Hydroelastic response of interconnected floating beams modelling longish very large floating structures

82 5.1.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ .... 103 5.2.1 Effects of Auxiliary Beam Length δ and Conne

HYDROELASTIC RESPONSE OF INTERCONNECTED

FLOATING BEAMS MODELLING LONGISH VLFS

MUHAMMAD RIYANSYAH

NATIONAL UNIVERSITY OF SINGAPORE

2009

HYDROELASTIC RESPONSE OF INTERCONNECTED

FLOATING BEAMS MODELLING LONGISH VLFS

MUHAMMAD RIYANSYAH

B Eng (Hons), Institut Teknologi Bandung, Indonesia

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF CIVIL ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

A CKNOWLEDGMENTS

This thesis is a result of four years of research work since I was admitted into the PhD programme in the Department of Civil Engineering, the National University of Singapore I have worked with a great number of people whose contributions in the research deserved special mention It is a pleasure to convey my gratitude to them all

in this acknowledgment section

In the first place, I want to show my utmost gratitude to Prof Wang Chien Ming for his supervision, advice, guidance, and above all, for his patience from the very early stage of this research At the times I had a hard time in my research study, he provided me encouragement and support in various ways so that I could continue the research and finally finish this thesis I am indebted to him more than he knows

I also want to record my gratitude to Prof Choo Yoo Sang whose vast knowledge and experience have triggered and nourished my intellectual maturity that I will benefit from for a long time to come I thank him for all the valuable suggestions that he made

to help me shaping up my ideas and research

I would like to thank Dr Chan Chun Tat for his patience in teaching me the mathematical formulation and solution technique for the fluid-structure interaction

problem His immense knowledge in computer coding has helped me to write my own computer code for the hydroelastic analysis of floating interconnected beams

To Mr Sit Beng Chiat, I would like to thank him for being the role model for the hard workers in the Hydraulics Laboratory I am also grateful to Mr Yip Kwok Keong who helped me whenever I had a problem with my computer To Mr Khrisna Sanmugam, Mr Shaja Khan, Mr Semawi bin Sadi, Mr Koh Seng Chee, and Mdm Annie Tan, I would convey my appreciation for their indispensable help in my experiments It is impossible for me to finish this research without their assistance

It is a pleasure to pay tribute also to the administrative staffs To Ms Lim Sau Koon, Ms Norela Bte Buang, and Ms Juliana Bte Miswan, I am thankful for their assistance in dealing with administration matters during my study in National University of Singapore

I gratefully thank my colleagues, Mr Tay Zhi Yung, Ms Emma Patricia Bangun, and Dr Pham Dhuc Chuyen for their advice and their willingness to share their bright thoughts with me It was great to collaborate with them

I was extraordinarily fortunate in having Dr Dradjat Hoedajanto as my adviser in Institut Teknologi Bandung I could never have embarked and started all of this without his support His teachings have encouraged me to grab this challenging research opportunity To Dr Iswandi Imran, Dr Made Suarjana, and Dr Bambang Budiono, I am thankful for showing me the fun-side of research work

My parents deserve a special mention for their inseparable support and prayers

My father, Damrin Lubis, is the person who always reminds me the importance of learning My mother, Surya Aziz, is the one who sincerely raised me with her never-ending caring and love Kak Yeyen, Bang Ipan, and Bang Andi, thanks for being

supportive and caring siblings

It is unfair if I did not express my appreciation to Ms Rosa Permata Sari Her passion and ambition have altered my perspective, my way of thinking, and made me the way I am right now I am grateful for all the support that she has given

I would like to thank everybody who has helped me, as well as expressing my apology that I could not mention personally one by one Finally, I would like to thank the National University of Singapore for providing the scholarship that enabled me to study here in Singapore

