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O n t h e a p p l i c a t io n o f s im i li t u d e t o in s t a l la t i o n o p e r a t i o n so f o f f sh o r e s t ee l j a c k e t sS U B R O T O K U M A R B H A T T A C H A R Y Y AOcean Engineering Centre, Indian Institute of Technology, Madras 600 036, IndiaTh e ro l e o f s im i l i t u d e i n scal ed s im u la t i o n o f m ajo r i n s t a l l a ti o n o p e ra t i o n s o f o f f sh o re s t ee l j ack e t s ,n am ely , l o ad o u t , l au n ch in g an d u p en d in g h as b een s t u d i ed . Th e p h y s i ca l m o d e l l i n g can b e l o o k edu p o n b o t h as an ad ju n c t t o n u m er i ca l m o d e l l i n g us in g co m p u te r s an d al so as an in d ep e n d en t t o o l o fi n v es t ig a t i o n . Th e p ro b l em s o f d es ig n o f m o d e l s an d o f ex p er im en t s a re d i scu ssed an d t h e p red i c t i o nequat ions based on s imi l i tude are g iven .

I N T R O D U C T I O NTh e i n s t a l la t i o n p ro cess o f o f f sh o re s t ee l j ack e t s i n v o lv esth ree m ajo r seq u en t i a l o p era t i o n s , n am ely , l o ad o u t , l au n ch -in g an d u p en d in g . To g e th er t h ey r ep resen t a p ro b l em in t h ea r e a o f r ig id b o d y h y d r o s t a t ic s a n d h y d r o d y n a m i c s . T h een g in eer in g o b j ec t i v e i s two - fo ld . F i r s t l y , t h e ap p rec i a t i o no f t h e p ro b l em in i t s en t i r e ty i s d es i red . S eco n d ly t h eab i l i t y to p red i c t t h e e f f ec t s t h a t wi ll r e su l t f ro m a ch an g eth a t i s im p o sed o n t h e sy s t em m u s t b e es t ab l ish ed wi thg o o d co n f id en ce . In o th e r wo rd s , t h e t r an s fe r fu n c t i o n s o ft h e sy s t em a re r eq u i r ed wi th r esp ec t t o d i f f e r en t sp ec i f i ccausal i t ies .Th e sy s t em i s d esc r i b ed co m p le t e ly i n t e rm s o f i t sg eo m et ry , m a t e r i a l p ro p er t i e s , i n i t i a l co n d i t i o n s an db o u n d a r y c o n d i t i o n s . T h e p e r t u r b a t i o n s e f f e c t i n g t h esy s t em ( say t h e j ack e t ) a r e d u e t o in t e r ac t i o n wi th e i th e rt h e as so c i a t ed sy s t em ( su ch as t h e l au n ch b arg e)o r t h ef i e ld s o r b o th .T h e j u s t if i c a t io n t o e m p l o y t h e t e c h n i q u e o f m o d e lan a ly s i s b ased o n s im i l i t u d e s t em s f ro m th e v e ry n a tu re o ft h i s p ro b l em . F i r s t l y , d u e t o t h e sh eer m ag n i tu d e an d sca l eo f o p era t i o n s i n v o lv ed i n t h e p ro b l em ren d er i t im p o ss ib l eto m ak e d i r ec t o b se rv a t i o n s o f i t s b eh av io u r an d t h u st r an s la t i n g t h em to t h e ab i l i t y t o p red i c t . S eco n d ly , t h en u m er i ca l m o d e l l i n g t ech n iq u es b ased o n a m ath em at i ca ld esc r i p t i o n o f t h e p ro b l em h av e i n h eren t l im i t a t i o n d u eto t h e as so c i a t ed as su m p t io n s an d ap p ro x im at io n s r esu l t i n gf ro m th e co m p lex i t y , vas tness an d n o n l in ea r n a tu re o f th ep ro b l em . C lea r l y p h y s i ca l m o d e l l i n g o f f e r s an ex ce l l en t' s eco n d o p in io n ' u n d er t h e c i r cu m s tan ces . An ex ce l l en tacco u n t o f t h e sco p e o f m o d e l m ech an i cs i s g iv en b yHo ssd o r f . 1T H E O P E R A T IO N S A N D M A J O R P A R A M E T E R STh e j ack e t a s a sy s t em fo rm s th e fo ca l p o in t o f s t u d y i n th ein s t a l l a t i o n o p era t i o n s . In l o ad -o u t , t h e l au n ch b arg e an dth e j ack e t a r e co u p l ed t o g e th er fo rm in g th e j ack e t -b a rg esy s t em . In l au n ch in g , th i s j ack e t -b a rg e sy s t em fu n c t i o n sto g e th er u n t i l t h e i n s t an t w h en t h e j ack e t l eav es t h e b a rg e .In u p en d in g , t h e d e r r i ck b a rg e- j ack e t sy s t em fo rm s t h et h e m e o f s t u d y . A n a c c o u n t o f t h e s e o p e r a t i o n s i sp r e s e n t e d b y G r a f t . 2Accepted February 1984. D iscussion closes Dece mbe r 1984.

