Innovative Solutions for Minimizing Differential Deflection and Heaving Motion in Very Large...

INNOVATIVE SOLUTIONS FOR MINIMIZING

DIFFERENTIAL DEFLECTION AND HEAVING MOTION

IN VERY LARGE FLOATING STRUCTURES

PHAM DUC CHUYEN

NATIONAL UNIVERSITY OF SINGAPORE

2009

INNOVATIVE SOLUTIONS FOR MINIMIZING

DIFFERENTIAL DEFLECTION AND HEAVING MOTION

IN VERY LARGE FLOATING STRUCTURES

PHAM DUC CHUYEN

B.Eng. (NUCE), M.Eng. (AIT)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

Acknowledgements

i

ACKNOWLEDGMENTS

First of all, my greatest thanks go to my supervisor Professor Wang Chien Ming, for

his education, ideas, inspiration, advises, prompt and clear comments about our work,

answers on many questions, and constant enthusiasm and interest. I consider myself as

a very happy person to work with such an outstanding researcher as a supervisor. And

I learned a lot about research, scientific philosophy and other things, for instance, how

to get joy from the equations and problems.

Next, I am grateful to my co-supervisor Professor Ang Kok Keng for his

valuable and useful discussions, information and comments through out the research. I

am also indeed grateful to Professor Tomoaki Utsunomiya from Kyoto University for

his advice and useful discussions on this research study.

I would also like to thank the chairman of my thesis advisory committee,

Professor Koh Chan Ghee for his valuable advice and suggestions on my research and

thesis. I am also thankful for the professors who examined my thesis. Their supportive

comments and suggestions were very much useful for the future improvement of the

thesis.

I’m grateful to the National University of Singapore for providing financial

support in the form of the NUS scholarship and facilities to carry out the research. The

Acknowledgements

ii

support provided by Mr. Krishna Sanmugam and technicians at Hydraulic Laboratory

in the use of equipments and computer facilities to carry out the experiment is also

appreciated.

In addition, I would like to extend my gratefulness to my colleagues in Civil

Engineering Department, especially Mr Tay Zhi Yung, Mr Muhammad Riyansyah, Ms

Bangun Emma Patricia, for their friendship, encouragement and valuable discussion

during the study.

Last but not least, I would like to express the deepest gratitude to my beloved

parents, wife and sisters for their eternal support, encouragement and love. I could not

finish the whole study without the great love and care from you.

iii

TABLE OF CONTENTS

Acknowledgements i

Table of Contents iii

Summary viii

List of Tables xi

List of Figures xii

Notations xix

Chapter 1 Introduction 1

1.1 Background information on VLFS 3

1.1.1 Types of VLFS 3

1.1.2 Advantages of VLFS 4

1.1.3 Applications of VLFS 5

1.1.4 VLFS components 12

1.2 Literature Survey 12

1.2.1 VLFS assumptions, shapes and models 13

1.2.2 VLFS and water interaction modeling 15

1.2.3 Minimizing differential deflection in VLFS 17

1.2.4 Minimizing motion in VLFS 18

1.3 Objectives and scope of study 21

1.4 Layout of thesis 22

Table of Contents

iv

Chapter 2 Minimizing Differential Deflection in Circular VLFS 25

2.1 Introduction 25

2.2 Problem definition 27

2.3 Basic assumptions and governing equations 29

2.4 Exact bending solutions and boundary conditions 31

2.4.1 VLFS regions without buoyancy force 31

2.4.2 VLFS regions with buoyancy force 32

2.4.3 Boundary and continuity conditions 33

2.5 Results and discussions on effectiveness of gill cells 34

2.5.1 Basic dimensions and properties of VLFS example, loading and

freeboard conditions 34

2.5.2 Verification of results of VLFS with gill cells 34

2.5.3 Effectiveness of gill cells in reducing the differential deflection and

stress-resultants 36

2.6 Optimal design/location of gill cells in circular VLFS 40

2.6.1 Case 1 – Varying loading magnitudes ql 41

2.6.2 Case 2 – Varying loading radius r0 43

2.6.3 Case 3 – Varying top and bottom plate thicknesses t 45

2.6.4 Case 1 – Varying freeboard hf 47

2.7 Comparing the effectiveness of gill cells with stepped VLFS 48

2.7.1 Case 1 – Varying loading magnitudes ql 49

2.7.2 Case 2 – Varying loading radius r0 50

2.7.3 Case 3 – Varying top and bottom plate thicknesses t 51

2.8 Optimal design/location of gill cells in annular VLFS 52

Table of Contents

v

2.8.1 Dimensions, properties of annual VLFS and proposed gill cells

location 52

2.8.2 Optimization formulation 53

2.8.3 Optimization results 54

2.9 Concluding remarks 56

Chapter 3 Minimizing Differential Deflection in Non-circular Shaped VLFS

Using Gill Cells 58

3.1 Introduction 58


3.3 Basic assumptions and FEM model for VLFS bending analysis 61

3.4 Results and discussions on effectiveness of gill cells 64

3.4.1 Dimensions and properties of VLFS examples, loading and freeboard

conditions 64

3.4.2 Verification of results 64

3.4.3 Effect of gill cells in reducing the differential deflection and von Mises

stress 65

3.5 Optimization problem 67

3.6 Optimization results 72

3.6.1 Optimization results for square VLFS 72

3.6.2 Optimization results for rectangular VLFS 78

3.6.3 Optimization results for I-shaped VLFS 84


Table of Contents

vi

Chapter 4 Minimizing Heaving Motion of Circular VLFS using Submerged

Plate 91

4.1 Introduction 91


4.3 Basic assumptions, equations and boundary conditions for circular VLFS 93

4.4 Model expansion of motion 96

4.5 Solution for radiation potentials 99

4.6 Solution for diffraction potentials 113

4.7 Equation of motion in modal coordinate 114

4.8 Results and discussions 117


Chapter 5 Minimizing Heaving Motion of Longish VLFS Using Anti-Heaving

Devices 130

5.1 Introduction 130

5.2 Experimental facility and instrumentation 133

5.2.1 Wave tank 133

5.2.2 Wave generating system 135

5.2.3 Data acquisition system 135

5.2.4 Testing condition 138

5.2.5 VLFS model 138

5.2.6 Types of anti-heaving devices 140

5.3 Experimental procedure 142

5.4 Experimental results 143

5.4.1 VLFS with submerged horizontal plate 143

5.4.2 VLFS with vertical plate 147

Table of Contents

vii

5.4.3 VLFS with L-shaped plate 151

5.4.4 VLFS with inclined plate 155

5.5 Most effective anti-heaving device 158


Chapter 6 Conclusions and Recommendations for Future Work 167

6.1 Conclusions 168

6.2 Recommendations for future work 171

References 174

List of Publications 186

viii

SUMMARY

In the new millennium, the world is facing new problems such as the lack of land, due

to the growing population and fast urban developments. Many island countries and

countries with long coastline have applied the traditional land reclamation method to

create land from the sea in order to decrease the pressure on the heavily used land

space. In recent years, an attractive alternative to land reclamation has emerged – the

very large floating structures technology. Very large floating structures (VLFS) can

and are already being used for storage facilities, industrial space, bridges, ferry piers,

docks, rescue bases, airports, entertainment facilities, military purpose, and even

habitation. VLFSs can be speedily constructed, exploited, and easily relocated,

expanded, or removed. These structures are reliable, cost effective, and

environmentally friendly.

VLFS may undergo large differential deflection under heavy nonuniformly

distributed loads and large motion under strong wave action. These conditions may

affect the smooth operation of equipments, the structural integrity and even the safety

of people on VLFS. The main objectives of this study are to develop (1) innovative

solutions to minimize the differential deflection of VLFS and (2) innovative solutions

to minimize the heaving motion of VLFSs for various shapes and dimensions. Various

alternative solutions are treated.

For minimizing the differential deflection in unevenly loaded circular VLFS,

two solutions were proposed. One solution makes use of the so-called “gill cells” and

Summary

ix

the other solution involves stepped VLFS (i.e. VLFS with varying height). Gill cells

are compartments in VLFS with holes or slits at their bottom floor that allow water to

flow in and out freely. As a result, the buoyancy forces are eliminated at the VLFS

region with gill cells. By suitable positioning the gill cells, one can reduce the

differential deflection of VLFS significantly. Although the stepped VLFS solution

reduces the differential deflection, it is not as effective as using the gill cells.

Furthermore, by reducing the differential deflection, the stress-resultants and bending

stresses in the VLFS are correspondingly reduced. In this study, the gill cells have

been used to minimize the slope in an annular shaped VLFS. In the circular and

annular shaped VLFS, the analytical bending solutions based on the Kirchhoff

hypothesis for plate bending analysis and the classical Newton optimization method

were used to seek the optimal design of gill cells.

Further investigation on the use of gill cells for minimizing the differential

deflection of arbitrary non-circular VLFS were made. As analytical solutions for

bending analysis of arbitrarily shaped VLFS are not available, finite plate element

analysis based on the Mindlin plate theory was employed. The Mindlin plate theory

was adopted in order to provide better accuracy of the computed stress resultants as

well as to include the effect of transverse shear deformation. For the optimization of

the position of gill cells in an arbitrarily shaped VLFS under an arbitrary loading

condition, Genetic Algorithms were employed because of its robustness in getting the

global optimum solution and the method allows the natural switching on and off the

gill cells in the optimization search. The effective of gill cells in reducing differential

deflection and stress-resultants in VLFS was confirmed from the extensive numerical

computations and examples considered.

Summary

x

In the investigation of the second topic of this study, i.e. the development of

anti-motion devices for VLFS under wave action, the author first considers circular

VLFS attached with a submerged horizontal annular plate strip that goes around the

edge of VLFS. Exact analytical solutions of hydroelastic were derived by making use

of the exact circular and annular plate solutions and exact velocity potential solutions

for circular domains. The effectiveness of this anti-motion device is demonstrated by

performing the analysis and comparing the results with and without the submerged

annular plate strip.

Further studies were investigated on minimizing the heaving motion of

rectangular VLFS under the action of waves by using various anti-motion devices

attached to its fore-end. These anti-motion devices include submerged horizontal plate,

vertical plate, L-shaped plate, and inclined plate. The effectiveness of these anti-

motion devices is confirmed by experimental results. Among these devices, inclined

plate seems to be the most effective device and it is recommended to be used in

appropriate VLFS applications.

xi

LIST OF TABLES

Table 2.1 Effect of gill cells in reducing maximum differential deflection and maximum stress-resultants 40 Table 2.2 Effectiveness of optimal gill cells in reducing the maximum slope and maximum stresses-resultants 55

Table 5.1 Pertinent information of model 140

Table 5.2 Anti-heaving devices considered in the experiment 141

xii

LIST OF FIGURES

Fig. 1.1 Types of VLFS 3

Fig. 1.2 Mega-float in Tokyo Bay, Japan 6

Fig. 1.3 Floating Oil Storage Base at Kamigoto, Japan 6

Fig. 1.4 Yumeshima-Maishima Floating Bridge in Osaka, Japan 6

Fig. 1.5 Floating Rescue Emergency Base at Osaka Bay, Japan 6

Fig. 1.6 Floating island at Onomichi Hiroshima, Japan 6

Fig. 1.7 Floating pier at Ujina Port Hiroshima, Japan 6

Fig. 1.8 Floating Restaurant in Yokohoma, Japan 7

Fig. 1.9 Floating Performance Stage, Singapore 7

Fig. 1.10 Floating helicopter pad in Vancouver, Canada 7

Fig. 1.11 Kelowna floating bridge in British Columbia, Canada 7

Fig. 1.12 Bergsøysund floating bridge, Norway 7

Fig. 1.13 Nordhordland Bridge Floating Bridge, Norway 7

Fig. 1.14 Hood Canal Floating Bridge in Washington States, USA 8

Fig. 1.15 Dubai Floating Bridge in Dubai, United Arab Emirates 8

Fig. 1.16 Marine Uranus by Nishimatsu Corporation 11

Fig. 1.17 Pearl Shell by Shimizu Corporation 11

Fig. 1.18 Osaka Focus A by Japanese Society of Steel Construction 11

Fig. 1.19 Osaka Focus B by Japanese Society of Steel Construction 11

List of Figures

xiii

Fig. 1.20 Components of a pontoon-type VLFS 12

Fig. 2.1 Pontoon-type circular VLFS with gill cells and varying heights 28

Fig. 2.2 Pontoon-type ring-shaped VLFS 29

Fig. 2.3 Design mesh of circular VLFS with gill cells used in SAP2000 analysis 35

Fig. 2.4 Comparison deflections w between analytical solutions and SAP2000 result of circular VLFS with gill cells 36 Fig. 2.5 Effect of gill cells on deflections w 38

Fig. 2.6 Effect of gill cells on shear force rQ 38

Fig. 2.7 Effect of gill cells on radial moment rM 39

Fig. 2.8 Effect of gill cells on circumferential moment θM 39

Fig. 2.9 Optimal design of gill cells when varying ql 42

Fig. 2.10 Effect of optimal gill cells when varying ql 42

Fig. 2.11 Optimal design of gill cells when varying r0 44

Fig. 2.12 Effect of optimal gill cells when varying r0 45

Fig. 2.13 Optimal design of gill cells when varying t 46

Fig. 2.14 Effect of optimal gill cells when varying t 46

Fig. 2.15 Optimal design of gill cells when varying freeboard hf 47

Fig. 2.16 Effect of optimal gill cells when varying freeboard hf 48

Fig. 2.17 Effect of optimal gill cells and optimal stepped VLFS when varying ql 50

Fig. 2.18 Effect of optimal gill cells and optimal stepped VLFS when varying r0 51

Fig. 2.19 Effect of optimal gill cells and optimal stepped VLFS when varying t 52

Fig. 2.20 Annular VLFS with gill cells 53

Fig. 2.21 Effect of optimal gill cells on slope dw 55

Fig. 3.1 Square, rectangular and I-shaped VLFSs 60

List of Figures

xiv

Fig. 3.2 Definitions of deflection and rotations 62

Fig. 3.3 Nondimensionalized deflection contour w in a quarter of the square VLFS without gill cells 66 Fig. 3.4 Nondimensionalized deflection contour w in a quarter of the square VLFS with gill cells 67 Fig. 3.5 Transformation of binary string to a quarter of the square VLFS with gill cells (shaded cells) in genetic algorithms rubric 70 Fig. 3.6 Optimization program flow-chart integrating GA and developed FEM model 72 Fig. 3.7 Stage 1 for a quarter of square VLFS 73

Fig. 3.8 Layouts of gill cells in quadrant of square VLFS for various generations in Stage 2 74 Fig. 3.9 Square VLFS with some possible optimal designs of gill cells 75

Fig. 3.10 Nondimensionalized deflection contour w of square VLFS 76

Fig. 3.11 Von Mises stress of a quarter of square VLFS 77

Fig. 3.12 Stage 1 for rectangular VLFS 79

Fig. 3.13 Layouts of gill cells in a quadrant of rectangular VLFS for various generations in Stage 2 79 Fig. 3.14 Rectangular VLFS with some possible optimal designs of gill cells 81

Fig. 3.15 Nondimensionalized deflection contour w in the rectangular VLFS 82

Fig. 3.16 Von Mises stress of a quarter of rectangular VLFS 83

Fig. 3.17 Stage 1 for I-shaped VLFS 84

Fig. 3.18 Layouts of gill cells in quadrant of I-shaped VLFS for various generations in Stage 2 85 Fig. 3.19 I-shaped VLFS with some possible optimal designs of gill cells 87

Fig. 3.20 Nondimesionalized deflection contour w in I-shaped VLFS 88

Fig. 3.21 Von Mises stress of a quarter of the I-shaped VLFS 89

Fig. 4.1 Geometry of uniform circular VLFS and coordinate system 93

List of Figures

xv

Fig. 4.2 Nondimensionalized deflection amplitude for circular VLFS with 2m width submerged plate 119 Fig. 4.3 Nondimensionalized deflection amplitude for circular VLFS with 10m width submerged plate 119 Fig. 4.4 Nondimensionalized deflection amplitude of VLFS without and with submerged plate of various widths 120 Fig. 4.5 Nondimensionalized bending moment amplitude of VLFS without and with submerged plate of various widths 120 Fig. 4.6 Nondimensionalized twisting moment amplitude of VLFS without and with submerged plate of various widths 121 Fig. 4.7 Nondimensionalized transverse shear force amplitude of VLFS without and with submerged plate of various widths 121 Fig. 4.8 Nondimensionalized deflection amplitude for circular VLFS with 8m width submerged plate at level d = 2m 123 Fig. 4.9 Nondimensionalized deflection amplitude for circular VLFS with 8m width submerged plate at level d = 19.5m 123 Fig. 4.10 Nondimensionalized deflection amplitude of VLFS without and with submerged plate at various depths 124 Fig. 4.11 Nondimensionalized bending moment amplitude of VLFS without and with submerged plate at various depths 124 Fig. 4.12 Nondimensionalized twisting moment amplitude of VLFS without and with submerged plate at various depths 125 Fig. 4.13 Nondimensionalized shear force amplitude of VLFS without and with submerged plate at various depths 125 Fig. 4.14 Nondimensionalized deflection amplitude for basic circular VLFS at fore-end 126 Fig. 4.15 Nondimensionalized deflection amplitude for circular VLFS with submerged plate at fore-end 127 Fig. 4.16 Nondimensionalized deflection amplitude of VLFS at fore-end 127

Fig. 4.17 Nondimensionalized deflection amplitude of VLFS at mid-position 128

Fig. 4.18 Nondimensionalized deflection amplitude of VLFS at back-end 128

Fig. 5.1 VLFS with box-shaped anti-motion device 130

List of Figures

xvi

Fig. 5.2 VLFS with submerged vertical plate anti-motion device 131

Fig. 5.3 VLFS with anti-motion devices treated by Ohta et al. (2002) 132

Fig. 5.4 Wave tank in Hydraulic Engineering Laboratory, NUS 133

Fig. 5.5 Experimental setup 134

Fig. 5.6 Wave generating system controller 136

Fig. 5.7 Wave data acquisition system 137

Fig. 5.8 Deflection data acquisition system 137

Fig. 5.9 Experimental model placed in wave tank 139

Fig. 5.10 Simple bending test to determine flexural rigidity EI of VLFS model 140

Fig. 5.11 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 0.8s 144 Fig. 5.12 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 0.9s 145 Fig. 5.13 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 1s 145 Fig. 5.14 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 1.1s 146 Fig. 5.15 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 1.2s 146 Fig. 5.16 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 1.3s 147 Fig. 5.17 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 0.8s 148 Fig. 5.18 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 0.9s 148 Fig. 5.19 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 1s 149 Fig. 5.20 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 1.1s 149 Fig. 5.21 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 1.2s 150

List of Figures

xvii

Fig. 5.22 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 1.3s 150 Fig. 5.23 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 0.8s 152 Fig. 5.24 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 0.9s 152 Fig. 5.25 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 1s 153 Fig. 5.26 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 1.1s 153 Fig. 5.27 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 1.2s 154 Fig. 5.28 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 1.3s 154 Fig. 5.29 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 0.8s 155 Fig. 5.30 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 0.9s 156 Fig. 5.31 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 1s 156 Fig. 5.32 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 1.1s 157 Fig. 5.33 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 1.2s 157 Fig. 5.34 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 1.3s 158 Fig. 5.35 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 0.8s 159 Fig. 5.36 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 0.9s 159 Fig. 5.37 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 1s 160

List of Figures

xviii

Fig. 5.38 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 1.1s 160 Fig. 5.39 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 1.2s 161 Fig. 5.40 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 1.3s 161 Fig. 5.41 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 0.8s 162 Fig. 5.42 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 0.9s 163 Fig. 5.43 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 1s 163 Fig. 5.44 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 1.1s 164 Fig. 5.45 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 1.2s 164 Fig. 5.46 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 1.3s 165

Notations

xix

NOTATIONS

A amplitude of incident wave (m)

a half width of annual submerged horizontal plate, length of cantilever test (m)

B horizontal dimension of anti-heaving device

b half length of VLFS testing model

Bχ curvature-displacement matrix

Bγ shear strain displacement matrix

D flexural rigidity of VLFS (kNm)

d distance from sea-bed to submerged horizontal plate (m)

Db bending stiffness matrix

DS shear stiffness matrix

dwdx

slope in annular VLFS

maxdwdx

maximum slope in annular VLFS with gill cells

0dwdx

maximum slope in annular VLFS without gill cells

E Young modulus of material (kN/m2)

Em equivalent Young modulus of VLFS (kN/m2)

Fi force matrix

G shear modulus (kN/m2)

Notations

xx

g gravitational acceleration (m/s2)

H water depth (m)

h overall depth of VLFS (m)

hf freeboard at the edge of VLFS (m)

h1 larger depth of stepped VLFS (m)

h2 smaller depth of stepped VLFS (m)

( )2 ( )nH • Hankel function of the second-kind

i imaginary unit i2 = -1

)(•nI modified first kind Bessel function

)(•nJ first kind Bessel function

K stiffness matrix

k wave number

6/5=sK shear correction factor for the Mindlin plate

( )nK • modified second-kind Bessel function

kw Winkler spring constant (kN/m3)

Mr radial bending moment (kNm)

Mθ twisting moment (kNm)

( )r rM M R D= non-dimensional bending moment

( )M M R Dθ θ= non-dimensional twisting moment

Mrmax maximum radial bending moment in VLFS with gill cells (kNm)

Mθmax maximum twisting moment in VLFS with gill cells (kNm)

Mr0 maximum radial bending moment in VLFS (kNm)

Mθ0 maximum twisting moment in VLFS (kNm)

Nw shape functions matrix

Notations

xxi

p total load acting on VLFS (kN/m2)

qs self weight of VLFS (kN/m2)

ql uniformly distributed loading magnitude acting on VLFS (kN/m2)

q0 basic loading magnitudes acting on VLFS (kN/m2)

Qr shear force (kN)

( )2r rQ Q R D= non-dimensional shear force

Qrmax maximum shear force in VLFS with gill cells (kN)

Qr0 maximum shear force in VLFS (kN)

R radius of circular VLFS, outer radius of annular VLFS (m)

r radial coordinate (m)

r0 loading radius of circular VLFS, inner radius of annular VLFS (m)

r1 inner radius of gill cells region in circular VLFS, inner radius of stepped region in circular VLFS, inner radius of the first gill cells region in annular VLFS (m) r2 outer radius of gill cells region in circular VLFS, outer radius of stepped region in circular VLFS, outer radius of the first gill cells region in annular VLFS (m) r3 inner radius of loading region in annular VLFS (m)

r4 outer radius of loading region in annular VLFS (m)

r5 inner radius of the second gill cells region in annular VLFS (m)

r6 outer radius of the second gill cells region in annular VLFS (m)

S non-dimensional plate rigidity

s number of sequence for each mode

t top and bottom plate thickness of VLFS (m)

t0 basic top and bottom plate thickness of VLFS (m)

w deflection of VLFS (m)

w displacement vector

we deflection at the edge of VLFS (m)

Notations

xxii

wc deflection at the center of VLFS (m)

wgm maximum deflection in gill cells region (m)

wma maximum deflection of VLFS (m)

wmi minimum deflection of VLFS (m)

/w w h= non-dimensional deflection

x horizontal coordinate (m)

( )nY • second kind Bessel function

z vertical coordinate (m)

2 1h hα = step ratio

Γ truncation number

γ shear strain vector

γm weight density of VLFS (kg/m3)

γw weight density of water (kg/m3)

wΔ maximum differential deflection in VLFS with gill cells or stepped VLFS (m)

0wΔ maximum differential deflection in VLFS (m)

( )∇ • Laplacian operator

ε curvature strain vector

θ circumferential coordinate (radian)

λ wave length (m)

njλ j -th positive root of the equation ( ) 0n njJ λ =

υ Poisson ratio

χ non-dimensional radial coordinate = r/R

φ velocity potential

nsφ radiation potential

Notations

xxiii

Dnφ diffraction potential

Iφ velocity potential of the undisturbed incident wave

xφ rotation about the y axe

yφ rotation about the x axe

ϖ wave frequency

1

CHAPTER 1

INTRODUCTION

The total land area of the Earth’s surface is about 148,300,000 square kilometers,

while the Earth’s surface area is 510,083,000 square kilometers. Thus, the main part of

the Earth’s surface is covered by sea, lakes, rivers, etc, which takes up 70 percent of

the Earth’s total surface area. Therefore, the land that we lived on forms only 30% of

the Earth’s surface. A large part of the Earth, which is the ocean, remains unexploited.

By the World Bank’s projection for global population, there will be 8.13 billion

people by 2030 and about 60% of this population will be living in urban areas. The

additional 2 billion people in a span of about 26 years will add tremendous pressure on

the land scarce cities. Thus in many island countries and countries with long coast

lines, the governments of these countries have resorted to land reclamation from the

sea in order to ease the expected pressure on the land space. There are, however,

constraints on land reclamation works such as the negative environmental impact on

the coastlines of the country and neighboring countries and marine ecological system,

as well as huge economic costs in reclaiming land from deep coastal water, especially

when the sand for reclamation has to be bought from other countries (Watanabe et al.

2004a).

In response to the aforementioned needs and problems, researchers and

engineers have proposed an interesting and attractive solution – the construction of

very large floating structures (VLFSs). In order to qualify for the VLFS status, Suzuki

and Yoshida (1996) proposed that the floating structure length to characteristic length

Introduction

2

(which is a function of the flexural rigidity to the buoyancy force) ratio and structural

length to wavelength ratio must be greater than unity. These two ratios imply a flexible

structure where the hydroelastic response dominates the rigid body responses under the

action of waves.

VLFS can be constructed to create floating airports, bridges, breakwaters, piers

and docks, storage facilities, wind and solar power plants, industrial space, emergency

bases, entertainment facilities, recreation parks, mobile offshore structures and even

for habitation. In certain applications of VLFS such as floating airports, floating

container terminals and floating dormitories where high loads are placed in certain

parts of the floating structure, the resulting differential deflections can be somewhat

large and may render certain equipment non operational. Therefore, it is important to

reduce the differential deflection in VLFS.

Generally, VLFS undergoes heaving, pitching and sway motion in the high seas.

Ohkusu and Nanba (1996) proposed an approach that treats the motion of VLFS as a

propagation of waves beneath a thin elastic-platform. Therefore, in order to reduce the

motion of VLFS, floating breakwaters as well as anti-motion devices on VLFS have

been proposed to reduce the wave transmitted under the VLFS. So far, there are many

research studies done on floating breakwaters. Recently, anti-motion devices have

been developed as alternatives for reducing the effect of waves on VLFS. An anti-

motion device is a body attached to an edge of VLFS so it does not need mooring

system like floating breakwaters and the time needed for construction is also shorter.

