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Directional wave effects on large offshore structures of arbitrary shape Sinha, Sanjay 1985

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DIRECTIONAL WAVE EFFECTS ON LARGE OFFSHORE STRUCTURES OF ARBITRARY  SHAPE  by SANJAY SINHA B.Sc.(Engg),  Ranchi U n i v e r s i t y ,  Ranchi, I n d i a ,  1981  M.Tech., I n d i a n I n s t i t u t e Of T e c h n o l o g y , M a d r a s , I n d i a , A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS MASTER  FOR THE DEGREE OF  OF APPLIED SCIENCE  in FACULTY OF GRADUATE  STUDIES  D e p a r t m e n t Of C i v i l E n g i n e e r i n g We a c c e p t t h i s  t h e s i s as conforming  to the required standard  THE UNIVERSITY OF BRITISH MAY ©  COLUMBIA  1985  SANJAY SINHA, 1985  1983  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the  requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference  and study.  I further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r her r e p r e s e n t a t i v e s .  It is  understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not be allowed without my w r i t t e n  permission.  Department O f  C i v i l Engineering  The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1 Y 3 Date  DE-6  <"*/fm  ^/xdjlr  ^rr^L^.  ABSTRACT A n u m e r i c a l method wave  effects  on  large  s h a p e , b a s e d on an theory  for  i s described offshore  extension  regular  waves.  developed  t o compute  loading  amplitude  operators  spectra  for  both  Cosine  powered  independent  and  and  in  loading assumed random loading  probabilistic  Gaussian and  transverse  the  and  response  random  functions  show  that  for  there and  the  is  a  i n the  o f t h e components  of the  waves.  Since  the sea s u r f a c e  a Gaussian d i s t r i b u t i o n ,  response  which a r e  hence  of  In  waves.  of t h e r e s u l t s f o r  loading,  properties  variables.  loading  horizontal  the  loading  these  short-crested  components  occur  both  is  are also  waves,  the  in-line  and  t o t h e p r i n c i p a l wave d i r e c t i o n . Thus t h e maximum  horizontal  describe  follow  response  spreading  and r e s p o n s e a r e d e s c r i b e d . to  has been  and  short-crested  seas  wave  functions  the  r e s p o n s e , due t o s h o r t c r e s t e d n e s s The  program  waves. C o m p a r i s i o n s  reduction  of a r b i t r a r y  have been u s e d t o a c c o u n t  short-crested  significant  directional  diffraction  computer  transfer  directional  of  linear  hence  long-  of f r e q u e n c y  shortcrestedness long-  and  study  structures  of A  to  and  r e s p o n s e may  occur  d i r e c t i o n . An a n a l y t i c a l method a l s o the p r o b a b i l i s t i c  components  and  the  is  properties  maxima  of  i n an a r b i t r a r y developed  to  of t h e maxima o f  their  horizontal  resultants. In freely  the present floating  study,  box.  results  Comparisons  i i  are  described  a r e made w i t h  for  a  published  results  and  are  found t o be  quite  favourable.  Table of Contents ABSTRACT  i  L I S T OF TABLES  i v  L I S T OF FIGURES  v  i  ACKNOWLEDGEMENTS  i i iX  NOMENCLATURE  X  1.  1  2.  INTRODUCTION 1 . 1 GENERAL  1  1.2 LITERATURE REVIEW  2  1 .3 SCOPE OF THE PRESENT STUDY  6  1.4 DESCRIPTION OF METHOD  7  LINEAR DIFFRACTION THEORY  11  2.1 INTRODUCTION  11  2.2 DESCRIPTION OF RANDOM WAVES  11  2 . 3 LONG-CRESTED REGULAR WAVES  3.  13  2.3.1 WAVE FORCES  17  2.3.2 BODY MOTIONS  19  2.3.3 NUMERICAL  22  INTEGRATION  2.4 LONG-CRESTED RANDOM WAVES  24  2.5 SHORT-CRESTED RANDOM WAVES  26  PROBABILISTIC PROPERTIES OF LOADING AND RESPONSE  30  3.1 INTRODUCTION  30  3.2 PROBABILISTIC PROPERTIES OF COMPONENTS  31  3.2.1 FREQUENCY  OF UPCROSSING OF COMPONENTS  3.3 PROBABILISTIC PROPERTIES RESULTANT OF COMPONENTS  OF  THE  33  HORIZONTAL  3.3.1 A SPECIAL CASE  34 38  3.4 PROBABILISTIC PROPERTIES OF RESULTANT MAXIMA i v  39  3.5 P R O B A B I L I T Y DISTRIBUTION R E S U L T A N T MAXIMA 3.6 HUNTINGTON AND 4.  OF  EXTREMES  OF 45  G I L B E R T ' S APPROACH  47  R E S U L T S AND D I S C U S S I O N 4.1  L O A D I N G AND  RESPONSE  4.2 P R O B A B I L I S T I C RESPONSE 4.2.1  49 SPECTRA  PROPERTIES  OF  LOADING  AND 54  L O A D I N G AND R E S P O N S E  4.2.2 H O R I Z O N T A L RESPONSE  49  COMPONENTS  RESULTANTS  OF  LOADING  55 AND 56  4.3 A D E S I G N PROCEDURE 4.3.1 5.  ..58  A WORKED E X A M P L E  CONCLUSIONS  61  REFERENCES  63  A P P E N D I X I . GREEN'S APPENDIX I I .  59  FUNCTION  67  P R O B A B I L I T Y D E N S I T Y F U N C T I O N OF R E S U L T A N T  V  ...71  L I S T OF TABLES Table 2.1  page Normalizing  f a c t o r of d i r e c t i o n a l  spreading  73  function. 4.1  4.2  Load  reduction  seas  for different  Response r e d u c t i o n short-crested  4.3  f a c t o r R^.  in short-crested  s values. f a c t o r R^j i n  seas f o r d i f f e r e n t  Expected values  74  75 s values.  of the extremes of t h e  76  maxima o f l o a d i n g . 4.4  Expected values  of t h e e x t r e m e s o f t h e  maxima o f r e s p o n s e .  vi  77  L I S T OF FIGURES FIGURE  page  2.1  S k e t c h o f a u n i - d i r e c t i o n a l wave s p e c t r u m .  2.2  S k e t c h o f a d i r e c t i o n a l wave s p e c t r u m .  78  2.3  D e f i n i t i o n sketch  79  component 2.4  of a f l o a t i n g  body  •  showing  78  motions.  D e f i n i t i o n sketch  o f t h e i n c i d e n t wave  79  direction. 2.5  D i r e c t i o n a l spreading values  D e f i n i t i o n sketch  4.2  Loading  transfer  short-crested  of a f l o a t i n g functions  80  box.  i n long-  80  and  81  seas.  Response a m p l i t u d e short-crested  4.4  for different  o f s.  4.1  4.3  function  operators  i n long-  and  84  seas.  U n i - d i r e c t i o n a l wave s p e c t r u m u s e d  in  87  computations. 4.5  Loading  spectra  i n long-  and s h o r t - c r e s t e d  88  seas. 4.6  Response s p e c t r a  i n long-  and s h o r t - c r e s t e d  91  seas. 4.7  Expected values of  loading  and r e s p o n s e  short-crested 4.8  o f t h e e x t r e m e o f t h e maxima  loading  i n l o n g - and  seas.  Expected values of  o f t h e e x t r e m e o f t h e maxima  and r e s p o n s e c o m p o n e n t s i n  short-crested  94  seas.  vii  95  4.9  E x p e c t e d v a l u e s of of  horizontal  the  e x t r e m e s of  resultants  in  the  maxima  96  short-crested  seas. 4.10  F r e q u e n c y of loading  and  upcrossing response  short-crested 4.11  F r e q u e n c y of resultants  4.12  Comparision  of  components of  in long-  97  and  seas. upcrossing  of  in short-crested  horizontal seas.  of  present  method w i t h t h a t  H u n t i n g t o n and  Gilbert  (1979) f o r  surface-piercing  circular  v i i i  97  a  cylinder.  of  98  ACKNOWLEDGEMENTS In p r e s e n t i n g t h i s  thesis,  gratitude  t o Dr M. de S t . Q.  guidance  and  suggestions  t h e s i s work. I a l s o indirectly  during  thank a l l t h o s e  assistantship  financial from  support  the  C o u n c i l , Canada,  to express  Isaacson  h e l p e d me i n c o m p l e t i n g  Finally,  Research  I wish  for h i s invaluable  the e n t i r e who this  course  have thesis  i n t h e form  Natural  my s i n c e r e  Sciences  of  directly  or  successfully. a  research  and E n g i n e e r i n g  i s v e r y much a p p r e c i a t e d .  ix  of the  NOMENCLATURE characteristic  body  length,  matrix  coefficients,  see e q u a t i o n  (2.26),  matrix  coefficients,  see e q u a t i o n  (2.31),  hydrostatic  s t i f f n e s s matrix  normalizing water  factor,  depth,  directional expected source zero  coefficients,  spreading  function,  value,  strength distribution  upcrossing  frequency  function,  frequency,  o f maxima,  maximum v a l u e  of f  or f , x z  frequency  of upcrossing  frequency  of u p c r o s s i n g of h o r i z o n t a l r e s u l t a n t ,  f o r c e o r moment gravitational  of  component,  components,  constant,  Green's f u n c t i o n , incident  wave  significant  height,  wave  height,  moment o r p r o d u c t  of  inertia,  modified  Bessel  f u n c t i o n of o r d e r  Jacobian  of t r a n s f o r m a t i o n ,  wave number, body mass, mass m a t r i x  coefficients,  zero,  n - t h s p e c t r a l moment o f x component, n - t h s p e c t r a l moment o f y component, number o f maxima, number o f f a c e t s , hydrodynamic  pressure,  probability  density  function,  probability  distribution function,  probability  density  function  of extreme  of maxima, probability of  d i s t r i b u t i o n function  o f extreme  maxima,  radius  of g y r a t i o n ,  reduction  f a c t o r , see e q u a t i o n  spreading  i n d e x , see e q u a t i o n  waterplane  area,  equilbrium  body  wave  (2.44),  surface,  spectrum,  waterplane  moments,  waterplane  moments,  in-line  loading  transverse cross  (2.51),  loading  loading  loading  or response  spectrum,  or response  or response  or response  spectrum,  spectrum,  spectrum,  time, peak p e r i o d transfer  o f wave  function  operator,  xi  spectrum,  or response  amplitude  X  random  (x,y,z)  coordinate  X  maxima  m  Gaussian  extreme  y  random  z  resultant  B  Z  G  z  m  z  m  1 <k  jk  in y  direction,  axes,  o f maxima Gaussian  o f x,  variable  of x and y  components,  of c e n t r e  of  buoyancy,  z coordinate  of c e n t r e  of  gravity,  maxima  o f z,  extreme vector  o f maxima of point  amplitude mass  damping  wave  o f z, ( £ , T } , $ ) .  o f body  motion,  coefficients,  coefficients,  Kronecker CcJ  direction,  z coordinate  added X  in x  o f x,  V Z  variable  delta,  angular  velocity  frequency,  potential,  P  water  density,  0  incident  0o  principal  [u]  correlation  X  correlation  wave wave  direction, direction,  matrix, coefficient  xi i  between  x and y,  1.  INTRODUCTION  1 . 1 GENERAL Considerable attention of  offshore  regions  world-wide  demand  beneath  the  seabed  world's  total  larger  water  development gravity in  design  for can  recent  hostile  of  of  variety  Sea  200  located  The  - 400  offshore  importance,  both  from  comply  huge  with  r e s e r v e s of o i l  f o r 20 p e r c e n t  meters  have  of s t r u c t u r a l  p r o v i d e one  such  to  development  of  the  environmental conditions  i n water  of  to the  years  alone account  depths a  been p a i d  energy.  need. The  platform  North  in  has  depths  led  to  and the  concepts. Concrete o f about  200  meters  s u c h example. The  more  efficient  structures  of  paramount  economic  as  is well  as  from  safety  viewpoints. Environmental d e s i g n and such  loading  wave l o a d i n g  loads.  is  the  i s often  Offshore  one  which  considered  to l i e i n the d i f f r a c t i o n  is  Isaacson, then  modify  1979),  generally  the  and  based  which  incident  contain  regime  o f wave  of  wave  diffraction  i n Sarpkaya  diffraction  wave t h e o r y assumes t h e waves t o be  both energy  random  and  Isaacson,  sinusoidal.  short-crested  propagating  1  over  linear  long-crested  real  a  loading loading  The  (multi-directional  simultaneously  are  wave t h e o r y  1981).  However,  of  large  field  (summarized  ( u n i - d i r e c t i o n a l ) and  and  linear  severe  wave  the computation on  component i n  o f t h e most  structures  components  (e.g.  critical  seas  are  w i t h wave range  of  2  directions). tends  This  to decrease  predicted design  for  loads,  directional  directional t h e wave  obtained  improvement  in design  develop  design  by  in  of waves g e n e r a l l y  comparison  waves.  This  incorporating  appears  to  procedures.  procedures  randomness of waves but of  loads  uni-directional  spreading,  new  spreading  be  Thus  w h i c h not  a l s o f o r the  in  effects  of  a very is  those  reduction  the  it  to  significant  essential  only account  directional  to  f o r the spreading  waves.  1 .2 LITERATURE REVIEW The  major . r e s e a r c h  offshore random  s t r u c t u r e s has waves.  S a r p k a y a and attention  A  been  spreading  of  to  absence  the  into  considered  thorough  Isaacson  has  effort  given  long-crested  summary  (1981).  On  the  t o the  wave l o a d i n g on  of  regular  this  other  effect  is  the  to  loading  reliable  i n c o r p o r a t e the and  On  measured  directional  and  basis  i n a storm  multi-directional  emphasized  the  spreading  of  importance  effects  of d i r e c t i o n a l  have  due wave been  spreading  on  response p r e d i c t i o n s .  the  velocities  little  t h e waves. T h i s l a c k o f a t t e n t i o n i s p a r t l y of  by  directional  s p e c t r a . However, more r e c e n t l y s e v e r a l a t t e m p t s made  or  given  hand v e r y of  large  of  of  comparisons  wave  measured  with p r e d i c t i o n s assuming waves, F o r r i s t a l l  importance waves.  of  of  Lovaas  et.  accounting (1984) has  directionality  on  particle  in turn uni-  al.  (1978) have  for  directional  a l s o emphasized the  loading  the and  3  response  of offshore  When form  taking  Borgman  directional  described from  a  against known  of  wave  for  records,  cross-spectrum  a s e t of computer  quite  (1972),  methods  measured  data  properties various  i s  use  offshore  in  on  a  He  simulated wave  expressions  spectra  Fourier  series  t h e method  records  spectrum  He  wave  checked  wave  primary  design.  directional  based  a  the  having  a  a n d i s shown t o  favourable.  Borgman reviewed  adopted  account,  different  data.  directional  into  provided  for estimating  o f wave  theoretical  spreading  spectrum  (1969)  spectra  method  an a r r a y  analysis  be  directional  of the d i r e c t i o n a l  consideration. for  structures.  and  more  of evaluating and  o f wave  methods  recently  directional of a n a l y s i s  energy.  