Ne x t , t h e m a jo r p a ram ete r s o f i n t e r es t in each in d iv id u a lo p e r a t i o n b o t h f r o m t h e p o i n t o f v ie w o f t h e m o d e l a n a l y s tand the designer wi l l be iden t i f ied . For the designer , thech arac t e r i s t i c s o f t h e p a ram ete r s o f i n t e r es t wi l l f o rm abasis o f answering fundamental design quest ions. In load-o u t , b a rg e b a l l a s t s , w in ch p u l l , f en d er r eac t i o n s , m o o r in gl in e fo rces , ro ck er a rm reac t i o n s a r e o f d es ig n i n te r es tsu b j ec t t o d ep th o f wa t e r f ro n t , t i d e r an g e an d r a t e , co-e f f i c i en t s o f f r i c t i o n b e twe en l au n c h t ru ss o f t h e j ack e twi th g ro u n d , ro ck e t a rm an d l au n c h g i rd e r s . To g e th e r ,t h e y c o n s t i t u t e t h e p r o b l e m u n d e r s t u d y.In l au n ch in g , ro ck er a rm reac t i o n s , ro ck er a rm ro t a t i o n s ,b a rg e su rge , t ~a j ec to ry an d d ep th o f p en e t r a t i o n o f t h ej ack e t , e f f ec t o f i n i t i a l co n d i t i o n as d e f i n ed b y t r im an dd r a f t o f t h e b a rg e a n d e f f e c t o f j a c k e t e n t r y v e l o c i t y o nthese pa ram eters are of design in terest . The f ield para-m ete r s o f t h i s o p era t i o n i n c lu d e v i sco u s , i n t e r t i a l an dg rav i t a ti o n a l fo rces ; s t a t i c an d d y n am ic pressu re fo rces ;f l ex u ra l fo rces an d fo rce d u e t o su r f ace t en s io n .In u p en d in g , t h e c r an e h o o k l o ad , h o o k e l ev a t io n ,j ack e t b a l la s t , jack e t o r i en t a t i o n , r ig h t i ng m o m en t an db ear in g l o ad wh e n t h e j ack e t se t t l e s d o w n to seab ed ar eimportan t design quer ies . S tat ic p ressure generat ingb u o y an cy fo rce an d jack e t i n e r ti a l f o rce co n s t i t u t e th ef i el d p a ram ete r s .Th e w a t e r d e p th a t si te r ep resen t s t h e b o u n d a ry co n -d i t i o n i n b o th l au n ch in g an d u p en d in g f ro m th e sa fe typ o in t o f v i ew. C l ea r l y en o u g h , t h e wh o le p ro b l em can o n lyb e assessed co m p le t e ly i n t e rm s o f a l arge n u m b er o f p a ra -m e te r s i n t e r r e l a t ed b y m an y cau sa l i t y re l a t io n sh ip s .

P R I N C I P L ES O F S I M I L I T U D ETh ro u g h s im i l i t u d e r e l a t i o n sh ip s o n e m u s t fo rm u la t e p r o p e rsy s t em eq u a t i o n s an d p red i c t i o n eq u a t i o n s i n o rd er t op ro j ec t t h e m o d e l t e s t r e su l t s t o p ro to ty p e sca l e . Th eg en era l p ro ce d u re fo r t h i s can b e d esc r i b ed as fo l l o ws .3 L e t

7 r , = f ~ l r 2 , ~ 3 . . . . . 7r.) ( 1 )b e t h e c h a r a c te r i st i c e q u a t i o n o f t h e p h e n o m e n o n i n w h i c hl r~ i s the d imensionless group contain ing the pred ic t ionq u an t i t y an d 7 r2 , r r3 , . . . , z rn are-d imensionless groups havings ig n if i cant b ea r i n g o n t h e p h en o m en o n . In g en era l f isu n k n o w n .

0141-1187/841040221-06 $2.00 1984 CML Publications Ap plie d Ocean Research, 1984, Vo l. 6, No . 4 2 2 1

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A p p l i c a t i o n o f s i m i l i t u d e t o i n s ta l l at i o n o p e r a t i o n s o f o f f s h o r e s t e e l ] a c k et s : S . K . B h a t t a c h a ~ . v a

C or r e s pond i ng t o ( 1 ) , t he m ode l l ed s y s t em w i l l have acha r ac t e r i st i c equa t i on o f the s am e f o r m w hi ch i s7 r i m = f m ( l r 2 m , * r am . . . . . r gn r n) ( 2 )

w her e t he s uos c r i p t m has r e f e r ence t o t he m ode l . I f t hep r o t o t y p e a n d t h e m o d e l p h e n o m e n a a r e id e n t ic a l ,f = f m (3 )

w h i ch t hen r e s u l t s i n t o t he s ys t em equa t i ons1r = t r im , i = 2 , 3 . . . . . n (4 )

( 3 ) and ( 4 ) obv i ou s l y r e s u l ts i nrr, = 7 r , , . ( 5 )

w hi ch can be us ed t o a r r i ve a t t he p r o t o t ype p r ed i c t i onva l ues f r om t he m ode l va l ue s . Th i s i s c a l l ed t he p r ed i c t i onequa t i on o f t he s ys t em . D ue t o p r ac t i c a l l i m i t a t i onsr e s u l ti ng f r om s ca le e f f ec t s it m ay h appe n t ha trrir ~ rr (6 )

i s chos en . I n o t h e r w or ds ,7rim = err (7 )

This i s so ca l led di s tor t ion model l ing which resul t s in re-wr i t ing (5) as7/" "~" ~/7r I m ( 8 )

Thus f o r p r ed i c t i on pu r pos e s one needs t o e s t ab l i s h 3 , - er e l a t ions h i p , w h i ch can be don e i n a num b er o f w ays . 3

S IM I L IT U D E I N L O A D - O U T A N D U P E N D I N GThes e t w o ope r a t i ons i nvo lve hydr o s t a t i c m ode l l i ng andt he r e f o r e a r e cons i de r ed t oge t he r . G eom e t r i c a l s i m i l a r i t y i sa s s um ed . I n l oad - ou t t he s i t ua t i on i s s i m p l e ; F w , F f , F mdepend d i r ec t l y on t a a s t he s e f o r ce s r e s u l t f r om s ys t embalance agains t f r i c t ional forces . In s impl i f i ed s imi l i tudes y m b o l i s m ,