The rest of this chapter is organized as follows. Firstly, the background

information on VLFS is presented. A literature review then gives the information on

problems studied by researches and engineers, methods developed, and results derived

Introduction

3

about minimizing differential deflection and motion in VLFS. Finally, the objectives

and scope of study are given.

1.1 Background information on VLFS

1.1.1 Types of VLFS

VLFS may be classified under two broad categories (Watanabe et al. 2004a), namely

the pontoon-type and the semi-submersible type (Fig. 1.1). The latter type has a ballast

column tubes to raise the platform above the water level and suitable for use in open

seas where the wave heights are relatively large. VLFS of the semi-submersible type is

used for oil or gas exploration in sea and other purposes. It is kept in its location by

either tethers or thrusters. In contrast, the pontoon-type VLFS is a simple flat box

structure and features high stability, low manufacturing cost and easy maintenance and

repair. However, it is only suitable in calm sea waters, often near the shoreline.

Pontoon-type VLFS is also known in the literature as mat-like VLFS because of its

small draft in relation to the length dimensions. The Japanese refers to them as Mega-

Floats.

Pontoon-type

Semi-submersible-type

Fig. 1.1 Types of VLFS

Introduction

4

1.1.2 Advantages of VLFS

Very large floating structures have the following advantages over traditional

land reclamation in creating land from the sea:

• They are easy and fast to construct (components may be made at shipyards

and then be transported to and assembled at the site), thus, the sea space can

be quickly exploited.

• They are cost effective when the water depth is large or sea bed is soft.

• They are environmentally friendly as they do not damage the marine eco-

system or silt-up deep harbors or disrupt the ocean currents.

• They can easily be relocated (transported), removed, or expanded.

• The structures and people on VLFSs are protected from seismic shocks

since VLFSs are inherently base isolated.

• They do not suffer from differential settlement as in reclaimed soil

consolidation.

• Their positions with respect to the water surface are constant and thus

facilitate small boats and ship to come alongside when used as piers and

berths.

• Their location in coastal waters provide scenic body of water all around,

making them suitable for developments associated with leisure and water

sport activities.

• Their interior spaces may be used for car parks, offices, etc.

• There is no problem with rising sea level due to global warming.

Introduction

5

1.1.3 Applications of VLFS

Very large floating structures have already been used for different purposes in Japan,

Canada, Norway, USA, UK, Singapore, Brazil, and Saudi Arabia. China, Israel, the

Netherlands, Germany, and New Zealand are going to employ VLFSs in the near

future.

As the world’s leader in VLFSs, Japan constructed the Mega-Float in the

Tokyo Bay (Fig. 1.2), the floating oil storage base Shirashima and Kamigoto (Fig. 1.3),

the Yumeshima-Maishima Floating Bridge in Osaka (Fig. 1.4), the floating emergency

rescue bases in Tokyo Bay, Osaka Bay and Yokohama Bay (Fig. 1.5), the floating

island at Onomichi Hiroshima (Fig. 1.6), the floating ferry piers at Ujina Port

Hiroshima (Fig. 1.7) and the floating restaurant in Yokohoma (Fig. 1.8).

The world largest floating performance platform (Fig. 1.9) was built in

Singapore. Canada has a floating heliport pad in Vancouver (Fig. 1.10) and the

Kelowna floating bridge on Lake On in British Columbia (Fig. 1.11). Norway has the

Bergsøysund floating bridge (Fig. 1.12) and the Nordhordland bridge (Fig. 1.13). The

United States has the Hood Canal floating bridge in Washington State (Fig. 1.14).

Recently, in 2007, Dubai opened a 300m long floating concrete bridge over the Dubai

Creek (Fig. 1.15). The United Kingdom, Saudi Arabia, Brazil and other countries also

use floating structures for bridges, oil storage and other purposes.

Introduction

6

Fig. 1.2 Mega-float in Tokyo Bay, Japan

Fig. 1.3 Floating Oil Storage Base

at Kamigoto, Japan

Fig. 1.4 Yumeshima-Maishima Floating

Bridge in Osaka, Japan

Fig. 1.5 Floating Rescue Emergency

Base at Osaka Bay, Japan

Fig. 1.6 Floating island at Onomichi Hiroshima, Japan

Fig. 1.7 Floating pier at Ujina Port

Hiroshima, Japan

Introduction

7

Fig. 1.8 Floating Restaurant in Yokohoma, Japan

Fig. 1.9 Floating Performance Stage,

Singapore

Fig. 1.10 Floating helicopter pad

in Vancouver, Canada

Fig. 1.11 Kelowna floating bridge

in British Columbia, Canada

Fig. 1.12 Bergsøysund floating bridge,

Norway

Fig. 1.13 Nordhordland Bridge Floating

Bridge, Norway

Introduction

8

Fig. 1.14 Hood Canal Floating Bridge

in Washington States, USA Fig. 1.15 Dubai Floating Bridge in Dubai, United Arab Emirates

Source: Fig. 1.2 ~ 1.8 and 1.10 ~ 1.14 from Watanabe et al. (2004b), Fig. 1.15 from website: www.dubaiiformer.com

Pontoon floating bridge is the earliest application of the pontoon-type VLFS.

The first floating bridge is King Xerxes’ floating boat bridge across the Hellespont

(about 480 BC). Watanabe et al. (2004b) and Watanabe (2003) described the history

and worldwide development of floating bridges. The most famous floating bridges

constructed in the world are: the Galata Floating Bridge (1912) in Istanbul, the First

(1940), the Second (1963), and the Third Bridge (1989, Lacey Murrow), on Lake

Washington, and Hood Canal Bridge (1961) near Seattle in Washington state, the

Kelowna Floating Bridge (1958) in British Colombia, Canada, the Bergsøysund Bridge

(1992) and the Nordhordland Bridge (1994) over the deep fjords in Norway, a West

India Quay Footbridge in the Docklands, London (1996), the Admiral Clarey Bridge in

Hawaii (1998), Seebrücke (2000) in Saxony-Anhalt, Germany, a new swing floating

arch bridge Yumemai Bridge, in Osaka (2000), an pontoon bridges on the rivers

Amudarya and Syrdarya rivers in Uzbekistan and Turkmenistan (1989-2005). A

complete list of the pontoon bridges still in use and demolished can be found on the

website (http://en.structurae.de/structures/stype/index.cfm?ID=1051).

Large floating offshore structures can also be used for floating docks, piers and

container terminals. Many floating docks, piers, and berths are already in use all over

Introduction

9

the world. Floating piers have been constructed in Hiroshima, Japan, and Vancouver,

Canada. In Valdez, Alaska, a floating pier was designed for berthing the 50000-ton

container ships. The main advantage of a floating pier is its constant position with

respect to the waterline. Therefore, loading and unloading of cargo between the pier

and ship/ferry can operate smoothly. Floating docks have been constructed in the USA

and other countries. In case of rather deep water, VLFS are a good alternative to

traditional harbor facilities. Research on floating harbor facilities, their design and

analysis is going on in many countries (Watanabe et al. 2004a).

One more application of very large floating structures is the floating storage

facility. VLFSs have already been used for storing fuel. An offshore oil storage facility

is constructed like flat box-shaped tankers connected to each other and to other

components of the VLFS system. Japan has two major floating oil storage systems

(the only two in the world so far) namely Kamigoto (1990, near Nagasaki) and

Shirashima (1996, near Kitakyusyu) with capacity of 4.4 and 5.6 million kiloliters,

respectively. Complete information on the design, experiments and mooring of the oil

storage bases are given by Yoneyama et al. (2004).

VLFSs are ideal for applications as floating emergency rescue bases in seismic

prone areas owing to the fact that their bases are inherently isolated from seismic

motion. Japan has a number of such floating rescue bases parked in the Tokyo Bay, Ise

Bay and Osaka Bay. Specifications of the floating rescue bases can be found in

Yoneyama et al. (2004) and Watanabe et al. (2004b).

Another advantage of VLFS is its attractive panoramic view of the water body.

Waterfront properties and the sea appeal to the general public. Thus, VLFSs are

attractive for used as floating entertainment facilities such as hotels, restaurants,

shopping centers, amusement and recreation parks, exhibition centers, and theaters.

Introduction

10

Some floating entertainment facilities have been constructed. For example, the world

largest floating performer platform was built on the Marina Bay in Singapore in 2007

(Fig. 1.9), Aquapolis exhibition center in Okinawa (1975, already removed), the

Floating Island near Onomichi which resembles the Parthenon, and floating hotels in

British Columbia, Canada, and floating restaurants in Japan and Hong Kong.

In addition, VLFS can also be used as floating power plants. Floating power

plants for various types of energy are already being used in Brazil, Japan, Bangladesh,

Saudi Arabia, Argentina, and Jamaica. There are proposals to use VLFS for wind and

solar power plants (Takagi 2003 and Takagi and Noguchi 2005) and studies on this are

already underway. The Floating Structure Association of Japan has presented concept

designs of a clean power plant.

One of the most exciting applications of VLFS is the floating airport. The first

contemporary floating airport is constructed in 1943 by US Navy Civil Engineers

Corps by connecting pontoons. Recently, with the growth of cities and increase in air

traffic as well as the rise in land costs in major cities, city planners are considering the

possibility of using the coastal waters for urban developments including having

floating airports. Japan has made great progress by constructing a large airport in the

sea. Kansai International Airport (1994), Osaka, Japan, is the first airport in the world

completely constructed in the sea, although on an artificial reclaimed island. Airport

with runways on reclaimed islands in the sea are Chek Lap Kok International Airport

(1998), Hong Kong, China; Incheon International Airport (2001), Seoul, Korea;

Changi airport, Singapore; Central Japan International Airport, Nagoya; Kobe airport,

Japan. However, the extensive research on floating airports is continually pursued in

Japan (Tokyo, Osaka), USA (San Diego) and many other countries. The first largest

floating runway is the one-km long Mega-Float rest model built in 1998 in the Tokyo

Introduction

11

bay (Fig. 1.2). This is award of the Guinness book of records in 1999 for the world’s

largest man-made floating island. For small size, floating helicopter ports have already

been constructed in Vancouver, Canada (Fig. 1.9) and other places.

The final and the most advanced application of VLFS is the floating city. It

could become reality soon as various concepts have been proposed for building

floating cities or huge living complex. Figs. 1.16-1.19 show artist impressions of some

floating cities that have been proposed by various Japanese corporations.

Fig. 1.16 Marine Uranus by Nishimatsu Corporation

Fig. 1.17 Pearl Shell

by Shimizu Corporation

Fig. 1.18 Osaka Focus A by Japanese Society of Steel Construction

Fig. 1.19 Osaka Focus B by Japanese

Society of Steel Construction

Source: Fig. 1.16~1.19 from Watanabe et al. (2004b)

Introduction

12

1.1.4 VLFS components

The components of a VLFS system (general concept) are shown in Fig. 1.20. The

system comprises (1) a very large pontoon floating structure, (2) an access bridge or a

floating road to get to the floating structure from shore, (3) a mooring facility or station

keeping system to keep the floating structure in the specified place, and (4) a

breakwater, (usually needed if the significant wave height is greater than 4 m) which

can be floating as well, or anti-heaving device for reducing wave forces impacting the

floating structure, (5) structures, facilities and communications located on a VLFS.

Fig. 1.20 Components of a pontoon-type VLFS

1.2 Literature survey

This survey covers books, papers, reports and abstracts that give the basic theory for

reducing the differential deflection of VLFS under heavy load and reducing the motion

of VLFS under the action of wave. Different methods used for the problem are

described such as VLFS models and shapes, wave and other forces, anti-motion

devices. Also the author reviews what has been done already, what is currently being

Introduction

13

investigated and the future directions to study for the problem of reducing the

differential deflection and motion of VLFS.

Watanabe et al. (2004a and 2004b) presented very detailed surveys on the

research work on pontoon-type VLFSs. The numerous publications reported in

offshore structures/VLFS conference proceedings, journals, books and websites

confirm the interest in and importance of these structures to engineers and scientists.

Many papers on the analysis of VLFS were published in the following international

journals: Applied Ocean Research, Engineering Structures, International Journal of

Offshore and Polar Engineering, Journal of Engineering Mathematics, Journal of Fluid

Mechanics, Journal of Fluids and Structures, Marine Structures, Ocean Engineering,

Wave Motion; in the Proceedings of the International Workshops on Water Waves and

Floating Bodies (IWWWFB), International Offshore and Polar Engineering

Conference (ISOPE) and other conferences, workshops and seminars. Many

publications about VLFS have been published in non-scientific or scientific-popular

journals and newspapers and on the internet. Thus, the attention to and interest in the

problems of the behavior of floating plates in waves have recently increased.

1.2.1 VLFS assumptions, shapes and models

In this subsection, the author discusses the basic assumptions, the different shapes of

very large floating structures and their models proposed and analyzed by researchers

and engineers worldwide. In the hydroelastic analysis of VLFS, the basic assumptions

are usually made (Newman 1997, Stoker 1957, Watanabe et al. 2004a & 2004b,

Wehausen and Laitone 1960):

• the VLFS is modeled as an isotropic/orthotropic plate with free edges;

Introduction

14

• the fluid is ideal, incompressible, inviscid, the fluid motion is irrotational,

so that the velocity potential exists;

• the amplitude of the incident wave and the motions of the VLFS are both

small, and only the vertical motion of the structure is considered;

• there are no gaps between the VLFS and the water surface;

• the sea bottom is assumed to be flat.

In practice, VLFS can take on any shape (Yoneyama et al. 2004 and Watanabe

et al. 2004a). The choice of VLFS shape depends on many considerations such as its

purpose, the ocean/sea currents, the wave behavior on site, etc. Mainly, the VLFS

having a rectangular planform has been studied because of practical reasons for this

shape and also it lends itself for the construction of semi-analytical methods for

solutions. Some of these are Andrianov and Hermas (2003), Endo (2000), Hermans

(2003), Korobkin (2000), Mamidipudi and Webster (1994), Ohkusu and Namba (1996),

Ohkusu (1999), Takagi et al. (2000), Takagi and Nagayasu (2007) and Utsunomiya et

al. (1998). There are very few studies on non-rectangular shape VLFS. Hamamota and

Fujita (2002) treated L-shaped, T-shaped, C-shaped and X-shaped VLFSs. The

Japanese Society of Steel Construction (JSSC 1996) suggested that hexagonal shaped

VLFSs be constructed for floating cities in the Osaka Bay for better revolving

resistance under the action of waves as well as for easy expansion of floating structure.

Circular pontoon-type VLFS is considered by Hamamoto (1994), Meylan and

Squire (1996), Zilman and Miloh (2000), Tsubogo (2000), Peter et al. (2004), Sturova

(2003), Watanabe et al. (2003b), Andrianov and Hermans (2004, 2005), Wang et al.

(2004), Le Thi Thu Hang et al. (2005) and Watanabe et al. (2006). A ring-shaped

floating plate has been considered by Adrianov and Hermans (2006). Hermans (2001),

Meylan (2001, 2002) and Takagi and Nagayasu (2001) considered the general

Introduction

15

geometry of a VLFS; several interesting shapes of the structures are discussed in these

papers. These non-rectangular VLFS can be used for floating airports, cities, storage

facilities, power plant, etc. (Watanabe et al. 2004a and 2004b), and the behavior of

such VLFS could be analyzed by the methods developed.

Researchers commonly model these pontoon-type VLFSs as isotropic thin

plates and applied the classical thin plate theory (Kirchhoff 1850; Reddy 1999) for

determining the structural response. In order to obtain more accurate stress resultants,

some researchers such as (Watanabe et al. 2003b, Wang et al. 2004, Le Thi Thu Hang

et al. 2005, Watanabe et al. 2006, and Zhao et al. 2007) have used the first-order shear

deformation plate theory (Mindlin 1951) to model VLFS so that computation of stress

resultants requires only at most the evaluation of the first derivatives of the

displacement functions.

1.2.2 VLFS and water interaction modeling

The analysis may be carried out in frequency domain or in time domain. For normal

interaction of VLFS and water, the frequency domain is employed due to its simplicity

and numerical efficiency. However, for transient responses and for nonlinear equations

of motion due to the effects of mooring system or nonlinear wave (in severe wave

condition), the time domain has to be used.

In the frequency domain, two commonly-used approaches are modal expansion

method and direct method. The modal expansion method consists of separating the

hydrodynamic analysis and dynamic response and the dynamic response analysis of

the plate. The deflection of the plate with free edges is first decomposed into vibration

modes that can be arbitrarily chosen. Next, the hydrodynamic radiation forces are

evaluated for unit amplitude motions of each mode. The Galerkin’s method, by which

Introduction

16

the governing equation of the plate is approximately satisfied, is then used to calculate

the modal amplitudes, and the modal responses are summed up to obtain the total

response. Some of the researches using modal expansion method are Wang et al.

(2001), Takaki and Gu (1996a, 1996b), Hamamoto and Fujita (1995, 2002) and

Hamamoto et al. (1996, 1997).

For the direct method, the deflection of the VLFS is determined by solving the

equations directly without expanding the plate motion into eigenmodes. The solution

can be derived for both a two-dimensional geometry and a three-dimensional geometry.

Mamidipudi and Webster (1994) pioneered this direct method for VLFS. In their

solution procedure, the potentials of the diffraction and radiation are established first,

and the deflection of VLFS is determined by solving the combined hydroelastic

equation using the finite difference scheme. Their model was applied by Yago and

Endo (1996) who applied the pressure distribution method and the equation of motion

was solved using the finite element method. In the direct method of Kashiwagi (1998),

the pressure distribution method is applied and the deflection is derived from the

vibration equation of the structure. Ohkusu and Namba (1996, 1998) proposed a

different type of direct method. Their approach is based on the idea that the thin plate

is part of the free water surface but with different physical characteristics than those of

the free surface. This approach was used to analyze similar problems of two-

dimensional ice floe dynamics by Meylan and Squire (1994), and a circular floating

plate by Zilman and Miloh (2000).

In the time-domain analysis, the commonly-used approaches are the direct time

integration method (Watanabe and Utsunomiya 1996, Watanabe et al. 1998) and

Fourier transform method (Miao et al. 1996, Endo et al. 1998, Ohmatsu 1998, Endo

2000, Kashiwagi 2000). In the direct time integration method, the equations of motion

Introduction

17

are discretized for both the structure and the fluid domain. In the Fourier transform

method, the frequency domain solutions are first obtained for the fluid domain and

then the results are inserted into the differential equations for elastic motion.

1.2.3 Minimizing differential deflection in VLFS

Although a lot of studies have contributed much to the theory and analysis of VLFS,

there is relatively little work carried out in studying the problem of reducing the

differential deflection and stress-resultants in VLFS when subjected to a heavy central

load. There are some ways to decrease the differential deflection (Wang et al. 2006).

One solution is to increase the flexural stiffness of the structure by increasing the

overall depth of the floating structures or the thickness of the top and bottom slabs.

This solution will lead to an increase in costs. Another solution is to use stepped VLFS

by increasing the depth of VLFS at the loaded regions and reducing it at the unloaded

regions. Stepped circular plates have been treated by some researchers such as Wang

(1992), Avalos et al. (1995), Xiang and Zhang (2004) and Le Thi Thu Hang et al.

(2005) to investigate the free vibration However, no studies have applied these two

methods to analyze the differential deflection of VLFS.

Recently, Wang et al. (2006) proposed an innovative solution to reduce the

differential deflection by having compartments in the floating structure with holes or

slits at their bottom floors that allow water to flow in and out freely. The free flowing

of water through these holes and slits resembles the gills of fish and thus these

compartments have been referred to as gill cells. At the gill cells, the buoyancy forces

are eliminated. By appropriate positioning of these gill cells, it is possible to reduce the

differential deflection significantly as well as the stress-resultants. It is worth noting

that the holes in the gill cells have negligible effect on the flexural rigidity of the

Introduction

18

floating structure. Of course, the presence of gill cells leads to a slight loss in

buoyancy for the floating structure, which is a small price to pay for the advantage

gained in minimizing the differential deflection and stress-resultants. Wang et al.

(2006) analyzed a rectangular super-large floating container terminal by using the gill

cells. The VLFS is analyzed with the sea state of Singapore. In the studies of Wang and

Wang (2005), the hydroelastic of a super-large floating container terminal under sea

state of Singapore was analyzed and they found that the deflections due to the wave

loads are very small when compared to the static deflections due to the large live loads

from the containers resting on the floating structure. Therefore, Wang et al. (2006)

neglected the action of wave and they found that when gill cells are appropriately

designed and located, the differential deflection and stress-resultants in VLFS are

reduced significantly.

1.2.4 Minimizing motion of VLFS

There are some ways to reduce the effect of wave on the VLFS. The traditional way is

using breakwaters which reduce the height of incident water waves on the leeward side

to acceptable level. However, in many cases such as for consideration of

environmental protection and economics in open sea, breakwaters with high wave

transmission are adopted and the response at the ends of VLFS may become an

obstacle to the facilities mounted on the floating structures. Recently, anti-motion

devices have been proposed as alternatives for reducing the effect of waves on VLFS

where the wave dissipation effect of breakwaters is small or there are no breakwaters.

An anti-motion device is a body attached to an edge of VLFS so it does not need

mooring system like floating breakwaters and the time needed for construction is also

shorter.

Introduction

19

Ohkusu and Nanba (1996) proposed an approach that treats the motion of VLFS

as a propagation of waves beneath a thin elastic-platform. According to this approach,

the motion of VLFS is presented as waves. That means the anti-motion coincides with

a reduction of wave-transmission from the outside to the inside of VLFS. Following

this idea, some simple anti-motion devices have been proposed and investigated.

Takagi et al. (2000) proposed a box-shaped anti-motion device and investigated its

performance both theoretically and experimentally. They found that this device

reduces not only the deformation but also the shearing force and moment of the

platform. The motion of VLFS with this device is reduced in both beam-sea and

oblique sea.

A horizontal single plate attached to the fore-end of VLFS was proposed and

investigated experimentally by Ohta et al. (1999). The experimental results showed

that the displacement of VLFS with this anti-motion device is reduced significantly not

only at the edges but also the inner parts. They suggested that it would be possible to

eliminate the construction of breakwaters in a bay where waves are comparatively

small. Utsunomiya et al. (2000) made an attempt to reproduce these experimental

results by analysis. The comparison of the analytical results with the experimental

results has shown that their simple model can reproduce the reduction effect only

qualitatively. A more precise model considering rigorously the configuration of the

submerged horizontal plate within the framework of linear potential theory is

constructed in the study of Watanabe et al. (2003a) and has successfully reproduced

the experimental results.

Takagi et al. (2000) and Watanabe et al. (2003a) formulated the diffraction and

radiation potentials using the eigen-function expansion method which was originally

proposed by Stoker (1957) for the estimation of the elastic floating break-water. This

Introduction

20

method has been widely applied in many studies such as the study of elastic

deformation of ice floes (e.g., Evans and Davies 1968, Fox and Squire 1990, Melan

and Squire 1993) and study of the oblique incidence of surface waves onto an

infinitely long platform (e.g., Sturova 1998, Kim and Ertekin 1998).

More experimental work was investigated by Ohta et al. (2002). Typical features

of anti-motion devices treated in their study are L-shaped, reverse-L-shaped and

beach-type plate. They concluded that L-shaped plate is more effective against long

waves whereas beach-type and reverse-L-shaped plates are more effective against

short waves.

There are some other ideas in reducing the motion of VLFS under wave action.

Maeda et al. (2000) proposed a hydro-elastic response reduction system of a very large

floating structure by using wave energy absorption devices with oscillating water

column (OWC) attached to its fore and aft ends. Their results show the effectiveness of

this system in reducing the hydroelastic response of VLFS. Ikoma et al. (2005)

investigated the effects of a submerged vertical plate and an OWC to a hydroelastic

response reduction of VLFS. They found that this system is effective especially at the

wave period of 14s, it is possible to reduce the hydroelastic response up to 45%. Hong

and Hong (2007) proposed a method using pin connection from fore-end of VLFS to

OWC breakwater. They derived analytical solutions and obtained results showed that

this anti-motion device is effective in reducing the deflections, bending moments and

shear force of VLFS.

With the idea to reduce vibration of VLFS under action of wave, Zhao et al.

(2007) analyzed theoretically a VLFS with springs attached from fore-end of VLFS to

sea bed. They found the motion of VLFS is reduced by adding this kind of anti-motion

Introduction

21

device. However this idea maybe difficult in applying to real VLFS placed at deep sea

condition.

1.3 Objectives and scope of study

The main aim of the present study is to minimize the differential deflection and motion

of VLFS. More specifically, the objectives of this thesis are to:

1) investigate and compare the effectiveness of some solutions such as gill

cells and stepped VLFS in reducing the differential deflection of VLFS

under heavy loads to find the most effective solution. This would be

valuable in providing a better understanding about reducing differential

deflection in VLFS. The best solution would be applied to real VLFS.

2) to develop an analytical model using the best solution found above to

minimize the differential deflection in circular and annular VLFS under

heavy loads. The obtained analytical results could serve as benchmark

solutions in designing circular and annular VLFS.

3) to develop a finite element model using the best solution found above to

minimize the differential deflection in non-circular VLFS under heavy

loads. This FEM model could be applied to any shapes of VLFS. This

model is simple and it should facilitate in the preliminary design stage of

arbitrarily shaped VLFS.

4) to develop an analytical model to investigate the submerged horizontal

plate anti-motion device for minimizing the motion of circular VLFS under

wave action. The obtained analytical results could serve as benchmark

solution in designing submerged horizontal plate anti-motion device for

VLFS.

Introduction

22

5) to investigate other types of anti-motion devices applying to non-circular

VLFS. This gives a better understanding of anti-motion devices. The best

anti-motion device will be judged based on comparison of the obtained

results. This will serve as guideline in applying anti-motion devices in

actual VLFS.

As it is very difficult in minimizing the differential deflection of VLFS if the

heavy loads were acting on arbitrary places, the present study would only consider the

heavy loads acting on the central portion of VLFS. There would be many kinds of anti-

motion devices for VLFS. However, it may not be possible to obtain analytical

solutions for complex anti-motion devices. Therefore, the present study would only

derive the analytical solution of a simple anti-motion device which was a submerged

plate attached to the edge of circular VLFS. For other types of anti-motion devices

applying to non-circular VLFS, the present study would investigate experimentally to

find the best device in reducing the motion of VLFS.