Pinkster  models  o f wave  numerical  Ochi wave  (1982)  have  spectra  from  of the d i r e c t i o n a l  (1984)  has reviewed  elevation  the  in directional  seas. The  above  studies  multi-directional  waves  considered  effects  & Thompson wave  and  linear  calculation  on  a  on  large  of loading random  multi-directional  their  Several  offshore  seas,  and  theoretical  waves w i t h  found  directional  cylinder.  They  waves t o  f o r both a  predictions  experimental  have  Huntington  f o r regular  functions  of  authors  studied  circular  theory  transfer  properties  structures.  (1979)  vertical  diffraction  between  the  in isolation.  and Huntington  wave  short-crested  comparison and  (1976),  effects  extended the  their  treat  long-  favourable for  results.  uni-  These  4 authors  studies  i n d i c a t e that  to d i r e c t i o n a l spreading (1977) s t u d i e d using  method  wave t h e o r i e s . Dean for the  the  case  force  reduction  the  total  force  concluded  that  a linear  may  occur  Dean  on  near  simulated  wave and  loading nonlinear  to  account  the  centre  into account, a  with of  a  further  using  reduction  on  calculated  equation.  al.  tends (1979)  to  be  also  wave  i n the l o a d i n g (1982) s t u d i e d loads  on  He  forces  somewhat  studied  o f waves i n t h e t i m e domain  structures. Battjes  shortcrestedness  Morison  wave o f time  o f t h e e x t r e m e peak  theories  et.  of s h o r t c r e s t e d n e s s a  the  prediction  Shinozuka  predicted  the  G a u s s i a n model and t h e r e b y  on a p i l e the  numerically  d i r e c t i o n a l sea as a f u n c t i o n  from d e t e r m i n i s t i c  conservative.  offshore  significant.  w h i c h combines l i n e a r  e f f e c t i s taken  i n a random  the b a s i s  also  as  (1979)  on  effects  due  i s predicted.  Hackley  obtained  effects  loading  i n w h i c h p r i n c i p a l wave d i r e c t i o n v a r i e s  When t h i s  kinematics  wave  in  (1977) e x t e n d e d t h i s method  wave f r e q u e n c y ,  hurricane.  of waves i s q u i t e  directional  a "hybrid"  the r e d u c t i o n  the and  and r e s p o n s e of the long  effects  of  horizontal  structures. Borgman spectra the  and Y f a n t i s  and c r o s s - s p e c t r a  forces  on  the  (1981) d e r i v e d of the  jacket  b a s e d on t h e M o r i s o n e q u a t i o n wave  spectrum.  horizontal  structures and  expressions  a  f o r the  components  of  i n d i r e c t i o n a l seas  specified  directional  5  Lambrakos movements  of  directional series, that  a seas  the  investigated  pipeline were  i n v o l v i n g 51  lying  of  reasonable  the  frequencies  not  the  wave  the  sea  by  a  and  floor.  The  double  Fourier  and  wave  found  response  are  strong  direction,  but  within  s t r o n g l y d e p e n d e n t on t h e d e g r e e o f  spreading. Teigen  (1983)  compared  t e s t s and computer platform  in  the r e s u l t s  simulated  long-  considerable  and  reduction  results  for  short-crested  i n the  total  of p h y s i c a l a  energy  on l o n g - c r e s t e d responses  seas.  However,  (maxima  significantly  or  the  minima)  were  et.al.  (1984)  diectional  spreading  platforms  on a b a r g e and c o n c l u d e d  directionality Tickell equation  compared  in irregular and  Elwany  to analyse  vertical  o f waves on  probabilistic  of  in  models w i t h  experimental  H u n t i n g t o n and G i l b e r t values  (1979)  members  of t h e the  based the  to  be  waves.  effects  transport  that  main  of  found  a  of  of  jackup  the e f f e c t s  o f wave  i s small.  the h o r i z o n t a l  structural  preliminary  waves  the  the  to those  not  studied  leg  found  values  a f f e c t e d by t h e d i r e c t i o n a l i t y  Dallinga  He for  extreme  model  tension  seas.  r e s p o n s e modes i n s h o r t - c r e s t e d s e a s compared  with  loads  a n d 21 d i r e c t i o n s . He  loads  principal  limits  on  represented  wave-induced  functions  wave  (1982)  have e x t e n d e d t h e M o r i s o n wave-induced short-crested numerical  forces seas.  simulations  on They and  data. (1979)  the extremes of i n - l i n e  predicted  the  expected  and t r a n s v e r s e  f o r c e and  6 moment  components  on  a vertical  seas.  They  then  developed  value  of  the  resultant  components be  in-line  and  i n terms  of  very  good  experimental  by  seas  may  force  agreement  or  in  short-crested  which  extremes  of the spectra  and transverse  reported  a method  in short-crested  obtained  cylinder  of  the  expected  force  o r moment  be c a l c u l a t e d .  This can  and c r o s s - s p e c t r a moment  between  of the  components.  theoretical  They  predictions  results.  1.3 S C O P E OF T H E P R E S E N T STUDY The  primary  directional  wave  large  offshore  this  using  and  aim  structures  that  of a structure.  those  based  offshore  methods  These  to predict  t o occur  of  significance  i n design  spectra  the  i s  random  amplitude  because . they  spectra  during  the  a r e then  of  compared t o  the  relate loads  seas. t o any  large  loading in  transfer  are of incident  and  The  operators  loading  operators  loads  operating  the  amplitude The  extend  t h e extreme  t o compute  seas.  study  and t o  a n d c a n be a p p l i e d shape  to  and response of  long-crested  and the response  and the response  to  loading  predictions  general  and short-crested  work  of a r b i t r a r y shape,  of general  functions  functions  the  assumption  i s quite  structure  transfer long-  on t h e  present  on  are likely  life  used  the  effects  statistical  responses  method  of  great wave  responses of  interest. The loading  probabilistic and response  properties  have  been  of t h e components  obtained  using  a  of the method  7  described method of  has  the  C l o u g h and  been d e v e l o p e d  and  can  be  applied  A  t o compute t h e the  all  analytical  e x t r e m e s of maxima  in-line  and  method  transverse  is quite  situations  general  requiring  o r t n o g o n a l components w i t h  Gaussian  distributions  in design  since  set  limits  structures life.  The  Gilbert  and  is significant  that  on  loading  are  likely  r e s u l t s are  response  to occur during  compared t o t h o s e  the  of  of and  designer offshore  their  operating  Huntington  t h e o r e t i c a l development  is described  3.  Results  are" p r e s e n t e d  are  and  4 i s given  part  problem  and  basically  the  Finally,  the  and  the  2  i n Chapter  4.  A brief  i s given  description  i n Appendix  probability  I.  The  used  in  parts.  The  theory  II.  above  wave  responses are  of  of  three  the  wave  uni-directional source  method  statistics  comprises  solution  regular  three-dimensional extends  the  5.  in Chapters  METHOD  involves for  of  used  i n Appendix  DESCRIPTION OF method  in Chapter  function  derivation  The  discussion  presented  Green's  detailed Chapter  resultant  of  The  the  loads  the  (1979).  Conclusions  first  and  of  the  amplitude  can  of  new  the  means. T h i s  1.4  in  properties  zero  of  (1975).  r e s p o n s e components. The  statistical two  Penzien  h o r i z o n t a l r e s u l t a n t s of  loading and  by  of  to the  method.  waves The  developed using  of  using  second  short-crested extremes  diffraction a  part  random waves.  the  horizontal  probability  theory.  8 The  solution  several  authors,  Faltinsen Garrison  and  be  with  which  the  been d e s c r i b e d  and  (1985).  assumes  for nonlinear fluid  The  the  motion  (1972),  Standing  (1974),  method  of  motion  satisfies  the  surface,  free  of  in various  the  linearized surface  by  waves  wave s o u r c e s  over  are the  associated  The  equation,  far  described  by  waves  velocity  the  field.  seabed  The  the  the  unknown  scattered"  distributions  equilbrium  be  velocity  boundary c o n d i t i o n s at  and  immersed  a  forced  body.  p o t e n t i a l components a s s o c i a t e d w i t h  forced  to  equations  components  Laplace  neglects  height  i s described of  by  Chow  wave  terms  i s a combination  b o u n d a r y c o n d i t i o n and  and  has  Garrison  Isaacson  The  mode  potential  velocity  part  i n c i d e n t waves, s c a t t e r e d waves, and  to each  body  first  ( 1 9 7 4 ) , Hogben and  and  small  neglected.  potential  due  Michelsen  effects  sufficiently  the  including  (1978) and  viscous  to  of  of  surface  point  of  the  body. The a  integral  known  are  quadrilateral constant  equations strengths The of  solved  discretizing the  into at  involve  f u n c t i o n . These  integral  the  body  of  centre the  of  into  strengths  transforms  s e t s of a l g e b r a i c e q u a t i o n s  the  Once  i n t h i s manner  source  over each element. T h i s  surface  elements.  by  Green's  elements, with  accuracy the  obtained  three-dimensinal  equations  be  equations  each element  assumed  the  with  and  improves  with  the  velocity  potentials  the  to  integral the  initially  s o l u t i o n d e p e n d s on  plane  source  unknown.  discretization  increasing are  number known,  of the  9 required  hydrodynamic  applying  the l i n e a r i z e d  coefficients Bernoulli  are  determined  equation  p r e s s u r e components and i n t e g r a t i n g  to  these over  obtain  by the  the structure  surface. The  above  short-crested random along  the  waves of u n i t  order  response  the  trains  of  applied  amplitude obtain  to  the  of  a  different  double  to  of  waves  and  components a r e t h e n  loading  transfer  with a s p e c i f i e d  of f r e q u e n c i e s  transfer  f u n c t i o n s and  waves,  frequency. the sea s t a t e i s  of  component  and d i r e c t i o n s . to these  transfer  short-crested  obtained  wave  waves, d i r e c t i o n a l  uni-directional  specfied directional frequency.  of  wave  in  response  direction. and  multiplying  the  In  wave s p e c t r u m spectrum  Thus t h e  loading  amplitude  spectrum.  spreading function  and  the  by  wave  components  f u n c t i o n s and  f u n c t i o n s and r e s p o n s e incident  wave  a range  cross-spectra  response  for  o f component  o p e r a t o r s as f u n c t i o n s of frequency spectra  long-crested  t o be c o n c e n t r a t e d  series  series  i s applied  l o n g - and  o n l y , and t h e t h e o r y  a  frequencies  to o b t a i n the loading  of  i s taken  of s h o r t - c r e s t e d  order  specified  case  loading  for regular  The  the  and s p a n n i n g  theory  amplitude  extended  o p e r a t o r s as f u n c t i o n s of  case  t o comprise  In  wave d i r e c t i o n  is  amplitude  In taken  to  then  wave e n e r g y  the p r i n c i p a l  trains  is  random waves.  waves,  regular  in  approach  the  operators case  i s taken  multiplied  which  of as a  by  a  i s independent  10 Finally, of  the  the p r o b a b i l i s t i c  loading  and  response  resultants are described. response  are  means. T h i s a  Gaussian  linear.  properties  described  their  by G a u s s i a n d i s t r i b u t i o n s w i t h  zero  with  surface  zero  elevation also  o f t h e e x t r e m e s o f maxima o f t h e  and  of  horizontal resultants for a specified  then  obtained.  spectra  This  loading  procedure  and c r o s s - s p e c t r a  response as the primary  has  mean and t h e a n a l y s i s i s  in-line their  horizontal and  The e x p e c t e d v a l u e s transverse  of  The components o f t h e l o a d i n g  i s because the water distribution  and  o f t h e components  and r e s p o n s e components a n d  requires  t h e moments o f t h e  o f components o f t h e  input  parameters.  sea s t a t e a r e  loading  and  2. LINEAR  2.1  DIFFRACTION  THEORY  INTRODUCTION In  the present  described offshore the  to  chapter  study  a general  directional  numerical  wave  example by G a r r i s o n  wave  source  a n d Chow  Garrison  Isaacson  (1985), which p r o v i d e s  long-crested  (1978),  regular  structures  of general  Hogben  wave l o a d s  t o both long-  linear  described  in  diffraction described  Section  2.2 DESCRIPTION  The  long-  wave  One energy  principal  is  the  (1974) a n d in  and s h o r t - c r e s t e d offshore  in  for regular  extension  Section waves i s  of  linear  a n d s h o r t - c r e s t e d waves i s  OF RANDOM WAVES  in  wave  energy  to  describing  terms o f l o n g - c r e s t e d  i s assumed  to  direction  f r e q u e n c y . The o t h e r which  Standing  2.4 a n d 2.5 r e s p e c t i v e l y .  T h e r e a r e two a p p r o a c h e s waves.  extends  and responses  i s given  theory  2.3.  t o both  i n Sections  large  shape.  diffraction  theory  and  i s a p p l i c a b l e t o any l a r g e  The d e s c r i p t i o n o f random waves The  is  described for  F a l t i n s e n & Michelsen  waves,  random waves. The method  on  method  method,  (1972),  (1974),  2.2.  effects  s t r u c t u r e s o f a r b i t r a r y s h a p e . The  three-dimensional  method  be and  concentrated to  i s i n terms of i s assumed  frequency and d i r e c t i o n .  11  seas  be  random  ocean  i n which t h e along  distributed  short-crested  seas  t o be d i s t r i b u t e d o v e r  the over in both  1 2  Long-crested uni-directional variety been  seas  are  generally  described  wave s p e c t r u m , as i n d i c a t e d i n F i g u r e  of u n i - d i r e c t i o n a l  summarized,  for  by  2.1. A  wave s p e c t r a a r e i n use and have  example,  by  Sarpkaya  and  Isaacson  (1981 ) . In  the r e a l  range  of  gives  rise  surface spatial  o c e a n e n v i r o n m e n t , waves p r o p a g a t e  directions, to a  and  has a t w o - d i m e n s i o n a l  directional frequency  and on  sketched  in  spectrum the  Figure  2.2.  