Fw- - = f ( t a ) ( 9 )WFf = f ( / a ) ( 1O)W6- - = f ( u ) ( i l )W

w hi ch a r e the cha r ac t e r i s t i c equa t i ons o f t he s ys t em . A l s oF f = Y F f i , i = 1 ,2 ( u s ua l l y ) ( 12 )

i

F t = ~ F t i , i = 1 t o 4 ( u su a ll y ) ( 1 3)i

w her e i r ep r e s en t s t he i t h f end e r o r t he it h m oor i ng l i ne . I nw r i t i ng t he s e equa t i ons one s hou l d r ecogn i s e t he ex i s t enceso f a m or e r i go r ous r ep r e s en t a t i on , l i ke s ay ,Fw- ~ = f ( /a , - '~ " , " - '~ ) ( 1 4 )

and s o on . H ow eve r , s i m p l e r ep r e s en t a t i ons a r e p r e f e r ab l ew hen caus a l i t i e s a r e no t i n t r ac t ab l e w i t hou t m a t hem a t i ca lm an i pu l a t i ons . I n i dea l s t a ti c c a s e ( 9 ) , ( 1 0 ) a nd ( 11 ) r ed uce

r e s pec t i ve l y t oF;

- - u ( 1 5 )W WFl- - = 0 ( 1 6 )W

B ut expe r i m en t a l l y , a m or e r ea l i s t i c s i m u l a t i on o f env i r on -m en t w i l l g i ve a f i r s t hand know l edge o f t he na t u r e o fva r i a t i on o f t he s e f o r ce s .For . barge b al las t and rock er arm hinge reac t ions , thecha r ac t e r i s ti c equ a t i ons a r e

w h e r e

w h e r e

f i a W aw , : z ' i r , ( 1 7 )o~ = f ( X ) ( 18 )

X = X l '

[3 = f ( X ) ( 20 )Here c~ and 3 are l inear funct ion s of the po s i t ion o f thej acke t w i t h r e s pec t t o t he ba r ge , w h i ch i s de f i ned by thej acke t t r ave l X . I n upend i ng , one o f t he s i m p l i fi ed cha r ac t e r -i s ti c s equa t i ons can be w r i t t en a s

H t f ( 21 )W \ W ' B 'Othe rs can be wr i t t e n on s im i lar l ines . Main poin t s ofi n t e r e s t ba s ed on equa t i ons ( 9 ) - ( 21 ) a r e r eco r ded be l ow .

( 1 ) O n l y t he r ig id bo dy pa r am e t e r s o f the s ys t em s a r eo f i n t e r e s t.( 2 ) T h e g ro u p s a , f l , h / l i nd i ca t e geo m e t r i c a l s i m i l a r i t yr e q u i r e m e n t .( 3 ) I n l oad - ou t on l y I ha s t o be s im i l a r.( 4 ) I n upend i ng t he g r oup W / B s igni fies tha t b oth Wand B m u s t be s i m i la r .(5) In load-o ut barge bal las t , F b , need to b e s imi lar .( 6 ) I n upend i ng , j a cke t ba l l a s t s, F , need t o be s im i l a r.Fu r t he r i t c an be conc l uded t ha t

(1) Jac ket iner t i a should be s imi lar .( 2 ) T i de , i n l oad - ou t , s hou l d f o l l ow geom e t r i c a l s im i -lar i ty .( 3 ) Wa t e r dep t h and hook e l eva t i on s hou l d f o l l owgeom e t r i c a l s i m i l a r it y i n upend i ng .( 4 ) J acke t o r i en t a t i on i n upend i ng s hou l d be i den t ic a lf o r b o t h t h e p r o t o t y p e a n d t h e m o d e l .( 5 ) D i s pos i t i on o f t he hook s li ngs in upend i ng s hou l d bei den t ic a l f o r bo t h t he p r o t o t yp e and t he m ode l .(6) Sl ing loads in upen ding shou ld be s imi lar .( 7 ) J a c k e t c e n t r e o f g ra v i ty a n d c e n t re o f b u o y a n c ys hou l d f o l l ow geom e t r i c a l s i m i la r i ty .

D E S I G N C R I T E R I A O F M O D E L ST h e d e p e n d e n c e o f a s u c c e s s f u l m o d e l s t u d y o n p r o p e rdes ign and f ab r i ca t i on o f t he m ode l s c anno t be ove r -em phas i s ed . The de s i gn c r i t e r i a m us t be ba s ed on appr o -p r i a t e s i m i l i tude . F r om t he f o r m ul a t i on o f s im i l i tude i n thel as t s ec t i on a s app l i cab l e t o t he p r e s en t p r o b l em , one can

2 2 2 A p p l i e d O c e a n R e s e a r c h , 1 9 8 4 , V o l . 6 , N o . 4

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Application o f similitude to installation operations o f of/sho re steel jackets: S. K. Bhattacharyyad i r ec t l y l i s t d o wn th e m ajo r d es ig n c r i t e r i a . F o r t h e b a rg ethese are as fo l lows:

(1 ) Geo m et r i ca l s im i l a ri t y as d i c t a t ed b y X m u s t b em ain t a in ed fo r t h e h u l l .( 2 ) As t h e m o d e l b a rg e m u s t su p p o r t b a l l a st i n g ( an dd eb a l l as ti n g ) t h e i n t e rn a l v o lu m es o f th e t an k s m u s to b e yX~t = X} Xp (2 2)

