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Hydrodynamic interactions between ice masses and large offshore structures Cheung, Kwok Fai 1987

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HYDRODYNAMIC INTERACTIONS BETWEEN ICE MASSES AND LARGE OFFSHORE STRUCTURES  by KWOK FAI CHEUNG B.A.Sc. U n i v e r s i t y  of Ottawa, 1985.  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  in FACULTY OF GRADUATE STUDIES Department of C i v i l  We accept  t h i s t h e s i s as conforming  to the r e q u i r e d  THE  Engineering  standard  UNIVERSITY OF BRITISH COLUMBIA March, 1987 ©  Kwok F a i Cheung, 1987  In p r e s e n t i n g  this  requirements f o r an  thesis  agree that  freely a v a i l a b l e for  scholarly  reference  for extensive  purposes  Department  or  by  the L i b r a r y and  copying of  be  granted  his  or  her  shall  not  1987  Columbia  of  the  University s h a l l make  I further  the  Head  of it  agree  this thesis  representatives.  be allowed  Engineering  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date: March,  by  or p u b l i c a t i o n of  permission.  Department of C i v i l  study.  may  understood that c o p y i n g f i n a n c i a l gain  fulfilment  advanced degree at the The  B r i t i s h Columbia, I  that p e r m i s s i o n  in p a r t i a l  of  for my  It  is  this thesis  for  without  my  written  ABSTRACT The  o b j e c t i v e of the work  to evaluate motion of  the an  Models changes i n  s i g n i f i c a n c e of ice  s t r u c t u r e and  mass  during for  flow  c u r r e n t near  described  p o t e n t i a l flow  the ambient  i n the  vicinity  field  drift  around  offshore theory.  are an  force  two-dimensional  first  ice  structure  The  offshore  model  separation  The in  investigated  effects  and  for  the  based  on  distances  ice the  up to  a by  current  i n t r o d u c i n g the added  coefficients  numerical  range of  an  the  reviewed.  element method i s developed to c a l c u l a t e these over a  f l u i d on  mass d r i f t i n g are  proximity  i n t e r a c t i o n s are g e n e r a l i z e d by and c o n v e c t i v e  of  is  the subsequent impact mechanism.  iceberg  an  in t h i s t h e s i s  mass  mass.  A  boundary  coefficients the p o i n t  of  contact. A numerical i s developed  model based on i c e p r o p e r t i e s and  to s i m u l a t e  s t r u c t u r e . Both the in t h i s t h e s i s and  added masses  impact model s e p a r a t e l y . The  cases are compared and  Finally, a  force a c t i n g  on  the  estimated  the t r a d i t i o n a l l y assumed f a r - f i e l d added  by the ambient f l u i d d u r i n g  further  impact  'contact-point'  masses are used i n the from the two  the  geometry  number of  the c r u c i a l r o l e s played  impact are  discussed.  related topics  studies.  ii  results  i s proposed  for  Table of  Contents  ABSTRACT  i  LIST OF TABLES  iv  LIST OF FIGURES  v  ACKNOWLEGEMENTS  viii  1.  2.  LITERATURE REVIEW  1  1.1  Introduction  1  1.2  F a r - F i e l d Phase  2  1.3  N e a r - F i e l d Phase  3  1.4  Contact Phase  4  1.5  Improvements  6  HYDRODYNAMIC FORCES ON TWO  7  2.1  Introduction  7  2.2  T h e o r e t i c a l Formulation  9  2.2.1  Governing  9  2.2.2  Hydrodynamic Force C o e f f i c i e n t s  2.3  2.4  3.  CYLINDERS  Numerical  Equations  Solution  12 16  2.3.1  D i s t r i b u t e d Source Method  16  2.3.2  Numerical Formulation  18  2.3.3  Singularities  20  R e s u l t s and D i s c u s s i o n s  22  2.4.1  Comparison with A n a l y t i c a l R e s u l t s  23  2.4.2  Kinematics and Dynamics of F l u i d Flow  25  2.4.3  Added Mass during Impact  27  2.4.4  D i s c u s s i o n of R e s u l t s  29  ICEBERG IMPACT LOAD  33  3.1  Introduction  33  3.2  Ice Crushing Model  34  iii  3.2.1  U n i a x i a l Compressive S t r e n g t h  35  3.2.2 C h a r a c t e r i s t i c S t r a i n Rate  37  3.2.3  38  Indentation F a c t o r  3.2.4 Contact 3.2.5  Factor  39  Shape F a c t o r  40  3.3 Impact Model 3.3.1  Foundation  40 Model  41  3.3.2 Added Masses with Both C y l i n d e r s Moving ...43 3.3.3  Added Masses: from 2-D t o 3-D  3.3.4 Mathematical Formulation  45  3.3.5  47  Numerical Procedure  3.4 R e s u l t s and D i s c u s s i o n s 3.4.1  Added Mass E f f e c t s  3.4.2 Parameter Study 4.  44  48 49 51  CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY ...54 4.1 C o n c l u s i o n s  54  4.2 Recommendations f o r F u r t h e r Study  55  APPENDIX A: LIST OF SYMBOLS  57  APPENDIX B: REFERENCES  61  iv  LIST OF TABLES 3.1  Basic case used i n impact a n a l y s i s  65  3.2  Soil properties  65  3.3  Comparison of i c e p e n e t r a t i o n s  3.4  Added mass e f f e c t s : energy and  (Adapted from Croteau, 1983)  v  66 impulse  66  LIST OF FIGURES 2.1  Two-cylinder problem: d e f i n i t i o n sketch  2.2  Two-cylinder problem: i l l u s t r a t i o n  of  67 numerical  67  procedure 2.3  Added mass c o e f f i c i e n t  for V  2.4  Added mass c o e f f i c i e n t  for U  x  and  68 68  A.  2.5  Added mass c o e f f i c i e n t  for U  69  2.6  Convective force c o e f f i c i e n t  for V  and V x y  69  2.7  Convective force c o e f f i c i e n t  for U  2  70  2.8  Convective force c o e f f i c i e n t  for U  2  70  2  2  y 2.9  Convective force c o e f f i c i e n t  for V U A  2.10  Convective force c o e f f i c i e n t  2.11  Convective force c o e f f i c i e n t V U x y  71 A  for V U y y for U U , V U x y y x  71 and  2.12  Comparison with a n a l y t i c  results: C  2.13  Comparison with a n a l y t i c  results: C 2  2.14  Comparison with a n a l y t i c  results: C  2.15  Comparison w i t h a n a l y t i c  r e s u l t s : C 2 and C ^  2.16  Distribution cylinder  1  73  g 1  a n c  * C^  to  i t s own  ?3 74  v 1  of normalized u n i t p o t e n t i a l due  72  motion  in  7  around the  ^ 75  x  d i r e c t ion 2.17  Distribution cylinder  of normalized u n i t  2 due  to the motion  potential of c y l i n d e r  around  76  1 in  the x d i r e c t i o n 3.1  Mathematical  model of i c e element  1 985)  vi  (Powell et a l ,  77  3.2  Normalized  uniaxial  compressive  s t r a i n rate  (Adapted from Bohon and  strength  vs  77  Weingarten,  1985) 3.3  I n d e n t a t i o n t e s t (Schematic sketch)  3.4  Indentation comparison  factor  for  78  brittle  failure:  between e x p e r i m e n t a l and  theoretical  78  results 3.5  Mathematical  model  of  foundation  (Elastic  79  half-space) 3.6  Added mass c o e f f i c i e n t f o r h o r i z o n t a l motions of a f l o a t i n g c i r c u l a r c y l i n d e r i n f i n i t e depth water  of  (Yeung,l98l)  3.7  Impact c o n f i g u r a t i o n  3.8  Mathematical model of impact not  80  81 (Hydrodynamic f o r c e  81  shown)  3.9  Added mass e f f e c t s : displacement of s t r u c t u r e  82  3.10  Added mass e f f e c t s : impact  82  3.11  Variation  f o r c e on s t r u c t u r e  of  i c e strength:  displacement  of  83  of  i c e strength:  impact  force  on  83  impact v e l o c i t y :  displacement  of  84  impact v e l o c i t y :  impact  force  on  84  displacement  of  85  structure 3.12  Variation structure  3.13  V a r i a t i o n of structure  3.14  V a r i a t i o n of structure  3.15  Variation  of  iceberg  draft:  structure vii  3.16  V a r i a t i o n of  iceberg  draft:  structure  viii  impact  force  on  85  ACKNOWLEGEMENTS I  wish  to  s u p e r v i s o r , Dr.  express  my  M. Isaacson,  sincere  gratitude  f o r h i s v a l u a b l e advice  guidance d u r i n g the p r e p a r a t i o n of t h i s appreciate his e f f o r t  thesis. I  and time i n r e v i e w i n g  reviewing The  this  Thanks a r e a l s o due  t o Dr.  my and  greatly  the p r e l i m i n a r y  d r a f t of t h i s t h e s i s and the v a l u a b l e suggestions the content.  to  to improve  N.D. Nathan f o r  thesis.  financial  a s s i s t a n t s h i p from  support  in  the N a t u r a l  the  form  Sciences  Research C o u n c i l of Canada i s g r a t e f u l l y  •ix  of and  a  research  Engineering  acknowledged.  1 . LITERATURE REVIEW  1.1  INTRODUCTION  The  collision  forces  mass on an o f f s h o r e rin  offshore  which a r i s e  design  location,  before  in  the  v e l o c i t y and  collision  i c e mass  and  coriolis  force  impact  s t r u c t u r e are an important Arctic  determine the impact load on the  during  region.  to  the s t r u c t u r e , a knowledge  of  acceleration  In  of the  i c e mass  These are i n f l u e n c e d by  s t r u c t u r e geometries, as  consideration order  i s necessary.  as w e l l  of an i c e  current,  the l o c a t i o n  wave,  of the  the wind,  i c e mass  r e l a t i v e t o the s t r u c t u r e . The  presence  of  significant effects masses. Since different  large the d r i f t  the c u r r e n t and distances  offshore motions  structure  has  of  ice  nearby  wave f i e l d s are m o d i f i e d  for  between the two o b j e c t s ,  the  f o r c e , wave-frequency  masses, c o n v e c t i v e be  on  separation  wave d r i f t  a  and zero-frequency  f o r c e s and l o c a l c u r r e n t  added  v e l o c i t y have t o  r e c a l c u l a t e d at d i f f e r e n t l o c a t i o n s of the i c e mass. With  a time stepping  procedure, such c a l c u l a t i o n s e v e n t u a l l y  to the impact angle and v e l o c i t y with which the impact is  load  estimated. A number of research  these  two-body  and  t e c h n i c a l l i t e r a t u r e on and  lead  s t u d i e s have been d i r e c t e d towards  component  these t o p i c s  t h e r e f o r e , a summary of  research  reports  problems.  However,  i s widely  some r e p r e s e n t a t i v e  i s presented  i n t h i s chapter. 1  the  scattered, papers The  and  studies  2 in t h i s area can be to  classified  i n t o t h r e e phases  according  the l o c a t i o n s of the i c e mass r e l a t i v e to the  namely, F a r - F i e l d , N e a r - F i e l d and Contact  structure:  Phases.  1.2 FAR-FIELD PHASE The F a r - F i e l d Phase  concerns i c e b e r g d r i f t  where the s e p a r a t i o n between  the i c e b e r g and the  i s l a r g e and the i n t e r a c t i o n s The t r a j e c t o r y  depends  geometry, and  between them are  mainly on  environmental  wave, wind and  the c o r i o l i s  iceberg  conditions effect.  models  kinematic,  can  and  broadly  statistical  approach adopted 1980;  be  (Hsiung  iceberg d r i f t  as  have been  current, numerical proposed.  as  nature  depending  Aboul-Azm,  1982;  was  and s t a t i s t i c a l  dynamic, on  the  Mountain,  models was  that  the  a s s o c i a t e d with c u r r e n t seldom a v a i l a b l e  concluded that iceberg d r i f t future.  with the accuracy  conducted by  found  methods to  developed by G a s k i l l and Rochester  A study concerned  are  such  and  Sodhi and El-Tahum, 1980). In a d d i t i o n , a h y b r i d model  which combines dynamic  they  and  negligible.  properties  classified  in  water,  structure  A number of  models, based on d i f f e r e n t assumptions, These  i n open  in  Arctic major  source  of  (1984).  of iceberg  drift  (1984), i n  which  uncertainty  i n f o r m a t i o n and i c e geometry, an a c c u r a t e manner. Therefore  an a c c u r a t e i s not  Sciences  predict  likely  model i n to  p r e d i c t i n g long  be a v a i l a b l e  i n the  is which they term near  3 1.3  NEAR-FIELD PHASE  The  r e a l i z a t i o n that  large  offshore  i c e mass  structure  environmental  conditions,  iceberg d r i f t  onto  motions of icebergs The caused  disturbance by  an  the are  has  not  depend  focussed  influenced  to  the  by  the  flow f i e l d  influences  of  cylinders  a c c e l e r a t i n g current  in  is  attracting  an  toward  the  added  varies  significantly  together.  The  fixed  two-body  i n v e s t i g a t e d by Ohkusu Loken (1981),  where  which  the  to  ice  motion  f o r c e s on an to  moving and  one  problem  as  wave  flow.  For  the  one  fixed,  it  acts  two wave  Oortmerssen  frequency  between them. The  are  was  and  Zibell  (1979)  and  mass  and  extended to p r e d i c t the motions  offshore  structure.  separation  i n waves was  of an  and  close was  added  (1986)  the  action  hydrodynamic i n t e r a c t i o n  s e v e r a l v e r t i c a l bodies of r e v o l u t i o n by Kokkinowrachos, Thanos  on  frequency  damping c o e f f i c i e n t s were found to vary with the distance  (1976)  uniformly  zero  the  i n pure  (1974), Van the  the  of  arbitary  a  always  and  mass  hydrodynamic  the d r i f t  force  f l o a t i n g one mass  on  around an  a two-dimensional  c y l i n d e r s , one  that  studies  structure.  subjected  s p e c i a l case of two found  on  more fundamental study, Yamamoto  circular  a  solely  in  leads  o b t a i n e d an a n a l y t i c s o l u t i o n f o r the number  the  Phase,  structure  eventually  i c e mass. In a  do  Near-Field  offshore  i n t e r a c t i o n s and the  motions i n the v i c i n i t y of  the  between studied model  i c e mass near  an  4  The and  problem of the motions of an  a c u r r e n t near an o f f s h o r e s t r u c t u r e was  Isaacson Dello  (1986) (See a l s o Hay  Stritto,  1986).  In  this  insignificant,  and  separately.  obtained  as a s o l u t i o n to  and  the d r i f t  drift  model,  and  time-stepping  e x p e r i e n t a l l y by t e s t s were made  s t u d i e s found no  nonlinear  fields  the two-body d i f f r a c t i o n  A  to simulate  and the  problem  Rojansky impact  a fixed caisson  CONTACT PHASE  The  Contact Phase i s c h a r a c t e r i z e d  were problem wave  through  was  a  studied  (1986),  where  of d i f f e r e n t  wave  type s t r u c t u r e .  Both  impact occured f o r small  1.4  obtained  to were  for c u r r e n t drag,  similar  Sakvalaggio  i c e b e r g s with  the  and  oscillations  added mass e f f e c t s , was  procedure.  