. A survey is conducted on the geometry of tubular

joints to recognise these ranges.

The dynamic experimental method adopted here, uses the measured natural frequencies of

the joints. The main piece of equipment in the experiments was a Fourier analyser which

was employed to measure the natural frequencies. Test results are reported and compared

with the theoretical values from Dynamic Deformation method when joints are considered

to be rigid. There is a significant difference observed between the measurements and

calculated natural frequencies by Dynamic Deformation method which is attributed to the

joint flexibility. Various methods, including computer analysis and modal analysis, were

used to confirm the correspondence between the natural frequencies and mode shapes of a

joint. W h e n natural frequency of a mode of vibration is measured, assurance should be made

that it corresponds to the desired mode shape.

The stiffness method was used to study the effect of support conditions on the natural

frequencies of T-joints in inplane bending and out of plane bending modes. The results

indicated that there is hardly any effects due to the support conditions on the natural

frequencies of a T-joint in inplane bending mode. Out of plane bending, however, requires

fixed support conditions, otherwise instability will occur for the joint specimen. Despite the

minimal effects of support conditions on the inplane bending vibrations, a special

arrangement was used to minimise the rotational fixity of the chord supports.

The dynamic method used for flexibility determination of tubular joints can be performed

easily and inexpensively. The shortcomings of this method are:

1) the need of a suitable arrangement for support conditions in some deformation

cases, and

2) fairly high capital cost of measurement equipment involved in order to conduct the

dynamic tests.

Chapter 9: Summary and Conclusions ^ ^ ^ ^ ^ ^ 1?Q

The advantages of the method are:

1) its ability to capture the stiffness and mass properties of a joint, including any

deficiencies and imperfections such as those created during manufacturing or those

caused after structure is being in service.

2) The speed of the test procedure can be counted as another advantage over the

static methods of testing.

The parametric formulae developed in this study for the stiffness of tubular T-joints are

based on the results of an extensive Finite Element analysis. Therefore, Chapter Five is

dedicated to the Finite Element method and the general steps used to model the T-joints.

There is a brief look at the different types of F E techniques to model the shell structures.

Plate elements and degenerated shell elements have been used successfully in the studies

before. The plate element is used herein to model the T-joints. The computer package

ALGOR used for conducting the F E analyses is introduced. A bench mark study on the plate

element in ALGOR is performed by modelling a cylinder under internal pressure. The results

of F E analyses by ALGOR show a good agreement with the calculated results from the

theory of elasticity. The performance of the three dimensional solid elements in modelling of

a T-joint was also investigated. It was found that plate elements can make more realistic

predictions of the natural frequencies of a T-joint compared to three dimensional elements.

The effect of support conditions on the frequencies of in plane bending mode was also

investigated through F E analyses. It was observed that the effects of support conditions are

even less than what stiffness method showed in chapter Four.

The results of the F E analyses of the test specimens are reported in Chapter Five. The mean

absolute error of the predicted natural frequencies by F E models were 3.1% compared to

the measurement results. This indicates that the Finite Element meshes used to model the

joints were appropriate. The agreement of out of plane bending results was not as good as

inplane bending for the T-joints with large brace diameters. The main reason is attributed to

the ground connection of the support assembly.

The description of the parametric study of the joint stiffness is given in Chapter Six. There

were totally 270 F E analyses, covering the c o m m o n ranges of joint parameters used in

offshore structures. The F E analyses included three modes of deformations in tubular joints,

i.e. inplane bending, out of plane bending and axial deformation of brace. The chord length

to diameter ratio was chosen to be 6.85. The results of the F E analyses for all three modes

indicate that the joints with large brace diameters or small chord diameter to thickness ratio

possess a relatively high stiffness. The results also show that these types of joints are most

sensitive to non-dimensional parameters such as dJD or DI1T in which D and T are diameter

and thickness of the chord, and d is the diameter of the brace. The effect of thickness ratio

(t = tIT) on the stiffness of T-joints is presently ignored. This effect is investigated for the

Chapter 9: Summary and Conclusions fjg

first time in this study. The results of the FE analyses indicate a significant change of joint

stiffness when x is taken into account.