multiplied  by a d i r e c t i o n a l  sea s t a t e random  by  assumed  process,  each  in  itself  significant direction. effect  on  considered  to  a  of  spreading  with  correspond considered  The l o n g  are  described  depends wave  both  a  on  the  propagation,  as  random  seas  wave  are  spectrum  function. stationary,  t e r m t h e wave  by a  conditions  to a s e r i e s of such sea s t a t e s , stationary  with  different  peak p e r i o d and t h e p r i n c i p a l  structures  study.  free  by  wave e l e v a t i o n d e s c r i b e d  term v a r i a t i o n  offshore  the  than a one-dimensional  uni-directional  In t h e l o n g  wave h e i g h t ,  in this  which  which  i s g e n e r a l l y assumed t o be a  Gaussian d i s t r i b u t i o n . are  seas  in  Short-crested  described  ergodic  rather  direction  generally  The  waves,  Short-crested  wave  a  t h e s u p e r p o s i t i o n o f s u c h waves  short-crested  variation.  over  of is  sea  states  a distinct  wave  and i t s  problem not  1 3 2.3 LONG-CRESTED The  following  based is  largely based  on t h a t on  wave h e i g h t  to  be  the  i s assumed  be  that  and t h e f l o w i s  condition  a  at  regular  small  The  the  theory  fluid  is  irrotational.  enables the f l u i d  velocity  potential.  enables l i n e a r i z e d  applied  waves i s  The  t o be s m a l l .  by  assumption  regular  by I s a a c s o n ( 1 9 8 5 ) .  assumptions  inviscid  described  for long-crested  given  irrotationality  amplitude to  WAVES  development  incompressible,  The  REGULAR  t h e body  surface,  The s m a l l  boundary free  motion wave  conditions  surface,  and f a r  field. A angular past in  f r e q u e n c y co p r o p a g a t e s i n w a t e r  a  large,  Figure  Let  system,  freely  Oxyz  d e g r e e s of freedom, and  motions  parallel  pitch  and  yaw.  yaw  level.  namely  Surge,  and  are rotational  of  body.  interaction  indicated Cartesian  and  z  measured  sway, heave  heave,  respectively, about  the f l u i d  of a r e g u l a r  with roll,  are traslational and  roll,  t h e same  three  t h e p r o b l e m c a n be t r e a t e d  by e a c h o f t h e s i x component m o t i o n s the  2.4  motions  seven problems:  d  moving a t an a n g l e 6  surge,  sway  H and  The body o s c i l l a t e s  t o t h e x, y , z a x e s  a x e s . Due t o l i n e a r i z a t i o n , superposition  i n Figure  water  body a s  right-handed  w i t h t h e wave t r a i n  from t h e s t i l l  pitch,  a  of height  of constant depth  floating  form  t h e x a x i s as i n d i c a t e d  upward six  arbitrary,  2.3.  coordinate with  a m p l i t u d e wave t r a i n  motion  as  produced  o f t h e body a s w e l l  wave t r a i n  a  as  with the r e s t r a i n e d  1 4 E a c h component m o t i o n equal  to  that  represented  k  =  the  incident  = $  k  k  4,5,6,  $  i  k  frequency  CJ a n d c a n t h u s be  waves  sought,  the  flow  only  a steady  of each state  for  component  solution  is  time dependence o c c u r s as exp(-icot) and t h e boundary v a l u e  problem  explicitly.  p o t e n t i a l i s made up o f components  t h e i n c i d e n t waves  (subscript  (2.1)  complex a m p l i t u d e  s  h e n c e does n o t e n t e r  with  a  f o r k = 1,2,3 and a r o t a t i o n  motion and t i s time. Since  The  with  exp(-iut)  i s a displacement  being  harmonic  as:  a  where  of  is  ( s u b s c r i p t o ) , the  7 ) , each of these b e i n g  wave h e i g h t ,  and f o r c e d  (subscript  1,...,6),  and  scattered  proportional  waves due t o each  each  associated waves  to incident  mode  of  proportional  c o r r e s p o n d i n g motion a m p l i t u d e . Thus t h e v e l o c i t y  motion to the  potential  0 may be w r i t t e n a s :  - i i )H <t> = [ ^(<t>o <t>i) +  z  ^ + £ -iwS^,,] k=1 K K  exp(-icjt)  where 4>^ , k = 0,1,...,7 i s g e n e r a l l y Each equation, surface. surface  of  the  potentials  must  The  linearized  i s g i v e n by:  boundary  complex. satisfy  and t h e boundary c o n d i t i o n s  (2.2)  the  Laplace  a t t h e seabed and f r e e  condition  on  the  body  15  n  where n d e n o t e s - d i s t a n c e normal  to  itself  body  surface. V  i n the d i r e c t i o n  V  = Z k=1  n  in  i n the d i r e c t i o n  equilbrium  f r o m t h e body  (2.3)  D  surface  i s the  n  of n and  of u n i t  vector  and d i r e c t e d  velocity i s given  of  outward  the  surface  by:  - icj$. n, e x p ( - i c j t ) K  (2.4)  K  which n. = n , 1 x' n n  =  2  3  =  n. = y n - z n , 4 z y' n . = z n - x n , 5 x z' = x n - y n . 1  n , n  z'  where  n , n  vector  n i n t h e x, y and z  Substituting (2.3),  we  and n  are the d i r e c t i o n  equations  cosines  of the normal  directions.  (2.2),  (2.4) and  6 = Z -icj$.n. k=1  exp(-icot)  (2.5)  into  equation  obtain:  90 0  UV7 30:  30^  9n  3n  3n  Separating problem  (2.5)  c  y  K  out the terms c o r r e s p o n d i n g  (k=0,7) and e a c h component  (k=1,...,6), e q u a t i o n  (2.6)  K  to  the  diffraction  of the r a d i a t i o n  (2.6) c a n be decomposed  into  problem  the  16  f o l l o w i n g  form:  f o r  k=1  , ..  .,  6  30, (2.7)  3n  30  (  for  k=7  9n  The  0  i n c i d e n t  0  ^ f s i n h k d ^  =  where  p o t e n t i a l  k  i s  the  0^  may  wave  each  o v e r  p o t e n t i a l  0^(x)  e x p r e s s e d  a s :  0 (x>  at  S  =  where  w h i c h  £  the  G r e e n ' s  i s  k  (  s a t i s f i e s  the the  r a d i a t i o n  due  g e n e r a l  £  }  G  (  p o i n t  f o r at  +  i s  g i v e n  a s :  (2.8)  y s i n f l ) ]  the  £.  to  e q u i l b r i u m  i n t e g r a t i o n  s t r e n g t h  the  the  f u n c t i o n  c o n d i t i o n , and  f  b  a n d  t h e o r y ,  as  £ ' i  r e p r e s e n t s  f u n c t i o n ,  i t  the a  S  k  u n i t  p o t e n t i a l  r e p r e s e n t e d  s o u r c e s  known  number.  to  be  i s  O  e x p [ i k ( x c o s 0  wave  A c c o r d i n g  0  The  a  }  d  c o n d i t i o n .  p o t e n t i a l s  d i s t r i b u t i o n  body  s u r f a c e  x  =  of  p o i n t  S^.  Thus  the  ( x , y , z )  may  be  (  s o u r c e on  s t r e n g t h body  s u r f a c e  p e r f o r m e d ,  a n d  G ( x , £ )  p o i n t  x  f u n c t i o n  e q u a t i o n , f r e e  due  s u r f a c e  A l t e r n a t i v e  t h e  i s  2  -  9  )  d i s t r i b u t i o n  t h e  G r e e n ' s  l i n e a r i z e d  unknown  s  g e n e r a l  L a p l a c e  a  p o i n t  (£,TJ,$) i s  t h e  to  a  o v e r i g  the  s o u r c e  of  chosen  so  t h a t  sea  b o u n d a r y  b o u n d a r y  c o n d i t i o n s  e x p r e s s i o n s  f o r  i t  17 are g i v e n , i n Appendix I . I t remains f o r the source functions condition (2.7)  f ^ t o be c h o s e n so t h a t t h e body g i v e n by e q u a t i o n  together  surface  with  integral  (2.7)  equation  equations  is  (2.9)  strength  s u r f a c e boundary  satisfied.  Equation  reduce  a s e t of  to  for f^:  D  for where  x lies  condition  is  integration (2.9),  on t h e body applied,  i s carried  the r i g h t  k = 1 ,..., 7  s u r f a c e a t t h e p o i n t where  boundary  n  and  is  out over  measured  from  x,  £ . From e q u a t i o n s  hand s i d e of e q u a t i o n  the  (2.8) and  (2.10) i s g i v e n a s :  f  2n b  k  f o r k =1,...,6  k  (2.11)  sinh(kd)  [n sinh(k(z d)) + z  +  i ( n cos0+n s i n 0 ) c o s h ( k ( z + d ) ) ] e x p [ i k ( x c o s 0 x y for  The  integral  equations  source  strengths  section  2.3.3.  2.3.1  f  k  + ysinfl)]  k=7  (2.10) a r e s o l v e d n u m e r i c a l l y f o r t h e and  this  solution  is  described in  WAVE FORCES Once  a l l the  hydrodynamic  potentials  p r e s s u r e p may  <p^  are  known,  be o b t a i n e d by t h e  the  18 linearized  Bernoulli  equation:  (2.12)  The  components  of  wave f o r c e  and moment a c t i n g  on t h e  body a r e t h u s g i v e n b y :  F. = -icjpj"- <j> n . dS J • b  f o r j = 1,... 6  (2.13)  f  3  F^,  where F ^ y,  z  directions  components be  denote the f o r c e  potentials the  (0  + o  into  Fg,  Fg  components  associated  potentials  with  decomposed  the  into  acceleration  forced  and  in  phase  (2.13)  may  with  the  with  each such  force.  The  components  in with  may f u r t h e r be  phase the  with  the  v e l o c i t y of the  total  fluid  force  be e x p r e s s e d a s :  (e)  6  = [F\ '+ I (" Mj e  2  k  + iwX . )$ ]exp(-iwt) ;  k  k  for The c o e f f i c i e n t s M ^ J a n d X ^ j are  t h e moment  first  c o r r e s p o n d i n g body m o t i o n . T h u s t h e  Fj  x,  The  potential  components  the  0 . 6  i s termed t h e e x c i t i n g  associated  can  denote  associated  0 7 ) a n d components  forced  component  F^,  in  a b o u t t h e x, y , z a x e s . E q u a t i o n  decomposed  of  and  components  taken  damping  as  are  frequency  (2.14) j = 1 ,...,6 dependent,  r e a l a n d a r e t e r m e d t h e a d d e d masses and  coefficients  respectively.  Substituting  19 equations  (2.2) a n d  collecting and  terms  damping  (2.14)  into  equation  (2.13)  f o r t h e e x c i t i n g f o r c e , added  coefficients,  We  obtain  the  and  masses  following  expressions:  F^  u.  X  in  e )  pw H Sc  =  = -pRe{ I  k  j k  which  jk  =  (2.15)  o  j  k  }  = -pcjlm{ I  j  k  (2.16)  }  (2.17)  Re{ } a n d Im{ } d e n o t e t h e r e a l  p a r t s and I j  J  U +07)n.dS  2  k  and imaginary  i s given a s :  's V j  d  s  { 2  b  -  1 8 )  2.3.2 BODY MOTIONS The in  ^  equation  o f m o t i o n o f t h e body may be w r i t t e n  the f o l l o w i n g form:  [  V  (  m  j k  +  *jk> -  i w X  jk  +  c  jk *k ]  for  where  i s t h e mass m a t r i x  =  j j=  * *  F  and C j  k  ( 2  1 9 )  1 ,..., 6  i s the hydrostatic  20 stiffness  m  m a t r i x . The mass m a t r i x  m  0  0  0  0  m  0  -mz  0  0  m  0  0  -m z  0  h  mz  G  0  in  0  0  0  0  which m i s  coordinate are  G  the  mz G  0  0  0  -I  -I xy  -I  I yz  mass  a n d may  4  = m(r*  +  z*)  1  5  = m(rj  +  zg)  1  6  = m(rj)  r  of  the  of g r a v i t y  the  x, y , z a x e s r e s p e c t i v e l y .  xy yz zx  where  ;  v  , r ^ and r  p  z  of i n e r t i a  b  J  v  p  b  /  v  p  fa  yz 6_J  body.  z  Q  o f t h e body.  is I  4  the , 1^, I  a b o u t t h e x, y , z  z g  axes  (2.21)  which  body's p r o d u c t  (2.20)  zx  be e x p r e s s e d a s :  in  x  -I  5  t h e body's moment o f i n e r t i a  1  0  G  0  X  of c e n t r e  respectively  i s given as:  are the r a d i i  of g y r a t i o n  I „ , I„_ a n d I xy x which a r e given a s :  about  are  the  xydv (2.22)  yzdv zxdv  i s the density  o f t h e body  and v i s t h e volume  21  of  the  body.  The h y d r o s t a t i c s t i f f n e s s  0  0  0  0  0  0  0  0  0  0  0  0  0  3 3  C  3 4  C  3 4  C  4 4  C  3 5  0  0  0  0  5 5  C  ( 2 . 2 3 )  0  0  0  0  0  0  3 5  C  as:  0  0  which  "9 ' s  3 3  =  = pgS  '44  C  2 2  = pgS,!  in z  C  0  0  C  i s given  0  0  in  matrix  which  3 4  ~  =  P  is  the  B  -  z)  + mg(z  n  -  z )  waterplane  t  ] ( 2 . 2 4 )  r  r  of c e n t r e  of  of c e n t r e of g r a v i t y  of  area  and  S, = /xdS,  = Jx dS 2  The above  = -pgs  3 5  S  1 r  S , 2  S , n  S  buoyancy. body. 2 2  S  are the  moments:  S = /dS, n  c  2 f  z„ i s t h e z c o o r d i n a t e  waterplane  S  S  + mg(z  i s the z c o o r d i n a t e  G  9  ,  S  2 2  integrals  =  S  2  = JydS  /y dS 2  are taken  ( 2 . 2 5 )  over  waterplane.  22 2.3.3 NUMERICAL The  INTEGRATION  submerged  surface  is  discretized  finite  number o f f a c e t s N. The i n t e g r a l  is  then  a p p l i e d a t each f a c e t  of  linear  algebraic  L A..f^ j= 1 - 'J  and  (2.10)  c e n t r e and we o b t a i n a s e t  k )  f o r i=1,...,N,  k=1,...,7  (2.26)  1  (k) Here  equation  a  equations:  = b|  k )  1  into  ( k)  f.  and b:  d e n o t e f,,(x_.) and b. (x. ) r e s p e c t i v e l y  X j i s the value  of x at the c e n t r e  of  j - t h facet.  The c o e f f i c i e n t s A ^ j a r e g i v e n a s :  A  i j  in  • "  5  i j  which  the area equation area  +  ^  A..  * i ' i  )  d  S  (  2  -  (2.27)  can  *Si  be  taken  A S j , and t h u s  2  7  )  f u n c t i o n and ASj i s  o f t h e j t h f a c e t . When i * j , t h e i n t e g r a n d t o be c o n s t a n t  in  over the  A ^ j i s approximated a s :  8G ^(x.,1.)  (2.28)  27T  When i = j , 5 ^ facet  (  i s the Kronecker d e l t a  of the facet  A,  By  /AS..H  h  = 1 accounts  f o r the i n f l u e n c e  of i - t h  on i t s own c e n t r e and we g e t :  = -1  the s o l u t i o n  (2.29)  of equation  (2.29),  the source  strength  23 may  be  source  evaluated  strengths  obtained  at  f ^ are  each  known,  = Z B. . f ^ j= 1  k)  J  k  are given  B  i i  Again over  (2.30)  *AS  G  j  (*i>V*  and  thus  B^j  the c o r r e s p o n d i n g /AST  The  coefficients  <' > 2  integrand  i s taken  i s approximated  as  constant  (2.32)  occurs  expression  in equation  for B ^  may  be  (2.31), given  and  A  A = — — / v/AST  A  above  and  as:  (2.33)  47T  excludes 1/R  31  as:  iL  where  The  ^(x^).  AS. A + — - G  = 47T  term  at  S  When i = j , a s i n g u l a r i t y  h  be  (2.9):  AS . = — 1 G(x-,!•)  B  B  may  )  when i * j , the  ASj  potentials  the  by:  4?  =  J  c e n t r e . Once  3  (k) where <j>^ d e n o t e s t h e v a l u e B^j  the  by e v a l u a t i n g e q u a t i o n  H>\  facet  contribution  i s given  A C S  any  i  1  from  the  singular  as:  dS  (2.34)  R  developement  and  expressions  for different  24 facet  shapes  (1974).  Standing facet  and  rectangular  A  are  = —  given,  f o r example,  In p a r t i c u l a r ,  A  =  A = 2/7F  for a  3.525  by Hogben and for a  square  circular  facet.  For a  facet:  {ln[b +  /b  1] +  +  2  b  ln[l  (2.35)  +  •b  b  where b i s t h e a s p e c t Finally,  ratio  of t h e f a c e t .  (2.15)  equations  and ( 2 . 1 8 )  c a n be w r i t t e n  i n d i s c r e t i z e d form:  F^ J  = - IpgHk Z • j 1  e )  N  'jk  2.4  u (  0  )  + * !  