Hydroelasticity of VLFS allowances for flexible connectors, gill cells, arbitrary shapes and stochastic waves

HYDROELASTICITY OF VLFS: ALLOWANCES FOR FLEXIBLE CONNECTORS, GILL CELLS, ARBITRARY SHAPES AND STOCHASTIC WAVES GAO RUIPING NATIONAL UNIVERSITY OF SINGAPORE 2012 HYDROELASTICITY OF VLFS: ALLOWANCES FOR FLEXIBLE CONNECTORS, GILL CELLS, ARBITRARY SHAPES AND STOCHASTIC WAVES GAO RUIPING (M.Eng., B.Eng., Southeast University, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL AND ENVIRONMENTAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2012 DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. GAO RUIPING 30 October 2012 Acknowledgements Acknowledgements This thesis is a result of four years of research work at the Department of Civil and Environmental Engineering of the National University of Singapore. First of all, my greatest thanks go to my supervisor Professor Wang Chien Ming, for his ideas, inspiration, advice, constant enthusiasm and interest, and continuous support on my research work. His immense knowledge nourished my intellectual maturity that I will benefit from in my life. I feel really happy to work with such an outstanding researcher as a supervisor. Also, I would like to thank my supervisor Professor Koh Chan Ghee for the guidance, valuable advices, discussions, and comments on my research work. Professor Koh teaches me to think out of the box and always look at the ‘big picture’ which saved me from confusion and doubts in research and in my life. I am grateful to all my friends and colleagues for their support and encouragement. Special thanks to Dr. Tay Zhi Yung for his help and valuable suggestions in my research study. I would also like to thank Dr. Iason Papaioannou for his collaboration during my visit to the Chair of Computing in Engineering of the Technische Universität München. The few months that I spent in Munich were very fruitful in my research work on stochastic hydroelastic analysis. I would like to thank Professor Ernst Rank, Dr. RalfPeter Mundani, and Professor Wang Chien Ming, who made this visit possible. i Acknowledgements I am grateful to the National University of Singapore for providing the scholarship that enabled me to read my PhD degree programme in Singapore. Finally, I would like to thank my family for their immense love and support. ii Table of contents Table of contents Acknowledgements i Table of contents iii Summary vii List of tables xi List of figures xiii List of notations xix Chapter Introduction 1.1 Very Large Floating Structures . 1.1.1 Definition of VLFS 1.1.2 Applications of VLFS 1.2 Literature review . 1.2.1 Background on hydroelastic analysis of VLFS . 1.2.2 Modeling of VLFS for hydroelastic analysis . 10 1.2.3 Ways for reducing hydroelastic response of VLFS . 12 1.2.4 Hydroelastic analysis of VLFS under stochastic waves 22 1.3 Objectives of research study . 23 1.4 Layout of thesis . 25 Chapter Hydroelastic analysis in frequency domain 27 2.1 Water–plate model 27 2.2 Mathematical formulation . 28 iii Table of contents 2.2.1 Equations of motion for the floating plate .29 2.2.2 Reduction to a single frequency problem .31 2.2.3 Equations of motion for water 32 2.3 Finite element method for solving floating plate motion 34 2.4 Boundary element method for solving fluid motion .39 2.4.1 Constant panel method .41 2.4.2 Higher order boundary element method .44 2.5 Method of decoupling and solutions .50 2.5.1 Modal expansion method .53 2.5.2 Direct method .58 2.6 Stress resultants of plate, and deflection response parameter .59 2.7 Summary .60 Chapter Hydroelastic behavior of VLFS with flexible connector system 63 3.1 Numerical model .63 3.2 Verification of numerical model .66 3.3 Floating plate with one flexible line connector system .71 3.3.1 Effect of connector stiffness and connector location .71 3.3.2 Effect of wave angle .74 3.3.3 Effect of water depth 80 3.3.4 Effect of aspect ratio 82 3.4 Floating plate with multiple flexible line connectors system 84 3.5 Concluding remarks 89 Chapter Hydroelastic behavior of VLFS with hybrid system 91 4.1 Introduction .91 4.2 Numerical model .92 4.3 Optimization of layouts of gill cells 94 4.4 Results and discussion .99 iv Appendix A A3 Strain–stress relations The strain–stress