wave load computation in direct strength analysis of ss platform structure

8/10/2019 Wave Load Computation in Direct Strength Analysis of Ss Platform Structure

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J o u r n a l o f M a r i n e S c i e n c e a n d A p p l i c a t i o n , Vo l . 3 , N o . 1 , J u n e 2 0 0 4

Wave load computa t ion in d i rec t s t reng th ana lys i s o fs emi subm er s ib le p l a t fo rm s truc tu re s

Z H A N G H a l - b i n , R E N H u i - l o n g , D A I Y a n g - s h a n , a n d G E F e i

School of Shipbuilding Engineering Harbin Engineering U niversity Harbin150001, China

Abstract: A wave load computation appr oach in direct strength analysis of semi-su bm ersib le platform s tructure s was presente d in th

paper. Considering the differences in shap e of pontoon, column and bea m , the combination o f accumulat ive chord length cubic param

ter spline theory and analytic m ethod was adopted for generating the w et surface mesh of platform. The hydrodynam ic coefficients

platform were calculated by the three-dimensional potential flow theory of the linea r hydrodynamic p roblem for platform w ith low forw

speed. The equation of platform motions was established and solved in frequ ency dom ain, and the responses of wa ve-ind uced loads

the platform can be obtained. With the interpolation method bein g utilized , the pressm e loads on shell elem ents for finite elem ent an

ysis (FE A ) were converted from those on the hydrodynamic computat ion me sh, which pave the basis for FEA w ith commercial sof twa

A com puter program based on this method has been developed , and a calculation example of semi-subm ersible platform was i l lustrateAnalysis results sho w that this m ethod is a satisfying approac h of wave loads com putation for this kind of platform.

Key w ords : serni-submersible p latform ; wave load s; m esh-g enera ting; direct strength analysis

CLC number: U 6 6 1 . 4 Document code:A A rt ide ID : 1671 - 9433 (20 04 ) 01 - 0007 - 07

I N T R O D U C T I O N

T h e d i r e c t s tr e n g th a n a l y s i s m e t h o d i s b e c o m i n g

t h e m o s t i m p o r t a n t w a y f o r d e s i g n a n d s t r e n g t h a s s e s s -

m e n t o f th e s e m i - s u b m e r s i b l e p l a t fo r m s t r uc t u r e s.

H o w e v e r , b e c a u s e o f t h e n u m e r o u s n e s s i n d e s i g n c o n -d i t i o n s a n d c o m p l e x i t y in o p e r a t i o n e n v i r o n m e n t , t h e r e

a r e m a n y d i f f i c u l t ie s d u r i n g t h e s e l e c t i o n o f c r i ti c a l l o a d

c a s e s , t h e c o m p u t a t i o n o f e n v i r o n m e n t a l l o a d s a n d t h e

s t r u c t u r a l a n a l y s i s o f p l a t f o r m , i n w h i c h t h e w a v e l o a d

c o m p u t a t i o n i s th e k e y p r o b l e m g o v e r n in g t h e r e s u l ts o f

s t r e n g t h a n a l y s i s . T h e r e f o r e , i t i s n e c e s s a r y t o p r o p o s e

a p r a c t i c a l a n d e f f i c ie n t a p p r o a c h t o c a l c u l a t e w a v e

l o a d s i n t h e d i r e c t s t r e n g t h a n a l y s i s o f th i s p l a t f o r m .

S t o c h a s t i c a n a l y s i s a p p r o a c h a n d d e s i g n w a v e a -

n a l y s i s a p p r o a c h a r e tw o m e t h o d s o f d e s i g n w a v e l o a d

c o m p u t a t i o n f o r s e m i - s u b m e r s i b l e p l a t f o r m s , i n w h i c h

t h e d e s i g n w a v e a n a l y s i s a p p r o a c h i s m o r e w i d e l y

u s e d . To s a t is f y th e n e e d f o r s i m u l t a n e i t y o f t h e r e -

s p o n s e s , t h e d e s i g n w a v e a p p r o a c h i s o f t e n a d o p t e d f o r

m a x i m u m s t r es s a n al y s i s o f s e m i - s u b m e r s i b l e s . T h e

m e r i t s o f t h e s t o c h a s t i c a p p r o a c h a r e r e t a i n e d b y u s i n g

t h e e x t r e m e s t o c h a s t i c v a l u e s o f s o m e c h a r a c t e r i s t i c r e -

s p o n s e p a r a m e t e r s i n t h e s e l e c t io n o f d e s i g n w a v e p a -

r a m e t e r s . W h e n c a l c u l a t in g w a v e l o a d r e s p o n s e s , t h e

a d o p t e d m e t h o d u s u a l l y i n c lu d e t h e m e t h o d b a s e d o n

