Very large floating structures

... 1.1.1 Definition and types of VLFS Floating structures are broadly classified as pontoon type and semi-submersible floating structures Semi-submersible type floating structures are partly raised above... porosity Hence instead of building large structures in deep water region and soft seabed conditions, floating structures are an economical option Also, due to their floating nature, the foundation... VLFS (e) Floating infrastructures: Two of the major infrastructural applications of floating structures have been covered above, viz bridges and airplane runways Other applications include floating

EFFECTS OF WAVELENGTH, WATER DEPTH, ASPECT RATIO AND SLOPING SEABED ON HYDROELASTIC RESPONSE OF VLFS               MRINALINI PATHAK NATIONAL UNIVERSITY OF SINGAPORE 2014 EFFECTS OF WAVELENGTH, WATER DEPTH, ASPECT RATIO AND SLOPING SEABED ON HYDROELASTIC RESPONSE OF VLFS             MRINALINI PATHAK B.Tech. (Civil Engineering), Sardar Vallabhbhai National Institute of Technology, Surat, India A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014   DECLARATION I hereby declare that the thesis is my original work and has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in this thesis. This thesis has not been submitted for any degree in any university previously. ___________________________________ MRINALINI MUKESHDUTTA PATHAK 9TH DECEMBER 2014 Dedicated to My parents Acknowledgements I wish to express my deep gratitude to Professor Wang Chien Ming for his guidance, encouragements and invaluable suggestions throughout this research. His enthusiasm and research acumen have inspired me immensely and greatly shaped my reasoning ability. Over the period of last two years, the knowledge gained and lessons learnt under his supervision are invaluable to me and I believe that they will enormously help me in my future endeavors. I would also like to extend my sincere thankfulness to Dr. Gao Ruiping for his patience and persistent cooperation and help throughout the research. He has been a great support during my course of study and helped me with the tiniest doubts I encountered with utmost patience. I will be always grateful to him for sharing his knowledge and experience with me. I would also like to extend my thanks to the faculty members of Civil Engineering department who have taught me various courses which have helped me in research. I would like to thank National University of Singapore for giving me the opportunity to study here and I appreciate the staff of Civil Engineering department for addressing and helping with administrative matters. Last but not least, I am immensely grateful to my parents for providing all the love and support. It is due to their belief on me and their constant encouragement that I have successfully completed my study in Singapore. I would also like to thank my friends and loved ones for their constant support and understanding. Mrinalini Pathak   Table of Contents Table of Contents Acknowledgements ............................................................................................................................... i Table of Contents ................................................................................................................................ iii Summary............................................................................................................................................... v List of Tables ...................................................................................................................................... vii List of Figures...................................................................................................................................... ix List of Notations ................................................................................................................................ xiii Chapter 1 INTRODUCTION ............................................................................................................... 1 1.1 Background information on VLFS ............................................................................................ 1 1.1.1 Definition and types of VLFS ............................................................................................ 2 1.1.2 Advantages of VLFS.......................................................................................................... 4 1.1.3 Applications of VLFS: Past to present and future ............................................................. 5 1.2 Literature survey ........................................................................................................................ 8 1.3 Research objectives .................................................................................................................. 13 1.4 Thesis layout ............................................................................................................................ 14 Chapter 2 HYDROELASTIC ANALYSIS OF VLFS ....................................................................... 17 2.1 Plate-water model .................................................................................................................... 18 2.2 Governing equations of motions .............................................................................................. 19 2.2.1 Equations of motion of plate ............................................................................................ 19 2.2.2 Equations of motions of water ......................................................................................... 22 2.3 Numerical solutions of fluid-structure system ......................................................................... 25 2.3.1 Finite element method for solving plate deflections ........................................................ 25 2.3.2 Boundary element method for solving velocity potentials............................................... 30 2.3.3 Constant panel method to calculate velocity potential ..................................................... 32 2.4 Modal expansion method ......................................................................................................... 34 2.5 Summary .................................................................................................................................. 38 Chapter 3 EFFECT OF WAVELENGTH ON HYDROELASTIC BEHAVIOUR OF VLFS ......... 39 3.1 Numerical model ...................................................................................................................... 39 3.2 Effect of wavelength on hydroelastic behaviour of VLFS ....................................................... 41 iii Table of Contents 3.3 Effect of water depth on hydroelastic behaviour of VLFS ....................................................... 43 3.4 Upstream and downstream deflections of VLFS ...................................................................... 45 3.5 Summary .................................................................................................................................. 47 Chapter 4 EFFECT OF ASPECT RATIO ON HYDROELASTIC BEHAVIOUR OF VLFS .......... 49 4.1 Numerical model ...................................................................................................................... 49 4.2 Effect of aspect ratio (B/L) on hydroelastic behaviour of VLFS .............................................. 51 4.3 Effect of water depth on hydroelastic behaviour of VLFS with different aspect ratios ........... 55 4.4 Effect of aspect ratios on upstream and downstream deflections of VLFS .............................. 59 4.4.1 Effect of water depth ........................................................................................................ 63 4.4.2 Effect of wavelength......................................................................................................... 65 4.5 Summary .................................................................................................................................. 66 Chapter 5 EFFECT OF SEABED SLOPE ON HYDROELASTIC BEHAVIOUR OF VLFS ......... 67 5.1 Numerical model ...................................................................................................................... 67 5.2 Effect of seabed slope on hydroelastic behaviour of VLFS ..................................................... 70 5.3 Summary .................................................................................................................................. 76 Chapter 6 CONCLUSIONS AND RECOMMENDATIONS ............................................................. 77 6.1 Conclusions .............................................................................................................................. 77 6.2 Recommendations .................................................................................................................... 79 References ........................................................................................................................................... 81 Appendix A Finite Element Formulation of Mindlin Plate ................................................................. 87 Appendix B Boundary Integral Equation ............................................................................................ 97 iv Summary Summary Very Large Floating Structures (VLFS) is a promising technology that facilitates ocean space colonization, a sustainable and environmental friendly technological innovation which enables creation of land on water without disturbing marine environment, polluting coastal waters and disrupting ocean currents. VLFS are becoming a popular choice in a wide range of applications like floating bridges, walkways, recreational structures, storage facilities and offshore platforms. Owing to their large size and small depth, the VLFS deforms under the action of waves. Hence, their hydroelastic deformations should be checked for serviceability limits. In order to understand the hydroelastic behaviour of VLFS under various sea states and seabed conditions, parametric studies are undertaken in this thesis. Owing to the purpose and function, a VLFS can take many shapes and orientations. In this thesis, the behaviour of the most commonly used, rectangular VLFS of different aspect ratios subjected to different wavelengths and water depths is studied in detail. Since a VLFS is generally connected to the land and deployed in near-shore regions, the effect of varying seabed topography also becomes significant. Therefore studies investigating the effect of sloping seabed topography are also presented herein. In addition to overall hydroelastic responses of VLFS, the end deflections which are usually largest in magnitude are also studied for different situations. The ends of a VLFS are particularly important because they are also the connecting interface between the VLFS and land-masses. To idealize the physical VLFS system, an equivalent numerical model is considered in terms of a plate floating on a fluid domain. The VLFS is modelled as an isotropic, elastic and flat Mindlin plate with free edges. The fluid is assumed to be ideal, incompressible, inviscid and irrotational so that the velocity potential exists. Linear wave potential theory is used to model the fluid-motion by using the velocity potential, i.e. single frequency velocity potential of water which satisfies the v Summary Laplace’s equation. The wave potential satisfies boundary condition at the seabed, linearized kinematic boundary condition on the fluid structure interface, linearized dynamic boundary condition on free surface of water and Somerfield radiation condition at artificial boundary condition at infinity. The main objective of the hydroelastic analysis is to determine the fluid velocity potentials and plate displacements. In order to decouple the fluid–structure interaction problem, the modal expansion method is adopted for the hydroelastic analysis which is carried out in the frequency domain. Boundary element method is used to solve the Laplace equation for the velocity potential and finite element method is employed for solving the equations of motion of the floating plate. In Chapter 1, a general introduction of VLFS and their past, present and future applications in different fields are presented. Chapter 2 describes the hydroelastic analysis of a floating structure and present method for solving the fluid-structure interaction problem. Chapters 3, 4 and 5 present and discuss the results obtained from the formulation as well as parametric studies to comprehend the behaviour of VLFS. Chapter 6 presents the summary and key points of this thesis and also the suggested future studies related to the present research. The studies carried out in the thesis provide an insight to the hydroelastic behaviour of a VLFS in different sea conditions and on constant and sloping seabed topographies. The solutions to the fluid structure interaction problems in this thesis may serve as benchmark solutions for structural and offshore engineers in analysis of VLFS. vi List of Tables List of Tables Table 3.1. Parameters of VLFS-water model used by Yago and Endo (1996) ................................... 40 Table 4.1. Parameters of VLFS-water model used by Yago and Endo (1996) ................................... 50 Table 5.1. Parameters of VLFS-water model used by Kyoung et al. (2005) ...................................... 69 vii viii List of Figures List of Figures Figure 1. 1. Components of VLFS system ............................................................................................ 3 Figure 1. 2. Two phase floating runway project, the Mega-Float, in Japan (Suzuki et al. 2005) ......... 6 Figure 1. 3. (a) World’s longest floating bridge, Governor Albert D Rossellini Bridge (aka Evergreen Point Bridge) (b) Yumemai Bridge, Japan (c) Dubai floating bridge, (source: bridge-info.org, Wikipedia) .................................................................................. 7 Figure 1. 4.(a) Ujina’s floating pier (b) Floating walkway made of High Density PolyEthylene (HDPE) modules ................................................................................................. 7 Figure 1. 5. (a) Keppel-built 6th Generation Semi-Submersible (b) FPSO vessel in Angola (source: energy-pedia news) (c) Kamigoto Floating Oil Storage Base, Nagasaki Prefecture, Japan (d) Floating nuclear power plant in Russia (source: RIA novosti) (e) Hexicon, a Swedish design for wind energy farm (source:Main(e) Consulting Ltd.) (f) 125 MWE OTEC Plant designed by Dr Alfred Yee........................................... 11 Figure 1. 6.(a) Performing stage at Marina Bay, Singapore (b) Floating breakwater, Monaco seawall, in Monaco (source: FCC const.) (c) Jumbo restaurant in Hong Kong (d) Prof Wang’s proposed floating crab restaurant (e) Lilipads – floating cities(sourc:vincent.callebaut.org) (f) Proposed Greenstar floating hotel and convention centre ............................................................................................................. 12 Figure 2. 1. Plan and elevation views of plate-water problem domain ................................................ 19 Figure 2. 2. Mindlin plate element in physical coordinates and its isoparametric transformation in natural coordinates ........................................................................................................ 27 Figure 3.1. Plan and elevation views of plate-water problem domain ................................................. 40 Figure 3.2. Centre-line deflections of VLFS in four different wavelengths (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6 in water depth and H=58.5m. ...................................................... 41 Figure 3.3. Centre-line deflection of VLFS subjected to different wavelengths, α=0.2,0.4,0.5 and 0.6 in water depths (a) H=20m (b) H=58.5m and (c) H=100m .................................. 42 Figure 3.4. Centre-line deflections of VLFS in three different water depths, H=20m, 40m, 58.5m and 100m for (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6 .................................... 44 ix List of Figures Figure 3.5. Variations of (a) upstream and (b) downstream deflections of VLFS with respect to water depth, H , in different wavelengths (α=0.1, 0.2, 0.3, 0.4, 0.5 and 0.6) ................... 47 Figure 4.1. Plan and elevation views of plate-water problem domain ................................................ 50 Figure 4.2. Centre-line deflections of VLFS with five aspect ratio, B/L= 1/5,1/3,1,3/2 and 3, in water depth, H=20m for (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6.............................. 52 Figure 4.3. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1/5 subjected to wavelength of α=0.2 and 0.6 in water depth, H=20m....................................................... 54 Figure 4.4. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1 subjected to wavelength of α=0.2 and 0.6 in water depth, H=20m....................................................... 54 Figure 4.5. Centre-line and edge-line response of VLFS of aspect ratio, B/L=3 subjected to wavelength of α=0.2 and 0.6 in water depth, H=20m....................................................... 55 Figure 4.6. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1/5 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m..................................................... 56 Figure 4.7. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m..................................................... 56 Figure 4.8. Centre-line and edge-line response of VLFS of aspect ratio, B/L=3 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m..................................................... 57 Figure 4.9. Centre-line deflection of VLFS of different aspect ratio (a) B/L=1/5 (b) B/L=1 (c) B/L=3, in four water depths, H=20m, 40m, 58.5m and 100m subjected to two wavelength α=0.2 and 0.6 ................................................................................................. 58 Figure 4.10. Displacement of upstream point of VLFS in different water depths and wavelengths for aspect ratio (a) B/L=1/5 (b) B/L=1 and (c) B/L=3 .................................. 61 Figure 4. 11.Displacement of downstream point of VLFS in different water depths and wavelengths for aspect ratio (a) B/L=1/5 (b) B/L=1 and (c) B/L=3 .................................. 