1.4 Layout of thesis

In this chapter, an introduction of VLFS including its system, advantages and

applications have been given. A complete overview of the literature for minimizing

differential deflection and motion in VLFS is presented pertaining to both

experimental investigations and theoretical modeling. The scope and objectives of the

present study are presented.

In Chapter 2, the author studies the use of two methods, i.e. gill cells and

stepped VLFS, for minimizing the differential deflection in circular VLFS. A

comparison between two proposed methods is presented to show that gill cells method

is more effective than stepped VLFS. The use of gill cells to minimize the slope in an

Introduction

23

annular VLFS is also treated. In this chapter exact solutions of bending analysis based

on Kirchhoff plate theory for circular and annular VLFS and traditional optimization

based on Newton method are presented to seek the optimal design of gill cells and

stepped VLFS.

In Chapter 3, an extensive analysis of using gill cells in minimizing differential

deflection in arbitrarily shaped VLFS is presented and discussed. Finite element

method based on the Mindlin plate theory for bending analysis of arbitrarily shaped

VLFS and Genetic Algorithms are presented to seek the optimal design of gill cells.

This chapter also shows the reduction Von Mises stress of VLFS by using optimal

design of gill cells.

In Chapter 4, the author proposed an innovative anti-motion device, i.e.

submerged horizontal annular plate attached to the edge of a circular VLFS to

minimize the motion of VLFS. Exact analytical solutions of hydroelastic response are

derived by making use of the exact circular and annular plate solutions and exact

velocity potential solutions for circular domains. The effectiveness of this anti-motion

device is demonstrated by performing the analysis and comparing the results of

circular VLFS with and without the submerged annular plate strip.

In Chapter 5, a simple anti-heaving device, which is a submerged inclined plate,

is proposed. The performances of this device as well as three devices proposed in other

previous studies, namely a horizontal plate, a vertical plate and an L-shaped plate, are

investigated experimentally in the present work. A small scale model of VLFS with

the plate attachments is tested in a wave-tank to assess their effectiveness in reducing

the heaving motion. The obtained results are compared and discussed to find the most

effective device in minimizing motion of VLFS.

Introduction

24

In Chapter 6, the conclusions drawn from the studies of the aforementioned

innovative solutions in reducing differential deflection and motion of VLFS are

presented. Moreover, some recommendations for future studies based on this research

work are presented

25

CHAPTER 2

MINIMIZING DIFFERENTIAL DEFLECTION IN

CIRCULAR VLFS

2.1 Introduction

In certain applications of VLFS, the resulting differential deflections must be kept

within a stringent limit so as not to render certain equipment non-operational. For

example, if the VLFS is used for a floating container terminal, the heavy container

loads placed in the stacking yard (central portion) of the floating structure can cause a

large difference in the deflection value at the mid centre and the deflections at the

edges of the floating structure. This differential deflection can cause the quay cranes

located at the edge of the floating container terminal to cease operation due to the

stringent gradient tolerances of the rails which must be satisfied. Therefore, it is

important to reduce the differential deflection in certain applications of VLFS.

There are some ways to decrease the differential deflection. One solution is to

increase the flexural stiffness of the structure by increasing the overall depth of the

floating structures or the thickness of the top and bottom slabs. Another solution is to

use varying depths of VLFS or stepped VLFS by increasing the depth of VLFS at the

loaded regions and reducing it at the unloaded regions. Recently, Wang et al. (2006)

proposed an innovative solution to reduce the differential deflection by using gill cells

which are compartments in the floating structure with holes or slits at their bottom

floors that allow water to flow in and out freely. At the gill cells, the buoyancy forces

Minimizing Differential Deflection in Circular VLFS

26

are eliminated. The holes in the gill cells have negligible effect on the flexural rigidity

of the floating structure. Wang et al. (2006) analyzed a rectangular super-large floating

container terminal by using the gill cells. They found that when gill cells are

appropriately designed and located, the differential deflection and stress-resultants are

reduced. The question is where the prescribed number of gill cells should be placed in

the floating structure for minimum differential deflection.

In this chapter, the author performs a thorough study on the effectiveness of the

gill cells in reducing the differential deflection and stress-resultants for a pontoon-type

circular VLFS with gill cells with respect to their sizes and locations. The optimal

design of gill cells that minimizes the differential deflection between deflection at the

center and deflection at the edge of the circular VLFS is investigated. For a specific

circular VLFS with prescribed dimensions, material properties and other conditions,

there is an optimal design of gill cells. The overall optimal designs of gill cells are

analyzed according to varying some parameters in VLFS and conditions on VLFS.

The author also analyzes the effectiveness of using stepped VLFS in reducing

the differential deflection for a pontoon-type circular VLFS by varying the size and

locations of stepped plate. Comparisons between using stepped plate and gill cells in

reducing the differential deflection in VLFS are carried out to find the best solution.

The same volume of VLFS in both methods is used for a fair comparison.

Finally, the author uses the best solution to analyze its effectiveness in

minimizing the slope in a pontoon-type annular VLFS. In this chapter, the exact

bending solutions of circular and annular VLFS based on Kirchhoff plate theory

(Kirchhoff 1850) are used. The classical optimization based on Newton’s method is

employed as an optimization tool to investigate the optimal differential deflection in


27

circular VLFS. For optimizing the slope in annular VLFS, a hybrid method combining

classical optimization and Genetic Algorithm (GA) is employed.

2.2 Problem definition

The author first considers the problem involving a flat box-shaped circular VLFS with

radius R, overall depth h and top and bottom thicknesses t as shown in Fig. 2.1a. The

considered VLFS is to carry a uniformly distributed load lq over its central portion

area bounded by the radius 0r . As the load is not uniformly distributed over the entire

floating structure, the induced deflection is not uniform. In fact, the structure will

deform into a bowl shape, with a maximum differential deflection occurring between

the centre of the structure and its outer free edge. The problem at hand is to minimize

the maximum differential deflection as well as to reduce the stress-resultants.

The author uses two methods, namely the gill cells solution and the varying

depth solution, to reduce the differential deflection problem. Owing to the

axisymmetric nature of the considered VLFS and the loading condition, it is proposed

that the gill cells be located as shown in Fig. 2.1b. The gill cells compartments are

bounded by the radii 1r and 2r such that 0 1 2r r r R≤ ≤ ≤ . These radii are to be

determined such that the maximum deferential deflection is minimized and the stress-

resultants are reduced. In the second method, the author uses varying depths with

smaller depth region (stepped region) located as shown in Fig. 2.1c with 2 1h h< . The

stepped region is bounded by the radii 1r and 2r . These radii are to be determined such

that the maximum deferential deflection is minimized. Then the author compares these

two solutions to find which of them is the best solution for reducing the differential

deflection and stress-resultants in VLFS.


28

Gill Cells

RrR

q

Rr

rrr

h

R

r0

0

rr12

0 0 1

2

q

r

z

r

r

θh

l l

(a) (b)

Step region

R rrr

h

R

r0

r

r

1

2

0 1

2

ql 1

h 2

(c)

Fig. 2.1 Pontoon-type circular VLFS with gill cells and varying heights

In addition to the circular VLFS problem, the author also considers an annular

VLFS with inner radius r0, outer radius R, overall depth h and top and bottom

thicknesses t as shown in Fig. 2.2. The annular VLFS is subjected to a uniformly

distributed load lq over an annular area bounded by the radii r3 and r4. As the load is

not uniformly distributed over the entire floating structure, it causes a deformed shape

with a large slope in the structure. The problem at hand is to minimize the maximum


29

slope as well as to reduce the stress-resultants by using the best solution found in the

first problem.

rR

q

r0

0

z

r

r

θ

h

l

rr 34

r0

3r

R

Loading region

r4

G

Fig. 2.2 Pontoon-type ring-shaped VLFS

2.3 Basic assumptions and governing equations

In the present axisymmetric bending analysis of the pontoon-type circular/annular

VLFS, the following assumptions have been made:

• The VLFS is modeled as an elastic classical thin circular/annular plate with free

edges.

• The buoyancy force is modeled by the Winkler-type spring system.

• There are no gaps between the VLFS and the water surface.

• The VLFS is considered to be placed in a benign sea environment. Therefore,

the influence of wave force on the vertical deflection of structure is assumed to

be negligible when compared to the deflection due to the heavy static load on


30

the floating structure. This assumption has been verified in the analysis by

Wang et al. (2006) for sea state in Singapore.

The formulation of the circular VLFS problem is best cast in the cylindrical

coordinate system (r, θ, z) with the origin at the center and mid-plane of the plate as

shown in Fig. 1a. Note that the r coordinate is taken radially outward from the center

of the plate, the z coordinate along the height of the plate and the θ coordinate is taken

along a circumference of the plate.

Based on the classical thin plate theory (Kirchhoff 1850, and Reddy 1999),

Hooke’s law and statical consideration, the axisymmetric bending of a thin circular

plate floating on water is governed by the fourth order ordinary differential equation

1 1w

d d d dwD r r k w pr dr dr r dr dr

⎡ ⎤⎧ ⎫⎛ ⎞ + =⎨ ⎬⎢ ⎥⎜ ⎟⎝ ⎠⎩ ⎭⎣ ⎦

(2.1)

where w is the transverse displacement of a point on the mid-plane (i.e., z = 0) of the

plate, ( ) ( )3 2

2 26 1 2 1Et Eh tD

ν ν= +

− − the flexural rigidity of the VLFS, t the thickness of

the top and bottom plates, E the Young’s modulus, and ν the Poisson ratio. The

Winkler spring constant that models the buoyancy force is given by

39.81 /wk kN m= in the region with buoyancy force (2.2a)

0wk = in the region without buoyancy force (2.2b)

while the load p is given by

l sp q q= + in the loaded region (2.3a)

sp q= in the unloaded region. (2.3b)

where sq is the self weight of the VLFS per square meter plan.


31

2. 4 Exact bending solutions and boundary conditions

2.4.1 VLFS regions without buoyancy force

For plate regions with gill cells or stepped plate with smaller depth above water level,

there is no buoyancy force acting on the floating plate. In view of Eq. (2.1), the exact

bending solutions for these regions are given by (Timoshenko and Woinowsky-Krieger

1959)

4

2 21 2 3 4log log

64prw C r r C r C r C

D= + + + + (2.4)

142r

C DprQr

= − − (2.5)

( )

( )

23

1 2 2

33

1 2

3 2 log 3 216

2 log 216

r

Cpr C r CD r

M DCpr C r r r C r

r D rν

⎡ ⎤+ + + −⎢ ⎥

⎢ ⎥= −⎢ ⎥⎧ ⎫+ + + + +⎨ ⎬⎢ ⎥

⎩ ⎭⎣ ⎦

(2.6)

( )

( )

23

1 2 2

33

1 2

3 2 log 3 216

1 2 log 216

Cpr C r CD r

M DCpr C r r r C r

r D r

θ

ν⎡ ⎤⎛ ⎞

+ + + −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥= − ⎢ ⎥⎧ ⎫⎢ ⎥+ + + + +⎨ ⎬

⎢ ⎥⎩ ⎭⎣ ⎦

(2.7)

in which 1 2 3, ,C C C , and 4C are unknown constants, rM the radial bending moment,

θM the circumferential bending moment and rQ the shear force.


32

2.4.2 VLFS regions with buoyancy force

In the view of Eq. (2.1), the exact solutions for the VLFS regions with buoyancy are

given by (Selvadurai 1979)

( ) ( ) ( ) ( )1 1 2 2 3 3 4 4w

pw B Z r l B Z r l B Z r l B Z r lk

= + + + + (2.8)

( ) ( ) ( ) ( )' ' ' '1 2 2 1 3 4 4 33r

DQ B Z r l B Z r l B Z r l B Z r ll⎡ ⎤= − − + −⎣ ⎦ (2.9)

( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )

1 2 2 1 3 4 4 3

' '1 1 2 22

' '3 3 4 4

1r

B Z r l B Z r l B Z r l B Z r lDM B Z r l B Z r lll

r B Z r l B Z r lν

⎡ ⎤− + −⎢ ⎥

⎧ ⎫= − +⎢ ⎥⎪ ⎪− − ⎨ ⎬⎢ ⎥+ +⎪ ⎪⎢ ⎥⎩ ⎭⎣ ⎦

(2.10)

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

1 2 2 1 3 4 4 3

' '2 1 1 2 2

' '3 3 4 4

1

B Z r l B Z r l B Z r l B Z r lDM B Z r l B Z r lll

r B Z r l B Z r lθ

ν

ν

⎡ ⎤− + −⎢ ⎥

= − ⎢ ⎥⎧ ⎫+⎪ ⎪+ −⎢ ⎥⎨ ⎬+ +⎢ ⎥⎪ ⎪⎩ ⎭⎣ ⎦

(2.11)

where 1 2 3, ,B B B and 4B are unknown constants, ( )1 4wl D k= and

( ) ( )( )

( )( )

( )( )

4 8 12

1 2 2 2

2 2 21 ...

2! 4! 6!

x x xZ x = − + − + (2.12)

( ) ( )( )

( )( )

( )( )

2 6 10

2 2 2 2

2 2 2...

1! 3! 5!

x x xZ x = − + − + (2.13)

( ) ( ) ( ) ( ) *3 1 1 2

1 2 log2 2

xZ x Z x R x Z x γπ⎡ ⎤⎧ ⎫= − + ⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

(2.14)

( ) ( ) ( ) ( ) *4 2 2 1

1 2 log2 2

xZ x Z x R x Z x γπ⎡ ⎤⎧ ⎫= + + ⎨ ⎬⎢ ⎥⎩ ⎭⎣ ⎦

(2.15)

with log * 0.57722 ECγ = = (Euler’s constant), and


33

( ) ( )( )

( )( )

2 6 10

1 2 2

3 5...

2 2 23! 5!x x xR x

φ φ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.16)

( ) ( )( )

( )( )

( )( )

4 8 12

2 2 2 2

2 4 6...

2 2 22! 4! 6!x x xR x

φ φ φ⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(2.17)

( )1

1n

mn

mφ

=

=∑ (2.18)

2.4.3 Boundary and continuity conditions

At each boundary between region of normal VLFS and region with gill cells or stepped

plate, between two parts of the smaller depth region of the stepped VLFS (submerged

part and the part above water level), or between loaded region and unloaded region,

there are four continuity conditions involving deflection w, slope dw dr , radial

bending moment rM and shear force rQ . At the free edge of the plate, the two edge

conditions are the radial bending moment 0rM = and the shear force 0rQ = . At the

center of the circular plate, the slope 0dw dr = and the shear force 0rQ = because of

axisymmetric bending. Using these boundary and continuity conditions, all the

unknown constants iC and iB (i = 1,2,3,4) in the exact solutions of each region can be

determined.


34

2.5 Results and discussions on effectiveness of gill cells

2.5.1 Basic dimensions and properties of VLFS example, loading and

freeboard conditions

For the example considered, the author will use the dimensions and properties of the

circular VLFS considered by Watanabe et al. (2006). The circular VLFS has R =

200m, h = 2m, t = 0.015m, v = 0.3, E = 2.06x1011N/m2 and γm = 0.25γw where γm is the

density of VLFS and γw the density of water. For the basic loading condition, the

VLFS is considered to carry a uniform load of ql = 30 kN/m2 on the central portion

area bounded by radius r0/R = 0.7. A freeboard that is equal or greater than 0.5h is

specified in order to prevent sea water on the top surface of the floating structure.

2.5.2 Verification of results of VLFS with gill cells

As no analytical results are available, the verification of the analytical model and

equations is established by comparing the analytical solutions written in MATLAB

(2007) against the finite element results obtained from SAP2000 (CSI 2006). For this

purpose, the author considers the gill cells spanning between r1/R = 0.8 and r2/R = 0.9.

A very fine mesh design as shown in Fig. 2.3 is used in SAP2000 in order to

get accurate results. In this mesh, the VLFS is divided into 12000 shell elements

connected by 12001 nodes. The elements connected by the node in the center of VLFS

are three-node triangular shell elements. The other elements are four-node quadrilateral

shell elements. All shell elements are plate type with thickness of 2m. This element

type combines plane stress and plate bending stress. The weight density of the element

is γm = 0.25γw. The rigidity of the element is equal to the rigidity of VLFS in the


35

analytical model. The buoyancy forces in the regions without gill cells are modeled in

SAP2000 by adding springs in vertical direction to the nodes in these regions. The

spring stiffness at each node is calculated by multiplying the Winkler spring constant

wk = 9.81kN/m3 with the associated plan area portions of the plate elements

contributing to that node.

Fig. 2.3 Design mesh of circular VLFS with gill cells used in SAP2000 analysis

It can be seen from Fig. 2.4 that the nondimensional deflections /w w h=

obtained from the two methods agree very well with each other. Therefore the

analytical modeling and equations are correctly set up.


36

0 0.2 0.4 0.6 0.8 1r/R

Analytical solution

Fig. 2.4 Comparison deflections w between analytical solutions and SAP2000 result of circular VLFS with gill cells

2.5.3 Effect of gill cells in reducing differential deflection and stress-

resultants

In order to study the effectiveness of gill cells quantitatively, the author analyzes the

VLFS without and with gill cells. The gill cells are placed in unloaded region with r1/R

= 0.8 and r2/R varying from 0.85 to 0.95.

The nondimensional deflection /w w h= curves in Fig. 2.5 show that VLFS

with gill cells has smaller deflection at its center and larger deflections at the free edge

when compared to its counterpart without gill cells. Therefore, with the introduction of

gill cells in this manner, the maximum differential deflection occurring between the

center and the edge of VLFS is reduced. As r2 increases, this reduction is even more

distinct.

Figure 2.6 shows that VLFS with gill cells has smaller nondimensional shear

force ( )2r rQ Q R D= when compared to its counterpart without gill cells. As r2

w

0

0.5

1

1.5

2


37

increases, the reduction in loaded region is more distinct. The shear forces at the

boundaries of the gill cells’ region are however noted to increase slightly.

Figures 2.7 and 2.8 show that nondimensional bending moments

( )r rM M R D= and ( )M M R Dθ θ= have been reduced when VLFS has gill cells

designed in this manner. As r2 increases, it is found that there are greater reductions in

the bending moments in the loaded region. However, near the edges of the gill cells

regions, the bending moments are observed to increase slightly.

Table 2.1 shows that VLFS with gill cells has smaller maximum differential

deflection ( wΔ ) as well as maximum stress resultants ( maxQ , max max,rM Mθ ) when

compared to VLFS without gill cells. As 2r increases, these reductions are more

distinct. Note that, 0wΔ , 0Q , 0rM and 0Mθ in Table 2.1 refer to the maximum

differential deflection, maximum shearing force and maximum bending moments of

VLFS without gill cells, respectively. It is to be noted that 2r can only increase up to a

certain value, otherwise the deflections at the edge of the VLFS will exceed the basic

freeboard. When gill cells span between r1/R = 0.8 and r2/R = 0.95, the percentage

reductions in maximum differential deflection, maximum shear force and maximum

moment are 25%, 18% and 28%, respectively. These reductions are significant and

they clearly show the effectiveness of gill cells in reducing the differential deflection

and stress-resultants in VLFS. Note that as these locations and sizes of gill cells are

still not optimal, the reductions are likely to be more when the gill cells are optimally

designed as will be demonstrated in the next section.


38

0r/R

00.2 0.4 0.6 0.8 1

R1r2r

Gill cells0rq

h

l

No gill cells

(r / R = 10.80

r / R = 2 0.85)

(r / R = 0.80r / R = 0.90)

(r / R = 0.80r / R = 0.95)

w

12

12

0.5

1

1.5

2 Fig. 2.5 Effect of gill cells on deflections w

-2

-1.5

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1r /R

Fig. 2.6 Effect of gill cells on shear force rQ

No gill cells r1/R = 0.8, r2/R = 0.85 r1/R = 0.8, r2/R = 0.90 r1/R = 0.8, r2/R = 0.95

Qr


39

-0.1

-0.05

0

0.05

0.1

0.15

0 0.2 0.4 0.6 0.8 1r/R

Fig. 2.7 Effect of gill cells on radial moment rM

-0.05

-0.025

0

0.025

0.05

0.075

0.1

0 0.2 0.4 0.6 0.8 1r/R

Fig. 2.8 Effect of gill cells on circumferential moment θM

Mr


Mθ



40

Table 2.1 Effect of gill cells in reducing maximum differential deflection and maximum stress-resultants

2 /r R For 1 0.80r R =

0.85 0.90 0.95

0/w wΔ Δ 0.9557 0.8839 0.7490

max 0Q Q 0.8989 0.8420 0.8220

max 0r rM M 0.9348 0.8470 0.8111

max 0M Mθ θ 0.9317 0.8466 0.7236

2.6 Optimal design/location of gill cells in circular VLFS

For a circular VLFS with prescribed dimensions, material properties, loading and

freeboard conditions, the optimization problem at hand is to optimally design the gill

cells, that is, to determine the optimal values of r1 and r2 (Fig. 2.1) that minimize the

differential deflection which is assumed to be the difference between the deflections at

the center wc and at the edge we of the VLFS. The constrained optimization problem

may be mathematically stated as

( )

( )1 2, c er r

Min w w−

subject to e fw h h≤ −

gmw h≤

0 1 2r r r R≤ ≤ ≤ (2.19)

where hf is the height of the freeboard at the edge of the VLFS, and wgm the maximum

deflection in the gill cell region.


41

The constrained optimization problem is solved using the MATLAB

Optimization Toolbox. The toolbox uses sequential quadratic programming (SQP)

implementation which is a generalization of Newton’s method. The SQP

implementation consists of three main stages including updating of the Hessian matrix

of the Lagrangian function (Powell 1978), quadratic programming problem solution

(Gill et al. 1984 and 1991), and line search and merit function calculation (Han 1997).

The optimal design of gill cells is defined by r1 and r2 and these radii depend

on various parameters including loading magnitude, radius of loading area, thickness

of top and bottom plates and the freeboard condition at the edge of VLFS. The optimal

design of gill cells for minimum differential deflection and reducing stress-resultants

will be investigated for four different cases, each case to assess the sensitivity of the

aforementioned parameters.

2.6.1 Case 1 - Varying loading magnitudes ql

The author adopts the dimensions and properties of VLFS and the freeboard condition

as in the part 2.5.1. In this case, the VLFS is considered to carry a uniform load ql with

magnitudes ranging from 0.6q0 to 1.6q0 (where q0 = 30 kN/m2) on the central portion

confined by the radius 0 0.7r R= .

Figure 2.9 shows that as ql increases, both r1 and r2 increase but r1 seems to

increase slightly faster than r2. This means that the optimal design of gill cells is

slightly smaller and the cells are located closer to the edge of the VLFS.


42

0.7

0.8

0.9

1

0.6 0.8 1 1.2 1.4 1.6q l/q 0

Opt

imal

r1/R

, r2/R

Fig. 2.9 Optimal design of gill cells when varying ql

0.4

0.6

0.8

1

0.6 0.8 1 1.2 1.4 1.6q l/q 0

Max

imum

diff

eren

tial d

efle

ctio

n te

an

d st

ress

-res

ulta

nts r

atio

s f g

Fig. 2.10 Effect of optimal gill cells when varying ql

The sensitivity of the optimal design of gill cells in reducing the maximum

differential deflection and stress-resultants is shown in Fig. 2.10. The 40% reduction

achieved for the maximum differential deflection is significant. As ql increases, this

percentage reduction decreases. For 0 00.6 0.7lq q q≤ ≤ , the optimal design of gill cells

r2/R r1/R

Δw/Δw0 Qmax/Q0 Mrmax/Mr0 Μθmax/Μθ0


43

led to a further reduction in the maximum bending moments as ql increases from 06.0 q

to 00.7q . However, for 0 00.7 1.6lq q q≤ ≤ , the reduction in the maximum bending

moment becomes smaller as ql increases from 00.7q to 06.1 q . Similarly, for

0 l 00.6 0.75q q q≤ ≤ , the presence of the optimal gill cells led to a further reduction in

the maximum shear force as ql increases from 06.0 q to 00.75q and when

0 l 00.75 1.6q q q≤ ≤ , the reduction in the maximum shear force becomes smaller as ql

increases from 00.75q to 06.1 q . In sum, the maximum percentage reduction in the

maximum shear force is approximately 55% when ql/q0 is 0.75 while the maximum

percentage reduction in the maximum bending moments is approximately 45% when

ql/q0 is 0.70.

2.6.2 Case 2 - Varying loading radius r0

The same dimensions and properties of VLFS and the freeboard condition are adopted.

In this case, the considered VLFS is to carry a uniform load of 30 kN/m2 over its

central area bounded by the radius 0r which the author shall vary from 0.4R to 0.85R.

Figure 2.11 shows that as the loading area radius r0 increases but is within

0.78R, both radii r1 and r2 increase with r1 increasing faster than r2. This means that

the optimal design of gill cells is smaller and located closer to the edge of the VLFS

for the loading configuration. When the loading radius becomes large, i.e r0 > 0.78R,

the optimal design of gill cells utilizes a relatively small area so as to ensure the

condition of freeboard at the edge of VLFS is satisfied.

As shown in Fig. 2.12, when the loading radius 0r increases, the reduction in

maximum shear force increases except for r0 > 0.82R. After 0r passes 0.82R, the


44

reduction in the maximum shear force decrease gradually. Figure 2.12 also shows that,

when the loading radius r0 < 0.78R, the reductions in the maximum differential

deflection and bending moments are significant and their maximum percentage

reductions are 28% and 37%, respectively. Considering loading radius r0 > 0.78R, any

increase in the loading radius leads to smaller percentage reductions in the maximum

differential deflection and bending moments. In summary, the optimal design of gill

cells seems to be most effective in reducing the maximum differential deflection and

the maximum stress-resultants when 0.5R < r0 < 0.75R.

0.5

0.6

0.7

0.8

0.9

1

0.4 0.5 0.6 0.7 0.8 0.9r 0/R

Opt

imal

r1/R

, r2/R

Fig. 2.11 Optimal design of gill cells when varying r0

r2/R r1/R


45

0.6

0.7

0.8

0.9

0.4 0.5 0.6 0.7 0.8 0.9r 0/R

Max

imum

diff

eren

tial d

efle

ctio

n df

an

d st

ress

-res

ulta

nts r

atio

s d

f

Fig. 2.12 Effect of optimal gill cells when varying r0

2.6.3 Case 3 - Varying top and bottom plate thicknesses t

In this case study, the same loading and freeboard conditions in the first example are

adopted. The same dimensions and properties of VLFS are also assumed except for the

top and bottom plate thickness t which now varies from 0.5t0 to 2.5t0 (in which t0 =

0.015 m).