( 3 ) A s t h e b u o y a n c y d e p e n d e n t f o r c e o n t h e m o d e lb a r g e d e p e n d o n t h e u n d e r w a t e r d i s p l a c e m e n t a n dd en s i t y o f t h e m ed iu m in wh ich i t is f l o a t in g , fo rb u o y an cy s im i l a r i ty i t m u s t h av e

(4 ) F o r d ra f t s im i l a r it y , wh ich i sXT = X ( 2 4 )

th e m o d e l b a rg e we ig h t ( l e t ,) m u s t sa t i s fyXwb = xsb = x] Xp (25)

an d a lso t h e d i s t r i b u t i o n o f l e t , sh o u ld b e su ch t h a t(24) i s sat i sf ied under s imi lar condi t ions.(5 ) Tan k b o u n d ar i es o f t h e b a rg e m u s t b e l o ca t ed a tg eo m et r i ca l l y sim il ar p o s i t i o n s i n t h e m o d e l .(6 ) Majo r s t ru c tu ra l f i t t i n g s l i k e l au n ch b eam s , ro ck era rm s e t c . i n t h e m o d e l sh o u ld a l so b e d i c t a t ed b y X .( 7 ) Oth e r le s s im p o r t an t fu n c t i o n a l f i t ti n g s li k e b o Uardey es , s t ru c tu ra l m em b er s , ro ck er a rm h in g es , d ecko p en in g s , win ch assem b ly e t c . m ay b e d es ig n ed o n lyf ro m p rac t i ca l co n s id e ra t i o n s o f co n v en i en ce . No wco m in g t o t h e j ack e t , wh ich i s t h e fo cu s o f t h i ss t u d y , i t i s seen t h a t t h e m o d e l l i n g r eq u i r em en t i nlo ad -o u t an d u p en d in g a re n o t t h e sam e . Th o u g hg eo m et r i ca l s im i l a r i t y i s r eq u i r ed i n b o th t h eo p era t i o n s , i n l o ad -o u t o n ly we ig h t s im i l a r i ty wil lsu f f i ce , wh i l e i n u p en d in g b o th we ig h t an db u o y an cy s im i l a r i t y m u s t b e s im u l t an eo u s ly sa t i s -f i ed . Th e l a t t e r wi l l b e co n s id e red as i t au to m at i -cal ly sat i sf ies the former .

Ov era l l g eo m et r i ca l s im i l a r i t y o f b o th t h e j ack e t an d t h eb arg e m u s t b e g iv en b y t h e sam e X .F o r t h e j ack e t , c l ea r l y ,Xw = XB = X~Xp (26a )

X c c = ~ B = ~ 't ( 2 6 b )~k ----Xf Xp ( 27 )

Th e j ack e t s t ru c tu re i s l a rge ly m ad e o f t u b u l a r s . Id ea l l y(2 6 ) sh o u ld b e sa t i s f i ed fo r each i n d iv id u a l m em b er . Th i sr eq u i r em en t wi ll n o w b e l o o k ed i n to i n d e t a i l .F o r a t u b u l a r , o n e h as , fo r b u o y an c y s im i l a r i t y , b y (2 6 )XaXtXp = X~ Xp (28)

a s a = l r / 4 , 1~ ' o = k , ( 2 9 )

F o r w e ig h t s im i l a r i t y , ag a in b y (2 6 ) , o n e h asXaXtkd = XaIXp

or. (30)

BU tx~ - (31 )

Co m b in in g (3 0 ) an d (3 1 )?'d

2om -- i ~ = kp~.~ (02p -- @Iv) (32 )F o r b o th we ig h t an d b u o y an c y sim i l a r i ty to g e th er , p u t(2 9 ) i n to (3 2 ) , wh ich g iv es

1 I : " Xa 11/:i,n=-~t lop-- -~p('2op--~p)j ( 3 3 )In l i g h t o f (3 3 ) , o n e h as t o ch o o se a m o d e l m at e r i a l , i. e .Xa su ch t h a t

~ _ Xd

o r , (3 4 )Xp

( i p / + o p ) = / > 1 - - - MP u t t i n g ip = Cop - - 2 tp i n to (3 4 ) i t f o l l o ws t h a t ,

~)tp l k pi - < - - - ( 3 5 )Cop Cop 4 Xa

Th i s i n eq u a l i t y m ay n o t b e sa t i s f i ed fo r a l l t u b u l a r s i zesfo r a l l ;Mr. F o r ex am p le , fo r Xp = 1 .0 25 (u su a l l y ) , i f m o d e lm ate r i a l i s p e r sp ex (dm = 1 .2) , then (35) g ives-Z ~> 24.3 9 (36 )tpt ak in g o n ly t h e p rac t i ca l in eq u a l i t y .

Equat ion (36) may not be sat i sf ied by al l s izes oft u b u l a r s . Ho wev er , fo r t u n a t e ly fo r t h i s u se fu l m o d e lm ate r i a l , t h i s i n eq u a l i t y i s sa t i s f i ed b y m o s t j ack e t t u b u l a rs izes . So , i t can be s tated that g iven a m ode l ma ter ial , (29)an d (3 3 ) wi l l d i c t a t e t h e m o d e l t u b u l a r s i zes su b j ec t t o(3 5 ) .No w co n s id e r in g (3 3 ) , i t m ay a t t im es b e im p rac t i cab l eto h av e a Oim as suggested by i t even when (2 9) i s sat i sfied .To g e t a b e t t e r l ev erag e o n t h e ch o i ce o f ~ira le t us considerth a t t h e t u b u l a r h o l l o w sp ace is fi l led u p b y a l iq u id o fspeci f ic g rav i ty d~n. In such a case,( ~ o p - ~ p ) z , d p = X ~ X p ( 3 7 )2(2o,. - ~ ) l . d , . + ~m t , . d , .