readily  )  n..AS. J 1  (2.36)  1  (k) * i  n  j i  A  S  i  (  above a p p r o a c h be  extended  for long-crested  to long-crested  a p p r o a c h o f Bendat a n d P i e r s o l wave  results,  obtained  at  together  with  the  spectra  and  cross-spectra  incident  components. The r e q u i r e d cross-spectra denotes  components of  )  2  '  3  7  )  LONG-CRESTED RANDOM WAVES The  S^j  7  =  regular  wave  uses  s e r i e s o f wave spectrum  of v a r i o u s  loading  derive  the  and response  a r e d e n o t e d by S ^ j ( w ) f o r i , j = 1 , . . . , 6 .  Here  of  response  frequencies,  and  cross-spectrum  and  to  the regular  spectra  the  loading  may  random waves u s i n g t h e  (1966). T h i s a  waves  the  i - t h and j - t h  o f l o a d o r r e s p o n s e a n d S^. d e n o t e s t h e s p e c t r u m  t h e i - t h component  of load or response. These  s p e c t r a and  25  cross-spectra as  are r e l a t e d  to the incident  wave s p e c t r u m  S(w)  follows:  S^fu)  where  T^((o,d)  amplitude ratio  | T \ (CJ,0) | | T J ( C J , 0 ) | S (CJ)  =  for a specified  loading  calculated the  i s a loading transfer  operator  of  or  response  for a sinusoidal  angle  (2.38)  of  incident  function  direction per  unit  wave t r a i n  8  or a  response  and  i s the  wave  amplitude  a t frequency  wave d i r e c t i o n  measured  u.  from  8  is  the x  axis. From are  equation  (2.15)  the transfer  given as:  T.(w,0) = - po) /-  (0 +0 )n.dS  2  o  7  for  and  f u n c t i o n s f o r the loads  from  operators  equations  (2.15) a n d ( 2 . 1 9 ) ,  f o r the motions -pcj / 2  Tj(u,0) =  j = 1,...,6  s  the response  of f l o a t i n g (0  + o  amplitude  body a r e g i v e n a s :  0?)njdS  -g k  (2.39)  (2.40)  Z [i  2 W  (m  j k + /  z  j k  )-i X W  : J k +  for  c  j k  ]  j = 1 ,..., 6  26  2.5 SHORT-CRESTED RANDOM In of  short-crested  WAVES  s e a s , t h e waves p r o p a g a t e  d i r e c t i o n s and may be c o n s i d e r e d a s  long-crested directions. S(co,8)  random  trains  The d i r e c t i o n a l  and  can  wave s p e c t r u m  Sico)  S(CJ,0)  where D(co,6)  wave  wave  a  o v e r a range  superposition  of  propagating in different spectrum  be e x p r e s s e d i n t e r m s  is  of a  denoted  by  uni-directional  as:  = S(OJ) D(CJ,0)  (2.41)  i s the d i r e c t i o n a l  spreading function  such  that  7T  S  Equation  D(CJ,<9) 66  (2.42) e n s u r e s t h a t  directional in  wave  spectrum  the  energy  assumed  in  directional  a  frequencies,  that  wave  so t h a t  i n the  spectrum.  spectrum  is  the d i r e c t i o n a l  the  It i s  o f wave same  energy  at a l l  s p r e a d i n g f u n c t i o n can  of frequency:  <* Did)  A common form of based  wave  the angular d i s t r i b u t i o n  taken as independent  D(u,0)  contained  i s equal t o the energy c o n t a i n e d  the corresponding u n i - d i r e c t i o n a l  often  be  (2.42)  = 1  (2.43)  directional  on a c o s i n e powered  spreading  distribution:  function  Did) i s  27  C(s)cos Die)  2 s  |6>-0 I <  (0-0 ) o  =  (2.44) 0-0 I o  where 0  O  */2  O  i s the  distribution to ensure  principal  i s centred.  that equation  C(s)  direction C(s)  >  w/2  about w h i c h  i s a normalizing  (2.42) i s s a t i s f i e d  the  angular  factor  and  chosen  i s given  =  by:  (2.45a) 2s J cos 0 17  d0  b  ~1t  The  normalizing  range  of  values  alternative be  written  of  i n t e r m s of  loading  F  or  long-crested directions  =  (  C(s) s  expression  C(s)  The  factor  and  f o r the  in  Table  2.1  for  i s sketched  in Figure  normalizing  factor  gamma f u n c t i o n r  the  2.5.  C(s)  +  spectra S^J(CJ)  waves  given  An may  (2.45b)  1  random  a  as:  | r(s+l) s  response  0 are  is listed  due  propagating  to  uncorrelated  in  different  as:  it  Sj.fu)  = /  | T ( c j , 0 ) | |T . ( C J , 0 ) | S(co.e) i  -it  for  where T J ( C J , 0 ) response and  are  the  amplitude  (2.40)  d0  (2.46)  3  loading operators,  respectively.  If  i , j=  1 ,. .. , 6  transfer  functions  defined  in equations  the  directional  or  the  (2.39)  spreading  28 function  D(o>,0) i s t a k e n a s i n d e p e n d e n t  equation  ( 2 . 4 6 ) c a n be w r i t t e n a s :  S -(o})  = { J |T.  i  Since is  | | T \ ( u , 0) | D ( 0 ) }  t h e i n t e g r a t i o n over  convenient  function  to  6  define  i s now  independent  a directionally  directin  0  i t  S(CJ),  averaged  d0 }  2  i;  of  between  the  form:  = |T .(w) |  case 8,  of  transfer  1  /  (2.48)  2  TT  i n t h e common  the  (2.47)  i , j = 1 , . . ., 6  enables the r e l a t i o n s h i p  S.j(u)  In  S(CJ  then  as:  T^^(CJ)  —  written  frequency,  for  T..(w) = { ] | T . ( u , 0 ) | |T.(o),0) |D(0)  This  of  spectra .  to •  S(u)  long-crested  the corresponding  be  (2.49)  waves  propagating  transfer  function  i n the is  given  as:  T  (a;)  = {Ti(ure0)T^(u,eQ)} ]/2  The  rms  i ; j  response an  values  i n long-  integration  (2.50)  of  t h e components  and s h o r t - c r e s t e d over  r e s p e c t i v e l y . The  rms  a)  of  values  s e a s may be o b t a i n e d  equations of  o f the l o a d i n g and  the  (2.38)  and  components  by  (2.47) of  the  29 l o a d i n g and compared seas.  We  response  to may  factor  R^j  values.  Thus oo  the  I  =  corresponding  r a t i o of  these  two  be  long-crested reduction rms  TT  |T. (w,0) | | T . ( w , e ) |  ° ~*  D(0)d0}s(cj)dw  •  (2.51)  °° l  i  | |T.(a>,0 ) | S(w)do) for i , j =  0  o  1  reduction component  factor,  loads  uni-directional  Since  the  assumed t o be in-line  and  then,  and  spreading  long-crested uncorrelated,  transverse  directional  may  be  responses  waves i n o r d e r  when d i r e c t i o n a l  the  in  then  characteristics  the  i\T (u 6  if  values  may  as  o  rms  rms  seas  response  j  This  short-crested  thus d e f i n e a l o a d i n g or a  ;{; R?.  in  a p p l i e d d i r e c t l y to predicted  to obtain  function  1,..., 6  by  assuming  coresponding  i s taken  into  values  account.  random waves f r o m d i r e c t i o n s l o a d i n g and  to the  principal  wave s p e c t r u m  response are  are  induced  wave d i r e c t i o n  i s symmetric.  the  even  3.  3.1  PROBABILISTIC PROPERTIES OF LOADING AND  INTRODUCTION  The  spectra  and  described  i n the  interest  to  cross-spectra previous  the  in  a certain  unlikely  structures. be  limit  from using  of  he  will  on t h e l o a d i n g  the  enable and  statistical  which of the  themselves  spectra  methods w h i c h a r e  chapter  first  describes  the  both  and s h o r t - c r e s t e d  long-  loads  wave  direction.  A method  and  described  turn  probabilistic  and r e s p o n s e f o r In  in-line  short-crested and  The  overall  which v a r i e s  randomly  i s described  properties  components a s w e l l  both  direction.  i s thus a vector  probabilistic  random s e a s .  and r e s p o n s e s o c c u r  the p r i n c i p a l  response  in  life  the  o f t h e components o f t h e l o a d i n g  and  designer  response  and response can of  to occur  the  the operating  properties  direct  needs t o know t h e  properties  to  little  chapter.  The  seas,  and r e s p o n s e  and r e s p o n s e w h i c h a r e l i k e l y  The maximum l o a d i n g  predicted  this  Rather,  t o be e x c e e d e d d u r i n g  cross-spectra in  are  storm c o n d i t i o n . T h i s  s e t the design  are  of the l o a d i n g  chapter  designer.  maxima o f t h e l o a d i n g  to  RESPONSE  here  loading  to  provide  described  the  of the h o r i z o n t a l  r e s u l t a n t of the  a s o f t h e maxima o f t h i s  r e s u l t a n t . These  in a specified  properties  or  i n magnitude  a r e u s e d t o p r e d i c t t h e maximum l o a d i n g  occuring  transverse  storm  duration.  The  o f t h e components o f t h e l o a d i n g in Section  3.2, w h i l e  30  those  of  and r e s p o n s e probabilistic  and r e s p o n s e a r e their  horizontal  31 resultants  and  described  in  related  the  Sections  probabilistic horizontal  of  3.3  properties  resultants  approach  summarized  maxima and  of  are  of  these  3.4  resultants  are  respectively.  The  t h e e x t r e m e s o f maxima o f t h e  described  g i v e n by H u n t i n g t o n  in  Section  and G i l b e r t  3.5.  A  (1979) i s  i n S e c t i o n 3.6.  3.2 PROBABILISTIC PROPERTIES OF COMPONENTS The  component  direction, Gaussian water  denoted  surface  "i s  parameter spectral  density Rayleigh  many  i s because  also  distribution,  x  to  the  possess  a  theory i s  a  narrow-band m  o f x. The  spectral  On  e  spectrum,  the  the  tail  Detailed  of  level  x  so t h a t  as above  entailed  in  the d i s t r i b u t i o n  t h e c a s e s when x p o s s e s s e s a  x  narrow-band  width  which  probability  reduces  o t h e r h a n d , a s e —>  becomes G a u s s i a n  to the high frequency  of  and f o u r t h 0,  o f x. F o r  o f t h e component  maxima below t h e mean  treatments  maxima  depends on t h e z e r o t h , s e c o n d  distribution. density  the  w h i c h d e p e n d s on t h e s p e c t r a l  e of t h e s p e c t r u m  o f t h e maxima x  probability  the  to possess a  w i t h z e r o mean, a n d a l i n e a r  moments o f t h e s p e c t r u m to  in  t h e l o a d s and r e s p o n s e s .  e itself  corresponds  response  assumed  have a d i s t r i b u t i o n  width parameter  the  w i t h a z e r o mean. T h i s  S i n c e x has a Gaussian themselves  or  simply as x i s taken  elevation  distribution  to predict  loading here  distribution  Gaussian used  of the  there  to  a  1, t h e are  as  i t , corresponding a  wide  spectrum.  o f t h e maxima f o r spectrum  and  an  32  arbitrary by  spectrum a r e g i v e n  Cartwright The  which and  and L o n g u e t - H i g g i n s  single  x » m  largest  13  to occur  a  The  specified  x m  o f t h e maxima x m  m  T,  is  independent  is  distribution  i n terms o f t h e d i s t r i b u t i o n expression  and  respectively.  duration  corresponding  resulting  Penzien,  o r extreme v a l u e  3  in  be d e f i n e d The  (1956)  (1952)  i n a number M o f maxima c o n s i d e r e d  occurs  requirement. may  by L o n g u e t - H i g g i n s  primary  function P(x ) m  of  given  a  the  as  maxima  (Clough  and  1975):  P(TJ)  = exp[- M exp(-i? /2)]  (3.1)  2  where rj i s d e f i n e d a s x //m^, a n d m m  0  i s the  variance  of  x  defined as:  m  The  0  = J S o  (GJ) dcj x  probability  differentiating Using expected (Clough  equation  of x  number  m  frequency  (3.1) w i t h  may  p(rj) c a n be o b t a i n e d  be  by  r e s p e c t t o rj.  developed  and  is  f o r the  given  as  1975):  = (21n(M))  of  function  ( 3 . 1 ) , an a p p r o x i m a t e e x p r e s s i o n  and P e n z i e n ,  Efx/v/mu)  The  density  equation  value  (3.2)  x  l / 2  + . 5772 (21n ( M ) ) "  maxima M may  of zero u p c r o s s i n g  be o b t a i n e d f  0  1/  2  (3.3)  by m u l t i p l y i n g t h e  by t h e s p e c i f i e d  duration T,  33 M = f  0  T .  One i m p o r t a n t and  (3.3) no l o n g e r  but  now  itself the is  p o i n t t o be n o t e d  depend  i n v o l v e the s p e c t r a l  on t h e f r e q u e n c y  it  will  be r e a l i z e d  number o f maxima i n t h e c a s e smaller  than  same peak  that  s i n c e the frequency  now be r e l a t i v e l y  3.2.1  defined  x  in  and  moments  of  spectrum.  duration the spectrum  is  shape w i t h t h e  of zero  upcrossing  small.  mean f r e q u e n c y  of u p c r o s s i n g  a t any l e v e l  = / xp(x,x)di o  denotes  (3.4)  a derivative  random  with  k are uncorrelated, their  function  1 m m 0  0  variables,  respect  joint  t o time.  probability  and  a  Since x density  i s given as:  p(x,x) =  where m  x is  a s ( P a p o u l i s , 1965) :  w h i c h x and x a r e G a u s s i a n  dot  which  FREQUENCY OF UPCROSSING OF COMPONENTS The  f  narrow-band spectral  e,  e q u a t i o n (3.3)  f o r a given  f o r an a r b i t r a r y  frequency,  spectral  of narrow-band  that  of a  parameter  upcrossing,  (1982) e m p h a s i z e s ,  e x a c t l y t h e same a s f o r t h e c a s e  However,  width  of zero  d e p e n d s on t h e z e r o t h and s e c o n d  s p e c t r u m o f x. As O c h i  will  i s t h a t e q u a t i o n s (3.1)  exp{-l(  i i + §i)} 0  (3.5)  2  2  i s t h e v a r i a n c e o f x, m  2  i s t h e second  spectral  34  moment o f x which and m  i s a l s o equal  are defined  2  t o v a r i a n c e o f k,  and  m  0  by:  CO  m  = ; w S ( a > ) dco o xx  Substituting obtain  f  As  x  equation  (3.5)  following expression  =  a  2¥  (^)  l / 2  obtained  t. -  value  f , w h i c h was 0  l^) ' 1  PROBABILISTIC  equation  (3.4),  x =  of  we  for f : x  exp(-xV2m )  by s e t t i n g  £  into  (3.7a)  0  specific  upcrossing  3.3  (3.'6)  n  n  f ,  the  frequency  r e q u i r e d i n the  of  zero  foregoing,  is  0.  (3.7b)  2  PROPERTIES OF THE  HORIZONTAL RESULTANT  OF  COMPONENTS Let  x and y be two G a u s s i a n  means, a c t i n g i n o r t h o g o n a l plane.  Their  p(x,y)=  joint  random v a r i a b l e s w i t h  directions  probability  — exp{/27rm m ( 1-X )  distribution  (  1  2  0  0  in  x  2(1-X )  ?  (  X  x  y  /m m 0  +  0  horizontal  i s given as:  m  2  2  the  JLl)}  m  0  zero  35  -°° < x, y < -1  where and  X  m  are  m  0  m  0  and  0  defined  0  variances  i n terms of  the  of  spectra  x and of  y  respectively  x and  y  as:  (3.9a)  = J S o  (3.9b)  (oj)dco y  the  y  correlation  = c //m m  2  0  itself  coefficient  between x and  y and  0  given  (3.9c)  o  of  the  cross-spectrum  of  x  and  y  and  (3.9d)  x  Let  z be  the  + / x + y . The 2  2  resultant variable p(z,0)  z  resultant  of  may  z  the  probability be  and  8  random v a r i a b l e s  density  developed  8 i s introduced of  is  as:  oo = ; S (co) du o ^  0  is  follows:  i s variance  c  the  (3.8)  = / S (w)do> o  g i v e n as  c  are  0  X < +1  xx  is  X  m  <  +00  and is  the  function  as  follows:  joint  obtained  x and p(z) An  i n t e r m s of  p r o b a b i l i t y density  a l o n e may  a  o b t a i n e d by  suitable  z =  of  the  auxilliary  probability  t r a n s f o r m a t i o n p r o c e d u r e . The t h e n be  y:  density  p ( x , y ) by p(z)  integration  of  of  a z  this  36 joint  probability  density  p(z)  may  distribution e x a m p l e , by In  (z,0)  X  as  to  Finally,  integrated  obtain  to  function P(z). This Bury  the  defined  be  density p(z,0).  