relations can be obtained from Hook’s law as  xx   yy   zz    E  yy   yy   xx   zz   E  zz   zz   xx   yy   E  xx   xy   xy 2G  xz  xz    xz 2G   yz  yz   yz 2G  xy  (A3a) (A3b) A4 Stress–strain relations Solving stress components from the Hook’s law (  zz  ) gives  E  1     x   y    x       E    x    D f     z   D f      y     y   1     xy   G xy      (A4a) w   y    xz  G xz  E 1   xz  x     G              Ds   (A4b)  w        xz   yz  G xz    x   y   where matrices  D f  and  Ds  are defined as follows 179 Appendix A 1  E   D f    1    0   , 1     Ds   E 1  1    0  (A4c) A5 Stress resultant–displacement relations Assuming the isotropic material of the plate obeys Hooke’s law, the stress resultant– displacement relations are obtained by integrating stresses M xx   h /2 M yy   h /2 h/  h /2 M xy   h /2  h /2 Qx     xy  zdz    h /2  h /2  h /2  y    x E         zdz D     yy xx  h /2  x   y (A5b) h /2 h /2  h/ h /2 h3 and 12 D 1     y  x     x   y (A5c) (A5d)  w  G yz dz   2Gh   x   h/ y   (A5e) h /2  yz dz    z dz  G xy  zdz  w   G xz dz   2Gh  y    h /2 x    xz dz     h/ Note that (A5a)  yy  zdz   h /2 Qy      y  x  E     zdz D       xx yy  h /2  y   x h /2  xx  zdz   h /2  h /2  h /2 dz  h . M xx , M yy and M xy are the bending moments and twisting moment per unit length of plate, Qx , Qy the transverse shear forces per unit length of plate,  xx ,  yy the normal stresses,  xy ,  xz ,  yz the shear stresses, h is the plate thickness, E the modulus of elasticity, G  E /  1    the shear modulus,  the Poisson ratio, D  Eh3 / 12 1    the flexural rigidity and  the shear correction factor to compensate for the error in assuming a constant shear stress throughout the plate thickness. 180 Appendix A A6 Governing equation of motion The Hamilton’s principle is used in deriving the governing equation of motion which requires (Clough and Penzien, 1993) t t t0 t0   T  V  dt    Wnc dt  (A6) where t0 denotes the initial time, t the final time,  the variational operator, T represents the kinetic energy, V the potential energy including both strain energy U and potential Vc of conservative external force (hydrostatic force), and Wnc the work done by non-conservative external forces. The strain energy functional U consisting bending strain energy and shearing strain energy are given as U 1 T T    D f    z dx       Ds   dx   2 (A7) The potential of hydrostatic force is given by Vc   w gw2 dAp  A (A8) Substituting Eq. (A2) into Eq. (A7), and integrating over the plate thickness, the total potential energy functional is obtained as 181 Appendix A V  U  Vc 2  x  y    y  x      y  x     D       1      A2  x  y  y  x  y  x         (A9) 2   w   w   2  Gh  y        gw   dAp    x x   y     where Ap is the plate area, and dAp  dxdy . The kinetic energy functional T of the vibrating Mindlin plate is given by  u   v 2  w 2  T    p          d   t   t   t   (A10) where  p is the mass density (per unit volume). By substituting Eqs. (A1a)–(A1c) into Eq. (A8), and integrating through the thickness dimension, the kinetic energy functional T may be expressed as  w  h    h   y   x  T   dAp        A  t  12  t  12  t    ph (A11) The work done Wnc by non-conservative external forces is given by Wnc   pnc  x, y  wdAp A 182 (A12) Appendix A where pnc  x, y  is the non-conservative hydrodynamic forces acting on the bottom of the plate due to the velocity potential of wave, calculated as    . t The substitution of Eqs. (A9), (A11) and (A12) into Eq. (A6) yields 2  D    x   x  y    y  x   y t0  A   x  y   1   y x   y  x      t 2  2Gh  w   w     y      gw     x   x   y   (A13)  p h  w 2 2  h   x  h   y              t  12  t  12  t   t    w dAp dt   By taking variations with respect to w , one obtains t    x  x  y  y  y  x  x  y D     A x x y x x y   y y  t0     y  x   y  x         y x  y x    w    w   w    w     2Gh  y     x      y      x  x   x   y   y    (A14)   gw w   w  w h  x  x h  y  y     w  ph      dAp dt  12 t t 12 t t  t  t t  Performing integration by parts on Eq. (A14), one gets 183 Appendix A    2 x  2 y  2 y  2 x D          t0 A   y x x y xy y xy x   t   2 y   2 y  2 x  2  y   x  2x  x     y  xy xy x  y    y  w 2w  w   y   w   2Gh  y y  x x x     x x    x w 2w  w   x   w  y y y    2w  h  2 x h2   y        gw w w   ph   