Isaacson  current  wave-induced  t r a j e c t o r y , accounting  f o r c e s and  driven  The  waves  the c u r r e n t were assumed  the wave  treated  to  i n v e s t i g a t e d by  & Company, 1986;  i n t e r a c t i o n s between waves and be  i c e mass due  i c e mass.  by the impact of an  ice  mass on an o f f s h o r e s t r u c t u r e . T h i s i n c l u d e s s t u d i e s of  ice  p r o p e r t i e s , c r u s h i n g mechanism as w e l l as numerical Depending on the complexity or dynamic a n a l y s i s may The treating  be  models.  of the problem, an energy method used to estimate  the  impact  load.  energy method i s the most fundamental procedure ice  mass  structure-foundation simplest approach,  impact  systems. the  initial  problems  In  the  most  k i n e t i c energy  on common  in  rigid and  of the  ice  mass i s equated to the energy d i s s i p a t e d by c r u s h i n g of  the  5 i c e . With  the  assumption  maximum impact f o r c e d i r e c t l y from  the  constant  energy  balance  equation  the  estimated  (Johnson  equation i s s o l v e d i n c r e m e n t a l l y ,  and  approach,  and the i c e  f u n c t i o n of i c e  indentation  (Cox, 1985).  A comprehensive Croteau  can be  1986). In a more r i g o r o u s  impact f o r c e i s o b t a i n e d as a distance  i c e strength,  and i c e i n d e n t a t i o n  Nevel, 1985; Gershunov, the energy  of  (1983)  flexibility  study using dynamic models was done  where  and  the  ice properties,  structure-foundation  modelled. With h i s numerical  models,  by  structural  interactions  were  three d i f f e r e n t  types  of i c e - s t r u c t u r e i n t e r a c t i o n s were i n v e s t i g a t e d :  earthquake  responses of an o f f s h o r e s t r u c t u r e surrounded by a t h i n sheet, m i g r a t i n g  of  a  f l e x i b l e member and the  large i c e floe impact of an  structure. Various r e s u l t s  across  ice  a  slender  i c e b e r g on a  massive  were presented  as f u n c t i o n s  of  time. An experimental scheme and a combined model concerning i c e mass impact on were developed by S a l v a l a g g i o parameters  in  the  model  an o f f s h o r e  and Rojansky were  kinematic-dynamic structure  (1986). The key  indentified  through  a  s e n s i t i v i t y a n a l y s i s and the r e s u l t s were c o r r e l a t e d w i t h probabilitistic The  a  model.  structure  estimated by Johnson  design  loads  f o r i c e impact  and Nevel (1985).  energy based mathematical  In t h e i r  model f o r p r e d i c t i n g  were  study,  an  i c e impact  f o r c e s was developed, and the design loads were o b t a i n e d  by  6 using the Monte C a r l o d i s t r i b u t i o n technique and  statistical  input d i s t r i b u t i o n . However, t h i s model t e l l s nothing the f o r c e v a r i a t i o n d u r i n g practical  1.5  the p e r i o d of contact  about  which i s of  importance i n d e s i g n .  IMPROVEMENTS  Nowadays, numerical  models  o f f s h o r e s t r u c t u r e and under  extensive  the  research  s t u d i e s of hydrodynamic are mainly d i r e c t e d  for  o f f s h o r e s t r u c t u r e and  motions  near  an  subsequent impact mechanism  are  and  the  development.  However,  i n t e r a c t i o n s between the two  to wave  f i e l d around a d r i f t i n g  iceberg  a c t i o n s . The  objects  changes i n  flow  i c e mass i n the v i c i n i t y of a during  the  impact process  are  large ignored  in most models. Some e m p i r i c a l r e l a t i o n s or a r b i t a r y assumed constants  are  u s u a l l y used  R e s u l t s obtained  to account  by such models  sometimes g i v e r i s e to m i s l e a d i n g It  is  attempted here  instantaneous the  hydrodynamic  interaction  between a  are not  flow  element method. The model to estimate  the  fully  design  to r e c t i f y  drifting  theory  and  reliable  this oversight.  ice  and  mass and  by  The  characterize  current are  solved  r e s u l t s are then  effects.  criteria.  c o e f f i c i e n t s which  l a r g e o f f s h o r e s t r u c t u r e i n a uniform using p o t e n t i a l  f o r these  a  fixed  formulated  the  a p p l i e d to an  boundary impact  i c e b e r g impact load on s t r u c t u r e .  2. HYDRODYNAMIC FORCES ON TWO CYLINDERS  2.1  INTRODUCTION  Hydrodynamic f o r c e s a r i s e whenever  there a r e motions between  an object and  In p o t e n t i a l flow,  forces  can  i t s ambient f l u i d .  be  broadly  namely, temporal  and  classified  convective  into  two  categories,  forces.  The  former are  induced by temporal a c c e l e r a t i o n of an o b j e c t f l u i d and  are l i n e a r l y  acceleration.  The  proportional  latter,  i n the ambient  t o the magnitude  associated  with  squared term i n the B e r n o u l l i equation,  a r e caused by the  f l u i d a t the s u r f a c e  body.  mass  coefficients,  the added  the hydrodynamic  of  the v e l o c i t y  r e l a t i v e motion of the ambient Introducing  these  and  forces  of the  convective can  thereby  force be  expressed i n terms of the v e l o c i t y and a c c e l e r a t i o n of the body. The added  mass can  c e r t a i n volume of f l u i d body. The  added mass  be i n t e r p r e t e d  that of a  i s a c c e l e r a t e d with an immersed general  body  dependent, s i n c e the amount of f l u i d body depends  as the mass of a  on the p r e c i s e flow  is  directionally  a c c e l e r a t e d with  geometry. In  every f l u i d p a r t i c l e a c c e l e r a t e s t o some extent a c c e l e r a t e s , and s u i t a b l y weighted  the added mass integration  can  the  principle,  as the  be c o n s i d e r e d  of  the e n t i r e  fluid  r e l a t i v e to  the ambient  fluid,  body as a mass  surrounding the body. As a  body moves  r e l a t i v e f l u i d v e l o c i t y v a r i e s along 7  the s u r f a c e  the  of the body  8 and so  does  the  pressure  associated  squared term i n the B e r n o u l l i such a  pressure  with  the  velocity  equation. The i n t e g r a t i o n  distribution  constitutes  the  of  convective  f o r c e . U n l e s s the flow geometry i s symmetric, the c o n v e c t i v e force i s usually the  relative  non-zero. Since t h i s  velocity  of  the  force i s r e l a t e d  object,  it  r e f e r r e d as a damping f o r c e . However, from the p o i n t of view, t h i s this  force i s c a l l e d  is  to  sometimes  hydrodynamic  a convective force  in  thesis. Because of the changing flow geometry when a body moves  r e l a t i v e to the  ambient f l u i d  i n the  v i c i n i t y of  s t a t i o n a r y body, the hydrodynamic f o r c e s markedly. Such  proximity  e x p r e s s i n g the  hydrodynamic  their  geometry. The  relative  stage or at since  the  the c o n t a c t drift  motions  effects  can  a c t i n g on i t be  c o e f f i c i e n t s as  of  a  by  functions  of  of p r a c t i c a l floating  body  at  coefficients  i n two-dimensions  flow theory and solution  the added mass  s o l v e d by  and  are formulated by  forces.  element method.  c y l i n d e r s , one f i x e d and one moving, i n an unsteady  A  circular uniform  i s then compared  that o b t a i n e d by Yamamoto (1976) on the b a s i s of the theorem.  force  potential  s p e c i a l case of two  c u r r e n t . The boundary element s o l u t i o n  the  and c o n v e c t i v e  the boundary  i s p r e s e n t e d f o r the  this  interest,  subsequent impact l o a d depend c o n s i d e r a b l y on these In t h i s c h a p t e r ,  vary  generalized  hydrodynamic f o r c e s  phase are  another  with  circle  9 2 . 2 THEORETICAL FORMULATION The  two-dimensional  problem c o n s i d e r e d  corresponds  to  two  c y l i n d e r s of a r b i t a r y s e c t i o n s , one f i x e d and one moving  in  an  fluid.  In  to correspond  to  free surface  to  unsteady  uniform  three-dimensions,  current  t h i s may  two v e r t i c a l c y l i n d e r s the seabed  i n water  of  an  infinite  be c o n s i d e r e d  extending  from the  of constant  depth when  free  surface  e f f e c t s are n e g l i g i b l e .  2.2.1  GOVERNING EQUATIONS The  fluid  i s assumed  i n c o m p r e s s i b l e and i n v i s c i d ,  the flow i s assumed to be i r r o t a t i o n a l . As f a r as r e a l i s concerned, separation  such assumptions  occurs  i n most  and fluid  a r e u n s a t i s f a c t o r y and  flow  situations.  flow  However,  s e p a r a t i o n on the downstream s i d e of the s t a t i o n a r y  cylinder  has no s i g n i f i c a n t e f f e c t on the hydrodynamic a s p e c t s of the approaching  c y l i n d e r . Furthermore,  when the moving  cylinder  moves c l o s e t o c u r r e n t speed, the e f f e c t s of flow s e p a r a t i o n are  negligible.  c y l i n d e r can theory, and  The  hydrodynamic  reasonably therefore  velocity potential,  be the  force  estimated flow  can  s a t i s f y i n g the  on  by be  the  moving  potential described  flow by  L a p l a c e equation  a in  the f l u i d domain, 0 : V<i> = 0 2  With  reference  stationary cylinder  to  (2.1)  within  Fig. 2 . 1 ,  the  centroid  of the  ( c y l i n d e r 2 ) i s f i x e d a t the o r i g i n of a  10 right-handed C a r t e s i a n cylinder  (cylinder  velocity V =  c o o r d i n a t e system,  1) i s  ^ ' y ^ * Both v  V  x  unsteady uniform current The  moving  with  potential $  u  the  due  s u b j e c t e d to  an  x  incident  to the  incident  moving c y l i n d e r  near the  $  i s made  up of a  current  $  current due  v  to  stationary  * = *  U  unsteady  u  a component  + c  p o t e n t i a l corresponding  known and  an  other  U = ( ,Uy).  c y l i n d e r s , and  The  f r e e l y with  c y l i n d e r s are  velocity potential,  associated  while the  *  a  scattered  acting  on  the  two  the motion  of  the  cylinder.  '+ $  u  ,  component  to the  (2.2)  v  incident  i s given in terms of the c u r r e n t  <i>  current  is  c  components U"  and  x  as: y $ The  on  potential  the c y l i n d e r  fluid  c  = U x + U y x y  i s a l s o subjected to boundary  s u r f a c e s and  i s taken as  (2.3)  J  in the  i n v i s c i d , the  far-field.  tangential  s u r f a c e s may  be non-zero. However, the  velocity  r e l a t i v e to  each c y l i n d e r  boundary c o n d i t i o n s  on c y l i n d e r  3$ V — = 0 d n  where n i s the  distance  on on  Since  the  v e l o c i t i e s on  cylinder  the  conditions  surface  normal  i s zero.  s u r f a c e s are  the  fluid Thus,  given  by:  T,  r  (2.4) 2  normal to the  surface  and  V  is  the  F i n a l l y , in the  far  n cylinder  v e l o c i t y in  the d i r e c t i o n n.  11  field,  the  diminish  influence  so  that  $  is  respectively,  components potential  equal  may b e  generated  by  subjected  to  with  unit  the  first  two  scattering the  unit  can  fluid  and The  in and  terms  + Uy ( 0u y + y ) v  incident incident  be  as  the  2.5  presence  of  four  i n terms  potentials fixed  magnitude.  a  with  can  y,  uniform y x  the  current  and  0 ^,  the  x and  is  are  v  otherwise  also  and  Together  x and  2 which  potentials. of  (2.5)  \*---J>  are  1 i n an  cylinder  of  unit  unit  associated  c o n d i t i o n E q . 2.4,  V , E q . 2.4  both  cylinders. 0  boundary  the  scattered  when  cylinder  that  - V y 0 vy  represent  potentials  n  flow  potentials,  motion of  these  so  current  two s t a t i o n a r y  in  into  of  around  the  the  decomposed  current  in Eq.  of  V x * v0 x  -  1  terms  expressing V  still  stationary.  be  expressed  Substituting Eq.  velocity  y  components,  V  2.5 and  becomes:  30  TJ ( JIJUX +  30  ) + u ( _Zuy.  n  3n  x  further  two c y l i n d e r s  scattered  components  to  c  interpreted  the an  required  # .  proportional U , U , V and V , x ^ x y # c a n be e x p r e s s e d a s :  0  and  x  is  v  and  Y  u  cylinders  to  u  $ = Ux ( 0ux +x) 0  the  <i> a n d <i> a r e e a c h p r o p o r t i o n a l t o U a n d  The p o t e n t i a l s V  of  y  x  V n =  x  3n + V n  +  ) _  n  Y on  30  _Zvx _  v  x  3n  30  J!Vy  V  Y  T, (2.6)  y y  x  3n  0 where  n  x  and  n^  are  the  direction  cosines  with  respect  to  12 the  x  and  y directions.  v a l u e s of V" , V^, U x  x  Since  n  the  n  F l  (2.7)  0  on T 2  30 — ^ 9n  -n  on T,  =  3</» — ^ 3n  - -n  on T, and T  2  (2.9)  = -n  on  2  (2.10)  (2.8)  y  o  on T  v  the c a l c u l a t i o n  unit  °  3n  ^ 3n  derivative  of  2  and T  added  masses,  of the v e l o c i t y p o t e n t i a l $ potentials  are f u n c t i o n s  space  0 , ux  r  evaluated,  the  velocity  d e r i v a t i v e are d e f i n e d .  0  , uy  0 and vx  potential  (x,y) o n l y ,  - V 0 y vy  1  t  potentials  Since  t o time g i v e s :  3$ — = U (0 +x) + U (<t> +y) - V 0 g x ux y uy x^vx y  the temporal  i s required.  of  d i f f e r e n t i a t i o n of Eq. 2.5 with respect  Once the u n i t  defined  i n t o four uncoupled e x p r e s s i o n s :  *vx _ " x  30  In  for arbitrary  and U^, the boundary c o n d i t i o n  i n Eq. 2.6 can be r e s o l v e d 9  t h i s holds  and  (2.11)  d> have vy its  been  temporal  The hydrodynamic f o r c e a c t i n g on the  moving c y l i n d e r can r e a d i l y be o b t a i n e d . 2.2.2  HYDRODYNAMIC FORCE As p o t e n t i a l  flow  COEFFICIENTS theory i s used,  p r e s s u r e a c t i n g on the moving surface by the unsteady B e r n o u l l i  equation  the  hydrodynamic  of c y l i n d e r  1 i s given  (Milne-Thomson, 1938):  13  p = " P  u w  3$ p — 9t 2  q  r  + C  2 r  where C i s a constant when the c y l i n d e r far  as the r e s u l t a n t  i s not r o t a t i n g .  f o r c e a c t i n g on the body i s  C w i l l not c o n t r i b u t e to the f i n a l as zero f o r convenience, f l u i d at the  (2.12)  q^ i s  body s u r f a c e  As  concerned,  r e s u l t s and can be  taken  the t a n g e n t i a l v e l o c i t y  r e l a t i v e to  the moving  of  surface  i t s e l f and so i s given by:  Or  ^  =  " "Vy (  where V n and V n correspond x y y x  +  V  y x n  )  ( 2  to the clockwise and  '  counter  clockwise t a n g e n t i a l v e l o c i t y components r e s p e c t i v e l y differentiation direction.  