On simpl i f icat ion , th is g ives' [ ' +i .0, ,+}]%Tw o se t s o f co n d i t i o n s a r e p o ss ibl e :1. (a) dr,, > d,n (3 9 )

d p- - i > 0 ( 40)\ ~ p l X p

Usi ng ~ ) i p = ~ ) o p - - 2tp in (40) and s impl i fy ing ,t p ( " ) 1 Xp

- - 1 - - t p < . - - - ( 4 1 )( ~ o p d C ) O p 4 k

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Applicat ion of s im il i tude to ins tal lat ion o perations o f of fshore s teel jackets: S . K. Bhattacha~_ .vaS o ( 3 9 ) a n d ( 4 1 ) f o r m s t h e f i rs t s e t.2 . ( a ) am > am ( 4 2 )

dp dptb) 2 - - ~ < 0 ( 4 3 )whi ch on s i mi l a r s i mp l i f i c a t i on g i ve s

tp ~1 tp 1 Xp- - I - - - - I / > ( 4 4 )Cop " Cop" 4 Xa

S o ( 4 2 ) a n d ( 4 4 ) f o r m t h e s e c o n d s e t.C l ea r l y be t w een t he se t w o se t s a ll t ubu l a r s c an betm o d e l l e d b y a l l m a t e r i a l s s u b j e c t t o c o r r e c t c h o i c e o f din.T h i s w a y t h e t h e o r e t i c a l d i f f i c u l t y p o s e d b y ( 3 5 ) m a y b eo v e r c o m e .

F r o m ( 3 8 ) i t c a n b e s ee n t h a t a su i t ab l e b a l a n c e b e t w e e nelm and d m i s pos s i b l e , sub j ec t t o Oim < Oom.E x a m p l e 1

S a y X = 5 0 , d m = 1.2, pp = 1 . 025 , P m= 1 , Oop = 1 3 8 4m m , t p = 4 4 m m . S o, (~orn= 2 7 . 6 8 m m a nd (~op/tp =3 1 . 4 4 .So ( 35 ) i s s a ti s f ied . B y ( 33 )

Oim --- 1 2 . 8 6 m mS a y , o n e r e q u i re s

~ l , n = 20 mmUs i ng t h i s va l ue i n ( 38 )

d~n = 0 . 7 04Such a l i qu i d i s pos s i b le .

E x a m p l e 2Ot h e r t h i ng s be i ng t he s ame a s i n exam pl e 1 , t he t ubu l a r

s ize i s g iven by o p = 6 6 0 m m , t p = 3 2 m m . S o (~o,n=1 3 . 2 m m a n d (~op/tp = 2 0 . 6 2 5 .So ( 35 ) i s no t s a t i s f i ed .A d o p t t h e c o n d i t i o n s e t g i v e n b y ( 4 2 ) a n d ( 4 4 ) . I n t h i s

case (44 ) i s sa ti s f ied . S o d~n > d m m u s t b e u s e d . F r o m ( 3 8 )3 6 . 6 6~ , . = 1.2 -d ; ,

rSe l ec t ~im = 7 . 2 m m ( s a y ) w h i c h g i v e s dm = 1 . 907 . Such al iquid i s poss ible .I t c an be r ead i l y s een t ha t , f o r a pa r t i cu l a r member , i f

Xa = X P - - ( 4 5 )2 __ 2~op e ~ i pt h e n t h e m e m b e r n e e d s t o b e a s o li d r o d o f c i r c ul a r c ro s ss e c t i o n . F o r u p e n d i n g o n e f u r t h e r r e q u i r e s

XFi = X~ Xp ( 4 6 )for ballast legs. Ballast CG a l s o m u s t b e g e o m e t r i c a l l ys i mi la r . No w i f li qu i d u sed i n t he m ode l f o r ba l l a s t ha sspec i f i c g r av i t y Pm t hen ,

Xeo = Xei = X (47 )Th i s i s a cond i t i on wh i ch may be d i f f i cu l t t o s a t i s f y i np r ac t i c e . A l t e r na t i ve l y i f a l iqu i d o f d i f f e r en t spec i f i cg r av i t y ( P~ n) i s u sed f o r mod e l ba l l a s t , on e ha s

tX6i= ~ k l P p / P r a = X~Xp

or .

~ i = Xt (P~n/Pm )1/2 ( 4 8 )!Cl ea r l y , su i t ab l e ba l ance be t ween Oim and Pra i s poss ible .

In g enera l , P~n > Pm wi l l p r ov i de a p r ac t i c a l so l u t i on .F o r a p l a t e i n t h e j a c k e t s t r u c t u r e , c o n d i t i o n f o rb u o y a n c y s i m i l a r i t y i s

Xt = Xt (49 )and t ha t f o r we i gh t s i mi l a r i t y i s

xt = x~xp/xa (50)Usi ng a s t e e l p l a t e i n t he mo de l i s c l e a r l y t he be s t cou r sebecause t h i s g ive s t he r a t i o o f Xp and X qu i t e c l o se t ou n i t y .

