HYDROELASTIC RESPONSE OF INTERCONNECTED FLOATING BEAMS MODELLING LONGISH VLFS MUHAMMAD RIYANSYAH NATIONAL UNIVERSITY OF SINGAPORE 2009 HYDROELASTIC RESPONSE OF INTERCONNECTED FLOATING BEAMS MODELLING LONGISH VLFS MUHAMMAD RIYANSYAH B. Eng. (Hons), Institut Teknologi Bandung, Indonesia A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgments ACKNOWLEDGMENTS This thesis is a result of four years of research work since I was admitted into the PhD programme in the Department of Civil Engineering, the National University of Singapore. I have worked with a great number of people whose contributions in the research deserved special mention. It is a pleasure to convey my gratitude to them all in this acknowledgment section. In the first place, I want to show my utmost gratitude to Prof. Wang Chien Ming for his supervision, advice, guidance, and above all, for his patience from the very early stage of this research. At the times I had a hard time in my research study, he provided me encouragement and support in various ways so that I could continue the research and finally finish this thesis. I am indebted to him more than he knows. I also want to record my gratitude to Prof. Choo Yoo Sang whose vast knowledge and experience have triggered and nourished my intellectual maturity that I will benefit from for a long time to come. I thank him for all the valuable suggestions that he made to help me shaping up my ideas and research. I would like to thank Dr. Chan Chun Tat for his patience in teaching me the mathematical formulation and solution technique for the fluid-structure interaction i Acknowledgments problem. His immense knowledge in computer coding has helped me to write my own computer code for the hydroelastic analysis of floating interconnected beams. To Mr. Sit Beng Chiat, I would like to thank him for being the role model for the hard workers in the Hydraulics Laboratory. I am also grateful to Mr. Yip Kwok Keong who helped me whenever I had a problem with my computer. To Mr. Khrisna Sanmugam, Mr. Shaja Khan, Mr. Semawi bin Sadi, Mr. Koh Seng Chee, and Mdm. Annie Tan, I would convey my appreciation for their indispensable help in my experiments. It is impossible for me to finish this research without their assistance. It is a pleasure to pay tribute also to the administrative staffs. To Ms. Lim Sau Koon, Ms. Norela Bte. Buang, and Ms. Juliana Bte. Miswan, I am thankful for their assistance in dealing with administration matters during my study in National University of Singapore. I gratefully thank my colleagues, Mr. Tay Zhi Yung, Ms. Emma Patricia Bangun, and Dr. Pham Dhuc Chuyen for their advice and their willingness to share their bright thoughts with me. It was great to collaborate with them. I was extraordinarily fortunate in having Dr. Dradjat Hoedajanto as my adviser in Institut Teknologi Bandung. I could never have embarked and started all of this without his support. His teachings have encouraged me to grab this challenging research opportunity. To Dr. Iswandi Imran, Dr. Made Suarjana, and Dr. Bambang Budiono, I am thankful for showing me the fun-side of research work. My parents deserve a special mention for their inseparable support and prayers. My father, Damrin Lubis, is the person who always reminds me the importance of learning. My mother, Surya Aziz, is the one who sincerely raised me with her neverending caring and love. Kak Yeyen, Bang Ipan, and Bang Andi, thanks for being ii Acknowledgments supportive and caring siblings. It is unfair if I did not express my appreciation to Ms. Rosa Permata Sari. Her passion and ambition have altered my perspective, my way of thinking, and made me the way I am right now. I am grateful for all the support that she has given. I would like to thank everybody who has helped me, as well as expressing my apology that I could not mention personally one by one. Finally, I would like to thank the National University of Singapore for providing the scholarship that enabled me to study here in Singapore. iii Table of Contents TABLE OF CONTENTS ACKNOWLEDGMENTS . i TABLE OF CONTENTS . iv SUMMARY . x LIST OF TABLES . xvi LIST OF FIGURES . xvii LIST OF NOTATIONS . xxv Chapter INTRODUCTION . 