T ABLE OF C ONTENTS

ACKNOWLEDGMENTS i

TABLE OF CONTENTS iv

SUMMARY x

LIST OF TABLES xvi

LIST OF FIGURES xvii

LIST OF NOTATIONS xxv

Chapter 1 INTRODUCTION 1

1.1 Very Large Floating Structures 3

1.1.1 Definition of VLFS 3

1.1.2 Advantages of VLFS over Land Reclamation 6

1.1.3 Applications of VLFS 8

1.2 Literature Review 13

1.2.1 Hydroelastic Analysis of VLFS 13

1.2.2 Factors Affecting the Hydroelastic Response of VLFS 16

1.2.3 Hydroelastic Response Reduction of VLFS 17

1.2.4 Floating Structure as Wave Energy Converter 23

1.3 Research Objectives 26

1.4 Layout of Thesis 30

Chapter 2 PROBLEM DEFINITION AND FORMULATION FOR HYDROELASTIC ANALYSIS OF INTERCONNECTED FLOATING BEAMS 33

2.1 Problem Definition 33

2.2 Formulation for Hydroelastic Analysis of Interconnected Floating Beams 41

2.2.1 Fluid Part 43

2.2.2 Floating Beam Part 47

Chapter 3 METHOD OF SOLUTION FOR HYDROELASTIC PROBLEM 50

3.1 Boundary Element Method (BEM) for Solving Fluid Motion Problem 51

3.2 Finite Element Method (FEM) for Solving Floating Beam Motion Problem 53

3.3 Coupled BEM-FEM Solution Method for Hydroelastic Problem 56

3.3.1 Radiation Boundaries 56

3.3.2 Fluid-structure Interface 61

3.4 Hydroelastic Response, Bending Moments, and Shear Forces 65

3.5 Compliance Parameter 65

Chapter 4 VERIFICATION OF FORMULATION AND METHOD OF SOLUTION 67

4.1 Convergence Study 67

4.1.1 Numerical Model 68

4.1.2 Computational Parameters for Converged Compliance Parameter 68

4.2 Validation of Formulation and Method of Solution with Experimental Test 72

4.3 Verification of Formulation and Method of Solution with Existing Numerical Results 79

4.4 Summary 80

Chapter 5 FLOATING MAIN BEAM WITH AUXILIARY BEAMS 82

5.1 Floating Main Beam with Auxiliary Beam at the Front End 82

5.1.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ 84

5.1.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating Main Beam Compliance Parameter χ 86

5.1.3 Comparison of Hydroelastic Response of Floating Main Beam with and without Auxiliary Beam 87

5.2 Floating Main Beam with Auxiliary Beam at the Rear End 103

5.2.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ 103

5.2.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating Main Beam Compliance Parameter χ 105

5.2.3 Comparison of Hydroelastic Response of Floating Main Beam

with and without Auxiliary Beam 106 5.3 Floating Main Beam with Auxiliary Beams at Both Ends 121 5.3.1 Effects of Auxiliary Beam Length δ and Connection

Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ 122 5.3.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating

Main Beam Compliance Parameter χ 123 5.3.3 Comparison of Hydroelastic Response of Floating Main Beam

with and without Auxiliary Beams 124 5.4 Summary 139

Chapter 6 LARGE FLOATING BEAM WITH MULTIPLE

CONNECTIONS 142 6.1 Effects of Connection Location β and Rotational Stiffness ξ on the

Hydroelastic Response of Floating Beam with Single Connection 143 6.2 Optimum Rotational Stiffnesses ξi for Prescribed Number of

Equally-spaced Connections 151 6.3 Optimum Design of Locations βi and Rotational Stiffnesses ξi for

Prescribed Number of Connections 160 6.4 Summary 168

Chapter 7 FLOATING ARTICULATED BEAM FOR WAVE ENERGY

CONVERTER 170 7.1 Optimum Connection Design for Maximum Total Work Done in the

Connections for Floating Articulated Beam System 171

7.2 Effect of Number of Connections on the Floating Articulated Beam

Response and its Wave Energy Capturing Efficiency 180

7.3 Summary 192

Chapter 8 CONCLUSIONS AND RECOMMENDATIONS 195

8.1 Summary of Results 195

8.2 Recommendations for Future Research Studies 205

8.2.1 Hydroelastic Response of Interconnected Floating Beams with Dynamic Loads 206

8.2.2 Hydroelastic Response of Interconnected Floating Beams in Oblique Sea 206

8.2.3 Hydroelastic Response of Interconnected Floating Beams in Three Dimensional 207

8.2.4 Hydroelastic Response of Interconnected Floating Plates 207

8.2.5 Interaction of the Hydroelastic Responses of Multiple Interconnected Floating Structures 207

REFERENCES 209

LIST OF AUTHOR’S PUBLICATIONS 218

Appendix A BOUNDARY ELEMENT METHOD 220

A.1 Analytical Formulation 220

A.2 Numerical Implementation 223

A.2.1 Boundary Discretization 223

A.3 Calculation Examples 230

Appendix B FINITE ELEMENT METHOD 234

B.1 Analytical Formulation 234

B.1.1 Internal Virtual Work 236

B.1.2 External Virtual Work 237

B.2 Matrix Assembly 241

S UMMARY

Lack of space has become a huge problem for most countries in the world, especially the developed ones The problem is getting worse every year as the population in those countries grows rapidly while the land area remains the same In the countries with long coastlines, one of the alternatives to overcome the problem is by creating a new land parcel from the sea A relatively new approach to create a new land parcel from the sea is through the very large floating structure (VLFS) technology Proposed by the Japanese and the Americans, this new approach has several advantages over the conventional land reclamation and given its broad applications it may become the key technology in human utilization of ocean space in the future