The optimal design of gill cells is shown in Fig. 2.13. It may be observed that

the optimal design of gill cells does not change much with varying top and bottom

plate thicknesses t. As the plate thickness t increases, r1 is observed to remain

approximately constant while r2 decreases gradually.

As shown in Fig. 2.14, when the plate thickness t increases, the reduction in

maximum shear force increases but the reductions in maximum differential deflection

and bending moments seem to decrease gradually. The maximum percentage



46

reductions in maximum differential deflection, bending moment and shear force are

28%, 35% and 26%, respectively.

0.75

0.8

0.85

0.9

0.95

1

0.5 1 1.5 2 2.5t /t 0

Opt

imal

r 1/R

, r2/R

Fig. 2.13 Optimal design of gill cells when varying t

0.6

0.7

0.8

0.5 1 1.5 2 2.5t /t 0

Max

imum

diff

eren

tial d

efle

ctio

n

df

and

stre

ss-r

esul

tant

s rat

ios

df

Fig. 2.14 Effect of optimal gill cells when varying t

r2/R r1/R



47

2.6.4 Case 4 - Varying freeboard hf

In this case study, the same dimensions and properties of VLFS and the loading

conditions are adopted. The freeboard height hf at the edge of VLFS is now varied

from 0.2h to 0.8h to study its effect on the optimal design of gill cells.

The optimal design of gill cells is shown in Fig. 2.15. As the freeboard height

increases, both gill cells radii r1 and r2 decrease with r2 decreasing faster than r1. This

means that the optimal size of gill cells is smaller and the cells are located closer to the

center of the VLFS.

Figure 2.16 shows the effectiveness of optimal design of gill cells on reducing

the maximum differential deflection and stress-resultants. As the freeboard height hf

increases, the reduction in maximum shear force increases but the reductions in

maximum differential deflection and bending moments decrease significantly. The

maximum percentage reductions in the maximum differential deflection, bending

moment and shear are 45%, 40% and 35%, respectively.

0.75

0.8

0.85

0.9

0.95

1

0.2 0.4 0.6 0.8h f /h

Opt

imal

r1/R

, r2/R

Fig. 2.15 Optimal design of gill cells when varying freeboard hf

r2/R r1/R


48

0.5

0.6

0.7

0.8

0.9

1

0.2 0.4 0.6 0.8h f /h

Max

imum

diff

eren

tial d

efle

ctio

n df

an

d str

ess-

resu

ltant

s ra

tios

df

Fig. 2.16 Effect of optimal gill cells when varying freeboard hf

2.7 Comparing the effectiveness of gill cells with stepped VLFS

The author will compare the effectiveness in reducing the differential deflection of

optimal design of gill cells in VLFS with that of optimal stepped VLFS (optimal

values of the radii 1r and 2r in Fig 2.1c). The volume of the stepped plate is kept the

same as the volume of the reference constant thickness plate. As shown in Fig. 2.1, the

constant thickness of the reference plate as h, the thickness of the stepped plate are

related to h, via the constant volume constraint, in the following manner



49

( )1 2 2

1 21 1

hhr rR R

α=

⎛ ⎞⎛ ⎞ ⎛ ⎞+ − −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

(2.20)

2 1h hα= (2.21)

where 2 1 1h h= <α controls the change in the thickness of the stepped plate.

2.7.1 Case 1 - Varying loading magnitudes ql

The author adopts the basic dimensions and properties of VLFS and the freeboard

condition as in section 2.5.1. For stepped VLFS, the author adopts basic

ratio 2 1 0.5h h= =α . In this case, the VLFS is considered to carry a uniform load ql

with magnitudes ranging from 0.5q0 to 1.5q0 (where q0 = 30 kN/m2) on the central

portion confined by the radius 0 0.7r R= .

Figure 2.17 shows the effect of optimal design of gill cells in VLFS and

optimal stepped. Both methods are effective in reducing the maximum differential

deflection of VLFS. We see clearly that using gill cells gives more reduction in

maximum differential deflection in VLFS than using stepped VLFS about 18% in

every loading conditions.


50

0.4

0.6

0.8

1

0.5 0.7 0.9 1.1 1.3 1.5q l/q 0

Stepped VLFSGill cells

Fig. 2.17 Effect of optimal gill cells and optimal stepped VLFS when varying ql

2.7.2 Case 2 - Varying loading radius r0

The same basic dimensions and properties of VLFS and the freeboard condition with

case 1 are adopted. In this case, the considered VLFS is to carry a uniform load of 30

kN/m2 over its central area bounded by the radius 0r which the author shall vary from

0.5R to 0.9R.




maximum differential deflection in VLFS than using stepped VLFS. Only when the

loading radius r0 > 0.88R, stepped VLFS is more effective than gill cells. However,

this loading radius condition is not common in VLFS as the edge area of VLFS is used

for quay crane.

Max

imum

diff

eren

tial d

efle

ctio

n ra

tio Δ

w/Δ

w0


51

0.6

0.7

0.8

0.9

1

0.5 0.6 0.7 0.8 0.9r 0/R


Fig. 2.18 Effect of optimal gill cells and optimal stepped VLFS when varying r0

2.7.3 Case 3 - Varying top and bottom plate thicknesses t

In this case study, the same basic loading and freeboard conditions are adopted. The

same dimensions and properties of VLFS are also assumed except for the top and

bottom plate thickness t which now varies from 0.5t0 to 2.5t0 (in which t0 = 0.015 m).




maximum differential deflection in VLFS than using stepped VLFS about 20%.

Max

imum

diff

eren

tial d

efle

ctio

n ra

tio Δ

w/Δ

w0


52

0.6

0.7

0.8

0.9

0.5 1 1.5 2 2.5t /t 0


Fig. 2.19 Effect of optimal gill cells and optimal stepped VLFS when varying t

2.8 Optimal design/location of gill cells in annular VLFS

2.8.1 Dimensions, properties of annular VLFS and proposed gill cells

location

As the load is not uniformly distributed over the entire floating structure as shown in

Fig. 2.2, it causes a deformed shape with a large slope in the structure. Owing to the

axisymmetric nature of the considered VLFS and the loading condition, it is proposed

that the gill cells be located as shown in Fig. 2.20. The gill cells compartments are

bounded in two regions by the radii r1, r2, r5, and r6 such that 0 1 2 3r r r r≤ ≤ ≤ and

4 5 6r r r R≤ ≤ ≤ .The problem at hand is to minimize the maximum slope as well as to

reduce the stress-resultants by finding the optimal values of these radii.

Max

imum

diff

eren

tial d

efle

ctio

n ra

tio Δ

w/Δ

w0


53

r

q

0h

l

rr

14

Rr6

r5

r4r3

r2 r1

r 0

Gill cells' regions

r2

r3

rrR

56

Fig. 2.20 Annular VLFS with gill cells.

For the example considered, the author shall use the dimensions and properties

of the thin circular VLFS considered by Watanabe, et al. (2006) and cut out a central

circular portion of 0 0.2r R= . The ring-shape VLFS has R = 200m, r0 = 40m, h = 2m, t

= 0.015m, v = 0.3, E = 2.06x1011 N/m2 and γm = 0.25γw where γm is the weight density

of VLFS and γw the weight density of water. The VLFS is considered to carry a

uniform load of ql = 25 kN/m2 on a portion area bounded by radii r3/R = 0.5 and r4/R =

0.7. An allowable draft at free edges that is equal or less than 0.7h is specified.

2.8.2 Optimization formulation

For an annular VLFS with prescribed dimensions, material properties, loading and

freeboard conditions, the optimization at hand is to optimally design the gill cells, that

is, to determine the optimal values of r1, r2, r5, and r6 that minimize the slope of the

VLFS. The constrained optimization problem may be mathematically stated as


54

( ) ( )

( )1 2 5 6 0, , ,r r r r r r RMin Max dw

= ÷

⎛ ⎞⎜ ⎟⎝ ⎠

subject to e fw h h≤ − , gmw h≤ ,

0 1 2 3 4 5 6andr r r r r r r R≤ ≤ ≤ ≤ ≤ ≤ (2.22)

where hf = 0.3h is the allowable freeboard at the free edges of the VLFS, we the

maximum deflection at the free edges of VLFS, and wgm the maximum deflection in

the gill cell regions.

The slope function in VLFS is continuous and differentiable. So at each design

(location and size) of gill cells, the maximum slope is found by using MATLAB

Optimization Toolbox in which classical optimization algorithms is fully updated.

Obviously, the problem of determining the maximum slopes at different designs of gill

cells is stochastic in nature and does not follow a continuous and differentiable

function. Therefore, the minimization of the maximum slope problem cannot be solved

by any classical optimization algorithm. So MATLAB Genetic Algorithm and Direct

Search Toolbox is used to seek the optimal number and locations of gill cells that

minimize the maximum slope.

2.8.3 Optimization result

By carrying out the optimization exercise, the author found the optimal design of gill

cells to be bounded by radii r1 = 40.9177m, r2 = 65.2682m, r5 = 171.6198m, and r6 =

196.9966m. The slope curves in Fig. 2.21 show that VLFS with optimal gill cells has a

significantly smaller slope when compared to its counterpart without gill cells. Table

2.2 shows that VLFS with optimal gill cells has significantly smaller maximum slope


55

(dwmax ) as well as maximum stress resultants (Qrmax, Mrmax, Mθmax) when compared to

VLFS without gill cells. Note that dw0 is the maximum slope in annular VLFS without

gill cells. The percentage reductions in maximum slope, maximum shear force and

maximum moment are 59%, 12% and 65%, respectively. These reductions clearly

show the effectiveness of optimal gill cells in reducing the slope and stress-resultants

in VLFS

-0.03

-0.01

0.01

0.03

0.2 0.4 0.6 0.8 1r /R

No gill cellsOptimal gill cells

Fig. 2.21 Effect of optimal gill cells on slope dw

Table 2.2 Effectiveness of optimal gill cells in reducing the maximum slope and maximum stresses-resultants

Optimal design of gill cells dwmax /dw0 0.4100 Qrmax /Qr0 0.8826 Mrmax /Mr0 0.5880 Mθmax /Mθ0 0.3546

2.9 Concluding remarks


56

The effect of gill cells in reducing differential deflection and stress-resultants in

pontoon-type, circular VLFS under a confined central uniform load has been

investigated. The optimal design of gill cells with the view to minimizing the

differential deflection and stress-resultants has been carried out. The study conducted

considered a number of design parameters such as the loading magnitude, the radius of

the loading area, the thicknesses of the top and bottom plates and the freeboard height

at the edge of VLFS. The results show that

1) Gill cells, when designed appropriately, are effective in reducing the maximum

differential deflection and stress-resultants. The percentage reductions in the

maximum differential deflection, bending moments and shear force can be up

to 45%, 45% and 55%, respectively, for the example considered.

2) When the thicknesses of the top and bottom plates and the freeboard height of

the circular VLFS are increased, the optimal design of gill cells is more

effective in reducing the maximum shear force but is comparatively less

effective in reducing the maximum differential deflection and bending

moments.

3) The optimal design of gill cells is most effective when the radius of the loading

area is bounded by the radius that varies from 0.5R to 0.75R and when the

loading magnitude is not too large.

This chapter also compared the effectiveness in reducing the differential

deflection in circular VLFS by using gill cells and varying heights of VLFS. In both

cases, the same volume was used. The obtained results show that both methods are

effective in reducing the maximum differential deflection in VLFS. However, using

gill cells in VLFS is found more effective than using varying heights of VLFS.


57

By using gill cells in an annular VLFS, the optimal design of gill cells with the

view to minimizing the slope and reducing the stress-resultants has been investigated.

The obtained results show that optimal design of gill cells in annular VLFS can reduce

the maximum slope, shear force and bending moment up to 59%, 12% and 65%,

respectively for the example considered. These reductions are significant and clearly

show the effectiveness of gill cells in reducing the slope and stress-resultants.

58

CHAPTER 3

MINIMIZING DIFFERENTIAL DEFLECTION IN

NON-CIRCULAR SHAPED VLFS USING GILL

CELLS

3.1 Introduction

In the previous chapter, we have minimized the differential deflection in a circular

VLFS and bending slope in an annular VLFS by using gill cells by analytical method.

In practice, VLFS can take on any shape (Watanabe 2004a) and an analytical method

for seeking the optimal design of gill cells is not possible. So in this chapter, we seek

the optimal layout (number and locations) of gill cells in an arbitrarily shaped VLFS

with the view to minimizing the differential deflection using a numerical technique. In

this technique, we developed finite plate elements based on the Mindlin plate theory

(Mindlin 1951) to model the VLFS so that more accurate bending results, especially

the stress resultants, may be obtained. The buoyancy forces are modeled as Winkler

springs. At the gill cell locations, the Winkler springs are removed since there are no

buoyancy forces.

Unlike the circular VLFS treated in Chapter 2, the objective function (which is

the differential deflection) of the optimization problem for an arbitrarily shaped VLFS

does not have an explicit analytical form because it has to be determined numerically.

Hence it lacks the continuous and differentiable function form. Therefore, the

59

optimization problem cannot be solved by any classical optimization algorithms that

require a continuous and differential objective function. Since the gill cell number and

locations are the variables in the optimization problem, we believe that genetic

algorithms (GA) are ideal for seeking these optimal variables that minimize the

maximum differential deflection.


Although the algorithm developed here could be used to handle any VLFS shape and

loading pattern, we shall consider some common shapes such as square, rectangular

and I-shape under symmetrical loading pattern for analysis and discussion. The I-shape

is considered as it provides more edge length for ship to berth. The flat box-like

VLFSs have an overall depth h, top and bottom plate thickness t and carrying a

uniform load ql on a central portion as shown in Fig. 3.1. Owing to the symmetry in

shape and loading conditions considered for these VLFS problems, we will analyze

only a quarter of the structure. As the load is not uniformly distributed over the entire

floating structures, the deflection surface is not uniform.

VLFS comprises many watertight compartments to provide the necessary

buoyancy. Gill cells are compartments without buoyancy because they have holes or

slits at their bottom floors that allow water to flow in and out freely. At the gill cells

locations, the buoyancy force is eliminated. The problem at hand is to seek the optimal

layout (number and positions) of gill cells that will minimize the maximum differential

deflection of VLFS under the non-uniform load.

60

30 120 30 30240

3018

030

240

Aea lq

30

Loading

Lc

Lc A

A

180 30240

3018

030

240

Aea lq

30

Loading

Lc

Lc A

A Note: all dimensions are in meter

180 30240

3012

030

180

Aea lq

30

Loading

Lc

Lc

A A

t=0.

03

180 30240

30

lq

Section A-A

h=3 VLFS

Fig. 3.1 Square, rectangular and I-shaped VLFSs

61

3.3 Basic assumptions and FEM model for VLFS bending analysis

In the bending analysis of the pontoon-type VLFS in this chapter, the following

assumptions have been made:

• The pontoon-type VLFS is modeled as an elastic solid Mindlin plate with free

edges.

• The buoyancy forces are modeled by Winkler-type springs.

• There are no gaps between the VLFS and the water surface.

• The VLFS is considered to be placed in a benign sea environment. Therefore,

the influence of wave force on the vertical deflection of structure is assumed to

be negligible when compared to the deflection due to the heavy static load on

the floating structure. This assumption has been verified in the analysis by

Wang et al. (2006).

The box-like VLFS is modeled as an equivalent solid Mindlin plate having the

thickness h taken to be equal to the height of the box-like structure. However, the

Young’s modulus of the solid plate has to be tweaked in order to get the same

equivalent stiffness as the real box-like structure. By equating the flexural rigidity of

the solid plate model to that of the box-like structure, the equivalent elastic modulus

mE is given by

123 3

212 2 12

−⎛ ⎞ ⎛ ⎞−⎛ ⎞= × × + × ×⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠m

t h t hE E t (3.1)

where E is the Young modulus of the material used in VLFS.

As all the cells in the VLFS are rectangular, it is expedient to use rectangular

finite elements for the analysis. At each node, there are three degrees of freedom

62

, ,x yw φ φ in which w is the vertical displacement, xφ and yφ denote rotations about the

y and x axes, respectively. The definitions of vertical displacement and rotations are

shown in Fig. 3.2.

xy

y x

w, z

φ

φ

Fig. 3.2 Definitions of deflection and rotations

The vector sum of displacement and rotation angles includes twelve degrees of

freedom in an element, i.e.

1 1 1 2 2 2 4 4 4w ⎡ ⎤= ⎣ ⎦i x y x y x yw w wφ φ φ φ φ φ (3.2)

From this vector, the displacement and strain vectors are expressed as:

w N w= w i (3.3)

ε B wx y xy iz χε ε γ⎡ ⎤= =⎣ ⎦ (3.4)

xzγ B w⎡ ⎤= =⎣ ⎦yz iγγ γ (3.5)

where Nw is the shape functions, Bχ the curvature-displacement matrix, and Bγ the

shear strain displacement matrix

According to the principle of virtual work, we can write

( )Tε σdv γ τdv w w dAT TS w

V V A

K p kδ δ δ+ = −∫ ∫ ∫ (3.6)

63

{ }B D B dA B D B dA N N dA N dA⎛ ⎞

+ + =⎜ ⎟⎝ ⎠∫ ∫ ∫ ∫T T T w T

b S w i wA A A A

k w pχ χ γ γ (3.7)

[ ]{ } { }K w F=i i (3.8)

where 6/5=sK is the shear correction factor for the Mindlin plate, Db the bending

stiffness matrix, DS the shear stiffness matrix, [K] stiffness matrix and {Fi} the force

matrix.

( )3

2

1 0D 1 0

12 1 10 02

b

vEt v

v v

⎡ ⎤⎢ ⎥⎢ ⎥

= ⎢ ⎥− ⎢ ⎥−⎢ ⎥⎣ ⎦

(3.9)

( )1 0

D0 12 1

SS

K Etv⎡ ⎤

= ⎢ ⎥+ ⎣ ⎦ (3.10)

The Winkler spring constant wk in Eqs. (3.6) and (3.7) that models the

buoyancy force is given by

39.81 /wk kN m= in the region without gill cells (3.11a)

0wk = in the region with gill cells (3.11b)

while the load p is given by

= +l sp q q in the loaded region (3.12a)

= sp q in the unloaded region (3.12b)

in which sq is the self weight of the VLFS per square meter.

Based on the foregoing equations, a finite element method (FEM) program is

developed in MATLAB (2007) for the bending analysis.

64

3.4 Results and discussions on effectiveness of gill cells

3.4.1 Dimensions and properties of VLFS examples, loading and

freeboard conditions

The dimensions of the considered VLFSs are shown in Fig. 3.1. The material

properties, adopted from Chen et al. (2003), are: Poisson’s ratio 0.13=v , Young’s

modulus 10 21.19 10 N m= ×mE and weight density 3256.25kg m=mγ . We assumed

that top and bottom plates of VLFSs are made of steel with thickness 0.03m=t and

Young’s modulus 11 22.0 10 N m= ×E . From Eq. (3.1), we can get the equivalent

Young’s modulus Em. The VLFSs are assumed to be subjected to a uniform distributed

load of 230kN m=lq over the central portion as shown in Fig. 3.1. A freeboard that is

equal or greater than 0.25h is specified in order to prevent green water on the top

surface of the floating structures.

3.4.2 Verification of results

The verification of the developed FEM model is established by comparing the FEM

results with the results obtained from the commercial software SAP2000 (CSI 2006).

For this purpose, we consider a quarter of the square VLFS without gill cells and with

gill cells as shown in Fig. 3.5b with the unshaded compartments denoting normal

VLFS and shaded compartments denoting the gill cells. A mesh design, as shown in

Fig. 3.5b, is used for both the present FEM model and the SAP2000 model. In this

mesh, a quarter of the VLFS is divided into 576 equal four-node quadrilateral plate

elements connected by 625 nodes. The buoyancy forces in the regions without gill

65

cells are modeled in SAP2000 by adding springs in the vertical direction to the nodes

in these regions. The spring stiffness at each node is calculated by multiplying the

Winkler spring constant wk with the associated plan area portions of the plate elements

that contribute to that node.

It can be seen from Figs. 3.3 and 3.4 that the nondimensionalized deflection

contours =w w h obtained from the two computer codes agree very well for both the

VLFS with and without gill cells cases. Therefore, the present FEM model is correctly

set up and will be used to analyze the bending of VLFSs.

3.4.3 Effect of gill cells in reducing differential deflection and von

Mises stress

In order to study the effectiveness of gill cells quantitatively, we analyze the square

VLFS with and without gill cells. The gill cells are placed in the unloaded region as

shown in Fig. 3.5b. Figures 3.3 and 3.4 show the deflection contours of the VLFS with

and without gill cells, respectively. It can be seen that the deflection contour of VLFS

with gill cells is significantly flatter. Furthermore, the percentage reductions in the

maximum differential deflection and the maximum von Mises stress are about 21%

and 10%, respectively. These reductions are significant and clearly show the

effectiveness of gill cells. Note that as these gill cells are still not optimal, the

reductions are likely to be greater when the gill cells are optimally placed as will be

demonstrated in the optimization exercise later.

66

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(a) Using current FEM model

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(b) Using SAP2000

Fig. 3.3 Nondimensionalized deflection contour w in a quarter of the square VLFS without gill cells

67

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(a) Using current FEM model

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(b) Using SAP2000

Fig. 3.4 Nondimensionalized deflection contour w in a quarter of the square VLFS with gill cells

3.5 Optimization problems

For a VLFS with prescribed dimensions, material properties, loading and freeboard

conditions, we seek the optimal layout (number and locations) of gill cells that

68

minimizes the maximum differential deflection. The constrained optimization problem

may be mathematically stated as:

( )x

−ma miMin w w (3.13)

subject to gmw h≤ (3.14)

e fw h h≤ − (3.15)

where x is the vector containing the information on the number and locations of gill

cells, maw the maximum deflection of VLFS, miw the minimum deflection of VLFS,

gmw the maximum deflection of VLFS in gill cell regions, ew the maximum deflection

at the free edges of VLFS, and fh is the allowable freeboard at the free edges of VLFS.

Note that <fh h .

The objective function given in Eq. (3.13) is discontinuous and non-

differentiable. Therefore, this optimization exercise cannot be solved by any classical

optimization algorithm. We shall adopt genetic algorithm (GA) which can be applied

to solve a variety of optimization problems that are not well suited for standard

optimization algorithms, including problems in which the objective function is

discontinuous, non-differentiable, stochastic, or highly nonlinear.

The constrained optimization problem is solved by using the MATLAB

Genetic Algorithm and Direct Search Toolbox in which GA is fully coded. This

toolbox uses the basic GA (Goldberg 1989) and a subproblem formulated in GA by

combining the objective functions and nonlinear constraint function using the

Lagrangian and the penalty parameters (Conn et al. 1991) to solve a nonlinear

optimization problem with nonlinear constraints, linear constraints, and bounds. The

69

GA repeatedly modifies a population of individual solutions. At each step, some

members of the current population are chosen as parents. From these parents, children

are produced by making random changes to a single parent (called mutation), by

combining the vector entries of a pair of parents (called crossover), or by keeping

some parents (called elite). The current population is replaced with the children to

form the next generation. Over successive generations, the population will evolve

towards an optimal solution.

The basic idea in implementing GA for the optimization exercise is to represent

the compartments in VLFS as a binary string. In this binary string, ‘0’ represents the

gill cell, which means that there is no buoyancy force under the compartment and ‘1’

represents the normal cell, which means that there is a buoyancy force under the

compartment. The advantage of doing so is that, in GA, the binary string is the

simplest form for reproduction (crossover and mutation). For the square VLFS

considered, a quarter of VLFS is analyzed due to geometry and loading symmetry. In

this quadrant there are 576 compartments. So we produce 1152 binary strings where

one binary string contains 576 arrangements of gill cells and normal cells. One

example of this binary string is shown in Fig. 3.5a. When this binary string is set in the

VLFS shape with compartments as shown in Fig. 3.5b, we have a VLFS with gill cells

denoted by ‘0’ and normal cells (or compartments) denoted by ‘1’. In subsequent

figures, the shaded compartments shall denote gill cells whereas the unshaded

compartments shall denote normal cells.

70

(a) an example

of binary

string for

analysis

00000000000011111111111100000000000011111111111100111 1111111111111111111001111111111111111111111001111111111 1111111111110011111111111111111111110011111111111111111 1111100111111111111111111111100111111111111111111111100 1111111111111111111111001111111111111111111111001111111 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

(b) binary

string set in

the form of

VLFS shape.

Fig. 3.5 Transformation of binary string to a quarter of the square VLFS with gill cells (shaded cells) in genetic algorithms rubric

The flow-chart for the algorithm in defining the optimal layout of gill cells is

shown in Fig. 3.6. It involves GA and the developed FEM model. To speed up the

optimization process, it is proposed that large cells (each large cell consists of four

cells) be used in the first stage of analysis. In the second stage, all the large cells are

replaced by four cells for refinement of optimal layout of gill cells. In summary, the

two-stage method for solution is:

• Stage 1: Combine four cells into one large cell and run optimization program

71

(Fig. 3.6) to identify the approximate position of the large gill cells. In this

stage, it is to be noted that:

o The initial population of large gill cells is randomly placed on the VLFS

domain.

o The total members of population are equal to ten times the total number

of large cells.

o Program is set to terminate if no improvement in objective function is

obtained after an additional run of 20 generations.

In each pattern of large gill cells, the number of variable reduces by four times

and the time needed to run FEM program is very short as compared to the case

when the normal cells are used. Therefore, it takes a short time for computer

program to execute entire Stage 1.

• Stage 2: Refine the layout of VLFS by splitting each large cell into four equal

cells. Run the program to find the optimal layout of gill cells. In this second

stage, note that:

o The initial population of gill cells is randomly placed on the region

which covers the shaded areas identified in stage 1.

o The total members of population are now made equal to two times the

total cells.

o Program is set to terminate if no improvement in objective function is

obtained after an additional run of 20 generations.

72

MutationCrossoverElite Yes

No

Solution found

Initial population

Meet the stop criteria

and constraints usingPopulation

Creating newpopulation

Calculate fitness

current FEM model

Fig. 3.6 Optimization program flow-chart integrating GA and developed FEM model

3.6 Optimization results

3.6.1 Optimization results for square VLFS

Figure 3.7 shows the mesh of a quarter of VLFS with a loaded area for Stage 1 and the

results for the 1st generation and 32nd (or final) generation. Note that, wΔ and 0wΔ are

maximum differential deflection in VLFS with and without gill cells, respectively. At

this stage a quarter of VLFS consists of 144 cells, each cell measuring 10mx10m. The

initial population is randomly placed in the VLFS domain and result shows a reduction

in the maximum differential deflection by 22% when compared to VLFS design

without gill cells. In the 32nd generation, it reduces the maximum differential

deflection by 53% and the gill cells evolve towards the corner of VLFS.