the the  procedure  probability probability  i s described,  for  (1975).  present  0 = tan  1  case,  the  ( y / x ) so  that  auxilliary the  variable 0 is  transformation  from  (x,y) i s :  = ZCOS0 (3.10)  y  The  =  zsin0  joint  expressed p(x,y),  probability  i n t e r m s of  given  p(z,0) =  where  by  is  transformation.  p(z)  By  equation  probability  (3.8),  as  p(z,0)  may  density  the'  determinant view  (3.10),  |J|  density,p(z)  probability  = J p(z,0)  v i r t u e of  of  function  follows:  of  the  the  Jacobian  transformation  =  z  in  may  be  obtained  density with  the  respect  present  to  by  of given  case.  the by The  integrating this  0:  d0  equations  be  (3.11)  In  probability joint  joint  function  | J | p(x,y)  |J|  equation  density  (3.12)  (3.10) and  (3.11) t h i s may  be  written  37  as:  p ( z ) = / z p ( Z C O S 0 , z s i n f l ) 66  Applying  equation  (3.8) t o e q u a t i o n  out  the i n t e g r a t i o n  2ff,  an e x p r e s s i o n  for  (3.13)  the r e s u l t a n t  with  (3.13) a n d t h e n  r e s p e c t t o 8, o v e r  f o r the p r o b a b i l i t y z may be d e v e l o p e d  r z  v/moirioO-X )  exp{-  2  the range  0 to  density function p(z)  as follows:  2  m +m  0  2m m  0  0  2(1-X ) 2  lot-  carrying  0  A}  }  f o r z>0  2(1-X ) 2  p(z) =  (3.14) f o r z<0  in  which A = /(m - m ) 0  and  I  0  + 4X m m /2m m  The  is carried probability  resultant  (3.15)  2  0  0  i s the modified Bessel's  derivation  density  2  0  0  0  function of order  zero. This  out i n Appendix I I . distribution  function  z may be o b t a i n e d by i n t e g r a t i n g  P(z) of the  the probability  p(z) a s :  P(z)  z = ; p(z)dz o  Substituting  equation  (3.16)  (3.14) i n t o e q u a t i o n  (3.16),  we g e t :  38  P(z)  where K,,  = J  K  2  I  exp(-K z  and K  2  :  )l (K z )dz 3  (3.17)  a r e c o n s t a n t s w h i c h a r e d e f i n e d as;  3  K,  = /  m m (l-X )  K  2  = (m  0  K  3  =  (3.18a)  2  0  0  + m )/4(1-X )m m  (3.18b)  2  0  0  0  A/2(1-X )  (3.18c)  2  Upon c a r r y i n g out t h e i n t e g r a t i o n i n respect obtain  f o r z>0  2  0  2  to  z,  and  using  a  equation  integral  (3.17)  equation  with  f o r I , we 0  the f l l o w i n g e x p r e s s i o n : . 7r e x p { - ( K + K c o s 0 ) z } 2  1  1 -  27TK,  2  J  =  for  30  (K +K cos0) 2  P(z)  3  (3.19) for  3.3.1  z^O  3  A SPECIAL When  the  CASE standard  d i r e c t i o n s are equal uncorrelated,  z<0  then  deviations  and a l s o  equations  the  in two  (3.14) and  two  component  components (3.19) r e d u c e  are  39 to:  z  / exp(-  2m ,  p(z)  % )  for  2  =  (3.20)  1 - exp(P(z)  )  for  z<0  for  z>0  2m?  =•  (3.21) for  where  z>0  2m,  2  m  is  0  expected,  the  these  variance  equations  of  either  z<0  component.  correspond  As  t o the Rayleigh  distribution.  3.4 PROBABILISTIC PROPERTIES OF RESULTANT The  probability  Section which in  3.3,  d e n s i t y o f t h e r e s u l t a n t z was d e v e l o p e d but  it  i s the d i s t r i b u t i o n  i s o f more p r a c t i c a l  turn  to  MAXIMA  interest,  since  develop the d i s t r i b u t i o n  o f t h e maxima z this  the d e r i v a t i o n of the p r o b a b i l i t y  of'  is  It  characteristic denotes  number  m a g n i t u d e . Thus successive  convenient,  frequencies  initially,  to  distribution define  maxima  is  c  the  average  regardless  number o f u p c r o s s i n g s  various  o f t h e random v a r i a b l e s z ( t ) .  o f maxima p e r u n i t t i m e r e g a r d l e s s 1/f  m be u s e d  will  o f e x t r e m e s o f z . We m  now c o n s i d e r z^.  in  per  unit  period  of magnitude, time  with  of t h e i r  between f  f  the  denotes the  respect  to  a  40 certain joint  threshold level probability  v a l u e s of  i s the  0  f  Finally, We  f  now  maximum z z <  0  of  m  the  as  maxima  equation  p(z  m  time  and  Thus, can  which  as  follows.  i s the  number of  i s the v a l u e f  z  with  of  z ( t ) occurs  function p(z ) m  of  by  may  be  at z =  z^.  when z = 0,  information  obtained  f  r e s p e c t to  distribution  the be  are  about  z. A and the  c o n s i d e r i n g the  d e n s i t y f u n c t i o n of z, z a n d  density  z,  and  the  d e f i n e d i n terms  ( L i n , 1976):  1  * ° | i/ -oo  C  joint  extremely  This  of  = " — f  )  S i n c e the is  unit  frequency,  random p r o c e s s  probability  m  z, where a dot  as:  which are encountered  z  simultaneously.  this  p ( z  i n terms of  (3.22)  c o n s i d e r the p r o b a b i l i t y  probability of  r e s p e c t to time,  i s t h e maximum v a l u e of  distribution joint  with  zero upcrossing  zero u p c r o s s i n g s per 0.  expressed  = J z p ( z , z ) dz — oo  z  Specific f  be  d e n s i t y f u n c t i o n of z and  denotes a d e r i v a t i v e  f  z. T h i s may  2  p ( z , z = 0 , z ) dz  probability difficult  to  (3.23)  d e n s i t y f u n c t i o n of obtain,  an  z, z  and  approximation  z to  (3.24) i s a p p l i e d :  1 ) = - _L f c  d  approximation  (  z ?_ dz f  )  i s v e r y good  (3.24)  i f t h e component  s p e c t r a are  41  n a r r o w - b a n d e d . In spectra, p(z ).  this  problem  to  function  of  The  The  that z,  z  use  the  leads  used  variables  probability  density by  to  This  approximate  of  p(z m  r  expression  i s somewhat  order  develop  cross-spectrum  band  overestimate  of  simplifies  S  m  )  on  to  described  6 are  the  is  derived ) of  introduced  suitable  is  taken  in  obtained  joint  joint  terms The  by  of  joint  suitable  i s then  applied  (3.24),  for  probability density  an  assumption  result  as  and  maxima z . m  i n t r a c t a b l e and  a  to  The  from equation  f o r the  used  obtained.  procedure.  the  of  3.3.  obtained  then  basis  in section  is first  then  the  that  probability density  formula,  p(x,y,x,y) to  z as  is  z  lines  p(x,y,x,y)  p(z,z)  u  A complete  wide  probability density  P(  transformation  joint  J  the  p(z,z,0,0)  p r o b a b i l i t y density  r  joint  obtain  of  6 and  a  density  integration.  an  (3.24)  f i n d i n g the  distribution  probability  to  equation  distribution  p(x,y,x,y)  y possessing  only.  auxilliary  the  x and  (3.24) p r o c e e d s a l o n g  joint  t o an  of  of  of  procedure  equation  case  approximation  However, t h e  m  obtain  the  zero.  is  This  made i n  that  i s true  the for  an  generally  a  xy axisymmetric resonable Since  structure  and  in  any  case  is  approximation. x,  zero  means,  given  as:  y,  i , y are  their  p(x,y,x,y)= — ! —  joint  Gaussian  random  probability  exp{-l[X]T  M  1  [X]}  variables  density  with  function  is  (3.25)  42 T where  [X]  random  v a r i a b l e s and i s :  [X] = T  [u]  is  the  transpose of the v e c t o r c o n t a i n i n g the  {x,y,x,y}  (3.26)  i s a correlation  matrix  w i t h components  given as:  Oo  = Re  {/ S, .(CJ) dcj} f o r i=x,y and j=ic,y o -  (3.27)  1  A i s the various  determinant spectra  of  the  correlation  occuring i n equation  matrix  n.  (3.27) a r e r e l a t e d  The as  follows:  S. •  xx  =  CJ2S  ,  S. .  xx'  Thus, t h e c o r r e l a t i o n known  m  =  yy  The c o r r e l a t i o n  x  dw,  matrix  (3.28)  yy  matrix  s p e c t r a l moments  = J u> n S ( C J ) o  CJ2S  M may  be e x p r e s s e d  o f x and y d e f i n e d a s  m'  = / w o  n  S  of  follows:  (w) dco y  i n terms  (3.29)  y  i s eventually given  i n terms of these  43 moments a s :  m  in]  =  0  0  0  0  m  0  0  0  0  m  0  0  0  0  0  0  2  m  (3.30)  2  As a r e s u l t  of the assumption  interaction  coefficients  correlation auxilliary tan  1  (y/x)  Cartesian  matrix  as  in this  reduces  variables before.  6  that  to  =0,  matrix a  v a r i a b l e s x, y, k,  non-diagonal  are a l l zero  diagonal  a n d 6 a r e now The  the  matrix.  introduced  transformation  and t h e The  where  between  0 = the  y a n d t h e p o l a r v a r i a b l e s z, z,  0, 0* i s :  x = ZCOS0 (3.31) y = zsin0  x = z c o s 0 - z0*sin0 (3.32) y = z s i n 0 - z0cos0  The  joint  probability  d e n s i t y p(z,0,z,0*) may be o b t a i n e d i n  t e r m s o f p ( x , y , x , y ) by t h e t r a n s f o r m a t i o n :  p(z,0,z,0) = | J | p(x,y,x,y)  (3.33)  44  where  |J| is  transformation. |J|  = z  determinant  On t h e b a s i s  i n the present  2  The is  the  joint  now  Jacobian  of  (3.31) a n d  the  (3.32)  case.  may  density  the  of e q u a t i o n s  probability density  required  probability  of  function  be o b t a i n e d  of z and z which  by i n t e g r a t i n g  p(z,z,0,0) over  the joint  the complete ranges of 6  and 8:  2TT  p(z,z) = J o  Combining  equations  oo f  = J  (3.34)  (3.22) a n d (3.33) we  2TT  0 0  /  / z z  O  — 00  obtain:  p ( x , y , x , y ) dz dd d$  2  (3.35)  — OD  Substituting density  .  0 0  { J | J | p(x,y,x,y)dd}d0 -°°  equation  (3.25)  and i n t e g r a t i n g  first  for with  the  joint  respect  to 6  probability then  z  we  obtain:  2TT  f z  = J o  kA(0)exp{-(cos 0/2m 2  + sin 0/2m )z } 2  o  2  o  dd  (3.36)  where k  = z/(27r)  Aid)  = /m sin 0 2  2  Substituting obtain function  3 / / 2  /m m 0  + m cos 0  equation  the following o f maxima:  (3.37a)  0  (3.37b)  2  2  (3.36) i n t o e q u a t i o n expression  (3.24) we  finally  f o r the p r o b a b i l i t y  density  45 p(z_)  J . kA(6>) [ z B ( 0 ) - 1 ] e x p { - B ( 0 ) z / 2 } dd  = —  2  r-  III  c  f  (3.38)  2  0  where B(0)  It  = cos 6>/m  + sin 0/m  2  should  equation  that  the assumption  (3.24) i m p l i e s t h a t d f /dz z g r e a t e r than  random p r o c e s s by  density  t o be  expected  z ( t ) . The  equation  probability (3.38),  o  be m e n t i o n e d h e r e  for a l e v e l  given  (3.39)  2  0  t h a t of  latter  (3.14).  evaluated  the expected  may  be  This  function  evaluated only  i s t o be  assumption given  m  f o r the  only  v a l u e of  evaluated  p(z ),  underlying  from  the  p(z),  allows by  the  equation  v a l u e s g r e a t e r than  the  v a l u e s of t h e p r o c e s s z ( t ) .  Finally,  the z  expression  distribution  of  probability  d e n s i t y g i v e n by  m  may  be  for  obtained equation  the by  probability  integrating  (3.24) and  the  i s thereby  evaluated using following r e l a t i o n :  P(z) m  3.5  =  f 1 - j2 t c  (3.40)  PROBABILITY DISTRIBUTION OF The  single occurs  probability  largest in  1  a  = P  distribution  or extreme a m p l i t u d e sample  (Longuet-Higgins,  P,(zJ m  EXTREMES OF  M  U J m  1952,  of  M  function z  m  P , ( z ) of  the  m  o f t h e maxima z  independent  Benjamin and  RESULTANT MAXIMA  m  that  maxima i s g i v e n  Cornell,  as  1970):  (3.41)  46  where P( ) c o r r e s p o n d s by  equation  function  The  ( 3 . 4 0 ) .  p, may  to the d i s t r i b u t i o n  then  corresponding  be o b t a i n e d  o f z^, g i v e n  probability  here  density  from:  dP (z) /- \ m P' m = — m M  ( z  )  d z  = M P  Substituting probability maxima may Now  M  _  1  ( z ) p ( z J m m  (3.42)  r  from  equations  density thus  be  that  function p,(z ) m  the  particular,  resultant  E(z  Since  m  )  probability  /  z  z  *  m m  r  should  dz  m  value E ( z ) ,  is  o f e x t r e m e o f maxima o f  following  relation:  (3.43)  for  the  probability  only  for levels  the  lower  than  of z  limit  the expected  greater  of  value  density than  integration o f t h e random  z (t).  The  calculation  integration  of  E  ^  m  )  requires  i n c o r p o r a t i n g the r e s u l t s  depends  on  is  by s p e c i f i e d  given  m  obtained.  m  obtained  a l s o be g r e a t e r  process  o f z may be m •* •  value  using  function P i ( z  m  p ( z ) was v a l i d  expected  f o r t h e extreme of the  density  values  expected  p,(z)  the expression  function the  the  z may be o b t a i n e d  =  the  ( 3 . 4 0 ) ,  evaluated.  known, v a r i o u s c h a r a c t e r i s t i c In  and  ( 3 . 3 8 )  both  v/m7  obtained  a  numerical  here.  E  and t h e number o f maxima M. The duration  T  multiplied  by  f  ^  m  )  latter m  , the  47 maximum v a l u e o f f  with  r e s p e c t t o z:  M  (3.44)  Hence t h e r e q u i r e d c o m p u t a t i o n s This  is  obtained  w i t h z, a s g i v e n  i n c l u d e t h e d e t e r m i n a t i o n of  n u m e r i c a l l y from  by e q u a t i o n  the v a r i a t i o n  of f  (3.38).  3.6 HUNTINGTON AND GILBERT'S APPROACH Huntington method  and G i l b e r t  by  resultant  which  the  (1979) have p r e s e n t e d extreme  o f two o r t h o g o n a l  values  of  components  an  alternative  maxima  may  be  of  the  estimated.  They s t a t e d t h e p r o b l e m a s f o l l o w s : Given spectra the  a  vector  and  level  cross-spectrum  of  probability  Gaussian  magnitude  p will  S  z  process  (x(t),y(t))  (co) , S (co) , S xx yy xy =  /x +y 2  which  2  n o t be e x c e e d e d d u r i n g  They s o l v e d t h i s p r o b l e m by a s s u m i n g the u p c r o s s i n g s a r e independent occur  with a Poisson d i s t r i b u t i o n  frequency event  events.  of  i n time  upcrossing  frequency  with  high  T ?  that f o r large  z,  Then t h e u p c r o s s i n g s  determined  Hence  (co), what i s  by  the  the probability  mean of no  T i s g i v e n by: p(z)  Thus t h e  f .  time  with  problem of  =  exp(-f T)  (3.45)  z  reduces upcrossing  to f  that  of  They  finding provide  the a  mean  lengthy  48  expression  for  (3.36), which  f ,  analogous  depends o n l y  but  different  on t h e s p e c t r a l  to equation  moments  of x  and  the  one  y. Their  approach  developed of  here.  t h e extreme  obtained  via  z(t).  