Interestingly, the same technology can also be adopted to harness the wave energy

to generate electricity It should be noted that wave energy is an attractive renewable energy source which has not been widely exploited due to the lack of technology available It was the British who introduced the use of very large floating articulated beam as wave energy converter The wave-induced motion of such floating beam is used to generate the electricity in generators equipped within the floating beam system Typically, a VLFS has large horizontal dimensions with relatively small depth In

some applications such as floating bridges, floating airplane runway, or floating causeway, only one of the horizontal dimensions is large (longish VLFS) The large horizontal dimensions when compared to the wavelength and small bending rigidity cause a VLFS to have a significant elastic deformation in addition to the rigid body motion under wave actions Therefore, the interaction between VLFS elastic deformation and the fluid flow field around it has to be considered in the VLFS response analysis This interaction is referred to as hydroelastic interaction and the response is referred to as hydroelastic response It is obvious that the hydroelastic analysis is critical in the VLFS assessment as well as for design improvement, such as

to find ways to reduce the hydroelastic response In the case the floating structure is used as wave energy converter in which the power is generated by the wave-induced motion of the floating structure, the design improvement aims to increase the response

to maximize the amount of electricity produced

Various approaches have been proposed to reduce the hydroelastic response of a VLFS Some of them include the use of breakwaters, active anti-motion devices, deep-draft or submerged wave reflectors, or simply by increasing the structural stiffness of the floating structure However, little had been studied the use sacrificial floating structure and/or semi-rigid connections as the way to reduce the hydroelastic response

of VLFS which actually is an interconnected floating structures, as for various reasons,

a VLFS is likely going to be built by connecting several smaller size floating modules

to form the complete VLFS Furthermore, it is possible that the same approach can also be adopted to maximize the efficiency of the floating beam when it is used as wave energy converter

This thesis is concerned with the hydroelastic response of interconnected floating

beams modelling longish VLFS and the way to reduce or to increase it through appropriate design of the floating beams and the connections It challenges the common assumptions that rigid connections in the VLFS are better, especially in conventional VLFS where small response is desirable As longish VLFS has large horizontal dimensions only in one direction, such problem can be modelled as floating beams on a two-dimensional fluid domain

For the hydroelastic analysis, a frequency domain numerical solution method for two-dimensional hydroelastic problem is developed The fluid is modelled as ideal fluid and the floating beams are modelled by Euler-Bernoulli beams The following assumptions are made: the fluid is assumed to be inviscid and incompressible, and the flow is assumed to be irrotational so that the fluid motion can be described by velocity potential which mush satisfy the Laplace equation; the wave amplitude is sufficiently small and hence, the linear wave theory is applicable to describe the wave; the interconnected floating beams are allowed to move freely only in the vertical direction and sufficiently small, are always in contact with the fluid surface, and have negligible draft (this is acceptable as the added mass is much larger than the mass of the floating structure itself) Boundary condition at the radiation boundaries is described by Sommerfeld condition which states that waves at infinity are outgoing The Boundary Element Method (BEM) with free space Green's function is used to solve the governing equations of the fluid motion, and the Finite Element Method (FEM) is used

to solve the governing equations of the beam motion The fluid-structure interaction problem is shown in the fluid-structure interface, as the fluid boundary condition contains the beam displacement and the governing equation of beam motion contains the fluid velocity potential It is solved by substituting the beam displacement in the

boundary condition of fluid motion with the beam displacement from the governing equation of beam motion and hence, the problem is reduced to linear problem with one unknown: fluid velocity potential and can be solved directly Once the velocity potential is known, the beam displacement can be easily determined by using the relationship in the fluid-structure interface boundary condition The validity and accuracy of the numerical solution method to solve the hydroelastic problem of interconnected floating beams are assessed by comparing the results obtained with those obtained from the experiment and by other researchers It was found that the results from the proposed formulation and method of solution are in agreement with those obtained from the experiments as well as by other researchers