62 Figure 4.12. Ratio of upstream and downstream deflections of aspect ratio B/L=1/5, 1/3, 1, 3/2 and 3 v/s water depth in different wavelengths (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6 ................................................................................................................................. 64 Figure 4.13. Ratio of upstream and downstream deflections of aspect ratio B/L=1/5, 1/3, 1, 3/2 and 3 v/s wavelengths in different water depths (a) H=20m (b) H=30m (c) H=50m and (d) H=100m ................................................................................................................ 65 x List of Figures Figure 5.1. Plan and elevation views of plate-water problem domain ................................................. 68 Figure 5.2. Validation against Kyoung et al. (2005) in three different cases of seabed slope configurations, (a) H1, H2=15m, 7.5m (b) H1, H2=30m, 15m (c) H1, H2=58.5m, 29.25m for wavelengths α=0.2, 0.6 .................................................................................. 69 Figure 5.3. Various seabed topographies used by Kyoung et al. (2005) (a) slope from upstream point of VLFS to mid-ship of structure (b) slope from mid-ship of structure to downstream point of VLFS (c) slope from upstream to downstream points of VLFS (d) seabed cosine hump ..................................................................................................... 70 Figure 5.4. Centre-line deflections of VLFS on sloping seabed when H1=20m, 30m, 58.5m and 100m and H2=20m in four wavelengths (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6 ...... 73 Figure 5.5. Variation in Wd with respect to the wavelength for (a) =20m (b) =30m (c) =40m and (d) =50m in different upstream water depths H1. .................................... 74 Figure 5.6. Variation in Wd with respect to the water depth for =20m, 30m, 40m and 50m in wavelength (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6. ................................................. 75 xi xii List of Notations List of Notations A Wave amplitude, m B Plate width, m D Plate flexural rigidity, Nm E Young’s modulus, N/m2 g Gravitational acceleration, 9.81m/s2 G Shear modulus, N/m2 h Plate thickness, m H Water depth, m Hw Non-dimensional water depth, H/L H1 Upstream water depth H2 Downstream water depth H1w Non-dimensional upstream water depth H2w Non-dimensional downstream water depth k Wave number, m-1 L Plate length, m N Number of modes considered in the modal expansion method p Number of plate elements in the mesh pw Wave pressure q Number of nodes in the mesh Sj Fluid domain boundaries T Wave period, s xiii List of Notations w, , ( Plate deflection, rotations about x and y axes ) Modal functions and complex amplitudes of the l-th mode of the plate x,y,z Cartesian coordinates x=(x,y,z) Field points ( ) Source points G(x, ) Three-dimensional free surface Green function R(x, ) Distance between source and field point α Incident wavelength-to-structure length ratio θ Incident wave angle Shear correction factor λ Incident wavelength, m ν Poisson’s ratio Plate material density, kg/m3 Water density, kg/m3 Velocity potential of the fluid Incident, diffraction, radiated and scattered potential Radiation potential corresponding to the unit-amplitude motion of l-th modal function Ω Fluid domain ω Wave frequency, rad/s Seabed slope Wu Upstream plate deflection per unit amplitude Wd Downstream plate deflection per unit amplitude Normal strains Shear strains xiv List of Notations Normal stresses Shear stresses The e-th element in the plate mesh (g,h) Natural coordinates of parametric element ( J [ ) Shape functions of 8-node serendipity element Jacobian matrix ] [ Flexural-strain-displacement matrix ] Shear-strain-displacement matrix due to assumed shear strain field [ ] Global flexural stiffness matrix [ ] Global shear stiffness matrix [ ] Global mass matrix [ ] Global hydrostatic stiffness matrix [ ] Global added mass matrix [ ] Global added damping matrix { } Exciting Force vector xv xvi Chapter 1 INTRODUCTION This chapter introduces Very Large Floating Structures (VLFS) as an emerging technology and solution for land creation from the sea. In this chapter, a brief introduction, literature survey and thesis outline are given. The brief introduction describes the VLFS system in general, its applications in the past and the present and its inherent advantages over the traditional land reclamation technique for birthing land from sea. Also, examples of prominent VLFS systems from around the world are mentioned as evidence of their emerging importance. A literature review on various studies, methods and results of VLFS by different researchers is presented. It also shows the gaps between the studies undertaken till now and the real world problems and also provides the backdrop for the present study as a solution to the problem. Lastly, the research objectives of the thesis and its layout are given. 1.1 Background information on VLFS Population explosion and lifestyle preferences for coastal areas are leading to lack of space and amenities. In order to alleviate these demands of coastal land pressure and land scarcity, near shore and ocean space are now emerging as potential frontiers for colonization. Also, oceans offer a plethora of opportunities in terms of renewable energy, food resources, minerals and hydrocarbons. Countries like Japan, Singapore, Monaco, the Netherlands have extended their land mass by land reclamation processes. These methods have numerous topographical limitations and resulted in boundary disputes, environmental disruptions and are cost inefficient. Therefore, VLFS emerged as a promising technology which facilitates ocean space colonization, a sustainable and environmental Chapter 1 friendly technological innovation which enables creation of land on water without disturbing marine environment, polluting coastal waters and disrupting ocean currents. With numerous inherent advantages, VLFS technology is adopted by various countries, viz. Japan, Singapore, United States of America, the Netherlands, Monaco, South Korea and many more densely populated cities and countries around the world. In view of inevitable growing needs of VLFSs, researchers and engineers have studied their behaviour, developed and implemented innovative ideas to make this technology popular and mainstream. Following sub-topics present a general description of VLFS, their applications in different fields and some of their inherent advantages. 1.1.1 Definition and types of VLFS Floating structures are broadly classified as pontoon type and semi-submersible floating structures. Semi-submersible type floating structures are partly raised above the sea level using column tubes or watertight ballast structural compartments at the bottom/hull to minimize the effects of waves while maintaining a constant buoyant force. Therefore they are suitably deployed in deep seas with large waves. Floating oil drilling platforms used for drilling and production of oil and gas and semisubmersible type floating wind farms are typical examples of semi-submersible-type floating structures. Some semi submersibles are transported using outside vessels such as tugs or barges, and some have their own propulsion system for transport and are then properly moored to the sea beds. When these floating structures are attached to the seabed using vertical tethers with high pretension as provided by additional buoyancy of the structure, they are referred as tension-leg platforms. This particular type of floating structures is currently believed to be one of the few serviceable solutions for areas further away from shorelines where waves are larger and have been exploited as early as the 1970s. 2 Introduction Pontoon-type floating structures are in direct contact to the water surface and utilize larger area than semi-submersibles, making them prone to wave motion. Thus the hydroelastic deformations are more significant than rigid deformations, making them more suitable in calm waters. They are basically simple box type structures which float on the surface of the water body and feature high stability for calm sea conditions, e.g., a cove or a lagoon or near the shoreline. In rough sea conditions, being prone to roll and pitch due to large waves and swells, they are installed along with breakwaters or other protective installations. To restrain their movements in horizontal direction, they are anchored to the seabed with the help of mooring lines which can be chains, ropes, sinkers, anchors or tension legs depending on the requirement of the structure. For greater restrain, either the pier/quay wall method or the dolphin-frame guide mooring system may be adopted. The pontoon type floating structures are very cost effective with low manufacturing costs and easy to repair and maintain. As mentioned above, a pontoon type VLFS typically has five major components (see Fig. 1.1), namely (1) an access bridge or a floating walkway from land (2) very large pontoon-type floating structure, VLFS (3) superstructure and facilities (4) a mooring system or a station keeping system (5) a breakwater. Figure 1.1.Components of VLFS system A floating structure can be classified as a VLFS depending on two parameters. First is its length with respect to the wavelength of the incoming wave and second is its characteristic length. 3 Chapter 1 Characteristic length, defined by Suzuki et al. (1996), is a parameter of the structure which is equivalent to the length of the structure influenced due to an equivalent point load applied on it. For a plate, it is expressed as √ where ( ) is the flexural rigidity of the plate and (1.1) is an equivalent spring constant of the fluid on which plate rests. Therefore, a floating structure can be considered as a VLFS if its structural length is larger than its characteristic length and also the wavelength of the incoming wave. Once the floating structure is classified as a VLFS, it is essential to conduct a hydroelastic analysis due to its flexibility under wave action. 1.1.2 Advantages of VLFS Apart from alleviating pressure on land demand, VLFS has many inherent advantages over the traditional land reclamation technique in creating land from sea. These advantages are (Wang et al. 2008): (a) Environmental friendly: Construction and installation of VLFS do not require reclamation and dredging of sand, hence saving the marine habitat and restoring the marine ecosystems. (b) Cost effective: The principle behind working of VLFS is buoyancy which is unaffected by the water depth and seabed porosity. Hence instead of building large structures in deep water region and soft seabed conditions, floating structures are an economical option. Also, due to their floating nature, the foundation systems needed for them are limited to foundations of moorings. (c) Easy and fast construction: VLFS can be constructed as a single module or multiple modules connected together. Different modules can be constructed at different locations simultaneously and can be towed to the desired location and assembled together thereby reducing time and 4 Introduction money. Also they can be dismantled or retracted or moved to facilitate other operations or transport them to different places. (d) Protection from seismic shocks: Superstructures and people on VLFS are insulated from seismic shocks as the VLFS is inherently base-isolated. (e) Internal spaces: Empty watertight spaces in the hull of VLFS create the necessary buoyancy of the VLFS which can be used as a storage space, parking spaces or offices. Next section is dedicated to some prominent VLFS systems around the world and their applications in numerous fields. 1.1.3 Applications of VLFS: Past to present and future Floating structures have been a part of various cultures in the forms of floating homes and villages and floating docks and bridges. They were an innovative idea to create and connect different landmasses or landmass to marine vessels, alleviate traffic pressures, enable movement of equipment and soldiers in the time of war etc. The use of floating structures began in offshore industry when a semi-submersible rig was accidently invented in Gulf of Mexico by Blue Water Drilling Company. Floating structures have evolved in last 50 years in terms of size and serviceability. VLFS technology is now opted other infrastructural areas like floating air base, emergency centre, recreational centre, performing stage, barges and FPSOs. Following is a summary of VLFS’s functional areas and their examples. (a) Floating Air-base: Japan adopted the VLFS technology to build the Mega-float, a one kilometre long and 60 m wide pontoon type floating airplane runway. The near to shore runway is cheaper in comparison to reclaimed landmass. Fig. 1.2 shows two phases of the project undertaken by Japan Government. 5 Chapter 1 Figure 1.2.Two phase floating runway project, the Mega-Float, in Japan (Suzuki et al. 2005) (b) Floating bridges: These longish VLFS enable both road and marine traffic as they are easy to construct and deploy and are an economical alternative in calm waters. Largest floating bridge in the world is in United States of America, Evergreen Point Bridge which is 4750m long and has four lanes of vehicular traffic (Fig. 1.3). Very recent example of floating bridge is in Dubai, which is a temporary arrangement for an upcoming mega-project and can handle 3000 vehicles per hour of vehicular traffic flow in each direction. It is 365m long and 22m wide. It was constructed in a record time of 300 days. An inherited advantage of floating bridges is that they can be constructed in segments which can be retracted, hence allowing marine traffic too. (c) Floating docks and bases: Floating docks have been an important part in army operations and as a connecting link between landmasses and marine vessels. Ujina’s floating pier (see Fig. 1.4(a)) in Hiroshima, Japan is a concrete floating pier which extends in the water and facilitates docking of vessels and movement of goods. Alaska’s floating concrete terminal in Valdez provide services to 50,000 tonnes capacity ships and 5000 tonnes capacity barges. Watertight containers in the concrete units of floating piers provide buoyancy and maintain required freeboard. Small scale floating walkways are used very frequently for recreational purposes on the shore and rivers as shown in Fig. 1.4b. Rescue emergency bases such as in Osaka, Tokyo and Ise bays in Japan, country which is in an earthquake-prone geographical region, are also floating structures which are immune to seismic forces and are designed to accommodate gravity and wave loads. 6 Introduction (a) (c) (b) Figure 1. 3. (a) World’s longest floating bridge, Governor Albert D Rossellini Bridge (aka Evergreen Point Bridge) (b) Yumemai Bridge, Japan (c) Dubai floating bridge, (source: bridgeinfo.org, Wikipedia) (a) (b) Figure 1. 4. (a) Ujina’s floating pier (b)Floating walkway made of High Density Poly-Ethylene (HDPE) modules 7 Chapter 1 (d) Floating offshore structures and facilities: With oil exploration moving to deeper waters, VLFS, both semi-submersible and pontoon types, have emerged like FPSO (floating production storage and offloading) vessels and oil storage units such as Kamigoto storage facility in Nagasaki in Japan which has storage capacity of 7,00,000 kl of fuel and oil storage facility at Pulao Sebarok in Singapore, which holds 3,00,000 cubic meters of fuel. Apart from offshore structure related to hydrocarbon industries, offshore structures like wind-farms, ocean thermal energy conversion (OTEC) platforms, floating wave-energy converters, floating nuclear power plants are also successfully in operation in different parts of world harnessing other sources of energy. Fig. 1.5 encompasses some major offshore VLFS. (e) Floating infrastructures: Two of the major infrastructural applications of floating structures have been covered above, viz. bridges and airplane runways. Other applications include floating entertainment facilities like Marina Bay performing stage in Singapore and Jumbo restaurant in Hong Kong, floating breakwaters like Monaco seawall in Monaco, serving as a breakwater and vehicle parking arena simultaneously. Many architects and engineers around the world have proposed various sustainable floating cities. Some examples are shown in Fig. 1.6. 