Received date :2003 - 10 - 21.

r u l e s , s t ri p m e t h o d a n d t h r e e - d i m e n s i o n a l m e t h o d . I n

t h is p a p e r , t h e l i n e a r A i ry w a v e t h e o r y w a s a d o p t e d ,

a n d t h e ra d i a t i o n a n d d i f f r a c t io n p r o b l e m c a n b e c o n -

s i d e r e d a s a t h r e e - d i m e n s i o n a l h y d r o d y n a m i c p r o b l e m

o f p l a t f o r m w i t h lo w fo r w a r d s p e e d , i n w h i c h t h e s p e e d

w a s t a k e n a s z e r o in t i m e o f o p e r a t i o n c o n d i t i o n a n d

s u r v i v a l c o n d i t i o n .

I t i s a b s o l u t e l y n e c e s s a r y t o g e n e r a t e t h e w e t s u r -

f a c e m e s h o f s e m i - s u b m e r s i b l e p l a t f o r m s w h e n c a l c u l a t -

i n g w a v e lo a d s w it h th e t h r e e - d i m e n s i o n a l h y d r o d y n a m -

i c t h e o r y. I t i s a h a r d w o r k b e c a u s e t h e p l a t f o r m c o m -

p r i s e s o f s e v e r a l c o m p o n e n t s s u c h a s p o n t o o n s , c o l -

u m n s a n d b e a m s , a n d e a c h c o m p o n e n t h a s i ts ow n

s h a p e . I n t h is p a p e r , t h e a c c u m u l a t i v e c h o r d le n g t h

c u b i c p a r a m e t e r s p l i n e t h e o r y i s u t i li z e d f o r t h e m e s h -

g e n e r a t i n g o f p o n t o o n , a n d t h e a n a l y t i c m e t h o d i s u t i -

l i ze d f o r t h e m e s h - g e n e r a t i n g o f o t h e r c o m p o n e n t s . A t

t h e s a m e t i m e , t h e i n t e rs e c t in g s i t ua t i on a m o n g t h e s e

c o m p o n e n t s i s c o n s i d e r e d c a r e fu l l y .

A f t e r t h e w a v e l o a d c o m p u t a t i o n w a s c o m p l e t e d ,

t h e d i r e c t s t r e n g t h a n a l y s i s fo r s e m i - s u b m e r s i b l e p l a t -

f o r m c a n b e c a r r ie d o u t . H o w e v e r , t h e m e s h o f h y d r o -

d y n a m i c c o m p u t a t i o n i s u s u a l l y n o t i d e n t i c a l w i t h t h a t

o f F E A . T h e r e f o r e , h o w to c o n v e r t t h e p r e s s u r e l o a d s

o n t h e h y d r o d y n a m i c c o m p u t a t i o n m e s h t o th o s e o n

s h e l l e l e m e n t s f o r F E A i s a n o t h e r k e y p r o b l e m . I n th i s

p a p e r , c o m b i n i n g t h e m e s h - g e n e r a t i n g m e t h o d o f h y -

d r o d y n a m i c c o m p u t a t i o n m e s h , t h e f l u id d y n a m i c p r e s -

s u r e fi e l d o f p l a t f o r n l w e t s u r f a c e w a s b u i l t , a n d t h e n

t h e p r e s s u r e l o a d s o n sh e l l e l e m e n t s f o r F E A w e r e c a l -


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9 8 9 Jou rna l o f Mar ine Sc i ence and App l i ca t i on ,Vo l . 3 ,No . 1 , June 2004

cu la t ed by t he i n t e rpo l a t i on me thod .

In or der to cer t i fy the feas ib i l i ty of th is metho d in

the d i r ec t s t r eng th ana ly s i s o f p l a t fo rms , t he com pu ta -

t ion and analys is were car r ied out for a semi-submers i -

b le p la t form.

2 T H E O R E T I C A L F O R M U L AT I O N

2 . 1 T h e m e s h g e n e r a ti n g m e t h o d o f s e m i

submers ib l e p l a t fo rm

The typ i ca l s emi - submers ib l e p l a t fo rm compr i s e s

pon toon s , co lumn s , beams and b rac ings . The pon toon

looks l i ke sh ip hu l l , so the accum ula t i ve cho rd l eng th

cub ic pa rame te r sp l i ne t heo ry can be u sed i n t he mesh -

gener a t ing ca lcula t ion [11 . As the colum n an d beam are

co lumni fo rm, t he ana ly t i c me thod i s u t i l i z ed i n t hemesh -gene ra t i ng o f t he se componen t s . I n add i t i on , t he

in t e r sec t i ng r eg ion among the se componen t s shou ld be

cons ide red . The b rac ings a r e u sua l l y neg l ec t ed i n t he

wave l oad compu ta t i on w i th t h r ee -d imens iona l hyd rody-

namic t heo ry a s t he i r d imens ions a r e sma l l enough .