A set of equations are established for the stiffness of T-joints based on F E results which

include X ratio in addition to other parameters. The curve fitting of the F E results has

maximum errors of 2 5 % , 1 8 % and 2 9 % error in inplane bending, out of plane bending, and

axial stiffness formulae, respectively. The stiffness of tubular T-joints varies with x ratio as

follows:

1) inplane bending stiffness: p(r«/m+i.s5)xo._s ^

2) out of plane bending stiffness: x° 21, and

3) axial stiffness: p(-Y/27.3+o.6T)To.9i

A good agreement was obtained when the results of the developed equations herein and

those from other studies were compared. The average absolute differences of 15.4%, 15.2%

and 27.4% were observed from the comparisons of the joint stiffness values in each mode of

deformation i.e. inplane bending, out of plane bending and axial deformation of brace,

respectively.

The concept of stress concentration factor in analysis and design of tubular joints is

presented in Chapter Seven. Various methods of investigations on the stress or strain

distribution of tubular joints are briefly described. The most popular procedures are strain

gauge reading and Finite Element method. There are variations observed and reported for

the calculated or measured stress concentration factors by other researchers. The reasons

for such variations are also presented. The reliability approach to the stress concentration

factor and the parameters involved are stated. In this approach different design values of

S C F are selected according to the level of confidence required in different applications e.g.

primary or secondary joint or structure (Tebbett and Lalani, 1984). Dynamic strain reading

is discovered as a factor that introduces some variations to the S C F of tubular joints. This

variable is presently ignored in design practice of offshore structures and even in research

works. A series of tests were conducted in this thesis to study the effects of load frequency

on the strain concentration factor of T-joints. The test results show a significant variation of

strain concentration factor with respect to load frequency. A series of analyses were also

carried out using the stiffness method to study the dynamic effects of loading on the strain

concentration factor. The findings of stiffness method analyses indicate a variation in strain

concentration, however it was not as high as the experimental results.

A work has been initiated on the relationship between joint stiffness and stress concentration

factor of tubular T-joints (Kohoutek and Hoshyari, 1993). This thesis has not included this

topic, but the data used in the published paper are taken from this study.

Chapter 9: Summary and Conclusions ^ ^ ^ ^ 180

The effects of considering joint stiffness on the behaviour of an offshore tower is studied in

Chapter Eight. The tower modelled is 100 meter high and represents the structures in the

North Sea. The deflections, axial load and bending moment distributions under static loads

are compared between two analyses, one with rigid joints and one with semi-rigid joints.

The first nine natural frequencies and mode shapes are also calculated and compared. The

geometry and specification of the tower and the applied loadings are taken from the report

U 2 2 by Underwater Engineering Group (1984). There are four load cases considered in the

analyses. Load case 1 consists of a horizontal point load at the top of the tower for studying

the deflection changes. Load cases 2 to 4 are derived from a 100 year storm wave passing

through the tower with different phase angles. The programs used in this chapter for

structural analysis and wave force calculation are written by the author. The theoretical

model described in Chapter Three is employed in the structural analysis program used

herein. A deck mass of 5000 tonnes and also added mass effects were included in the

dynamic analyses.

A fatigue analysis was also carried out on several joints of the towers, and the results were

presented. A wave specification in a particular sea state was used to calculate the maximum

stress range per unit wave height of the joints under study. The results indicate a significant

change of fatigue life estimation in tubular joints when the stiffness of the joints are

introduced in the analysis.


9.2 Conclusions

In connection with the studies carried out in this thesis there are a number of conclusions

that can be made as follows:

1) At present, flexibility of joints is not included in the analysis of offshore structures, due to

the lack of an efficient method of joint modelling.