i s performed  Substituting  the  in  the  counter  expression  1 3 )  while  clockwise  f o r the  flow  p o t e n t i a l <t> given i n Eq. 2.5 g i v e s 30  ^r " V — =  30  "v> " V  x>  n  3<6 + U( — ^ ar  + n  30 - n ) + U ( — + y y 3r  x  The r e s u l t a n t hydrodynamic  n) x  (2.14)  f o r c e components a c t i n g  on  the moving body i n the i - t h d i r e c t i o n are given by: „(i>  = -  I pn <dr i  i  i  p i w  /• 3$ P f — n.dr + -H I q n.dT / 3t 2 /, 2  1  r i  where i=1,2  denotes the  r  1  i = 1,2  (2.15)  r  f o r c e components  d i r e c t i o n s r e s p e c t i v e l y . The f i r s t  i n the  x and  y  and second terms on the  14 right-hand  side  convective  forces r e s p e c t i v e l y .  convective  terms with Eqs. 2.11 and 2.14, Eq. 2.15 becomes:  F  (0  = A [-Ci? lPw  of  in  the temporal and  Expanding the temporal and  V, - C « V + (eg + C®)V, + (CU + y  + RiP.[{C$  -  Eq. 2.15 represent  2(eg V,U  V? +  z  +  2cg V,V,  y  V  + 2<7<? tf.tf,  + Cj{ V») + ( C «  eg V,ff, + C« 7 ff, +  C$)U }  C<J, V,U,)]  + c"  tf ) 2  y  {  2  ]  e  which  (2.17)  cg =— r  f  yn,dT  (2.18)  (0 _  (2.19) J " f 4>vyflidT  (2.20) (2.21 ) (2.22)  1  I  ,dd>  2Ri J  Tl  L / cl;' = _ 2i?j J  v  r i  m  ny)2n,(fr  dT  (  dT  _ n  J L / (£*2  +  2i?! /  v F  l  ar  '  ) (  ar  (2.23) (2.24)  +  nz)2n,-<fT  (2.25)  i 'r , a ^ 2i?! J 1  (2.26)  dT  [ Ti  /" , 3 ^ / ar 1  F l  2^1 Jpj ar v  n  '  ) (  ar  n,)2n,-<fT  +  ni)  n fr )(  (2.27) (2.28)  )  15  ^^Mf^-^*^ where the C ' s and  c  a  force  v  ' s are the  coefficients  -  (2 30)  added mass and  corresponding  to  convective  the  appropriate  a c c e l e r a t i o n and v e l o c i t y terms. In Eq. 2.16, the C to the  c o e f f i c i e n t s represent  Q  incident current  presence of the  if  two c y l i n d e r s .  c o e f f i c i e n t s corresponding associated  i t were  with  the  not modified  On the other  to U"  and  x  scattered  due  by  the  hand, the  represent  potential.  f o r c e s due to the i n c i d e n t c u r r e n t  forces  The  C  forces  resultant  are due to both terms  so  that p r o p o r t i o n a l to (C +C ) which i s sometimes r e f e r r e d o a  to  as the i n e r t i a For  coefficient.  i representing  the x and y d i r e c t i o n s , a t o t a l of 8  added mass and 20 c o n v e c t i v e •to d e s c r i b e  f o r c e c o e f f i c i e n t s are r e q u i r e d  the hydrodynamic  c y l i n d e r . However, because  force  a c t i n g on  the  of the o r t h o g o n a l i t y  moving  properties  of p o t e n t i a l f u n c t i o n s , the t o t a l number of c o e f f i c i e n t s may be g r e a t l y reduced i f the  flow geometry i s symmetric  any of the axes. Under any circumstance, the C  Q  can  or  readily  integrat ions.  be  obtained  by  closed-form  about  coefficients numerical  16 2.3 As  NUMERICAL SOLUTION long  as  the  two  cylinders  touching each other, the  problem  of c l a s s i c a l methods, such as 1847)  and  method of  the  circle  images can  circular cylinders  are c i r c u l a r can  be a p p l i e d  to solve  same  Yamamoto (1976) a p p l i e d  the  circle  solve  multiple  cylinders  problem  size  most  and  so  s i t u a t i o n s . For more methods may  be  solutions are  not  domain, i t  1938).  The  problem of  two  moving  arbitary  by  colinearly.  moving  and  circular  flow.  classical  suitable  lack  numerical  used  methods, the  and  first Lamb  discretization  i s sometimes  distributed  i n f l u i d mechanics and  method was  Karman (1927)  for  methods  g e n e r a l geometries, s e v e r a l  used. Among these  the  (Kelvin,  practical  used in t h i s t h e s i s . The  method i n v o l v e s  number  most  source method i s f r e q u e n t l y  context by von  not  theorem to formulate  in an u n i f o r m l y a c c e l e r a t i n g  However, generality  of  images  the  are  by a  (Miln-Thomson,  the  the  solved  the method of  theorem  of  be  and  of  r e f e r r e d as  described  is  in  this  (1932). Since  this  the the  boundary  of  a  boundary  element  flow problem i n which the  solution  method.  2.3.1  DISTRIBUTED SOURCE METHOD Consider a p o t e n t i a l  has  to  satisfy  condition.  Since any  sources w i l l condition,  the  it  Laplace  equation  i n d i v i d u a l source  s a t i s f y the is natural  and  far-field  or combinations  L a p l a c e equation and to represent  the  the  the  of  far-field  boundary  by  a  17 continuous  distribution  distribution  of  sources.  i s d e f i n e d i n such a way  The  source  strength  that the one  remaining  boundary c o n d i t i o n on the body surface i s a l s o s a t i s f i e d . Representing  the boundary by a continuous  distribution  of sources, the u n i t p o t e n t i a l at any point x = (x,y) in the domain can  be o b t a i n e d  each source and  by summing  the c o n t r i b u t i o n s  from  i s given by the f o l l o w i n g i n t e g r a l over  the  e n t i r e boundary, T 0(x)  = — 2TT  I  Jr  f(*)G(x,£) dr  (2.33)  in which G(x,£) = l n ( s ) = In /(x-£) + (y-r?) 2  where j[ =  (£,17)  i s the c o o r d i n a t e of a point in the boundary  T, G(x,j[) i s the Green's f u n c t i o n corresponding of u n i t  (2.34)  2  strength  located  at  £ and  to a  source  the  source  f(ji) i s  s t r e n g t h at £. Eq.  2.33  satisfies  f a r - f i e l d conditions.  The  the  Laplace  remaining  equation  and  the  boundary c o n d i t i o n  the body s u r f a c e can be obtained by d i f f e r e n t i a t i n g Eq.  on 2.33  with respect to n, d i s t a n c e normal to the boundary, 30 - — 3n  1 1 r 3G (x) = - - f ( x ) + — I f(£) — (x,£) dr 2 2ir Jr 3n  where x i s now condition  is  side of Eq.  d e f i n e d as the p o i n t e n f o r c e d . The  2.35  each source to the first  term i s due  at which the  second term  r e p r e s e n t s the  sum  on the  (2.35)  boundary right-hand  of c o n t r i b u t i o n s  u n i t normal v e l o c i t y  at x» whereas  to the source at x i t s e l f and a r i s e s  from the from  18  a careful Brebbia,  i n t e g r a t i o n of t h i s s i n g u l a r i t y  (See, f o r example,  1980).  The  boundary  expression as  condition  i n Eqs.  2.7  equates to 2.10.  3<£/3n  to  The o n l y  Eq. 2.35 i s the source  strength d i s t r i b u t i o n  which can be obtained  n u m e r i c a l l y through  a  known  unknown  function,  a  in  f(£),  discretization  procedure.  2.3.2 NUMERICAL.FORMULATION The  i n t e g r a l equation,  approximate numerical strength d i s t r i b u t i o n  Eq. 2.35, i s used t o develop  procedure function.  i n t o N segments each of  to s o l v e  The boundary  ( F i g . 2.2). Thus,  Eq. 2.35  each segment  the i n t e g r a l  linear  source  T i s divided  l e n g t h Ar.., and the unknown  s t r e n g t h f(J.j) i s assumed to be constant  and  f o r the  source  w i t h i n each segment  i s s a t i s f i e d at equation  an  the c e n t r e  i s reduced  to  of N  equations:  1 1 N 3G d<p - - f(x.) + — L f(£.) — (x.,£.) A r . = - — (x.) 2 2TT j=i 3 3n 3 J 3n -  1  or i n matrix  _  1  _  (2.36)  1  form [B]{f}  = {b}  (2.37)  in which B., l x  = -  5.. 2  A r . 3G + -J- — (x.,£.) 2TT 3n 1  and bj are the r i g h t - h a n d s i d e s 5. . i s the Kronecker d e l t a  (2.38)  1  given  i n Eqs. 2.7 t o  and s. . i s the d i s t a n c e  2.10.  between  19 and J_j . 9G/9n  points  i s given simply as  - ( 9G/9s)n^ • 1 ^ ,  in which n^ i s the normal v e c t o r a t x ^ , and l ^ j vector  f o r the l i n e ,  i s the  s ^ , j o i n i n g p o i n t s x^ and  as  unit shown  in F i g 2.2. By o b t a i n i n g the d e r i v a t i v e of G, Eq. 2.38 can be w r i t t e n i n a more convenient form f o r programming: 6.. Ar. 1 B. . = -. - -Jl 2 2TT  n.-l. .  x  When i = j ,  the  term  segment c e n t e r due  5^  accounts  t o the source  segment and thus the  second  otherwise be s i n g u l a r , Once the matrix  f o r the v e l o c i t y  at a  d i s t r i b u t i o n on the  term of  Eq. 2.39, which  same would  i s i n s t e a d taken as z e r o . [B] has  strength d i s t r i b u t i o n  (2.39)  ^  1  been e v a l u a t e d ,  the  source  {f} can be s o l v e d from Eq. 2.37 by any  standard matrix s o l u t i o n potential distribution  r o u t i n e . With {f}  i s then  now known,  obtained from the  the  numerical  form of Eq. 2.33 [<t>] = [A]{f}  (2.40)  in which Ar.  - A. . = —=- G(x.,£.) l 2n ~ 1 x  l  for i ? j .  (2.41)  When i = j , a s i n g u l a r i t y occurs i n Eq. 2.41, but the value of A^  can s t i l l  up of  be e v a l u a t e d . Consider each element being made  a continuous  strength.  The  d i s t r i b u t i o n of  contribution  p o t e n t i a l a t the c e n t e r c l o s e d form  integration  from  sources with  these  of the element  sources  constant to  can be o b t a i n e d  i n t h i s case and i s given by:  the by  20  Ar.  /Ar. (In —=• 2  A. . = —i1 1  2TT  - 1 )  With the a p p r o p r i a t e e x p r e s s i o n s boundary c o n d i t i o n s  in  potentials, 0 , 0 , 0 ^ ^ux' uy added mass readily  and  0,  a c t i n g on a  the  four  unit  be determined.  force c o e f f i c i e n t s  may  The  then  be  (2.43)  U  0  Ftn.HAr}  {  =  ;  V  2  2R,  9T  are  the  to those  9T  appropriate  d e f i n e d i n Eqs.  determined, the  stationary  (2.44)  1  unit 2.19  potentials  to 2.32.  With  t o t a l hydrodynamic  c y l i n d e r moving r e l a t i v e  near another  c y l i n d e r can  force  to i t s ambient be  fluid  calculated  from  2.16.  SINGULARITIES Theoretically for  all  c y l i n d e r s are  0.02,  just  the f o r m u l a t i o n given so f a r i s  2.37  through  touching.  cylinders is  where R,  [B] i n Eq. obtained  speaking,  separation distances  between the two than  can  r  the  T  coefficients  valid  0 vy  2.10,  1  and  corresponding  2.3.3  to  given by  = — { 0 , } [ n . HAD A,  ( i )  v  Eq.  and  convective  a  C  all  vx  2.7  f o r {b}  obtained: C  where  Eqs.  (2.42)  when  However, when very s m a l l , say  i s the r a d i u s of c y l i n d e r  tends to be the  except  s i n g u l a r . The  i n v e r s i o n of  the hydrodynamic c o e f f i c i e n t s are  [B] may  the  the  two  gap,  e,  f o r e/R, 1, the  unit  less matrix  potentials  not be unique,  unstable.  and  21 Examining the f o r m u l a t i o n , a l l the d i a g o n a l  terms, c o n t a i n  e n t r i e s i n [B],  terms l i k e  1  / ij  except  ( I«  s  2.39),  Ec  where s ^ j i s the d i s t a n c e between a source and the p o i n t which boundary  c o n d i t i o n i s enforced  (Fig  s e p a r a t i o n d i s t a n c e i s very s m a l l , there small  values  of  s^  across  c y l i n d e r s . Consequently, have r e l a t i v e l y This gives  the gap  2.2). When  e x i s t a few between  the corresponding  r i s e to  difficulties  of s i n g u l a r i t i e s  i n the matrix f o r the  the boundaries  and  s i n g u l a r when  the gap  p l a c e d on  an  a u x i l i a r y boundary  boundaries  T, the minimum value of  The  i s very  shape of the a u x i l i a r y  s e n s i t i v e t o the method i n v o l v e s a  entries. inversion occurrence  away  [B] becomes  the sources  from  s ^ j can be  controlled.  boundary can be v a r i e d u n t i l As the r e s u l t s are  m i n i m i z a t i o n procedure  are  the p h y s i c a l  l o c a t i o n of the a u x i l i a r y boundary,  scheme must be adopted (see,  in t h i s  two  (source p o i n t s )  therefore  small. If  optimum c o n f i g u r a t i o n i s obtained.  event,  very  in [B],  In the above f o r m u l a t i o n , s i n g u l a r i t i e s on  the  e n t r i e s i n [B]  l a r g e values compared to a l l other  r o u t i n e and c o n s t i t u t e s the main reason  are p l a c e d  the  at  and an  an very this  iterative  f o r example, Han, 86). In any  the use of an a u x i l i a r y boundary has not been adopted thesis.  22 2.4 RESULTS AND DISCUSSIONS A FORTRAN program, HYDYFC, based on the above formulation i s developed to  calculate  the  8  added  masses  and  the  20  c o n v e c t i v e f o r c e c o e f f i c i e n t s f o r one c y l i n d e r moving i n the v i c i n i t y of a second f i x e d c y l i n d e r . In the program, a with v a r i a b l e  facet  since  f a c e t s are u s u a l l y  smaller  s i z e s can  when the s e p a r a t i o n  distance  be generated required  automatically,  at some  between the  mesh  locations  two c y l i n d e r s  is  small. In the r e s u l t s  presented, both  s t r u c t u r e are assumed equal to R, and R applied  2  the i c e mass and  to be c i r c u l a r  c y l i n d e r s with  the radii  r e s p e c t i v e l y . Although the program may be  i n general  to two c y l i n d e r s  of a r b i t r a r y  section,  a t t e n t i o n here i s given to a more fundamental case. This intended  to  contribute  hydrodynamics developed offshore  structure.  orthogonality  to  the  f o r an In  properties  understanding  i c e mass  order  to  of p o t e n t i a l  the  i c e mass has been  t h i s case,  the  reduced to 13  nonzero  and the r e l a t i o n s  functions  force  C < ' = C < > a1 a2 1  2)  of so  as  the c e n t r o i d  2  <: i ) v1  =  = 2C < v8  c  2)  (1 ) v3 = 2C < > v9 2  the an the to of For  coefficients is  among them are  as:  C < v5  use  c o n s i d e r e d to l i e on the x - a x i s .  number of  c  d r i f t i n g near  make  reduce the number of c o e f f i c i e n t s i n v o l v e d ,  of  is  summarized  23  C  a3  , }  '  C  a4 '' 2  while a l l  C  v  ' v6  1 >  '' v 7  C  4  other  C  1 >  a  coefficients  n  d  C  virj  are  1  )  r  zero.  e  m  a  i  n  °  n  ^o  n z  Because  of  the  p r o p e r t i e s of the mesh layout scheme, t h i s arrangement  also  y i e l d s more accurate r e s u l t s than o t h e r s . Rearranging  Eq. 2.16, the r e s u l t a n t hydrodynamic  a c t i n g on c y l i n d e r  p  U  )  -  [  C  f  (  Y  )  =  v2 x U  V i  [  C  +  The 3 added plotted  C  U  C  C  +  v3 y  - a1^y  "w i v6 R  1 i s given by the f o l l o w i n g e q u a t i o n s .  - a1^x  2  ( 2 U  +  U  (C  +  a2 <V +1)  "  2  ( C  x y  mass and  i n F i g s . 