2 . 1 . 1 T h e m e s h - g e n e ra t in g o f p o n t oo n

The mesh -gene ra t i ng o f pon toon can be ca r r i ed ou t

by two s teps , the f i rs t s tep i s the m esh-ge nera t ing of

the midd le pa r t o f pon toon , and t he s econd s t ep i s t he

mesh -gene ra t i ng o f bow and s t e rn a s w e l l a s t he i r con -nect ions wi th the middle par t .

The mesh o f midd le pa r t o f pon toon was gene ra t ed

by c rea t i ng nodes on i t , wh ich i nc ludes two phases .

The f ir s t phase i s t he n ode -gene ra t i ng i n each t r ans-

ve r se s ec t i on o f midd le pa r t o f pon toon wi th t he cub i c

pa rame te r sp l i ne i n t e rpo l a t i on me thod , and t he s econd

one i s i t s node-genera t ing in the longi tudina l d i rec t ion .

In the f i r s t ph ase , the offse ts of a sec t ion of mid-

dle par t of pontoon are sugges ted to be g iven as fo l -

l ows ,P x ~ , r~ ) i = o , 1 , . . . , n ) . 1 )

Accord ing t o t he pa rame te r sp l i ne func t ion t heo -

ry, t he cub i c pa rame te r sp l i ne cu rve can be exp re s sed

wi th the offse t va lues ,

P t ) = x t ) , y t ) ) . ( 2 )

The num ber and l oca t ion o f t he i n t e rpo l a t ed node

can be de t e rmined acco rd ing t o needs . Gen e ra l l y, t he

sca l e o f two ve r ti c a l ne ighbor ing meshes r eq u i r e s nea r ly

iden t i ca l , so t he un i fo rm cho rd l eng th d iv i s ion me thod

i s adop ted . Acco rd ing t he i npu t numb er N o f ve r t i c a l

mesh o f each ha l f s ec t i on , t he pa ram e te r coo rd ina t e s

c a n b e c o m p u t e d ,

t k = k . t m / N k = 0 , 1 , 2 , . . . , N ) , ( 3 )

wh ere , t,n = ~ l j , l j i s the chord length b e tween twoj l

ne ighbor ing po in t s , andP x m , ym ) i s the offse t of the

top poin t . Th en the coordina te informat ion of in terpola-

ted nodes in th is sec t ion wi l l be obta ined by subs t i tu-

t i n g E q . ( 3 ) i n t o E q . ( 2 )

P x k , Y k ) ( k = 0 , 1 , ' " , N ) . ( 4 )

Discre t iza t ions of the sur face o f the m iddle par t o f

pontoon wi l l be f in ished wi th the method ment ioned a-

bove . The m esh i s usual ly very rough. In order to ob-

t a in more accu ra t e mesh g r id , t he s econd phase i s nee -

essary to be car r ied out on the bas is of rough mesh. In

th is ph ase , the aspe ct ra t io of mesh i s in t rodu ced to

def ine longi tudina l in terpola t ion parameter in s tead of

un i fo rm cho rd l eng th . The accum ula t i ve cho rd l eng th

cub ic pa ram e te r sp l i ne cu rve P ( t ) - - ( x ( t ) , y ( t ) , z( t ) ) c an be bu i l t a cco rd ing t o r e l evan t nodes o f each

sec ti on o f midd le pa r t o f pon toon , and t he p a rame te r

coo rd ina t e t c an be compu ted b y t he a spec t r a t i o o f

mesh , and t hen t he l ong i tud ina l mesh nodes w i l l be

c rea t ed . F ina l l y, we can ob t a in t he mesh o f midd le

par t of pontoon sa t i s fy ing a g iven asp ect ra t io .

Fo r the m esh -gene ra t i ng o f bow and s t e rn , we

cons ider not only the par t icu lar i ty in shape of bow and

s t e rn , bu t a l so t he cons i s t ency be tween midd le pa r t

and bow or s te rn . The f i r st aspect i s ensured by theoffse ts of the m, and the o ther aspect wi l l be rea l ized

wi th t he connec t ion cond i t i on be tween midd le pa r t and

bbw or s te rn . The offse ts of longi tudina l prof i le curve

beyond the bow sec t i on and s t e rn s ec t i on a r e r equ i r ed ,

which descr ibe the bow and s tern shape wi th the com-

binat ion of offse ts of bow and s tern sec t ion and the con-

nec t i on cond i ti on be tween midd le pa r t o f pon toon and

bow or s te rn .