In the first phase of this research study, the hydroelastic response of a floating main beam with auxiliary beams is investigated The auxiliary beams can either be attached to the front end, the rear end, or both ends of the floating main beam A parametric study is carried out to study the effects of the lengths and the flexural rigidities of the auxiliary beams as well as the rotational stiffnesses of the connections

on the hydroelastic response of the main floating beam The main objective in this part

of the study is to find the appropriate properties of the auxiliary beams and the connections for effective reduction in the hydroelastic response of the main floating beam Significant reductions in floating main beam displacement, bending moment, and shear forces are observed when the auxiliary beam is attached to the front end of the floating main beam with semi-rigid connection The rotational stiffness of the semi-rigid connection depends mainly on the length of the auxiliary beam However, this configuration is not suitable when the incident waves come from the opposite direction (i.e the auxiliary beam is attached to the rear end of the floating main beam)

It was shown that in the case the auxiliary beam is attached to the rear end, the floating main beam response may increase significantly Therefore, we propose the use of two auxiliary beams at to both ends of the floating main beam connected by semi-rigid connections The response reduction by this configuration is as effective as having auxiliary beam at the front end only; and it is suitable not only for the case when the incident waves come from one direction, but also from the opposite direction

In the second phase of the study, the hydroelastic response of large floating beam with multiple connections is investigated These connections are readily available as VLFS are usually built by connecting several smaller floating modules The effects of the connections locations and the rotational stiffnesses on the hydroelastic response of the floating beams system are studied The main objective in this part of the study is to find the optimum locations and rotational stiffnesses of the connections for minimum hydroelastic response of the interconnected floating beams system Alternating variable method is adopted for the optimization procedure The results show that the hydroelastic response of the floating beam system can be reduced significantly if semi-rigid connections are used It challenges the common perception among engineers that the complete floating structure would have a smaller hydroelastic response if the modules are rigidly connected Further improvement in the reduction can be achieved

by placing the connections at their optimum locations in addition to having optimum rotational stiffnesses

As the third phase of the study, the hydroelastic response of floating articulated beams for wave energy converter is investigated The problem considered in this part

of the study is similar to that of the second phase but with a different objective Here, instead of reducing the hydroelastic response, we maximize the hydroelastic response

in order to maximize the floating beams wave energy capturing efficiency by optimizing the connections design (i.e the number of connections, the rotational stiffnesses, and the locations) In this part of the study, we challenge the design used in the Pelamis machine, the example of large floating beam system for wave energy converter Our results suggest that the current Pelamis design can be improved by introducing the ability to change its connections locations This feature would increase the machine efficiency significantly

Lastly, the conclusions and recommendations for future research works on the interconnected floating beams are presented

L IST OF T ABLES

Table 4.1 Convergence of compliance parameter χ with respect to κ and

em

M for various α for floating continuous beam in Fig 4.1 71

Table 4.2 Convergence of compliance parameter χ with respect to l with

150

=

κ and M em =10 for floating continuous beam in Fig 4.1 72

Table 6.1 Optimum location β and rotational stiffness ξ of connection for

various incident wavelengths α 146 Table A.1 BEM calculation example for P and Q are in different elements 232

Table A.2 BEM calculation example for P and Q are in the same element 233

L IST OF F IGURES

Figure 1.1 Mega-float in Tokyo Bay 9

Figure 1.2 Kamigoto oil storage facility in Nagasaki, Japan 10

Figure 1.3 Shirashima oil storage facility in Kitakyusyu, Japan 10

Figure 1.4 Floating bridge in Dubai 11

Figure 1.5 Ujina’s floating concrete pier in Hiroshima, Japan 12

Figure 1.6 Lilypad by Vincent Callebaut 12

Figure 1.7 Pelamis Wave Energy Converter 24

Figure 1.8 Pelamis power conversion module 25

Figure 1.9 Plan view of wave farm comprises of interlinked Pelamis machines 26

Figure 2.1 Floating main beams with auxiliary beams 34

Figure 2.2 Floating beam system cross-section 35

Figure 2.3 Floating two-beam system 36

Figure 2.4 Large floating beam with equally-spaced connections 37

Figure 2.5 Large floating beam with optimum locations and rotational stiffnesses for the connections 38

Figure 2.6 Floating articulated beam with equally-spaced connections for wave energy converter 40