O n c e t h e j a c k e t m o d e l i s d e s i g n e d b a s e d o n s u c h ap r o c e d u r e , t h e CG a n d C B wi l l be geome t r i c a l l y s i mi l a r and( 26 ) , ( 27 ) w i l l be au t oma t i ca l l y s a t i s f i ed .N o w a g o o d l o o k i s n e e d e d t o f i n d i f t h e r e is a n y ' l e sst h a n i d e a l ' m e t h o d t o m o d e l t h e j a c k e t i n o r d e r to ge tr ea sonab l e r e su l t s f r om t he mode l t e s t s .O o v i o u s l y , t h e f u n d a m e n t a l e q u a t i o n s a s g i ve n b y ( 2 6 )a n d ( 2 7 ) m u s t b e s a t i sf i ed .D e p e n d i n g u p o n t h e m a t e r ia l c h o s e n a n d s iz e s a v a il a b le ,o n e c a n c h o o s e e i t h e r ( 4 7 ) o r ( 4 8 ) i n o r d e r t o a rr iv e a t a

su i t ab l e X . I t r ema i ns t o b e dec i de d on a s cheme o fm e m b e r d e s ig n w i t h r e l a ti o n t o t h e o v er a ll g e o m e t r y s u c ht ha t f i r s tl y ( 26 ) and ( 27 ) a r e s a ti s f ied and s econ d l y , t hem o d e l j a c k e t m a y b e c o n s i d e r e d a s a 'c l o s e ' c o p y o f t h ep r o t o t y p e w h i c h c a n b e e x p e c t e d t o p r o d u c e r e a so n a b led a t a l e a d i ng t o r e a s o n a b l e p r e d i c t i o n o f p r o t o t y p e p a r a -me t e r s o f i n t e r e s t . A pos s i b l e s cheme i s deve l o ped be l o ws t ep by s t ep .

( 1 ) Ove r a l l geome t r i c s i mi l a r i t y g i ven by X t o bem a i n t a i n e d .( 2 ) F o r a n y m e m b e r , l e n g t h m u s t b e as s u g g e s te d b y Xb u t ( 2 9 ) a n d ( 3 0 ) n e e d n o t b e s a t i s f i e d a n d t h e r e -f o r e ( 3 3 ) a n d ( 3 4 ) a n d s u b s e q u e n t e q u a t i o n s a l s o d on o t c o m e i n t o t h e p i c t u r e .

( 3 ) H o w e v e r , d e v i a ti o n f r o m ( 2 9 ) s h o u l d n o t b e t o og r e a t a n d t h e m e m b e r s w h i c h c a n n o t b e m o d e l l e deven by the smal les t tube s izes avai lable ( i . e . devia-t i o n f r o m ( 2 9 ) i s l a rg e ) m a y b e o m i t t e d a l t o g e t h e r.( 4 ) T h e j a c k e t is m a d e o f d i sc r e te m e m b e r s a n d in i tsgeo me t r y , t he l evel s ( a t d i f f e r en t e l eva t i ons ) and t hef ace s a r e c l ea r l y d i s c r enab l e a s d i s c r e t e bu t m a j o rc o n t r i b u t o r s t o w e ig h t a n d b u o y a n c y ( d e p e n d in gu p o n t h e m o d e o f f l o a t a t i o n ) . T h e r e f o r e , i f w e i g h ta n d b u o y a n c y o f t h e i n d iv i d u a l le v el s a n d f a c e ssa t i s f y ( 26 ) and ( 27 ) an d so does t he j a cke t i no v e r al l t e r m s , t h e n i t ca n b e a s s u m e d t h a t t h e m o d e lr e p r e s e n ts t h e p r o t o t y p e c l o s e ly e n o u g h . T h escheme desc r i bed above r equ i r e s i t e r a t i ve p r oces st o be ca r r i ed ou t f o r e ach l eve l and each f ace un t i la s a t i s f ac t o r y con f i gu r a t i on i s r e ached . I t i s l i ke l yt ha t i n o r de r t o ma i n t a i n ( 26 ) and ( 27 ) a t e ach l eve la n d f a c e , c e r t a i n a d d i t i o n a l m e m b e r s m a y b ei nc l uded i n t he f o r m o f f l a t p l a t e s , t ubes o r so l i dr o d s , a n d c e r t a in m e m b e r s m a y b e o m i t t e d d u e t othei r smal lness in m od el sca le . I t i s advisable toi n c l u d e a d d i t i o n a l m e m b e r s o n l y a t t h e l e v e l s a n dno t a t t he f ace s .I t m u s t b e r e al is e d t h a t e v e n a f t er p r o p e r m a t c h i n g o fl evel s and f ace s , t he r e ma y be some d i s c r epa ncy i n ove r al lw e i g h t , b u o y a n c y , i n e r t i a , CG and CB . T h i s c a n o n l y b e

2 2 4 App lied Ocean Research, 198 4, Vol. 6 , No. 4

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App lication o f similitude to installation operations o f offshore steel jackets: S. K. Bhattacharyyar em o v ed b y l u m p in g o f su i t ab l e we ig h t s a t so m e co n v en i en tl o ca t i o n s . T h e o b j ec t i v e , c l ea rl y , sh o u ld b e t o u se m in im u mn u m b e r o f s u ch l u m p e d w e i g h ts . O n e g e n er a l r e q u i r e m e n to f su ch an i t e r a t i v e p ro ce d u re i s t h a t fo r a m e m b er (m o s to f t h e m , i n f a c t ) ,

W /a < 1 (5 1 )so t h a t l u m p in g o f we ig h t t o ach i ev e (2 6 ) an d (2 7 ) fo r t h ej ack e t i n o v era l l sen se can b e e f f ec t i v e . Eq u a t i o n (5 1 ) canb e wr i t t en as

x. < X~Xp/Xa (52)i n li g h t o f (2 8 ) t o (3 0 ) . I n t h e p resen t ap p ro a ch l e t