Fi rs t ly, the uni form chord length d iv is ion i s car-

r ied out for the longi tudina l prof i le curve beyond thebow sec t ion and s t e rn s ec t i on , whose num ber o f d iv i-

s ion i s equal to tha t of midd le par t of pontoon. Th en ,

the longi tudina l in terpola t ion i s going on be tween nodes

o f l ong i tud ina l p ro f i l e cu rve and bow-and- s t e rn s ec t ion

wi th t he accumula t i ve cho rd l eng th cub i c pa rame te r

sp l i ne t heo ry, i n wh ich t he pa ram e te r t c an a lso be de -

termin ed by the aspect ra t io of mesh. In order to en-

su re t he cons i s t ency be tween midd le pa r t o f pon toon

and bow or s te rn in th is s tage , a cer ta in connect io n

condi t ion for the bu i l t sp l ine cu rve should b e sa t i s f iedon the bow and s t e rn s ec t ion . The re fo re , t he accum u-

l a t i ve cho rd l eng th cub i c pa ra lne t e r sp l i ne func t ion

shou ld be bu i l t be tween m esh nodes o f midd le pa r t o f


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ZHANG Hai - biin,et l : Wave oad computation n direct strength analysis of semi-submersibleplatform tructures 9 9 9

pon toon and those o f long i tud ina l p ro f il e cu rve bey ond

bow and s te rn s ec ti on . And the n , t he cons i s t en t mesh

nodes o f bow and s t e rn w i th t hose o f midd le pa r t o f

pontoon wi l l be obta ined .

2 . 1 . 2 T h e m e s h - g e n e ra t i ng o f c o l u m n a n d it s i n te r -

sec t ing region wi th pontoon

F i r s t l y, t he m esh o f pon toon is gene ra t ed up t o t op

edge o f i t , and t h en t he t op su r f ace o f pon toon i s to be

meshed . The l ong i tud ina l dens it y o f mesh i s de t e r-

mined by t ha t o f pon toon ou t e r su r f ace mesh , and

t r ansve r se dens i t y o f mesh l i e s on inpu t da t a . Th en ,

the who le su r f ace m esh o f pon toon wi ll be ab t a ined .

Howeve r, t he i n t e r s ec t i ng r eg ion be tween co lumn

and top su r f ace o f pon toon shou ld be r econs ide red .

Wi th t he a s sumpt ion t ha t t he t r ansve r se d iv i s ion num-

be r o f ha l f t op su r f ace mesh o f pon toon is n t , and t he

long i tud ina l d iv is ion numb er o f t op su r f ace mesh o f

pon toon on wh ich t he co lumn s t ri de s ove r is n l , and

then t he c i r cumfe ren t i a l cu rve o f t he co lum n wi l l be eq -

uab ly d iv ided in to n t + n I segments . The mesh on the

in t e r sec ti ng r eg ion be tween co lum n and top su r f ace o f

pon toon can be ob t a ined a f t e r the se nodes con nec t t o

the sur rounding nodes as shown in Fig . 1 . The mesh -

gene ra t i ng o f t op su r f ace o f pon toon wi l l be co mple t ed

a f t e r t he en t i r e i n t e r s ec ti ng r eg ions have been cons id -

e r ed .

Fo r the m esh -gene ra t i ng o f co lumn i t s e l f , t he ana -

ly t i c me thod i s adop ted w i th cons ide r ing t he co lumn i s

co lumni fo rm. Af t e r t he c i r cumfe ren t i a l d iv i s ion num ber

and ve r t i c a l d iv i sion numb er o f co lumn mesh a r e de t e r-

min ed , t he su r f ace mesh o f t he co lum n wi ll be ac -

qu i r ed .

Fig. 1 M esh of intersecting region betweenpontoon and column

2 .1 .3 The mesh -gene ra t i ng o f beam and i ts i n t e r s ec -

t ing region wi th column

Gene ra l l y speak ing , t he beam i s i n t e r s ec t ed t o t he

co lumn . Af t e r t he mesh o f co lumn i s ob t a ined , t he i n -

t e r s ec t ing r eg ion can be de t e rmined b y t he d imens ionsand in tersec t ing pos i t ion of the beam .