1.2 Literature survey Hydroelastic analysis of a pontoon type VLFS can be regarded as a fluid-structure interaction problem. Therefore both the fluid part and the structure part have to be modelled in order to simulate the VLFS system. The fluid part, water, is usually assumed to be an ideal fluid, i.e. incompressible, inviscid and has irrotational motion, thereby a velocity potential exists. The motion of the water is represented by velocity potential which is governed by Laplace’s equation. The structure part, VLFS, can be modelled as a one dimensional beam or two dimensional plate structure. Its deflection and stresses denotes its hydroelastic response to wave forces. Early studies to solve the motion of rigid plate using boundary value problem were undertaken by John (1949; 1950). He used Green’s function 8 Introduction for boundary integral formulation to account for wave scattering by rigid bodies. Wehausen and Laitone (1960), on the other hand have studied in detail the linear wave theory and published their study in the famous article ‘Surface Waves’. Their article provided one of the pioneering solutions of the wave-structure interaction problems. Significant work by Bishop et al. (1986) in hydroelastic analysis of floating and fixed offshore structures included three dimensional fluid-structure modelling using of Green’s function method and finite element method. These developments made the feasibility and popularity of VLFS possible. In recent times, major contributors to the development of hydroelastic theory of VLFS are Etekin et al. (1993), Suzuki (1996; 2005), Yago and Endo (1996), Kashiwagi (1998; 2000), Utsunomiya et al. (1998), Ohmatsu (1998; 1999) and Belibassakis (2008). Similar ice-floe problems were studied by Meylan and Squire (1996). Early research includes approximation of a longish VLFS structure as a one-dimensional structure. Longish VLFS denotes that one horizontal dimension of the VLFS is significantly larger than the other one. Such simplistic model can be used to model runways, bridges, walkways and piers. However, such models are overestimated as the wave effects from the edges perpendicular to wave direction is neglected (Yamashita et al. 2003). Also, such models are inadequate to model square type VLFS. More refined model consists of modelling a two-dimensional plate structure on a three dimensional fluid domain. This model has the structure based on Kirchhoff’s plate theory and zero draft assumption (Watanabe et al. 2004). Kirchhoff’s theory assumes that the horizontal dimensions of the structure are very large compared to its depth. The use of such model is proven to be accurate by various researchers like Kashiwagi (1998), Meylan (2001) and Watanabe et al. (2000). But the drawback of this method is that the stress resultants are not accurately predicted at the free edges as they are calculated from approximate deflection derivatives. Also the effects of shear deformation and rotary inertia are neglected. These may lead to erroneous predictions in high frequency vibration and shorter wavelengths. In order to overcome these difficulties, researchers 9 Chapter 1 turned to Mindlin plate theory to model the structure (Wang et al. 2001). In this thesis, the Mindlin plate theory is used. Boundary element method (BEM) and finite element method (FEM) have been widely used to model the fluid part. BEM is more suitable and accurate for linear problems and give almost same level of accuracy as FEM. BEM uses lesser elements and nodes in hydroelastic analysis than FEM which reduces computation time and data storage capacity. Researchers, like Becker (1992), Hermans (2000), popularly use BEM whereas others like Sannasiraj et al. (2000), Kyoung et al. (2005), have used FEM in their formulation of fluid part. Due to linear nature of problem, BEM is used in this thesis as it gives fairly accurate results in a lesser time. Moreover, Belibassakis and Athanassoulis (2005) used coupled-mode model for hydroelastic analysis of large floating bodies. Belibassakis (2008) used hybrid technique of solving the problem using boundary element method and coupledmode model. While designing the VLFS, it is important to fulfil its functional requirements and serviceability, which are directly related to the hydroelastic response of VLFS. Hence understanding the hydroelastic response of the structure in different conditions is important. The behaviour of semiinfinite and strip VLFP (Very Large Floating Platform) in shallow, finite and infinite water depths has been studied by Andrianov and Hermans (2003). They investigated the behaviour of VLFP with different flexural rigidities in different wavelengths and water depths. Xia et al. (2000) studied the effects of incoming wave frequencies on multi-module infinite and semi-infinite two dimensional VLFS. They also investigated the displacements of the plate at the free edges and found that they edge displacements were much larger than that in the middle of the plate. However, main drawback is their assumption of a semi-infinite plate to represent a finite VLFS which may be inaccurate.Yago and Endo (1996) have also studied the effect of different wavelengths and wave angles on the hydroelastic behaviour of a finite two dimensional plate modelled as VLFS resting on two dimensional fluid domain. 10 Introduction (a) (b) (c) (d) (e) (f) Figure 1. 5. (a)Keppel-built 6th Generation Semi-Submersible (b)FPSO vessel in Angola (source: energy-pedia news) (c)Kamigoto Floating Oil Storage Base, Nagasaki Prefecture, Japan (d)Floating nuclear power plant in Russia (source:RIAnovosti) (e)Hexicon, a Swedish design for wind energy farm (source: Main(e) Consulting Ltd.) (f)125 MWE OTEC Plant designed by Dr Alfred Yee 11 Chapter 1 (a) (b) (c) (d) (e) (f) Figure 1. 6. (a) Performing stage at Marina Bay, Singapore (b) Floating breakwater, Monaco seawall, in Monaco (source: FCC const.) (c) Jumbo restaurant in Hong Kong (d) Prof Wang’s proposed floating crab restaurant (e) Lilipads – floating cities(sourc:vincent.callebaut.org) (f) Proposed Greenstar floating hotel and convention centre 12 Introduction The aforementioned studies are largely based on constant seabed configuration. Studies on variable seabed topographies have been done by various researchers like Kyoung et al. (2005), Belibassakis (2008), Wang and Meylan (2002) in both time and frequency domains. Kyoung et al. (2005) studied the effects of different seabed topographies on two dimensional finite VLFS resting on three dimensional fluid domain and used FEM for both structure and fluid part to solve the problem. Studies involving BEM for hydrodynamic analysis of floating bodies in two and three dimensional fluid domains with variable bathymetry region are undertaken by Belibassakis (2008) and Belibassakis and Athanassoulis (2005). However, they considered a semi-infinite thin plate to idealize a floating structure. A more refined model which comprises modelling of the floating structure as a finite thick plate resting on a fluid domain modelled using hybrid FE-BE method (finite elementboundary element method) in frequency domain is presented in this thesis. Such model makes the problem more practical and realistic. 1.3 Research objectives In line with the above studies, the objective of the present research is to refine the VLFS model to make it more generic and carry out parametric studies to understand the hydroelastic behaviour of VLFS in more depth. Therefore the main objectives of this research are:  to investigate the effect of wavelength, water depth and aspect ratios of VLFS on hydroelastic behaviour of pontoon-type VLFS resting on three dimensional fluid domain with constant seabed;  to solve the hydroelastic problem of pontoon-type longish VLFS resting on fluid domain with sloping seabed topography. The parametric study in this thesis involving different parameters like wavelength, water depth and aspect ratios is very important to understand the behaviour of VLFS in a given set of sea state conditions. This study can be used as a benchmark to predict the hydroelastic response of a VLFS in a given set of parameters. Studies relating to the most sensitive regions of a VLFS are also explained. 13 Chapter 1 Also, the hydroelastic behaviour of a longish VLFS modelled as a Mindlin plate on a sloping seabed topography will provide researchers and engineers a deeper insight to design near-shore longish VLFS like runways, bridges and walkways. 1.4 Thesis layout In Chapter 1, the definition and concept of VLFS are explained and the major advantages and their applications are discussed. A brief literature review on different VLFS problems and their modelling and computational methods is given thereafter. Finally, the objectives of the research in this thesis are presented. In Chapter 2, a description of the numerical model is given and the assumptions adopted for the model as stated. Formulations of the floating plate and water are presented and the method of solution, i.e. finite element method for solving the plate deflections and boundary element method for solving the velocity potential are given afterwards. The constant panel method is used to implement boundary element method on the fluid boundaries. Formulations for both uneven and constant seabed topographies are presented and accordingly the solutions are derived. Also, the modal expansion method to decouple and solve the coupled plate-water interaction problem is explained. In Chapter 3, results are obtained from the mathematical formulation. The hydroelastic responses of a longish VLFS in a constant water depth are studied and interpreted to reveal the effects of different wavelengths and water depths. Then, the hydroelastic response at the most sensitive regions of VLFS, i.e. upstream and downstream ends, are studied. In Chapter 4, the hydroelastic behaviours of VLFS with different aspect ratios are studied. The responses of longish, square and widish VLFS differ from each other. The variations of these differences with various wavelengths of incoming wave and different water depths are studied. The overall responses of all VLFS which include their edge deflection and upstream and downstream deflections are studied in detail. 14 Introduction In Chapter 5, the numerical model with irregular seabed condition is used to derive results for a sloping seabed condition. This type of seabed condition is very pertinent to near-shore zones and hence in the design of floating structures like bridges and walkways. The effect of different depths and wavelengths on hydroelastic response of VLFS on sloping seabed is studied. Chapter 6presents the conclusions of the present studies and recommendations for future research work relating to the present problem. 15 16 Chapter 2 HYDROELASTIC ANALYSIS OF VLFS This chapter presents the assumptions and the mathematical formulations of the interaction problem between the VLFS and the water waves. The plate-water model for solving the interaction problem is introduced in the first section. The governing equations of motion and boundary conditions of the VLFS and water are presented. Next, the method of solution for solving the coupled problem of fluidstructure interaction is described. The boundary element method is used to solve the velocity potential and the finite element method is used to obtain the vertical displacement of the VLFS. Modal expansion method is used to decouple the problem. We wish to determine the hydroelastic response of the VLFS under the action of waves. The governing equations and boundary conditions of the structure and fluid parts are described and then a general mathematical formulation to solve the motion of VLFS stimulated by the water waves is presented. Since this fluid-structure interaction problem is coupled in nature, the modal expansion method is adopted to decouple the problem and solve for the water’s velocity potential and VLFS’s vertical deflection. The velocity potential is solved by transforming the decoupled governing equation and boundary conditions of the water into boundary integral equations. The solution of velocity potential can be expressed in terms of the three-dimensional free-surface Green’s function by using the constant panel method in solving the boundary integral equations. Once the velocity potential is obtained, the water pressure acting on the floating structure is known. Then, the VLFS’s vertical deflection is obtained from the decoupled governing equation with the known water pressure. This chapter also includes the numerical solution of the problem using the hybrid finite element and boundary element methods. Therefore, this chapter is the basis of all the subsequent analysis in this thesis. Chapter 2 2.1 Plate-water model The physical system includes a pontoon type VLFS floating on open water body and subjected to water waves. To idealize the VLFS system, an equivalent model is considered in terms of a thick plate floating on a fluid domain (see Fig. 2.1). The length of the plate is , width is and thickness is . For simplicity, we have assumed a zero draft for the floating plate. The system is set in Cartesian coordinates system where the origin is located at the middle of the fluid-structure interface. Thus the plate spans from – to in direction and from of the water region is defined as ( ). The depth variation along to in direction. The free surface and the irregular seabed is expressed as a function of , direction is approximated as step-like variation. Water domain, Ω, is assumed to be closed within the radiation boundaries , seabed SB, fluid-plate interface SHB and free water surface SF. The incident wave of wavelength λ, circular frequency ω and wave amplitude A, is supposed to travel from structure at a wave angle of θ. The deflection to . The incident wave impacts the of the plate is measured from , i.e. free surface of water. In the hydroelastic analysis is of a pontoon-type VLFS, the following assumptions are made (Watanabe et al. 2004):  The VLFS is modelled as an isotropic and elastic thick Mindlin plate with free edges.  The plate is always in contact with the free surface and hence there is no gap between them.  The fluid is ideal, incompressible, inviscid and irrotational so that a velocity potential exists.  The amplitude of the incident wave and VLFS motions are small. 18 Hydroelastic analysis of VLFS SHB SF Ω SB Figure 2.1. Plan and elevation views of plate-water problem domain Based on foregoing assumptions, hydroelastic analysis of VLFS is carried out in frequency domain. Frequency domain analysis is chosen over time domain analysis due to its computational efficiency and its ability to extract steady state responses for appreciable physical insight. However time domain analysis should be adopted when transient responses due to mooring systems or nonlinear waves arise. In subsequent sections, governing equations and boundary conditions of the plate-water model are described. Then numerical solutions of the coupled problem to obtain hydroelastic response of VLFS are presented thereafter. 2.2 Governing equations of motions 2.2.1 Equations of motion of plate The VLFS is modelled as a thick plate governed by the Mindlin plate theory (Mindlin 1951). VLFSs have been modelled based on Euler-Bernoulli beam theory and Kirchhoff thin plate theory before. In 19 Chapter 2 these theories, the effect of transverse shear deformation and rotary inertia are not included. The Mindlin plate theory includes these effects but it needs to introduce a shear correction factor to compensate for the error of not having zero shear stress at the free surfaces. The displacement components in the Mindlin plate theory are transverse displacement ( ) and ⁄ ( and ) respectively. When the bending rotations ⁄ ( ) and bending rotations, and tend to take the values , the plate assumes the classical Kirchhoff plate theory. The governing equations of motion of an isotropic Mindlin plate are as follows (Liew et al. 1998) [( [( ) )( ( ) ( )] ( )( ) (2.1a) ( )] ) (2.1b) [( )( ) ( )( ( )] ) (2.1c) where is shear correction factor, taken as 5/6, density of the plate, Poisson’s ratio, and the shear modulus, ⁄[ the flexural rigidity, as ( ⁄[ ( )], the mass )], E the Young’s modulus, is the circular frequency of the harmonic motion of water waves, ( ) is the pressure from water waves which is the summation of hydrostatic and hydrodynamic pressures, i.e. ( ) | 20 (2.2) Hydroelastic analysis of VLFS where is the mass density of water, the gravitational acceleration and ( ) the fluid velocity potential. Since the plate has free and straight edges, the bending moments, the twisting moments and the transverse shear forces should vanish at the edges (Liew et al. 1998), i.e. ( ( ) ) ( ) ( where and (2.3a) (2.3b) ) (2.3c) are the normal and tangential directions to the plate edge, normal to the edge, the twisting moment at the edge and is the bending moment is the shear force in the direction. Since the problem is solved in frequency domain, we are concerned with the steady state responses of the plate excited by a single frequency sine wave. We assume that the plate vibrations and the water motions have the same frequency, ω. Therefore, the whole system can be treated as a single frequency fluid-interaction problem. The parameters can be written in terms of real part of the product of temporal and spatial variables as ( ) [ ( ) ] (2.4a) ( ) [ ( ) ] (2.4b) ( ) [ ( ) ] (2.4c) ( ( ) ) [ ( ) ] ) [( Ω (2.4d) ] (2.4e) Therefore the governing Eq. (2.1a-c) after substituting above relations and omitting the temporal part, , become 21 Chapter 2 [( [( ) ( )( )] ( ) (2.5a) )( ( )] ) (2.5b) [( )( ( ) )( ( )] ) (2.5c) 2.2.2 Equations of motions of water It is assumed that water is an ideal fluid which is inviscid and incompressible and its motion is irrotational which creates a velocity potential. While considering steady state motions of the harmonically excited fluid-structure system, the velocity potential and the fluid surface elevation, are represented as ( ) ( where subscripts potential [( ) and ( ) ( ) ( ( ) [ ( ) ] )) ] (2.6a) (2.6b) denote incident, diffraction and radiation respectively. Diffracted is the sum of incident wave potential and scattered wave potential generated due to the presence of motionless floating body and radiated potential is the wave potential generated due to the waves generated by moving floating body. The linear wave potential theory is used to model the fluid-motion by using velocity potential, i.e. single frequency velocity potential of water satisfies Laplace’s equation (Wehausen and Laitone 1960) 22 Hydroelastic analysis of VLFS ( ) (2.7) Since there is no flow perpendicular inwards to the impervious seabed, the boundary condition at the seabed is ( ( )) (2.8) where n is the normal unit vector to seabed. The normal vector is perpendicular to the boundary surfaces and has three components in , and direction, namely nx, ny and nz which satisfy the condition nx2+ny2+nz2=1. On the fluid structure interface, linearized kinematic boundary condition which relates the velocity potential perpendicular to the surface boundary displacement is given by ( ) ( ) ( ) (2.9) Linearized kinematic boundary condition denotes zero gaps between plate bottom surface and water surface just below it or in other words it means that the fluid particles do not move across the free water surface. On the free water surface, pressure is constant with respect to time. The linearized dynamic boundary condition on free surface of water is ( ) ( ) (2.10) The velocity potential must also satisfy Somerfield radiation condition at artificial boundary condition at infinity ( ) (Wehausen and Laitone 1960) which is given by 23 Chapter 2 √ where ( ) ( ) (2.11) is the horizontal distance between the origin and the field point and boundary at infinity. The undisturbed incident velocity potential is the artificial fluid due to the incident wave is the solution of boundary conditions Eqs. (2.8)-(2.11) described above. Based on small wave amplitude theory, the velocity potential ( where ) for the incident wave is expressed as ( ( )( ( ))) ( ( ) ( )) ( )( ) (2.12) is the wave amplitude, θ is wave angle, ( ) is the varying water depth, ( ) is the wave number which is related to wavelength ( ) of the wave by ⁄ ( ) ( ) and the circular frequency of water wave ω is related to wave number by the following dispersion relation (Athanassoulis and Belibassakis 1999) ( ) ( ( ) ( )) (2.13a) Depending on the water depth, the dispersion relations takes following forms (Wang et al. 2008) (2.13b) ( ) ( ) ( ) (2.13c) ( ( ) ( )) (2.13d) 24 Hydroelastic analysis of VLFS The effect of surface tension in the dispersion equation has been neglected while modelling the fluid motion. This section has covered a detailed description of governing equations of structure and fluid part. It is evident from Eq. (2.5a) and (2.9) that the unknown parameters, plate’s vertical deflection and velocity potential of fluid, are coupled. To solve this coupled system, the modal expansion method is adopted which is described in a Section 2.4. Next section is dedicated to the numerical methods used for solving the unknown parameters. 