73

area lqLoaded

120m30m 90m

30m

90m12

0m

Mesh and loading area

1st generation 0 0.7830w wΔ Δ =

32nd generation 0 0.4719w wΔ Δ =

Fig. 3.7 Stage 1 for a quarter of square VLFS

For Stage 2, a quarter of VLFS now consists of 576 cells, each measuring

5mx5m. The decrease in the maximum differential deflection and patterns of gill cells

at various generations for the first 80 generations of Stage 2 are shown in Fig. 3.8. The

initial population of gill cells is randomly placed in the region that covers the pattern

established from Stage 1. We can see from the results that after the 50th generation, the

maximum differential deflection hardly decreases. The gill cells seem to coalesce at

the corner of VLFS in a right-angled isosceles triangular shape. In practical

considerations, the gill cells layout should be a simple pattern in order to be fabricated

easily. We consider three design options (locations and number) of gill cells as shown

in Fig. 3.9 to study the sensitivity of the gill cell patterns in reducing the differential

deflection. Options 2 and 3 are very simple while Option 1 is a little more complicated

and arises from the optimization exercise. These three options satisfy the constraints

and hence are feasible solutions. The ratios of the maximum differential deflections of

VLFS with gill cells and without gill cells are 0.4663, 0.4945 and 0.5381 for Option 1,

Option 2, and Option 3, respectively. These numbers clearly show the effectiveness of

gill cells in reducing the maximum differential deflection, even when gill cell layout is

very simple, the reduction in the maximum differential deflection is still more than

74

46%. Option 1 has the minimum differential deflections among the three options and

this differential deflection is also very close to the value obtained in the 80th generation

of the optimization program. Therefore, we choose Option 1 as the optimal design of

gill cells for the square VLFS.

Initial generation 0 0.5093w wΔ Δ =

10th generation 0 0.4876w wΔ Δ =





Fig. 3.8 Layouts of gill cells in quadrant of square VLFS for various generations

in Stage 2

75

30m

30m

180m

30m

240m

30m

Lc

Lc

240m

180m

Option 1 0w wΔ Δ = 0.4663

Lc

Lc30m180m

240m30m

30m

180m

30m

240m


Lc

Lc30m30m

240m180m

30m

180m

30m

240m


Fig. 3.9 Square VLFS with some possible optimal designs of gill cells

76

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

(a) Without gill cells

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

(b) With optimal design of gill cells

Fig. 3.10 Nondimensionalized deflection contour w of square VLFS

0.14 0.14

0.14 0.14

1.23

0.696 0.696

0.696 0.696

1.29

77

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

(a) Without gill cells (in MPa unit)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

(b) With optimal design of gill cells (in MPa unit)

Fig. 3.11 Von Mises stress of a quarter of square VLFS

6.3

0.4

0.4

6.5

0.5

78

Figure 3.10 shows the non-dimensionalized deflection contours for this VLFS

without gill cells and with the optimal gill cells. It is clear that the deflection surface of

the VLFS with gill cells (Fig. 3.10b) is significantly flatter over the entire structural

domain as compared to the deflection surface of the VLFS without gill cells (Fig.

3.10a). Figure 3.11 shows the von Mises stresses in a quadrant of this VLFS without

gill cells and with optimal design of gill cells, respectively. The maximum von Mises

stress moves from the region near the center of VLFS without gill cells (Fig. 3.11a) to

the region near the boundary between gill cell region and normal VLFS (Fig. 3.11b)

and its value decreases. Therefore, the optimal design of gill cells reduces the

maximum differential deflection by as much as 54% and it also reduces the maximum

von Mises stress by about 3%.

3.6.2 Optimization results for rectangular VLFS

Figure 3.12 shows the mesh of a quarter of VLFS with the loaded area for Stage 1 and

results for the 1st generation and the final generation (31st generation). At this stage a

quarter of VLFS consists of 108 cells, each cell measuring 10mx10m. The initial

population is randomly placed in the VLFS domain and the result reduces the

differential deflection by 21% when compared to VLFS without gill cells. By the 31st

generation, the differential deflection reduces by as much as 54% and the gill cells

move towards the corner of VLFS.

79

area lqLoaded

120m90m30m

30m

60m90

m




Fig. 3.12 Stage 1 for rectangular VLFS

For Stage 2, we have a quarter of VLFS comprising 432 cells, each cell

measuring 5mx5m. Figure 3.13 shows the decrease in the maximum differential

deflection and patterns of gill cells at various generations for the first 60 generations.

The initial population of gill cells is randomly placed in the bounded corner region as

shown in Fig. 3.13 just like in the case of the square VLFS, but with the bounded

longer side on the longer edge of the VLFS. After the optimization process, the gill

cells coalesce around the corner of VLFS in a right-angled triangular shape. After

examining the gill cells locations and based on practical considerations, we consider

three design options (locations and number) of gill cells as shown in Fig. 3.14. The

ratios of the maximum differential deflection of VLFS with gill cells and without gill

cells are 0.4468, 0.4767, and 0.4839 for Option 1, Option 2, and Option 3, respectively.

These ratios clearly show the effectiveness of gill cells in reducing the maximum

differential deflection. Even when the gill cell layout is very simple, the reduction in

the maximum differential deflection can be more than 51%. Option 1 has the minimum

differential deflections among the three options and this differential deflection is also

very close to the value obtained in the 60th generation. Therefore, we choose Option 1

80

as the optimal design of gill cells for the rectangular VLFS. Figure 3.15 shows the

non-dimensionalized deflection contour in this VLFS with the optimal gill cells and

without gill cells. VLFS with the optimal design of gill cells (Fig. 3.15b) is

significantly flatter over the entire structural domain as compared to the VLFS without

gill cells (Fig. 3.15a). Figure 3.16 shows the von Mises stresses in a quadrant of this

VLFS with optimal design of gill cells and without gill cells, respectively. VLFS with

optimal design of gill cells (Fig. 3.16b) has smaller maximum von Mises stress than

that of the VLFS without gill cells (Fig. 3.16a). Therefore, in this example, the optimal

design of gill cells reduces the maximum differential deflection by about 55% as well

as the maximum von Mises stress by 24%.


10th generation

0 0.4747w wΔ Δ =

20th generation

0 0.461w wΔ Δ =




Fig. 3.13 Layouts of gill cells in a quadrant of rectangular VLFS for various

generations in Stage 2

81

Lc

Lc30m180m

240m30m

30m

120m

30m

180m

Option 1 0 0.4468w wΔ Δ =

Lc

Lc30m

240m180m30m

30m

120m

180m

30m


Lc

Lc30m180m30m

240m

30m

120m

30m

180m


Fig. 3.14 Rectangular VLFS with some possible optimal designs of gill cells

82

0.200.250.300.350.400.450.500.550.600.650.700.750.800.850.900.951.001.051.10

(a) Without gill cells

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

(b) With optimal design of gill cells

Fig. 3.15 Nondimensionalized deflection contour w in rectangular VLFS

1.133

0.222 0.222

0.222 0.222

1.101

0.694 0.694

0.694 0.694

83

0.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.5

(a) Without gill cells (in MPa unit)

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

(b) With optimal design of gill cells (in MPa unit)

Fig. 3.16 Von Mises stress of a quarter of rectangular VLFS

0.47 0.26

6.027.94

84

3.6.3 Optimization results for I-shaped VLFS

Figure 3.17 shows the mesh of a quarter of VLFS with the loaded area for Stage 1 and

the results for the 1st generation and 33rd generation (final generation). At this stage, a

quarter of VLFS consists of 117 cells, each cell measuring 10mx10m. The initial

population is randomly placed in the VLFS domain and the result reduces the

differential deflection by 16%. By the 33rd generation, the differential deflection has

been reduced by 66% and the gill cells move towards the corner of VLFS.

area lqLoading

60m30m30m120m

30m

90m12

0m



33rd generation 0 0.3422w wΔ Δ =

Fig. 3.17 Stage 1 for I-shaped VLFS

For Stage 2, we have 468 cells in the quarter of VLFS. The decrease in the

maximum differential deflection and patterns of gill cells at various generations for the

first 100 generations are shown in Fig. 3.18. The initial population is randomly placed

in the bold bound region which covers the gill cell locations found in the 33rd

generation of Stage 1.

85


10th generation

0 0.3759w wΔ Δ =

20th generation

0 0.3465w wΔ Δ =


70th generation

0 0.3421w wΔ Δ =


Fig. 3.18 Layouts of gill cells in quadrant of I-shaped VLFS for various generations

in Stage 2

After examining the trend of movement of gill cells in the optimization

exercise (Fig. 3.18), we consider three practical design options for the locations and

number of gill cells as shown in Fig. 3.19. The ratios of the maximum differential

deflection for VLFS with and without gill cells are 0.3440, 0.3658, and 0.3909 for

Option 1, Option 2, and Option 3, respectively. These ratios clearly demonstrate the

effectiveness of gill cells in reducing the differential deflection. Even with a simple gill

cell layout, the reduction can be more than 60%. We choose Option 1 as the optimal

design of gill cells because it has the minimum differential deflection among the three

options and this differential deflection is also very close to the value obtained in the

100th generation of the optimization program. Figure 3.20 shows the non-

86

dimesionalized deflection contour in this VLFS with the optimal gill cells and without

gill cells. VLFS with the optimal design of gill cells is significantly flatter as compared

to the VLFS without gill cells. Figure 3.21 shows the von Mises stresses in a quadrant

of this VLFS with optimal design of gill cells and without gill cells, respectively. The

maximum von Mises stress moves from the center of VLFS without gill cells (Fig.

3.21a) to the reentrance corner of VLFS (Fig. 3.21b) and its values increases. In this

example, the optimal design of gill cells reduces the maximum differential deflection

by about 66%. However, it increases the maximum von Mises stress by 26% near the

re-entrant corner. This is caused by the sudden change in the bending curvature at the

re-entrant corner. By placing suitable steel reinforcements in the re-entrant, one can

accommodate for these von Mises stresses. From this result, in order to avoid

increasing the von Mises stresses in VLFS with gill cells, one can provide smooth

transition at these re-entrant corners.

87

Lc

Lc

30m

180m

30m

240m

30m 30m 120m 30m 30m240m


Lc

Lc

30m

180m

30m

240m

30m 30m 120m 30m 30m240m


Lc

Lc

30m

180m

30m

240m

30m30m120m30m30m240m


Fig. 3.19 I-shaped VLFS with some possible optimal designs of gill cells

88

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

1.10

(a) without gill cells

0.70

0.73

0.75

0.80

0.85

0.90

0.95

1.00

1.05

(b) with optimal design of gill cells

Fig. 3.20 Nondimesionalized deflection contour w in I-shaped VLFS

1.08 1.13

0.72

0.72

0.08 0.08

0.08 0.08

89

0

1

2

3

4

5

6

7

(a) without gill cells (in MPa unit)

0

1

2

3

4

5

6

7

8

9

(b) with optimal design of gill cells (in MPa unit)

Fig. 3.21 Von Mises stress of a quarter of the I-shaped VLFS

9.95

0.11 0.02

7.89

90


In this chapter, a FE computer code employing Mindlin plate element is developed for

the bending analysis of VLFS. Genetic algorithms, fully coded in MATLAB, are

employed to seek the optimal layout (number and locations) of gill cells in VLFS. The

modeling and optimization technique are illustrated with examples of various shapes

of VLFSs such as square, rectangular and I-shape. The results of these example VLFSs

show that, with the optimal design of gill cells, the differential deflections in square,

rectangular and I-shaped VLFSs can be significantly reduced by about 55%, 54%, and

66%, respectively. The VLFSs deflection shapes become significantly flatter over the

entire of VLFS domains. Furthermore, with optimal gill cells, the von Mises stresses

are also reduced in the square and rectangular VLFS. However for I-shaped VLFS, the

maximum von Mises stress increases at the re-entrant corners. Therefore, we can

conclude that gill cells may be applied to VLFS to minimize the differential deflection.

But for the VLFS with re-entrant corners, reinforcement must be introduced since the

von Mises stress may increase in these regions. Other way to avoid increasing the von

Mises stresses in VLFS with gill cells, one can provide smooth transition at these re-

entrant corners. The FEM model and optimization technique can be applied to any

shape of floating structure under any loading configuration so as to minimize the

differential deflection.

91

CHAPTER 4

MINIMIZING HEAVING MOTION OF

CIRCULAR VLFS USING SUBMERGED PLATE

4.1 Introduction

A submerged horizontal single plate attached to the fore-end of a rectangular VLFS

was proposed and investigated experimentally by Ohta et al. (1999). The experimental

results showed that the displacement of a VLFS model with this anti-motion device is

reduced significantly not only at the edges but also in the inner domain of the floating

structure. They suggested that by using this anti-motion device, it would be possible to

eliminate the construction of breakwaters in a bay where waves are comparatively

small. Utsunomiya et al. (2000) made an attempt to reproduce these experimental

results by analysis. The comparison of the analytical results with the experimental

results has shown that their simple model can reproduce the reduction effect only

qualitatively. A more precise model considering rigorously the configuration of the

submerged horizontal plate within the framework of linear potential theory is

constructed in a study by Watanabe et al. (2003a) and they validated their model with

experimental results.

In this chapter, we investigate the effectiveness of a submerged horizontal

annular plate strip as an anti-heaving device in reducing vertical deflections and stress-

resultants of a circular VLFS. In earlier studies of such an anti-motion device attached

to the fore-end of a rectangular VLFS, the vertical displacement at the aft-end is noted

Minimizing Heaving Motion of Circular VLFS using Submerged Plate

92

to be still relatively large. Therefore, we will analyze a circular VLFS with this device

attached around its edge. We formulate the diffraction and radiation potentials using

the eigen-function expansion method which was originally proposed by Stoker (1957)

for the estimation of the elastic floating breakwater. This method has been widely

applied in many studies such as the study of elastic deformation of ice floes (e.g.,

Evans and Davies 1968, Fox and Squire 1990, Meylan and Squire 1993) and the study

of the oblique incidence of surface waves onto an infinitely long platform (e.g.,

Sturova 1998, Kim and Ertekin 1998). Takagi et al. (2000) and Watanabe et al. (2003)

also applied this method to analyze the box-shaped and single plate anti-motion device

to a rectangular VLFS. We will compare obtained results of a VLFS with submerged

horizontal plate anti-motion device with the results of the VLFS alone to show the

efficiency of this device in a circular VLFS in reducing vertical deflection and stress-

resultants.


The fluid-structure system and the cylindrical coordinates system are shown in Fig. 4.1.

In this figure, region 1 is within VLFS and the submerged plate. Region 2 is beneath

the submerged plate. Region 3 is outside VLFS and the submerged plate. Region 4 is

outside VLFS and on top of the submerged plate. Region 5 is confined from the top of

the submerged plate to the bottom of VLFS. The origin of the coordinates system is on

the flat sea-bed and the z- axis is pointing upwards. The undisturbed free surface is on

the plane z H= , and the sea-bed is assumed to be flat at 0=z . The floating, flat,

circular plate has a radius of R and a uniform thickness h . The zero draft is assumed

for simplifying the fluid-domain analysis. The horizontal annular plate strip of width

2a is attached around the edge of the VLFS at a submerged level z = d. The sinusoidal


93

plane wave is assumed to be incident to VLFS at 0=θ . The problem at hand is to

determine the effectiveness of the horizontal annular plate strip in reducing the

deflections and stress-resultants of the uniform circular plate under the wave action.

x

r,

θ

y

χ

z

R

2a1

5 4

2

3z=d

z=0

h

z=H

Sea bed

Incident Wave

Fig. 4.1 Geometry of uniform circular VLFS and coordinate system

4.3 Basic assumptions, equations and boundary conditions for

circular VLFS

The hydroelastic analysis is performed in the frequency domain. Considering time-

harmonic motions with the complex time dependence tie ϖ being applied to all first-

order oscillatory quantities, where i represents the imaginary unit, ϖ the angular

frequency of the wave and t the time, the complex velocity potential ),,( zr θφ is

governed by the Laplace’s equation:


94

0),,(2 =∇ zr θφ (4.1)

in the fluid domain. The potential must satisfy the following boundary conditions on

the free surface, on the sea-bed, on the wetted bottom surface of the floating circular

body and on the submerged annular plate:

),,(),,( 2

zrgz

zr θφϖθφ =∂

∂ on , z H r R= > (4.2)

0),,( =∂

∂z

zr θφ on 0=z (4.3)

),(),,( θϖθφ rwiz

zr =∂

∂ on , z H r R= ≤ (4.4)

( , , ) ( , )r z i w Rz

φ θ ϖ θ∂ =∂

on , z d R a r R a= − ≤ ≤ + (4.5)

where ),( θrw is the vertical complex displacement of the VLFS, and g the

gravitational acceleration.

The radiation condition for the scattering and radiation potential is also applied

at infinity.

0)()(lim =⎥⎦⎤

⎢⎣⎡ −+

∂−∂

∞→ II

rik

rr φφφφ as ∞→r (4.6)

where r is the radial coordinate measured from the center of the circular VLFS, k the

wave number, and Iφ the potential representing the undisturbed incident wave:

1/ 20

00

cosh ( ) ( ) coscosh cosh

ikx nI n n

n

igAMigA kz e Z z i J kr nkH kH

φ ε θϖ ϖ

∞

=

= = ∑ (4.7)


95

where 0 1, 2 ( 1)n nε ε= = ≥ ; A the amplitude of the incident wave; nJ the Bessel

function of the first kind of order n ; and

2

tanhk kHg

ϖ= (4.8)

1/ 20 0( ) coshZ z M kz−= (4.9)

01 sinh 212 2

kHMkH

⎛ ⎞= +⎜ ⎟⎝ ⎠

(4.10)

By assuming the circular VLFS to be an elastic, isotropic, shear deformable

plate, the motion of the floating circular body is governed by the following Mindlin

plate equations (Mindlin 1951):

012

1 32 =+−

−+

∂∂

+∂

∂rr

rrr hQr

MMMrr

M ψγϖθ

θθ (4.11)

012

21 32 =+−+

∂∂

+∂

∂θθ

θθθ ψγϖθ

hQr

MMrr

M rrr (4.12)

),(1 2 θγϖθ

θ rphwr

QQrr

Q rr =++∂

∂+

∂∂ (4.13)

where w is the modal displacement, θψψ ,r are the modal rotations. The bending

moments θMM r , , twisting moment θrM , and the shear forces θQQr , can be

calculated from the following relations:

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛

∂∂++

∂∂=

θψψνψ θ

rr

r rrDM (4.14)

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛

∂∂

++∂

∂=

θψψψν θ

θ rr

rrDM 1 (4.15)


96

( )⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ −

∂∂

+∂

∂−= θ

θθ ψ

θψψν r

r rrDM 112

(4.16)

r S rwQ K Ghr

ψ ∂⎛ ⎞= +⎜ ⎟∂⎝ ⎠ (4.17)

1S

wQ K Ghrθ θψ

θ∂⎛ ⎞= +⎜ ⎟∂⎝ ⎠

(4.18)

in which D is the plate rigidity, γ the mass per unit area of the plate, ν the Poisson

ratio, SK the Mindlin shear correction factor, and ),( θrp the fluid pressure on the

bottom surface of the plate.

The pressure ),( θrp is related to the velocity potential ),,( zr θφ by

Utsunomiya et al. (1998):

( , ) ( , , ) ( , )wp r i r H gw rθ ρϖφ θ γ θ= − − (4.19)

where wgγ is the hydrostatic restoring force factor and wγ the density of the water.

The floating body is free to move in the vertical direction along its edges and at

its free edges, the bending moment, twisting moment and shear force must vanish, i.e.

0 ,0 ,0 === rrr QMM θ (4.20)

4.4 Modal expansion of motion

The fluid-structure interaction is decomposed into the hydrodynamic problem in terms

of the velocity potential and the mechanical problem for the vibration of the circular

plate, the motion of the plate is expanded by the modal functions which consist of the

product of the natural dry modes of circular Mindlin plates with free edges.


97

Irie et al. (1980) derived the exact vibration solutions for circular Mindlin plates

with free edges. The natural frequency parameters 2 hRDγξ ω= may be calculated

from setting the determinant of the matrix for the following homogeneous system of

equations to zero, i.e.

{ }11 12 13 1

21 22 23 2

31 32 33 3

k k k Ak k k Ak k k A

⎡ ⎤ ⎧ ⎫⎪ ⎪⎢ ⎥ =⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎣ ⎦ ⎩ ⎭

0 (4.21)

where the elements of the matrix are given by:

)]()()()[1( 12

1'

1''

111 δνδνδσ nnn JnJJk −+−= (4.22)

)]()()[1(2 22'

222 δδσ nn JJnk −−−= (4.23)

)]()()[1( 33'

13 δδν nn JJnk −−= (4.24)

)]()()[1(2 11'

121 δδσ nn JJnk −−−= (4.25)

)]()()[1(2 22'

222 δδσ nn JJnk −−−= (4.26)

'' ' 223 3 3 3( ) ( ) ( )n n nk J J n J= − δ + δ − δ (4.27)

)( 1'

131 δσ nJk = (4.28)

)( 2'

232 δσ nJk = (4.29)

)( 333 δnnJk = (4.30)

and

( )2 2 2 22 1 2 1

1 2 22 2 23

2

( , ) 2( , ), 16(1 )

12

δ δ δ δσ σδ ντ ξ ν κ

τ

= =−⎡ ⎤−−⎢ ⎥

⎣ ⎦

(4.31)


98

22 2 2 2 22 2

1 2 2 2 2

4, 2 12 6(1 ) 12 6(1 )

ξ τ τ τ τδ δν κ ν κ ξ

⎡ ⎤⎛ ⎞⎢ ⎥= + ± − +⎜ ⎟⎢ ⎥− −⎝ ⎠⎣ ⎦ (4.32)

( )2 2 2

23 2

2 6(1 )1 12

τ ξ ν κδν τ

⎡ ⎤−= −⎢ ⎥− ⎣ ⎦ (4.33)

Rh=τ (4.34)

The modal displacement w and the modal rotations θψψ ,r are given by:

21 Θ+Θ=w (4.35)

∂θ∂

∂∂σ

∂∂σψ 32

21

11)1()1(

Θ+

Θ−+

Θ−=

rrrr (4.36)

rrr ∂∂

∂θ∂σ

∂θ∂σψ θ

322

11

1)1(1)1(Θ

−Θ

−+Θ

−= (4.37)

where

( ) θχ nRA n cos111 Δ=Θ (4.38)

( ) θχ nRA n cos222 Δ=Θ (4.39)

( ) θχ nRA n sin333 Δ=Θ (4.40)

in which Rr /=χ ; n ( ),...,1,0 N= is the number of nodal diameters in the vibration

mode; and

( ) 3,2,1,0Im0

2

2

=⎩⎨⎧

<≥

=Δ jifif

jj

jjj δδ

δδ (4.41a)

( ) ( )( ) 3,2,1,

00

2

2

=⎩⎨⎧

<Δ≥Δ

=Δ jifIifJ

Rjjn

jjnjn δχ

δχχ (4.41b)


99

and )(•nJ and )(•nI are the first kind and the modified first kind Bessel functions of

order n.

By using the superposition of the natural dry modes and the two rigid-body

motions (heave and pitch), the final solution of the plate deflection is given by:

∑∑= =

++=N

n

M

snsns nrwwwrw

0 110100000 cos)(cos),( θζθζζθ (4.42)

∑∑= =

=N

n

M

snsrnsr nrr

0 1, cos)(),( θψζθψ (4.43)

∑∑= =

=N

n

M

snsns nrr

0 1, sin)(),( θψζθψ θθ (4.44)

where nsnsrnsw ,, , , θψψ describe the natural dry modes (mode shape of free vibration);

s the sequence number for a given n value (i.e. );,...,2,1 Ms = and

Rrww /2 , 2 1000 == (4.45)

Note that 00w represents the heaving motion while 10w the pitching motion. The

complex modal amplitudes nsζ are the unknowns which are to be determined. When

N and M equal to infinity, Eqs. (4.42) to (4.44) provide the exact solution of a circular

Mindlin plate.