With the assumpt ion tha t the c i rcumferent ia l d iv i -

s ion num ber o f beam su r f ace mesh i s nh , and t he ve r t i -

ca l d iv i s ion number o f co lumn su r f ace mesh on wh ich

the beam s t r ide s ove r is nv, and t hen t he c i r cum fe ren -

t i al c u rve o f t he be am can be equab ly d iv ided i n to 2 ( n~

+ n h - 2 ) s egmen t s . The mesh o f t he i n t e r s ec t i ng r e -

g ion be tween beam and co lum n can be ob t a ined a f t e r

t he se nodes connec t t o t he su r round ing nodes a s shown

in F ig . 2 . Un l ike t he ca se o f co lumn , t he i n t e r s ec ti ng

cu rve be tween beam and co lumn i s no t a p l ane cm-ve ,

but a c losed curve spacia l . The ver t ica l and longi tudi -

na l coo rd ina t e s can be ob t a ined i n t he c ro s s - sec t i on o f

beam by the c i r c l e equa t ion , and t r ansve r se coo rd i -

na t e s can be compu ted i n t he c ro s s - sec t i on wh ich pas -

ses th is no de an d i s ver t ica l to c ross-sec t io n of the

beam by p ro j ec t i on o f long i tud ina l node coo rd ina t e and

rad ius o f co lumn .

In t he s ame w ay, t he mesh -gene ra t i ng o f beam i t -

s e l f c an a l so adop t t he ana ly t ic me th od , and t hen t he

su r f ace mesh o f t he beam wi l l be ob t a ined .

Fig. 2 M esh of intersecting region between column and beam

2 . 2 T h e c o m p u t a t i o n a l m e t h o d o f w a v e l o a d s

f o r s e m i s u b m e r s i b le p l a t fo r m

At p re sen t , t he t h r ee -d imens iona l po t en t i a l f low

theory I31 i s a prefere nt ia l metho d of ca lcula t ing wave

loads w hen the des ign wave analys is ap proa ch E21 i s ap-

p l i ed t o de t e rmine t he des ign wave l oads o f s emi - sub -

mers ib le p la t forms.2 .2 .1 The equa t ion o f p l a t fo rm mot ions

Accord ing to t he dynam ics o f r i g id body, t he e -

quat ion of p la t form mot ions wi th the center of gravi ty

EMB ED Equa t ion . 3 be ing t he cen t e r o f mom en t can be

exp re s sed a s ,

[ M ] - { ~ /( t) } = I F ( t ) } = {Y } " e ~ , ( 5 )

whe re [ M] i s t he gene ra l i zed mass ma t r i x o f p l a t fo rm,

{ F ( t ) t i s the vec to r o f f l u id l oads on p l a t fo rm, wh ich

exc lude t he s t il l wa t e r buoyan t fo r ce i n ba l ance w i th t he

gravi ty of p la t form.The f l u id l oads ac t i ng on t he p l a t fo rm su r f ace can

be d iv ided i n to t he hyd ros t a t i c re s to r ing l oads t FS ( t ) }

induced by t he d i sp l acemen t o f t he p l a t fo rm from the


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mean pos i t i on , an d t he hyd rody namic loadsF V ( t ) }

depended on t he waves and p l a t fo rm mot ions , name ly

I F ( t ) } = { F S ( t ) } + I F D ( t ) } , ( 6 )

The hyd ros t a ti c r e s to r ing l oads can be ob t a ined by f l u id

s ta t ics , wr i t ten as ,

I F S ( t ) } = - [ C I{ ~ / ( t ) } , ( 7 )

wh ere [ C 1 i s the matr ix o f hydros ta t ic res tor ing force

coeff ic ients .

The hyd rodynamic l oads can be ob t a ined by i n -

t eg ra ti ng t he h yd rodynam ic p re ssu re s on t he we t su r f ace

of the p la t form. Acco rding to the d iv is ion of the ve loc i -

t y po t en t ia l o f f l ow , the h yd rodynam ic loads can be de -

composed i n to i nc iden t wave fo rce , d i f f r ac t i on wave

fo rce and r ad i a t i on fo rce , name ly

{ F ' ( t ) } = { F , ( t ) } + { F, ~ ( t ) } + I F R ( t ) t .

( 8 )

The i nc iden t wave fo rce and d i f fr ac t ion wave fo rce

can be wr i t t en i n to t he combined fo rm a s t he so -ca l l ed

wave exci ta t ion loads ,

{ f ( t ) l = { F, ( t ) / + { F D ( t ) } 9 ( 9 )

The r ad i a t i on fo rce can be exp re s sed a s ,

I F R ( t ) } = - [ A 1{ i} ( t ) } - [B ] {~ / ( t ) } ,

( l O )

where [ A ] , [ B ] a r e t he ma t r ix o f t h r ee -d imens iona l hy -

d rodynamic coe ff i c i en t s and can be w r i tt en a s

Aij + iw = p ox lS

( 1 1 )

( i , j = 1 , 2 , . . - , 6 ) ,

whe re ~b i s uni t rad ia t ion pote nt ia l , U i s the forward

speed o f p l a t fo rm( t aken a s ze ro i n ope ra t i on and su r-

v iva l cond i t i on , w i s t he c i r cu l a r f r equency o f encoun-

ter, and n i i s the gen era l ized no rmal vec tor of body

surface .