2.3 Numerical solutions of fluid-structure system A combination of finite element method for solving plate deflections and boundary element method for solving velocity potential is adopted. Section 2.3.1 describes finite element method and formulation of equation of motion of plate used for solving the plate displacements and Section 2.3.2 describes boundary element method and formulation of boundary integral equation (BIE) for solving fluid velocity potential. 2.3.1 Finite element method for solving plate deflections Finite element method (FEM) is adopted to solve the plate’s equation of motion to obtain plate deflections and rotations. The governing equation of plate motion, Eq. (2.1a-c), is transformed into a variational form using Hamiltonian’s principle (Liew et al. 1998) as described in Appendix A. The derived variational form of the plate motion is 25 Chapter 2 ∫ { [( ( ) [( ) ) ( ( ) ]} ] ) ] ∫ ∫ [ ∫ (2.14) The plate is divided into p number of elements. The element adopted to discretize the floating plate is a rectangular 8-node serendipity Mindlin plate element with four nodes at the corners and four nodes at the mid-points of the edges. Each node has three degrees of freedom, viz. vertical displacement rotation about and axis, and , . Figure 2.2 shows the Mindlin plate element used in this thesis. To make the problem formulation easy, isoparametric concept is applied. The plate elements are transformed to the natural coordinate system, ( ( ) from its physical coordinate system, ) As stated earlier, the motion of a plate is represented by three components, i.e. displacement and two rotations. The displacement field vector ( ) ( ) { ( ) of the plate can be defined as ( ) ( ) ( )} ( ) ( ) [ ( )]{ 26 ( ) ( )} (2.15) Hydroelastic analysis of VLFS ( Physical element, ) Master element, ( ) Figure 2. 2.Mindlin plate element in physical coordinates and its isoparametric transformation in natural coordinates The nodal displacement vector ( ) ( ) has all the degrees of freedom of one plate element as expressed in Eqs. (2.16a-d) { ( ) ( )} { } (2.16a) ∑ ( ) (2.16b) ∑ ( ) (2.16c) ∑ ( ) (2.16d) [ ( )] is the FEM shape function matrix of an 8-node serendipity element in natural coordinate system. The shape function matrix and its nodal components are given below (Petyt 1990) 27 Chapter 2 [ ( )] ( ) ( )] ( ) [ [ ] (2.17) where ( ) ( )( )( )( ( ) ( ( ) ( ) (2.18a) ) (2.18b) ) (2.18c) )( Similarly, the velocity potential can also be approximately calculated in a similar way over the panels which divide the boundary surfaces. The discretization of the boundary surfaces is explained in Section 2.3.3. ∑ ( ) (2.19a) ∑ ( ) (2.19b) The equation of motion, Eq. (2.14), can be condensed by substituting nodal displacement vectors, ( ) and velocity potential [ ] from Eqs. (2.15), (2.16a-d), (2.19), as shown below (Wang and Wang 2008) ([ where [ ], [ ], [ ] [ ] [ ] [ ]){ } [ ][ ] (2.20) ] and [ ] are the global flexural stiffness matrix, shear stiffness matrix, global hydrostatic stiffness matrix, and global mass matrix, respectively. If the FEM elements, the size of the matrices in Eq. (2.20) is 28 is the total number nodes in all . The corresponding local matrix Hydroelastic analysis of VLFS of one element is of size . The R.H.S of the Eq. (2.20) denotes the excitation force imparted by fluid to the structure. ] and [ ] are obtained following procedure explained in ] ,[ The elemental matrices,[ the book by Petyt(1990). [ ] ∫ [ ] [ ] [ [ [ The term [ ] ] [ ] [ ] ∫ ∫ ] (2.21a) ( ) [ ] [ ] is the material constant matrix and [ ][ ( ) (2.21b) ( ) ] (2.21c) ] is the elemental flexural-strain displacement matrix, which is given by [ where ( ) ] [ ] represents partial derivatives of shape functions [ ] and [ , ] (2.23) [ The derivation of[ ] , [ (2.22) ] ] is shown in Appendix A. The elemental shear stiffness matrix [ ] is not as direct to derive as other matrices because of the shear-locking phenomenon encountered while calculating transverse shear strains of plate 29 Chapter 2 element. To address this issue, assumed shear strain field method is used where shear strain is calculated on Gaussian points instead of nodes (Bathe and Dvorkin 1985). A detailed formulation is shown in Appendix A. The final expression of shear stiffness matrix is given by [ where [ [ ( ) ] ( ) ∫ [ ] [ ][ ( ) ] (2.24) ] is the elemental shear-strain displacement matrix due to assumed shear strain field and ] is the material constant matrix. 2.3.2 Boundary element method for solving velocity potentials The boundary element method is used to determine the fluid motion by solving the Laplace equation and the boundary conditions. The three dimensional fundamental solution of Laplace’s equation (Becker 1992) is ( where ) [ ( ] ) (2.25) is a constant representing the strength of the source point in the computational domain, the three dimensional distance between source and field points and ( ) and ( is ). The Green function should satisfy the free water surface boundary condition, seabed boundary condition and Somerfield’s radiation condition at infinity. Therefore, a complimentary solution which satisfies these boundary conditions and governing equations is added to the fundamental solution. The eigen-function expansion form of the three-dimensional free surface Green function for fluid domain of finite water depth, , is given by Linton (1999), which is given by ( ) ∑ ( ) ( ( 30 )) ( ( )) (2.26) Hydroelastic analysis of VLFS ( here ), is the modified Bessel function of the second kind, distance between the source and field points and is the horizontal (m=1,2,3,...) are the positive roots which satisfy the following relation ( while , where ( )) (2.27) is the wave number satisfying the relation, ( ) ⁄ . Equation (2.26) is the total solution that is used to form the following Boundary Integral Equation (BIE) to solve the velocity potential ∫ ∫ (2.28) where [ ] [ ] (2.29) The equations are derived in Appendix B. The free surface Green’s function is the solution to find the velocity potential generated due to the interaction between the wetted plate bottom surface and the seabed. The Green’s function and its derivatives have four components as shown in Eq. (2.29). represents the part of the solution when the source points lie on the surface A and field points lie on surface B. For example, the Green’s function component, , contributes to the solution of velocity potential generated when the source points are on the seabed and field points are on the plate bottom. In the following section, constant panel method to solve the BIE is explained. 31 Chapter 2 2.3.3 Constant panel method to calculate velocity potential As mentioned in Eq. (2.6a), total potential can be divided in radiation potential potential and diffraction based on linear wave theory. Therefore, Eq. (2.28) can be rewritten as ∫ ∫ ∫ (2.30a) ∫ (2.30b) After applying the boundary conditions, the final equation becomes ∫ ∫ (2.31) { The above equation can be solved for radiation and diffraction potential using constant panel method (Brebbia et al. 1984). The derived fluid potential is used to obtain wave forces on the floating body. The normal derivative of the three-dimensional free surface Green’s function is calculated on the plate bottom surface and seabed surface. The four components of the Green’s function and derivative of Green’s function encompass the effect when the source points and field points are on seabed and plate bottom. The normal derivatives of three-dimensional free surface Green’s function are calculated as follows ( where and are the and ) [ ] [ ] (2.32) components of the normal vector to the field surface pointing outwards with respect to the source point. The expression for ( 32 ), ( ) and ( ) are as follows Hydroelastic analysis of VLFS ( ) ( ) ( ∑ ( ∑ ( ) ∑ ( ) ( ) ( ( ( )) ( ( ))) | | (2.32a) ( ( )) ( ( ))) | | (2.32b) ) ( ( )) ( ( )) (2.32c) In the constant panel method, the fluid velocity potential and its derivative are calculated on the rectangular panels which divide the boundary surfaces. The boundary surfaces S HB and SB are discretized into panels in the same way the plate bottom is discretized using FEM. The fluid potential is assumed to be constant within each panel. It is calculated on the centre of the panels and is integrated over the area of the panel. The two integrals in Eq. (2.31) are approximately calculated as where the subscript ∫ ∑ ∫ ∑ ( ) (2.33a) ∫ ( ) (2.33b) ∫ denotes -th boundary element whose centroid acts as source point and is the total number of panels on the boundary surface. The Green’s function and its derivative can be written in the form of matrix. The number of rows in the matrices is equal to the number of source points and number of columns is equal to the number of field points in the fluid domain. For example, the Green matrix component, calculated from number of source points on the seabed on number of field points on the plate bottom, can be represented as 33 Chapter 2 ∫ ( ( ) ∫ ( ) ) (2.34) ∫ ( ∫ ( ) ) [ ] Likewise, the derivatives of the Green’s function are calculated on every panel on the boundary surfaces SHB and SB. After calculating all four components of three dimensional free surface Green’s function matrices and their derivatives, the radiation and diffraction velocity potentials on the plate bottom are solved. As stated earlier, the problem which we are dealing with is a coupled problem as the plate deflection ( ) and velocity potential are interdependent. Therefore decoupling methods are employed to decouple the problem and solve the equation of motion of plate for plate deflection ( 2.4 ). Modal expansion method Modal expansion method is employed to decouple the plate-water interaction problem. In modal expansion method, the plate deflection and fluid velocity potential are expressed as a series expansion of terms, shown in Eqs. (2.35a-b). The plate deflection is a sum of modal function ( ), and complex amplitude ; and radiation potential, of series of product of unit amplitude of -th modal function Taylor and Waite 1978). The complex amplitude terms of series of product of , is a sum of , and complex amplitude, terms (Eatock is assumed to be same for both plate displacement and radiation potential (Newman 1994). ( ) ∑ ( 34 ) (2.35a) Hydroelastic analysis of VLFS ( The modal function [ problem([ ] [ ) ∑ ( ) (2.35b) ] is obtained from free vibration analysis of the plate by solving the eigen- ]){ } [ ]{ }. The diffracted potential is the sum of incident and scattered velocity potential as stated before. They represent the incident and outgoing waves from the structure body respectively. The expanded forms of plate deflection and velocity potential, when substituted to the Laplace’s equation and boundary conditions of the fluid domain, yield the following set of equations (Wang et al. 2008) ( ( ) ) Ω ( { ) ( ( √ ( (2.36b) ( )) ) (2.36a) (2.36c) ( ) ) (2.36d) (2.36e) By substituting the decoupled boundary condition Eq. (2.36b) in Eq. (2.31), the boundary integral equation can be rewritten as ∫ ∫ (2.37) { 35 Chapter 2 By solving Eq. (2.37) using constant panel method as discussed Section 2.3.3, the radiation potential and diffracted potential related to the plate motion can be solved. The solutions can be expressed in terms of four components of the Green’s function matrix and its derivative matrix as [( [ ̃] ) ] { } ( { } [( ) ( [( ) ) ( ( ) ) ] ) ( ( ) ) ]{ } [ ] { } (2.38a) [( ) ] ( { } where [( ) ( is the number of nodes on the plate, discretized, [ ) ) ( ) [ ] (2.38b) is the number of panels in which plate bottom is is the number of panels in which the seabed is discretized, ]is the matrix of ] is the identity matrix, eigen-vectors found from the free-vibration analysis of floating plate, corresponding to the plate deflection. { } is the displacement vector. { } and { } are the global matrix required to proportionally distribute the physical quantities calculated at centroid to the whole area of the panel and nodes of the panel, respectively. We have used uniform 8-node serendipity element to discretize the boundaries. Therefore, the value of elements in matrices { } and { } is (1/8). In the case where the plate is resting on fluid domain with constant seabed depth, the expressions for radiation and diffraction potential reduce to the following 36 Hydroelastic analysis of VLFS [( [ ̃] ) ] { } [( { } ) [( { } ) ] where ( ] [( ) [( ) ]{ } [ ] ] [ ] { } (2.39a) (2.39b) ) is easily calculated as (Meylan and Squire 1996) ( ( ) The total velocity potential,[ ] [ ] [ ) (2.39c) ], is substituted in the plate’s equation of motion to solve for the plate displacement{ }. The matrix representation of the plate equation is ([ [ ] ] [ ] [ ]){ } ([ ] [ ]) (2.40) After substituting the decoupled versions of radiation and diffracted potential in above equation, it can be re-written as [ ] ([ ] [ ] [ ] [ ] [ ] [ ]) [ ] { } [ ] { } { } (2.41) where [ ], [ ], [ ], [ ], [ ] and [ ] are the global flexural stiffness matrix, global stiffness matrix, global hydrostatic stiffness matrix, global mass matrix, global added mass matrix and global added damping matrix respectively. [ ] is modal function matrix obtained by free-vibration analysis of the Mindlin plate and is equal to [ ] . The terms of the matrix contain 37 eigen- Chapter 2 vectors corresponding to the plate deflection, rotation about and axes. { } is the excitation force matrix. The added mass matrix, added damping matrix and excitation forces are calculated as [ [ [ ([ ̃ ] ] ([ ̃ ] ] ] ) [ ] [ ) ] (2.42a) (2.42b) (2.42c) where[ ̃ ] is the unit-amplitude radiation potential. Upon solving the coupled water-plate equation (2.41), we obtain the complex amplitudes { } and then we back substitute the amplitudes into Eq. (2.35a) to obtain the deflection and rotations of the plate, { }. 2.5 Summary This chapter presents the assumptions and governing equations of the fluid and structure part of the fluid-structure interaction problem. In the hydroelastic analysis, water is modelled as an ideal fluid and its motion is assumed to be irrotational so that a velocity potential exists. The VLFS is modelled as a plate structure according to the Mindlin plate theory. Finite element method is adopted to solve the plate motion and boundary element method is used to solve the fluid velocity potential or the water motion. The constant panel method is used to derive the velocity potential corresponding to plate motion. In the hydroelastic analysis, modal expansion method is used to decouple the fluidstructure interaction problem and eventually the equation of plate motion is solved for plate displacements. In subsequent chapters, present plate-water model is used to study the hydroelastic response of the VLFS. In particular, the effects of water depth, wavelength, aspect-ratio of floating structure and sloping seabed topography on the hydroelastic behaviour of VLFS are analysed and compared. 38 Chapter 3 EFFECT OF WAVELENGTH ON HYDROELASTIC BEHAVIOUR OF VLFS This chapter presents the effect of wavelength on the hydroelastic response of VLFS. A numerical plate- water model is adopted to study the behaviour of VLFS. The numerical model is based on the hydroelastic analysis formulation as described in Chapter 2. The numerical model is verified by comparing the deflections of VLFS on constant seabed with existing results of the same model. Then, the verified model is used to study the hydroelastic behaviour of VLFS subjected to different wavelengths and water depths. 3.1 Numerical model The plate-water model explained in Chapter 2 is used for the hydroelastic analysis. Consider a rectangular VLFS of length , width , and height as shown in Fig. 3.1. The VLFS is modelled as a plate with free edges resting on a fluid domain Ω, with constant water depth . In non-dimensional form, the water depth reads , i.e. . The water is assumed to be an ideal fluid, i.e. it is inviscid, incompressible and its motion is irrotational so that a velocity potential, domain is bounded by radiation boundary ).The hull bottom is represented as , free water surface . An incident wave ,( exists. The fluid ) andseabed, , of wave period , ( and wave amplitude , is assumed to enter from the left side of the domain. It strikes the VLFS at an angle θ and its wavelength is represented in terms of length of the structure, , where α is the non- dimensional wavelength. The particulars of the model is adopted from Yago and Endo (1996)and tabulated in Table 3.1. Chapter 3 In the hydroelastic analysis of the floating plate, the size of the BEM elements is kept smaller than ⁄ ⁄ (Gao et al. 2011) to obtain converged results. The deflection results are expressed as deflections per unit amplitude, |w|/A, and centre-line deflections, i.e. the deflections of the plate in direction at . Also, to denote the deflections of the VLFS at end points, upstream deflection Wu, and downstream deflection, Wd are used. Wu is the deflection of the VLFS at the end where the wave strikes it, the wave leaves it, and and . Wd is the deflection of the VLFS at the end where . SHB SF Ω SB Figure 3.1.Plan and elevation views of plate-water problem domain Table 3.1. Parameters of VLFS-water model used by Yago and Endo (1996) Parameter Total length Total width Total height Density Young’s modulus of elasticity Poisson’s ratio Water depth Wavelength to structure length ratio Symbol ρ Unit m m m kg/m3 GN/m2 ν α 40 m m/m Value 300 60 2 0.25625 11.9 0.13 20m-150m 0.2,0.4,0.5,0.6 Effect of wavelength In order to validate the present model, we study the response of a VLFS under constant water depth and compared with existing results. Figure 3.2 shows the centre-line deflection along the longitudinal axis of the continuous plate obtained from the present numerical model and the experimental results obtained by Yago and Endo (1996). It can be seen that both the results are in well agreement with each other. Thus it validates and confirms the accuracy of the formulation and the numerical model presented in this thesis. It is worth noting that the plate-water model adopted by Yago and Endo (1996) used a two-dimensional structure and a two-dimensional fluid whereas a three-dimensional plate-water model is used herein. 