2.3  v4 x x  2 C  V  a3  +1  )  0  " y x V  U  y  U  ]  "  "  to 2.11 as  v5 y y  2 C  V  U  ]  (  2  >  4  5  )  +  VV  6 convective  s e p a r a t i o n d i s t a n c e s between radius r a t i o  force  ( 2  '  force c o e f f i c i e n t s  f u n c t i o n s of the  the two  4 6 )  are  relative  c y l i n d e r s , e/R  and  1f  R /Ri. 2  2.4.1 COMPARISON WITH ANALYTICAL RESULTS The boundary element s o l u t i o n s obtained i n t h i s for  the case  of two  circular cylinders  those o b t a i n e d by Yamamoto the problem using  i n an unsteady  are compared  with  (1976), who formulated and s o l v e d  the c i r c l e  the two c i r c u l a r c y l i n d e r s  thesis  theorem.  In h i s f o r m u l a t i o n ,  are represented by two  uniform flow,  while an  infinite  doublets number  of  24 tiny  ( a u x i l i a r y ) d o u b l e t s are  conditions solution the  on  the  40  except when the  infinite  convergence  two  near  r e s p e c t i v e l y . The 1 induced by  cylinder are  are  totally different  2  and  compared  compared the  i n the  l e s s than  in  two  the  convective  approaches are  expressed i n  comparison has  been  above two  an  in Figs.  to  accelerating 2.12  2 and  Figs.  cases.  be  its  and  cylinder  by a  2.14  to the  2.13  current  and  2.15  problem  are  both r e s u l t s  agree  distance  is  boundary  element  solutions  the mesh  in t h i s  force c o e f f i c i e n t s different  the range  touching. in  the  formats and  f o r them.  small,  l a y o u t . On  c y l i n d e r s are  accuracy of the  force c o e f f i c i e n t s could of the  to  1 due  separation  two  attempted  fast  touching.  r e s u l t s also diverge  become undefined when the  the  due  methods,  dependent on  other hand, Yamamoto's  expected that  i s very  cylinder  approaches  0.02,  become unstable and  other  to  cylinder  i d e n t i c a l l y . However, when the  The  close  and  required  c o n v e c t i v e f o r c e c o e f f i c i e n t s for  r e s p e c t i v e l y . Although  and  analytic  obtained,  of t h i s s e r i e s  i t s motion near  them  for e/R,  boundary  c y l i n d e r s . An  s e r i e s was  c y l i n d e r s are  p a s s i n g them  passing  two  added mass c o e f f i c i e n t s f o r  acceleration current  of an  to enforce the  terms of t h i s were used to compute the  c o e f f i c i e n t s . The  The  s u r f a c e s of the  i n terms  first  used  However,  whole set of  r e f l e c t e d through the  two  so  no  it  is  convective comparison  25 2.4.2 KINEMATICS AND DYNAMICS OF FLUID FLOW With the numerical i n t e g r a t i o n s 2.32, two  defined  by Eqs. 2.17 to  the complicated hydrodynamic i n t e r a c t i o n s between cylinders  are g e n e r a l i z e d  c o e f f i c i e n t s . The  by  a set  r e s u l t a n t hydrodynamic  the moving c y l i n d e r are simply  of  the  dimensionless  forces acting  given by Eq. 2.16. In  to i n t e r p r e t these c o e f f i c i e n t s adequately, i t i s  on  order  desirable  to look i n t o the a c t u a l kinematics and dynamics of the flow. For  purposes of i l l u s t r a t i o n ,  the case when  cylinder  c y l i n d e r 2 and both  the d i s c u s s i o n  i s directed  1 i s moving along the  c y l i n d e r s are of  x axis  to near  the same r a d i u s .  The  c o r r e s p o n d i n g u n i t p o t e n t i a l s are normalized by the c y l i n d e r radius  and are shown i n F i g s . 2.16 and 2.17.  As mentioned i n S e c t i o n  2.1, the hydrodynamic p r e s s u r e s  i n such i n t e r a c t i o n s can be c l a s s i f i e d  i n t o two  categories,  namely, temporal and c o n v e c t i v e p r e s s u r e s . The former can be i n t e r p r e t e d as the f l u i d pressure r e s i s t i n g the a c c e l e r a t i o n of a body and i s p r o p o r t i o n a l surface.  The l a t t e r  i s induced by  f l u i d p a r t i c l e s on the body the  to the p o t e n t i a l on the  square of the r e l a t i v e  body  the r e l a t i v e motions  of  and i s p r o p o r t i o n a l  to  surface  tangential  v e l o c i t y on the  body  surface. When the distribution  two c y l i n d e r s on  the  s i n u s o i d a l from which coefficients  are  r e s p e c t i v e l y . This  are f a r a p a r t ,  surface  of  cylinder  the added mass  approximately agrees  with  equal the  the 1  potential  is  and c o n v e c t i v e to values  one  almost force  and  predicted  zero by  26 c l a s s i c a l methods for an i s o l a t e d c i r c u l a r When the two c y l i n d e r s  are c l o s e  a c t s as  a blockage  t o the  Instead  of  in  moving  cylinder  2.  cylinder  1 to  far-field.  Therefore,  experiencing a  higher  amount of f l u i d .  the a c c e l e r a t i o n  of  of for  i s i n the  cylinder  pressure i n accelerating  but a c t s  1.  as i t does i n the  upstream surface  force  2  fluid  i s required  the a c c e l e r a t i o n  The i n t e g r a t i o n  g i v e s an a d d i t i o n a l  the  along the p e r i p h e r y  same motion  In other words, when  by c y l i n d e r  directions,  h i g h e r energy  produce the  p o s i t i v e d i r e c t i o n , the  increase  usual  t o flow a  together, c y l i n d e r  flow generated  the  p a r t i c l e s are now f o r c e d  cylinder.  the same  of the increased  proportional  1 is  pressure  to the magnitude  of  i n the opposite d i r e c t i o n . Such an  i n temporal f o r c e  i s manipulated as an increase  in  added masses. As  the  cylinder  forward  2, the  fluid  normal t o the motions the  two  cylinders  motion  of the  flow  i s forced  of c y l i n d e r  decreases,  pronounced and the t a n g e n t i a l increase increases locations the  of  The  these  i n t o the  directions  When the gap effects  gradients  convective  by  between  become  more  v e l o c i t i e s on the two s u r f a c e s  two c y l i n d e r s ,  of  1.  i s blocked  phenomenon i s represented by  potential  of the  increase  cylinders  local  r a p i d l y . This  fluid  at  which force  the  sharp  corresponding  eventually attracting  leads the  to two  together. i n t e r a c t i o n s between the  effects.  The  sharp  two c y l i n d e r s are  increases  i n the  mainly  potential  27 d i s t r i b u t i o n s are mainly two  confined  i n the  region where  the  boundaries a r e c l o s e . The p o t e n t i a l d i s t r i b u t i o n s on the  other  s i d e s of the two c y l i n d e r s remain f a i r l y constant  various separation  distances.  Although the  s i n g u l a r when the two boundaries a r e touching,  for  model  becomes  the  temporal  f l u i d pressure and the t a n g e n t i a l v e l o c i t y a t the s u r f a c e of the  two  cylinders  are  d i s c o n t i n u i t y at the  finite  be  steady  p o i n t of c o n t a c t .  the hydrodynamic c o e f f i c i e n t s reasonably  and  estimated  by  despite  Therefore,  at the p o i n t  the  most  of c o n t a c t  e x t r a p o l a t i o n of  known  of can  values  over a range of s e p a r a t i o n d i s t a n c e s .  2.4.3  ADDED MASS DURING IMPACT As  the  impacts,  decceleration  term  i s much  the magnitude of c o n v e c t i v e  higher  during  f o r c e s compared to t h a t  of temporal f o r c e s i s s m a l l . T h e r e f o r e ,  i t i s reasonable  ignore the  impacts.  convective  forces  problem i s s e t up i n numerical evaluate after  the  mathematical  contact.  d e f i n e d at the boundary of  boundary  V X  «  the to  masses can  be  the two merging c y l i n d e r s and  a  by the o u t l i n e d procedure.  Under t h i s  volume of the two bodies  added  conditions  Such an approach i s s a t i s f a c t o r y f o r 0 but not f o r 0  Since  form, i t i s not d i f f i c u l t  s o l u t i o n f o r the  Appropriate  s o l u t i o n may be obtained  during  to  U X  r  #  uv  and  mathematical model, the  0  y  v  >  total  decreases as they merge. The  rate  of decrease of the s o l i d volume i s p r o p o r t i o n a l t o the  rate  of i n d e n t a t i o n .  In  a c t u a l impacts,  despite  some  initial  28 e l a s t i c compression, constant  i n the  system. The  only c r e a t e l o c a l major problem volume of  the t o t a l volume of i c e remains c r u s h i n g and  d i s t u r b a n c e s i n the  of the  the i c e  model  mass,  induced by d e c r e a s i n g  i s not  s p a l l i n g of  ice  f l u i d . However,  the  with any  but rather  decrease  with the  fluid  into  c y l i n d e r 2,  s i m u l t a n e o u s l y , and  the  f l u i d must  merging  cylinders  Consequently,  the  unit  model r e l a t e s to  two  motions of c y l i n d e r towards  the  1, and  system  i n t e g r a t i o n of 0  the corresponding  <P ,  external Therefore,  as  a  obtained  VX  by  this  induced by  the other corresponding the  sink.  lost  the  to a flow  volume.  the boundary of c y l i n d e r  The  1 leads  added mass c o e f f i c i e n t . Since such a  regime i s u n r e a l i s t i c ,  to r e p r e s e n t a r e a l i s t i c value for  to an  model.  magnitude s m a l l e r f e a t u r e being  ice penetration i s than  the  considered.  Therefore,  c o n f i g u r a t i o n remains f a i r l y c o n t a c t . Due  to  ambient f l u i d ,  about two  o v e r a l l geometry  the decrease the added mass  the  c l o s e to that  is  application  C a l c u l a t i o n s have i n d i c a t e d that for the c o n d i t i o n s i n t e r e s t here, the  to flow  the added mass c a l c u l a t e d from i t  not expected impact  cylinder  the  part  components: one  replenishing  over  v x  potential,  in  an  diminishes  the l o s t volume.  behave  flow  volume  flow through  boundary i n order to r e p l e n i s h the  solid  in flow  i c e volume i n the system. Consider  e x t e r n a l boundary c o n t a i n i n g the two c y l i n d e r s . As 1 indents  fairly  orders  of  overall of the  the  of of ice  impact initial  i n contact  s u r f a c e with  i s expected  to drop  the  slightly  29 during  indentation.  As  a  conservative  approach, the added mass a t the p o i n t  and  reasonable  of i n i t i a l  contact  is  used i n impact models.  2.4.4 DISCUSSION OF RESULTS In order t o i l l u s t r a t e two-cylinder  problem,  c y l i n d e r s placed  the p r o x i m i t y e f f e c t s f o r  the  solutions  along the x - a x i s  f o r two  are studied.  this  circular For t h i s  c  p a r t i c u l a r case,  the number  force c o e f f i c i e n t s already  reduces  i n d i c a t e d . The  on c y l i n d e r  of added to  3  mass and  and 6  convective  respectively  r e s u l t a n t hydrodynamic f o r c e  as  acting  1 i s given by Eqs. 2.45 and 2.46. In order  i n t e r p r e t the p h y s i c a l  s i g n i f i c a n c e of the two  force equations,  fundamental cases  two  to  hydrodynamic  are discussed  in  detail. Case cylinder  I: in  equations  One cylinder  moving  an otherwise  for this  still  case  convenient form by vector £ The  first  =  " w P  A l C  a1 ^  +  fluid.  another  stationary  The hydrodynamic  can be  expressed  force  -in a  P  w  i t s acceleration  mass c o e f f i c i e n t which  Rl  C  1<|V| ,0)  (2.47)  2  V  near c y l i n d e r  second term  2. C  i s independent of  a c c e l e r a t i o n and i s a f u n c t i o n only. The  more  notation.  term i s the temporal f o r c e a c t i n g on c y l i n d e r  resisting  cylinder  near  of  a 1  1  in  i s the added  the d i r e c t i o n  the s e p a r a t i o n  i s the c o n v e c t i v e f o r c e  1 due t o the asymmetric d i s t r i b u t i o n of  of  distance acting  on  tangential  30 v e l o c i t y on  i t s s u r f a c e and always  c y l i n d e r s together solution for a  r e g a r d l e s s of  obtained  the  the motion d i r e c t i o n .  The  c y l i n d e r moving near  mathematically equivalent symmetrically  acts to a t t r a c t  to two  about the l i n e of  a s t a t i o n a r y plane i s equal  cylinders  moving  symmetry and can be  simply  by the method of s u p e r p o s i t i o n , or approximated  by  a large radius r a t i o , R / P M . 2  Considering  F i g s . 2.3 and 2.6, C , and C , i n c r e a s e a1 v1  3  as  3  the d i s t a n c e between the two c y l i n d e r s decreases and as the radius r a t i o  increases.  Thus, the e f f e c t s  of a  stationary  c y l i n d e r or plane boundary on the hydrodynamic behaviour a nearby d r i f t i n g and  to  induce  stationary Case  c y l i n d e r are to an  attracting  increase  force  on  of  i t s added  mass  i t toward  the  object. II:  Two stationary  This i s equivalent the negative  to  cylinders  in  the case when  a uniform  current.  two c y l i n d e r s move i n  d i r e c t i o n p l u s an unscattered  incident  current  in the p o s i t i v e d i r e c t i o n . The r e s u l t a n t hydrodynamic  forces  a c t i n g on c y l i n d e r 1 a r e given by:  p  U  p ( Y >  The  " "« '  )  A  =  terms  ^ ' (  (  C  a  2  c  ( C  a2  a 3 + 1  + 1  ^  K  + 1 )  °y  )  a  n  d  +  +  2 p  V ' w i R  ^ a3 ^ C  + 1  [ C  C  v2 x U  2  +  C  v3 y U  2 ]  ( 2  v6 x y U  a  r  e  U  k  n  ( 2  o  w  n  a  s  t^e  '  '  4 8 )  4 9 )  inertia  coef f i c i e n t s . When the flow c y l i n d e r s are  i s i n the x  c l o s e enough,  direction  the f l u i d  and the two  motion i n the gap  31 between  the  two  cylinders  d i s t r i b u t i o n of f l u i d cylinder  is  velocity  stagnated.  i s found  An  asymmetric  on the s u r f a c e  of  1 and so i s the c o n v e c t i v e p r e s s u r e a s s o c i a t e d with  t h i s v e l o c i t y d i s t r i b u t i o n . The i n t e g r a t i o n of such  pressure  d i s t r i b u t i o n g i v e s r i s e to a f o r c e p u l l i n g the two c y l i n d e r s apart and  i s represented  coefficient,  by  a negative  convective  On the other hand, the f o r c e induced by an  a c c e l e r a t i n g c u r r e n t i n the x d i r e c t i o n decreases cylinders  get  coefficient,  force  closer  (C  + a 2  and  the  1 ) , decreases  as the two  corresponding  inertia  from i t s f a r - f i e l d value  of  two. When the  flow  c u r v a t u r e of the gap  is  in  direction,  two c y l i n d e r s , the  due  fluid velocity  to the i n the  i n c r e a s e s markedly and g i v e s r i s e t o a c o n v e c t i v e  a t t r a c t i n g the  two  cylinders together.  