2) According to the literature survey, in spite of the importance of fatigue criteria in design

of tubular joints in offshore structures, the research on the joint stiffness has not covered the

fatigue life changes. Furthermore, the adverse effect of fatigue fracture on the stiffness of

tubular joints and subsequentiy its influence on the behaviour of whole structure have not

been investigated before.

3) The literature survey shows that there is no experimental study into the effects of joint

stiffness on the behaviour of offshore structures.

4) Although the joints with p = 1 are extensively used in offshore structures, most

researchers in this field have concentrated on the joints with a range of P ratio between 0.2

and 0.8.

5) The F E analyses indicate that there is effectively no influence from support conditions on

the natural frequencies of a T-joint under inplane bending vibration.

6) The fixity of supports was critical for the joints with p = 1 in out of plane bending mode.

The worst correlation between F E analysis and test results was also for this type of joints.

Support conditions of the test specimens became most critical for the stiffer joints. These

conditions should be as close as possible to those assumed in the theoretical model.

7) The inclusion of X ratio in the parametric formulae for stiffness of tubular joints is

recommended, especially for axial loading case. The thickness ratio x makes a difference of

up to approximately 4 0 % for out of plane and inplane bending stiffness, and 2 3 0 % for axial

stiffness of brace.

8) The parametric equations, which are presently used to determine the hot spot stress

values at a tubular joint, are all based on the databases from static experiments. These

equations do not differentiate between dynamic loads, which are applicable to the offshore

platforms, and static loads.

9) There is a variation of strain or stress concentration factor in tubular joints when

measurements or calculations are carried out in a dynamic fashion. For the range of

frequencies examined in this study (0.5 to 50 Hz), the following variations observed on the

strain concentration factors of tubular T-joints:


inplane bending mode: 100%,

out of plane bending mode: 120%.

These values are measured at the frequencies much higher than the natural frequency of an

offshore platform. However, they indicate the variation which exist for stress concentration

factor due to the load frequency.

10) It is recommended that a frequency dependent factor or conservatively a constant factor

to be included in the parametric formulae for stress concentration factors of tubular joints to

take the effect of dynamic loading into account

11) The results of the analyses carried out on two towers, one with semi-rigid joints and the

other with rigid joints, to recognise the effects of joint flexibility are as follows:

a) The change in the maximum lateral deflection is negligible when wave loading is

applied to the towers. However, when a point load is acting, the flexibility of the

joints in the vicinity of the load can generate extra deflections which may be up to

1 0 % of the maximum deflection of the tower.

_>) Axial forces were almost the same for the towers analysed.

c) Bending moments had a maximum variation of 2 7 % at the brace sections and 2 5 %

at the chord sections. The overall change in bending moments were more significant

for the leg members than for the brace members.

d) The flexibility allowance in the dynamic analysis altered the higher natural

frequencies more significantly than the lower frequencies. A n extra mode shape was

also generated when the joints were assumed to be semi-rigid in the analysis.

12) A mean increase of 3 0 % resulted for the fatigue life estimates of tubular joints when

joint stiffness was considered in the analysis.

Chapter 9: Summary and Conclusions ^ 183

9.3 Future research work

Further investigation is required on the following topics to achieve more insight to the

stiffness of tubular joints and its effects on the behaviour of offshore steel structures:

1) Investigation of the effects of fatigue fracture on the stiffness of tubular joints and

consequently the whole structure.

2) Further study on the stiffness of the joints with diameter ratios greater 0.8. Diameter

ratio varies between 0.2 and 1.

3) In order to verify and substantiate the works already conducted on the stiffness of tubular

joints, there is a need to investigate the effects of joint flexibility on the different aspects of

structural behaviour by conducting experiment on reasonable sized frame specimens.

4) Further investigation is required to be able to draw conclusions on the relationship

between stress concentration factor and joint stiffness of tubular joints.