1.4 1.4 α = 0.2 1.2 present study Yago and Endo (1996) |w|/A (m/m) |w|/A (m/m) 1 0.8 0.6 0.4 present study Yago and Endo (1996) 1 0.8 0.6 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 0 1 1.4 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0.9 1 1.4 α = 0.5 1.2 present study α = 0.6 1.2 Yago and Endo (1996) |w|/A (m/m) 1 |w|/A (m/m) α = 0.4 1.2 0.8 0.6 0.4 present study Yago and Endo (1996) 1 0.8 0.6 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x/L Figure 3.2.Centre-line deflections of VLFS in four different wavelengths (a) α=0.2 (b) α=0.4 (c) α=0.5and (d) α=0.6 in water depth and H=58.5m 3.2 Effect of wavelength on hydroelastic behaviour of VLFS In this section, the effect of wavelengths on hydroelastic behaviour of VLFS in different water depths is studied. Figure 3.3 shows the centre-line deflections of the VLFS in three different water depths, = 20m, 58.5m and 100m subjected to wavelengths α = 0.2, 0.4, 0.5 and 0.6. As expected, irrespective 41 Chapter 3 of water depth, the response of the VLFS increases as the wavelength of the incoming wave increases. Also, the downstream deflections, Wd, are larger when long waves (α = 0.5 and 0.6) are considered and lesser in short waves (α = 0.2 and 0.4). The reason for this behaviour is that a longer wave, which has low frequency, transmits most of its wave energy across the VLFS more than a shorter wave, which has a high frequency (Xia et al. 2000). (a) 1.4 H = 20m 1.2 α = 0.2 α = 0.4 |w|/A (m/m) 1 α = 0.5 0.8 α = 0.6 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.8 0.9 1 0.8 0.9 1 x/L (b) 1.4 α = 0.2 α = 0.4 α = 0.5 α = 0.6 Yago and Endo (1996) H = 58.5m 1.2 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (c) 0.1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 1.4 H = 100m 1.2 α = 0.2 α = 0.4 |w|/A (m/m) 1 α = 0.5 0.8 α = 0.6 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x/L Figure 3.3.Centre-line deflection of VLFS subjected to different wavelengths, α=0.2,0.4,0.5and 0.6 in water depths (a) H=20m (b) H=58.5m and (c) H=100m 42 Effect of wavelength 3.3 Effect of water depth on hydroelastic behaviour of VLFS The effect of water depth on the hydroelastic behaviour of VLFS is studied by considering four water depths, = 20m, 40m, 58.5m and 100m. Figure 3.4 shows the centre-line deflection amplitudes of the VLFS in the four water depths for four cases of wavelengths of incoming wave, α = 0.2, 0.4, 0.5 and 0.6. Figure 3.4 shows the dependence of hydroelastic response of the VLFS on water depth. The response profile of VLFS for = 58.5m and = 100m are very close, whereas some differences can be seen in the VLFS response in shallower water depth, = 20m and 40m. Therefore, the effect of water depth is significant when VLFS is in a shallow and intermediate water depth range. It is also observed that in the case of shorter wavelengths, the centre-line deflection of VLFS in shallow water depths is smaller than that in deeper water depths (see Fig. 3.4(a)-(b)). This can be attributed to the motion of water particles in deep and shallow water depths. Fluid particles in deeper water depths can complete their circular paths whereas in shallow water depths, their motions are suppressed to elliptical paths. In contrast to this behaviour, the response of VLFS in shallow water depths for larger wavelength cases is higher than that in deep water depths (see Fig. 3.4(c)-(d)). This behaviour can be attributed to the transfer of energy from the particles which move in elongated elliptical paths. The energy of the particles in elliptical paths is transferred to the fluid particles of neighbouring crest. Such interaction and transfer of potential energy result in increased response of VLFS. This type of interaction is absent in deeper water waves as the motion of the particles is completely circular. This is in contrast to the generalization by Tavana and Khanjani (2013) that the response of VLFS increases with the water depth. In summary, the effect of water depth on the hydroelastic response of the VLFS is coupled with the effect of wavelength of the incoming wave. However, a more significant effect on hydroelastic response by the water depth is observed only in the downstream region of the VLFS. As for the upstream region of the VLFS, hydroelastic response is negligibly small by changing water depth by changing water depth. 43 Chapter 3 (a) 1.4 α = 0.2 1.2 H=20m H=40m |w|/A (m/m) 1 H=58.5m 0.8 H=100m 0.6 0.4 0.2 0 0 (b) 0.1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 1.4 α = 0.4 1.2 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (c) 0.1 1.4 α = 0.5 1.2 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (d) 0.1 0.2 1.4 α = 0.6 1.2 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 Figure 3.4. Centre-line deflections of VLFS in three different water depths, H=20m, 40m, 58.5m and 100m for (a) α=0.2 (b) α=0.4 (c) α=0.5and (d) α=0.6 44 Effect of wavelength 3.4 Upstream and downstream deflections of VLFS While studying the effect of water depth in Section 3.3, it is observed that the hydroelastic response of the VLFS is affected significantly in the downstream region and is negligible in the upstream side of the VLFS. Also, the downstream deflections of a longish VLFS are always less than upstream deflections. Figure 3.5 shows the variation of deflection magnitude of the upstream and downstream sides of the VLFS, Wu and Wd with respect to water depth. The upstream deflections of VLFS become constant in deeper waters, irrespective of the wavelengths it is subjected to (see Fig. 3.5(a)). But the variation is notable in the shallow water region. The upstream deflection gradually increases as the water depth increases when the wavelengths of the incident wave is very small (α 0.1), whereas the upstream deflections decrease with increasing water depth, when the wavelengths of the incident wave are long (α 0.2). The reason for this is that when shorter waves are considered, the energy imparted by incident wave on the upstream portion of the VLFS is governed by the motion of the fluid particles as interaction between neighbouring particles is comparatively lesser in shorter waves than longer waves. Therefore, as the water gets deeper, the motion of the particles tends to get circular from elliptical which results in higher response of VLFS. When larger wavelengths are considered, the interaction of neighbouring fluid particles become dominant phenomenon which reduces with increasing water depths. After the deep water condition is attained, i.e. ⁄ , the upstream deflection becomes constant as the motion of the fluid particles become completely circular and stable and hence remain unaffected by deeper water depths. Similar to the upstream deflections, the downstream deflections, Wd, are also affected in shallow and intermediate water depth regions and become constant in deeper water depth regions. An interesting observation from Fig. 3.5(b) is that the downstream deflection in short and intermediate wavelengths, (0.1 α 0.4) follow a same pattern which is a gradual increase with increasing water depth. This can be accounted for by the loss of energy of the short water wave while navigating through the length of VLFS. However, the downstream deflections of VLFS subjected to larger 45 Chapter 3 wavelengths, α = 0.5 and 0.6, gradually decrease in same fashion as their corresponding upstream deflections. The trend-line of Wd in longer waves (α = 0.5 and 0.6) is higher than the trend lines of Wd in shorter waves (α = 0.1 and 0.2) as more energy is transmitted by longer waves and dissipated by VLFS in case of shorter waves. Lower values of Wd in α = 0.5 than those in α = 0.3 and 0.4 in deeper water depths is due to the VLFS property and this trend may change when the flexural rigidity of the VLFS is altered. Lower values of Wd denote comparatively stable downstream region in a longish VLFS. In practice, the stability of the VLFS is crucial to maintain its functionality and serviceability. Based on the results presented in Sections 3.2-3.4, it can be found that the most sensitive regions of a VLFS are its upstream and downstream regions. A longish VLFS is comparatively stable in the centre region with mild crests and troughs. It is necessary to study the end displacement in a VLFS as they are usually the connecting interfaces with the land-masses or vessels. Also, stability of the structure in end regions is necessary in structures like emergency base, performing stage or a floating pier used for loading and unloading. The parametric studies in this chapter will also help to take proper measures to mitigate the response of the VLFS at appropriate locations. 46 Effect of wavelength (a) 1.2 α = 0.1 |Wu| /A (m/m) 1 α = 0.2 0.8 α = 0.3 0.6 α = 0.4 0.4 α = 0.5 0.2 α = 0.6 0 0 20 40 60 80 100 120 140 120 140 Water depth, H (m) (b) 0.4 |Wd| /A (m/m) 0.3 0.2 0.1 0 0 20 40 60 80 100 Water depth, H (m) Figure 3.5.Variations of (a) upstream and (b) downstream deflections of VLFS with respect to water depth, H , in different wavelengths (α=0.1, 0.2, 0.3, 0.4, 0.5 and 0.6) 3.5 Summary Hydroelastic behaviour of a longish VLFS is investigated by a parametric study. Effects of wavelengths and constant water depths on the global behaviour of VLFS are investigated in detail. Major observations and conclusions are:  The hydroelastic response of the VLFS increases as the wavelength of incident water wave increases. 47 Chapter 3  Water depth also has significant impact on the behaviour of VLFS. The behaviour of VLFS in a given water depth depends on the wavelength of the incident wave. When VLFS is subjected to shorter wavelength, the response of VLFS in shallow water depths is smaller and increases as the wavelength increases. However, in large wavelengths, the response of VLFS in shallow water is larger than that in deep water depths. This behaviour depends on two factors: the motion of excited fluid particles and transfer of energy between neighbouring particles. The former is dominant in short wavelengths and latter is more significant in larger waves.  The most affected areas in a VLFS are the upstream and downstream regions and therefore they are studied to understand the global behaviour of VLFS. Again, the variation in upstream and downstream deflections depends more on the wavelength than water depth. 48 Chapter 4 EFFECT OF ASPECT RATIO ON HYDROELASTIC BEHAVIOUR OF VLFS This chapter presents the effect of aspect ratio on the hydroelastic response of VLFS. The validated numerical plate-water model in Chapter 2 is adopted to study the behaviour of VLFS. Hydroelastic analyses are carried out for VLFS having different aspect ratios, ranging from longish VLFS to square VLFS and widish VLFS. Their behaviours in different water depths and wavelengths are studied. Also, their upstream and downstream deflections are analysed. 4.1 Numerical model The plate-water model presented in Chapter 2 is adopted for analysis. Consider a rectangular VLFS of length , width and height . The VLFS is modelled as a plate with free edges resting on a fluid domain Ω, with constant water depth . In non-dimensional form, the water depth reads , i.e. . In order to study the effect of aspect ratio, the length of the VLFS, , is assumed constant and the width , is altered to make the VLFS longish, square and widish. The water is assumed to be an ideal fluid, i.e. it is inviscid, incompressible and its motion is irrotational so that a velocity potential, exists. The fluid domain is bounded by radiation boundary ), and seabed period ,( ). The hull bottom is represented as , free water surface . An incident wave ,( , of wave and wave amplitude , is assumed to enter from the left side of the domain. It strikes the VLFS at an angle θ and its wavelength is represented in terms of length of the structure, .A schematic diagram is shown in Fig. 4.1. The particulars of the model is adapted from Yago and Endo (1996) and tabulated in Table 4.1. Chapter 4 SHB SF Ω SB Figure 4.1.Plan and elevation views of plate-water problem domain Table 4.1. Parameters of VLFS-water model used by Yago and Endo (1996) Parameter Symbol Total length Total width Total height Density Young’s modulus of elasticity Poisson’s ratio Water depth Wavelength to structure length ratio ρ Unit Value m 300 m m kg/m3 GN/m2 60,100,300,450,900 2 0.25625 11.9 0.13 20m-150m 0.2,0.4,0.5,0.6 ν m α The hydroelastic behaviour of VLFS is expressed in terms of displacements per unit amplitude, |w|/A. Centre-line deflection is the deflection of VLFS in deflection of VLFS in direction at direction at and edge-line deflection is the . The deflections at both edges of VLFS are assumed to be same as the motion of VLFS is symmetrical in head sea condition. Also, upstream and 50 Effect of aspect ratio of VLFS downstream deflections, Wu and Wd are used to denote end point deflection of VLFS at ⁄ 4.2 ⁄ and , respectively. Effect of aspect ratio (B/L) on hydroelastic behaviour of VLFS To study the effect of aspect ratio on VLFS, the width VLFS is changed to make the aspect ratio, constant, square and (dimension of VLFS in y direction) of the = 1/5, 1/3, 1, 3/2 and 3, while keeping the length = 300m. The VLFS with aspect ratio, < 1 are geometrically longish, = 1 are > 1 are geometrically widish. The centre-line response of VLFS with the aforementioned aspect ratios are shown in Fig. 4.2 in water depth, = 20m and four wavelengths, α = 0.2, 0.4, 0.5 and 0.6. As shown in Fig. 4.2(a), it is observed that in shorter wavelength, α = 0.2, the response of longish VLFS is very different from square and widish VLFS. Larger deflections are obtained when 1. Also, the crests and troughs formed in the widish VLFS are significantly larger than longish VLFS. These large deflections signify lesser stability and hence lesser serviceability in widish VLFS. Moreover, the upstream and downstream deflections, Wu and Wd increase with increasing aspect ratios. There is a large increase in the hydroelastic response of VLFS when approaches values greater than or equal to one. The values of Wu and Wd in square and widish VLFS are approximately two times than that of longish VLFS. The reason of these higher values of deflections can be contributed to the diffraction of waves. In longish VLFS, the waves are diffracted away when they strike the edge of the structure. But as the width of the structure becomes larger, the diffraction of waves in the middle part along the width of the VLFS is lesser. Hence the waves travel beneath the floating structure, increasing its overall response. Higher Wd values in higher ratio are because there is very less diffraction of short waves by square and widish VLFS and hence most of the wave energy is transmitted through the plate. 51 Chapter 4 (a) 1.2 α = 0.2 B/L=1/5 B/L=1/3 B/L=1 B/L=3/2 B/L=3 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (b) 0.1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 1.2 α = 0.4 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (c) 0.1 0.2 1.2 α = 0.5 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (d) 0.1 0.2 1.2 α = 0.6 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 0.1 0.2 Figure 4.2. Centre-line deflections of VLFS with five aspect ratio, B/L=1/5,1/3,1,3/2 and 3, in water depth, H=20m for (a) α=0.2 (b) α=0.4 (c) α=0.5and (d) α=0.6 As the wavelength increases to α = 0.4, 0.5 and 0.6, the response of the VLFS increases. The upstream deflections become closer for all aspect ratio, whereas the downstream deflections are 52 Effect of aspect ratio of VLFS different irrespective of the wavelength of incident wave. It is because longish VLFS diffract the waves and less energy is transmitted to its downstream end, but in square and widish VLFS, there is less to negligible diffraction of waves due to which the wave transfers its energy to the structure’s central and downstream regions. As stated above, less diffraction of waves causes different hydroelastic behaviour of square and widish VLFS. It is also observed that the edge-line deflection is different than the centre-line deflection of the VLFS with ratio more than and equal to one. Figures 4.3 to 4.5 shows the hydroelastic response of the VLFS in water depth = 20m and wavelengths α = 0.2 and 0.6 for aspect ratios 1/5, 1 and 3 respectively. The associated centre-line and edge-line deflections of the VLFS are extracted and compared in Figs. 4.3 to 4.5. As shown in Fig. 4.3, the centre-line and edgeline deflections in longish VLFS are nearly the same in the two wavelengths considered when subjected to head sea wave condition. For , the difference between the centre-line and edge- line deflections becomes significant, as shown in Figs. 4.4 and 4.5. This difference is because for a given wavelength of incoming water waves, there is less diffraction of waves by square and widish VLFS than longish VLFS. Therefore, the un-diffracted wave is transmitted through the length of VLFS, increasing its response in and near centre-line. On the edges of the VLFS, the incoming waves are diffracted thereby reducing the transmission and hence edge-line deflections of VLFS. This difference between the centre-line and edge-line deflections increases as the width of the VLFS increases or in turn the diffraction of waves decreases. The difference between the centre-line and edge-line deflections is more significant in shorter waves when compared to large wavelength. This is because transmission of wave energy is more in larger wavelengths than shorter wavelengths. Therefore the edge-line deflections of square and widish VLFS are higher in larger wavelengths than shorter wavelengths and therefore closer to centre-line deflections. 