convective force c o e f f i c i e n t gap  the y  between the two  two c y l i n d e r s  C ^ increases  corresponding r a p i d l y as  the  c y l i n d e r s d e c r e a s e s . However, when  the  are touching,  them i s stopped  The  force  the passage  of f l u i d  between  and the p o s i t i o n of the maximum v e l o c i t y and  minimum pressure switch to a p o i n t f a r t h e s t l o c a t i o n . Consequently,  the  corresponding  from the contact convective  force  a c t s i n the o p p o s i t e d i r e c t i o n and C ^ changes s i g n when the v  c y l i n d e r s are i n contact  (Yamamoto, Nath and S l o t t a ,  With the two fundamental idea about the drifting  in  illustrated.  cases mentioned, the  hydrodynamic f o r c e s  the  vicinity  Obviously,  of the  an  a c t i n g on offshore  interactions  1974). general  an i c e mass structure i s become  more  32 complicated, if  and a l l of the c o u p l i n g terms w i l l be  the i c e mass i s moving with a c u r r e n t . Such  i n t e r a c t i o n s are important the  increases  relatively before  with  small  decreasing  model i n  predicting  offshore structue  added mass  In c o n t r a s t value  increase  increase  iceberg j u s t before  i s a l s o an  from the f a r - f i e l d  far-field  value  to the  of 1.0 to 1.6 c u r r e n t design  contact  For  increases  f o r R /Ri 2  an  equal  procedure where  to the  of added mass i s used i n an impact model,  in  load.  a  impact on  important c o n s i d e r a t i o n .  c o e f f i c i e n t at  added mass  i n impact energy and  the d e s i g n  distance,  occur.  added mass of an  example, the  separation  added  i c e mass may slow down and change d i r e c t i o n  any impact c o u l d  The  60%  hydrodynamic  impact l o c a t i o n and v e l o c i t y . For example, as the  mass  two.  to a d r i f t  involved  gives  r i s e to  a  a  substantial  may lead to m o d i f i c a t i o n s  of  3. ICEBERG IMPACT LOAD  3.1  INTRODUCTION  Collisions  between  gravity-type  large  offshore  ice  features  structures  in  the  present an important e n g i n e e r i n g problem. i c e b e r g impact  load  overall stability Although  yet  problem  such  i c e b e r g s have  is  Arctic  region  In many cases, the and  governs  being  completely s a t i s f a c t o r y  been o b t a i n e d .  parameters  design  massive  as the not been  various empirical  In  most impact  crushing force  extensively  d e s i g n method models, and added  adequately s i m u l a t e d ,  formulae  derived  from  and  sometimes  give  rise  to  has  important mass  and  of  instead  r e l a t e d cases  u s u a l l y used. R e s u l t s o b t a i n e d by these models are not reliable  the  of the s t r u c t u r e .  this  investigated, a not  a f f e c t s the  and  misleading  are fully  design  criteria. An attempt  i s made here to p r e d i c t the i c e impact loads  on o f f s h o r e s t r u c t u r e s u s i n g a more r e a l i s t i c  approach.  The  r e d e f i n i t i o n of  by  and  Weingarten  the  ice  (1985) i s f i r s t  crushing  formula  Bohon  addressed. With the r e d e f i n e d i c e  c r u s h i n g formula, a simple "head-on type" impact model based on the dynamic d u r i n g impact The  e q u i l i b r i u m of  input to the  mass and  structure  mass c o e f f i c i e n t s  estimated  i s developed.  ' c o n t a c t - p o i n t ' added  in Chapter Two  the i c e  are  c o r r e l a t e d with 3-dimensional cases  impact model.  The 33  results  are then  and  compared  34 with those obtained by using f a r - f i e l d added mass d u r i n g the i n d e n t a t i o n process. The damage to the s t r u c t u r e i n the cases  i s quantified  absorbed  by c a l c u l a t i n g the  impulse  and  two  energy  by the s t r u c t u r e during the impact.  3.2 ICE CRUSHING MODEL The  crushing  f o r c e of  parameters i n crushing  impact  load  formula  of the  In a  i s necessary.  interaction  Korzhavin  models.  adequately,  c r u s h i n g model such an  i c e i s one  order  most  to  realistic  important  estimate  and  consistent  T r a d i t i o n a l l y , the  i s given by  force  the e x p e r i m e n t a l l y  c  failure  (Korzhavin, 1962)  ice-structure  rate,  s t r e n g t h of i c e at  the  c h a r a c t e r i s t i c width of  the  t represents  The v a r i a b l e s  M, I  the t h i c k n e s s  and K  are the  i n d e n t a t i o n f a c t o r and c o n t a c t f a c t o r  These are  dimensionless parameters  u n i a x i a l strength multi-axial As  D i s the  i n t e r f a c e and  the i c e f e a t u r e . factor,  (3.1)  i s the u n i a x i a l compressive strain  in  based  F = M I K D t a where a  the  of i c e with the  used  to  of  shape  respectively. correlate  crushing force  the  due  to  impact.  implied  by  the  structure  of  the  formula,  the  p r e s s u r e d i s t r i b u t i o n over the c o n t a c t area i s assumed to be uniform.  It i s  obvious that the  Korzhavin  formula  is  a p p l i c a b l e to head-on impacts, but most authors s t i l l it  to e c c e n t r i c  impacts, which correspond  only apply  to a skewed r a t h e r  35 than a uniform for the l o c a l  p r e s s u r e d i s t r i b u t i o n . In i c e pressure  reasonably modelled  distribution,  by the s o - c a l l e d  or the n o n - l i n e a r f i n i t e element et a l ,  1985).  Since  Although i t s parameters  the i c e should  formula  and  are  not  section,  Korzhavin  collisions,  Eq. 3.1 i s s i m p l e - l o o k i n g , the d e f i n i t i o n s clear.  set of u n i v e r s a l l y accepted this  the i c e  thesis.  Many  discussions  l i t e r a t u r e are c o n t r a d i c t o r y , and i s d i f f i c u l t  In  be  (Powell et a l , 1985; Bercha  r e s u l t s f o r head-on type  the former i s used i n t h i s  account  " i c e element" ( F i g 3.1)  the Korzhavin  element give s i m i l a r  order to  formula  an  i n the  to o b t a i n  a  d e f i n i t i o n s of these parameters.  ice crushing  and  of  the most  model  based  up-to-date  on  literature  the is  summarized.  3.2.1 UNIAXIAL COMPRESSIVE STRENGTH The  u n i a x i a l compressive  of g r a i n type, g r a i n w e l l as the s t r a i n  s t r e n g t h of i c e i s a  o r i e n t a t i o n , temperature,  salinity  that the  i c e ( g r a n u l a r t e x t u r e ) , Bohon  p l o t s of  u n i a x i a l strength  s t r a i n r a t e f o r i c e specimens of d i f f e r e n t  occur approximately  at a  s t r a i n r a t e of 2  x 10"  transition  rate  i s known  as  the  and  specimens 3  same o b s e r v a t i o n s f o r sea i c e were a l s o reported by strain  versus  temperatures  s a l i n i t i e s are s i m i l a r . The peak s t r e n g t h s of a l l  (1971). T h i s  as  rate.  Based on laboratory-grown (1984) showed  function  s ~ . The 1  Schwarz  reference  s t r a i n r a t e which d e f i n e s the t r a n s i t i o n  or  between  36 ductile  and b r i t t l e  f a i l u r e s . With i n c r e a s e s of s t r a i n  beyond the r e f e r e n c e  s t r a i n r a t e , the  rate  specimen f a i l s  in a  b r i t t l e mode a t a lower p r e s s u r e . On the other hand, i f the i c e specimen  i s subjected  specimen creeps  and f a i l s  to a  low r a t e  in a ductile  Based on t h e i r experimental strain  r a t e of  2 x  proposed a curve  10"  3  s  _ 1  to d e f i n e  c  (1985)  strength for  a l l  r a t e e ( F i g 3.2): for e > 2X10"  a [ 0 . 3 + 0.351og(5xl0 e)]  for e < 2X10"  3  s"  3  s"  1  =  (3.2) 4  o  where  reference  and Weingarten  a [ 0 . 3 - 0.451og(14.29e)] o  a  and the  the u n i a x i a l  kinds of i c e as f u n c t i o n of s t r a i n  the  mode.  results  , Bohon  of l o a d i n g ,  a  i s the  0  corresponding  unaxial  compressive  1  strength  of  i c e specimen measured a t the r e f e r e n c e  the strain  r a t e of 2 x 10" s " . 3  Although  Eq.  1  3.2  was  developed  experiments using laboratory-grown i c e of a l l  different  the curve  together  columnar  sea  types has  data  results  points  bounded  considered  to form an upper l i m i t kinds of i c e .  basis  been v a l i d a t e d by  adequately  different  by the  the  of  ice, i t s a p p l i c a b i l i t y to  with p u b l i s h e d  i c e . The  on  curve,  so  plotting  for granular  were  found  that Eq.  to  3.2 can  and be be  t o the u n a x i a l s t r e n g t h of  37 3.2.2  CHARACTERISTIC STRAIN RATE In a  u n i a x i a l compressive  obviously  defined  as the  length d i v i d e d  by  ( F i g . 3.3),  indentor  an  f i n i t e thickness,  r a t e of  that  and  t h i s case, the s t r a i n  test,  strain  change of  length. is  the  In  an  a  rate  specimen's  indentation  pushed i n t o  an  ice  m u l t i - a x i a l compression  test  sheet  of  arises.  In  r a t e i s t r a d i t i o n a l l y given  by:  V e = — 4D where V and Unlike indentation strain  D are the the test  are  rate should  (3.3)  v e l o c i t y and  unaxial  test,  the  depend on the  d e f i n e d by  Eq.  3.3,  indentation  t e s t was  diameter of the  multiaxial  as w e l l as the width of  the  the  indentor.  interactions  and  the  thickness  of the  strain  the  characteristic  i n d e n t o r . With the  transition  in  L  ice  sheet  strain  rate  rate  for  found to be a f u n c t i o n of aspect  T h i s causes a s e r i o u s  i s based on  the constant  the  ratio.  drawback, s i n c e the a p p l i c a b i l i t y  the Korzhavin equation  is  of  reference  s t r a i n r a t e of the u n a x i a l t e s t . To  overcome  characteristic was  this  strain  problem,  r a t e obtained  proposed by Bohon and ' V/(4D) e =< VD/t  2  , 2V/t where t  is  a  the t h i c k n e s s  new i n an  Weingarten  for  indentation  the test  (1985). T h i s i s  f o r D/t  <  for 0.5  < D/t  f o r D/t  >  2.0  the  ice  of  formula  0.5 < 2.0  sheet.  (3.4)  Using  this  38 formula, the different  peak i n d e n t a t i o n  aspect  approximately reference  2  strain  ratios x  10~  pressures  have  been  s" ,  which  3  1  rate for  for  found  specimens to  occur  at  with  the  coincides  unaxial strength.  of  Consequently,  the c o r r e l a t i o n between the m u l t i a x i a l and u n i a x i a l  strength  i n the Korzhavin equation becomes more c o n s i s t e n t . An examination of  Eq. 3.4  e f f e c t s f o r d i f f e r e n t aspect failure  is restricted  occur  hand,  in  characteristic For  rate i s proportional  out-of-plane strain  intermediate  characteristic  r a t i o s . For small aspect  f o r l a r g e aspect  the  physical ratios  to the plane of the i c e sheet, so that  the c h a r a c t e r i s t i c s t r a i n the other  reveals d i f f e r e n t  ratios, direction  to V/D.  On  f a i l u r e tends  to  so  that  rate i s a f u n c t i o n of t r a t h e r  aspect  lengths  r a t i o s , both D and t are so  that the s t r a i n  the  than D.  appropriate  r a t e depends  on  both of them.  3.2.3  INDENTATION FACTOR The i n d e n t a t i o n  interaction  is  f a c t o r accounts  multiaxial  rather  u s u a l l y expressed as a f u n c t i o n ice-indentor  f o r the f a c t than  uniaxial  of the aspect  and  r a t i o of  the is the  i n t e r f a c e . P h y s i c a l l y , i t can be i n t e r p r e t e d as  the r a t i o of the peak i n d e n t a t i o n pressure r a t i o to the  that  peak u n i a x i a l  d e f i n i t i o n of s t r a i n  a t a given  aspect  s t r e n g t h of i c e . With the  new  r a t e , both peaks occur approximately at  the s t r a i n r a t e of 2 x 10~  3  s" . 1  39 On the b a s i s  of a p l a s t i c  yielding c r i t e r i a ,  Croteau  i n d e n t a t i o n f a c t o r as ductile  failure,  l i m i t a n a l y s i s and  (1983) expressed  a f u n c t i o n of  an  in-plane  the  assumed  theoretical  the aspect  Prandtl-type  ratio.  For  failure  is  assumed, and the i n d e n t a t i o n f a c t o r can be approximated by: 0.80 I = 3.0 +  < 4.5  (3.5)  D/t For b r i t t l e and t r a n s i t i o n f l a k i n g type  failure  s t r a i n r a t e s , an out-of-plane  i s assumed, and the i n d e n t a t i o n  or  factor  i s given by the f o l l o w i n g approximate e x p r e s s i o n : 1 = 1.2+ A similar experimentally  result by  0.32 D/t  for  Bohon  < 3.0  brittle and  (3.6) failure  Weingarten  was  obtained  (1985).  The  comparison between the two r e s u l t s i s shown at F i g 3.4.  3.2.4 CONTACT FACTOR The c o n t a c t the  f a c t o r accounts  ice-structure  interface.  quantified evaluation mechanism. Although and  of the  as a f u n c t i o n of s t r a i n  ductile failure, in perfect  the  More  contact  precisely,  b r i t t l e n e s s of  t h i s f a c t o r depends  other environmental  Obviously,  f o r incomplete  it  the  is  a  crushing  on the type of  conditions, i t is usually  at  ice  expressed  rate only.  contact  factor  i s unity  for  creeping  i n which the i c e sheet and the indentor are  contact.  proposed f o r b r i t t l e  However, d i f f e r e n t failure.  values  have  Bohon and Weingarten  been  defined  40 the  contact  factor  as  i n d e n t a t i o n pressure  to the  with which the c o n t a c t to vary  from  suggested  0.2  to  transition  failures  ratio  of  normalized  factor 0.3.  that the contact  for b r i t t l e  the  uniaxial  normalized strength,  for b r i t t l e  f a i l u r e was  Michel  Toussaint  and  f a c t o r ranges  found (1977)  from 0.3 t o  0.35  for ductile  and  and i s equal t o 0.6  failures.  As f a r as impact fail  the  i n the  problems a r e concerned, i c e masses  b r i t t l e mode  and a  contact  factor  of 0.3 i s  commonly used i n most impact models.  3.2.5 SHAPE FACTOR The  shape f a c t o r accounts  and a p p l i e s p r i m a r i l y piers.  