Figure 2.7 Floating articulated beam with optimum locations and rotational

stiffnesses for the connections for wave energy converter 40

Figure 2.8 Problem domain 42

Figure 3.1 Boundary element method on discretized boundary 52

Figure 3.2 Forces and displacements at nodes of beam element 54

Figure 4.1 Floating continuous beam used for convergence study 68

Figure 4.2 Boundary and finite element for floating beam problem 69

Figure 4.3 Experimental model 74

Figure 4.4 Simple bending test to determine beam model flexural rigidity EI 74

Figure 4.5 Hinge connection in the experimental model 75

Figure 4.6 Experimental setup in the wave flume for floating two-beam experiment 76

Figure 4.7 Schematic diagram of the experimental setup 76

Figure 4.8 Normalized maximum vertical displacement along the length of floating two-beam system connected by hinge (ξ =0) for (a) α =0.4, (b) α =0.6, (c) α =0.8 77

Figure 4.9 Numerical model by Khabakhpasheva and Korobkin (2002) 79

Figure 4.10 Normalized maximum vertical displacement along the length of floating two-beam system connected by (a) hinge (ξ =0) and (b) semi-rigid connection (ξ =625) 81

Figure 5.1 Floating main beam with auxiliary beam at the front end 83

Figure 5.2 Variation of the normalized compliance parameter χ/χm with respect to ξ and δ for γ =1.00 and α =0.10 90

Figure 5.3 Variation of the normalized compliance parameter χ/χm with respect to ξ and δ for γ =1.00 and α =0.20 91

Figure 5.4 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.30 92 Figure 5.5 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.40 93 Figure 5.6 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.10 94 Figure 5.7 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.20 95 Figure 5.8 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.30 96 Figure 5.9 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.40 97 Figure 5.10 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.10 98 Figure 5.11 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.20 99 Figure 5.12 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.30 100 Figure 5.13 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.40 101 Figure 5.14 Variation of the normalized compliance parameter χ/χm with

respect to ξ and α for δ =0.25 and γ =1.00 102 Figure 5.15 Floating main beam with auxiliary beam at the rear end 103 Figure 5.16 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.10 108

Figure 5.17 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.20 109 Figure 5.18 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.30 110 Figure 5.19 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.40 111 Figure 5.20 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.10 112 Figure 5.21 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.20 113 Figure 5.22 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.30 114 Figure 5.23 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.40 115 Figure 5.24 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.10 116 Figure 5.25 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.20 117 Figure 5.26 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.30 118 Figure 5.27 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.40 119 Figure 5.28 Variation of the normalized compliance parameter χ/χm with

respect to ξ and α for δ =0.25 and γ =1.00 120

Figure 5.30 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.10 126 Figure 5.31 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.20 127 Figure 5.32 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.30 128 Figure 5.33 Variation of the normalized compliance parameter χ/χm with

respect to ξ and δ for γ =1.00 and α =0.40 129 Figure 5.34 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.10 130 Figure 5.35 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.20 131 Figure 5.36 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.30 132 Figure 5.37 Variation of the normalized compliance parameter χ/χm with

respect to ξ and γ for δ =0.25 and α =0.40 133 Figure 5.38 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.10 134 Figure 5.39 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.20 135 Figure 5.40 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.30 136 Figure 5.41 Hydroelastic response of the floating beam system with various ξ

for δ =0.25, γ =1.00, and α =0.40 137

Figure 5.42 Variation of the normalized compliance parameter χ/χm with

respect to ξ and α for δ =0.25 and γ =1.00 138 Figure 6.1 Floating beam with a single connection 143 Figure 6.2 Variation of the normalized compliance parameter χ/χ∞ with

respect to ξ and β for α =0.10 147 Figure 6.3 Variation of the normalized compliance parameter χ/χ∞ with

respect to ξ and β for α =0.20 148 Figure 6.4 Variation of the normalized compliance parameter χ/χ∞ with

respect to ξ and β for α =0.30 149 Figure 6.5 Variation of the normalized compliance parameter χ/χ∞ with

respect to ξ and β for α =0.40 150 Figure 6.6 Floating beam with equally-spaced connections 152 Figure 6.7 Hydroelastic response of floating beam with optimum rotational