Thus,  turn  the  significantly  In t h e p r e v i o u s a p p r o a c h ,  the  distribution  the d i s t r i b u t i o n  distribution of the  used here  i s able  of  the  random  maxima  z . m  the  o f t h e maxima z ^ , and f i n a l l y  to  describe  process  distribution  in  z(t), of  the the  the expected v a l u e of  Huntington and G i l b e r t ' s  (1979) a p p r o a c h  the  probability  of the extremes  distribution  been  o f maxima o f random p r o c e s s  z « m  to  o f any number o f i n d e p e n d e n t maxima has  the approach  distribution extremes  differs  involves z .  only  4. RESULTS AND  4.1  LOADING In  AND  RESPONSE  the present  freely  floating  DISCUSSION  SPECTRA  chapter,  results  are  box i n d i c a t e d i n F i g u r e  described  4.1  y,  z a x e s a r e 33.04 m,  (x,y,z)  coordinates  (0,0,10.62m). considered in  present  with  conditions  compute  using  loading  the  same  t h e x,  one  as  that  (1974) and i s c h o s e n  provide  obtained  i s quite  transfer  in  long-  functions  suitable  for  and  general  functions offshore  comparisons  2.48)  and  4.2  and c a n  in  the  4.3  RAO'S  wave  seas.  with  used  to  amplitude  of  general  The  loading  operators  in  those published  by  directionally  averaged  and r e s p o n s e a m p l i t u d e  operators  short-crested  for long-crested and  be  and r e s p o n s e  short-crested  ( 1 9 7 4 ) . The  transfer functions  water  t h e box s u r f a c e . The  and t h e r e s p o n s e a m p l i t u d e  and M i c h e l s e n  t o those  deep  structures  s e a s have been compared  Figures functions  about  o f g r a v i t y o f t h e box a r e  48 f a c e t s t o r e p r e s e n t  equation,  compared  i s the  32.92 m r e s p e c t i v e l y and t h e  is  to  (RAO'S) f o r l a r g e  long-crested Faltinsen  been  method used  both  transfer  (see  have  loading  operators shape  i n order  B  results.  Results  numerical  example  of g y r a t i o n  F a l t i n s e n and M i c h e l s e n  study  their  32.09 m,  of the c e n t r e  This  by  The r a d i i  a  of s i z e L x B x  D = 90 m x 90 m x 40 m where L i s t h e box l e n g t h , beam, and D i s t h e d r a f t .  for  have  been  seas.  present i n long-  49  seas  the  loading  transfer  and s h o r t - c r e s t e d  seas,  50  defined  i n equations  short-crested RAO'S  have  functions  values  for  show t h e  results three  One  for  interesting  roll  and  absent  i n the case whereas  symmetrical  (2.44) w i t h 10  the  that be  x axis  (6  symmetrical  they  are  of  f u n c t i o n s and  the  all  spreading  spreading  the  index  influence  of  a s s e s s e d . However, i n seas  =  0  the  components o f  of  the  are  2,  3)  In a l l c a s e s ,  f e a t u r e of  yaw  case  the  f u n c t i o n s (s=1,  presented in order  the  to  principal  0*).  figures  is  l o a d i n g and loading  in  response  are  long-crested  loading i n s h o r t - c r e s t e d seas. This supports  the  occur  in-line  direction,  and  even  if  t h a t the  transverse the  in  the  of  made i n S e c t i o n 2.5  present  that  case  statement both  so  short-crested  results.  i s along  In  directional  o f waves may  spreading  sway,  using  2,...,  t r e n d of t h e  wave d i r e c t i o n  seas,  1,  (2.39).  loading transfer  equation  spreading  figures  only  the  and  obtained  g i v e n by  directional the  seas, been  s t a k i n g on  (2.38)  the  l o a d i n g and  to  the  directional  response  principal  wave  wave  spectrum  is  symmetric. For  increasing  -concentrated in-line in  directional  of  s, w h i c h c o r r e s p o n d s  wave s p e c t r u m ,  components of l o a d i n g and  corresponds  these  in-line other  components,  to  increasing  components w i t h  hand,  the energy  sway and  roll,  the  response  s h o r t - c r e s t e d seas approach those  This  the  values  t o a more  results  for  ( s u r g e and  for long-crested  energy  the  pitch) seas.  being concentrated  increasing  s, as e x p e c t e d .  a s s o c i a t e d w i t h the  in short-crested  seas  in On  transverse decreases  51 with be  increasing  easily  s, a g a i n  drawn from t h e  Comparisons in long-  T^J(CJ)  out  with  force  and  sway  short-crested are  exciting  the  88%  sway  and  seas w i t h  wave  are  energy  and same  due  the  to  other  wave  water,  seas. These & Thompson  the  case  hand, t h e r e  i s no  pitch  The  wave for  in  roll  in  distribution  in  59%  of  long-crested these  (1976) and a  of  in-line  in-line  r e s u l t s compare  of  in  exciting  corresponding  energy or  as  directionality f o l l o w s : For  the  radiated  coefficients. coefficients  soley  the  exciting  values  moment  angular  of  carried  Dean  well  (1977)  surface-piercing  cylinder.  explained  t o an  for  energy.  respectively)  total  of H u n t i n g t o n  w h i c h were o b t a i n e d  On  pitch  in  corresponding  same  52%  be  d i s t r i b u t i o n of  exciting  the  functions  the  moment  same t o t a l  in short-crested  those  the  the  about  (surge the  of the  short-crested  circular  that  2  in  with  show  seas w i t h a c o s 0 a n g u l a r  force  components  transfer  s e a s , w h i c h may  and  can  4.3.  loading  4.2,  about  with  and  to Figure  seas w i t h  seas  the  4.2  short-crested  long-crested  components  expected. These o b s e r v a t i o n s  Figures  between  reference in  energy  as  By can  the  any  independent  terms of  of  can  Haskind  in turn  heave  be  Figure  structure  waves  i n c i d e n t wave t r a i n . in  (see  e f f e c t on  be  4.2c). This heaving  related  relations,  related to  Hence t h i s  heave  the  i n c i d e n t wave d i r e c t i o n .  to  can  still  damping  heave f o r c e  problem  be  be  damping  these  f o r c e can  radiation  in  forces  due  expressed  and  thus i s  52  The is  yaw moment  about  1.2%  long-crested yaw  (see Figure  of  the  seas with  value  moment r e m a i n s same w i t h  that  given  The  f o r heave  reduction  definitions  of the l o a d i n g  the  transverse  seas,  to the i n - l i n e  with  and  the  respect  out  discussed  short  crested-seas  in  and  52%  and  short-crested  in  However,  i n the case of  4.1  short-crested with  in  respect  long-crested  f a c t o r f o r yaw i s d e f i n e d i n long-crested  seas,  seas. operators  w h i c h may be c a r r i e d  amplitudes  in  87% a n d 85% r e s p e c t i v e l y o f t h e  long-crested  57%  respectively seas.  in  Table  The  4.3, a r e s i m i l a r t o t h o s e f o r  seas  amplitudes  respectively  and p i t c h ) components 67%  4.1.  factors  The s u r g e a n d p i t c h  a r e about  directional  response amplitude  e n e r g y . The sway a n d r o l l  (surge 60%  the  to Figure  already.  values  spreading  Table  factors are defined  reduction  between  reference  seas a r e about  in  a n d p i t c h ) components  loading  loads  total  (2.51).  t o the p i t c h loading  corresponding  due t o o t h e r  reduction  i n l o n g - and s h o r t - c r e s t e d  with  the  by a s i m i l a r argument  summarized  reduction  Comparisons T^J(CJ)  are  equation  (surge  i n the  However,  directional  (sway a n d r o l l ) components  the loading  seas;  different  i n the loadings  to  energy.  seas  forces.  functions  according  t h e p i t c h moment  be e x p l a i n e d  spreading  are  of  t h e same t o t a l  f u n c t i o n s . T h i s may a g a i n as  4.2e) i n s h o r t - c r e s t e d  of  with  in-line  same  in short-crested of  in long-crested the  the  the  seas,  in-line o r about  components  in  53 There amplitude 4.3e)  i s no e f f e c t  (see F i g u r e 4.3c).  is  seas.  o f wave d i r e c t i o n a l i t y  about  index  s.  The  The yaw a m p l i t u d e  2.7% o f t h e p i t c h  I t r e m a i n s same w i t h absence  on t h e heave  amplitude  different  of  (see  Figure  in long-crested  values  of  wave d i r e c t i o n a l  spreading  effects  on t h e  heave and yaw a m p l i t u d e s  i s as expected.  varying  a s t h e a d d e d mass c o e f f i c i e n t s , t h e  parameters  such  damping c o e f f i c i e n t s yaw,  which  incident  influence  wave  The  and t h e e x c i t i n g the  reduction  corresponding  study  factors  and  The  sketched  height  15m  = s  of  spectra  definitions to  due  those  to  of  of the  In  a. s p e c i f i e d the  wave s p e c t r u m  i n F i g u r e 4.4, w i t h  and  spreading  factors.  wave c o n d i t i o n s a r e now p r e s e n t e d .  H  and  independent  due t o o t h e r  similar  t h e example u n i - d i r e c t i o n a l  spectrum  are  4.2.  are  response  ISSC s p e c t r u m  peak p e r i o d T  p  present  used  i s the  significant  wave  = 1 5 s e c . The ISSC  i s given as:  S(CJ)  = a exp(-pya> )/w 9  where a = 488 H|/T , 0 = p  Figures  i n Table  loading reduction  load  incident  responses,  i n the responses  f u n c t i o n s a r e summarized  The  l o a d i n g s i n heave  direction.  reduction  response  T h i s i s because the  4.5  and  4.6  (4.1)  s  1948/T . p  present  s p e c t r a . T h e s e have been o b t a i n e d  the by  loading  and  multiplying  response the  wave  54  spectral  density  functions  and  (equation that  transfer  i n Figures  the loading  and  a specified  transverse  and  amplitude  These  figures  the s p e c t r a  f o r long-  f o r the  loading  amplitude  operators  4.2 and 4.3 and a r e d i s c u s s e d  AND have  already.  RESPONSE been  obtained,  t h e n be u s e d t o p r e d i c t t h e extreme of and r e s p o n s e t h a t  are l i k e l y  s e a s t a t e . In s h o r t - c r e s t e d  occur  transfer  response  response  and r e s p o n s e s p e c t r a  p r o p e r t i e s may  response  the  PROPERTIES OF LOADING  maxima o f t h e l o a d i n g during  and  seas a r e s i m i l a r t o those  4.2 PROBABILISTIC  their  2.39)  loading  2.40) a t a l l f r e q u e n c i e s .  functions  presented  corresponding  t h e r e l a t i o n s h i p between  short-crested  Once  the  (equation  operators indicate  by  both  in-line  (sway and r o l l )  (surge  seas,  and  t o the p r i n c i p a l  to.  occur  loading  pitch)  wave  direction.  Thus, t h e e x t r e m e o f maxima o f t h e r e s u l t a n t o f i n - l i n e transverse  components  random d i r e c t i o n w i t h contains  information  of the l o a d i n g  about  two components  direction  section,  the  results  properties  of the l o a d i n g  the  horizontal  components  and r e s p o n s e o c c u r  random m a g n i t u d e . T h i s  are  given  for  and r e s p o n s e  resultant  of the l o a d i n g  random  of  In the  in a  i n the present  probabilistic  components  in-line  and r e s p o n s e .  the  and  quantity  c h a n c e s o f maxima o c c u r i n g simultaneously.  and  and  and  for  transverse  55 4.2.1  LOADING AND For  the  presents maxima  RESPONSE COMPONENTS  case  of  the expected of  long-crested  figure  4.7  v a l u e E ( x ) / / m ^ o f t h e extreme  of  m  components  of  the  loading  corresponding  normalized  with  respect to  the  deviation  /m^  of  a  x,  as  maxima M. T h i s c o r r e s p o n d s applies  equally  seas,  to  and  response, standard  f u n c t i o n o f t h e number o f  simply  to equation  a l l components  of  (3.3)  and  loading  and  response. Figure  4.8  short-crested of  the  Figures  between  results for  values  E(x )/t/m^ m  and  v a l u e s of t h e s p r e a d i n g  4.8(b)  provide  the  ( s u r g e and p i t c h ) and t r a n s v e r s e  components  respectively.  the r e s u l t s  expected been  deviation in  and shows e x p e c t e d  for different 4.8(a)  in-line  have  seas  the corresponding  extreme o f maxima a s f u n c t i o n s o f t h e number o f  maxima M,  the  presents  for long-  values  0  and  i n both  normalized v/m  To p r o v i d e  with  of the i n - l i n e  index  results  (sway and  suitable  s. for  roll)  comparisons  short-crested  seas,  F i g u r e s 4.8(a) and  4.8(b)  respect  to  the  standard  ( s u r g e and p i t c h )  component  long-crested seas. As  discussed  in  l o a d i n g and r e s p o n s e spreading  in  4.1,  a r e independent  t h e heave and of the  both  statistical  l o a d i n g and r e s p o n s e  spectra  moments o f are  l o n g - and s h o r t - c r e s t e d s e a s . Hence  p r o p e r t i e s a r e u n a f f e c t e d due  yaw  directional  o f waves. Thus t h e v a r i o u s s p e c t r a l  t h e heave and yaw same  Section  to  the their  spreading  56 of waves i n s h o r t - c r e s t e d s e a s . Comparisons short-crested expected and  in  Huntington  amplitude  and  relation  maxima  loading  of  Gilbert  horizontal presents  now  (1979)  f o r the expected  components function  loads  f o r the  s e a s . The  expected  normalized  with  in-line  circular  values  Section  4.1.  extreme  of  cylinder.  AND  RESPONSE  p r o p e r t i e s of  motions,  of i n - l i n e case  the  Figure  4.9  o f t h e extreme o f  m  number  s  =  and t r a n s v e r s e 1,  again  as  maxima M i n s h o r t - c r e s t e d in  Figure  4.9  have  been  respect t o the standard d e v i a t i o n V m ^ i n s h o r t - c r e s t e d seas. Figure  e q u a l l y to the r e s u l t a n t t o surge  as  also predicted a  v a l u e s E(z )//STQ"  component  corresponding  in-line  functions  v a l u e s of the  and  the r e s u l t a n t  the  and  and t r a n s v e r s e components o f  f o r the p a r t i c u l a r of  have  to the p r o b a b i l i s t i c  the expected of  in  RESULTANTS OF LOADING  resultant  maxima  applies  operators  for a surface-piercing  Turning  the  relation  of l o a d i n g t r a n s f e r  the i n - l i n e  4.2.2 HORIZONTAL  the  long-  o f t h e l o a d i n g and r e s p o n s e  the context  response  similar  for  s e a s have shown a s i m i l a r  t r a n s v e r s e components  and  results  v a l u e s o f t h e e x t r e m e o f t h e maxima o f  discussed  of  between  o f l o a d s and  of 4.9  responses  a n d sway a n d t o p i t c h and r o l l i n  turn. The number o f maxima M d i s c u s s e d i n t h e c o n t e x t the  v a r i o u s expected  of  v a l u e s o f e x t r e m e s may be o b t a i n e d  57 by  equation  duration. at  T i s the  of t h e components,  so t h a t  in equation  (3.36).  