1.1 Very Large Floating Structures . 1.1.1 Definition of VLFS 1.1.2 Advantages of VLFS over Land Reclamation . 1.1.3 Applications of VLFS 1.2 Literature Review 13 1.2.1 Hydroelastic Analysis of VLFS . 13 1.2.2 Factors Affecting the Hydroelastic Response of VLFS 16 1.2.3 Hydroelastic Response Reduction of VLFS 17 1.2.4 Floating Structure as Wave Energy Converter 23 iv Table of Contents 1.3 Research Objectives . 26 1.4 Layout of Thesis . 30 Chapter PROBLEM DEFINITION AND FORMULATION FOR HYDROELASTIC ANALYSIS OF INTERCONNECTED FLOATING BEAMS . 33 2.1 Problem Definition . 33 2.2 Formulation for Hydroelastic Analysis of Interconnected Floating Beams 41 2.2.1 Fluid Part 43 2.2.2 Floating Beam Part . 47 Chapter METHOD OF SOLUTION FOR HYDROELASTIC PROBLEM . 50 3.1 Boundary Element Method (BEM) for Solving Fluid Motion Problem 51 3.2 Finite Element Method (FEM) for Solving Floating Beam Motion Problem 53 3.3 Coupled BEM-FEM Solution Method for Hydroelastic Problem . 56 3.3.1 Radiation Boundaries . 56 3.3.2 Fluid-structure Interface . 61 3.4 Hydroelastic Response, Bending Moments, and Shear Forces 65 3.5 Compliance Parameter . 65 Chapter VERIFICATION OF FORMULATION AND METHOD OF SOLUTION . 67 v Table of Contents 4.1 Convergence Study 67 4.1.1 Numerical Model . 68 4.1.2 Computational Parameters for Converged Compliance Parameter . 68 4.2 Validation of Formulation and Method of Solution with Experimental Test 72 4.3 Verification of Formulation and Method of Solution with Existing Numerical Results . 79 4.4 Summary . 80 Chapter FLOATING MAIN BEAM WITH AUXILIARY BEAMS . 82 5.1 Floating Main Beam with Auxiliary Beam at the Front End 82 5.1.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ 84 5.1.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating Main Beam Compliance Parameter χ 86 5.1.3 Comparison of Hydroelastic Response of Floating Main Beam with and without Auxiliary Beam . 87 5.2 Floating Main Beam with Auxiliary Beam at the Rear End . 103 5.2.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ 103 5.2.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating Main Beam Compliance Parameter χ 105 vi Table of Contents 5.2.3 Comparison of Hydroelastic Response of Floating Main Beam with and without Auxiliary Beam 106 5.3 Floating Main Beam with Auxiliary Beams at Both Ends . 121 5.3.1 Effects of Auxiliary Beam Length δ and Connection Rotational Stiffness ξ on Floating Main Beam Compliance Parameter χ . 122 5.3.2 Effects of Auxiliary Beam Flexural Rigidity γ on Floating Main Beam Compliance Parameter χ . 123 5.3.3 Comparison of Hydroelastic Response of Floating Main Beam with and without Auxiliary Beams 124 5.4 Summary 139 Chapter LARGE FLOATING BEAM WITH MULTIPLE CONNECTIONS 142 6.1 Effects of Connection Location β and Rotational Stiffness ξ on the Hydroelastic Response of Floating Beam with Single Connection . 143 6.2 Optimum Rotational Stiffnesses ξ i for Prescribed Number of Equally-spaced Connections 151 6.3 Optimum Design of Locations β i and Rotational Stiffnesses ξ i for Prescribed Number of Connections . 160 6.4 Summary 168 Chapter FLOATING ARTICULATED BEAM FOR WAVE ENERGY CONVERTER 170 7.1 Optimum Connection Design for Maximum Total Work Done in the Connections for Floating Articulated Beam System 171 vii Table of Contents 7.2 Effect of Number of Connections on the Floating Articulated Beam Response and its Wave Energy Capturing Efficiency . 180 7.3 Summary . 