The d i ff rac t ion wave force can be w r i t ten in to thefo l lowing in tegra l form

{ F z ) ( t ) / = - l ( p ( ~ ( i w - U O ) 4 ) 7 { n j l d s . e i~S

( 1 2 )

whe re 1])7 i s uni t d i ff rac t ion potent ia l .

The pan e l e l em en t g r id s on t he we t su r f ace o f the

p l a t fo rm can be gene ra t ed w i th t he me thod p roposed a -

bo ve , and th en the uni t rad ia t ion p otent ia l qb~ i = 1 ,2 ,

9 . , 6 ) and the un i t d i ff rac t ion potent ia l~ 7 c a n b e

so lved w ith t he combina t ion o f Gree n ' s Func t ion me th -od and pane l me thod .

The re fo re , t he equa t ion o f p l a t fo rm mot ions i n

r egu la r waves Eq . (5 ) can be wr i tt en a s ,

{ Q ' } =

( [ M I + [ A I { i }( t) } + [ B l I / / ( t ) / +

[ C ] { ~ 7 ( t ) } = { f ( t ) } = { f t e j~t. ( 1 3 )

2 .2 .2 The f l u id dynamic p re s su re on p l a tfo rm

Af te r t he so lu t ion o f ve loc i t y po t en t i al @ ( j = 1 , 2 ,

9 . , 7 ) and t he s t eady - s ta t e so lu t ion rb ( j = 1 ,2 , " .. , 6 )of p la t form mot ion responses in regu lar waves be ing ob-

ta in ed , the radia t ion poten t ia l 4~R and the d i ff rac t ion

potent ia l 4}0 can be de te rm ine d,6

c b R ( x , z , z ) = ~ [ io o~b 9 + : ( x , y , z ) ] , ( 14 )j = l

D X , y , z ) : a ~ 7 x , y , z ) . 1 5 )

W ith the co nt r ibut ion of the hyd ros ta t ic res tor ing

fo rce be ing i nc luded , t he t o t a l f l u id dynamic p re s su re

can be ob t a ined by l i nea r ized Be rn ou l l i ' s Equa t ion ,

P ( x , y , z , t ) = R e [ p ( x , y , z ) C ~ ' l

p ( x , y , z ) = p s ( x , y , z ) - p . i w . [ q ~ , ( x , y , z ) l tJ+ +~ x,y,z) + +R x,y,z)

( 1 6 )

w h e r e p ,~ , ( x , y , z ) -- - p g ( m + Y m - x m ) , 4 , i s i n c i-

d e n t wave potent ia l .

Thus , t he f l u id dynamic p re s su re l oads on t he

con t ro ll i ng po in t o f e ach pane l e l emen t can be ob -

t a ined , and t he f l u id dynamic p re s su re d i s t r i bu t ion on

the wet sur face o f p la t form can be g iven .

2 .2 .3 The t yp i ca l s ec t iona l l oads o f p l a t fo rm

For a regular wave wi th a cer ta in c i rcular f requen-

cy w , a f te r the p la t form mot ions and the f lu id dynamic

p re s su re l oads be ing ob t a ined , t he wave - induced fo rce s

and mom en t s i n t he s ec t i ons o f p l a t fo rm can be ca l cu -

l a t ed by u s ing D ' A lember t ' s P r inc ip l e , wh ich i nc lude

the ver t ica l and hor izonta l shear forces and moments as

wel l as tors ional moments .

L S '

SF;

: _ f f p . y.z ) I , l - I , ITM' s~B M

B M ~

1 7 )

In t he above exp re s s ion , t he l ong i tud ina l shea r

force and hor izonta l sp l i t force should exclude the grav-

i ta t ional com pone nt of p la t form in coordin a te sys tem

fixed on p la t form due to i ts mot ions . The tors ional mo-

men t shou ld be t r ans l a t ed to shea r cen t e r. And the ve r-

t i c a l and ho r i zon t a l bend ing momen t s shou ld exc ludethe componen t s occu r r ed by t he ve r t i c a l and ho r i zon t a l

shear forces . Then, the f ina l sec t ional loads can be ex-

p re s sed a s ,


5/7

Z H A N G H a i - b i i n ,et l : Wa v e l o a d c o m p u t a t i o n i n d i r e c t s t r e n g t h a n a l y s i s o f s e m i s u b m e r s i b l e p l a t t b n n s t r u c t u r e s " 11 9

L s ] [L s ]

B M z B M s j - x S f y

w h e r e , L S i s longi tudina l shear force ,S F v is horizontal

sp l i t force , S F z i s ver t ica l force , T M is torsional mo-

m e n t , B M v i s ve r t i c a l bend ing momen t ,B M z is hori-

zonta l bend ing m om ent , S~ i s par t ia l wet sur face of

p la t form, p i s en t i re f lu id dynamic pressure , z~~h) is the

ve r t i c al coo rd ina te o f shea r cen t e r, [ M) i s gene ra l i zed

mass matr ix of par t ia l p la t form, and /~ i s mass of par t ia l

platform.