stiffnesses ξi for n c =3 and α =0.20 156

Figure 6.8 Hydroelastic response of floating beam with optimum rotational

stiffnesses ξi for n c =5 and α =0.20 157

Figure 6.9 Hydroelastic response of floating beam with optimum rotational

stiffnesses ξi for n c =7 and α =0.20 158

Figure 6.10 Hydroelastic response of floating beam with optimum rotational

stiffnesses ξi for n c =9 and α =0.20 159

Figure 6.11 Floating beam with variable connection locations and rotational

stiffnesses 161 Figure 6.12 Hydroelastic response of floating beam with optimum locations βi

and optimum rotational stiffnesses ξi for n c =3 and α =0.20 164

Figure 6.13 Hydroelastic response of floating beam with optimum locations βi

and optimum rotational stiffnesses ξi for n c =5 and α =0.20 165

Figure 6.14 Hydroelastic response of floating beam with optimum locations βi

and optimum rotational stiffnesses ξi for n c =7 and α =0.20 166

Figure 6.15 Hydroelastic response of floating beam with optimum locations βi

and optimum rotational stiffnesses ξi for n c =9 and α =0.20 167

Figure 7.1 Floating articulated beam system for wave energy converter 174 Figure 7.2 Hydroelastic response of floating articulated beam with optimum

connection design for maximum total work done in the connection for n c =3 and α =0.20 177

Figure 7.3 Hydroelastic response of floating articulated beam with optimum

connection design for maximum total work done in the connection for n c =3 and α =0.40 178

Figure 7.4 Hydroelastic response of floating articulated beam with optimum

connection design for maximum total work done in the connection for n c =3 and α =0.60 179

Figure 7.5 Comparison of hydroelastic response of floating articulated beam

with optimum connection design for maximum total work done in the connections for various number of connections n and c α =0.20 183

Figure 7.6 Comparison of hydroelastic response of floating articulated beam

with optimum connection design for maximum total work done in the connections for various number of connections n and c α =0.40 186

Figure 7.7 Comparison of hydroelastic response of floating articulated beam

with optimum connection design for maximum total work done in the connections for various number of connections n and c α =0.60 189

Figure A.1 Boundary Element Method 222

Figure B.1 Virtual displacements and nodal forces of a beam element 235 Figure B.2 Example of floating beam problem with semi-rigid connection 241 Figure B.3 Discretization of floating beam system with semi-rigid connection 242

L IST OF N OTATIONS

x , y , z Cartesian coordinates system

n Unit vector outward normal to fluid domain boundary

d , d2, d Water depth at the left boundary of the fluid domain, in the fluid domain, 3

and at the right of the fluid domain, respectively, m

*

l Distance between from the free ends of the beam to the radiation

boundaries of the fluid domain, m

i

Φ, φ Fluid velocity potential

w

W, w, w* Fluid surface and beam displacement, m

p Fluid total pressure, Pa

c

λ Floating beam characteristic length, m

L, L * Floating beam system total length, m

b Floating beam cross-section width, m

t Floating beam top and bottom plates thickness, m

h Floating beam cross-section thickness, m

b

ρ Floating beam material density, kg/m3

EI Floating beam flexural rigidity, Nm2

α Incident wavelength to beam length ratio

δ Auxiliary beam length to main beam length ratio

β Locations parameters of the connection

i

β Locations parameters of i -th connection

γ Auxiliary beam thickness to main beam thickness ratio

ξ Rotational stiffness parameter of the connection

{F Assembled external load force vector

N Total number of elements on the boundary

[T1 Transformation matrix to relate the fluid velocity potential φ to the

external force vector {F}

]

[T2 Transformation matrix to relate the beam displacement w to the normal

derivative of the velocity potential

χ Compliance parameter of continuous floating beam

κ Number of elements in beam system

em

W Work done in the connection

W Virtual work done by the external pressure within a beam element

ζ , η Local coordinate system for boundary element

c

J Jacobian of coordinate transformation

x

n , n z Unit outward normal inx - and z -direction, respectively

U Number of standard Gaussian integration points

l

U Number of logarithmic Gaussian integration points

u

ζ , v u Gaussian coordinate and its associated weight function for standard

Gaussian quadrature integration

l

u

η , v u l Gaussian coordinate and its associated weight function for logarithmic

Gaussian quadrature integration

Chapter 1

The lack of space in major cities is getting worse each year as the population expands and industrial and urban development increases In order to overcome the space crunch, engineers make use of the airspace and subterranean space High rise buildings with interconnected sky bridges, underground network of roads, railways and utility tunnels, and multi-tier roads are some examples However, not all the facilities can be built vertically and land space is needed For example, an airport requires free space both horizontally and vertically for safety reasons