192 Chapter CONCLUSIONS AND RECOMMENDATIONS 195 8.1 Summary of Results 195 8.2 Recommendations for Future Research Studies 205 8.2.1 Hydroelastic Response of Interconnected Floating Beams with Dynamic Loads 206 8.2.2 Hydroelastic Response of Interconnected Floating Beams in Oblique Sea . 206 8.2.3 Hydroelastic Response of Interconnected Floating Beams in Three Dimensional 207 8.2.4 Hydroelastic Response of Interconnected Floating Plates 207 8.2.5 Interaction of the Hydroelastic Responses of Multiple Interconnected Floating Structures 207 REFERENCES . 209 LIST OF AUTHOR’S PUBLICATIONS . 218 Appendix A BOUNDARY ELEMENT METHOD . 220 A.1 Analytical Formulation . 220 A.2 Numerical Implementation . 223 A.2.1 Boundary Discretization . 223 A.2.2 Numerical Integration . 225 viii Boundary Element Method Now, all the components of the matrices [ A] and [ B] in Eq. (A.24) can be determined. They are given by the following equations, ∂G[( x a (0), z a (0)), ( xb (ζ ), z b (ζ ))] H c (ζ ) J (ζ ) dζ for (b ≠ a ) ∂n c =1 −1 Aa ,b = ∑ ∫ (A.31a) N Aa ,a = −∑ Aa ,b (A.31b) Ba ,b = ∑ ∫ G[( x a (0), z a (0)), ( xb (ζ ), z b (ζ ))]H c (ζ ) J (ζ ) dζ for (b ≠ a ) (A.32a) b =1 b≠a c =1 −1 1  1 1 = ∑  ∫ f (η ) ln dη + ∫ f (η ) ln (1)dη  c =1  η  η   Ba ,a (A.32b)   H c (−η ) J (−η ) ln 2π  rPQ (−η )  f (η ) = ln η (A.32c)   ln H c (η ) J (η ) 2π  rPQ (η )  f (η ) = ln η where a is the row index, and b the column index. A.3 Calculation Examples Consider the load point P is in the first element. The coordinates for the end nodes of the first element (load point element index a = ) are (0,0) and (0,−1) . On the other 230 Boundary Element Method hand, the field point Q is in the second element (field point element index b = ) with end nodes coordinates of (0,−1) and (1,−1) . By using the Gauss quadrature with four integration points, the calculations for Eqs. (A.31a) and (A.32a) are presented in Table A.1. From the calculations shown in Table A.1, we obtain A1, = −0.1762 and B1, = 0.0533 . The other off-diagonal components of [ A] and [ B] can be obtained using the same procedure. After the off-diagonal terms of matrices [ A] and [ B] are obtained, we proceed to determine the diagonal ones. By using Eq. (A.31b), the diagonal terms of [ A] can be easily obtained. We then use Eq. (A.32b) to obtain its diagonal terms for [ B] . As an example, we consider the load point P and the field point Q to be in the same element, i.e. the first element. As previously mentioned, the coordinates for the end nodes of the element are (0,0) and (0,−1) . The calculation is carried out using the special logarithmic Gauss quadrature and presented in Table A.2. From Table A.2, we obtain B1,1 = 0.1325 + 0.1325 = 0.2650 . 231 232 vu (2) 0.3479 0.6522 0.6522 0.3479 vu (2) 0.3479 0.6522 0.6522 0.3479 ζ (1) -0.8611 -0.3400 0.3400 0.8611 ζ (1) -0.8611 -0.3400 0.3400 0.8611 H1(ζ) (4) 0.9306 0.6700 0.3300 0.0695 (3) 0.1088 0.0815 0.0285 -0.0087 0.0695 0.3300 0.6700 0.9306 (4) H1(ζ) G(P,Q) -0.0713 -0.1139 -0.2217 -0.3123 (3) ∂ G(P,Q) / ∂ n 0.9306 0.6700 0.3300 0.0695 (5) H2(ζ) 0.9306 0.6700 0.3300 0.0695 (5) H2(ζ) 0.5 0.5 0.5 0.5 (6) J(ζ) 0.5 0.5 0.5 0.5 (6) J(ζ) Table A.1 – BEM calculation example for P and Q are in different elements ∑ ∑ 0.0533 -0.0015 0.0093 0.0266 0.0189 (7) ([(3) × (4) × (6)] + [(3) × (5) × (6)]) × (2) -0.1762 -0.0124 -0.0371 -0.0723 -0.0543 (7) ([(3) × (4) × (6)] + [(3) × (5) × (6)]) × (2) Boundary Element Method vlu (2) 0.3835 0.3869 0.1904 0.0392 vlu (2) 0.3835 0.3869 0.1904 0.0392 η − left (1) 0.0415 0.2453 0.5562 0.8490 η − right (1) 0.0415 0.2453 0.5562 0.8490 0.8490 0.5562 0.2453 0.0415 (3) ζ -0.8490 -0.5562 -0.2453 -0.0415 (3) ζ 0.1364 0.2037 0.3340 0.6168 (4) G(P,Q) 0.1364 0.2037 0.3340 0.6168 (4) G(P,Q) 0.0755 0.2219 0.3774 0.4793 (5) H1(η) 0.9245 0.7781 0.6227 0.5208 (5) H1(-η) 0.9245 0.7781 0.6227 0.5208 (6) H2(η) 0.0755 0.2219 0.3774 0.4793 (6) H2(-η) 