53 Chapter 4 1.4 edge-line 1 1.2 1 0.8 0.8 0.6 |w|/A |w|/A (m/m) centre-line α = 0.2 1.2 0.4 0.6 0.4 0.2 0.2 0 0 30 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 0 -30 0 y/(m) 1 250 200 50 100 300 150 x/(m) 1.4 edge-line 1 1.2 0.8 1 0.6 0.8 |w|/A |w|/A (m/m) centre-line α = 0.6 1.2 0.4 0.2 0.6 0.4 200 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 0 30 0 -30 0 y/(m) 1 50 100 250 300 150 x/(m) Figure 4.3.Centre-line and edge-line response of VLFS of aspect ratio, B/L=1/5 subjected to wavelength of α=0.2and 0.6 in water depth, H=20m 1.4 1.2 1 1 0.8 |w|/A |w|/A (m/m) centre-line edge-line α = 0.2 1.2 0.6 0.4 0.6 0.4 0.2 0.2 0 150 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 1.4 0.8 0.9 75 1 200 0 y/(m) -75 -150 0 50 100 250 300 150 x/(m) centre-line edge-line α = 0.6 1.2 1.2 1 1 0.8 |w|/A |w|/A (m/m) 0.8 0.6 0.4 0.8 0.6 0.4 0.2 0.2 0 150 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 75 200 0 y/(m) -75 -150 0 50 100 150 x/(m) Figure 4.4.Centre-line and edge-line response of VLFS of aspect ratio, B/L=1 subjected to wavelength of α=0.2 and 0.6 in water depth, H=20m 54 250 300 Effect of aspect ratio of VLFS 1.4 1.2 |w|/A |w|/A (m/m) centre-line edge-line α = 0.2 1.2 1 0.8 0.4 0 450 0.8 0.6 300 0.4 150 0 y/(m) 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 -150 -300 1 -450 0 -450 0 100 200 x/(m) 100 200 x/(m) 300 1.4 α = 0.6 1.2 centre-line edge-line |w|/A |w|/A (m/m) 1.2 1 0.8 0.4 0 450 0.8 0.6 300 150 0.4 0 y/(m) 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 -150 -300 1 300 Figure 4.5.Centre-line and edge-line response of VLFS of aspect ratio, B/L=3 subjected to wavelength of α=0.2 and 0.6 in water depth, H=20m 4.3 Effect of water depth on hydroelastic behaviour of VLFS with different aspect ratios The coupled effect of aspect ratio and water depth on the hydroelastic behaviour of VLFS of three aspect ratio, = 1/5, 1 and 3 is studied by considering four water depths, = 20m, 40m, 58.5m and 100m. As mentioned in Section 4.2, the centre-line and edge-line deflection patterns are different in square and widish VLFS. Figures 4.6 to 4.8 show the response of VLFS with aspect ratios 1/5, 1 and 3 in water depth = = 100m for wavelengths α = 0.2 and 0.6. It can be seen that there is significant difference in the VLFS response when compared to their counterpart in shallower water depth, = 20m as shown in Figs. 4.3 to 4.5, especially for shorter wavelength, e.g. α = 0.2. 55 Chapter 4 1.4 1 1.2 0.8 1 0.6 0.8 |w|/A |w|/A (m/m) centre-line edge-line α = 0.2 1.2 0.4 0.2 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 1.4 0.9 0 30 1 0 -30 0 y/(m) 50 100 250 200 150 x/(m) 300 centre-line edge-line α = 0.6 1.2 1 1.2 0.8 1 0.6 0.8 |w|/A |w|/A (m/m) 0.6 0.4 0.2 0.6 0.4 200 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 0 30 1 0 -30 0 y/(m) 50 100 250 300 150 x/(m) Figure 4.6. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1/5 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m 1.4 α = 0.2 centre-line edge-line 1 1.2 1 0.8 |w|/A |w|/A (m/m) 1.2 0.6 0.4 0.8 0.6 0.4 0.2 0.2 0 150 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 75 1 200 0 y/(m) -75 -150 0 50 100 250 300 150 x/(m) 1.4 α = 0.6 centre-line edge-line 1 1.2 1 0.8 |w|/A |w|/A (m/m) 1.2 0.6 0.4 0.8 0.6 0.4 0.2 0.2 0 150 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 75 200 0 y/(m) -75 -150 0 50 100 150 x/(m) Figure 4.7. Centre-line and edge-line response of VLFS of aspect ratio, B/L=1 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m 56 250 300 Effect of aspect ratio of VLFS 1.4 α = 0.2 1.2 centre-line edge-line |w|/A |w|/A (m/m) 1.2 1 0.8 0.4 0 450 0.8 300 0.6 150 0.4 0 y/(m) 0.2 -150 0 -300 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 -450 0 100 200 x/(m) 300 1.4 α = 0.6 centre-line edge-line 1 1.2 |w|/A |w|/A (m/m) 1.2 0.8 0.8 0.4 0 450 0.6 300 0.4 150 0 y/(m) 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 -150 -300 1 -450 0 100 200 x/(m) 300 Figure 4.8. Centre-line and edge-line response of VLFS of aspect ratio, B/L=3 subjected to wavelength of α=0.2 and 0.6 in water depth, H=100m Figure 4.9 shows the centre-line deflection amplitudes of the VLFS in four water depths (i.e., = 20m, 40m, 58.5m and 100m) and two wavelengths (i.e., α = 0.2 and 0.6). For longish VLFS, as stated in Chapter 3, the hydroelastic response is slightly lower in shallow water if short wavelengths are considered. Inversely, the response in shallow water depth is higher than deep water depth when larger wavelengths are considered as shown in Fig. 4.9 (a). It can be observed from Figs. 4.9 (b)-(c) that in short wavelength, α = 0.2, for square and widish VLFS, with hydroelastic response is higher in = 1 and 3, the overall = 20m water depth than in deeper water depths, = 40m, 58.5m and 100m. The gradual change can be noticed in the figure. This is opposite to the behaviour of longish VLFS as shown in Fig. 4.9(a). Also, it is evident that like longish VLFS, the hydroelastic behaviour of both the square and widish VLFS increases as the wavelength of incoming wave increases. Andrianov and Hermans (2003) studied the effect of water depth and wavelength on the hydroelastic behaviour of a strip VLFS (infinite in the 57 direction). They found that the response of a Chapter 4 strip VLFS increases as the wavelength increases this finding is in agreement with the present results for the widish VLFS. 1.4 1.4 α = 0.2 α = 0.6 H=20m 1.2 1.2 H=40m 1 H=58.5m 0.8 |w|/A (m/m) |w|/A (m/m) 1 H=100m 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 x/L 0.5 0.6 0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1 0.6 0.7 0.8 0.9 1 x/L (a) 1.4 1.4 α = 0.2 α = 0.6 H=20m 1.2 1.2 H=40m 1 H=58.5m 0.8 |w|/A (m/m) |w|/A (m/m) 1 H=100m 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 x/L 0.5 x/L (b) 1.4 1.4 α = 0.2 α = 0.6 H=20m 1.2 1.2 H=40m 1 H=58.5m 0.8 |w|/A (m/m) |w|/A (m/m) 1 H=100m 0.6 0.8 0.6 0.4 0.4 0.2 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x/L 0 0.1 0.2 0.3 0.4 0.5 x/L (c) Figure 4.9.Centre-line deflection of VLFS of different aspect ratio (a) B/L=1/5 (b) B/L=1 (c) B/L=3, in four water depths, H=20m, 40m, 58.5m and 100m subjected to two wavelength α=0.2 and 0.6 58 Effect of aspect ratio of VLFS The upstream deflection, Wu, is higher in shallow water depth and reduces as the water depth increases as seen in Fig. 4.9(b)-(c). It is because the interaction between fluid particles reduces in deep water depths, as they follow circular paths, and hence there is negligible transfer of energy to the fluid particles below upstream region of VLFS. However, the downstream deflections, Wd are higher in all water depths unlike Wu as the un-diffracted waves travel along the VLFS amplifying its response. In case of large wavelengths, α = 0.6, the response of VLFS follows same trend like longish VLFS. It is higher in shallow water depth and lowers in deep water depth. The transfer of energy taking place in neighbouring particles in large wavelengths and shallower water depths results in larger response. This response of widish VLFS at its upstream end in shorter wavelength (see Fig. 4.9(c)) is in agreement with the results ofAndrianov and Hermans (2003) who studied the hydroelastic behaviour of strip VLFS (infinite in the direction). They found that the upstream response of the strip VLFS in short wavelength and in infinite water depth greatly decreases when compared to finite water depth. They also noted that the deflections of the structure increases as the water depth increases. This observation may be true if the VLFS is widened infinitely so that it approximates the strip VLFS model treated by Andrianov and Hermans (2003). It is evident from the results in Fig. 4.9 that the behaviour of a longish VLFS is very different from a widish VLFS. Therefore a longish VLFS should not be assumed as an infinitely wide VLFS. 4.4 Effect of aspect ratios on upstream and downstream deflections of VLFS From the results discussed in Sections 4.2 and 4.3, it is observed that the variations of upstream and downstream deflections of square and widish VLFS in different water depths and wavelengths are interestingly very different from longish VLFS. Figure 4.10 shows the upstream deflection of VLFS with aspect ratios = 1/5, 1 and 3 in different water depths. From Fig. 4.10, it is confirmed that for longish VLFS, the hydroelastic response is increased as wavelength increases. Wu becomes constant after deep water condition, α 2, is attained. As the aspect ratio, 59 approaches values more Chapter 4 than and equal to unity, the trends are different than that of longish VLFS. For wavelengths, α = 0.2, maximum Wu is observed in shallow water depth, 1, in shorter = 20m, and it decreases gradually and become constant in deep water depths as shown in Figs. 4.10(b)-(c). When wavelength increases from α = 0.4 to α = 0.6, it can be seen that there is a shift in values of Wu in square and widish VLFS, when water depth increases from = 20m to = 60m. For depths greater than 60m, maximum upstream response is seen when wavelength α = 0.4. In widish VLFS, the response decreases as the wavelength increases after 0.4. Such trend depends on the flexural rigidity of the VLFS and may change if the flexural rigidity of VLFS is altered. For the present numerical model of VLFS, maximum upstream response of square and widish VLFS is recorded at α = 0.4 in deeper water depths. Also, in deeper water depths, the values of Wu in square and widish VLFS are less than longish VLFS. It is because in deeper water depths, the width to wavelength ratio is lesser for longish VLFS as compared to square to widish VLFS which implies more diffraction and hence higher response of VLFS. Conversely, in = 20m, Wu increases as the increases which is because the transfer of energy to the VLFS through the fluid particles of a less diffracted wave is more. (a) 1.4 1.2 α = 0.2 |Wu| /A (m/m) 1 0.8 α = 0.4 0.6 α = 0.5 0.4 α = 0.6 0.2 0 0 20 40 60 80 100 Water depth, H (m) 60 120 140 Effect of aspect ratio of VLFS (b) 1.4 |Wu| /A (m/m) 1.2 1 α = 0.2 0.8 α = 0.4 0.6 α = 0.5 0.4 α = 0.6 0.2 0 0 20 40 60 80 100 120 140 120 140 Water depth, H (m) (c) 1.4 1.2 |Wu| /A (m/m) 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 Water depth, H (m) Figure 4.10.Displacement of upstream point of VLFS in different water depths and wavelengths for aspect ratio (a) B/L=1/5 (b) B/L=1and (c) B/L=3 Figure 4.11(a)-(c) shows the variation of downstream deflection of VLFS with three aspect ratios, = 1/5, 1 and 3 in different water depths. It is evident from the figure that the variation in downstream deflection of VLFS with different aspect ratios is lesser with respect to changing wavelengths and water depths as compared to upstream deflections. For longish VLFS, maximum downstream deflection occurs at maximum wavelength considered. For square and widish VLFS, the downstream deflections in most of the water depths considered in wavelengths more than 0.2 are in the same range. 61 Chapter 4 (a) 1.2 1 |Wd| /A (m/m) α = 0.2 0.8 α = 0.4 0.6 α = 0.5 0.4 α = 0.6 0.2 0 0 20 40 60 80 100 120 140 120 140 120 140 Water depth, H (m) (b) 1.2 |Wd| /A (m/m) 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 Water depth, H (m) (c) 1.2 |Wd| /A (m/m) 1 0.8 0.6 0.4 0.2 0 0 20 40 60 80 100 Water depth, H (m) Figure 4. 11. Displacement of downstream point of VLFS in different water depths and wavelengths for aspect ratio (a) B/L=1/5 (b) B/L=1 and (c) B/L=3 62 Effect of aspect ratio of VLFS 4.4.1 Effect of water depth It is observed from Fig. 4.12 that Wu/Wd is higher for longish VLFS and closer to unity for square and widish VLFS. For longish VLFS subjected to shorter and intermediate wavelengths, α = 0.2 and 0.4, the ratio, Wu/Wd decreases gradually as the water depth decreases and become constant as deep water condition is attained. For longer wavelengths, α = 0.5 and 0.6, the ratio, Wu/Wd, increases gradually with increasing depth and become constant in deep waters. This result is in agreement with the observations made in Chapter 3 for longish VLFS. For aspect ratio, 1, the ratio, Wu/Wd is close to, but greater than unity in shallow water depths, indicating higher upstream deflections. As the depth increases, the ratio becomes less than one as the upstream deflections become less than downstream deflections. Also, when deep water condition is attained, the ratio becomes constant. This trend is observed in all wavelengths. Hence higher values of ratio Wu/Wd in longish VLFS indicate higher diffraction of waves or higher dissipation of wave energy in upstream region of VLFS, thus ensuring comparatively higher stability in central and downstream regions of VLFS. (a) |Wu/Wd| (m/m) 3 B/L=1/5 2 B/L=1/3 B/L=1 1 B/L=3/2 B/L=3 0 0 20 40 60 80 100 Water Depth, H (m) 63 120 140 Chapter 4 (b) 8 |Wu/Wd| (m/m) 7 6 B/L=1/5 5 B/L=1/3 4 B/L=1 3 B/L=3/2 2 B/L=3 1 0 0 20 40 60 80 100 120 140 120 140 120 140 Water Depth, H (m) (c) 6 |Wu/Wd| (m/m) 5 4 3 2 1 0 0 20 40 60 80 100 Water Depth, H (m) (d) |Wu/Wd| (m/m) 2 1 0 0 20 40 60 80 100 Water Depth, H (m) Figure 4.12.Ratio of upstream and downstream deflections of aspect ratio B/L=1/5, 1/3, 1, 3/2 and 3 v/s water depth in different wavelengths (a) α=0.2 (b) α=0.4 (c) α=0.5and (d) α=0.6 64 Effect of aspect ratio of VLFS 4.4.2 Effect of wavelength In the Section 4.4.1, the variation of ratio, Wu/Wd in different water depths are presented. In this section, the relation between Wu/Wd and wavelength is discussed. It can be seen in Fig. 4.13 that irrespective of water depth and wavelength, Wu/Wd is higher in longish VLFS. As the VLFS becomes square or widish, the value of Wu/Wd becomes closer to one. In short wavelengths, the value of Wu/Wd for all aspect ratios are similar and are in same range. The same is true for large wavelengths. Highest difference between upstream and downstream deflections occurs in intermediate wavelengths, e.g. α = 0.3 to 0.5. It is also observed from Fig. 4.13 that the Wu/Wd curve for longish VLFS becomes smaller as the water depths increase whereas the curve is similar for widish VLFS. 8 8 H = 20m |Wu / Wd| (m/m) 7 H = 30m 7 B/L=1/5 6 6 5 5 4 4 3 3 B/L=3/2 2 2 B/L=3 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6 B/L=1/3 B/L=1 0 0.1 0.2 Wavelength, α (m/m) (a) 0.4 0.5 0.6 0.5 0.6 (b) 8 8 H = 50m 7 |Wu / Wd| (m/m) 0.3 Wavelength, α (m/m) 6 6 5 5 4 4 3 3 2 2 1 1 0 0.1 0.2 H = 100m 7 0.3 0.4 0.5 0.6 0 0.1 Wavelength, α (m/m) 0.2 0.3 0.4 Wavelength, α (m/m) (c) (d) Figure 4.13.Ratio of upstream and downstream deflections of VLFS with aspect ratio B/L=1/5, 1/3, 1, 3/2 and 3 v/s wavelengths in water depths (a) H=20m (b) H=30m (c) H=50m and (d) H=100m 65 Chapter 4 4.5 Summary The effects of wavelengths and water depths are studied for different aspect ratios of VLFS and a comparative study is made. The hydroelastic behaviour of a square and widish VLFS is very different from a longish VLFS. Major observations are enlisted below:  The overall response of square and widish VLFS is very different and higher as compared to that of longish VLFS. The upstream and downstream deflections and intermediate crests and troughs of square and widish VLFS are larger than those of longish VLFS for a given water depth and wavelength. Major contributor to this behaviour may be due to the less diffraction of waves by square and widish VLFS.  In comparison to the centre-line deflections, the edge-line deflections of a VLFS are nearly same in longish VLFS but lower in square and widish VLFS when subjected to head sea condition, for a given water depth and wavelength.  In shallow water depths, the hydroelastic response of square and widish VLFS increases as the wavelength of the incoming wave increases. The overall deflections are significantly large in short wavelengths as compared to longish VLFS whereas in larger wavelengths, downstream deflections are significantly large. The number of crests and troughs also decreases for longer waves.  In deep water depths, the upstream deflection of square and widish VLFS decreases drastically in short wavelengths. However there is not much difference in their behaviour for larger wavelengths.  The difference between the upstream and downstream deflections of a square or widish VLFS is less when compared to a longish VLFS, which has significantly higher difference between upstream and downstream deflections, indicating lesser response in downstream region of VLFS.  The present result of the hydroelastic behaviour of a widish VLFS in short wavelength is corresponding to results by Andrianov and Hermans (2003). Also, their observation about increasing hydroelastic response of their strip VLFS model when wavelength increases, agrees with the present corresponding results of the widish VLFS. 66 Chapter 5 EFFECT OF SEABED SLOPE ON HYDROELASTIC BEHAVIOUR OF VLFS This chapter presents the effect of seabed slope on the hydroelastic response of VLFS. A numerical plate-water model is adopted to study the behaviour of VLFS. The numerical model is based on the hydroelastic analysis formulation as described in Chapter 2. The model is validated by comparing the present results with the literature results of the same VLFS model. The validated model is then used to study the effect of different sloping seabed topographies on the hydroelastic responses of VLFS in different wavelengths and water depths. 