4.5 Solutions for radiation potentials

The velocity potential φ is then decomposed into diffraction and radiation potentials

by using the same modal amplitudes as in Newman (1994)


100

( ) ( ) ( )( )( )

0 0 0 0,11 2

, , , cos , cosN N M

Dn ns nsn n s n

s n

r z r z n i r z nφ θ φ θ ϖ ζ φ θ= = = =

= ≥

= +∑ ∑ ∑ (4.46)

The boundary conditions on the free surface and wetted bottom surface of the

floating body, i.e. Eqs. (4.4-4.5), are modified to

( ),0Dn r z

zφ∂

=∂

on , and , z d R a r R a z H r R= − ≤ ≤ + = ≤ (4.47)

( ) ( ),ns

ns

r zw R

zφ∂

=∂

on , z d R a r R a= − ≤ ≤ + (4.48)

( ) ( ),ns

ns

r zw r

zφ∂

=∂

on , z H r R= ≤ (4.49)

Referring to Hamamoto and Tanaka (1992), in which the general solution for

circular VLFS is shown, the general solution for the boundary value problems can be

derived for each region from 1 to 5 defined in Fig. 4.1 as follows:

( ) ( ) ( ) ( )( )( )

( ) ( )( )( )

( )( )

( )( ) ( ) ( )

11 1 1 1(1)

,0 0 , 11

21 01

,

/ cosh /2 /sinh /

nn j

ns ns ns j jj n j

Rn nj nj

ns n njj nj n nj nj

I rrr z C Z C ZR a I R a

J r R z Rw r J r R rdr

R J H R

γφ

γ

λ λλ

λ λ λ

∞

=

∞

= +

⎛ ⎞= + +⎜ ⎟−⎝ ⎠ −

⋅ ⋅

∑

∑ ∫

(4.50)

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )

( ) ( )( )( )

( )( )( )

( ) ( )( )( )

( )( )( )

( )( )( )( )( ) ( ) ( )( )

2

2 2 2 2 2 2 2,0 0 ,0 0 , 2

1

2

2 2, 22

1 1 1

0

,

/2

cosh //

sinh /

n nn j

ns ns ns ns j jj n j

n j n njns j j

j j nj n njn j

R anj

ns n njnj

I rr R ar z C Z D Z C ZR a r I R a

K r J r R aD Z

R a JK R a

z R aw R J r R a rdr

d R a

γφ

γ

γ λλ λγ

λλ

λ

∞

=

∞ ∞

= = +

+

−⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠ +

++ + ⋅

+−

+⋅ +

+

∑

∑ ∑

∫

(4.51)


101

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( )( ) ( )

( )( )( ) ( )( )

( )2 3 3

03 3 3 3 3,0 0 ,2 3 3

10

,n n j

ns ns ns j jjn n j

H k r K k rr z C Z C Z

H k R a K k R aφ

∞

=

= ++ +

∑ (4.52)

If ( ) ( )2nj 1R a gω λ+ < , ( ) ( )( )2

njatanhb R a gω λ= +

( ) ( ) ( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( ) ( )( )

4 40 04 4 4 4 4

,0 0 ,0 04 40 0

4 44 4 4 4, ,4 4

1 1

21 1

0

,

/ cosh /2sinh /

/

n nns ns ns

n n

n j n j

ns j j ns j jj jn j n j

n nj nj

j nj n nj nj

R a

ns n nj

J k r Y k rr z C Z D Z

J k R a Y k R

I k r K k rC Z D Z

I k R a K k R

J r R a z H R a bR a J d H R a b

w R J r R a rdr

φ

λ λλ λ λ

λ

∞ ∞

= =

∞

= +

+

= + ++

+ ++

+ − + +⋅

+ − + +

⋅ +

∑ ∑

∑

∫

(4.53a)

If ( ) ( )2nj 1R a gω λ+ ≥ , ( ) ( )( )( )2

njatanhb g R aλ ω= +

( ) ( ) ( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( ) ( )( )

4 40 04 4 4 4 4

,0 0 ,0 04 40 0

4 44 4 4 4, ,4 4

1 1

21 1

0

,

/ sinh /2cosh /

/

n nns ns ns

n n

n j n j

ns j j ns j jj jn j n j

n nj nj

j nj n nj nj

R a

ns n nj


J k R a Y k R

I k r K k rC Z D Z

I k R a K k R

J r R a z H R a bR a J d H R a b

w R J r R a rdr

φ

λ λλ λ λ

λ

∞ ∞

= =

∞

= +

+

= + ++

+ ++

+ − + +⋅

+ − + +

⋅ +

∑ ∑

∑

∫

(4.53b)


102

( ) ( ) ( ) ( ) ( ) ( ) ( )( )( )( )( )

( )

( )( )( )

( ) ( )( )( )

( )( )( )

( )( ) ( )( )( ) ( )( ) ( ) ( )( )

( )

5

5 5 5 5 5 5 5,0 0 ,0 0 , 5

1

5

5 5, 25

1 1 1

0

,

/2

cosh //

sinh /

/2

n nn j

ns ns ns ns j jj n j

n j n njns j j

j j nj n njn j

R anj

ns n njnj

n nj

nj

I rr R ar z C Z D Z C ZR r I R

K r J r R aD Z

R a JK R a

z H R aw R J r R a rdr

d H R a

J r RR

γφ

γ

γ λλ λγ

λλ

λ

λλ

∞

=

∞ ∞

= = +

+

−⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

++ + ⋅

+−

− +⋅ + +

− +

∑

∑ ∑

∫

( )( )( )

( )( ) ( ) ( )21 01

cosh //

sinh /

Rnj

ns n njj n nj nj

z d Rw r J r R rdr

J H d R

λλ

λ λ

∞

= +

−⋅ ⋅

−∑ ∫

(4.54)

( )2 ,n nH I and nK represent Hankel function of the second-kind, modified Bessel

function of the first and second-kind of order n , respectively; njλ is the j -th positive

root of the equation ( ) 0n njJ λ = ; ( ),

ins jC and ( )

,i

ns jD are the unknown coefficients to be

determined; and

( )( )1

1

1

cos( )jj

j

zZ

Nγ

= (4.55)

( )( )2

2

2

cos( )jj

j

zZ

Nγ

= (4.56)

( )( )3

3

3

cos( )jj

j

k zZ

N= (4.57)

( )( ) ( )4

4

4

cos( )jj

j

k z dZ

N−

= (4.58)

( )( ) ( )( )5

5

5

cos jj

j

z dZ

N

γ −= (4.59)

( )1j

jHπγ = (4.60)


103

( )2j

jdπγ = (4.61)

( )5j

jH d

πγ =−

(4.62)

( ) ( )( )2

3 3tanj jk k Hg

ϖ− = ( )( )3 0; 1jk j> ≥ (4.63)

( ) ( ) ( )( )2

4 4tanj jk k H dg

ϖ− − = ( )( )4 0; 1jk j> ≥ (4.64)

( ) ( ) ( )2

4 4tanhk k H dg

ϖ− = (4.65)

1 j jN H ε= (4.66)

2 j jN d ε= (4.67)

( )5 j jN H d ε= − (4.68)

( )( )( )

3

3 3

sin 2

2 4j

jj

k HHNk

= + ( )1j ≥ (4.69)

( ) ( )( )( )

4

4 4

sin 2

2 4j

jj

k H dH dNk

−−= + ( )1j ≥ (4.70)

Note that when j = 0, we have ( )30k ik= − and Eq. (4.63) reduces to the dispersion

relation given in Eq. (4.8), ( ) ( )4 40k ik= − and Eq. (4.64) reduces to the dispersion

relation given in Eq. (4.65). The situation where 1≥j implies that we are referring to

evanescent wave.

In order to ensure the formulation of independent equations, the following

orthogonal relationships are satisfied:


104

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 1 2 2 3 3

0 0 0

4 4 5 5

H d H

i j i j i j

H H

i j i j ijd d

Z Z dz Z Z dz Z Z dz

Z Z dz Z Z dz δ

= = =

= =

∫ ∫ ∫∫ ∫

(4.71)

where ⎩⎨⎧

≠== )( 0

)( 1jiji

ijδ (4.72)

The continuity of the potential nsφ on the boundary between regions 2 and 3

( )r R a= + can be expressed as:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( )( )( ) ( )( )

( )

3 3 2 2 2 2 2 2, ,0 0 ,0 0 ,

0 1

2

2 2, 2

1

n

ns j j ns ns ns j jj j

n jns j j

j n j

R aC Z C Z D Z C ZR a

K R aD Z

K R a

γ

γ

∞ ∞

= =

∞

=

−⎛ ⎞= + +⎜ ⎟+⎝ ⎠

++

−

∑ ∑

∑ (4.73)

By applying ( )2

0

d

lZ dz∫ to both sides of the foregoing equation, we obtain:

( ) ( ) ( )

( )

( ) ( ) ( )

( ) ( )( ) ( )( )( ) ( )( ) ( )2 3

2 2,0 ,0

3 2 32, 0

0 2 2, , 2

for 0

for 0lj

n

ns nsd

ns j l jn lj

Z ns l ns ln l

R aC D lR a

C Z Z dz K R aC D l

K R a

γ

γ−

∞

=

⎧ −⎛ ⎞+ =⎪ ⎜ ⎟+⎝ ⎠⎪⎪= ⎨ +⎪ + >⎪ −⎪⎩

∑ ∫ (4.74)

Similarly, by considering the continuity of the potential nsφ on the boundary

between regions 4 and 3 ( )r R a= + and applying ( )4H

ldZ dz∫ to both sides of the

continuity equation, we obtain:


105

( ) ( ) ( )

( )

( ) ( )( ) ( )( )

( )( ) ( )

( ) ( )( ) ( )( )

( )( ) ( )4 3

404 4

,0 ,0 403 4 3

, 404 4, , 4

for 0

for 0lj

n

ns ns

nH

ns j l jdj

n lZ

ns l ns l

n l

Y k R aC D l

Y k RC Z Z dz

K k R aC D l

K k R−

∞

=

⎧ +⎪ + =⎪⎪= ⎨

+⎪+ >⎪

⎪⎩

∑ ∫ (4.75)

The continuity of the horizontal velocity ns rφ∂ ∂ on the boundary of the regions

2, 4 and 3 ( )r R a= + with ( ) ( )2nj 1R a gω λ+ < gives

( )( ) ( ) ( ) ( )( )

( ) ( ) ( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )

( )( ) ( ) ( )

( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )( )( )

( )( )( )( ) ( ) ( )( )

3 2 ' 3 3 3'0 03 3 3 3

,0 0 ,2 3 310

2 2'2 2 2 2 2 2,0 0 ,0 0 ,1 2

1

'

2 201

cosh /2 /sinh /

n j n jns ns j j

jn n j

nj n j

ns ns ns j jnj n j

n nj njns n nj

n nj nj

k H k R a k K k R aC Z C Z

H k R a K k R a

I R an R anC Z D Z C ZR a R a I R a

J z R aw R J r R a rdr

J d R aR a

γ γ

γ

λ λλ

λ λ

∞

=

∞

+=

+

+ ++ =

+ +

+−− +

+ + +

++ ⋅ ⋅ +

++

∑

∑

( )( ) ( ) ( )( )

( ) ( )( )( ) ( )

( )( ) ( ) ( )( )

( ) ( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )( )( )

( ) ( )( )( ) ( )( )

( ) ( )( )

1

2 2'2 2, 2

1

4 4 4 4' '0 04 4 4 4

,0 0 ,4 410

'

2 21 1

0

for 0

cosh /2sinh /

/

R a

j

j n jns j j

j n j

n j n j

ns ns j jjn n j

n nj nj

j n nj nj

R a

ns n nj

K R aD Z z d

K R a

k J k R a k I k R aC Z C Z

J k R a I k R a

J z H R a b

J d H R a bR a

w R J r R a rdr D

γ γ

γ

λ λλ λ

λ

+∞

=

∞

=

∞

=

∞

= +

+

++ ≤ <

−

+ ++ +

+ +

− + +⋅ ⋅

− + ++

+ +

∑ ∫

∑

∑

∑

∫ ( )( ) ( ) ( )( )

( )( )( )

( )( ) ( ) ( )( )

( )( )( ) ( )

4 4'0 04 4

,0 040

4 4'4 4, 4

1for

n

nsn

j n jns j j

j n j

k Y k R aZ

Y k R

k K k R aD Z d z H

K k R

∞

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪

+⎪+⎪

⎪⎪⎪ +⎪ ≤ ≤⎪⎩∑

(4.76)

Applying ( )3

0

H

lZ dz∫ to both sides of the above equation, we obtain:


106

( )( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( )( ) ( ) ( )( )

( ) ( )( ) ( )

( )

( )( ) ( ) ( )

( )( )

( )( ) ( ) ( )( )

( ) ( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )

3 2 ' 30 03

,0 2 30 2 2 3 2 2 3

,0 0 ,0 013 3'3, 3

2 2 2 2' '2 2 3 2 2 3, ,2 2

1 1

2

for 0

for 0

2

n

nsn

n

ns l ns ln

l n lns l

n l

j n j j n jns j jl ns j jl

j jn j n j

k H k R aC l

H k R a n R anC Z D ZR a R ak K k R a

C lK k R a

I R a K R aC Z D Z

I R a K R a

R a

γ γ γ γ

γ γ

− −+

∞ ∞− −

= =

⎫+⎪=⎪+ −⎪ = −⎬ + ++ ⎪

> ⎪+ ⎪⎭

+ ++ + +

+ −

+

∑ ∑

( )( )

( )( )( )( )

( )

( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )( )( )

( )( ) ( ) ( )( )

( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )( ) ( ) ( )( )

'3

2 01 1

4 4'0 04 4 3

,0 040 0

4 4 4 4' '0 04 4 3 4 4 3

,0 0 ,4 410

4 4'4,

cosh /

sinh /

/

dn nj njl

j n nj nj

R an

ns n nj ns l

n

n j n j

ns l ns j jljn n j

j n j

ns j

n j

J z R aZ dz

J d R a

k J k R aw R J r R a rdr C Z

J k R a

k Y k R a k I k R aD Z C Z

Y k R I k R a

k K k R aD

K k

λ λλ λ

λ

∞

= +

+−

∞− −

=

+⋅ ⋅

+

++ +

+

+ ++ +

+

++

∑ ∫

∫

∑

( )( )( )

( )( )( )

( ) ( )( )( ) ( )( )

( ) ( ) ( )( )

'4 3

2 241 1 1

3

0

2

cosh //

sinh /

n njjl

j j n nj

R aH nj

l ns n njdnj

JZ

JR aR

z H R a bZ dz w R J r R a rdr

d H R a b

λλ

λλ

λ

∞ ∞−

= = +

+

+ ⋅+

− + +⋅ +

− + +

∑ ∑

∫ ∫(4.77a)

Similarly, considering the continuity of the horizontal velocity ns rφ∂ ∂ on the

boundary of the regions 2, 4 and 3 ( )r R a= + with ( ) ( )2nj 1R a gω λ+ ≥ , and

applying ( )3

0

H

lZ dz∫ to both sides of the continuity equation, we obtain:


107

( )( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( )( ) ( ) ( )( )

( ) ( )( ) ( )

( )

( )( ) ( ) ( )

( )( )

( )( ) ( ) ( )( )

( ) ( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )

3 2 ' 30 03

,0 2 30 2 2 3 2 2 3

,0 0 ,0 013 3'3, 3

2 2 2 2' '2 2 3 2 2 3, ,2 2

1 1

2

for 0

for 0

2

n

nsn

n

ns l ns ln

l n lns l

n l

j n j j n jns j jl ns j jl

j jn j n j

k H k R aC l

H k R a n R anC Z D ZR a R ak K k R a

C lK k R a

I R a K R aC Z D Z

I R a K R a

R a

γ γ γ γ

γ γ

− −+

∞ ∞− −

= =

⎫+⎪=⎪+ −⎪ = −⎬ + ++ ⎪

> ⎪+ ⎪⎭

+ ++ + +

+ −

+

∑ ∑

( )( )

( )( )( )( )

( )

( ) ( )( ) ( )( ) ( ) ( )( )

( ) ( )( )( )

( )( ) ( ) ( )( )

( )( )( ) ( )

( ) ( ) ( )( )( ) ( )( )

( )

( )( ) ( ) ( )( )

'3

2 01 1

4 4'0 04 4 3

,0 040 0

4 4 4 4' '0 04 4 3 4 4 3

,0 0 ,4 410

4 4'4,

cosh /

sinh /

/

dn nj njl

j n nj nj

R an

ns n nj ns l

n

n j n j

ns l ns j jljn n j

j n j

ns j

n j

J z R aZ dz

J d R a

k J k R aw R J r R a rdr C Z

J k R a

k Y k R a k I k R aD Z C Z

Y k R I k R a

k K k R aD

K k

λ λλ λ

λ

∞

= +

+−

∞− −

=

+⋅ ⋅

+

++ +

+

+ ++ +

+

++

∑ ∫

∫

∑

( )( )( )

( )( )( )

( ) ( )( )( ) ( )( )

( ) ( ) ( )( )

'4 3

2 241 1 1

3

0

2

sinh //

cosh /

n njjl

j j n nj

R aH nj

l ns n njdnj

JZ

JR aR

z H R a bZ dz w R J r R a rdr

d H R a b

λλ

λλ

λ

∞ ∞−

= = +

+

+ ⋅+

− + +⋅ +

− + +

∑ ∑

∫ ∫(4.77b)

Similarly, considering the continuity of the potential nsφ on the boundary

between regions 2 and 1 (r = R-a), and applying ( )2

0

d

lZ dz∫ to both sides of the



108

( )( )( )

( )( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )

( )( ) ( )( )( ) ( )( )

( ) ( )

( )

2 1

22 0

1 01

2 2,0 ,0

1 2 12, 0

0 2 2, ,2

/ cosh /2 /sinh /

for 0

for 0

2

lj

Rdn nj nj

l ns n njj nj n nj nj

n

ns nsd

ns j l jn lj

Z ns l ns l

n l

n

nj

J R a R z RZ dz w r J r R rdr

R J H R

R aC D lR a

C Z Z dz I R aC D l

I R a

JR a

λ λλ

λ λ λ

γ

γ

λλ

−

∞

= +

∞

=

−⋅ ⋅

⎧ −⎛ ⎞ + =⎪ ⎜ ⎟+⎝ ⎠⎪⎪+ = ⎨ −⎪ + >⎪ +⎪⎩

++

∑ ∫ ∫

∑ ∫

( ) ( )( )( )

( )( )( )( )

( )

( ) ( )( )

22 0

1 1

0

/ cosh /

sinh /

/

dnj njl

j n nj nj

R a

ns n nj

R a R a z R aZ dz

J d R a

w R J r R a rdr

λλ λ

λ

∞

= +

+

− + +⋅ ⋅

+

+

∑ ∫

∫

(4.78)

Similarly, considering the continuity of the potential nsφ on the boundary

between regions 5 and 1 (r = R-a), and applying ( )5H

ldZ dz∫ to both sides of the


( )( )( )

( )( )

( ) ( ) ( )

( ) ( ) ( )

( )

( ) ( ) ( )

( )( ) ( )( )

( )( )( ) ( )

( )

5 1

52

1 01

5 5,0 ,0

1 5 15,

0 5 5, ,5

/ cosh /2 /sinh /

for 0

for 0

/2

lj

RHn nj nj

l ns n njdj nj n nj nj

n

ns nsH

ns j l jd n ljZ ns l ns l

n l

n nj

nj


R J H R

R aC D lR

C Z Z dz I R aC D l

I R

J R aR

λ λλ

λ λ λ

γ

γ

λλ

−

∞

= +

∞

=

−⋅ ⋅

⎧ −⎛ ⎞ + =⎪ ⎜ ⎟⎝ ⎠⎪⎪+ = +⎨ −⎪ + >⎪

⎪⎩

−

∑ ∫ ∫

∑ ∫

( )( )

( )( )( )( )

( )

( ) ( ) ( )( ) ( )( )

( )( ) ( )( )( ) ( )( )

( ) ( ) ( )( )

52

1 1

210 1

5

0

cosh /

sinh /

/2/

cosh //

sinh /

H njld

j n nj nj

Rn nj

ns n njj nj n nj

R aH nj

l ns n njdnj

R z d RZ dz

J H d R

J R a R aw r J r R rdr

R a J

z H R aZ dz w R J r R a rdr

d H R a

λλ λ

λλ

λ λ

λλ

λ

∞

= +

∞

= +

+

−⋅ ⋅

−

− ++

+

− +⋅ ⋅ +

− +

∑ ∫

∑∫

∫ ∫

(4.79)


109

The continuity of the horizontal velocity ns rφ∂ ∂ on the boundary of the regions

1, 2 and 5 ( )r R a= − , and the application of ( )1

0

H

lZ dz∫ yield:

( )

( )

( )( ) ( ) ( )( )

( ) ( )( ) ( )

( )( )( )

( )( )

( ) ( ) ( )

( ) ( )( )

( )( )

( ) ( )( ) ( ) ( )( )

( ) ( )( )

1,0

1 1'1

, 1

'1

2 2 01 01

2 2'1 22 2 1 2 1 2,0,0 0 0 , 2

for 0

for 0

/ cosh /2 /sinh /

ns

l n l

ns ln l

RHn nj nj

l ns n njj n nj nj

nj n jns

ns l l ns jnn j

nCl

R aI R a

C lI R a


R J H R

I R anDn R aC Z Z C

R aR a I R a

γ γ

γ

λ λλ

λ λ

γ γ

γ

∞

= +

−− −

⎧=⎪

−⎪+⎨ −⎪ >⎪ −⎩

−⋅ =

−−− +

−+ +

∑ ∫ ∫

( )

( )( ) ( ) ( )( )

( ) ( )( )( )

( )( ) ( )( )

( )( )( )( )( )

( ) ( ) ( )( )

( ) ( )( )

( )( )

( ) ( )( ) ( ) ( )( )

( )

2 1

1

2 2' '2 2 1, 2 22

1 1 1

1

00

5 5'1 55 5 1 5 1 5,0,0 0 0 , 5

/2

cosh //

sinh /

jlj

j n j n njns j jl

j j n njn j

R ad nj

l ns n njnj

nj n jns

ns l l ns jnn j

Z

K R a J R a R aD Z

JR aK R a

z R aZ dz w R J r R a rdr

d R a

I R anDn R aC Z Z C

R aR I

γ γ λλγ

λλ

λ

γ γ

γ

∞−

=

∞ ∞−

= = +

+

−− −

+

− − ++

+−

+⋅ ⋅ + +

+

−−− +

−

∑

∑ ∑

∫ ∫

( )( )

( )( ) ( ) ( )( )

( ) ( )( )( )

( )( ) ( )( )

( )( ) ( )( )( ) ( )( )

( ) ( ) ( )( )

( )( )( )

( )( )( )( )

( )

5 1

1

5 5' '5 5 1, 2 25

1 1 1

1

0

'1

2 21 1

/2

cosh //

sinh /

/ cosh /2sinh /

jlj

j n j n njns j jl

j j n njn j

R aH nj

l ns n njdnj

n nj njld

j n nj nj

ZR

K R a J R a R aD Z

JR aK R a

z H R aZ dzw R J r R a rdr

d H R a

J R a R z d RZ dz

R J H d R

γ γ λλγ

λλ

λ

λ λλ λ

∞−

=

∞ ∞−

= = +

+

∞

= +

+

− − ++ ⋅

+−

− ++ +

− +

− −⋅

−

∑

∑ ∑

∫ ∫

∑ ( ) ( )0

/R

H

ns n njw r J r R rdrλ∫ ∫

(4.80)

The continuity of the potential nsφ on the boundary between regions 4 and 5

( )r R= in the case ( ) ( )2nj 1R a gω λ+ < , and the application of ( )4H

ldZ dz∫ yield:


110

( )( )( )

( ) ( )( )( ) ( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( )

( ) ( )( ) ( ) ( ) ( )

( )4 50

404 4

,0 ,040

44 4, ,4

42

1 1

5 4 5,0 0

0

for 0

for 0

/ cosh /2sinh /

/

l

nns ns

n

n l

ns l ns ln l

Hn nj nj

lj nj dn nj nj

R aH

ns n nj ns ld

Z

J k RC D l

J k R a

I k RC D l

I k R a

J R R a z H R a bZ dz

R a J d H R a b

w R J r R a rdr C Z Z dz

λ λλ λ λ

λ−

∞

= +

+

⎧⎪ + =⎪ +⎪ +⎨⎪

+ >⎪+⎪⎩

+ − + +⋅ ⋅

+ − + +

+ =

∑ ∫

∫ ∫

( ) ( ) ( )

( )

( ) ( ) ( )

( )

( )( )( )

( ) ( )( )( ) ( )

( )( )

( )( )( )

( ) ( )( )( ) ( )( )

( )

4 5 4 50

4 5

5 4 5 5 4 5,0 0 ,

1

5

5 4 5, 25

1 1 1

4

/2

cosh /

sinh /

l lj

lj

nH H

ns l ns j l jd dj

Z Z

Hn j n njns j l jd

j j nj n njn jZ

Hnj

ld nj

R aD Z Z dz C Z Z dzR

K R J R R aD Z Z dz

R a JK R a

z H R aZ dz

d H R a

γ λλ λγ

λλ

− −

−

∞

=

∞ ∞

= = +

+

−⎛ ⎞ + +⎜ ⎟⎝ ⎠

++

+−

− +⋅ ⋅

− +

∑∫ ∫

∑ ∑∫

∫ ( ) ( )( )0

/R a

ns n njw R J r R a rdrλ+

+∫ (4.81a)

In case ( ) ( )2nj 1R a gω λ+ ≥ , we have


111

( )( )( )

( ) ( )( )( ) ( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( )

( ) ( )( ) ( ) ( ) ( )

( )4 50

404 4

,0 ,040

44 4, ,4

42

1 1

5 4 5,0 0

0

for 0

for 0

/ sinh /2cosh /

/

l

nns ns

n

n l

ns l ns ln l

Hn nj nj

lj nj dn nj nj

R aH

ns n nj ns ld

Z

J k RC D l

J k R a

I k RC D l

I k R a


R a J d H R a b

w R J r R a rdr C Z Z dz

λ λλ λ λ

λ−

∞

= +

+

⎧⎪ + =⎪ +⎪ +⎨⎪

+ >⎪+⎪⎩

+ − + +⋅ ⋅

+ − + +

+ =

∑ ∫

∫ ∫

( ) ( ) ( )

( )

( ) ( ) ( )

( )

( )( )( )

( ) ( )( )( ) ( )

( )( )

( )( )( )

( ) ( )( )( ) ( )( )

( )

4 5 4 50

4 5

5 4 5 5 4 5,0 0 ,

1

5

5 4 5, 25

1 1 1

4

/2

cosh /

sinh /

l lj

lj

nH H

ns l ns j l jd dj

Z Z

Hn j n njns j l jd

j j nj n njn jZ

Hnj

ld nj

R aD Z Z dz C Z Z dzR

K R J R R aD Z Z dz

R a JK R a

z H R aZ dz

d H R a

γ λλ λγ

λλ

− −

−

∞

=

∞ ∞

= = +

+

−⎛ ⎞ + +⎜ ⎟⎝ ⎠

++

+−

− +⋅ ⋅

− +

∑∫ ∫

∑ ∑∫

∫ ( ) ( )( )0

/R a

ns n njw R J r R a rdrλ+

+∫ (4.81b)

The continuity of the horizontal velocity ns rφ∂ ∂ on the boundary between

regions 4 and 5 ( )r R= in the case ( ) ( )2nj 1R a gω λ+ < , and the application

( )4H

ldZ dz∫ yield:


112

( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )( )( ) ( )

( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )( )( ) ( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( )

( )

4 4 4 4' '0 0 0 04 4

,0 ,04 40 0

4 4 4 4' '4 4, ,4 4

'4

2 21 1

for 0

for 0

/ cosh /2sinh /

/

n n

ns nsn n

l n l l n l

ns l ns l

n l n l

Hn nj njld

j n nj nj

ns n nj

k J k R k Y k RC D l

J k R a Y k R

k I k R k K k RC D l

I k R a K k R


J d H R a bR a

w R J r

λ λλ λ

λ

∞

= +

⎧⎪ + =⎪ +⎪ +⎨⎪

+ >⎪+⎪⎩

+ − + +⋅ ⋅

− + ++∑ ∫

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( )( )

( )( )( ) ( )

( ) ( )( )( ) ( )( )

( )

( )( )

( )( )( )( )

( ) ( ) ( )

( )

5 4 5 5 4 5,0 0 ,0 01

0

5 5 5 5' '5 4 5 5 4 5, ,5 5

1 1

'4

2 21 01

2

cosh /2 /sinh /

2

nR a

ns l ns ln

j n j j n j

ns j lj ns j ljj jn j n j

RHn nj nj

l ns n njdj n nj nj

n R anR a rdr C Z D ZR R

I R K RC Z D Z

I R K R a

J z d RZ dz w r J r R rdr

R J H d R

J

R a

γ γ γ γ

γ γ

λ λλ

λ λ

+− −

+

∞ ∞− −

= =

∞

= +

−+ = −

+ + +−

−⋅ ⋅

−

++

∫

∑ ∑

∑ ∫ ∫

( )( )( )

( ) ( )( )( ) ( )( )

( )

( ) ( )( )

'4

21 1

0

/ cosh /

sinh /

/

Hn nj njld

j n nj nj

R a

ns n nj

R R a z H R aZ dz

J d H R a

w R J r R a rdr

λ λλ λ

λ

∞

= +

+

+ − +⋅ ⋅

− +

+

∑ ∫

∫ (4.82a)

In the case ( ) ( )2nj 1R a gω λ+ ≥ , we have


113

( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )( )( ) ( )

( )( ) ( )( )

( ) ( )( )( )

( ) ( )( )( )( ) ( )

( )( )( )

( )( ) ( )( )( ) ( )( )

( )

( )

4 4 4 4' '0 0 0 04 4

,0 ,04 40 0

4 4 4 4' '4 4, ,4 4

'4

2 21 1

for 0

for 0

/ sinh /2cosh /

/

n n

ns nsn n

l n l l n l

ns l ns l

n l n l

Hn nj njld

j n nj nj

ns n nj

k J k R k Y k RC D l

J k R a Y k R

k I k R k K k RC D l

I k R a K k R


J d H R a bR a

w R J r

λ λλ λ

λ

∞

= +

⎧⎪ + =⎪ +⎪ +⎨⎪

+ >⎪+⎪⎩

+ − + +⋅ ⋅

− + ++∑ ∫

( )( ) ( ) ( ) ( ) ( ) ( )

( )( ) ( )( )

( )( )( ) ( )

( ) ( )( )( ) ( )( )

( )

( )( )

( )( )( )( )

( ) ( ) ( )

( )

5 4 5 5 4 5,0 0 ,0 01

0

5 5 5 5' '5 4 5 5 4 5, ,5 5

1 1

'4

2 21 01

2

cosh /2 /sinh /

2

nR a

ns l ns ln

j n j j n j

ns j lj ns j ljj jn j n j

RHn nj nj

l ns n njdj n nj nj

n R anR a rdr C Z D ZR R

I R K RC Z D Z

I R K R a

J z d RZ dz w r J r R rdr

R J H d R

J

R a

γ γ γ γ

γ γ

λ λλ

λ λ

+− −

+

∞ ∞− −

= =

∞

= +

−+ = −

+ + +−

−⋅ ⋅

−

++

∫

∑ ∑

∑ ∫ ∫

( )( )( )

( ) ( )( )( ) ( )( )

( )

( ) ( )( )

'4

21 1

0

/ cosh /

sinh /

/

Hn nj njld

j n nj nj

R a

ns n nj

R R a z H R aZ dz

J d H R a

w R J r R a rdr

λ λλ λ

λ

∞

= +

+

+ − +⋅ ⋅

− +

+

∑ ∫

∫ (4.82b)

With the above procedures, the continuity equations for both potential and

velocities have been satisfied. By solving the system of 8 sets of linear equations, we

can determine the 8 sets of unknown coefficients

( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 2 4 4 5 53, , , , , , , ,, , , , , , ,ns j ns j ns j ns j ns j ns j ns j ns jC C D C C D C D .