0.5 0.5 0.5 0.5 (7) J(η) 0.5 0.5 0.5 0.5 (7) J(-η) Table A.2 – BEM calculation example for P and Q are in the same element 0.1637 0.5866 1.4053 3.1821 (8) ln (1/η) 0.1637 0.5866 1.4053 3.1821 (8) ln (1/η) ∑ ∑ 0.1325 0.0163 0.0331 0.0460 0.0372 (9) ([(4) × (5) × (7) / (8)] + [(4) × (6) × (7) / (8)]) × (2) 0.1325 0.0163 0.0331 0.0460 0.0372 (9) ([(4) × (5) × (7) / (8)] + [(4) × (6) × (7) / (8)]) × (2) Boundary Element Method 233 Finite Element Method Appendix B FINITE ELEMENT METHOD In this appendix, we present the finite element method (FEM) for solving the equation of motion of the floating beam. As described in Chapter 2, the floating beam is modelled by the Euler-Bernoulli beam theory. In the present FEM formulation, the floating beam is represented by interconnected beam elements, the buoyancy force is modelled by Winkler springs, and the wave-induced load is represented by the external pressure. The mass, stiffness, and hydrostatic stiffness matrices as well as the external load vector of an individual element are derived by utilizing the principle of virtual displacements. The global matrix of the entire floating beam system is obtained by assembling the matrices of the elements. B.1 Analytical Formulation According to the Euler-Bernoulli beam theory, the governing equation of motion for the floating beam is given by (Smith, 1988) − ω mb w + EI ∂4w =p ∂x (B.1) where w is the displacement, mb = ρ b bb h the mass per unit length of the beam, 234 Finite Element Method I = (1 / 12)bb h the second moment of area of the beam section, E the Young’s modulus, ρ b the beam density, bb the width of the beam, h the depth of the beam, and p the external pressure. The floating beam is represented by interconnected beam elements. Each element has two end nodes with two variables, namely the vertical and rotational displacements (see Fig. B.1). ~ / ∂x ∂w ~ V3 ~ / ∂x ∂w ~ M2 ~ w ~ w ~ V1 ~ M4 ~ / ∂x ∂w dx ~ w Figure B.1 – Virtual displacements and nodal forces of a beam element ~ at a point x from the left node in the element can be expressed The displacements w in the form of cubic polynomial, i.e. ~ = a + a x + a x + a x = [ x]{a} w (B.2a) where [ x] = [1 x {a}T = {a1 a2 x2 x3 ] a3 a4 } (B.2b) (B.2c) 235 Finite Element Method The coefficients {a} can be determined from the nodal displacements of the beam element. The nodal displacements shown in Fig. B.1 ( x = for the left node and x = s for the right node) can be written in the following matrix form, {w~} = [V ]{a} (B.3a) where {w~} = w~1  1 0 [V ] =  1  0 ~ ∂w ∂x ~ w ~  ∂w  ∂x  T     2 2s 3s  0 s s2 0 s3 (B.3b) (B.3c) In order to obtain the general relationship between the forces and displacements, we use the principle of virtual displacements. This approach states that the total virtual work must vanish if the system is in equilibrium. B.1.1 Internal Virtual Work The internal virtual work in a beam element is obtained by integrating the internal work done in a small portion of the beam element dx defined as (Smith, 1988), dWi = ~ d 2w  d 2w  dx − EI dx  dx  (B.4) where w is the virtual displacements. By substituting Eq. (B.2) into Eq. (B.4), one obtains 236 Finite Element Method ( )( ) ~} dx dWi = − EI [U ][V ] −1 {w } [U ][V ]−1 {w (B.5a) [U ] = [0 x] (B.5b) Transposing, rearranging, and integrating Eq. (B.5a) over the length of the beam element, the total internal virtual work within an element is given by (Smith, 1988) s  T T ~} Wi = − EI {w } [V ] −1  ∫ [U ]T [U ] dx  [V ] −1 {w 0  ( ) (B.6) B.1.2 External Virtual Work The total external virtual work for a beam element is obtained from the sum of the product of all the forces and the corresponding virtual displacements. The forces are the nodal forces (as the internal forces are required to connect the element to the adjacent elements), the inertia forces (from the acceleration