When de t e rmin ing t he des ign wave pa rame te r s i nthe d i rec t s t rength analys is of the sem i-subm ers ib le

pla t form, the typica l sec t ional loads to be cons idered

inc lude ve r t i c a l bend ing momen tB M v , tors ional mo-

m e n t T M , longi tudina l shear forceL S and hor izonta lsp l i t forceS F v.

2 . 3 T h e p o s t t r e a t m e n t fo r p r e s s u r e l o a d s o f

s e m i s u b m e r s i b l e p l a t f o r m

Af te r ob t a in ing t he des ign wave pa ram e te r s , t he

s ix-deg ree- f reedo m iner t ia l acce lera t io ns and f lu id dy-

namic pressure d is t r ibut ion a long wet sur face of the

p l a tfo rm fo r FEA can be ob t a ined . How eve r, t he ob -

ta ined pressure loads are those of cont ro l l ing poin ts of

hyd rodynamic compu ta t i on pane l e l eme n t s , and some

measu res mu st be take n to conver t them to those of ge-

ometr ic centers of she l l e lements for FEA.

Accord ing t o t he shape o f pon toon , co lumn and

beam, t he f l u id dynamic p re s su re f i e ld can be bu i l t by

the pa rame te r o f con t ro l l i ng po in t s o f hyd rodynamic

compu ta t i on pane l e l emen t s .

On the assum pt ion tha t the cross sec t ion of pon-

toon , co lumn and beam i s i ny - z p l a n e , a n d t h e n t h ef lu id dynamic p re s su re i n t h i s p l ane can be exp re s sed

by the accum ula t i ve cho rd l eng th cub i c p a rame te r

sp l i ne ,

p c ( t ) = ( p c ( y , z ) , t ) , 1 9 )

where , t i s a ccumula t i ve cho rd l eng th pa rame te r de t e r-

mine d by y and z coordina tes o f cont ro l l ing poin t ac-

co rd ing t o Eq . 3 ) .

In the l ong i tud ina l d i r ec t ion o f pon toon , co lunm

and beam, t he f l u id dynamic p re s su re w i l l be ex -

p re s sed i n l i nea r fo rm,p , ( ~ ) - * - * ~ ( ' ~ t ' -~ ~ ) p c ( t ) ) + p ~ ( t ) ,

X i 1 - - X i

2 0 )

wh ere , i , i + 1 refers to two neighbo r ing cross sec t ions

o f hyd rodynamic compu ta t i on mesh .

Then , t he en t i r e f l u id dynamic p re s su re f i e ld o f

p la t form wet sur face wi l l be bui l t wi th com binat ion of

E q . 1 9 ) a n d E q . 2 0 ) .

In order to obta in the pressure loads on she l l e le -

men t s i n fi n i te e l emen t mesh , t he spa t i a l coo rd ina te s o f

geome t r i c cen t e r s o f she ll e l emen t s shou ld be de t e r-

mined f ir s t ly. T hen the pressure in terpola t io n can be

carr ied out in the f lu id dynamic pressure f ie ld bui l t as

above , and t he i n t e rpo l a t ed p re s su rewil l be t aken a s

the un i fo rm p re s su re l oads on she l l e l emen t s .

3 N U M E R I C A L E X A M P L E A N D A N A L Y S I S

3 . 1 T h e e x p l a n a t io n a n d p a r a m e t e r s o f a

s e m i s u b m e r s i b l e p l a t f o r m

In t h i s pape r, a wave l oad compu ta t i on app roach

in d i rec t s t rength analys is of sem i-subm ers ib le p la t form

s t ruc tu re s was p re sen t ed , i nc lud ing mesh -gene ra t i ng o f

p l a t fo rm we t su r f ace , wave l oad compu ta t i on and con -

vers ion of pressure loads for FEA . In ord er to tes t i fy

the f ea s ib i l i ty and accu racy o f t h i s me thod , t he comp u-

t a t ion fo r a s em i - submers ib l e p l a t fo rm was c a r r i ed ou t ,

and the analys is and va l ida t ion for the resul t s were a lso

preformed . Consider ing the analys is method for opera-

t ion , surv iva l and t rans i t condi t ion is s imi lar exce pt forthe de t e rmina t ion o f de s ign wave pa rame te r s , t he com-

puta t ional resul t s a re only se t for th for surv iva l condi -

t ion.