f  specified  i s a maximum  f = f . On t h e o t h e r hand, i n m of the h o r i z o n t a l resultants, f must be m n u m e r i c a l l y from t h e v a r i a t i o n o f f w i t h z a s z  obtained  Figure 0  m  level  the case  f / f  M = f r , where  In the case  the zero  given  (3.44),  4.10  presents  of components  applicable  to both  respect  which  is different  the frequency  of u p c r o s s i n g  o f t h e l o a d i n g and r e s p o n s e  and  l o n g - and s h o r t - c r e s t e d s e a s .  because the frequency with  0  to  of  the  upcrossing frequency  f  of z e r o  f o r e a c h component  response  in  long-  presents  the  resultant  of the i n - l i n e  is  This i s  normalized  upcrossing  of  upcrossing  f / f  0  0  4.11  of  the  ( s u r g e or p i t c h ) and t r a n s v e r s e  (sway or r o l l )  components  of t h e l o a d i n g a n d r e s p o n s e  short-crested  seas  the p a r t i c u l a r  frequency  f  o f "the l o a d i n g a n d  and s h o r t - c r e s t e d s e a s . F i g u r e  frequency  is  for  of u p c r o s s i n g f  in this  in  case  s = 1. The  is  normalized  case  Z  with the  respect in-line  response equally  t o the frequency  component  in  (surge or p i t c h )  short-crested  to  the  corresponding  of z e r o u p c r o s s i n g f  resultant  to surge  seas. of  of  Figure  loads  loading 4.11  and  and sway and t o p i t c h  0  of and  applies responses  and r o l l  in  turn. C o m p a r i s o n s o f t h e f r e q u e n c i e s of u p c r o s s i n g o f t h e components resultants  (Figure  4.10)  and  i n s h o r t - c r e s t e d seas  of  the  ( F i g u r e 4.11)  horizontal show t h a t  58  f  x  is  a maximum a t t h e z e r o  exponentially zero  at  before  in-line  (1979)  , and d e c a y s  for a  and  by H u n t i n g t o n  seas.  component and  =  4.12(b)  1,  we  present  that of  The  a  Huntington  surface-piercing  has been t a k e n  transverse  fors  a t z/Vnvo" = 0.78.  method w i t h  short-crested  component  in-line  occurs  4.12(a)  of t h e p r e s e n t  in  the  given  level  Figures  Gilbert  cylinder  0  hand, f , i s z a n d i n c r e a s e s t o t h e maximum f m  = 1.58 and t h i s  comparison  of  u  i t d e c a y s e x p o n e n t i a l l y . In f a c t  Finally,  and  f = f m  a s z i n c r e a s e s . On t h e o t h e r  the zero  fm/fo  have  level,  circular  v a r i a n c e of t h e  t o be t h r e e  times  that  a s s t a t e d i n t h e example  Gilbert  (1979).  Further,  the  and t r a n s v e r s e components a r e u n c o r r e l a t e d . The  comparison  i s q u i t e f a v o u r a b l e a n d goes some way t o w a r d s  establishing  the v a l i d i t y  of the present  method.  4.3 A DESIGN PROCEDURE Finally,  i n order  of  the present  to  calculate  short-crested procedure 1.  method, a d e s i g n design seas  takes  Linear  to illustrate  operators  in a  specified  sea  applicability  i s presented both  here  i n l o n g - and  state.  The  design  the f o l l o w i n g s t e p s :  transfer in  corresponding function.  procedure  l o a d s and responses  diffraction  loading  the p r a c t i c a l  theory functions  both to  is  used  and  to  c a l c u l a t e the  response  amplitude  l o n g - as w e l l as s h o r t - c r e s t e d seas a  specified  directional  spreading  59 amplitude and  operators  response  Suitable spectra  t o o b t a i n the c o r r e s p o n d i n g  loading  s p e c t r a and c r o s s - s p e c t r a .  integration  o f t h e v a r i o u s l o a d i n g and  and c r o s s - s p e c t r a a r e c a r r i e d  r e q u i r e d moments,  including  response  out t o o b t a i n  the z e r o t h , f i r s t  and  the  second  moments. These  spectral  moments a r e t h e n  expected  values  various  loading  horizotal chapter The  and t h e f r e q u e n c y and  response  resultants  estimate  the  of u p c r o s s i n g of the  components  using the theory  number o f maxima M i s o b t a i n e d  storm  to  and  their  presented  i n the  3.  maximum  The  used  frequency  of  by  upcrossing  f  multiplying with  the  the s p e c i f i e d  d u r a t i o n T.  various expected  components resultant  v a l u e s o f e x t r e m e s o f maxima o f t h e  o f l o a d i n g and r e s p o n s e in  a  corresponding  specified-  to  5), are obtained  and t h e i r  storm  horizontal  duration  (i.e.  t h e number o f maxima o b t a i n e d  in step  using equations  (3.3) and  (3.36).  4.3.1 A WORKED EXAMPLE Results are presented box  described  short-crested are  given  directional storm  in  below  Section  4.1  s e a s . The r e s u l t s  for  the p a r t i c u l a r  spreading  duration  is  12  function hours  f o r the f r e e l y for  both  floating  long-  for  short-crested  case  of a cosine  (s=1). and  the  and seas  squared  The  specified  sea  state  is  60  directional storm  duration  described 15  spreading  by  is  12  hours  and  the  t h e ISSC s p e c t r u m w i t h  sec, sketched The  (S=1).  function  in Figure  specified  wave c o n d i t i o n  Spreading  Step  1  Figures  s  carried  out  to give  wave s p e c t r u m ,  to  condition  loading the  then  S t e p 6 was c a r r i e d  obtain  under results  out  to give  the  12 h o u r s  extreme storm  are given  and response  expected values  components a n d t h e i r  shown i n  using  the  the r e s u l t s out  to  s p e c t r a l moments o f t h e l o a d i n g a n d  4.10 a n d 4.11, t o o b t a i n  4.10  The  the r e s u l t s  4.5 a n d 4.6. S t e p 3 was c a r r i e d  various  hours d u r a t i o n .  hours  =1  r e s p o n s e s p e c t r a . S t e p 5 was Figures  =  = 1 5 sec  directional  the  p  = 15 m  g  p  shown i n F i g u r e s  is  = ISSC  4.2 a n d 4.3. S t e p 2 was c a r r i e d  specified  obtain  H  state  i s thus as f o l l o w s :  spectrum  wave h e i g h t ,  index,  was  wave  sea  = 15 m and T  =12  Uni-directional  Peak p e r i o d , T  g  specified  4.4.  Storm d u r a t i o n  Significant  H  The  carried  out,  using  number o f maxima i n 12 out u s i n g loading  Figures  4.7  and response  duration.  i n Tables  4.3 a n d 4.4  r e s p e c t i v e l y . These t a b l e s  f o r the provide  o f t h e e x t r e m e o f t h e maxima o f t h e horizontal resultants.  5. 1.  Linear  diffraction  estimate  of  short-crested are  components  of l o a d i n g  in  values  long-crested  sea.  degree  to a  i n these  quite  for  5.  cross-spectra  a greater  (surge  and p i t c h )  based The  on  o f waves,  greater  the  with  degree  these  in  these  depends  a greater  of  a  in-line on  the  reduction  spreading.  The  o f t h e l o a d i n g and  significant  t o be z e r o in  i n design.  seas approach  i n long-crested  short-crested  transverse  short-crested  of  components  The  are greater  o f waves.  waves,  those  seas.  seas,  the  results  for long-crested  o f wave d i r e c t i o n a l i t y  for seas.  on t h e heave  and r e s p o n s e .  The e x p e c t e d v a l u e s components  than the  (sway a n d r o l l ) o f t h e l o a d i n g  degree o f spreading  no e f f e c t  loading  assumption  of  i n - l i n e components  spreading  is  this of the  of the  seas a r e l e s s  reduction  For decreasing  There  from  and  and r e s p o n s e .  components  of  or yaw l o a d i n g 6.  and  significant  magnitidues  long-  spectra  response, p r e d i c t e d  are  both  results  i s e x p e c t e d t o be q u i t e  The t r a n s v e r s e and  in  offshore  The  spreading  corresponding  response  large  of the l o a d i n g and response  of  reduction  of  shape  short-crested  corresponding  been e x t e n d e d t o  seas.  The i n - l i n e components  components  4.  the  has  response  arbitrary random  response  theory  and  procedure  and  3.  wave  the loading  structures  2.  CONCLUSIONS  (surge  o f t h e e x t r e m e o f maxima o f and p i t c h )  61  of loading  in-line  and response i n  62 a  specified  values 7.  The  b a s e d on  response  values  components in  long-crested  9.  statistical  and  response  An  response method two  as 10.  of  as a t o o l analytical  of  seas. maxima  of  loading  and  t o be  zero  significant  in  the  yaw  unaffected  been d e v e l o p e d  of  the  the  roll)  and  heave and  are  has  of  the  in-line  (surge  components of  can  seas  be  used  can  first  and  seas. the  loading  or  or p i t c h )  and  loading  and  described.  in a l l situations Gaussian with  standard  the  which  the be  loading by  by  horizontal  deviations.  s e c o n d moments of  The with zero The  spectra  parameters. required  large  or  to  measure  slender  structures in directional  underlying  of  predicted  components w h i c h a r e  input  Experiments are  offshore  of  short-crested  requires  response  roll)  quite  w h i c h have d i f f e r e n t  primary  corresponding  waves i n s h o r t - c r e s t e d  properties  orthogonal  method  of  method  i s general  means and  extreme  are  properties  (sway or in  the  long-crested  duration,  seas,  resultant  transverse  the  components  analytical  response  of  of  (sway and  spreading  probabilistic  than  seas.  The  directional  less  a specified  short-crested 8.  are  the a s s u m p t i o n  expected  transverse  in  duration  c u r r e n t s . On for verifying or n u m e r i c a l  the the  other  the  (fixed  loading or  seas w i t h  floating) or  without  hand, t h i s w i l l  p r e d i c t i o n s b a s e d on  procedures.  and  serve  various  REFERENCES 1.  Abramowitz, M. and Stegun, I.A., 1964, Handbook of mathematical functions, National Bureau of Standard M a t h e m a t i c s S e r i e s , Dover p u b l i c a t i o n s I n c . , New York.  2.  B a t t j e s , A. J . , 1982, Effects of shortcrestedness on wave loads on l o n g s t r u c t u r e s , Applied Ocean Research, Vol. 4, No. 3, pp. 165-172.  3.  Bendat, analysis  4.  Benjamin, J.R. and Cornell, s t a t i s t i c s and decision for H i l l , New Y o r k .  5.  Borgman, L.E., 1969, Directional spectra models f o r design use, Proc. Offshore Technology Conference, H o u s t o n , Paper No. OTC 1069, V o l . 1, pp. 721-746.  6.  Borgman, L . E . , 1972, S t a t i s t i c a l models and wave f o r c e s , Advances in Hydroscience, 139-181 .  7.  Borgman, L.E. and Yfantis, V., 1981, spectral density of forces on fixed Confer ence on Directional Wave Spectra B e r k l e y , pp. 315-332.  8.  B u r y , K.V., 1975, S t a t i s t i c a l J . W i l e y & Sons, New Y o r k .  applied  science,  9.  Cartwright, D.E. and L o n g u e t - H i g g i n s , M.S., statistical distribution of the maxima of function, Proc. Roy. Soc. , S e r . A, Vol. 212-232.  1956, The a random 237, pp.  J . S . and P i e r s o l , A.G., 1966, Measurement of random data, J . W i l e y & S o n s , New Y o r k . A.C., civil  models  1970, engineers,  in  10. C l o u g h , W.R. and Penzien, J . , 1975, structures, McGraw H i l l , New Y o r k .  and  Probability, McGraw  f o r o c e a n waves V o l . 8, pp.  Directional platforms, Applications,  Dynamics  of  11. D a l l i n g a , R.P., Aalbers, A.B., V e g t v e r d e n , J.W.W., 1984, D e s i g n a s p e c t s f o r t r a n s p o r t o f Jackup platforms 63  64 on b a r g e , Proc. Offshore Technology Paper No. OTC 4733, p p . 195-202.  Conference,  Houston,  12. Dean, R.G., 1977, H y b r i d method of computing wave loading, Proc. Offshore Technol ogy Confer e nee, H o u s t o n , Paper No. OTC 3029, p p . 483-492. 13. F a l t i n s e n , O.M. and M i c h e l s e n , F.C., 1974, M o t i o n s o f large structures i n waves a t z e r o F r o u d e number, Proc. Int. Symp. on the Dynami cs of Marine Vehicles and Structures in Waves, University C o l l e g e , London, p p . 97-140. 14. F o r i s t a l l , G.Z., Ward, E.G., C a r d o n e , V . J . and Borgman, L.E. , 1978, D i r e c t i o n a l spectra and k i n e m a t i c s o f s u r f a c e g r a v i t y waves i n T r o p i c a l s t o r m D e l i a , /. Phys. Oceanography, V o l . 8, p p . 808-909. 15. G a r r i s o n , C . J . and Chow, P.Y., 1972, Wave f o r c e s on submerged b o d i e s , /. Waterways, Harbours and Coastal Eng. Div., ASCE, V o l . 98, No. WW3, pp. 375-392. 16. G a r r i s o n , C . J . , 1978, H y d r o d y n a m i c loading on l a r g e offshore structures: Three-dimensional source distribution methods, I n Numerical Methods in Offshore Engineering, e d s . , O.C. Zienkiewicz, R.W. Lewis, and K.G. S t a g g , J . W i l e y & Sons, E n g l a n d , pp. 97-140. 17. H a c k l e y , M.B., 1979, Wave f o r c e s i m u l a t i o n s t u d i e s i n d i r e c t i o n a l s e a s , Proc. Conf. Behaviour of Offshore St ruct ures, BOSS'79, London, V o l . I , p p . 187-219. 18. Hogben, N. a n d S t a n d i n g , R.G., 1974, Wave l o a d s on l a r g e b o d i e s , Proc. Int. Symp. on Dynami cs of Marine Ve hi cles and St ruct ur es in Waves, U n i v e r s i t y C o l l e g e , London, p p . 258-277. 19. H u n t i n g t o n , S.W. and Thompson, D.M., 1976, F o r c e s on large cylinder i n m u l t i d i r e c t i o n a l random waves, Proc. Offshore Technology Conf er ence, H o u s t o n , Paper No. OTC 2539, V o l . I I , pp. 169-183. 20. H u n t i n g t o n , S.W., 1979, Wave l o a d i n g on l a r g e c y l i n d e r s in short c r e s t e d seas. I n Mechanics of Wave Induced Forces on Cyli nders, e d . T. L . Shaw, P i t m a n , London, PP.  65  636-649. 