part of the element), the hydrodynamic pressure, and the hydrostatic pressure. External Virtual Work Done by Nodal Forces The external virtual work done by the nodal forces is the product of the nodal forces and the corresponding virtual displacements (see Fig. B.1) and is given by (Smith, 1988) {} T ~ Wenodal = {w } S {S~} = {V~ T ~ ~ M V3 (B.7a) ~ M4 } (B.7b) 237 Finite Element Method External Virtual Work Done by Inertia Forces The external virtual work done by the inertia forces is the product of the inertia forces and the displacement. The work done within a small portion of the beam element dx is given by dWeinertia = w ( x) f inertia ( x) dx (B.8) where the inertia force f inertia is defined as (Smith, 1988) ~ ( x) ∂2w mb ∂t (B.9a) ~( x)m f inertia ( x) = −ω w b (B.9b) f inertia ( x) = − By substituting Eq. (B.2) into Eq. (B.8), one obtains ( )( ) ~} dx dWeinertia = −ω mb [ x][V ] −1 {w } [ x][V ] −1 {w (B.10) Transposing, rearranging, and integrating over the length of the beam element, the total external virtual work done by the inertia forces within an element is given by (Smith, 1988) inertia e W 238 ( = −ω mb {w } [V ] T ) −1 T s T  ~}  ∫ [ x] [ x] dx  [V ] −1 {w   0  (B.11) Finite Element Method External Virtual Work Done by Pressure The external virtual work done by the pressure is the product of the pressure and the displacement. The work done within a small portion of the beam element dx is given by dWe pressure = w ( x) f where the external pressure force f f pressure pressure pressure (x ) dx (B.12) is defined as the following (Smith, 1988) ( x) = p ( x) + p HS ( x) (B.13) where p (x) is the hydrodynamic pressure and p HS (x) the hydrostatic pressure given ~( x )] . By substituting Eq. (B.2) and (B.13) into Eq. (B.12), one by [ p HS ( x) = − ρgw obtains ( )( ) ~} dx dWe pressure = [ x][V ]−1 {w } p ( x) − ρ g [ x][V ] −1 {w (B.14) Transposing, rearranging, and integrating over the length of the beam element, the total external virtual work done by the pressure within an element is given by  T We pressure =  {w } [V ] −1  ( s ) ∫ [ x] T T  T −  ρ g {w } [V ] −1   (  p( x) dx   ) T s T  ~}  ∫ [ x] [ x] dx  [V ]−1 {w    0   (B.15) 239 Finite Element Method As the total virtual work must vanish, we have Wi + Wenodal + Weinertia + We pressure = (B.16) By substituting Eqs. (B.6), (B.7), (B.11), and (B.15) into Eq. (B.16) and canceling out the virtual displacements {w }, we obtain the discretized version of the governing equation of beam motion, i.e. (− ω {} ) ~} = S~ [mb ] 4×4 + [k S ] 4×4 + [k HS ]4×4 {w 4×1 4×1 + { f }4×1 − 13s  13s − 3s  156 − 22s   − 22s 4s  22s  156  22 s 4s ms [mb ] = b  13s 420  54  − 13s − 3s 54 6s − 12 6s   12  6s s − 6s 2s  EI  [k S ] =  s − 12 − 6s 12 − s    − 6s 4s   6s s − 13s  13s − 3s  156 − 22s   − 22s 4s  22s  156  22s 4s ρgs  [k HS ] = 13s 420  54  − 13s − 3s 240 (B.17b) (B.17c) 54 {S~} = {V~ ~ ~ M V3  { f } = iωφ  s 2 s2 12 (B.17a) s ~ M4 } T (B.17d) (B.17e) T − s2   12  (B.17f) Finite Element Method B.2 Matrix Assembly The discretized version of the equation of motion for the entire floating beam system is obtained by simply adding the mass, stiffness, and external forces terms of adjacent elements wherever they have a degree-of-freedom in common. We now present example of matrix assembly procedure. Consider a floating two-beam system that are connected to each other as shown in Fig. B.2. Both beams have the same properties, and they are connected together by a semi-rigid connection with rotational stiffness kT . Hence the rotational displacement at the left of the connection w3 is different from the rotational displacement at the right w4 (see Fig. B.2). The beam system is subjected to uniform external pressures (note that the formulation can be easily modified if the external pressures on an element is different from the adjacent elements). ∂w2 / ∂x ∂w3 / ∂x ∂w5 / ∂x ∂w7 / ∂x Figure B.2 – Example of floating beam problem with semi-rigid connection As the first step, the beam system is discretized into elements, thus we have two beam elements and three nodes (see Fig. B.3). 