4.6 Solutions for diffraction potentials

The general solution for diffraction potential can be obtained by the following

equations:

( ) ( ) ( ) ( ) ( )( )( )

( ) ( )( )( ) ( )

1

1 1 1 1(1),0 0 , 1

1

,n

n j

Dn n n j jj n j

I rrr z C Z z C Z zR a I R a

γφ

γ

∞

=

⎛ ⎞= +⎜ ⎟−⎝ ⎠ −∑ (4.83)


114

( ) ( ) ( ) ( ) ( ) ( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( ) ( )( )

( )

2 2 2 2 2,0 0 ,0 0

2 22 2 2 2, ,2 2

1 1

,n n

Dn n n

n j n j

n j j n j jj jn j n j

r R ar z C Z D ZR a r

I r K rC Z D Z

I R a K R a

φ

γ γ

γ γ

∞ ∞

= =

−⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟+⎝ ⎠ ⎝ ⎠

++ −

∑ ∑ (4.84)

( ) ( ) ( )( ) ( )( )

( ) ( ) ( )( )( ) ( ) ( )

( )( )( ) ( )( )

( ) ( )

( ) ( ) ( )

2 3 303 3 3 3 3

,0 0 ,2 3 310

30

,n n j

Dn n n j jjn n j

In

H k r K k rr z C Z z C Z z

H k R a K k R a

r Z z H

φ

φ

∞

=

= ++ +

+

∑ (4.85)

( ) ( ) ( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

( )( )( )

( ) ( )( )( ) ( )

( )( )( )( )

( )

4 40 04 4 4 4 4

,0 0 ,0 04 40 0

4 44 4 4 4, ,4 4

1 1

,n n

Dn n n

n n

n j n j



J k R a Y k R

I k r K k rC Z D Z

I k R a K k R

φ

∞ ∞

= =

= + ++

++

∑ ∑ (4.86)

( ) ( ) ( ) ( ) ( ) ( )

( )( )( )( )( )

( ) ( )( )( )

( ) ( )( )( )

5 5 5 5 5,0 0 ,0 0

5 55 5 5 5, ,5 5

1 1

,n n

Dn n n

n j n j


r R ar z C Z D ZR r

I r K rC Z D Z

I R K R a

φ

γ γ

γ γ

∞ ∞

= =

−⎛ ⎞ ⎛ ⎞= + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+−

∑ ∑ (4.87)

where

1/ 20 ( )

coshn

In n nigAM i J kr

kHφ ε

ϖ= (4.88)

8 sets of unknown coefficients ( ) ( ) ( ) ( ) ( ) ( ) ( )( )1 2 2 4 4 5 53, , , , , , , ,, , , , , , ,n j n j n j n j n j n j n j n jC C D C C D C D can be

determined by applying the continuity equations for the potential and velocities on the

boundaries of each region with the same procedure as shown in part 4.5.

4.7 Equation of motion in modal coordinates

In order to derive the equations of motion in modal coordinates, we consider the

kinetic energy T , the strain energy U and the energy associated with the pressure V


115

( )22 2 2 2 2

0 0

12 12

R

m rhT h w rdrd

π

θγ ϖ ψ ψ θ⎧ ⎫= + +⎨ ⎬

⎩ ⎭∫ ∫ (4.89)

222

20 0

2 2 2

2 2

1 1{ [ 22

1 1] [ ]}2

Rr r

r r

rS r

U Dr r r r

w wr K Gh r rdrdr r r r

πθ θ

θθ θ

ψ ψψ ψν ψ ψθ θ

ψ ψν ψ ψ ψ θθ θ

∂ ∂∂ ∂ ⎛ ⎞ ⎛ ⎞⎛ ⎞= + ⋅ + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∂ ∂− ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞⋅ − − + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠

∫ ∫ (4.90)

( ) ( )

( ){ ( ) ( ) }2 21 5

0 0 0

2 2 22 5 4

0 0 0

( , ) ( , )

( , ) ( , ) ( , )

R a R

R a

R a R R a

RR a R a R

V p r wrdrd p r wrdrd

p r drd p r drd p r drd w

π π

π π π

θ θ θ θ

θ θ θ θ θ θ

−

−

+ +

− −

= − − −

⎡ ⎤− +⎢ ⎥⎣ ⎦

∫ ∫ ∫ ∫

∫ ∫ ∫ ∫ ∫ ∫ (4.91)

The Hamilton’s principle can be given as:

0=++− VUT δδδ (4.92)

By substituting ,w rψ and θψ which are represented by Eqs. (4.42)-(4.44), and

applying the Rayleigh-Ritz method (i.e. integration can be approximated by a linear

combination of elements), we obtain:


116

( )2

,,2, , , ,

0( 0,1) 0 01( 2

,,, , , ,

,, , , , ,2 2

{ } { [12

( ) ( )

1 1( ) ( ) (2

R RMr npr ns

ns m ns np r ns r np ns nps ns n

r npr nsnp r np ns r ns

nsns r ns np r np ns

hh w w rdr Dr r

n nr r r r

n n rr r

θ θ

θ θ

θθ θ θ

ψψζ γ ϖ ψ ψ ψ ψ

ψψν νψ ψ ψ ψ

ψνψ ψ ψ ψ ψ

= == ≥

∂⎡ ∂− + + + ⋅⎢ ∂ ∂⎣

∂∂+ ⋅ + + +

∂ ∂∂−+ + ⋅ + + −

∑ ∫ ∫

( )

( ) ( )

,

,, , , ,

12, ,2

0

5 22 2,

0

)

( )] [( ) ( )

1 ( )( )]} ] ( , )

( , ) ( ,

r ns

np npnsnp r np S r ns r np

R a

ns ns np np w ns np

R R

w ns np w ns np w np R nsR a

nr

wwr n K Ghr r r

nw r nw r rdr r H w rdrr

r H w rdr g w w rdr w r d

θθ

θ θ

ψ

ψψ ψ ψ ψ

ψ ψ γ ϖ φ

γ ϖ φ γ γ ϖ φ

−

−

+∂

∂ ∂∂⋅ − + + + ⋅ +∂ ∂ ∂

+ − + − + −

− + −

∫

∫ ∫

( ) ( )

( ) ( ) ( )

( ) ( )

5 42 2, ,

1 5 2,

0

5 4, ,

)

( , ) ( , )

( , ) ( , ) ( , )

( , ) ( , )

R a

R a

R R a

w np R ns w np R nsR a R

R a R R a

w Dn np w Dn np w np R DnR a R a

R R

w np R Dn w np R DnR a R

rdr

w r d rdr w r d rdr

i r H w rdr i r H w rdr i w r d rdr

i w r d rdr i w r d rdr

γ ϖ φ γ ϖ φ

γ ϖ φ γ ϖ φ γ ϖ φ

γ ϖ φ γ ϖ φ

+

−

+

−

− +

− −

+

−

+

⎤+ ⎥

⎦

= − − −

+ +

∫

∫ ∫

∫ ∫ ∫

∫a

∫

(4.93)

In view of the fact that normal modes satisfy Eqs. (4.11)-(4.18) and Eq. (4.20) , Eq.

(4.93) can be simplified to:

( )

( ) ( ) ( ) ( )

( )

12 2 2 2 2

0( 0,1) 01( 2)

5 2 5 4,

1

0 0

{ [ ( , )

( , ) ( ( , ) ( , ) ( , ) )]

} ( , )

R aM

ns m ps ns m ps w ns nps ns n

R R a R R a

ns np np R ns ns nsR a R a R a R

R R

w ns np w Dn np

hR hR r H w rdr

r H w rdr w r d rdr r d rdr r d rdr

g w w rdr i r H w rdr

ζ ϖ γ δ ϖ γ δ ϖ γ φ

φ φ φ φ

γ γ ϖ φ

−

= == ≥

+ +

− − −

−

− + − +

+ − −

+ = −

∑ ∫

∫ ∫ ∫ ∫

∫ ( )

( ) ( ) ( )

5

2 5 4,

( , )

( , ) ( , ) ( , )

a R

Dn npR a

R a R R a

np R Dn Dn DnR a R a R

r H w rdr

w r d rdr r d rdr r d rdr

φ

φ φ φ

−

+ +

− −

⎡+⎢

⎣⎤⎛ ⎞

+ − − ⎥⎜ ⎟⎥⎠⎝ ⎦

∫ ∫

∫ ∫ ∫

(4.94)

where the normalization of the modal vectors is made such that


117

( ) ps

R

npnsnprnsrnpns Rrdrww δψψψψτθθ

2

0,,,,

2

12=

⎭⎬⎫

⎩⎨⎧

++∫ (4.95)

and nsϖ represents the natural frequency. Equation (4.94) may be represented in a non-

dimensional form by:

( )

( ) ( ) ( ) ( )

( )

2 3 212 2

0( 0,1) 01( 2)

5 2 5 4,

1

0

{ [ ( , )

( , ) ( ( , ) ( , ) ( , ) )]

} [ ( , )

R aMm

ns ps ns ps ns nps n ws n

R R a R R a

ns np np R ns ns nsR a R a R a R

R

ns np Dn np

R S R r H w rdrg g

r H w rdr w r d rdr r d rdr r d rdr

iw w rdr r H w rdg

γϖ ϖζ τ δ λ δ φγ

φ φ φ φ

ϖ φ

−

= == ≥

+ +

− − −

⎛ ⎞− + −⎜ ⎟

⎝ ⎠

+ + − −

+ = −

∑ ∫

∫ ∫ ∫ ∫

∫ ( )

( ) ( ) ( )

5

0

2 5 4,

( , )

( ( , ) ( , ) ( , ) )]

R a R

Dn npR a

R a R R a

np R Dn Dn DnR a R a R

r r H w rdr

w r d rdr r d rdr r d rdr

φ

φ φ φ

−

−

+ +

− −

+

+ − −

∫ ∫

∫ ∫ ∫

(4.96)

where 4/( ), ,wS D gR h R r Rγ τ χ= = = . Note that Eq. (4.96) can be solved separately

for each n (number of nodal diameters or Fourier modes). Also, the frequency

parameters nsλ and the corresponding mode shapes nsw are available in the literature

(e.g. Wang et al. 2004).

4.8 Results and discussions

Consider the circular VLFS and sea state condition treated by Watanabe et al. (2006).

The circular VLFS has a radius R = 200m, thickness h = 2m, density ratio

0.25m wγ γ = , non-dimensional rigidity S = 0.000433, Poisson’s ratio 0.3υ = , and

shear correction factor SK = 5/6. It is assumed that the water depth H = 20m and the

incident wave length λ = 50m. For numerical calculation, we also consider the number

of nodal diameters N = 14, the number of sequence M = 5 for given N as recommended

by Watanabe et al. (2006). In their study, they used the truncation number for infinite


118

sums Γ =20. However, for the circular VLFS with an attached submerged annular

plate considered herein, the truncation number Γ may not be the same and it needs to

be established from a convergence study on the results.

In order to study the sensitivity of the width of the submerged horizontal plate

on the hydroelastic response of the VLFS, we analyze the circular VLFS without and

with a submerged horizontal annular plate having various widths ranging from 2m to

10m (a = 1~5m) and positioned at a depth d = 15m. The convergence check for the

truncation number Γ is done for two cases such as a VLFS with the smallest and

largest widths of the submerged annular plate. The convergence studies in terms of the

truncation of infinite sums for these two cases are shown in Figs. 4.2 and 4.3,

respectively, for Γ = 20, 40 and 60. It can be seen clearly that Γ = 40 will suffice for

converged results. So we have chosen the truncation number Γ = 40 for the subsequent

calculations. Figures 4.4 to 4.7 show the deflections and stress-resultants of a VLFS

without and with the submerged annular plate having various widths ranging from 2m

to 10m (a = 1~5m) and positioned at a depth d = 15m. We can see that as the width of

the submerged plate increases from 2m to 8m, the deflection and stress-resultants in

the VLFS decrease as compared with its counterpart without an attached submerged

plate. However when the width of the plate increases to 10m, the deflection and the

twisting moment near the fore end of the VLFS increase. From these results, the

effectiveness of the submerged horizontal plate in reducing the deflection and stress-

resultants is shown to be significant when the plate is not too wide. The submerged

annular plate with 8m width gives the best reduction.


119

Fig. 4.2 Nondimensionalized deflection amplitude for a circular VLFS with 2m width submerged plate

Fig. 4.3 Nondimensionalized deflection amplitude for a circular VLFS with 10m width submerged plate

Incoming wave

Incoming wave


120

Fig. 4.4 Nondimensionalized deflection amplitude of a circular VLFS without and with submerged plate of various widths

Fig. 4.5 Nondimensionalized bending moment amplitude of a circular VLFS without and with submerged plate of various widths

Incoming wave

Incoming wave


121

Fig. 4.6 Nondimensionalized twisting moment amplitude of a circular VLFS without and with submerged plate of various widths

Fig. 4.7 Nondimensionalized transverse shear force amplitude of a circular VLFS without and with submerged plate of various widths

In order to study the sensitivity of the level of the submerged plate, we analyze

the circular VLFS without and with submerged horizontal plate having a width of 8m.

The submerged plate is placed at various depths d = 2m to 19.5m. The convergence

Incoming wave

Incoming wave


122

studies for the required truncation number Γ are done for two cases, i.e. VLFS with

the submerged plate placed at the highest and lowest levels. The convergence

characteristics in terms of the truncation of infinite sums for these two cases are shown

in Fig. 4.8 and 4.9, respectively. It can be seen that Γ = 40 will suffice for converged

results for the case d = 19.5m whereas Γ = 60 is necessary for the case d = 2m. So we

have chosen the truncation number Γ = 60 for all calculations in this section. Figures

4.10 to 4.13 show the deflections and stress-resultants. Note that the free-edge

boundary conditions for the stress-resultants (Figs. 4.5 to 4.7 and 4.11 to 4.13) are

exactly satisfied because of the employment of the analytical Mindlin plate solutions.

It is generally difficult to observe the satisfaction of the natural boundary conditions

when using numerical methods based on the work-energy approach (see Wang et al.

2004 for a discussion on this matter). As the submerged plate moves nearer to the

bottom surface of the VLFS, its effectiveness becomes more apparent. However when

it moves too close to the VLFS (as the case for d = 19.5m), the motion of VLFS is

magnified near the fore-end. It may be due to the fact that the incoming wave makes

the submerged plate vibrate more and this motion gets transferred to the fore-end of

VLFS. We can see that the submerged annular plate with an 8m width and placed at d

= 18m seems to give the best reduction.


123

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x/R

Non

-dim

ensi

onal

ized

def

lect

ion

|w(x

)/A|

Γ = 20Γ = 40Γ = 60Γ = 80

Fig. 4.8 Nondimensionalized deflection amplitude for a circular VLFS

with 8m width submerged plate at level d = 2m

Fig. 4.9 Nondimensionalized deflection amplitude for a circular VLFS with 8m width submerged plate at level d = 19.5m

Incoming wave

Incoming wave


124

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

x/R

Non

-dim

ensi

onal

ized

def

lect

ion

|w(x

)/A|

Without attached plate2a=8m, d=2m2a=8m, d=4m2a=8m, d=8m2a=8m, d=12m2a=8m, d=16m2a=8m, d=18m2a=8m, d=19.5m

Fig. 4.10 Nondimensionalized deflection amplitude of a circular VLFS without and with submerged plate at various depths

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

5

10

15

20

x/R

Non

-dim

ensi

onal

ized

ra

dial

ben

ding

mom

ent |

MrR

/(DA

)|


Fig. 4.11 Nondimensionalized bending moment amplitude of a circular VLFS without and with submerged plate at various depths

Incoming wave

Incoming wave


125

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

2

4

6

8

10

x/R

Non

-dim

ensi

onal

ized

tw

istin

g m

omen

t |M

r θR

/(DA

)|


Fig. 4.12 Nondimensionalized twisting moment amplitude of a circular VLFS without and with submerged plate at various depths

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

r/R

Non

-dim

ensi

onal

ized

she

ar fo

rce

|QrR

2 /(DA

)|


Fig. 4.13 Nondimensionalized shear force amplitude of a circular VLFS without and with submerged plate at various depths

In order to study the effectiveness of the submerged plate on the hydroelastic

response of a VLFS under other wave conditions, we analyze the VLFS without and

with a submerged plate of 8m width and placed at d = 18m and consider the variation

Incoming wave

Incoming wave


126

of wavelengths λ from R/25 to infinity (R/λ -> 0). The convergence behaviors of the

results in terms of the truncation of infinite sums for VLFS without and with

submerged plate are shown in Fig. 4.14 and 4.15, respectively. In both cases, shorter

waves require larger truncation numbers for convergence. It can be seen that Γ = 40 is

adequate for converged results of the basic VLFS while Γ = 60 is necessary for

converged results of VLFS with a submerged plate.

Figures 4.16 to 4.18 show the vertical deflections at the fore-end, mid-position

and back-end of the VLFS without and with horizontal submerged annular plate of 8m

width and placed at d = 18m as a function of R/λ. As it can be seen from these figures,

the effect of the submerged annular plate in reducing the hydroelastic response is

clearly observed.

Fig. 4.14 Nondimensionalized deflection amplitude for a basic circular VLFS at fore-end

Incoming wave


127

0 5 10 15 20 250

0.2

0.4

0.6

0.8

1

1.2

1.4

R/λ

Non

-dim

ensi

onal

ized

def

lect

ion

|w(x

)/A|

Γ = 40Γ = 60

Γ = 80

Fig. 4.15 Nondimensionalized deflection amplitude for a circular VLFS with submerged plate at fore-end

Fig. 4.16 Nondimensionalized deflection amplitude of a circular VLFS at fore-end

Incoming wave

Incoming wave


128

Fig. 4.17 Nondimensionalized deflection amplitude of a circular VLFS at mid-position

Fig. 4.18 Nondimensionalized deflection amplitude of a circular VLFS at back-end


In this chapter, the hydroelastic problem of a circular VLFS with a submerged

horizontal annular plate strip attached to its perimeter is analyzed in an exact manner

Incoming wave

Incoming wave


129

for both the structure part and the fluid part. Although the formulations are given in

explicit formula, we have truncated the infinite sums. However, by using a suitable

truncation number Γ, one may still obtain very accurate converged results. The

significant reduction effect of the response with the attachment of the horizontal

submerged annular plate has been confirmed by the analysis.

It is important to note that the Mindlin plate theory has been adopted instead of

the commonly used classical thin plate theory in order to obtain more accurate stress-

resultants. Therefore the presented accurate hydroelastic responses should be useful to

VLFS engineers as benchmark solutions for verification of the numerical programs

based on the boundary element method and the finite element method.

130

CHAPTER 5

MINIMIZING HEAVING MOTION OF LONGISH

VLFS USING ANTI-HEAVING DEVICES

5.1 Introduction

In the previous chapter, we have investigated the effectiveness of a simple anti-heaving

device which is a submerged horizontal plate in reducing the heaving motion of VLFS

by applying it to a circular VLFS. However, there are many other proposed anti-

heaving devices for VLFS. For example, Takagi et al. (2000) proposed a box-shaped

anti-motion device (see Fig. 5.1) and investigated its performance both theoretically

and experimentally. They found that this device reduces not only the deformation but

also the moment and shearing force of the platform. The motion of VLFS with this

device is reduced in both beam-sea and oblique sea conditions. However, under action

of strong wave, this device needs to be relatively large and hence would be costly.

WaveVLFS

Box-shaped anti-motion device

Fig. 5.1 VLFS with box-shaped anti-motion device

131

VLFS Wave

Submerged vertical plate anti-motion device

Fig. 5.2 VLFS with submerged vertical plate anti-motion device

Ohta et al. (1999), on the other hand, proposed a submerged vertical single

plate attached to the fore-end of VLFS as shown in Fig. 5.2. They investigated the

performance of such anti-motion device by conducting some experiments in a water

basin. The experimental results showed that the displacement of VLFS with this

submerged vertical plate is reduced significantly at the edges and in the inner parts as

well.

Continuing in this line of investigation, Ohta et al. (2002) considered other

anti-motion body forms such as an L-shaped plate, a reverse-L-shaped plate and a

beach-type plate (see Fig 5.3). They concluded that L-shaped plate is more effective

against long waves whereas beach-type plates and reverse-L-shaped plates are more

effective against short waves.

WaveVLFS

L-shaped plate anti-motion device

(a) VLFS with L-shaped plate anti-motion device

132

WaveVLFS

Reverse-L-shaped plate anti-motion device

(b) VLFS with reverse-L-shaped plate anti-motion device

WaveVLFS

Beach typeanti-motion device

(c) VLFS with beach type anti-heaving device

Fig. 5.3 VLFS with anti-motion devices treated by Ohta et al. (2002)

In this chapter, we wish to investigate a new plate attachment configuration in

the form of a submerged inclined plate for mitigating heaving motion and to assess its

performance with respect to the other types of proposed anti-heaving devices such as a

horizontal plate, a vertical plate and an L-shaped plate. Different dimensions and

submerged depths of these plated attachments were considered to better understand

their behavior. The investigation is carried out by performing experiments on a longish

VLFS model attached with the aforementioned motion devices. Under different wave

periods, the heaving displacements were measured for the VLFS model with various

plate attachments and compared in order to establish the best among the considered

four anti-heaving devices.

133

5.2 Experimental facility and instrumentation

5.2.1 Wave tank

The experimental study was carried out in the two dimensional wave tank as shown in

Fig. 5.4 in the Hydraulic Engineering Laboratory of the National University of

Singapore. The experimental setup for the present study is depicted in Fig 5.5. The

wave tank measures 36m in length, 2m width and 1.3m depth. To meet the size of the

testing model, the tank was partitioned to 0.8m width. The tank has a smooth rigid

horizontal concrete base and side walls. Glass panels of about 20mm thickness are

provided on both sides of the tank for about one eighth length of the tank and placed in

the middle of the tank to facilitate photographic and video recordings and observation

during the test. A spending beach provided at the other end with an average slop of 1:6

made of gravel absorbs the wave energy.

Fig. 5.4 Wave tank in Hydraulic Engineering Laboratory, NUS

134

Wave direction

(b) Plan view

(a) Longitudinal section

36m

2m

0.8m

0.5m

Partition wall

Model of VLFS

16.38m0.5m0.5m12.38m4m

Waveprobe

Anti-motiondevice 0.

95 m

Beach

Model of VLFS

Wavemaker

Waveprobe Potentiometers

= 2.44m2b

Fig. 5.5 Experimental setup

135

5.2.2 Wave generating system

The wave generating system consists of a servo controlled piston type wave maker

fixed at one end of the wave tank. This provides a facility to generate both

monochromatic and random waves according to the input signal.

The wave maker paddle motion was controlled by the input signal received

through the servo controller. The motion of the paddle generates a wave of certain

amplitude and frequency corresponding to the input signal. The frequencies of input

signal are the frequencies of the generated waves. However, the amplitudes of the

input signal are not the amplitudes of the generated waves. So the relationship between

the input signal and the actual values of amplitudes of the generated wave has to be

determined a priori. This relationship was carried out without the model in the channel.