5) Experimental study on the axial stiffness of tubular joints.

6) Application of the dynamic method of joint stiffness determination used in this thesis for

other types of tubular joints such as Y, K, etc..

7) Further investigation into the effects of dynamic loading on the stress concentration

factor in tubular joints.

8) Further investigation into the effects of joint stiffness on the behaviour of offshore towers

with various configurations and geometries. This shall include the structural response of

three dimensional frames and also fatigue life estimate.

References 754

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APPENDICES

Appendices: Appendix A 193

Appendix A

Stiffness matrix of beam with semi-rigid ends

The following sign conventions have been used for bending deformations and moments to

derive the stiffness matrix of a beam with semi-rigid ends.

c-M

--)

V

vory

Bending moments and shear forces Bending moments and deflections

Figure Al. Positive sign conventions for beam element

Considering an infinitesimal beam element under external loads and inertia forces, as shown

below:

V+dV) M+dM

dx

Figure A.2. Beam element loaded and being under equilibrium conditions

The equilibrium condition requires that:

3 V d2v

dx dt2

which by using the following two equations for the beam members:

BM .. . rTd2y _ u

= V and EI—^r = M dx dx2

the differential equation for dynamic deformation of a beam element can be obtained as:

(A.1)

(A.2)


-£~g+,.o (A.3)

Assuming \(x,t) is separable as \(x,t) = y(jc)sin(_or) and no intermediate load is acting on the

beam, then:

i4 t4

(EI—T~ma>2y)sincor = 0, therefore EI-——m(02y = 0 dx* dx* y

(A.4)

Assuming A, = /(mo)2/__-)l/4, Equation (A.4) can be simplified to:

dx4 (A.5)

The solution for the above equation is:

ten , retell y(x) = CicosQudl) + C_sin(AJc/Z) + G e ^ ' + C<e (A.6)

Considering the following boundary conditions:

8, = y(0) 5, = y(/)

ei=__m+/(o) e..=+-7^+/(/) (A.7)

in a matrix form will result:

A = <_>C (A.8)

where A ^

fS/1

8* C_

Aj) VC4J

and


<_> =

1 COSA

^ 2

aX(y) COSA- (y)sinA

0

sinA

X /

Assuming that the positive directions for the degrees of freedom of a beam element are as

shown below:

6j(3)

8i(1)

Figure A.3. Degrees of freedom for beam element

the end forces will be:

Vi = V(0) = _-/ym,(0)

Mi = -M(P) = -EIy"(0)

Vy = -V(/) = -EIym(l)

Mj = M(l)=EIy"(t) (A.9)

which can be shown in a matrix form as:

P = _i/QC (A. 10)

r-\r^\

)Vj[ in that E and / are Modulus of Elasticity and moment of inertia, respectively. P =n M f'

V, V Mt MjJ

C = c_ C3

*» and Q =

0

* 3 ,

(y) sinA

¥

-¥ * 3 ,

(y)cosA

0

•¥ A 3 •*

(7)e

42

».3

<7> -<7)V

4>2 L -(y)2«KA -(7)2sinA (y)V ( 7 ) ^ j


Substituting from Equation (A.8) into (A. 10) yields:

P = F/QO1 A (A.11)

from which the stiffness matrix can be obtained as:

k = EIQQrx (A. 12)

Matrix k can be shown in a similar form to the stiffness matrix of a beam with rigid joints as

given by Equation (A. 13). This is the form which Kolousek (1973) used to show the

stiffness matrix of a beam using continuous mass model. The coefficients F, are called

frequency functions.

k =

frF_<X) JTFM ffF,(A) f £ F A )

^F2(X) fF5(X) |fF6(X) fF7(A)

FT FT FT FT

fF3(X) -JiFsiX) ~F%(X) -JT^CX) FT FT FT FT

_ yrF4(X) TTF7(X) TTF9(A) — F 1 0 ( A )