For  to narrow s t r u c t u r e s  ice-structure  greater than  5 or  shape f a c t o r  is  f o r the shape of the indentor  interfaces  f o r indentors 1.0. For  f a c t o r can be taken  a  with  circular  approximately  equal  such as  bridge  with  aspect  ratios  flat  surfaces,  surface, to  the  the shape  0.9, and f o r a  90° wedge-shaped i n d e n t o r , i t i s about 0.7.  3.3 IMPACT MODEL The  impact f o r c e e x e r t e d by i c e on an o f f s h o r e s t r u c t u r e i s  u s u a l l y a f u n c t i o n of time and i s dynamic i n n a t u r e . cases, a s t a t i c a n a l y s i s i s not s u f f i c i e n t i n t e r a c t i o n s adequately  and  for i c e mass impact  on an  to d e s c r i b e  t h e r e f o r e dynamic  u s u a l l y used. G e n e r a l l y speaking,  In most  models  a complete dynamic  offshore structure  such are model  involves  a  41 reasonable  r e p r e s e n t a t i o n of the i n t e r a c t i o n between the i c e  mass, the  structure  ambient f l u i d . The step-by-step  and  its  resulting  as  the  n o n - l i n e a r problem r e q u i r e s  analysis  parameters l i s t e d above was  was  as w e l l  a  s o l u t i o n of the equations of motion.  A comprehensive  In h i s  foundation,  approach, the  involving  carried  the  first  out by Croteau  ice-structure-foundation  three (1983).  interaction  r e p r e s e n t e d by an e l a b o r a t e numerical model, whereas the  e f f e c t s of the ambient f l u i d were represented by a  constant  added mass, c o r r e s p o n d i n g  floating  c y l i n d e r . The impact  same  i s not  a relatively  illustrate  the  made i n most  simple  model,  but  dynamic model and  use  of  the  hydrodynamic  impacts.  FOUNDATION MODEL General  speaking,  the motions of a s t r u c t u r e s i t t i n g  a n o n - r i g i d f o u n d a t i o n are composed of three modes, translational,  r o c k i n g and  the head-on c o l l i s i o n structure  of an  t o r s i o n a l . In the present i c e mass on a massive  taken as n e g l i g i b l e  the s t r u c t u r e i s assumed to occur only. This i s s u f f i c i e n t  model, offshore  the e x t e n s i o n to take  and the response  i n the t r a n s l a t i o n a l  to i l l u s t r a t e the general account  of t i l t i n g ,  on  namely,  i s c o n s i d e r e d , so that the t i l t i n g and bending  the s t r u c t u r e are  and  other  literature.  importance  e f f e c t s during ice-structure  3.3.1  isolated  intended here to extend Croteau's  r a t h e r to adopt to  an  assumption has been  models d e s c r i b e d i n the  It  this  to that of  of of mode  procedure  which may  be  42 j u s t as important,  is straight  forword  From continuum mechanics, i s modelled  as a  massless  h a l f - s p a c e as i n d i c a t e d h a l f - s p a c e can added s o i l mass,  a dynamic foundation  rigid  disk  in F i g .  be e l a s t i c  in p r i n c i p l e .  r e s t i n g on a  3.5.  The  uniform  m a t e r i a l of  the  from which  the  s p r i n g system  are  or v i s c o e l a s t i c  e q u i v a l e n t dashpot and  system  e v a l u a t e d . With the assumption of an e l a s t i c m a t e r i a l , a set of  e q u i v a l e n t s o i l parameters independent of the  frequency  have  (1971) and  been d e s c r i b e d  Clough  and  by Newmark  Penzien  equivalent spring constant,  (1975).  and  Rosenblueth  They  v i s c o u s damper  excitation  express  the  and added  soil  mass as: k  = 18.2G  s  s  \-v  R  ( -„)  = 1.08/k p R  m  = 0.28p R  (3.8)  3  (3.9)  3  where R i s the r a d i u s of the p l a t e , G of  the s o i l ,  (3.7)  2  2  c  e  2  v i s the Poisson's  g  i s the shear  modulus  r a t i o of the s o i l and  p  is s  the  soil  density.  More  refined  foundation  models  with  viscoelastic  foundation m a t e r i a l have been given by V e l e t s o s  and  (1973),  Verbic  equivalent excitation The of  soil  and  Luco  parameters  (1974,  were  1976),  where  the  as  functions  of  given  frequency.  added s o i l mass i s  zero pressure  a c t i n g on  d e r i v e d through  the  the foundation  s o i l , with  s t r u c t u r e represented by a r i g i d  assumption  w e i g h t l e s s d i s k . The  the total  43 mass of  the  structure-foundation  system  is  then  usually  taken as the sum of the mass of s t r u c t u r e and the added mass. In  other  pressure  words,  the  overburden  on the added s o i l mass i s assumed to be  negligible  ( V e l e t s o s and V e r b i c ,  the  influence  of  soil  1973; Clough and Penzien,  1975).  3.3.2 ADDED MASSES WITH BOTH CYLINDERS MOVING The  added  offshore  masses  structure  assumptions of a  of  are  an  iceberg  derived  drifting  i n Chapter  two-dimensional flow and  near  Two of a  with  the  stationary  s t r u c t u r e . For a s t r u c t u r e with a n o n - r i g i d foundation, added masses  of the  masses a r e r e q u i r e d  s t r u c t u r e as  w e l l as  an  coupling  the added  i n the impact model.  There a r e two sets of c o u p l i n g added masses. These  are  the added mass on the iceberg due to the s t r u c t u r e ' s motions and  the added  mass on the  s t r u c t u r e due to  the i c e mass'  motions. P h y s i c a l l y , they can be i n t e r p r e t e d as the temporal forces acting  on  one object  another nearby  object.  Although some new necessary generate existing  to  modify  them.  to the  acceleration  parameters are r e q u i r e d , the  Indeed,  r e s u l t s by  due  existing  they  computer  can  be  used i n s t r u c t u r a l mechanics. For  i t i s not program  obtained  reciprocal relations  from  s i m i l a r to  example, given  to  a  the those  certain  s t r u c t u r e t o i c e mass radius r a t i o R / R i , and the c e n t r e 2  centre  distance  s t r u c t u r e i s given  L,  the  added  mass  of  coefficient  of  by that of the i c e mass f o r the r a t i o  to the of  44  R,/R  2  and the same L.  S i m i l a r l y , the added mass of the i c e  mass due t o the motion of  the s t r u c t u r e i n the x  direction  can be obtained by s u b t r a c t i n g C , from C f o r the r a t i o of a i az 0  R /R,. The other c o u p l i n g term, the added mass on 2  due  structure  to the motion of the i c e mass i n the x d i r e c t i o n can be  obtained i n the same manner but with the r a t i o  R,/R . 2  3.3.3 ADDED MASSES: FROM 2-D TO 3~D In  Chapter  2, the treatment  a two-dimensional  flow.  of added mass was based  In order t o  use the impact  on  model,  added masses f o r a t h r e e - d i m e n s i o n a l  flow, t a k i n g account of  f i n i t e water  drafts,  Although a  can  be  and  i c e mass  three-dimensional  implement approach,  depths  the task, the added obtained  as  an  model  approximate  masses f o r by  can  cylinder  multiplying  mass c o e f f i c i e n t s  in a finite  Yeung (1981) and  for  developed  but  their  to  reasonable case  two-dimensional  factors.  for a  depth of water  are used  be  the t h r e e - d i m e n s i o n a l  c o u n t e r p a r t s by a p p r o p r i a t e c o r r e l a t i o n The added  are r e q u i r e d .  floating  circular  have been computed  t h i s purpose.  These  by  results  are summarized i n F i g . 3.6. The t h r e e - d i m e n s i o n a l added mass coefficient  f o r a c y l i n d e r extending  i s e q u i v a l e n t t o the two-dimensional between the c y l i n d e r and  t o seabed i s u n i t y and case. As the c l e a r a n c e  the seabed  i n c r e a s e s , the added  mass c o e f f i c i e n t decreases a c c o r d i n g l y . In  both  the two-dimensional  cases, the added mass c o e f f i c i e n t  and  three-dimensional  f o r an i s o l a t e d  structure,  45 r e p r e s e n t e d as a c i r c u l a r c y l i n d e r extending from the seabed to the f r e e s u r f a c e , nearby,  the  increases  i s u n i t y . As an  added  from  mass  for a  coefficient  unity.  correlation calculation s t r u c t u r e , only  i c e mass i s  Therefore, to the the  of  the  when  drifting structure  applying  the  two-dimensional added  quantity  in  excess of  mass  one  is  affected.  3.3.4 MATHEMATICAL  FORMULATION  In most i c e mass impact problems, the i c e mass and offshore  structure  are  c y l i n d e r s , one f l o a t i n g from the external  seabed to  represented  and the other  the f r e e  surface  two  circular  f i x e d and  extending  (see F i g . 3.7). The  f o r c e s a c t i n g on the i c e mass during  convective forces,  wave f o r c e s ,  e t c . Although a l l  of these are  these f o r c e s are i c e impact The  by  generally  the  current  drag,  important  insignificant  impact  include  wind  in d r i f t  drag, models,  when compared  to  forces.  ice-structure  interactions  mathematical model as i n d i c a t e d of dynamic e q u i l i b r i u m  in  are represented F i g . 3.8. The  by  a  equations  of the i c e mass and the s t r u c t u r e are  g i v e n r e s p e c t i v e l y by: (M,+ ,)x, + M I x M1  M21X1 where  +  2  (M +m +/i j)x, + c x 2  2  M21 are the  + F = 0  2  coupling  2  + k x  2  (3. 10)  - F = 0  added masses, and M  (3.11)  1 1 f  M22  46 are  the added  t h e i r own of the  masses on  motions. The  the  ice crushing  r e l a t i v e v e l o c i t y and  which can d e v i c e . As  be c o n s i d e r e d  k i n e t i c energy of  the  of the  i c e mass.  to o b t a i n a  expressed i n matrix  is a  ignored, is  to  function  internal  objects damping  the  driving  from the  initial  Such energy i s  i s c h a r a c t e r i z e d by the  i c e mass j u s t before  In order  are  indentation  throughout the process and  s t r u c t u r e due  f o r c e , F,  non-linear  forces  impact and  the  displacement of the two  as a  a l l external  f o r c e f o r the  i c e and  dissipated velocity  impact. s o l u t i o n , Eqs.  3.10  and  3.11  are  form:  [M]{x} + [C]{x] + [K]{x} =  {F}  (3.12)  in which Mi  [M]  2  =  (3.13)  M21  M +m +M22 2  [C] =  (3.14)  [K] =  (3.15)  t  -F  \  (F} =  (3.16) F  From the t e c h n i c a l p o i n t  of view, the  i c e crushing  f o r c e , F,  47 i s now  c o n s i d e r e d as  an e x t e r n a l l y  applied force  to the  system rather than as an i n t e r n a l damping term. Eq. 3.12 n o n - l i n e a r and a time-stepping procedure i s adopted  is  for i t s  solut ion.  3.3.5  NUMERICAL PROCEDURE As d i s c u s s e d i n S e c t i o n s 3.1 and 3.2, the i c e c r u s h i n g  f o r c e i s h i g h l y n o n l i n e a r and so i s Eq. 3.12. Many numerical schemes  involving  time-stepping  developed f o r the However, most of  analysis  of  procedures such  have  nonlinear  these methods r e q u i r e  problems.  a knowledge of the  a p p l i e d f o r c e as an input to the a n a l y s i s . S i n c e the f o r c e i s unknown beforehand, an i t e r a t i v e with the time-stepping change of impact As  a  simple  time  impact  scheme i s combined  procedure i n order  to estimate  the  f o r c e d u r i n g each time s t e p .  a c c e l e r a t i o n method advanced  been  but  accurate  i s used to  procedure, give  the  the s o l u t i o n  linear at  an  (t+At) i n terms of the s o l u t i o n up to time  t.  S i n c e the change i n the a p p l i e d f o r c e , AF, i s unknown a t the first  iteration  of  each time  step,  an  arbitary  v a l u e , u s u a l l y zero, i s used. At the end of each the  displacements,  velocities  and  can be e v a l u a t e d , l e a d i n g the d i f f e r e n c e between the  to AF d u r i n g initially  iteration,  accelerations  system a r e c a l c u l a t e d , from which the impact  assumed  of the  f o r c e at (t+At)  t h i s time s t e p . I f  assumed value and the  c a l c u l a t e d value i s g r e a t e r than a p r e s c r i b e d t o l e r a n c e , the new AF  i s used  as an  input to  the next  iteration  until  48 s a t i s f a c t o r y convergence  i s obtained  and one then  proceeds  to the next time s t e p . Since a l l minor  f o r c e s such as v i s c o u s , c o n v e c t i v e  drag f o r c e s are n e g l e c t e d i n good when the impact  the f o r m u l a t i o n , the model  governing  terms  of  c a l c u l a t i o n s by t h i s model time-stepping procedure stopped i n energy  is  f o r c e i s r e l a t i v e l a r g e . While the i c e  mass i s not i n c o n t a c t with the s t r u c t u r e , the minor become  and  f r o n t of  motions  and  any  forces  subsequent  are u n r e a l i s t i c . Therefore,  stops when  the i c e mass i s  the s t r u c t u r e  a f t e r a l l the  the fully  kinetic  i s dissipated.  3.4 RESULTS AND DISCUSSIONS A FORTRAN program, and  formulation,  ICEIMP, based on has  displacement of the time  domain.  contact-point  been  developed  s t r u c t u r e and  Results added  the above  obtained masses  calculate  the impact  by  using  are  i m p l i c a t i o n s on d e s i g n c r i t e r i a a r e  to  assumptions the  load i n the  far-field  compared,  and  and  the  discussed in d e t a i l .  s e n s i t i v i t y a n a l y s i s i s then c a r r i e d out to i n v e s t i g a t e r e l a t i v e e f f e c t i v e n e s s of a  number of important  A the  parameters  on the response of the system. Since the complete  number of  dimensionless  variables  analysis  i n e f f i c i e n t . In order t o c u t unless stated  involved i s on  large,  a l l parameters  down the number of  Otherwise, a l l a n a l y s e s w i l l  a is  variables,  be done  b a s i c case as given i n Table 3.1. A l s o , two d i f f e r e n t  on  a  types  49 of foundation to  corresponding  t o weak and f i r m s o i l s a r e  i n v e s t i g a t e the e f f e c t s of  the u n d e r l y i n g  soil  used  on the  response of the system (Table 3.2).  3.4.1 ADDED MASS EFFECTS Ice p e n e t r a t i o n s mass and  the two  compared i n Table two  orders of  mass and during  for  sets  the two assumed of  assumed  soil  3.3. The p e n e t r a t i o n  magnitude smaller  fairly  c o n t a c t - p o i n t added mass  added  properties are  i n each case i s about  than the s i z e  the s t r u c t u r e . T h e r e f o r e ,  impact remains  v a l u e s of  of the i c e  the o v e r a l l  constant  geometry  and the use of the  throughout the  impact process  is  justif ied. An  i n s p e c t i o n of the r e s u l t s  indicates  that  the  structure  displacement  i c e penetration ( F i g . 