w       dAp dt x y  2  t 12 12 t t t          x  y  y   x       D dx dy dx dy       x y y x    y t0  x y x   y       y  x dx  x  x dy   y dx  x  y dy    y x  y x   t   w w   2Gh  y wdy   wdy  x wdx   wdx   dt  x y   (A15) where  is the boundary path. By grouping the terms in Eq. (A15) with respect to the variation terms, we have  2 y  D 1     2 x  2 y   w   p h  2 x       x t0 A  D  y  xy    x  xy    Gh  x  y   12 t  x        t    2 y  2 x  D 1      y  2 x  w   p h   y     D     y      Gh  y   2   xy      12 y x y x  t     x        w  w  y  x  2w        2Gh           h gw  p     w dAp dt y x y  t  t     x   t t0  y  y   x    x  x dx   y dy   y dx    x dy  y x y x    D    y       y  y dx  x  y dy   x dx  x  x dy   x  y y x    w w   2Gh  y wdy   wdy  x wdx   wdx   dt  x y    (A16) 184 Appendix A By equating the coefficients of the variation terms to zero for the functional over the plate area, and assuming free harmonic motion ( e it ), the following three governing equations of motion (after omitting the time factor e it ) are obtained   w  w    y  x  Gh       y   x y  x      p h w    gw  i   1     2 x  2 x D   y   x (A17)  1      2 x   y    w   p h 2  Gh   x           x   y xy   y  12   (A18)  1     2 y  2 y D   y   x  1     2 y  2 x   w   p h   Gh   y 0           y  12 x x y x           (A19) A7 Computation of shear stiffness The shear stiffness of the Mindlin plate is computed following the method of assumed shear strain field (Bathe and Dvorkin, 1985; Hinton and Huang, 1986) which prevents the spurious phenomenon of shear locking. Based on the definition of shear strain, the shear strains at an arbitrary point X ( rX , s X ) within element are evaluated from the displacement and normal rotation fields  xz( X )   yz( X )  w x  y X w  x y X X  X  i 1 i 1 N i  rX , s X  x N i  rX , s X  y wi   N i  rX , s X  y ,i (A20) i 1 wi   N i  rX , s X  x ,i (A21) i 1 185 Appendix A where w / x , w / y ,  x , and  y are approximated using basis function. The assumed shear strain fields are given by nx  xz   Ni x xzi (A22) i 1 ny  yz   Njy yzj (A23) j 1 where nx and ny are sampling points for  xz and  yz respectively, as shown in Fig. A2. Ni x and Njy are appropriate interpolation functions given by (Hinton and Huang, 1986) 1 r N1x  r , s    s   1  s  4 c 1 r N2x  r , s    s   1  s  4 c N3x  r , s   1  s 1  s  (A24) 1 r N4x  r , s     s   1  s  4 c 1 r N5x  r , s     s   1  s  4 c and Njy  r , s   Ni x  s, r  where the constant c  / . 186 (A25) Appendix A s s (2) (1) (4) (1) r r (3) (3) (5) (5) (4) (a) Sampling points for γxz (2) (b) Sampling points for γyz Figure A2 Sampling points for assumed shear strain filed: (a)  xz and (b)  yz . Interpolating the shear strain at the sampling points gives   yz N k  rX , s X   wk   N k  rX , s X  y ,k  x i 1 k 1  k 1  n  N  r , s     Njy   k X X wk   N k  rX , s X  x ,k  y j 1 k 1  k 1  m  xz   Ni x   (A26) is given by Thus, the elemental shear strain–displacement matrix  Bsa e   2 24  Bsa e     224   Bsa1 Bsa  Bsa  (A27a) where m i N k  ri , si    N x  r , s  x  Bsai    i n1 N k  ri , si    Njy  r , s  y  j 1 m i 1 n  Njy  r , s  N k  rj , s j  j 1   N  r , s  N  r , s   i x k i i     (A27b) 187 Appendix A In order to integrate the matrix in natural coordinate system for quadrilateral elements, the derivatives w.r.t. global coordinates need to be transformed to derivatives w.r.t. natural coordinates by means of Jacobian  x   J    xr   s y  r   y  s  (A28) Thus,  N i   N i   x    1  r   N    J     i  N i   s   y  188 (A29) Appendix B Appendix B BOUNDARY INTEGRAL EQUATION The Green’s second identity is used to transform the Laplace equation (2.7) together with the boundary conditions (Eqs. (2.8)–(2.10)) into a surface boundary integral equation (John, 1949, 1950; Newman, 1977; Sarpkaya and Isaacson, 1981; Meylan and Squire, 1996). Further details on boundary integral equation can be found in books by Brebbia et al. (1984) and Becker (1992). The Green’s second identity is given as  G     G  G  d      n  G n  dS   (B1) S where S is the boundary surface of the fluid domain, n the unit outward normal, G the fundamental solution taken as Green’s function, and  the velocity potential. The  satisfies  2  everywhere in the solution domain. The fundamental solution