Corn = 6 o p (5 3 )T h e n ( 5 2 ) r e d u c e s to

Xa (54)lm < 6 2 2 o p - - 4 t p ( o p - - t p ) )t~)tOClear ly o n e m u s t h av e , i n l ig h t o f (5 3 )

2 ~ . i ~ 2

I t i s seen that (54) and (55) can easi ly be sat i sf ied inp rac t i ce b ecau se o f t h e v e ry n a tu re o f t h e i n eq u a l it i e s .

r e q u i r e m e n t s o f w h i c h c a n n o t b e m e t s i m u l ta n e o u s ly .R e l a ti v e i m p o r t a n c e o f F r o u d e a n d R e y n o l d s f o r ce s c a n b ead ju s t ed ( t h ro u g h se l ec t i o n o f j ack e t v e lo c i t y ) , a we l lk n o w n t e c h n i q u e i n m a n y p r o b l e m s , s o t h a t l a u nc hm o d e l l i n g b e c o m e s p o s s i b l e . L a u n c h p h e n o m e n o n i sessen t ia l l y a s l o w sp eed o n e an d t h u s m ay b e ex p ec t ed t oresu l t i n l am in ar f l o w co n d i t i o n , t h e r eg im e i n wh ichR e y n o l d s f o r c e c a n n o t b e i g n or e d . B u t f o r t u n a t e l y , b y t h ev e r y n a t u r e o f ja c k e t g e o m e t r y , t u r b u l e n t f lo w i s g en e r a te d .Th i s o f co u r se can b e m ad e d o u b ly su re b y accep t edm eth o d s o f t u rb u l en ce s im u la t i o n as d o n e i n sh ip m o d e l s .In su ch a case , t h e e f f ec t o f v i sco s it y can b e co n s id e red t oh av e o n ly a v e ry m o d es t e f f ec t o n t r a j ec to ry , th e f l o w b e in gab o v e c r i t i ca l v a lu e o f Rey n o ld s n u m b er . Th e r esu l t i n gt r a j ec to ry can b e co n s id e red accu ra t e en o u g h fo r a l lp rac t i ca l ap p l i ca t i o n s .Based o n t h ese co n s id e ra t i o n s o n e can w r i t e ,

Tx ( r ) , T y ( r ) = f ( F ) ( 5 7 )su b j ec t t o

R m > R c r ( 5 8 )Th e 3 ' - e r e l a t i o n sh ip as p o in t ed o u t i n ( 7 ) an d (8 ) can o n lyb e f o u n d b y l o ng t e rm c o r r e l a t io n w i t h p r o t o t y p e m e a s u r e -m en t s . Th i s i s p ro b ab ly n o t n ecessa ry f ro m ap p l i ca t i o np o in t o f v iew.

S I M I L IT U D E I N L A U N C H I N GMo d el l in g o f j ack e t t r a j ec to ry i s o f fu n d am en ta l im p o r -t an ce i n l au n ch in g . F i r s t ly t h e t r a j ec to ry m u s t b e g eo -m e t r i ca l l y sim i l ar an d seco n d ly t h e a t t i t u d e o f t h e j ack e ta l o n g t h e t r a j e c t o r y i n b o t h t h e p r o t o t y p e a n d t h e m o d e lm u s t b e i d en t i ca l . It i s we l l k n o wn th a t an y d y n am icm o d e l l in g o f m o t i o n o f a b o d y i n f l ui d d e p e n d u p o n an u m b er o f fo rce f i e l d s . Th ese a r e i n t e r t i a l , v i sco u s , g r av i .t a t i o n a l , su r f ace t en s io n , f l ex u ra l , s t a t ic p ressu re an dd y n am ic p ressu re fo rces an d fo rces o f co m p ress ib i l i t y . I ti s k n o w n t h a t t h e p h e n o m e n o n c a n n o t b e m o d e l le d f o rd y n am ic s im i l a r it y o f a ll t h ese fo rces . Th e r e l a t iv eim p o r t an ce o f t h ese fo rces wi ll b e b r i e f l y rev i ewed h ere .In g en era l o n e can wr i t e ,

T , Ty = f ( F , R , o , S , M , ) t t ) ( 5 6 )based on d imensional analysis .As t h e re i s a v i r t u a l l ack o f im p o r t an ce o f su r f acet en s io n fo rces o n t h e sh ap e o f o cean su r f ace , S , th e Web ern u m b e r c a n b e e l i m i n a t e d f r o m ( 5 6 ) .Th i s p ro b l em i s n o t i n t h e d o m ain o f h y d ro e l as t i c i t y an dth ere fo re t h e f l ex u re n u m b er n eed n o t b e co n s id e red . Co n -s id e ri n g t h e v e lo c i t y r an g e o f t h e j ack e t (o rd er o f 5 m s-1 )i t can b e sa id t h a t fo rces o f co m p ress ib i l i t y wi ll n o t b eco m ep red o m in an t d u e t o n ea r ab sen ce o f g rav i t a ti o n a l fo rces .T h u s , M , t h e M a c h n u m b e r d o e s n o t c o m e i n t o p la y i nm o d e l l i n g .Becau se o f t h e sh ap e o f t h e en t ry f ace o f t h e j ack e t ,t h e re i s n o t r ace o f an en c i r c l i n g cav i t y fo rm ed o n im p ac t(wh ich g ro ws i n s i ze su b seq u en t l y ) a s h ap p en s i n t h e caseo f a h y d ro b a l l i s t i c m i ss il e, a Th e fo rce eq u i l i b r i u m o fcav i t y , t h e re fo re n eed n o t b e s im u la t ed i n l au n ch m o d e l l i n gwi th o u t cau s in g s i g n i f ican t e r ro r i n t r a j ec to ry p red i c t i o n .Bo th v i sco u s an d g rav i t a t i o n a l fo rces a r e co m p ared wi thin e r t i a l f o rce b ecau se i n e r t i a i s a lway s im p o r t an t i n t h ev e lo c i t y r an g e t o b e co n s id e red . Th ese co m p ar i so n s a r et h r o u g h R ( R e y n o l d s n u m b e r ) a n d F ( F r o u d e n u m b e r ) ,