3.9)  c o n t a c t - p o i n t added mass contact  i s larger  f o r the same type of i s larger i s smaller  i s used. Since  than the f a r - f i e l d  that the former g i v e s r i s e  t o a higher  t h e r e f o r e the p e n e t r a t i o n has dissipate additional  t o be  energy by  but the when  the added mass  value,  ice crushing.  i n order At the  time, as the i c e mass d e c c e l e r a t e s due t o the r e s i s t a n c e the s t r u c t u r e , the surrounding a way that  a resultant  structure against  its  fluid  i s accelerated in  temporal f o r c e always displacement  (Figs.  the at  i t i s obvious  incoming energy greater  soil  and to same of such  a c t s on the  2.16 and  2.17)..  T h i s f o r c e i s p r o p o r t i o n a l to the d e c e l e r a t i o n of the i c e mass during  impact and i s i n c o r p o r a t e d  i n t o the impact model  50 through the  c o u p l i n g added  These hydrodynamic e f f e c t s of the foundation  mass as  described  previously.  i n c r e a s e the e f f e c t i v e  stiffness  and l e a d t o a smaller displacement  of  the  structure. Comparisons i n  F i g . 3.10 i n d i c a t e  impact f o r c e s a r e e s s e n t i a l l y the assumptions of  added,  mass  and  d u r a t i o n of c o n t a c t and the in each  case.  Therefore,  c r u s h i n g and the  that  the  same under the foundation.  maximum  different  However,  the  extent of i c e p e n e t r a t i o n the  energy  dissipated  impulse t r a n s m i t t e d to  vary  in ice  the s t r u c t u r e  are  d i f f e r e n t . The energy and impulse (Table 3.4) c a l c u l a t e d using the c o n t a c t - p o i n t added mass are about 20% higher those  obtained  structure the  u s i n g the  far-field  considered  impulse  to serve  damage. In  other  of  the  underestimation  and  maximum  as measures words,  to  added  mass  foundation The and  its  design  of  (Fig.  mass. As  the  force  can  s e v e r i t y of  far-field contact  by as much  overall  local  added  leads  mass  to  as 20% i n  an underestimation  the  be  an this  may  be  structure  and  be r e l a t i v e l y  high  system.  c o n t a c t - p o i n t added effects relatively  mass i s s m a l l e r than the d r a f t of  of the  at  of l o c a l damage  the  impact  using the  p a r t i c u l a r case. Very o f t e n such crucial  added  than  i s assumed t o be a r i g i d body i n the impact model,  energy,  instead  by  by  the i c e mass  3.6). I f  mass w i l l  more s i g n i f i c a n t  when the i c e  s t r u c t u r e ( F i g . 2.3) and when becomes c l o s e  the d e s i g n  case  falls  to the i n t o one  water  the depth  or both  of  51 these c a t e g o r i e s , the added mass in an  impact  model  in  at c o n t a c t should be  order to  obtain a  more  used  realistic  result.  3.4.2 PARAMETER In order  STUDY to  parameters on  study  the  the r e l a t i v e  response  of the  s i g n i f i c a n t parameters are t r e a t e d at  a  time, to  obtained by  the b a s i c  varying  effects system,  various  some  of  the  as v a r i a b l e s , taken  case c o n s i d e r e d  each  of  parameter i n  so f a r . turn  one  Results  are  briefly  discussed. Foundation  Strength.  the i c e p e n e t r a t i o n  For a given c h o i c e of added  and energy d i s s i p a t e d  mass,  in crushing  are  l e s s f o r a weak foundation, s i n c e more energy i s devoted  to  rigid  of  body motions of the s t r u c t u r e r a t h e r than c r u s h i n g  the i c e . While  the type of  e f f e c t on the  maximum impact  s t r u c t u r e depends (Figs 3.9 and  foundations has no force,  c o n s i d e r a b l y on  case  i s four  foundation compared to the f i r m Uniaxial the use  of  the displacement the f o u n d a t i o n  3.10). The displacement  this particular  Strength the  of  Ice.  times  higher  time  a  strength  for  the  in  weak  foundation. For the case c o r r e s p o n d i n g to  contact-point  with  of  of the s t r u c t u r e  added  mass  and  foundation, F i g s . 3.11 and 3.12 show displacement force v a r i a t i o n s  significant  for different  the  firm  and impact  values  of  the  u n i a x i a l s t r e n g t h of the i c e . Since the s i z e and the i n i t i a l v e l o c i t y of the i c e mass are c o n s t a n t , the k i n e t i c energy of  52 the  incident  i c e mass  is  the  same  in  a l l cases,  t h e r e f o r e the energy d i s s i p a t e d i n c r u s h i n g and the t r a n s m i t t e d t o the s t r u c t u r e a l s o remain the same. the s t r u c t u r e displacement  and impact  i c e s t r e n g t h so t h a t the stronger higher  r a t e and over  more i n t e n s e  However,  force increase  with  i c e d i s s i p a t e s energy at a  a shorter d u r a t i o n ,  Velocity.  which leads to  For the same basic case based on  of the c o n t a c t - p o i n t added mass and the f i r m  F i g s . 3.13  impulse  and  3.14  d i f f e r e n t values  show the  corresponding  the  foundation, results  for  of impact v e l o c i t y of the i c e mass. As  the  k i n e t i c energy of the i c e mass at impact i s p r o p o r t i o n a l the square of the response of the the  a  impact.  Initial use  and  impact v e l o c i t y ,  system i s very  i t i s obvious that  to the  s e n s i t i v e to v a r i a t i o n s  of  impact v e l o c i t y . Response parameters such as the contact  time,  impact  force,  displacement  a l l increase  v e l o c i t y , emphasizing impact  ice  the  penetration appreciably  need to  and with  structure the  estimate  a  impact  reasonable  velocity.  Iceberg  Draft.  Results  obtained  by a p p l y i n g  different  i c e b e r g d r a f t s t o the b a s i c case are shown i n F i g s . 3.15 and 3.16.  The i n c r e a s e  impact energy but mass. As relatively  the  i n iceberg  a l s o increases  iceberg  small  d r a f t not only  aspect  first  causes i c e f a i l u r e c o n f i n e d to r e l a t i v e l y high f a i l u r e  the s t r e n g t h  touches  ratio  increases  of  the  the  of the i c e  structure,  contact  the  the  interface  the h o r i z o n t a l plane, and  pressure  i s then  r e q u i r e d . As  a  the  53 penetration i n c r e a s e s and  progresses, the i c e  d i r e c t i o n at a lower  the tends  draft  to  pressure.  of i c e s t r e n g t h i s accounted through  width fail  the  contact  the ratio  dependence  Therefore,  expected  in  the  a  higher  interaction  ice and  formula  an i c e b e r g with a  on an o f f s h o r e s t r u c t u r e , the aspect  process.  the  large  ratio  s m a l l d u r i n g the crushing  area  out-of-plane  f o r i n the i c e c r u s h i n g  the contact i n t e r f a c e remain f a i r l y  intense.  in  The aspect  the i n d e n t a t i o n f a c t o r . As  impacts  of  whole  pressure  response  is  of  is more  4. CONCLUSIONS AND RECOMMENDATIONS FOR FURTHER STUDY  4.1 CONCLUSIONS The  hydrodynamic  interactions  stationary cylinder potential  flow  in close  theory and  between  a  drifting  proximity are  s o l v e d by  and  formulated  the boundary  a by  element  method. These p r o x i m i t y e f f e c t s on the d r i f t i n g c y l i n d e r a r e generalized  by  8  added  mass  and  20  convective  c o e f f i c i e n t s , which i n turn may be expressed  force  as f u n c t i o n s of  the s e p a r a t i o n d i s t a n c e between the two c y l i n d e r s . For the case of two x-axis,  only  3  c o e f f i c i e n t s are  c i r c u l a r c y l i n d e r s l o c a t e d on the  added  mass  required  to  and  6  convective  describe  force  the i n t e r a c t i o n s .  These c o e f f i c i e n t s are found to vary s i g n i f i c a n t l y when separation distance  i s about  d r i f t i n g c y l i n d e r and two  cylinders  are  twice the  the i n t e r a c t i o n in  contact.  when  the  instance,  for  two  For  approximately  mass and the  y component i n e r t i a  coefficients  approximately  50%  cylinders  touching  in relation  far apart. coefficient  At  the  becomes  the  same s i z e ,  two  to the values same less  of the  culminates  c y l i n d e r s of  when  the  diameter  the  both the  added  increase  are  close  by to  when the c y l i n d e r s are  time,  the  x  than  one  in  component  inertia  relation  to the  f a r - f i e l d value of two. When the s e p a r a t i o n d i s t a n c e  is  l e s s than about 2%  the r a d i u s of the i c e mass, the numerical  procedure  becomes  u n s t a b l e . The added masses at t h i s stage a r e estimated 54  of  by an  55 e x t r a p o l a t i o n of  known v a l u e s  d i s t a n c e s . Three-dimensional m u l t i p l i n g two-dimensional f a c t o r . Although expected  to  reasonable approach,  added  mass a f t e r  added  as  separation  are obtained  masses  slightly, the  range of  added masses  the added  decrease  over a  by  a  correction  initial a  mass  by  contact  is  conservative at  initial  and  contact  should be used i n impact models. A numerical model i s developed to s i m u l a t e the response of an  an o f f s h o r e s t r u c t u r e s u b j e c t e d to the head-on impact ice  mass.  In  contrast  to  t r a d i t i o n a l l y assumed f a r - f i e l d  results  based  added mass,  on  a more  the severe  impact and s m a l l e r s t r u c t u r e displacement are obtained the  c o n t a c t - p o i n t added mass i s  more pronounced or  when the i c e mass s i z e  s m a l l e r than that of  draft  is  study  c l o s e to  identifies  the  Further s t u d i e s  impact  in  when become  i s of the same  depth. F i n a l l y , velocity  i n i c e impact  RECOMMENDATIONS FOR  effects  order  the s t r u c t u r e , or when the  the water  i n f l u e n t i a l factor  4.2  used. These  to  of  a  iceberg parameter  be  the  most  problems.  FURTHER STUDY  ice  mass  interactions  with  offshore  s t r u c t u r e s c o u l d be made i n s e v e r a l a r e a s . In  this  thesis,  the t h e o r e t i c a l  formulation  of  hydrodynamic  i n t e r a c t i o n between the two c y l i n d e r s i s  on p o t e n t i a l  flow t h e o r y . As  desirable  to  compare  the  experimental work. A l s o , the  ideal  fluid  results  the based  i s assumed, i t  obtained  here  use of the c o r r e l a t i o n  is  with factor  56 to  c o n v e r t an added mass c o e f f i c i e n t  three-dimensions  needs  to be  from two-dimensions  v e r i f i e d by  a more  to  rigorous  approach. In  most design  v e l o c i t y of an and  is  used  velocity  i c e mass i s directly  is  near a  in  the most  model, i t should Work has  procedures  a s p e c t s of  use, the  impact  be estimated  models.  As  impact  an  impact  parameter i n  in a  structure  more r a t i o n a l  these s t u d i e s  c o u l d be  (Isaacson,  data  the  p r e d i c t i n g the motion of an  offshore  impact  obtained from s t a t i s t i c a l  influential  been done i n large  in current  manner. ice  mass  1986).  Many  extended, such  as  m o d i f i c a t i o n of the model with the r e s u l t s o b t a i n e d i n t h e s i s , the wave input  inclusion and the  of m u l t i - d i r e c t i o n a l  comparison with  and  the this  irregular  experimental work  and  f i e l d measurements. In  most circumstances, the impact  offshore structure is  local  i c e mass on  e c c e n t r i c . The p r e s s u r e  on the c o n t a c t area i s equation can not  of an  distribution  no longer uniform and the  be a p p l i e d .  In order to  Korzhavin  account  i c e p r e s s u r e d i s t r i b u t i o n , the i c e should be  by the s o - c a l l e d  " i c e element"  et  a l , 1985). The  foundation  to  i n c l u d e the t o r s i o n a l mode  t a n g e n t i a l component  of the  slender  bending  important  structures, and  for  model should then be  and  tilting  For modes  should be i n c l u d e d i n the a n a l y s i s .  Bercha extended  of motion to account force.  