(A. 13)

The Frequency Functions are very long and cumbersome since the matrices C and O have

lengthy trigonometric and hyperbolic expressions. Frequency functions of a beam with semi

rigid joints modelled with a continuous mass property is given in Table A.l. The variables

used in Table A. 1 are defined as follwes:

X = l ' mO)2V/4

El (A. 14)

"'-Jul' J kji (A.15)


Table A.1. Frequency Functions of beam with semi-rigid joints (shear and axial deformation effects not included)

Den = --^-Ôi+c^XcosA+sinAHAe2^

Ft = -A3[2^(XiAH<MnA-cosAXl-eutt^ Den

F2= X3[(e2X-l)(l + CMXjA2) + A(cti + otj)(l + e2* + 2excosX) + 2e_iX(l -ctiOjX2]/Den

F3 = -X2(- a,XcosX + eÔjAcosX - sinX + e^sinX + ctj XsinX + eÔjXsinX)/ Den

F4= -X2(l + e2*- - ctiX + e^ctiX - 2excosX + 2^-CXMDX)/ Den

F5 = -X't-^cXjXâr-X-cosXXl-e^cx^ Den

F6= X2(l + e2* - ctjX + e^ctjX - 2excosX + 2exajXsinX)/ Den

F7 = X2(- ctiXcosX + e^ctiXcosX - sinX + e^sinX + ctiXsinX + euaiXsinX)/ Den

Fg = -X[(l - en)(cosX - 2ctjXsinX)+ sinX(l + e2*)]/ Den

F9= -X(-l + e2X-2e_iX)/D_«

Fw= -X[(l • e^XcosX - 2aiXsinX>i- sinX(l + e2*-))/ Pen ,

Appendices: Appendix B 198

Appendix B

Stiffness matrix of truss member with semi-rigid ends

The following sign conventions have been used to derive the stiffness matrix of an axially

loaded member with semi-rigid ends:

P<— 4** • =

r H

Figure B.l. Positive sign convention for axial deformation

The differential equation of deformation for a member axially loaded can be written by

applying the equilibrium equation on an element shown in Figure B.l. Therefore:

¥- = EA?^ (B.l) dx dx

jSA|_*-m^*=_0 (B.2) dx2 dt2

in which E is the Modulus of Elasticity, A is the sectional area and m is the mass per unit

length. h(x,t) describes the longitudinal displacement of the member. Assuming a solution

for Equation (B.2) as h(x,t) = K(x)sin(G>0 and also \jr = * J^Tj-, it can be written that:

^ + ( ^ ) 2 u = 0 (B.3) dx2 I

The particular solution of the above equation is:

u(x) = C5sin(\|tt//) + C_cos(\|tt//) (B-4)

Assuming the positive degrees of freedom for as shown in Figure B.2 below:


n,(l) 11,(2)

Figure B.2. Degrees of freedom for a truss member

The boundary conditions for a truss member with semi-rigid joints are:

T H = « ( 0 ) -k'

^=1/(7)4 P(D

(B.5)

in which k[ or k- refers to the axial stiffness the member ends. Equation (B.5) can be

written in a matrix form as:

A_ = <D_C_ (B.6)

where A = | M , *a = . ,„ _„ , and C = { CJ L x|j J a L sm \f cos \|/ J' a I Q J

The relationships between end forces and displacements are:

p{ = -P(0) = -EAu\0) Fj = P(0 = F_4u'(/) (B.7)

which in a matrix form, they will be:

P_ = __AQ_C_ (B.8)

in which Pa = j p J, and Q,, = /

0

^cos \jr ^ i n \j/

Substituting from Equation (B.6) into (B.8) yields:

P =EAQa*;lA (B.9)

from which the stiffness matrix can be obtained as:

k_ = E4Q_<&« -I (B.10)


The coefficients of matrix k_ are given in Table B.l.