Creating new land parcels from the sea is an alternative solution for solving the scarcity of land Until 20th century, land reclamation technology has enabled the creation of new land from the sea for countries with many islands or long coastlines Countries like the Netherlands, Japan, Monaco and Singapore have been successfully applied this approach to increase their land area for decades However, the land reclamation technology has some limitations and disadvantages For example, it can only be economically carried out in shallow water environments, with depths of no more than 20 meters For larger water depths and soft seabed conditions, land reclamation is too expensive and not economically viable The negative impact that

land reclamation works have on the coastal environment and marine life in the coastal waters as well as the waters in the neighbouring countries cannot be ignored Moreover, settlement problem of newly reclaimed land may pose a major problem for infrastructural development over many years

Proposed by the Japanese and the Americans, Very Large Floating Structure (VLFS) technology is relatively a new solution for creating land from the sea This technology has several advantages over the conventional land reclamation approach such as its environmentally friendliness to the marine ecosystem, its cost effectiveness when water depth is large, and its insulation from seismic shocks since it is inherently base isolated Furthermore, it can be used for various applications such as floating airports, floating bridges, floating oil storage, floating energy plants, floating piers and floating hotels Given its broad applications, VLFS is the key technology for mankind

to colonize the oceans for space, food, and energy

Typically, a VLFS has large horizontal dimensions with a relatively small depth With small depth-to-length ratio, a VLFS has relatively small bending rigidity The large horizontal dimensions when compared to the wavelength of the sea and the small bending rigidity cause a VLFS to have a significant elastic deformation in addition to the rigid body motion under wave actions Therefore, the interaction between VLFS elastic deformation and the fluid flow field around it has to be considered in the VLFS response analysis This interaction is referred to as hydroelastic interaction and the response is referred to as hydroelastic response Hydroelastic analysis is critical in the VLFS assessment as well as for design improvement, such as to find ways to reduce the hydroelastic response As a reduction in the hydroelastic response translates to a higher level of safety and serviceability levels of the VLFS

In the next section, the definition of VLFS to distinguish it from other conventional floating structures (such as ships or floating oil platforms), its advantages over conventional land reclamations, and its applications will be discussed further Subsequently, the hydroelastic analysis of VLFS and various approaches proposed to reduce the hydroelastic response of VLFS will also be discussed in more detail The objectives of the present study followed by the layout of the thesis will be presented at the end of this chapter

1.1 Very Large Floating Structures

is simpler in design and construction; hence it will be less expensive as the size of

Typically, a VLFS system comprises several components, such as the main floating structure, the station keeping system to keep the floating structure in place, an access bridge or causeway from the mainland for a nearshore VLFS, and a surrounding breakwater to protect the floating structure by reducing the incoming waves The surrounding breakwater may not be needed if the floating structure is located in calm waters, where the significant wave height is relatively small

The horizontal dimensions of a pontoon-type VLFS can be several hundred meters

to several kilometers while the thickness is only several meters Owing to its small depth-to-length ratio and large length dimensions when compared to the wavelength, the global response of a VLFS cannot be solely described by rigid body dynamics; but

it must involve the interaction between its elastic deformation and the fluid flow field around it This interaction has been referred to as hydroelastic interaction, and the response is referred to as hydroelastic response Suzuki and Yoshida (1996) proposed that in order to qualify for the status of a VLFS, a floating structure must satisfy two conditions The first condition is that the length of the structure must be greater than the wavelength, the latter of which is defined by the wave period and the water depth (for example, for a wave period of 6 seconds and the water depth of 20 meters, the wavelength is 55 meters) The second condition requires the length of the floating structure to be larger than the characteristic length defined by its flexural rigidity and the fluid density The characteristic length λc is defined as

where EI is the flexural rigidity, ρ the fluid density, and g the gravitational

acceleration In their formulation, the floating structure is modelled as a uniform beam model and the hydrostatic restoring force is modelled as an elastic foundation These two conditions are the unique features that distinguish a VLFS from a conventional floating platform used for oil production or ship In a conventional floating oil production platform, the horizontal dimensions are relatively small when compared to the wavelength of the sea On the other hand, a ship may have large horizontal dimensions (for example, an aircraft carrier), but due to its relatively large flexural rigidity, the elastic deformation is negligible Moreover, VLFSs are meant to support ocean resource development rather than ocean transportation and hence they need not have a streamline shape