241 Finite Element Method ~1 ~ ~ ∂w ∂w2 , f2 , M 21 , f 21 ∂x ∂x w1 , f1 ~1 ,V~1 , ~ w 1 f1 ~ ~ ~ ∂w ∂w ∂w , M 41 , f 41 , , f3 , f5 ∂x ∂x ∂x ~1 ,V~1 , ~ w 3 f3 w4 , f ~2 ~ ~ ∂w , M 22 , f 22 ∂x ~ ,V~ , ~ w f12 1 ~ ~ ~ ∂w ∂w , M 42 , f 42 , f7 ∂x ∂x ~ ,V~ , ~ w f 32 3 w6 , f Figure B.3 – Discretization of floating beam system with semi-rigid connection As shown in the figure, wi and ∂wi / ∂x are the global vertical and rotational ~ j and ∂w ~ j / ∂x the element’s vertical and rotational displacements, displacements, w i i ~ Vi j ~ and M i j the shear forces and bending moments due to beam element ~ displacements, f i the global load forces, and f i j the external load forces of the element (due to the pressure). Note that subscript j is the global displacement index and superscript j is the element index. The mass, stiffness, and hydrostatic stiffness matrices, together with the external load vector for each element are obtained using Eq. (B.17). Given that the properties of both beam elements are the same ( mb = 420 kg/m, EI = Nm2/m and s = 1) with p = 1, ρ = 420, and g = 1, the matrices are given by 156 22 54 − 13   22 13 −  j  [ mb ] =  54 13 156 − 22    − 13 − − 22 242 (B.18) Finite Element Method − 12   12  −6  j  [k S ] = − 12 − 12 − 6   −6 4  (B.19) 156 22 54 − 13   22 13 −  j  [k HS ] =  54 13 156 − 22    − 13 − − 22 (B.20) {} ~ f j 1 = 2 12 − 1  12  T (B.21) where j = and 2. As an example of the matrix assembly procedure, we present the procedure for the stiffness matrix [ K S ] of the complete floating beam system. The assembly procedure for the other matrices is similar. The stiffness matrix for a complete floating beam system is obtained by simply adding the stiffness terms of adjacent elements wherever they have a degree-of-freedom in common. In view of equilibrium condition at the nodes, the boundary conditions at the free ends, and the assumed element ~ with the global displacements w , we obtain displacements w 12 w1 + ∂w ∂w2 ~ − 12 w4 + = f 11 = f1 ∂x ∂x (B.22) w1 + ∂w ∂w2 ~ − w4 + = f 21 = f ∂x ∂x (B.23) 243 Finite Element Method ∂w  ∂w   − 12w1 − + 12w4 −  ∂x ∂x   ∂w ∂w  ~ ~  + 12w4 + − 12w6 +  = f 31 + f 12 = f ∂x ∂x   − 12 w1 − (B.24a) ∂w ∂w ∂w ∂w2 − + (12 + 12) w4 + − 12 w6 + = f ∂x ∂x ∂x ∂x (B.24b) ∂w ∂w    ∂w ∂w  ~  w1 + 2 − 6w4 +  + k T  −  = f = f ∂x ∂x  ∂x    ∂x w1 + (B.25a) ∂w ∂w2 + (4 − k T ) − w4 + kT w5 = f ∂x ∂x (B.25b) ∂w ∂w  ∂w  ~   ∂w  w4 + − w6 +  − k T  −  = f = f ∂x ∂x  ∂x   ∂x  kT (B.26a) ∂w3 ∂w ∂w + w4 + (4 − k T ) − w6 + = f ∂x ∂x ∂x w4 + (B.26b) ∂w5 ∂w − w6 + = f ∂x ∂x (B.27) Equations (B.22) to (B.27) can be written in the following matrix form [ K S ]{w} = {F } (B.28a) where      [K S ] =       12 6 − 12 0 {w} =  w1  244 −6 0 ∂w2 ∂x − kT −6 kT 0 ∂w3 ∂x − 12 −6 −6 − 24 − 12 w4 0 kT − kT −6 ∂w5 ∂x w6 0 − 12 −6 12 −6 ∂w7   ∂x  0 −6            (B.28b) T (B.28c) Finite Element Method {F } = { f1 f2 f3 f4 f5 f6 f7 } T (B.28d) Note that Eq. (B.28) can be specialized for the cases that the beams are connected by a simple hinge or by a rigid connection. This is easily done by setting the rotational stiffness kT in Eq. (B.28) to or ∞ for the hinge connection and rigid connection, respectively. 245 [...]... Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.10 116 Figure 5.25 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.20 117 Figure 5.26 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.30 118 Figure 5.27 Hydroelastic response of. .. response of a floating main beam with auxiliary beams is investigated The auxiliary beams can either be attached to the front end, the rear end, or both ends of the floating main beam A parametric study is carried out to study the effects of the lengths and the flexural rigidities of the auxiliary beams as well as the rotational stiffnesses of the connections on the hydroelastic response of the main floating. .. Variation of the normalized compliance parameter χ / χ m with respect to ξ and γ for δ = 0.25 and α = 0.40 97 Figure 5.10 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.10 98 Figure 5.11 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.20 99 Figure 5.12 Hydroelastic response of the floating. .. Figure 5.39 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.20 135 Figure 5.40 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.30 136 Figure 5.41 Hydroelastic response of the floating beam system with various ξ for δ = 0.25 , γ = 1.00 , and α = 0.40 137 xxi List of Figures Figure... 156 Figure 6.8 Hydroelastic response of floating beam with optimum rotational stiffnesses ξ i for nc = 5 and α = 0.20 157 Figure 6.9 Hydroelastic response of floating beam with optimum rotational stiffnesses ξ i for nc = 7 and α = 0.20 158 Figure 6.10 Hydroelastic response of floating beam with optimum rotational stiffnesses ξ i for nc = 9 and α = 0.20 159 Figure 6.11 Floating beam... stiffnesses 161 Figure 6.12 Hydroelastic response of floating beam with optimum locations β i and optimum rotational stiffnesses ξ i for nc = 3 and α = 0.20 164 xxii List of Figures Figure 6.13 Hydroelastic response of floating beam with optimum locations β i and optimum rotational stiffnesses ξ i for nc = 5 and α = 0.20 165 Figure 6.14 Hydroelastic response of floating beam with optimum locations... stiffnesses of the connections for minimum hydroelastic response of the interconnected floating beams system Alternating variable method is adopted for the optimization procedure The results show that the hydroelastic response of the floating beam system can be reduced significantly if semirigid connections are used It challenges the common perception among engineers that the complete floating structure... the same approach can also be adopted to maximize the efficiency of the floating beam when it is used as wave energy converter This thesis is concerned with the hydroelastic response of interconnected floating xi Summary beams modelling longish VLFS and the way to reduce or to increase it through appropriate design of the floating beams and the connections It challenges the common assumptions that rigid... Hydroelastic response of floating articulated beam with optimum connection design for maximum total work done in the connection for nc = 3 and α = 0.40 178 Figure 7.4 Hydroelastic response of floating articulated beam with optimum connection design for maximum total work done in the connection for nc = 3 and α = 0.60 179 Figure 7.5 Comparison of hydroelastic response of floating articulated... total work done in the connections for various number of connections nc and α = 0.20 183 Figure 7.6 Comparison of hydroelastic response of floating articulated beam with optimum connection design for maximum total work done in the connections for various number of connections nc and α = 0.40 186 Figure 7.7 Comparison of hydroelastic response of floating articulated beam with optimum connection design . Interconnected Floating Beams in Oblique Sea 206 8.2.3 Hydroelastic Response of Interconnected Floating Beams in Three Dimensional 207 8.2.4 Hydroelastic Response of Interconnected Floating Plates. Summary of Results 195 8.2 Recommendations for Future Research Studies 205 8.2.1 Hydroelastic Response of Interconnected Floating Beams with Dynamic Loads 206 8.2.2 Hydroelastic Response of Interconnected. HYDROELASTIC RESPONSE OF INTERCONNECTED FLOATING BEAMS MODELLING LONGISH VLFS MUHAMMAD RIYANSYAH NATIONAL UNIVERSITY OF SINGAPORE 2009 HYDROELASTIC