21. H u n t i n g t o n , S.W. and G i l b e r t , G., 1979, E x t r e m e f o r c e s in short crested seas, Proc. Offshore Technology Conference, Houston, Paper No. OTC 3595, V o l . I l l , pp. 2075-2084. 22.  I s a a c s o n , M. de S t . Q., 1979, Wave i n d u c e d f o r c e s i n t h e diffraction r e g i m e , In Mechanics of Wave Induced Forces on Circular Cylinders, ed. T. L. Shaw, Pitman, London, pp. 68-89.  23.  Isaacson, M. de St. Q., 1985, Wave e f f e c t s on l a r g e o f f s h o r e s t r u c t u r e s of arbitrary shape, Coastal/Ocean Engineering Report, Department of C i v i l Engineering, U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , C a n a d a .  24.  Lambrakos, K.F., 1982, M a r i n e p i p e l i n e d y n a m i c response to waves from directional wave spectra, Ocean Engineering, V o l . 9, No. 4, pp. 385-405.  25.  L i n , Y.K., dynamics, Florida.  26.  Longuet-Higgins, M.S., 1952, On the statistical distribution of the h e i g h t s of s e a waves, J. Marine Research, V o l . 11, pp. 245-266.  '1976, Probabilistic Robert E. K r i e g e r  theory of structural P u b l i s h i n g Company, M a l b a r ,  27. L o v a a s , J.H., 1984, Hydrodynamic l o a d s and response of marine structures, Proc. Symp. on Description and Modelling of Directional Seas, Paper No. D-1, Technical U n i v e r s i t y , Copenhagen, Denmark. 28. O c h i , M.K., 1982, S t o c h a s t i c a n a l y s i s p r e d i c t i o n o f random seas, Advances V o l . 13, pp. 217-375. 29.  30.  Papoulis, stochastic  A., 1965, Probability, processes, McGraw H i l l ,  New  and in  probabilistic Hydroscienee,  random York.  variables,  P i n k s t e r , J.A., 1984, N u m e r i c a l m o d e l l i n g o f d i r e c t i o n a l seas, Proc. Symp. on Description and Modelling of Directional Seas, Paper No. C-1, Technical University,  66 Copenhagen, Denmark.  31.  S a r p k a y a , T. and I s a a c s o n , M., forces on offshore structures, New Y o r k .  1981, Mechanics of wave Van N o s t r a n d R e i n h o l d ,  32.  S h i n o z u k a , M., F a n g , S.L.S. and Nishitani, A., 1979, Time-domain s t r u c t u r a l response i n a short c r e s t e d sea. Journal of Energy Resources Technology, T r a n s a c t i o n s of ASME, V o l . 101, pp. 270-275.  33. T e i g e n , P:S., 1983, The r e s p o n s e o f TLP i n s h o r t c r e s t e d waves, Proc. Offshore Technology Conference, Houston, P a p e r No. OTC 4642, V o l . I I I . 34. T i c k e l l , R.G., and E l w a n y , M.H.S., 1979, A p r o b a b i l i s t i c d e s c r i p t i o n of a member in short crested seas, In Mechanics of Wave Induced Forces on Cylinders, e d . T.L. Shaw, P i t m a n , L o n d o n , pp. 561-576.  APPENDIX I . GREEN'S FUNCTION  The  Green's  function  either  be  expressed  series.  The i n t e g r a l  G(x,£) =  where PV d e n o t e s infinite  i n t e r m s o f an i n t e g r a l  0  i C cosh[k($+d)]cosh[k(z+d)]J (kr) 0  0  0  value of i n t e g r a l .  ]cosh[k(z+d)]H  ( 1 ) 0  (kr)  + 4 Z C cos[M_(S+d)]cos[u (z+d)]K (u r) . 4 1 11 ill 111 ill m= l 0  where  = [ (x-£) + 2  R =  =  P t  ( y - T ? )  [r +(z-$) ] 2  2  2  ]  l  /  2  l / 2  [r +(z+2d+$) ] 2  (AI.1)  expression i s  G(x,£) = -iC cosh[k(S+d)  r  infinite  J (nr)6n  2  the p r i n c i p a l  series  o r an  expreesion i s  1. 1 g,+ 2 PV / F , F  -  The  f o r the three-dimensional case can  2  l / 2  ^  MtanhMd-v  67  (AI.2)  68  F  {-Md)cosh^( S+d) ) c o s h ( A i ( z + d )  = C3cp  2  )  cosh(^d)  _ -  r  *~o  2TTU2-K2)  (k -v )d+i> 2  2  u +v m 2  _ *~m " r  2  (u^+v )a+v 2  v °= k t a n h ( k d )  and  M  m  =  ^4  a r e t h e r e a l  p o s i t i v e  r o o t s  o f  t h e  e q u a t i o n  ju tan(n d) +^ = 0 m  taken  i nascending J  the  (AI.3)  m  order.  i s the Bessel's  0  Hankel  Bessel's  function function  functions  of thef i r s t of  first  of f i r s t  kind,  kind,  K  0  i s  a l lthese  kind, the being  H ^ ^ 1  0  i s  modified of order  zero.  The  gradient  3G  a t x i s also  I< '> • H v If » 2r  t  required  a n d i sg i v e n a s :  <*•'«>  where  n (x-$) n (y-r?) +  x  n  r  =  y  (AI.5)  69  T h u s t h e two e x p r e s s i o n s ( A I . 1 ) and  (AI.2) a r e g i v e n [n  9G 9  7\C  for  corresponding  +  equations  as:  (y-Tj)+n  (x-$) n  to  (z-$)]  f_  x  R3  n  [n (x-U+n x  y  (y-T/)+n (z+2d+t) ] 2  R  + 2/MF, [ F J o 3  (*ir)n  0  I  3  -F J,  (nr)n_]dn  2  r  z  ikC cosh[k(t+d)][sinh(k(z+d))J (kr)n 0  0  +  cosh(k(z+d))J,(kr)n ]  (AI.6)  r  and  !§•  -  cosh(k(z+d))]H(  ikcosh(k(t+d))[  -  sinh(k(z+d))H  ( 1 J 0  1 }  (kr)n  n  r  ] z  4 Z M C cos(/i ($+d)) [cos(ju (z+d)) m= 1 m  m  m  m  + sin(M (z d))K (M r)n ] +  m  0  m  z  K,(  M m  r)n  r  respectively  (AI.7)  where  sinh(/i(z+d)) F  3  = exp(-Md)  (AI.8)  cosh(Md)  Where k r i s l a r g e , t h e s e r i e s f o r m c o n v e r g e s r a p i d l y b e c a u s e  70  of  the b e h a v i o u r of K ( j x r ) .  Bessel's form c a n  function not  convenient. function  When  0  be  K0(MC)  used.  treatment  Isaacson  tends  to i n f i n i t y  In t h i s c a s e ,  A detailed  i s g i v e n by  tends  kr  integral  to  zero,  and  the  form  o f above m e n t i o n e d  (1985).  the  series  is  more  Green's  APPENDIX Equation  p(z)  I I . PROBABILITY DENSITY FUNCTION OF RESULTANT  (3.13) i n c h a p t e r  = j z p(zcos0,zsin0)  Substituting integrating  p(z)  with  z  =  equation  2  27TK,  3 i s given as:  d0  (AII.1)  (3.8) i n t o  r e s p e c t t o 8 from  * fexp[  -z (m cos 0 2  equation  and  0 t o 2TT, we g e t :  + m sin 0  2  2  o  (AII.1)  o  - 2X y / m m s i n 0 c o s 0 ) o  o  O  /2(1-X )m m ]d0 2  o  o  (All.2)  where K, = / m m ( 1 - X ) 2  0  Applying  0  trigonometrical  identities,  (All.2)  can  be  expressed as:  z = j ^ -  p(z)  2TT  exp(-K z )  /exp[-z (K cos20  2  2  2  c  where  K  = (m +m )/4(1-X )m m  0  =  (m -m )/4(l-X )m m  0  K_ = 2 X / m m / 4 ( 1 - X ) m m  0  K  2  2  c  0  0  0  2  0  0  0  2  0  0  0  71  -  K sin20)]d0(AII.3) g  72 Equation  p(z)  (All.3)  z = — — 27TK,  c a n be w r i t t e n  as:  2 7 r  exp(-K z )  /exp{-z A  2  c o s ( 2 0 + 0 ) }d0  2  2  (All.4)  o  where  A = /K  2  + K  '  2  (All.5)  and  0 = t a n ( K /K 1  Substituting  )  (All.6)  X = 0+0  into  equation  (All.4),  we  get:  0+47T  p(z)  = j^-  exp(-K z )  p(z)  0  (All.7)  c a n be w r i t t e n  z ' = rr- exp(-K z )  -jr-  2  2  1  Using  p(z)  where I  =  0  (All.7)  ^  (9.6.16) can  exp(-K z ) 2  i s the m o d i f i e d  4TT fexp{-z h  c o s X ) dX  from Abramowitz  finally  2  1  as:  2  o  4 7 r  identity  equation  (All.7)  2  2  1  Equation  Jfexp(-z A cosX)dX  2  be w r i t t e n  and S t e g u n ( 1 9 6 4 ) ,  as:  I (z A)  (All.9)  2  0  Bessel's  (All.8)  function  of o r d e r  zero.  73  s  C(s)  1  2/rr  2  8/37T  3  16/57T  4  1 28/35*  5  256/63*  6  1 024/231 TT  7  2048/4297T  8  32768/6435*  9  65536/12155TT 2621 44/461 89ir  10  Table  2.1  Normalizing  factor  of  functions  directional  spreading  74  s  SURGE  SWAY  1  0.880  0.516  2  0.924  3  HEAVE  ROLL  PITCH  YAW  1 .0  0.515  0.888  0.016  0.426  1.0  0.427  0.930  0.014  0.945  0.372  1.0  • 0.374  0.949  0.013  4  0.956  0.333  1.0  0.337  0.961  0.012  5  0.964  0.307  1.0  0.310  0.967  0.012  6  0.969  0.288  1.0  0.291  0.972  0.012  7  0.971  0.272  1.0  0.276  0.974  0.012  8  0.972  0.260  1.0  0.265  0.975  0.012  9  0.972  0.251  1.0  0.255  0.975  0.012  10  0.972  0.242  1.0  0.246  0.974  0.012  TABLE 4.1  L o a d i n g  r e d u c t i o n f o r  f a c t o r  d i f f e r e n t  s  R^..  i n  v a l u e s .  s h o r t - c r e s t e d  s e a s  75  s  SURGE  SWAY  1  0.874  0.520  2  0.919  3  ROLL  PITCH  YAW  1 .0  0.567  0.847  0.040  0.427  1 .0  0.456  0.898  0.030  0.941  0.371  1.0  0.391  0.923  0.030  4  0.957  0.333  1.0  0.348  0.938  0.029  5  0.961  0.305  1.0  0.317  0.947  0.028  6  0.966  0.285  1 .0  0.295  0.954  0.028  7  0.969  0.271  1 .0  0.279  0.957  0.028  8  0.970  0.258  1 .0  0.265  0.959  0.027  9  0.970  0.249  1 .0  0.255  0.960  0.027  10  0.970  0.241  1 .0  0.246  0.960  0.026  Table  4.2 M o t i o n  reduction for  HEAVE  f a c t o r R. ^ i n s h o r t - c r e s t e d  different  s  values.  seas  EXTREME LOADING  LONG-CRESTED  SHORT-CRESTED  Surge  630143 kN  554522 kN  Sway  327674 kN  Resultant  570506 kN  (surge  and sway)  Heave  282966 kN  Roll 7411021  Pitch  kN-m  Resultant (roll  3807432  kN-m  6549120  kN-m  6772205  kN-m  and p i t c h ) 615837  Yaw  Table  282966 kN  4.3  Expected values  of the extremes  loading.  kN-m  o f t h e maxima  EXTREME RESPONSE  LONG-CRESTED  SHORT-CRESTED  Surge  7.13m  6.26 m  Sway  3.70 m  Resultant  6.44 m  (surge  and sway)  Heave  12.90  m  12.90  Roll  0.433 r a d  Pitch  0.766 r a d  Resultant (roll  0.646 r a d 0.670 r a d  and p i t c h )  yaw  Table  m  0.12 r a d  4.4  Expected  values  of the extremes  response.  o f t h e maxima  f/f  Figure  0  2.1 S k e t c h o f a u n i - d i r e c t i o n a l  Figure  2.2 S k e t c h o f a d i r e c t i o n a l  wave  wave  spectrum.  spectrum.  79  i  |t,(heove)  V /  1  v*  Figure  2.3 D e f i n i t i o n  Figure  2.4  Definition  y  £ ( pitch) 5  _' ^  ^—X  /  W//////P/////A  C,Csur .) Q  ^  sketch of a floating component m o t i o n s .  sketch  of the i n c i d e n t  «  —  body  wave  showing  direction.  e - 0 Figure  2.5  Directional  O  (deg.)  spreading function values.  for  different  .L.  n  Figure  i  i n  u n  n  n  4.1 Definition  a  t m  sketch  n  /n  of a  n  n  floating  box  s  81  0.2  0.4  —i  0.6  0.8  u> r a d / s e c  o  (b) EXCITING FORCE IN SWAY 3.5  — —  S = 1 s = 2 s = 3  3H  2.5  H  rsi EH  2  1.5  —i  0.4  0.2  Figure  4.2  Loading  iii  0.8  0.6  rad/sec  transfer functions s h o r t - c r e s t e d seas.  in  long-  and  82 5  o  (c)  EXCITING  FORCE I N HEAVE  4.5  (all  E  3.5 H  2 •*  3  s)  "3 "ro ro  2.5  EH  2H 1.5  1 0.5 0.2  i 0.4  — i 0.8  r— 0.6  1  oi r a d / s e c «>  o  0.40  (d)  E X C I T I N G MOMENT I N R O L L  0.35  »  e i  s  S = 1 S = 2 — 3  0.30 H  0.25  E-  0.20 H  0.15 ^  0.10  0.2  0.4  0.6  u> r a d / s e c Figure  4.2  0.8  •  (cont.) Loading transfer functions short-crested seas.  i n long- and  83 0.8  o  (e) EXCITING MOMENT IN PITCH  x  long-crested s = 1  0.7  0.6  E I  2 ^  0.5-  m  in E->  0.4  0.3 0.2  9  o — X  —i— 0.4  0.6  w  0.8  rad/sec  15-| -  ( f ) EXCITING MOMENT IN YAW  14-  -  13-  ( a l l s)  12116  10-  ai  9-  2  87-  3  6vo  5432100.2  Figure  0.4  u> r a d / s e c  0.6  4.2 ( c o n t . ) L o a d i n g t r a n s f e r f u n c t i o n s i n short-crested seas.  —i 0.8  long-  and  84  O.H  •  0.2  ,  .  0.4  u) Figure  4.3  Response  ,  ,  ,  0.6  0.8  rad/sec  amplitude operators s h o r t - c r e s t e d seas.  in  long-  and  85  ( c ) HEAVE  . (all  u  rad/sec  (d)  Figure  4.3  s)  ROLL  ( c o n t . ) Response a m p l i t u d e o p e r a t o r s short-crested seas.  i n l o n g - and  86 co I  8.O-1  o  (e)  x  PITCH  7.0-j  -long-crested -s = 1 s = 2 s = 3  6.0  g  5.0 H  (0  4.0-  3  3.0H  in in  2.0 -\  LO  H  0.0  -* I  0.2  0.9  —i  0.4  OJ r a d / s e c  n  o X  0.8  0.6  ( f ) YAW  0.8  (all  s)  0.7 O  0.6  \ u  -r-  3  E-i  o.4-| 0.3 H 0.2 H  o.H 0.2  Figure  4.3  •  —I  0.4  CJ  1—  0.6  0.8  rad/sec  ( c o n t . ) Response a m p l i t u d e o p e r a t o r s short-crested seas.  i n l o n g - and  87  co r a d / s e c  Figure  4.4  Uni-directional wave computation.  spectrum  used  in  88  oH 0.2  1  •  1  0.4  i  0.6  >  1  0.8  co r a d / s e c o  55-1  co r a d / s e c Figure  4 . 5 Loading  spectra  in long-  and  short-crested  seas.  Figure  4.5  (cont.) Loading short-crested  spectra seas.  in  long-  and  90  25  n  (e) EXCITING MOMENT IN PITCH  o  long-crested s = 1  X  20 A  S  =  2  s = 3 (0  u  u 15 a;  V) i  E  I  z  10  5  ID  3  0.4  0.2  o  o X  m  0.8  0.6  CJ r a d / s e c ( f ) EXCITING MOMENT IN YAW  40  ( a l l s)  35 30  VJ \ U  cu 25  tn  I  £ I  20 ^  z  15 H 10  5H  0.2  Figure  4.5  CJ (cont.)  rad/sec  i 0.6  Loading spectra s h o r t - c r e s t e d seas.  "o.8  in  long-  and  91  0-| 0.2  ,  , 0.4  ,  —, 0.6  ,  , 0.8  co r a d / s e c Figure  4.6  Response s p e c t r a  in long-  and  short-crested  seas.  92  o.oo-l  .  0.2  1  .  0.4 w  Figure  4.6  rad/sec  (cont.) Response short-crested  1  0.6  spectra seas.  .  1  0.8  in  long-  and  93 0.45-1  (e)  PITCH  0.40  •long-crested s = 1 s = 2 s = 3  0.35  o <u (A I  <o  0.30 0.25  3  0.20 H  co.  0.15 H 0.10  in in  0.05 H 0.00 0.2  0.4  co r a d / s e c  0.6  0.8  (f)  YAW (all  1.75 H  S)  o  (/) 1.50 H 0)  I  m  u ~.  3  1-25  CO  H  0.75 H  0.50  —i  0.2  >  0.4  1  —  0.6  0.8  co r a d / s e c Figure  4.6  (cont.)  Response spectra s h o r t - c r e s t e d seas.  in  long-  and  94  Figure  4.7 E x p e c t e d v a l u e s of the e x t r e m e o f t h e maxima l o a d i n g and r e s p o n s e i n l o n g - c r e s t e d s e a s .  of  95  5.2 IN-LINE COMPONENT  4.8 H 4.4 H  e  H  3.6  3.2 H 2.8  1  — S  =  - s -s  = 2 = 3  2.4 H  2  5  4  log (M) 1 Q  3-i  2.8-  TRANSVERSE COMPONENT  2.62.42.22\  W  1.81.61.41.21-  0.8-  -r 4  -  2 log  Figure  1  (  J  5  6  (M)  4 . 8 E x p e c t e d v a l u e s o f t h e e x t r e m e o f t h e maxima l o a d i n g and r e s p o n s e i n s h o r t - c r e s t e d s e a s .  of  96  F i g u r e 4.9 E x p e c t e d v a l u e s o f t h e e x t r e m e o f t h e maxima h o r i z o n t a l r e s u l t a n t s i n s h o r t - c r e s t e d seas.  of  97  F i g u r e 4.10 loading  1.6  Frequency of u p c r o s s i n g of the components of and r e s p o n s e i n l o n g - and s h o r t - c r e s t e d s e a s .  H  N  Figure  4.11 F r e q u e n c y o f l o a d i n g and  of u p c r o s s i n g of h o r i z o n t a l r e s u l t a n t s response i n s h o r t - c r e s t e d seas.  98  (a) FREQUENCY OF UPCROSSING  0.8  P r e s e n t method H u n t i n g t o n and  0.4-  -r  Gilbert  -  5  4  (b) EXPECTED  VALUE  o  e <N  P r e s e n t method H u n t i n g t o n and  3-  o  i  2  3  4  5  Gilbert  6  iog  ( M ) l 0  Figure 4.12 Comparison of present method with t h a t of Huntington and Gilbert (1979) for a surface-piercing circular cylinder.  

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