Apart from its hull-type as described earlier in this subsection, a VLFS can also be modelled in various ways based on its design concepts From its horizontal dimensions, a VLFS can be modelled as a ‘one-dimensional’ or ‘two-dimensional’ structure A VLFS is modelled as a one-dimensional beam structure when one of its horizontal dimensions is significantly larger than the other Some examples of such longish VLFS are floating airplane runways and floating bridges The width of these floating structures may be less than 100 meters but the length may reach thousands of meters Obviously, a one-dimensional VLFS is the simplest form of VLFS, as the whole structure behaves like a floating beam On the other hand, a two-dimensional VLFS is when both the horizontal dimensions are significantly large, and therefore the floating structure behaves like a floating plate

type or a multi-module-type For a single-module VLFS, the whole complete floating structure is constructed in one piece However, for various reasons, it is unlikely that a VLFS is built this way It is more likely that a VLFS is going to be built by assembling several smaller modules together Smaller modules are easier and less expensive to construct and to transport to site Moreover, such modular VLFS allows for easy expansion or contraction in future applications

1.1.2 Advantages of VLFS over Land Reclamation

As a relatively new alternative solution for creating land from the sea, VLFS technology has several advantages over the conventional land reclamation approach

Cost Effective for Large Water Depth and Soft Seabed Conditions

While laterally restrained in the horizontal directions, VLFS depends on the buoyancy force of the water in the vertical directions to support it Therefore, it is not affected by large water depths or soft seabed condition On the other hand, land reclamation is only economically viable when the water depths are less than 20 meters The cost may increase in soft seabed conditions as soil improvement is needed Furthermore, land reclamation requires sand as fill materials For some countries, like Singapore, sand has to be purchased from other countries and this makes land reclamation too expensive

Environmentally Friendly

It is clear that land reclamation works have significant negative impacts on the environment The marine ecosystem underneath the reclaimed land is destroyed

Furthermore, the reclaimed land creates disturbances to the natural flow of the sea currents The disturbances may affect the life cycle and the feeding behaviours of marine life as well as the water quality in the surrounding region These problems could be avoided by using VLFS technology As VLFSs do not need any fill materials, the marine ecosystem underneath the floating structures will not be destroyed and it can continue to live in peace The presence of floating structures may disturb the sea currents, but the disturbance occurs only at the water surface At lower water depths, where most of the marine life is, the effect is small and thus the negative impacts are negligible

Easy and Fast Construction, Expansion, and Removal/Relocation

Settlement is a common problem in newly reclaimed lands Therefore, reclamation work often requires consolidation period for the reclaimed land to settle before further construction can be carried out The consolidation period may take two to five years and still there is a possibility that further settlements could occur in the future This also applies to the expansion of the already reclaimed land Another problem with land reclamation is that it cannot be readily and cheaply removed or relocated if the sea space is needed in future On the other hand, a VLFS can be instantly occupied once the whole construction process is completed It can also be expanded easily to accommodate the requirements for additional area by attaching new portions of floating structure onto the existing floating structure The new portion can also be instantly occupied Furthermore, it is possible to completely remove or to relocate the floating structure in future when it is not needed anymore in its current location

Inherently Base Isolated

Since floating structures are not directly connected to the earth, they are inherently base isolated and therefore they are protected from seismic shocks It is an advantage especially for countries that are in the earthquake-prone region With such advantage, floating structures are ideal for use as emergency bases as they can also be towed to the disaster site However, they are not protected from tsunami waves Therefore, such facility should be located in protected waters

Availability of Interior Spaces

As the floating structure have many watertight compartments to achieve the buoyancy, these interior spaces may be used to support the main facilities on the top side of the floating structure For example, they can be used for car parks, office space, and storage rooms

1999 The Mega-Float consisted of six modules, which were welded together into one

huge floating structure of 1000 m in length and 60 m (partially 121 m) in width, with the deck area of 84,000 m2 (see Figure 1.1)

Figure 1.1 – Mega-float in Tokyo Bay (www.jasnaoe.or.jp and www.marinetalk.com)

VLFSs are also ideal for floating oil storage facilities as its offshore location keeps the explosive and inflammable fluid away from populated land area Examples of large floating oil storage bases are the Kamigoto oil storage facility in Nagasaki (Figure 1.2) and Shirashima oil storage facility in Kitakyusyu, Japan (see Figure 1.3) Currently, Singapore which is the world’s 3rd largest oil trading hub, is planning to increase its oil storage capacity by building a floating oil storage facility (Periscope, 2007)