Table B.l. Stiffness matrix of a truss with semi-rigid ends

Den = \ircos\|/ (ft +ft ) + sin\|f (1- pifty2), pi = EAIk'J, ft = EAlk)l

ku = (£A/0v(cos\|/ - pjsin\|r)/Den ku = -(£A//)v/Den

k2i = -(EAJl)\\r/Den kn = (2_A//)\|/(cos\{/ - pi_in\|r)/Den

Appendices: Appendix C 201

Appendix C

M o d e shapes 1, 2, and 3 for the tested T-joint specimens

The mode shapes of a series of models corresponding to the tested T-joints were calculated

using ALGOR Package. Therefore ID beam elements and stiffness method were used to

model the joints. Although the joints are assumed to be rigid in ALGOR, yet the overall

picture of the mode shapes are correct

mode 1 mode 2 mode 3

Figure C I . First three mode shapes ofTl for IPB mode.


Figure C.2. First three mode shapes ofT2for IPB mode.



Appendices: Appendix C 202




Figure C.5. First three mode shapes ofTSfor IPB mode.





Appendices: Appendix C


203

Figure C.8. First three mode shapes of 78 for IPB mode.




Figure CIO. First three mode shapes of TIO for IPB mode.


Figure C I 1. First three mode shapes of Til for IPB mode.

Appendices: Appendix D 204

Appendix D

List of sub-programs in A L G O R (used in the analysis of joints)

AEDIT program for editing the ASCII data files.

DECODS generates an ASCII file from a graphic file.

SD2 (Superdraw) C A D program.

SSAPO static analysis module.

SSAP1 dynamic analysis module which performs eigenvalue analysis.

SUBSTRUC program for gluing two structure together.

SVIEW program for viewing a model, its deformed shapes and stresses.

Appendices: Appendix E 205

Appendix E

Description of T B C 3 and G C S 8 finite elements (Ashwell and Gallagher, 1976)

TBC3

Description:

Flat shell element oriented in the global cartesian system.

Number of Nodes: 3 at the vertices

Nodal coordinates: x, y, z

Degrees of freedom:

X, Y, Z, RX, RY, RZ at each node.

Geometric properties required:

ti, ti, ts nodal thicknesses.

Material properties required:

(i) Isotropic- E Young's Modulus

v Poisson's ratio

Loading:

Nodal point loads or moments may be applied. Constant or linearly varying normal pressure

loads may be applied.

Stress Output-

Three membrane stresses Gxx, O > and Gxy and three moments per unit length M « , M„ and

M,y are output at each node in the local axis system. The membrane stresses are constant

across the element

Figure E.l. TBC3 Element

GCS8

Description:

Arbitrary curved thin shell quadrilateral element with varying ftiOa^^c^y^^).

Number of Nodes: 3 (Vertices and midsides) *

Nodal coordinates: x, y, z

Degrees of freedom:

a) At corner nodes: X, Y, Z.

b) At midsides: X, Y, Z, RX, RV.

Geometric properties required:

tu th to A, ts, ft, ti, h (Thickness at the nodes). Figure E.2. GCS8 Shell element

Material properties: (i) Isotropic- E Young's Modulus

being determined by the node ordering on the element topology card.

Appendices: Appendix E 206

vPoisson's ratio

a-(Linear Expansion coefficient)

p-(Density of Material)

Loading:

Nodal point loads, pressure load and temperature loads may be applied.

Stress Output:

Membrane stresses (a«, < % and Gxy) and Bending Moments (A/_x, Myy and Mxy) in local

directions only, are output at each node.

Appendices: Appendix F 207

Appendix F

Effect of chord length (Efthymiou 1985)

The local joint stiffness obtained from a F E model depends on the length of chord modelled.

If the chord length is small, the restrained chord-ends will tend to stiffen the joint and hence

lead to higher stiffness coefficients. In offshore platforms the chord length between nodes is

typically 20 diameters or more, so the proper stiffness coefficients are those relating to large

chord lengths.