At each frequency, different values of input signal amplitude ranging from 50 – 500

ml/PP were sent to the wave maker and the amplitudes of generated waves at the test

section in the wave tank were measured by wave probes to generate the desired wave

heights.

5.2.3 Data acquisition system

The instrumentations for wave generating system are shown in Fig. 5.6. Frequency and

amplitude signals are set in the HP 33120A function generator as shown in Fig. 5.6a

and are then transferred through the servo controller as shown in Fig. 5.6b, to the wave

paddle to generate the waves.

136

(a) HP 33120A function generator

(b) Servo controller

Fig. 5.6 Wave generating system controller

The time histories of water surface elevation were measured using three wave

probes as shown in Fig. 5.5. The wave probes Kenek CH-4, which includes insulated

coating wire as a conductor along its length, were used. By putting the wave probe into

water and running electric current, a kind of condenser between water and conductor is

formed. Its static electrical capacity is proportional to the water depth. Figure 5.7

shows the instrumentations for wave data acquisition. Each Kenek wave probe was

connected to a Kenek controller (on the right hand side of Fig. 5.7) to set the range of

measurement. Then, all the data is transferred to a personal computer through an

amplifier (on the right hand side of Fig. 5.7).

137

Fig. 5.7 Wave data acquisition system

In order to measure the vertical displacements of the VLFS model, 13 linear

motion potentiometers were installed along the center line of the model at a spacing of

0.2m. The deflection of VLFS is collected using deflection data acquisition system as

shown in Fig. 5.8. On the right hand side is the power supply and the other is the

oscilloscope. Each potentiometer was connected to both the power supply and

oscilloscope. We used potentiometer type with measure range of 10cm and set the

power supply at 10V so that the collected data was in cm unit.

Fig. 5.8 Deflection data acquisition system

138

5.2.4 Testing condition

For regular waves, the wave period is varied from 5s to 10s and the wave height from

0.5m to 1.5m, thereby simulating varying sea conditions. By using a scale of 1:50, the

testing values of wave height range from 1cm to 3cm. The testing values of the wave

period range from 0.8s to 1.3s. The values of wave period were obtained by using the

same Froude number of the testing VLFS model and the real VLFS.

5.2.5 VLFS model

Consider a longish VLFS of length 122m, width 25m and height 3m. By using a scale

of 1:50, we fabricated a model for the VLFS of length 2.44m, width 0.5m, and height

0.06m by pasting a 57mm thick polyethylene foam (specific gravity = 0.019) to the

bottom of a 3mm thick aluminum plate (Young’s modulus = 70GPa, specific gravity =

2.3). Figure 5.9 shows the experimental model. At the fore-end of the model, there are

8 ready bolts installed to attach the anti-heaving device. The floating model was

allowed to move freely in the vertical direction but was restrained from moving in the

horizontal plane by a mooring system consisting of 4 angle plates placed at the 4

corners of the plate model. Oil was put on surface of angle plates so that the friction

between the corner of VLFS and angle plate is negligible.

139

Fig. 5.9 Experimental model placed in wave tank

Table 5.1 gives the pertinent information of the model. The draft of the model

was calculated from the specific gravities of the materials used and validated with the

actual draft value measured from the experiment. The flexural rigidity EI was

calculated from a simple test as shown in Fig. 5.10 which simulates a cantilever beam

under its own weight. This simple test was repeated three times with different

cantilever lengths to check the repeatability of the calculated EI. We measured the

deflection at the free end of the cantilever then used this value to calculate the flexural

rigidity EI of the model by using the deflection relation given by Ye (2008)

4

8sq awEI

= (5.1)

where w is the measured deflection at the free end of the cantilever, qs the self weight

of the model, and a is the cantilever length.

VLFS Model

Ready bolts Angle plate

Potentiometer

Holding system

140

Table 5.1 Pertinent information of model

Full scale 1:50 Model

Length (m) 122 2.44

Width (m) 25 0.5

Depth (m) 3 0.06

Draft (m) 0.8 0.016

EI per unit width (Nm) 1.17x109 185

Fig. 5.10 Simple bending test to determine flexural rigidity EI of VLFS model

5.2.6 Types of anti-heaving devices

Table 5.2 shows the four types of anti-heaving devices tested, namely a submerged

horizontal plate, a vertical plate, an L-shaped plate and an inclined plate, respectively.

For each type of anti-heaving device, we considered four cases as shown in Table 5.2.

All devices were fabricated from polypropylene plate of 5mm thick and possessing a

specific gravity = 0.9. Note that, the submerged inclined plated proposed herein differs

from the beach type anti-motion device proposed by Ohta et al. (2002) (Fig. 5.3). In

that the latter device uses more materials and is tapered as can be seen in Fig. 5.3.

VLFS model

141

The connection of the anti-heaving device to the fore-end of the model was

made possible by using two jigs made of aluminum (2.5cm width and 3mm thickness)

and 8 bolts on top surface of the fore-end (see Fig. 5.9). The anti-heaving device was

bolted to the jigs.

Table 5.2 Anti-heaving devices considered in the experiment

Type of anti-heaving device Case No.

Dimensions and submerged level

Case A1

B = 10cm and dd = 10cm

Case A2


Case A3


(A) Submerged horizontal plate

B (10, 20cm)

WaveAttaching jigsModel

Attachedhorizontal plate

H=9

5cmdd

(10,

20c

m)

Case A4


Case B1

dd = 10cm without slits

Case B2

dd = 10cm with slits of diameter 1cm and spacing 5cm

Case B3

dd = 20cm without slits

(B) Submerged vertical plate

WaveModel

Attachedvertical plate

(with and without slits)

dd (1

0, 2

0cm

)

H=9

5cm

Case B4

dd = 20cm with slits of diameter 1cm and spacing 5cm

Case C1


Case C2


Case C3


(C) Submerged L-shaped plate

Attached L-shaped device

B (5, 10cm)

WaveModel

H=9

5cmdd

(10,

20c

m)

Case C4


142

Case D1

B = 10cm without slits

Case D2

B = 10cm with slits of diameter 1cm and spacing 5cm

Case D3

B = 20cm without slits

(D) Submerged inclined plate (α = 45o)

Attached inclined plateB = 10, 20cm

(with and without slits)

Wave

α

Model

H=9

5cm

Case D4

B = 20cm with slits of diameter 1cm and spacing 5cm

5.3 Experimental procedure

The VLFS model and the various anti-motion devices were fabricated in the Hydraulic

Laboratory. On the top surface of the model, 13 nuts were fixed at an equal spacing of

0.2m. A holding system was prepared near the middle and on top of the wave tank in

order to fix the model and potentiometers. 3 wave probes were positioned in the wave

tank to measure the wave surface elevation.

The model was placed in the wave tank and restrained from horizontal

movement by 4 angle plates each at a corner of VLFS from the holding system. Finally

all 13 potentiometers were connected on top of the model by fixing each tip of the

potentiometer to 1 nut on top of the model and the other end of the potentiometer to

the holding system.

The VLFS model without anti-heaving device was tested first. Then one by one

anti-heaving device was connected to the for-end of the VLFS model and tested. Each

test was repeated three times to check the repeatability of the collected data and to

compute the statistical characteristics of the data. Each model was tested with various

wave period conditions ranging from 0.8s to 1.3s.

143

The time history of water surface elevation was sampled at a rate 100 data pts/s.

The time history of deflection of VLFS at each collected point was sampled at a rate

1600 data pts/s. To measure the time history of the deflection of VLFS under action of

incoming waves without the effect of the reflected waves from the end of the wave

tank, data were collected from the time the waves arrived at the fore-end of the model

up to the time when the reflected waves returned to the aft-end of the model. This time

period includes the steady-state response. The data collected were analyzed and only

the data in the steady-state response were used to study the effectiveness of anti-

heaving devices.

5.4 Experimental results

5.4.1 VLFS with submerged horizontal plate

Figures 5.11 to 5.16 show the vertical response distribution of VLFS without and with

the four cases (A1, A2, A3 and A4) of submerged horizontal plate anti-heaving device

mentioned in Section 5.2.6 for various wave periods ranging from 0.8s to 1.3s,

respectively. The ordinate represents the dimensionless values obtained by dividing the

amplitude w of vertical displacement on the center line of the model by the amplitude

A of the incident wave, while the abscissa represents the position of measuring points

in the longitudinal direction of the model. The wave propagates from right to left.

We can see that all these four anti-heaving devices are effective in reducing the

deflection near the aft-end of the model with more significant for longer anti-heaving

devices A2 and A4 (B = 20cm). According to Ohkusu and Nanba (1996), the motion of

VLFS is proportional to the propagation of waves beneath it. That means all these four

devices are effective in reducing the wave so that the obtained deflection is smaller

144

when compared to that of the basic VLFS. The longer plates have been observed to

provide better reduction effect near the aft-end and this tendency was also presented in

the experimental results by Ohta et al. (1999). However, near the fore-end of the model,

only anti-heaving devices placed at level dd = 10cm (i.e. Cases A1 and A2) are

effective in reducing the deflection while anti-heaving devices placed at level dd =

20cm (i.e. Cases A3 and A4) magnify the deflection instead since these devices placed

at a deeper level have a negative damping effect on the fore-end of the model. Among

these heaving devices, Case A2 seems to be the best device in reducing the deflection

of VLFS.

Fig. 5.11 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 0.8s

Incoming wave

145


Fig. 5.13 Nondimensionalized deflection amplitude of VLFS without and with horizontal plate at wave period 1s

Incoming wave

Incoming wave

146



Incoming wave

Incoming wave

147


5.4.2 VLFS with vertical plate


four cases (B1, B2, B3 and B4) of vertical plate mentioned in Section 5.2.6 for various

wave periods ranging from 0.8s to 1.3s, respectively. We can see clearly that all these

four devices are effective in reducing the displacement along the model. Apparently,

plates installed at a greater depth dd = 20cm (i.e. Cases B3 and B4) are more effective.

When the wave period is longer, the reductions become less significant since the wave

energy passes through the model under the bottom of the vertical plate. The tendency

of increasing heaving reduction along the model as the installation depth of vertical

plate increases and the waves become shorter was also observed by Ohta et al. (1999).

For a shorter plate dd = 10cm, plate with slits (i.e. Case B2) performs slightly better as

compared to the plate without slits (i.e. Case B1) whereas for the longer plate dd =

Incoming wave

148

20cm, the plate without slits (i.e. Case B3) is better than that with slits (i.e. Case B4).

Among these 4 devices, Case B3 provides the largest reduction in heaving motion.

Fig. 5.17 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 0.8s


Incoming wave

Incoming wave

149

Fig. 5.19 Nondimensionalized deflection amplitude of VLFS without and with vertical plate at wave period 1s


Incoming wave

Incoming wave

150



Incoming wave

Incoming wave

151

5.4.3 VLFS with L-shaped plate


four cases (C1, C2, C3 and C4) of L-shaped plate for various wave periods from 0.8s

to 1.3s, respectively. The maximum deflection of VLFS always occurs near the ends of

VLFS. We can see that all these four devices are effective in reducing the displacement

along the model. Apparently, the smallest L-shaped plate (i.e. Case C1) is the least

effective. For a shorter wave (i.e. wave period < 1.1s), Case C2 is the most effective in

reducing the deflection near the fore-end whereas Case C3 is the most effective in

reducing the deflection near the aft-end. As the wave period becomes longer (i.e. wave

period ≥ 1.1s), the effectiveness of the largest L-shaped device (i.e Case C4) becomes

dominant along the model. Ohta et al. (2002) pointed out that L-shape device is

effective in reducing the heaving motion of VLFS especially under action of long

waves. We can see that our results also follow this tendency especially for Case C4

(with the largest L-shaped plate).

152

Fig. 5.23 Nondimensionalized deflection amplitude of VLFS without and

with L-shaped plate at wave period 0.8s

Fig. 5.24 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 0.9s

Incoming wave

Incoming wave

153

Fig. 5.25 Nondimensionalized deflection amplitude of VLFS without and with L-shaped plate at wave period 1s


Incoming wave

Incoming wave

154



Incoming wave

Incoming wave

155

5.4.4 VLFS with inclined plate


four cases (D1, D2, D3 and D4) of inclined plate mentioned in Section 5.2.6 for

various wave periods from 0.8s to 1.3s, respectively. We can see that all these 4

devices are effective in reducing the displacement along the model. The maximum

deflection of VLFS with shorter inclined plates (i.e. Cases D1 and D2) occurs at fore-

end or aft-end of the model while that of VLFS with longer inclined plates (i.e. Cases

D3 and D4) occurs only at the aft-end of the model since the longer inclined plates

provide more damping effect on the fore-end of the model. The damping effect of the

slits can be seen clearly in the longer plate at the fore-end. As the wave period

becomes longer (i.e. wave period ≥ 1.1s), the effectiveness of the longer plates

becomes dominant along the model. For shorter wave (i.e. wave period <1.1s), the

effectiveness of shorter plates becomes dominant especially at the aft-end of the model.

Fig. 5.29 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 0.8s

Incoming wave

156


Fig. 5.31 Nondimensionalized deflection amplitude of VLFS without and with inclined plate at wave period 1s

Incoming wave

Incoming wave

157



Incoming wave

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158


5.5 Most effective anti-heaving device

In order to find the most effective anti-heaving device, we will compare the

performances of the 4 types of aforementioned anti-heaving device using the same

amount of materials. For the same length of 20cm, we consider the submerged

horizontal plate (Case A2), submerged vertical plate (Case B3), submerged L-shaped

plate (Case C2), and submerged inclined plate (Case D4). Figures 5.35 to 5.40 show

the vertical response distribution of VLFS without and with an anti-heaving device for

various wave periods from 0.8s to 1.3s, respectively. As the wave period becomes

longer (i.e. wave period ≥ 1.2s), the effectiveness of the inclined plate become

dominant along the model. For a shorter wave, the inclined plate is the most effective

device in reducing the deflection near the fore-end while the vertical plate is the most

effective device in reducing the deflection near the aft-end. Therefore, in general, the

Incoming wave

159

inclined plate may be regarded as the most effective device among the four considered

designs for this VLFS model.

Fig. 5.35 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 0.8s


Incoming wave

Incoming wave

160

Fig. 5.37 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 20cm length at wave period 1s


Incoming wave

Incoming wave

161



For the same length of 10cm, we consider submerged horizontal plate (Case A1),

submerged vertical plate (Case B2) and submerged inclined plate (Case D1). We also

Incoming wave

Incoming wave

162

make comparison with a submerged L-shaped plate (Case C1). Figures 5.41 to 5.46

show the vertical response distribution of VLFS without and with an anti-heaving

device for various wave periods from 0.8s to 1.3s, respectively. We can see clearly that

inclined plate is the most effective device in reducing the deflection along the VLFS.

Even when compared with an attached L-shaped device that uses more materials, the

inclined plate is still more effective. Therefore, we may deduce that the inclined plate

is the most effective anti-heaving device for such a longish floating structure.


Incoming wave

163


Fig. 5.43 Nondimensionalized deflection amplitude of VLFS without and with anti-heaving device of 10cm length at wave period 1s

Incoming wave

Incoming wave

164



Incoming wave

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165



In this chapter, we have proposed a submerged inclined plate attached to the fore-end

of VLFS to mitigate the heaving motion. The performance of this anti-heaving device

was tested by attaching it to the fore-end of a longish flexible floating structure. In

order to ascertain the effectiveness of the submerged inclined plate, we compare its

performance against three other types of anti-heaving devices (that previous

researchers have considered), namely the submerged horizontal plate, the vertical plate

and the L-shaped plate.

The effectiveness of all these devices in reducing the heaving motion has been

confirmed by the test results. Among these devices, the newly proposed submerged

inclined plate attachment is the most effective. The reductions of the deflection of the

longish floating structure at the fore-end and aft-end when using this device are noted

to be very significant. Therefore, by using this device, it would be possible to eliminate

Incoming wave

166

the need for an expensive breakwater. In this study, we have only used an inclined

plate of 45o. There should exist an optimal angle of inclination that maximizes the

reduction of the heaving motion of the longish floating structure and future studies

should consider developing a simulation model to seek such an optimal design of

submerged inclined plate attachment.

167

CHAPTER 6

CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE WORK

The main objective of this thesis is to minimize the differential deflection and heaving

motion of Very Large Floating Structure (VLFS). This study was prompted by the fact

that for certain applications of VLFS, there is a need to keep the floating platform calm

and flat so that the uneven heavy loads and wave forces do not affect the smooth operation

of certain equipment on VLFS as well as the structural integrity of VLFS. For example, in

the application of VLFS as a floating container terminal, the larger loading at the stacking

yard in the middle portion of the VLFS causes severe differential deflection that violates

the tight tolerances of the quay crane rails gradient and prevents the smooth operation of

the quay cranes. Other applications of VLFS, such as floating dormitories or hotels,

require minimum heaving, sway and pitching motions.

The innovative solution of using gill cells for minimizing the differential

deflection of VLFS under unevenly heavy load was studied in the first part of the thesis

(see Chapters 2 and 3). The second part investigates the innovative solutions of submerged

plates for minimizing the heaving motion of VLFS under wave action (see Chapters 4 and

5). This chapter summarizes the work and pertinent findings and discusses several

directions for future work in these aforementioned areas.

Conclusions and Recommendations for Future Work

168

6.1 Conclusions

The significant findings in this thesis can be summarized as follows:

1. The optimal designs of gill cells with the view to reduce the differential deflection

and bending stresses in circular VLFSs were analyzed analytically. The VLFS was

modeled as a classical thin (Kirchhoff) plate for the bending analysis and the

optimization exercise was performed using classical optimization algorithms based

on Newton’s method. The study considered a number of design parameters such as

loading magnitudes, loading radius, thickness of top and bottom plates and free

board at the edge of VLFS. The results show that optimal gill cells are very

effective in reducing the maximum differential deflection, bending moments and

shear forces. The percentage reductions in the maximum differential deflection,

bending moments and shear force can be up to 45%, 45% and 55%, respectively,

for the example considered. The optimal design of gill cells is most effective when

the loading radius is bound by 0.5R to 0.75R and the loading magnitude is not too

large. When the thicknesses of the top and bottom plates and the freeboard height

of the circular VLFS are increased, the optimal design of gill cells is more

effective in reducing the maximum shear force but is comparatively less effective

in reducing the maximum differential deflection and bending moments.

2. Also investigated herein is the effect of using stepped plate VLFS in reducing the

differential deflection of VLFS under heavy loads in pontoon-type, circular VLFS

under a central uniform partial load. The results show that this solution is also

effective in reducing the differential deflection in VLFS. By comparing with the


169

solution of using gill cells, the results indicate that the solution of using gill cells is

more effective than using stepped plate. This may be attributed to the fact that by

using gill cells, the buoyancy forces are eliminated and this allows uneven

buoyancy forces acting at the bottom hull of the VLFS to somewhat

counterbalance the applied uneven loading. These findings are of crucial

importance in enhancing our understanding on the effectiveness of some possible

solutions in reducing the differential deflection in VLFS and suggesting the best

solution to use in the offshore industry.

3. In addition to circular VLFS, the optimal designs of gill cells with the view to

reducing the slope and bending stresses in the annular VLFS were analyzed by

using a hybrid method of classical optimization algorithms based on Newton’s

method and genetic algorithm. The obtained results show that optimal design of

gill cells in annular VLFS can reduce the maximum slope, shear force and bending

moment up to 59%, 12% and 65%, respectively for the example considered. These

reductions are significant and clearly show the effectiveness of gill cells in

reducing the slope and stress-resultants. Together with the analysis of gill cells in

circular VLFS mentioned above, the findings provide useful benchmark solutions

for designing circular and annular VLFS with gill cells.

4. The optimal design of gill cells and its effectiveness in non-circular VLFS were

also studied by developing an appropriate FEM model for VLFS bending analysis

and using genetic algorithms as an optimization tool. The modeling and

optimization technique were illustrated with examples considering square,

rectangular and I-shape VLFSs. The results show that, with optimal gill cells, the

differential deflection is significantly reduced and the VLFS deflection shape is


170

significantly flatter over the entire of VLFS domain. However, for VLFSs with

reentrant corners, reinforcement may be necessary at these corners because of

increase in the maximum von Mises stress in these regions. This is due to the fact

that by using gill cells, the curvature of VLFS is changed. With optimal gill cells,

the curvature of VLFS is much flatter that makes the differential deflection reduce

and deflection shape flatter. At the reentrant corners, the curvature changes

abruptly and makes the bending stresses increase. The FEM model and

optimization technique developed herein can be applied to any shape of floating

structure under any loading configuration to minimize the differential deflection.

In the offshore industry, this model can be employed for the preliminary design

stage of arbitrarily shaped VLFS.

5. Also developed herein is an analytical model to investigate the submerged plate

anti-heaving device for minimizing the motion of circular VLFS under wave

action. This is a precise model considering rigorously the configuration of the

submerged horizontal plate within the framework of linear potential theory. The

obtained results show the effectiveness of this anti-heaving device in reducing not

only the heaving motion but also the stress-resultants of circular VLFS. This may

be due to the fact that the submerged plate reduces the wave height so that the

wave under the VLFS has lower amplitude when compared with the VLFS without

the attached plate.

6. To investigate anti-heaving device for longish VLFS (which are used as floating

runways or piers), it is proposed that a submerged inclined plate be attached to the

fore-end of VLFS. We have tested this device as well as three other types of anti-

heaving devices proposed in previous studies by other authors, namely submerged


171

horizontal plate, vertical plate and L-shaped plate. The experiments were carried

out with anti-heaving device attached to the fore-end of a longish flexible floating

structure. The effectiveness of these devices in reducing the heaving motion has

been confirmed by the test results. Among these devices, the newly proposed

inclined plate attachment seems to be the most effective one. The reductions of the

deflection of the longish floating structure at the fore-end and aft-end when using

this device are noted to be very significant. Therefore, by using this device, it

would be possible to eliminate the need for an expensive breakwater

6.2 Recommendations for future work

Based on the findings presented above, the study of innovative solutions for minimizing

differential deflection and heaving motion of VLFS can be extended to widen their

applicability. Some recommendations for future work are listed below:

1. In this study, the investigation of the best innovative solutions for minimizing

differential deflection in VLFS was carried on circular VLFS. Future

investigations should consider other shapes of VLFS.

2. With regard to gill cells for minimizing differential deflection in circular and

annular VLFS, analytical bending solutions used were based on the classical thin

plate theory. In future study, analytical solutions for bending analysis for circular

and annular VLFS can be developed based on Mindlin plate theory for better

prediction response of VLFS especially the stress-resultants.


172

3. Some interesting shapes of VLFSs were studied using gill cells to minimize the

differential deflection. The current research only considered the loading area to be

confined in the central portion of the VLFSs. Since the present numerical model is

capable of dealing with arbitrarily shaped VLFS and locations of loaded area, this

study can be extended to investigate other shapes and locations of loaded area.

4. Investigation on minimizing differential deflection in VLFS in this study

considered VLFS in a benign sea state condition where the influence of wave is

neglected. If the VLFS is located in a more severe sea state environment, it is

necessary to perform hydroelastic analysis of the VLFS under wave action as well

and determine the optimal design of gill cells allowing for this hydroelastic

component. Future studies are recommended to investigate the dynamic

hydroelastic problem of VLFS with gill cells.

5. In this study, an analytical model was developed to investigate the submerged

plate anti-heaving device for minimizing the motion of circular VLFS under wave

action was developed for the first time. However, this model is based on 2-D

configuration of VLFS and is restricted to only circular VLFS. Future works are

needed to investigate the effectiveness of submerged plate used in other shapes of

VLFSs. For arbitrarily shaped VLFSs, a 3-D configuration model is required.

Therefore, it is necessary to extend the 2-D model to 3-D model to analyze more

accurately the behavior and response of the real structure.

6. By performing experimental tests on 4 types of anti-heaving devices including a

submerged horizontal plate, a submerged vertical plate, a submerged L-shape plate

and a submerged inclined plate attached to the fore-end of a longish flexible

floating structure, it has been found that the inclined plate is the best anti-heaving


173

device. We have only considered the case of an inclined plate of 450. Future

studies should investigate the effect of the angle of inclination on the capability of

the anti-heaving device. In particular, it would be useful to carry out an

optimization study to determine whether there exists an optimal angle of

inclination of the inclined plate as an anti-heaving device. It is also proposed that

future studies focus on developing a numerical technique and computer code for

predicting the heaving motion of arbitrary shaped anti-heaving device and

experimental studies carried out to validate the numerical solutions.

174

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LIST OF PUBLICATIONS

INTERNATIONAL JOURNAL

Wang C.M., Pham D.C. and Ang K.K. (2006). Effectiveness and Optimal Design of

Gill Cells in Minimizing Differential Deflection in Circular VLFS. Engineering

Structures 29(8), 1845-53.

D.C. Pham, C.M. Wang and T. Utsunomiya (2008). Hydroelastic Analysis of a

Pontoon-Type Circular VLFS with an Attached Submerged Plate. Applied Ocean

Research 30(4), 287-96.

Pham D.C. and Wang C.M. (2009). Optimal Layout of Gill Cells for Very Large

Floating Structures. Journal of Structural Engineering. (In Press)

Pham D.C., Wang C.M. and Emma P.B. (2009). Experimental Study on Anti-Heaving

Devices for a Longish VLFS. The Journal of the Institution of Engineers,

Singapore, 02(4).

INTERNATIONAL CONFERENCE

Pham D.C., Wang C.M. and Ang K.K. (2005). Minimizing Differential Deflection and

Stresses in VLFS Using Gill Cells. Proceedings of the Eighteenth KKCNN

Symposium on Civil Engineering, Kaohsiung, Taiwan, 829-34.

List of Publications

187

Pham D.C., Wang C.M. and Ang K.K. (2007). Optimal Design of Gill Cells for Very

Large Floating Structures. The 7th International Conference on Optimization:

Technique and Application (ICOTA7), Kobe, Japan, 255-36.

Pham D.C. and Wang C.M. (2007). Effectiveness and Optimal Design of Gill Cells in

Minimizing Slope in annular VLFS. Proceedings of the Twentieth KKCNN

Symposium on Civil Engineering, Jeju, Korea, 255-8.

Pham D.C., Wang C.M. and Emma P.B. (2008). Anti-Heaving Devices for VLFS.

Proceedings of the Twenty First KKCNN Symposium on Civil Engineering,

Singapore, 286-89.

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