G, however, satisfies  2G  everywhere except at the source point ξ where it is singular. To deal with this problem, we can surround the source point ξ by a very small sphere of radius  and surface S , and examine the solution in the limit as   . By excluding this small sphere, the new volume is      and the new surface is  S  S  , hence, Eq. (B1) becomes      G G    dS S  S n   n  G  G  d     (B2) 189 Appendix B Within the volume      ,  2   2G  everywhere, which makes the lefthand side of Eq. (B2) equal to zero. The boundary surface S can be further decomposed into S  S where S is a small sphere of radius  around the source point ξ (see Fig. B1) Figure B1 Decomposition of a three-dimensional fluid domain. The surface integral in Eq. (B2) can now be split into two surface integrals resulting in the following equation      G  G     G dS     G   dS S S n  n   n  n (B3) Referring to the sphere centered at ξ , the second integral in Eq. (B3) can be written as   G      G   G dS     S  n n  0   n  G n  2 sin  d 190 (B4) Appendix B where  is the angle measured anticlockwise from the x-axis at source point ξ . By substituting for G from Eq. (2.26) and using  / n   / r on the surface S , the integral of Eq. (B4) becomes              2 sin  d  n  r  r n                2 sin  d r r r n        1        2 sin  d n      (B5)          2 sin  d n   Taking each term in the limit as   results in the following     2 sin  d  2   cos    4 (B6) Substituting this result into Eq. (B3) and rearranging the terms, we obtained the boundary integral equation for the fluid G   dS   G dS S n S n 4   (B7) If the source point ξ in Fig. B1 is on the boundary S, S  will become a small hemisphere with a surface area of 2 . Hence, the boundary integral equation for the fluid is given by 191 Appendix B G   dS   G dS S n S n 2   (B8) Free surface Green function which satisfies the boundary conditions of the fluid domain at the seabed, free surface and the Sommerfeld condition is used as a particular solution for the Laplace equation (2.7) with boundary conditions given by Eqs. (2.8)– (2.11). By decomposing the boundary surface S into SF, SHB, SB and S  , and applying boundary conditions on these surfaces, Eq. (B8) can be written as 2     G G   dS  n  n   S F  S HB  S B  S       G  G     G G  dS F      dS HB n n  n n  SF  S HB       G  G     G G  dS B      dS n n  n n  SB  S     G    G  G  dS F     G  dS HB n n  SF S HB  (B9)    G       G    dS B     G  dS n n  SB S   G   n dS S HB HB       G n  dS S HB HB    G     G  dS n n  S  By further decomposing the velocity potential  into the incident potential I , the scattered potential S and the radiated potential R , the integral on the right hand side of Eq. (B9) over the boundary surface at infinity S  can be written as 192 Appendix B  G    I   S   R   G    I  S  R  G  dS n n S         G  G  G    I  G I  dS    S  G S  dS    R G R n n  n n  n n S  S  S       n  G n  dS S    dS  (B10) As the scattered potential and radiated potential satisfy the Sommerfeld radiation condition (Eq. (2.11)), the integral on the right hand side of equation (B10) that involves S and R vanish at S  . Hence, Eq. (B10) can be written as  G      n  G n  dS S    G    I G I n n S    dS  (B11) I and G in Eq. (B11) are harmonic everywhere at S  , with exception at the point where the wave source ξ is located. Consider a small hemisphere of radius  around ξ with surface S (see Fig. B2) Figure B2 Decomposition of two-dimensional surface S  into S   S (a) plan view (b) side view. In this case  / n   / r . 193 Appendix B Equation (B11) is then given by     S I G        G  G  G I  dS    I  G I  dS    I  G I  dS n n  n n  n n  S  S  S     G    I  G I  dS n n  S    I G  dS   G I dS n n S   I  1 I dS   dS   r  r  r n S S S (B12)       I    dS   I   dS   n  S    S      I   2   I   n    2I   2  Note that the second term in the right hand side of Eq. (B12) vanished because  is very small. By substituting Eq. (B12) back into Eq. (B9), we obtain the final boundary integral equation 2  G   n dS S HB 194 HB    G n dS S HB HB  2I (B13) [...]