A long chord (length » diameter) loaded in an arbitrary manner at some sections OTigure

F.l) will become locally distorted. These distortions decay (in some manner) with distance

from the load. The decay length, x^, depends on the type of loading. Disturbances due to

axisymmetric loading (Figure F.lb) decay very fast (xj = V4D) but non-symmetrical load

(Figure F.lc), such as that caused by brace loading leads to inextensible bending of the

chord, which could take much longer to decay. In the finite element model it is important to

choose a correct chord length (at least equal to the decay length for the particular

geometry); otherwise the stiffness coefficients may be grossly overestimated.

Supported end

ry load

a)

b) c)

Figure F.l. a) Circular cylinder loaded at end-section, b) axisymmetric load and c) non-axisymmetric bad

Appendices: Appendix F 208

The decay length for some relevant idealised situations can be estimated analytically using

thin shell theory as follows:

a) Axisymmetric loading on a circular cylinder

The displacements, stresses, bending moments etc. decay according to e_Xx, where x is the

distance from the load and

„___^!_. (R.) R2t2

Decay is sufficient when XJC > 3, leading to x = 2.33 jRt. For typical Rlt ratios, say 20, the

decay length is x = 0.52/?.

b) Non-axisymmetric load (inextensional bending)

Displacements, stresses etc. decay according to <TRc(p) , where x is the distance from the

load and p is given by the fourth order equation:

12(l-v2)/?2p4 + m*(m

2 - 1 ) 2 = 0 (F.2)

The parameter m indicates the harmonic order round the cylinder periphery. For m - 2,

Equation (F.2) reduces to:

"-^p'tU.0 (F.3)

from which the first real solution for p is obtained as:

n _ V2r 12 t2 m (F 4)

P" 2 S-v2 R2)

Considering decay to be sufficient when Re(p)x/_?>3 and using (F.4) leads to 233R>jR/t for

Rlt =20, x = 10/?, i.e. 20 times more than the case of axisymmetric load.

The above result relates to m = 2, which has a severe ovalising effect like that found in a

double X-joint with two pairs of equal and opposite forces. It can be considered an upper

bound on the decay length for all tubular joints. This suggests that in an F E model of, say, a

T/Y joint the total length of chord required will be a little less than 2x10* = 10D. It was

established in task I that modelling a length of chord equal to 6D is adequate.

Appendices: Appendix G 209

Appendix G

Parametric SCF Formulae for inplane bending of T-joints

SCF for chord SCF for branch

Kuang et al. (1975)

-0.04-.0.60-.0.86 0.702p-OO4Ya60T

Wordsworth andSmedley (1878)

0 75<vo.6T o.8 (16po.25_ 07r32)

DNV (1977)

2i ,_ 0.38 -.1.05 [1.65-l.l(j_-0.42)2]Y038T

UEG* (1985)

0 75y o.6 x o.8 (1>6po.25 .0.7P2)Q,m

Efthymiou andDurkin (1985)

I 45RT0.85y(l-0.68P)

Hellier (1990)

2.31 a00033 Y a326 x °95 exp[(0.00154Y 0.0323)/P2-0.0248]

Lloyd's Register (1991)

1.22T°'8PY(1"a68P)

-0.38 «, 0_»3 ~ 0.38 l^oip-03^0-23*

1 + 0.63 SCFc

[0.95-0.65(p-0.4)2]Y°'39T°29

1 + 0.63 SCFc

I + O ^ P T 0 - ^ 1 0 9 0 - 7 7 ^

0 332 a00053 Y a37 x a296

exp(- 0.00436/p2 + 1.4)

l+xa2YP(0.26-0.21p)

Q' =1 for Y < 20 and 480/7(40 - 0.833Y) for 20 < Y 40.

1993 dynamic investigation of semi-rigid tubular t- joints

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