5.1 Numerical model The plate-water model explained in Chapter 2 is used for analysis. A rectangular VLFS of length , width , height is modeled as a plate with free edges resting on a fluid domain Ω, as shown in Fig. 5.1. The seabed has a slope µ, which is defined as ( where ) is the water depth at upstream point of VLFS and is the water depth at downstream point of VLFS, as shown in Fig. 5.1. The varying water depth in the sloping seabed region can be written in terms of , ( ) direction and the water depths where and , since it is constantly varying in can be expressed as ( ) ( ) Chapter 5 and are non-dimensionalized water depths, i.e. and . The water is assumed to be an ideal fluid, i.e. it is inviscid, incompressible and its motion is irrotational so that a velocity potential surface wave ,( exists. The fluid domain is bounded by radiation boundary ), and seabed , of wave period ,( ( )). The hull bottom is represented as , free water . An incident and wave amplitude , is assumed to enter from the left side of the domain from upstream region of VLFS. It strikes the VLFS at an angle θ and its wavelength is represented in terms of length of the structure, . A schematic diagram is shown in Fig. 5.1. The particulars of the model are adopted from Kyoung et al. (2005) and tabulated in Table 5.1. The hydroelastic behaviour of VLFS is expressed in terms of displacements per unit amplitude, |w|/A. Centre-line deflection is the deflection of VLFS in direction at . Also, upstream and downstream deflections, Wu and Wd are used to denote end point deflection of VLFS at ⁄ and ⁄ , respectively. SHB SF H2 H1 SB Figure 5.1. Plan and elevation views of plate-water problem domain 68 Effect of sloping seabed Table 5.1. Parameters of VLFS-water model used by Kyoung et al. (2005) Parameter Symbol Total length Total width Total height Density Young’s modulus of elasticity Poisson’s ratio Water depth, Upstream Downstream Wavelength to structure length ratio ρ Value m m m kg/m3 GN/m2 300 60 2 0.25625 11.9 0.13 20, 30, 40, 50 20 to 100 0.2,0.4,0.5,0.6 ν m m α 1.4 1.4 α = 0.2 1.2 present study Kyoung et al. (2005) 1 α = 0.6 1.2 |w|/A (m/m) |w|/A (m/m) Unit 0.8 0.6 0.4 0.2 present study Kyoung et al. (2005) 1 0.8 0.6 0.4 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 (a) 1.4 1.4 α = 0.2 α = 0.6 1.2 1 |w|/A (m/m) |w|/A (m/m) 1.2 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0 0.1 0.2 (b) 1.4 1.4 α = 0.2 α = 0.6 1.2 1 |w|/A (m/m) |w|/A (m/m) 1.2 0.8 0.6 0.4 0.2 1 0.8 0.6 0.4 0.2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 x/L 0.7 0.8 0.9 1 0 0.1 0.2 (c) Figure 5.2.Validation against Kyoung et al. (2005) in three different cases of seabed slope configurations, (a)H1, H2=15m, 7.5m (b) H1, H2=30m, 15m (c)H1, H2=58.5m, 29.25m for wavelengths α=0.2 and 0.6 69 Chapter 5 5.2 Effect of seabed slope on hydroelastic behaviour of VLFS Studies related to different seabed topographies have been mentioned in many research studies like that of Kyoung et al. (2005), Wang and Meylan (2002), Belibassakis and Athanassoulis (2005) and Belibassakis (2008). Kyoung et al. (2005) used finite element formulation for both plate and fluid parts and studied different seabed topographies including three different seabed slope configurations and a cosine seabed hump. They found that the effect of seabed topography is in the downstream region and significant in shallower water depths. Also the effect of seabed configuration in Fig.5.3(a) is the most and that of configuration in Fig. 5.3(c) is the least when compared to other mentioned topographies. VLFS Seabed (a) (b) (c) (d) Figure 5.3.Various seabed topographies used by Kyoung et al. (2005) (a) slope from upstream point of VLFS to mid-ship of structure (b) slope from mid-ship of structure to downstream point of VLFS (c) slope from upstream to downstream points of VLFS (d) seabed cosine hump Wang and Meylan (2002) also studied the effect of seabed hump on the hydroelastic behaviour of a two-dimensional thin plate structure using FEM formulation and found that the reflection of waves increases significantly in variable depth variable when compared to constant seabed case. Also the bending of the plate increases. 70 Effect of sloping seabed Belibassakis and Athanassoulis (2005) studied the effect of variable seabed topography on large floating bodies in time domain. They assumed an infinitely wide thin plate of uniform thickness as VLFS, resting on three seabed topographies, viz. flat, shoaling and corrugated. They found that shoaling seabed topography results in a significant increase in the deflection in upstream region of the floating body and a slight decrease in deflections of downstream region. A recent paper by Papathanasiou and Belibassakis (2014) has used hybrid couple mode-FEM formulation for hydroelastic analysis of VLFS of constant and varying thickness in sloping seabed topography. They found that the wavelength of hydroelastic wave affecting the structure decreases as it travels from upstream to downstream side. In this chapter, a detailed study on the effects of changing seabed slopes and wavelengths on the hydroelastic behaviour of a finite longish VLFS is studied using a three dimensional model solved using hybrid BEM-FEM techniques. The sloping seabed topography is very common in near shore areas and generally VLFS are deployed in such regions, for example, bridges, walkways, emergency bases, floating piers and jetties. Therefore, it is necessary and practical to study its impact on the hydroelastic behaviour of VLFS. To study the effect of sloping seabed on the hydroelastic behaviour of VLFS, different water depths in upstream and downstream regions are considered. The water depth in the downstream point of VLFS, is assumed to be constant and different upstream water depths, are considered. The wavelength of the water wave changes according to dispersion relation as the water depth changes when the wave travels. Simultaneously, due to the sloping topography of the seabed, waves are reflected back in the fluid domain. These two factors affect the response of the VLFS. Fig. 5.4 shows the centre-line response of VLFS in four wavelengths, α = 0.2, 0.4, 0.5 and 0.6 in four upstream water depths, = 20m, 30m, 58.5m and 100m and downstream water depth, The upstream water depth, = 20m. ranges from 20m to 100m, i.e. constant seabed to maximum slope, respectively. It can be observed that the most sensitive region to the change in slope of seabed 71 Chapter 5 is the downstream region of VLFS. Kyoung et al. (2005) also noted that the middle and downstream regions of VLFS are most affected by different seabed topographies. The results from Belibassakis and Athanassoulis (2005) suggest that the deflection for an infinitely wide plate significantly increases in its upstream region whereas its downstream region is less affected. It can be seen that in shorter wavelengths, α = 0.2 and 0.4, the response of VLFS increases as the upstream depth increases or in other words, seabed slope increases (see Fig. 5.4(a)-(b)). As the wavelength increases to α = 0.5 and 0.6, the response of VLFS decreases significantly in central and downstream regions of VLFS as increases (see Fig. 5.4(c)-(d)). In summary, a larger slope of the seabed in shorter wavelengths increases the hydroelastic response of the VLFS while it decreases its response for larger wavelengths. Figure 5.5 shows the variations of downstream deflection Wd of the VLFS for four cases of downstream water depths, = 20m, 30m, 40m and 50m. In each case one downstream water depth, and four upstream water depths, are considered. It is noticed that in short wavelengths, downstream deflection of VLFS, Wd increases slightly as the slope of the seabed increases, i.e. the upstream water depth, increases. Conversely, Wd decreases significantly as slope increases in larger wavelengths. Maximum change in response is noted in largest wavelength considered in this study, α = 0.6. Also, the decrease in the VLFS response, Wd becomes more significant when the downstream water depth, is shallower. This implies that the effect of seabed slopes in shallower water depths is more significant than in deeper water depths. 72 Effect of sloping seabed (a) 1.2 α = 0.2 H1 = 1 20m, constant depth |w|/A (m/m) 30m 0.8 58.5m 100m 0.6 0.4 0.2 0 0 (b) |w|/A (m/m) 1.2 0.1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 y α = 0.4 1 B=60m 0.8 z 0.6 H1 x L=300m H2=20m µ 0.4 0.2 0 0 (c) 0.1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 x/L 0.6 0.7 0.8 0.9 1 1.2 α = 0.5 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 (d) 0.1 1.2 α = 0.6 |w|/A (m/m) 1 0.8 0.6 0.4 0.2 0 0 0.1 Figure 5.4.Centre-line deflections of VLFS on sloping seabed when H1=20m, 30m, 58.5m and 100m and H2=20m in four wavelengths (a) α=0.2 (b)α=0.4 (c) α=0.5 and (d) α=0.6 73 Chapter 5 (a) 0.8 H1 = H2 = 20m 20m, constant 30m 58.5m 100m |Wd | /A (m/m) 0.6 0.4 0.2 0 0.2 0.3 0.4 0.5 0.6 0.5 0.6 0.5 0.6 0.5 0.6 Wavelength, α (m/m) (b) 0.8 H1 = |Wd | /A (m/m) H2 = 30m 30m, constant 40m 58.5m 100m 0.6 0.4 0.2 0 0.2 0.3 0.4 Wavelength, α (m/m) (c) 0.8 H1 = |Wd | /A (m/m) H2 = 40m 40m, constant 50m 58.5m 100m 0.6 0.4 0.2 0 0.2 0.3 0.4 Wavelength, α (m/m) (d) 0.8 H1 = |Wd | /A (m/m) H2 = 50m 50m, constant 0.6 58.5m 100m 0.4 0.2 0 0.2 0.3 0.4 Wavelength, α (m/m) Figure 5.5. Variation in Wd with respect to the wavelength for (a) =20m (b) and(d) =50m in different upstream water depths, H1 74 =30m (c) =40m Effect of sloping seabed Figure 5.6 shows the effect of water depth on the hydroelastic behaviour of VLFS. In the wavelength cases considered, the decrease in the values of Wd in comparison to constant seabed case, is maximum when the downstream water depth, is 20m. In other words, maximum change in the response of VLFS is observed when the change of slope occurs in shallower water domain. This is because shallow water depth in downstream region implies more interaction of waves with the seabed and hence larger reflection of waves. 0.8 0.8 α = 0.2 α = 0.4 H2 = 0.6 |Wd | /A (m/m) |Wd | /A (m/m) 0.6 0.4 0.2 20m 30m 0.4 40m 50m 0.2 0 0 0 20 40 60 80 100 0 Upstream Water Depth, H1 (m) 20 40 60 (a) 100 (b) 0.8 0.8 α = 0.5 α = 0.6 0.6 |Wd | /A (m/m) 0.6 |Wd | /A (m/m) 80 Upstream Water Depth, H1 (m) 0.4 0.2 0.4 0.2 0 0 0 20 40 60 80 100 0 Upstream Water Depth, H1 (m) 20 40 60 80 100 Upstream Water Depth, H1 (m) (c) (d) Figure 5.6. Variation in Wd with respect to the water depth for =20m, 30m, 40m and 50m in wavelength (a) α=0.2 (b) α=0.4 (c) α=0.5 and (d) α=0.6 Moreover, there is very less difference in the response of VLFS if the downstream water depth, increases (from 20m to 50m) in shorter wavelengths, α = 0.2 and 0.4. However, the 75 Chapter 5 difference becomes significant in larger wavelengths, α = 0.5 and 0.6. It may be attributed to transfer of energy between fluid particles as their circular paths gradually transform to elliptical paths in a given water depth as wavelength increases, combined with the reflection of waves from the seabed as the slope of the seabed increases. When the downstream water depth, condition, ⁄ satisfies deep water , the response of the VLFS becomes constant and does not change if any further increment in either upstream or downstream depths are made. 5.3 Summary To study the effect of sloping seabed on the hydroelastic behaviour of a longish VLFS, different water depths in upstream and downstream points of VLFS are considered and the response of VLFS is studied in different sets of water depths. Major observations are enlisted below:  Downstream region of the VLFS is the most sensitive part of the VLFS which is affected by sloping seabed topography. The downstream response of a longish VLFS on a sloping seabed is lesser than the response of VLFS in the corresponding constant water depth. In shorter wavelengths, the downstream response of the VLFS increases as the upstream water depth increases whereas in larger wavelengths, opposite is observed.  Maximum decrease in the downstream response of VLFS occurs in large wavelengths and shallow water depths. This conclusion agrees with the findings by Kyoung et al. (2005). The effect of wavelength diminishes if the seabed slope is in deeper water regions. 76 Chapter 6 CONCLUSIONS AND RECOMMENDATIONS This chapter presents the summary and key findings drawn from the parametric studies of VLFS under wave action. Also, future studies related to present study are suggested. 6.1 Conclusions The objective of this thesis was to understand the hydroelastic behaviour of VLFS and to study the effect of different parameters like wavelengths, water depth, aspect ratios and sloping seabed topography on its hydroelastic response. To achieve the objective, results from parametric study of a validated model are analysed and conclusions are presented in Chapters 3, 4 and 5. The results are expressed as deflections per unit amplitude of water wave. In Chapter 3, the effect of wavelength is investigated on the hydroelastic response of longish VLFS. The overall response of the VLFS increases as the wavelength increases in any given water depth. It is also observed that increasing water depth has different effects on the hydroelastic response of VLFS in different wavelengths. For shorter wavelengths, α = 0.2 and 0.4, the response of the VLFS increases as the water depth increases. Conversely, for larger wavelengths, α = 0.5 and 0.6, the hydroelastic response of VLFS decreases as water depth increases. Deflections of most sensitive regions of VLFS, viz. upstream, Wu, and downstream, Wd, deflections are also studied. Both upstream and downstream regions are more sensitive to variation in wavelength than water depth. Also, the variation in Wu with respect to changing wavelengths is more than that in Wd. It is observed that the response of VLFS becomes constant once the deep water condition is attained and further increase in water depth does not affect the VLFS response. Chapter 6 In Chapter 4, the effect of wavelengths and water depths on VLFS of different aspect ratios is investigated. The hydroelastic behaviour of longish VLFS was discussed in Chapter 3. The hydroelastic behaviour of square and widish VLFS is studied and compared. The edge-line deflections of square and widish VLFS are less than centre-line deflections unlike longish VLFS where both are same. It is observed that square and widish VLFS have higher overall responses than longish VLFS. Also, the undulations in the hydroelastic responses for square and widish VLFS are more prominent which mean lesser overall evenness of VLFS surface. The upstream and downstream deflections of VLFS with different aspect ratios are also studied. There is a drastic drop in upstream deflections of square and widish VLFS in short wavelengths as the depth increases. The variation in larger wavelengths is very small. Also, the parameter, Wu/Wd, is also studied to study the overall response and stability of VLFS. It is observed that the ratio Wu/Wd is closer to unity in square and widish VLFS and has higher values for longish VLFS which indicates lower deflection in downstream regions of longish VLFS than square and widish VLFS. In Chapter 5, the effect of sloping seabed topography on the hydroelastic behaviour of longish VLFS is studied. For the study, different upstream and downstream water depths are considered, where the upstream water depths are always deeper than downstream water depths. It is observed that seabed slope greatly affect the VLFS downstream region. For a given downstream depth, the downstream response of the VLFS decreases as upstream depth increases or in other words, when slope increases. Also it is observed that the effect of seabed slope is more in shallow water depths than the same slope in deeper water depths. In sum, the studies provided a better understanding of the hydroelastic behaviour of VLFS under different sea state and seabed conditions. The results would be useful to offshore structural engineers for preliminary analysis and can serve as benchmark solutions for numerical analysis. 78 Conclusions and Recommendations 6.2 Recommendations In this thesis, we have studied the hydroelastic response of VLFS for different parameters such as wavelength, water depth, aspect ratio of VLFS and sloping seabed. This study can be extended to following areas for a more elaborate understanding and realistic scenarios:  The VLFS is approximated as Mindlin plate with zero draft. Zero draft assumption is valid as the depth of the plate is very small as compared to the length of the VLFS. The effect of draft becomes significant when the ratio of thickness to length of plate is greater than 0.005 (Petyt, 1990). Thus to refine the model, the effect of draft can be included.  The present parametric study can be extended to analyse the hydroelastic behaviour in different seabed topographies like a mild hump, stepped bottom and varying slopes.  The response of VLFS with single or multiple elastic connectors on varying seabed topography can be studied. 79 80 References Andrianov, A. I. and Hermans, A. J. (2003). The influence of water depth on the hydroelastic response of a very large floating platform. Marine Structures 16(5):355–371. Athanassoulis, G.A., Belibassakis, K.A. (1999). A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. Journal of Fluid Mechanics, 389: 275-301. Bathe, K. J. and Dvorkin, E. N. (1985). A 4-node plate bending element based on Mindlin Reissner plate-theory and a mixed interpolation. International Journal for Numerical Methods in Engineering 21(2): 367–383. Becker, A. A. (1992). The Boundary Element Method in Engineering: A Complete Course. London; New York, McGraw-Hill. Belibassakis, K.A. (2008). A boundary element method for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Engineering Analysis with Boundary Elements 32: 796-810. Belibassakis, K. A. and Athanassoulis, G. A. (2005). A coupled-mode model for the hydroelastic analysis of large floating bodies over variable bathymetry regions. Journal of Fluid Mechanics 531: 221-249. Belibassakis, K. A. and Athanassoulis, G. A. (2004). Three-dimensional Green’s function for harmonic water waves over a bottom with different depths at infinity. Journal of Fluid Mechanics 510: 267-302. Bishop, R. E. D., et al. (1986). A General Linear Hydroelasticity Theory of Floating Structures Moving in a Seaway. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 316(1538): 375-426. Brebbia, C. A., Telles, J. C. F. and Wrobel, L. C. (1984).Boundary Element Techniques: Theory and Applications in Engineering. Berlin; New York, Springer-Verlag. References Clough, R. W. and Penzien, J. (1993). Dynamics of Structures. New York, McGraw-Hill. Eatock Taylor, R. and Waite, J. B. (1978). The dynamics of offshore structures evaluated by boundary integral techniques. International Journal for Numerical Methods in Engineering 13(1): 73–92. Ertekin, R. C., Riggs, H. R., Che, X. L. and Du, S. X. (1993). Efficient methods for hydroelastic analysis of very large floating structures. Journal of Ship Research 37(1): 58–76. Gao, R. P., Tay, Z. Y., Wang C. M. and Koh, C. G. (2011). Hydroelastic response of very large floating structure with a flexible line connection. Ocean Engineering 38(17-18): 1957-1966. Hermans, A. J. (2000). A boundary element method for the interaction of free-surface waves with a very large floating flexible platform. Journal of Fluids and Structures 14(7): 943–956. Hinton, E. and Huang, H. C. (1986).A family of quadrilateral Mindlin plate elements with substitute shear strain fields. Computers & Structures 23(3): 409–431. John, F. (1949). On the motion of floating bodies-I. Communications on Pure and Applied Mathematics 2(1): 13–57. John, F. (1950). On the motion of floating bodies-II. Simple harmonic motions. Communications on Pure and Applied Mathematics 3(1): 45–101. Kashiwagi, M. (1998). A b-spline galerkin scheme for calculating the hydroelastic response of a very large floating structure in waves. Journal of Marine Science and Technology 3(1): 37–49. Kashiwagi, M. (2000). Research on hydroelastic responses of VLFS: Recent progress and future work. International Journal of Offshore and Polar Engineering 10(2):81–90. Kyoung, J. H., Hong, S. Y., Kim, B.W., Cho and Cho, S. K. (2005). Hydroelastic response of a very large floating structure over a variable bottom topography. Ocean Engineering 32(17-18): 2040-2052. 