... hydroelastic analysis of VLFS, ways for reducing hydroelastic response of VLFS, hydroelastic analysis of the VLFS under stochastic waves are presented Finally, the objectives of the thesis are articulated and layout of the thesis given to assist reading 1.1 Very Large Floating Structures As nearly half of the industrialized world is now within a kilometer of the coast, the demand on land resources and space is... method for hydroelastic analysis of VLFSs with arbitrary shapes The shaping of the front edges and end edges of longish VLFS is explored with the view to reduce the hydroelastic response of VLFS Both the hydroelastic response and stress resultants can be effectively reduced by having appropriate end shapes in the studied VLFSs Based on the linear random vibration theory, Chapter 6 proposes a framework for. .. centerline of VLFS with optimum  for   0.1 ,   0.2 ,   0.3 , and   0.4 subjected to head sea wave condition   0 88 Figure 4.1 Rectangular VLFS with a flexible line connector and gill cells: (a) plan view and (b) elevation view 93 Figure 4.2 Transformation of binary string to matrix representation of VLFS with gill cells 97 Figure 4.3 Center of mass of initial... VLFS benchmark for (a)   0 and (b)   45 117 Figure 5.3 Geometry of a uniform circular VLFS and coordinate system 118 Figure 5.4 Deflection amplitude along the centerline of a circular VLFS 118 Figure 5.5 Longish VLFS with various fore- and aft-ends shapes 120 Figure 5.6 Deflection amplitude, bending moments and shear forces along the longitudinal centerline of VLFS subjected... serviceability performance and also to advance the hydroelastic analysis under random water waves In this study, the considered ways in reducing hydroelastic response of VLFSs are the use of flexible connectors, gill cells and appropriate shapes vii Summary For the hydroelastic analysis, the VLFS is modeled as a giant floating plate The Mindlin (or first order shear deformable) plate theory is adopted for a more... response of VLFSs The extended higher order boundary element method can be applied to predict the hydroelastic response of floating plates with arbitrary shape The proposed framework of stochastic hydroelastic analysis may be adopted for reliability assessment of VLFSs The results reported herein would be useful information to offshore and marine engineers working on VLFSs ix List of tables List of tables...Table of contents 4.4.1 Determination of percentages of gill cells 100 4.4.2 Effectiveness of gill cells in reducing hydroelastic response 101 4.4.3 Optimal position of hinge connector and distribution of gill cells for maximum reduction in hydroelastic response 105 4.4.4 Effect of incident wave angles 108 4.5 Concluding remarks 112 Chapter 5 Hydroelastic analysis of VLFS with arbitrary. .. centerline of VLFS having different rotational stiffness (a)  = 30 m and (b)  = 60 m Note that the configurations of the VLFS shown in the second row of Fig 4.6 are used 104 Figure 4.10 Normalized deflection parameter results for four configurations of VLFS (a)  = 30 m and (b)  = 60 m 106 Figure 4.11 Optimal configurations of gill cells layout and hinge connector position for VLFS. .. use of flexible connector and gill cells are investigated in Chapter 4 Gill viii Summary cells are compartments in VLFS that allow free passage of water It is found that a significant reduction in both the hydroelastic response and the stress resultants of the studied VLFS can be achieved with the combined presence of a suitably positioned flexible line connector and an appropriate distribution of gill. .. symmetries m for (a)  = 30 m and (b)  = 60 m 128 Figure 5.11 Hydroelastic response of triangular VLFS (m = 3) and square VLFS (m = 4) under  = 60 m for two wave angles (a)   0 and (b)    / m 129 Figure 5.12 Hydroelastic response of circular VLFS and diagon VLFS (a)  = 30 m and (b)  = 60 m 130 Figure 6.1 Schematic diagram of a coupled water–plate model: (a) plan view and (b) elevation . NATIONAL UNIVERSITY OF SINGAPORE 2012 HYDROELASTICITY OF VLFS: ALLOWANCES FOR FLEXIBLE CONNECTORS, GILL CELLS, ARBITRARY SHAPES AND STOCHASTIC WAVES . HYDROELASTICITY OF VLFS: ALLOWANCES FOR FLEXIBLE CONNECTORS, GILL CELLS, ARBITRARY SHAPES AND STOCHASTIC WAVES GAO. method for hydroelastic analysis of VLFSs with arbitrary shapes. The shaping of the front edges and end edges of longish VLFS is explored with the view to reduce the hydroelastic response of VLFS.