D E SIGN O F E X PE R IM E N TSS im u la t i o n o f l o ad .o u t , l au n ch in g an d u p en d in g m u s t b e i nacco rd an ce wi th p r i n c ip l es o f s im i l i t u d e . Th i s m ean s t h a ta p a r t f r o m t h e m o d e l s , t h e o t h e r i m p o s e d e x p e r i m e n t a lp a ram ete r s sh o u ld a l so b e g u id ed b y s im i l it u d e .In l o ad -o u t o n e m u s t h av e

~tX = ) tT =- ) t t ~ - ) t l ( 5 9 )/~m = gp (60 )

In l au n ch in g o n e m u s t h av eXv = XY2 (6 1 )

~im = ~ip ( 6 2 )In u p en d in g o n e m u s t h av e

) t h e ~ " ) t h = ) k l ( 6 3 )PR E D IC TIO N E Q U A TIO N SOn ce t h e b as i s o f m o d e l l i n g fo r l o ad -o u t , l au n ch in g an du p en d in g h as b een es t ab l i sh ed an d t h e m o d e l s a r e m ad e i nacco rd an ce t o t h e l a i d d o wn d es ig n c r i t e r i a , o n e can wr i t ed o w n a ll p o ss ib l e p red i c t i o n eq u a t i o n s i n th e fo rm o f (5 ) .S o m e o f t h e m o r e i m p o r t a n t o n e s a r e g i v e n b e l o w . O t h e r scan be wri t ten on s imi lar l ines .S o m e p r e d i c t i o n e q u a t i o n s o f l o a d - o u t a r e ,

) t F w = ) t F f i = ) t F l i = ) t F r = ) t F b = )t~ )tp ( 6 4 )S o m e p r e d i c t i o n e q u a t i o n s o f l a u nc h i n g a r e ,

) t r x ( r ) = ) t r y ( r ) = ) tt ( 6 5 )~ r ( T x ) = ) tr ( T y ) = XY2 (6 6 )

) tv ( r ) = ) t}/2 (67 )) tb ( r ) ,= 1 (68)

Xo(r ) = 1 = ) t~ ( r ) (6 9 )

App l ied Ocean Research. 19 84 V ol 6 No d ~9~

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Ap p l i c a t i o n o f si m i l i tu d e t o i n s ta l la t io n o p e r a t i o n s o f o f f s h o r e s t e e l /a c k e t s : S . K . B h a t t a c h a ~ y aX$ (7") = A} 1/2 (7 0)Xg(r ) = X71 (71 )

Som e p r ed i c t i on equa t i ons f o r upend i ng a r eXH = XFj = X~ Xp (7 2)

X - = 1 (73 )

C O N C L U D I N G R E M A R K SU s e o f s ca l ed m ode l s i n s i m u l a t i on o f i n s t a l l a t i onope r a t i ons ba s ed on p r i nc i p l e s o f s i m i l i t ude can be con-s i de red bo t h a s an ad j unc t t o n um er i ca l m od e l l i ng and a s ani ndepen den t t oo l o f i nves t i ga ti on . The r o l e o f s i m i l it udeand t he a s s oc i a t ed p r ob l em s have been d i s cus s ed . I t c an beus ed w i t h advan t age f o r t he s pec i a l p r ob l em s l i keop t i m i s i ng t he upend i ng op e r a t i ons , e f f ec t o f in i t ia l con -d i t ions on t r a j ec t o r y i n l aunch i ng e t c . i n ve r y s ho r t t i m e a scom par ed t o t ha t r equ i r ed i n com pu t e r s i m u l a t i on . Thepe r f o r m ance o f num er i ca l m ode l l i ng can a l s o be i m pr ovedi n l i gh t o f s uch a m ode l s t udy . The m os t a t t r a c t i ve f ea t u r eo f any s uch s t ud y f o r t he p r ac t is i ng eng i nee r w i ll p r obab l ybe t he unde r s t and i ng ga i ned by v i s ua l obs e r va t i on o f t heopera t ions . I t i s qui te l ike ly tha t the vi sual na ture ofobs e r va t i on w i l l l ead t o app r ec i a t i on o f s om e f i ne r po i n t so f t he op e r a t i ons w h i ch i s no t pos s i b le o t he r w i s e .

R E F E R E N C E S1 Hossdorf, H. Model A nalysis of Structures, Van NostrandReinhold Co., 197 42 Graph, W. J. Introduction to Offshore Structures, Gulf Publish-ing Co., Houston, T exas, 19813 Sed ov, L. I . Similarity and D imensional Methods in M echanics,MIR P ublishers, Moscow , 1982

N O M E N C L A T U R Et i c r os s s ec t i ona l a r ea o f t ubu l a r l o t c a l cu l a ti ng buo yanc ya cross sec t ional a rea of tubu lar for ca lcula t ing weightB b u o y a n c y o f j a ck e tB a b u o y a n c y o f b a r g eC B c e n t re o f b u o y a n c yC G cen t r e o f g r av i tyd s pec i f ic g r av i t y o f m od e l m a t e r i a lf a func t ion

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2 2 6 Ap p l i e d O c e a n Re s e a r c h , 1 9 8 4 , Vo l. 6 , No . 4