82 References Liew, K. M., Wang, C. M., Xiang, Y. and Kitipornchai, S. (1998). Vibration of Mindlin Plates: Programming the p-version Ritz Method. Oxford, UK, Elsevier. Linton, C. M. (1999). Rapidly convergent representations for Green’s functions for Laplace’s equation. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 455(1985): 1767–1797. MATLAB (2012). Mathworks. www.mathworks.com. Meylan, M. H. (2001). A variational equation for the wave forcing of floating thin plates. Applied Ocean Research 23(4): 195–206. Meylan, M. H. and Squire, V. A. (1996).Response of a circular ice floe to ocean waves. Journal of Geophysical Research 101(C4): 8869–8884. Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. Journal of Applied Mechanics 18: 31–38. Newman, J. N. (1994). Wave effects on deformable bodies. Applied Ocean Research 16(1): 47–59. Newman, J. N. (1977). Marine Hydrodynamics. Cambridge, Mass., USA, MIT Press. Ohmatsu, S. (1998). Numerical calculation of hydroelastic behavior of pontoon type VLFS in waves. Proceedings of 17th International Conference on Offshore Mechanics and Arctic Engineering, Lisbon, Portugal, pp.5–9. Ohmatsu, S. (1999). Numerical calculation method of hydroelastic response of a pontoon-type VLFS close to a breakwater. Proceedings of Third International Workshop on Very Large Floating Structures, Honolulu, Hawaii, USA, pp.805–811. Papathanasiou, T. K. and Belibassakis, K. A. (2014). Hydroelastic analysis of VLFS based on consistent coupled-mode system and FEM. The IES Journal Part A: Civil & Structural Engineering, 7:3, 195-206. 83 References Paulling, J. R. and Tyagi, S. (1993). Multi-module floating ocean structures. Marine Structures 6(2– 3): 187–205. Petty, M. (1990). Introduction to Finite Element Vibration Analysis. Cambridge (UK); New York, Cambridge University Press. Price, W. G. and Wu, Y. (1985). Hydroelasticity of Marine Structures. Theorectical and Applied Mechanics. F. I. Niordson and N. Olhoff, Elsevier Science Publisher: 311–337. Sannasiraj, S. A., Sundar, V. and Sundaravadivelu, R. (1995). The hydrodynamic behaviour of long floating structures in directional seas. Applied Ocean Research 17(4): 233–243. Sarpkaya, T. and Isaacson, M. (1981). Mechanics of Wave Forces on Offshore Structures. New York, Van Nostrand Reinhold Co. Seto, H. and Ochi, M. (1998).A hybrid element approach to hydroelastic behavior of a very large floating structure in regular wave. Proceedings of 2nd International Conference on Hydroelasticity in Marine Technology, Fukuoka, Japan, pp.185–194. Stoker, J. J. (1957). Water Waves: The mathematical theory with applications. New York, Interscience Publishers Suzuki, H. (2005). Overview of megafloat: Concept, design criteria, analysis, and design. Marine Structures 18(2): 111–132. Suzuki, H. and Yoshida, K. (1996). Design flow and strategy for safety of very large floating structures. Proceedings of International Workshop on Very Large Floating Structures, Hayama, Japan, pp.21–27. Tavana, H. and Khanjani, M. J. (2013).Reducing hydroelastic responses of very large floating structure: A Literature Review. International Journal of Computer Applications 71(5): 0975– 8887. 84 References Utsunomiya, T., Watanabe, E. and Eatock Taylor, R. (1998). Wave response analysis of a box-like VLFS close to a breakwater. Proceedings of the 17th International Conference on Offshore Mechanics and Arctic Engineering. Lisbon, Portugal, pp.1–8. Wang, C. D. and Meylan, M. H. (2002). The linear wave response of a floating thin plate on water of variable depth. Applied Ocean Research 24: 163-174 Wang, C. D. and Wang, C. M. (2008). Computation of the stress resultants of a floating Mindlin plate in response to linear wave forces. Journal of Fluids and Structures 24(7): 1042–1057. Wang, C. M., Watanabe, E. and Utsunomiya, T. (2008). Very Large Floating Structures. London; New York, Taylor & Francis. Wang, C. M., Xiang, Y., Utsunomiya, T. and Watanabe, E. (2001). Evaluation of modal stress resultants in freely vibrating plates. International Journal of Solids and Structures 38(36–37): 6525–6558. Watanabe, E., Utsunomiya, T. and Kubota, A. (2000). Analysis of wave-drift damping of a VLFS with shallow draft. Marine Structures 13(2000): 383–397. Watanabe, E., Utsunomiya, T. and Wang, C. M. (2004a). Hydroelastic analysis of pontoon-type VLFS: A literature survey. Engineering Structures 26(2): 245–256. Watanabe, E., Wang, C. M., Utsunomiya, T. and Moan, T. (2004b). Very large floating structures: Applications, analysis and design. CORE Report. National University of Singapore, Singapore. Xia, D. W., Kim, J. W. and Ertekin, R. C. (2000). On the hydroelastic behavior of two-dimensional articulated plates. Marine Structures 13(4–5): 261–278. Yago, K. and Endo, H. (1996). On the hydroelastic response of box shaped floating structure with shallow draft. Journal of the Society of Naval Architects of Japan 180: 341–352. 85 References Yamashita, T., Yonezawa, M., Shimamune, S. and Kinoshita, T. (2003). Joining technology for construction of very large floating structures. Proceedings of 4th International Workshop on Very Large Floating Structures, Tokyo, Japan, pp.229–236. 86 Appendix A Finite Element Formulation of Mindlin Plate The derivation of governing equations based on constitutive relations between plate displacements, strains and stresses is explained in this section. Later the computation of shear stiffness matrix of the Mindlin plate is explained. A detailed formulation can be found in book by Petyt (1990) and Liew et al. (1998). Section A.1 presents the strain-displacement, stress-strain and stress displacement relations of Mindlin plate and sign conventions. The derivation of governing equations of Mindlin plate based on the constitutive relations is discussed in Section A.2. Finally, in Section A.3, the computation of shear stiffness matrix is discussed. A.1 Constitutive relations Components of displacement and sign conventions The VLFS is governed by Mindlin plate theory. The assumptions of Mindlin plate theory are:  No deformations occur in the mid-plane of the plate.  Normal to the undeformed mid-plane remains straight and unstretched in length but not necessarily normal to the deformed mid-plane.  The effect of rotary inertia is considered. The displacements according to the assumptions can be expressed as ( ) ( ) ( where is time, and , , ( ( ) ( the in-plane displacement in are the bending rotations about ) and (A.1a) ) (A.1b) ) and (A.1c) direction, the transverse displacements axes respectively (see Fig. A.1). Appendix A Figure A. 1. Deflection and rotations of a Mindlin plate element Strain-Displacement relations The problem is considered as a plane stress problem, i.e. stress in the direction is zero. The engineering strain can be expressed as { } { { } } [ { } { ] [ , ] } (A.1e) [ where (A.1d) are the nominal strains, ] , [ , section curvature. Strain- Stress Relations 88 ] the shearing strains and { } is the plate’s Appendix A Following Hooke’s law, the strain stress relations for isotropic material for plate are as follows [ ( )] [ ( )] [ ( )] (A.1f) And, (A.1g) Similarly, using Hooke’s law, the normal stresses, , and shear stresses , , can be expressed as follows { } ( ) ( ) [ { } { ]{ } [ ]{ } (A.1h) ] } ( [ where [ is the modulus of elasticity, [ ) [ ][ ] [ ]{ } (A.1i) ] ] and [ ] are the flexural and shear stiffness matrices. They are defined as follows 89 Appendix A [ ] [ [ )] ( ] ( ) [ ] (A.1j) Stress resultants-displacement relations Stress resultants, transverse shear forces per unit length, and and , twisting moments per unit length , bending moments per unit length , are obtained by integrating the stresses mentioned above. ⁄ ∫ ⁄ ∫ ⁄ ⁄ ∫ (A.1k) ∫ ⁄ ( ⁄ ) ⁄ ∫ ∫ ( ⁄ ) ( ⁄ ⁄ ) ⁄ ∫ ⁄ ( ⁄ ⁄ ∫ where ) ⁄ ⁄ ∫ ( ⁄ ) ⁄ ∫ ⁄ ( ( ) ⁄ is flexural rigidity of the plate, ( is the shear modulus and ) (A.1l) ) is the shear correction factor for assuming constant shear stress along the plate thickness, h. ( ) ( 90 ) (A.1m) Appendix A A.2 Governing equation of motion of Mindlin plate Governing equation of motion of plate is derived from Hamilton’s principle (Clough and Penzien, 1993) ∫ ( where denotes initial time, ) ∫ (A.2a) is the final time, is the variational operator, is the potential energy which is the summation of hydrostatic force, strain energy, and is the kinetic energy, (conservative force) and is the work done by non-conservative external forces. The potential energy is given by ∫ { } [ ∫ ]{ } ∫ { } [ ]{ } (A.2b) After substituting (A.1j) in (A.2b), total potential energy functional becomes ∫ { [( ) ( ) ( ) ] (A.2c) [( where ) is the area of plate and kinetic energy functional ( ) ] and } is the mass density per unit volume. Similarly, of the plate is given by ∫ [( ) ( ) ( ) ] (A.2d) ∫ [( ) ( ) 91 ( ) ] Appendix A The work done by non-conservative external forces ( ∫ where ( is given by ) (A.2e) ) is the non-conservative hydrodynamic force acting on the plate bottom due to velocity potential which is equal to . Substituting (A.2c-e) in (A.2a), we get following equation ∫ ∫ { [( ( ) [( [( ) ) ) ( ( ( ) ] ) ] ) ( ) ] } (A.2f) By taking variations with respect to w, ∫ ∫ { [ ( ) ( ) ] ) ( [( ) [ ( )( ) ] ] } (A.2g) 92 Appendix A After integrating by parts and grouping the terms together, eq. (A.2g) takes following form ∫ ∫ {[ ( ( ) [ ( [ ) ( ) ( ( ) ) ( ( ) ) ) ( ( ) )] ∫∮ { ( ] } ) ( ) ( where ] )} (A.2h) is the boundary path. By equating the coefficients of the variation terms to zero for the functional over the plate area, and after omitting the time factor [( [( ) )( , three governing equations of motion are obtained ( )] ) ( (A.2i) )( )] ( ) (A.2j) [( )( ) ( )( )] ( ) (A.2k) A3 Computation of shear stiffness 93 Appendix A The shear stiffness matrix is calculated assuming shear strain field (Bathe and Dvorkin, 1985; Hinton and Huang, 1986) which avoids shear locking phenomenon. The shear strain at any point ( ) within element are evaluated from the displacement and normal rotation fields where ( ) | ( ) | and ∑ | ∑ ( ) ( ) ∑ ( ) (A.3a) ∑ ( ) (A.3b) are approximated using shape functions. Assumed shear strain fields are expressed as where nx and ny are sampling points for ∑ (A.3c) ∑ (A.3d) and respectively. The shape function in (A.3c-d) are given by (Hinton and Huang, 1986) ( ) ( )( ) (A.3e) ( ) ( )( ) (A.3f) ( ) ( )( 94 ) (A.3g) Appendix A ( ) ( )( ) (A.3h) ( ) ( )( ) (A.3i) And (A.3j) ( ) ( ) where the constant c = √ ⁄ . (b) Sampling points for (a) Sampling points for Figure A. 2. Sampling point for assumed shear strain field: (a) and (b) Interpolating the shear strain at the sampling points gives ∑ (∑ ∑ (∑ ( ) ( ) ∑ ( ) ) (A.3a) ∑ ( ) ) (A.3b) The shear strain-displacement matrix for an element is given by 95 Appendix A [ ( ) [ ] ] (A.3c) where [ ] ∑ ( ) ( ) ∑ ( ) ( ) (A.3d) ∑ [ ( ) ( ) ∑ ( ) ( ) ] The transformation of matrices from global coordinates to natural coordinates are done by means of Jacobian, expressed as follows [] [ ] (A.3e) Therefore, [] { { } 96 } (A.3f) Appendix B Boundary Integral Equation Laplace equation and the boundary conditions of the fluid domains are transformed into a boundary integral equation (John, 1949, 1950; Newman, 1977; Sarpkaya and Isaacson, 1981; Meylan and Squire, 1996) using Green’s second identity, which is given by: ∫( ) ∮( ) (B.1) where is the velocity potential, G is the Green’s function which is the fundamental solution, S is the boundary surface of the fluid domain and n is the unit outwards normal, perpendicular to S. More details can be referred in the books by Brebbia et al. (1984) and Becker (1992). The velocity potential satisfies Laplace equation, everywhere in the solution domain. G is the fundamental solution which satisfies the equation, everywhere in the solution domain except at source point , where it is singular. To find the singular solution at the source point, the source point is surrounded by a small sphere of radius ε and surface . We shall examine the result in the limit . After excluding this small sphere, the equation (B.1) becomes ∫( ) ∮ ( ) is the new volume excluding small sphere and volume , (B.2) is the new surface (see Fig. B1). In the , which makes the L.H.S of eq. (B.2) equal to zero. Therefore eq. (B.2) becomes ∮( ) ∮( ) ∮( ) (B.3) Appendix B Figure B. 1.Three dimensional fluid domain decomposition where α is the angle measured anticlockwise from the x-axis at source point . After substituting the fundamental solution of G by solving Laplace equation, using radius and taking limit if the , the equation (B.3) becomes ∮( ) ∮( ∮( ) ( ) ∮( ( ) ) ) (B.5) ∮( ) Taking limits, [ ∮( ) ] Equation (B.3) transforms into boundary integral equation for the fluid 98 Appendix B ∮ ∮ (B.6) By decomposing the boundary surface S into SF, SHB, SB and S and applying boundary conditions on these surfaces, we get following set of equations ∮ ∮( ) ( ∮( ) ) ∮( ∮( ) ∮( ) ∮( ) ) ∮( ) ) ∮( ∮( ) ∮( ) ∮( ) (B.7) The boundary integral equation derived above can be further decomposed and written in terms of different components of velocity potential, viz. radiation potential, scattered potential, incident potential. The term involving radiation boundary at infinity can be simplified in the following steps: ∮( ) ∮ (( ) 99 ( ) ) (B.8) Appendix B The scattered and radiated potential satisfy the Somerfield radiation condition, therefore terms including and vanish. The rest of the terms in the equation are as below ∮( ) ∮( The above equation is valid everywhere at ) (B.9) , but at the source point. Considering a small hemisphere of radius around the point , the eq. (B.9) is solved as follows ∮( ) ∮ ( ) ∮( ∮( ) ) ∮( ) ∫( ) (B.10) ∮ ( ) ∫ ( )∮ ( ( ) ( 100 )∫ ) Appendix B Figure B. 2. Plan view and side view of decomposition of two dimensional surface into After substituting eq. (B.10) back in eq. (B.7), a final expression of boundary integral equation is obtained ∮ ∮ 101 (B.11) [...]... VLFS Floating structures are broadly classified as pontoon type and semi-submersible floating structures Semi-submersible type floating structures are partly raised above the sea level using column tubes or watertight ballast structural compartments at the bottom/hull to minimize the effects of waves while maintaining a constant buoyant force Therefore they are suitably deployed in deep seas with large. .. dolphin-frame guide mooring system may be adopted The pontoon type floating structures are very cost effective with low manufacturing costs and easy to repair and maintain As mentioned above, a pontoon type VLFS typically has five major components (see Fig 1.1), namely (1) an access bridge or a floating walkway from land (2) very large pontoon-type floating structure, VLFS (3) superstructure and facilities... industries, offshore structures like wind-farms, ocean thermal energy conversion (OTEC) platforms, floating wave-energy converters, floating nuclear power plants are also successfully in operation in different parts of world harnessing other sources of energy Fig 1.5 encompasses some major offshore VLFS (e) Floating infrastructures: Two of the major infrastructural applications of floating structures have... waves Floating oil drilling platforms used for drilling and production of oil and gas and semisubmersible type floating wind farms are typical examples of semi-submersible-type floating structures Some semi submersibles are transported using outside vessels such as tugs or barges, and some have their own propulsion system for transport and are then properly moored to the sea beds When these floating structures. .. Past to present and future Floating structures have been a part of various cultures in the forms of floating homes and villages and floating docks and bridges They were an innovative idea to create and connect different landmasses or landmass to marine vessels, alleviate traffic pressures, enable movement of equipment and soldiers in the time of war etc The use of floating structures began in offshore... are also floating structures which are immune to seismic forces and are designed to accommodate gravity and wave loads 6 Introduction (a) (c) (b) Figure 1 3 (a) World’s longest floating bridge, Governor Albert D Rossellini Bridge (aka Evergreen Point Bridge) (b) Yumemai Bridge, Japan (c) Dubai floating bridge, (source: bridgeinfo.org, Wikipedia) (a) (b) Figure 1 4 (a) Ujina’s floating pier (b )Floating. .. referred as tension-leg platforms This particular type of floating structures is currently believed to be one of the few serviceable solutions for areas further away from shorelines where waves are larger and have been exploited as early as the 1970s 2 Introduction Pontoon-type floating structures are in direct contact to the water surface and utilize larger area than semi-submersibles, making them prone... Summary Summary Very Large Floating Structures (VLFS) is a promising technology that facilitates ocean space colonization, a sustainable and environmental friendly technological innovation which enables creation of land on water without disturbing marine environment, polluting coastal waters and disrupting ocean currents VLFS are becoming a popular choice in a wide range of applications like floating bridges,... days An inherited advantage of floating bridges is that they can be constructed in segments which can be retracted, hence allowing marine traffic too (c) Floating docks and bases: Floating docks have been an important part in army operations and as a connecting link between landmasses and marine vessels Ujina’s floating pier (see Fig 1.4(a)) in Hiroshima, Japan is a concrete floating pier which extends... system 3 Figure 1 2 Two phase floating runway project, the Mega-Float, in Japan (Suzuki et al 2005) 6 Figure 1 3 (a) World’s longest floating bridge, Governor Albert D Rossellini Bridge (aka Evergreen Point Bridge) (b) Yumemai Bridge, Japan (c) Dubai floating bridge, (source: bridge-info.org, Wikipedia) 7 Figure 1 4.(a) Ujina’s floating pier (b) Floating walkway made of High Density