the  modelled  (Powell et a l , 1985;  impact  an  for  the  relatively may  be  APPENDIX A: LIST OF SYMBOLS : c r o s s - s e c t i o n a l area  of i c e mass  : c r o s s - s e c t i o n a l area  of s t r u c t u r e  : c o e f f i c i e n t as d e f i n e d  by Eqs. 2.17 and 2.18  : added mass c o e f f i c i e n t  in i-th direction  : s o i l damping c o e f f i c i e n t  ( h o r i z o n t a l mode)  : Convective f o r c e c o e f f i c i e n t  in i-th direction  : damping matrix i n impact model : water depth : width of i c e - s t r u c t u r e i n t e r f a c e : gap between s t r u c t u r e and i c e mass i f p o s i t i v e ; indentation  of  structure  into  i c e mass  if  negative : source s t r e n g t h d i s t r i b u t i o n f u n c t i o n : i c e impact  force  : force vector  i n impact model  : r e s u l t a n t hydrodynamic f o r c e i n i - t h d i r e c t i o n : Green's f u n c t i o n : shear modulus of s o i l : d r a f t of i c e mass : indentation : contact  factor  (Korzhavin  f a c t o r (Korzhavin  : equivalent  spring  formula)  formula)  constant  ( h o r i z o n t a l mode) : s t i f f n e s s matrix : unit vector  i n impact model  f o r l i n e s.. 1  57  D  for  foundation  58  M  : shape f a c t o r (Korzhavin  Formula)  M,  : mass of i c e b e r g  M  2  : mass of s t r u c t u r e  M  s  : added s o i l mass ( h o r i z o n t a l mode)  [M]  : mass matrix  N  : number of elements  n^  : u n i t normal v e c t o r a t p o i n t i  n x  /  n v  i n impact model  : d i r e c t i o n c o s i n e s of  normal with  r e s p e c t to  x  and y axes p q  : hydrodynamic  pressure  : r e l a t i v e t a n g e n t i a l v e l o c i t y of f l u i d  r  structure  surface  R  : radius of foundation  R,  : r a d i u s of i c e mass  R  : r a d i u s of s t r u c t u r e  2  a t i c e or  s^j  : d i s t a n c e between sources  t  : t h i c k n e s s of i c e sheet  x'lTy  : x and y v e l o c i t y components of i n c i d e n t c u r r e n t  U  : v e l o c i t y v e c t o r of i n c i d e n t c u r r e n t  U  i and j  in indentation  test  V ,Vy  : x and y v e l o c i t y components of i c e mass motion  V  : v e l o c i t y v e c t o r of i c e mass motion  {x}  : displacement  x  vector  in  ice-structure  impact  model x,,x  2  : displacements colinear  x  of  i c e mass  impact model  : p o s i t i o n vector  and  structure  in  : p o s i t i o n vector  of source i  : Kronecker d e l t a : l e n g t h of element i : strain  r a t e of i c e  : added mass of i c e due to i t s own motion : added mass of i c e due to s t r u c t u r e ' s motion : added mass of s t r u c t u r e due to i c e ' s motion : added mass of s t r u c t u r e due to i t s own motion : Poisson's r a t i o of foundation  soil  : p o s i t i o n vector : p o s i t i o n vector  of source j  : d e n s i t y of foundation  soil  : d e n s i t y of water : u n i a x i a l compressive s t r e n g t h of i c e measured at reference  strain  r a t e of 2 X 1 0 "  : u n i a x i a l compressive s t r e n g t h a given  strain  3  s~'  of i c e measured at  rate  : velocity potential : velocity  potential  associated  with  incident  current : s c a t t e r e d p o t e n t i a l a s s o c i a t e d with : potential associated  current  with motion of i c e  : s c a t t e r e d u n i t p o t e n t i a l s a s s o c i a t e d with x  and  y components of c u r r e n t : unit  potentials  associated  components of i c e ' s motion : fluid  domain  with  x  and  y  boundaries of domain boundary of i c e mass boundary of s t r u c t u r e f i r s t and second d e r i v a t i v e with r e s p e c t  to time  APPENDIX B: REFERENCES A r c t i c S c i e n c e s L t d . (1984) "Iceberg M o d e l l i n g o f f East Coast: A Review and E v a l u a t i o n " .  Canada's  Bercha, F.G., Brown, T.G. and Cheung, M.S. (1985), "Local Pressure i n I c e - S t r u c t u r e I n t e r a c t i o n s " , Proc. Conf. A r c t i c '85, San F r a n c i s c o , pp.1243-1251 . Bohon W.M. (1984), "The C a l c u l a t i o n of Ice Force on Arctic Structure", Proc. 3rd Intl. Offshore Mechanics and Arctic E n g i n e e r i n g Symposium, New Orleans, Vol.3, pp.187-195. Bohon, W.M. and Weingarten, J.S. (1985), "Forces E x e r t e d by Ice F a i l u r e i n C r u s h i n g " , Proc. Conf. A r c t i c '85, San F r a n c i s c o , pp.456-464. Brebbia, C.A. (1980), "The Boundary Method f o r Pentech P r e s s , London, 189 pp.  Engineers",  Cheema, P.S. and Ahuja H.N. (1977), "Computer S i m u l a t i o n of Iceberg D r i f t " , Proc. O f f s h o r e Tech. Conf., Houston, Paper No. OTC 2951, pp.565-572. Clough, R.W. and Penzien, S t r u c t u r e s " , McGraw H i l l , New  J. (1975),"Dynamics York, 634pp.  of  Croteau, P. (1983), "Dynamic I n t e r a c t i o n s Between Floating i c e and Offshore S t r u c t u r e s " , Earthquake Engineering Research Center, University of C a l i f o r n i a , Berkeley, C a l . , Report No. UCB/EERC-83/06. F a l t i n s e n , O.M. and M i c h e l s e n , F.C. (1974), "Motion of Large Bodies at Zero Froude Number", Proc. Intl. Sym. on Dynamics of Marine V e h i c l e s and Offshore Structures, Univ. C o l l e g e , London, pp. 91-106. G a s k i l l , H.S. and Rochester, J . (1984), "A New Iceberg D r i f t Prediction", Cold Regions Technology, V o l . 8, pp.223-234.  Technique Science  for and  Gershunov, E.M. (1986), " C o l l i s i o n of Large Floating Ice Feature with Massive O f f s h o r e S t r u c t u r e " , J . Waterway, P o r t , C o a s t a l and Ocean D i v i s i o n , ASCE, Vol.112, No.3, August, pp.390-401. Han,  P.S., Olson, M.D. and Johnston, R.L. (1984), "A Galerkin Boundary Element Formulation with Moving S i n g u l a r i t i e s " , E n g i n e e r i n g Computations, V o l . 1 , No.3, September, pp.232-236.  Hay  & Company C o n s u l t a n t s Inc. (1986), "Motion and Impact of 61  62 Icebergs", Funds.  Canadian  Environmental  Studies  Revolving  Hsiung, C.C. and Aboul-Azm A.F. (1982), "Iceberg Drift A f f e c t e d by Wave A c t i o n " , Ocean E n g i n e e r i n g , Vol. 9, pp.433-439. Lamb, H. (1932), "Hydrodynamics", 6th E d i t i o n P u b l i c a t i o n s , New York, 738 pp.  (1945),  Dover  Isaacson, M. de St. Q. (1978), "Vertical C y l i n d e r s of A r b i t a r y S e c t i o n i n Waves", J . Waterway, P o r t , Coastal and Ocean D i v i s i o n , ASCE, Vol.104, No.WW4, Paper 13973, August, pp.309-324. Isaacson, M. de S t . Q. (1985), "Iceberg I n t e r a c t i o n s Offshore S t r u c t u r e s " , Proc. Conf. A r c t i c '85, F r a n c i s c o , CA, pp.276-284.  with San  Isaacson, M. de S t . Q. (1986), "Ice Mass Motions Near an O f f s h o r e S t r u c t u r e " , Proc. 5th I n t l . O f f s h o r e Mechanics and Arctic Engineering Symposium, Toyko, Vol.1, pp.441-447. Isaacson, M. de St. Q. and Dello S t r i t t o , F.J. (1986), "Motion of an Ice Mass Near a Large O f f s h o r e S t r u c t u r e " , Proc. O f f s h o r e Tech. Conf., Houston, Paper No. OTC 5085, pp.21-28. Johnson R.C. and Nevel D.E. (1985), "Ice Impact Structural Design Loads", Proc. 8th Intl. Conf. Port and Ocean E n g i n e e r i n g Under Arctic Condition, Narssarssuaq, Greenland. Kokkinowrachos, K., Thanos, I. and Zibell, H.G. (1986), "Hydrodynamic Interaction between Several Vertical Bodies of R e v o l u t i o n i n Waves", Proc. of the 5th Intl. O f f s h o r e Mechanics and Arctic Engineering Symposium, Tokyo, pp.194-205. Korzhavin, K.N. (1962), "Action of Ice on Engineering S t r u c t u r e s " , U.S. Army CRREL, CRREL T r a n s l a t i o n TL260, 1971 . Loken, E.A. (1981), "Hydrodynamic Interaction Between Several F l o a t i n g Bodies of A r b i t a r y Form i n Waves", Proc. Intl. Symposium on Hydrodynamics in Ocean E n g i n e e r i n g , Trondheim, V o l . 2, pp.745-779. Luco, J.E. (1974), "Impedence F u n c t i o n for a Rigid Foundation on a Layered Medium", Nuclear Engineering Design, Vol.31, No.2, January, pp.204-217. Luco, J.E.  (1976),  " V i b r a t i o n of  a R i g i d Disk on a  Layered  63  V i s c o e l a s t i c Medium", Nuclear E n g i n e e r i n g Vol.36, No.3, March, pp.325-340.  and  Design,  M i c h e l , B and N. T o u s s a i n t (1977), "Mechanism and Theory Indentation of Ice P l a t e s " , J. Glaciology, Vol.19, No.81, pp.285-300. Milne-Thomson, L.M. (1938), " T h e o r e t i c a l Hydrodynamics", E d i t i o n (1968), MacMillan, 600 pp. Mountain, D.G. (1980), "On P r e d i c t i n g Iceberg D r i f t " , Region S c i e n c e and Technology, V o l . 1, pp.273-282.  5th Cold  Newman, N.J. (1977), "Marine Cambridge, Mass., 402 pp.  Hydrodynamics",  Newmark, N.M. and Rosenblueth, Earthquake E n g i n e e r i n g " , C l i f f s , N.J., 640 pp.  E. (1971), "Fundamentals of Prentice Hall, Englewood  MIT  Press,  Newton, R.E. (1975), " F i n i t e Element A n a l y s i s of 2-D Added Mass and Damping". In F i n i t e Element i n F l u i d s , Vol.1, V i s c o u s Flow and Hydrodynamics, e d i t e d by R.H. G a l l a g h e r et a l , John Wiley, N.Y. pp.219-232. Ohkusu, M. (1974), "Hydrodynamic Forces on Multiple C y l i n d e r s i n Waves", Proc. I n t l . Symposium on Dynamics of Marine Vehicles and Structures i n Waves, Univ. C o l l e g e , London, pp.107-112. Powell, G et a l . ( l 9 8 5 ) , "Ice-Structure Interaction of an Offshore Platform", Proc. Conf. Arctic '85, San F r a n c i s c o , CA., pp.230-238. S a l v a l a g g i o , M.A. and Rojansky, M. (1986), "Importance of Wave-Driven Icebergs Impacting an O f f s h o r e S t r u c t u r e " , Proc. O f f s h o r e Tech. Conf., Houston, Paper No. OTC 5086, pp.29-38. Sarpkaya, T and Isaacson, M. (1981), "Mechanics of Wave Forces on O f f s h o r e S t r u c t u r e s " , Van Nostrand Reinhold, New York, 651 pp. Schwarz, J . (1971), "The Pressure of F l o a t i n g I c e - F i e l d on P i l e s " , I.A.H.R. Symposium 1971, Paper 6.3, Reykijavik, Iceland. Schwarz, J . and Weeks W.F. (1977), "Engineering P r o p e r t i e s of Sea I c e " , J . G l a c i o l o g y , Vol.19, No.81, pp.499-531. Sodhi, D.S. and El-Tahum, M. (1980), Iceberg D r i f t Trajectory During a G l a c i o l o g y , V o l . 1, pp.77-82. Van  Oortmerssen,  G.  (1979),  "Prediction of Storm", Annals  "Hydrodynaimc  an of  Interaction  64 Between Two S t r u c t u r e F l o a t i n g i n Waves", Proc. 2nd I n t . Conf. on the Behavior of O f f s h o r e S t r u c t u r e s , BOSS '79, London, V o l . 1, pp.339-356. Veletsos, A.S. and Verbic B. (1973), "Vibration of Viscoelastic Foundations", International Journal Earthquake E n g i n e e r i n g and S t r u c t u r e Dynamics, Vol.2, No.1, pp.87-102. Yamamoto, T. (1976), "Hydrodynamic Forces on Multiple Circular Cylinders", J . Hydraulics Division, ASCE, Vol.102, No. HY9, pp. 1193-1210. Yamamoto, T., Nath, J.H. and S l o t t a , L.S.(1974), "Wave Forces on C y l i n d e r s Near Plane Boundary", J . Waterways, Harbors and C o a s t a l E n g i n e e r i n g D i v i s i o n , ASCE,Vol.100, NO.WW4, November, pp.345-358. Yeung, R.W. (1981), "Added Mass and Damping of a V e r t i c a l C y l i n d e r i n F i n i t e - D e p t h Water", A p p l i e d Ocean Research, Vol.3, No.3, pp.1 19-133.  65  Radius of i c e mass  50 m  Draft of i c e mass  60 m  Radius of s t r u c t u r e  50 m  Height of s t r u c t u r e  125 m  Depth of water  100 m  Density of water  1025 kg/m 3  Density of s t r u c t u r e  1800 kg/m  3  Impact v e l o c i t y  2.0 m/s  U n i a x i a l strength of i c e  4.0 MPa  Table 3.1  Basic case used i n impact a n a l y s i s  Weak s o i l Mass d e n s i t y  (kg/m )  Shear modulus  (MPa)  Poisson's r a t i o  Table 3.2  3  1900 75 0.50  Firm s o i l 2200 350 0.33  S o i l properties (adapted from Croteau, 1983)  66  Far-field added mass (m)  Contact-point added mass (m)  Firm s o i l  1.43  1.59  Weak s o i 1  1.33  1.53  Table 3.3  Comparison of i c e penetrations  Far-field added mass I n i t i a l energy  Contact-point added mass  (MJ)  1623  1886  (MJ)  1590  1865  1600  1860  Fi rm s oiI  Energy d i s s i p a t e d  Impulse t r a n s m i t t e d (MNs) Weak  Soil  Energy D i s s i p a t e d  (MJ)  1430  1765  Impulse t r a n s m i t t e d  (MNs)  1480  1810  Table 3.4  Added mass e f f e c t s : energy and impulse  67  Y  incident current, U Fig 2.1  cylinder Fig 2.2  Two-Cylinder Problem: Definition Sketch  1 (drifting)  Two-Cylinder Problem: Illustration of Numerical Procedure  68  Fig 2.4  Added Mass Coefficient for  U  x  10-'  T 3  1  I 5  1  I 7  1  1  I 10-  1  1—' 3  I ' I " I 5 7 10'  Relative separation  Fig  Fig  2.6  2.5  1  1—> 3  distance,  I i I I 5 7 10' 1  1  1  v  Convective Force Coefficient for V£ and V  1  3  e/R,  Added Mass Coefficient for U  1—  y  I  1  5  I  1  1  7  I  10  J  70  Fig 2.8  Convective Force Coefficient for UJ.  73  • R  l/R|  +  Boundary  element  Yamamoto  (1976)  method  » 10.0  *a1  -|—' 3  I ' I ' M — 5 7 10-'  IO-«  Relative  Fig 2.12  +  +  Boundary C C  R  forU  a 3  f o r Uy  x  1  I 'I'M 5 7 10'  I' I 7 10* 1  separation  distance,  H 3  "—I ' I ' M 5 7 10'  e/R|  Comparison with Analytic Results:  element  a 2  I 5  C j a  method  (Yamamoto,  1976)  (Yamamoto,  1976)  » 10.0  2/R,  'a2 or -a 3  10.0  ~~l 3  ' I'I'M 5 7 10Relative  Fig 2.13  I 'I'M 5 7 10* separation  ~l 3  distance,  '—I ' I ' M 5 7 10' e/R,  Comparison with Analytic Results: C ^ and C ^  -1 3  '—I ' I ' I 5 7 10' 1  74  Fig  2.14  Comparison with Analytic Results: Cyj  o  Relative  Fig  2.15  separation  distance,  e/R,  Comparison with Analytic Results: Cy2 and C 3 v  75  Fig 2.16  Distribution of Normalized Unit Potential its Own Motion in the x Direction  around Cylinder 1 due  to  76  >  4-1  c o a  c  o T3  N  (0  e  u O 2  i  160.0  Angle Fig  2.17  r  180.0  (degrees)  Distribution of Normalized Unit Potential around the Motion of Cylinder 1 in the x Direction  Cylinder 2 due  to  ice  Node on Structure  Node on Ice F l o e c  k  Initial Gap  Fig 3.1  d  1ce  E l a s t i c Spring With L i n e a r Damper  NonlInear Dashpot (Rate Dependent)  Mathematical Model of Ice Element (Powell et al, 1985)  a 2  HI  cr W Q LLt N —I <  cr O 2  STRAIN RATE - 1 / SEC  Fig 3.2  Normalized Uniaxial Compressive Strength Vs Strain rate (Adapted from Bohon and Weingarten,  1985)  FORCE INDENTER ICE SHEET.  Fig 3.3  Indentation Test (Schematic Sketch)  — +  CROTEAU. 1983 + BONON ET AL, 1985  \ +  +  0.0  3.4  0.5  1 .0  1 .5  2.0 ASPECT  2.5 RATIO  3.0 (D/T)  3.5  4.0  4.5  5.0  Indentation Factor for Brittle Failure: Comparison Between Experimental and Theoretical Results  massless r i g i d disk  Fig 3.5  Mathematical Model of Foundation  81  -5-  H  D  Fig 3.7  Fig 3.8  Impact Configuration  Mathematical Model of Impact (Hydrodynamic Force not Shown)  82  1 - Contact-point  o o .  2 - far-field  added mass ( f i r m  3 - Contact-point 4 - far-field  added mass (firm  soil)  soil)  added mass (weak s o i l )  added mass (weak s o i l )  LUo  (_) cr . _j CO°. —rr'  o  o rsj'  o o"  0.0  0.2  0.4  0.6  TIME Fig 3.9  _  far-field far-field  o ~o •o  0.0  added mass (firm  added mass ( f i r m  Contact-point  X  1.0  1.2  1A  Added Mass Effects: Displacement of Structure  Contact-point  o  0.8  (SECOND)  soil)  soil)  added mass (weak s o i l )  added mass (weak s o i l )  0.2  0.4  0.6  TIME Fig 3.10  0.8  (SECOND)  1 .0  1.2  Added Mass Effects: Impact Force on Structure  1 .4  83 6.0 MPa  2.0 MPa  o.o  -i 0.2  1  1 0.4  Fig 3.11  1  1 0.6  1  1 0.B  1  1 1.0  1  1 1.2  TIME (SECOND)  1  1 1.4  1  1 1.6  1  1  1  1.8  1 2.0  Variation of Ice Strength: Displacement of Structure  6.0 MPa 5.0 MPa  2.0 MPa  Fig 3.12  Variation of Ice Strength: Impact Force on Structure  r  2.2  84  4.0 m/s  Fig 3.13  Variation of Impact Velocity: Displacement of Structure  X  4.0  m/s  LJO x.—  -1  2.0 Fig 3.14  Variation of Impact Velocity: Impact Force on Structure  1—  2.2  85 oo  Fig 3.15  Variation of Iceberg Draft: Displacement of Structure  O  o  1 .8  Fig 3.16  Variation of Iceberg Draft: Impact Force on Structure  

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