The d imens ion pa rame te r s o f t he s em i - submers ib l e

pla t form can be seen in Table . 1 .

Table 1 The dimen sion parameters of thecalculat ion platform

P a r a m e t e r s S m w i v a l c o n d i t io n

Length overa ll /m 92. 350Distance b etween forward and aft

54. 864column eenterl ines/m

Distance between port and starboard45. 720

column centerl ines/mPontoon heig ht/m 7. 620Pontoon width / m 15. 240

Diam eter of middle colunm s/ m 11. 125Diame ter of corner column s/ in 9. 754

Draught from B. L. / m 13.72Displacement/ t 21370

3 . 2 T h e m e s h g e n e r a t i n g o f th e s e m i s u b m e r s i b l e

p l a t f o r m

With t he me thod p re sen t ed i n t h i s pape r be ing


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9 12- Journa l o f Mar ine Sc ience and Appl ica t ion , Vol . 3 , No . 1 , June 2004

use d , the we t su r face mesh was genera ted fo r the p la t-

fo rm under su rv iva l cond i t ion as shown in F ig. 3 . The

f igure shows tha t th i s mesh-genera t ing method can mee t

the needs o f mesh ing th i s k ind o f p la t fo rm for hydrod y-

namic computa t ion .

Fig. 3 Discretizations on the wet surface of the

semi-submersible platform under

survival condition ( 1616 patches)

3 3 The compara t ive ana lys i s o f p la t fo rm mot ions

Heave and ro l l r e sponses have been computed re -

spec t ive ly fo r the semi-su bmers ib le p la t fo rm in head

waves and in beam waves , and compar i sons wi th mode l

tes t ing resu l t s in the opera t ion manu a l o f p la t fo rm a re

car r i ed ou t , a s shown in F ig . 4 and F ig . 5 , which i s

nondim ens ion a l i zed by wave ampl i tude . On the whole ,

the method in th i s paper can g ive sa t i s fy ing resu l t s fo r

pred ic t ing the mot ion responses o f semi-submers ib le

platform.

Because o f the l ack o f the mod e l t e s t ing resu lt s fo r

w a v e l o a d s , s u c h c o m p a r i s o n c a n ' t b e p e r fo r m e d .

Never the less , i t can be be l i eved tha t th i s method can

pred ic t wave loads o f the semi-subm ers ib le p la t fo rm

wel l , s ince i t has acqu i red good agreemen ts fo r those o f

sh ips I< and can comp ute p la t fo rm m ot ions we l l e -

n o u g h .

2.2-2.0s1.8s1.6 s1 4 s1.2

~> 1.0s~z 0.80.6 20.420 .2o.o~

-o.2 i

Fig. 4

o measuredcomputed

5 10 15 20 25 30Y s

Transfer function for heave response in

head w aves

3 4 The convers ion of p ressure loads

The regu la r wave per iod and phase fo r the c r i t i ca lload case can be de te rmined by the des ign wave ana ly -

s i s approac h . Af te r these des ign wave paramete rs have

been de te rmine d , the s ix -degree- f reed om iner t i a l ae -

0 . 7 -

0.6 ~

0.5-

~.~ 0.4-

-g 0.3-0.2-

0.1

0.0

Fig. 5

o measured-- computed

) on o

O

, 9 , . , . , 9 , . ,

5 10 15 20 25 30T s

Transfer function for foil response in

beam waves

ce le ra t ions and f lu id dynamic p ressure d i s t r ibu t ion a -

long wet su r face o f the p la t fo rm can be c ompu ted und er

these pa ramete rs . The quas i - s ta t i c d i rec t s t reng th ana l -

ysis with FEA software could go through af ter the un i-

fo rm pressure loads have been conver ted on she ll e l e -

ments o f f in i t e e lement mesh .

Accor d ing to the p ressure loads on the hydrody-

namic computa t ion pane l e lements , the un i fo rm pres -

sure loads on shel l e lements in f ini te element mesh wil l

be ob ta ined wi th the in te rpo la t ion method presen ted a -

bove.

The wet su r face d i sc re t i za t ions o f the semi-sub-

mersible platform in FE A are sho wn in Fig. 6 .

In ord er to test i fy the exa ctness of the interpolat ionmetho d , the re - in tegra l fo r in te rpo la ted p ressure loads

was ca r r i ed ou t , and com pared wi th the o r ig ina l va l -

ues , a s shown in Tab le 2 . I t can be conc lu ded tha t th i s

convers ion method fo r p ressure loads i s f eas ib le and ac -

curate .

Fig. 6 Wet surface discretizations of the semi-sub-

mersible platform in FEA


7/7

wave load computation in direct strength analysis of ss platform structure

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