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Proceedings o f the Eleventh (2001) hlternatio nai Offshore an d Polar E ngineering ConferenceStavanger,Norway, Ju ne 17-22 , 2001

Copyrigh t © 2001 by The In terna t iona l Soc ie ty o f O f fshore and Po lar Eng ineers

IS BN 1-8806 53-51-6 (Set); IS BN 1-880653-54-0 (VoL II1); IS SN 1098-6189 (Set)

Inundat ion Ef fec t and Quart ic A pprox imat ion o f Mor ison-Type Wa ve Loading

X . Y . Z h e n g a n d C Y L i a w

National University of S i n g a p o r e

Singapore

ABSTRACT

A q u a r t i c a p p r o x i m a t i o n o f t h e n o n l i n e a r i n u n d a t i o n e f f e c t o f

M o r i s o n - t y p e w a v e l o a d i n g i s p r o p o s e d . U s i n g T a y l o r s e r i e s

e x p a n s i o n , t h e i n u n d a t i o n f o r c e d u e t o v a r y i n g f r e e w a t e r s u r f a c ec a n b e r e p r e s e n t e d b y a ' c o n c e n t r a t e d l o a d ' , a c t i n g o n t h e s t r u c t u r e

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

m o d e l l e d b y a p o l y n o m i a l o f t h e f o u r t h o r d e r . P o l y n o m i a l

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

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

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

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

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

p a r t i c l e v e l o c i t y a t t h e m e a n w a t e r l e v e l .

KEY WORDS: Inundation effect, Morison wave force, quartic

approximation, least squares method, joint probability density function.

INTRODUCTION

Wave forces can have very significant nonlinear effects on the

response of offshore structures, especially for structures subjected to

Morison-type wave loading. There are mainly two types of nonl inear

effect attributable to wave forces. One is the effect of the nonlineardrag

force, which is usually the predominant wave force component for the

slender structural members of an offshore structure and can be

evaluated using the wel l-known Morison formula. Based on the method

of least squares approximation (Bendat, 1997), the nonl inear drag force

per uni t length can be represented by a polynomial expansion, and the

corresponding frequency response functions can then be derived using

the Volterra series method (Schetzen, 1980; Rugh, 1981). Borgman

(1969) studied a cubic representation of the drag term without current

and Gudmestad, et aL (1983) included the effect of current via fourth

order expansion. Li et al . , (1995) and Tognarelli e t a l . (1997) presented

the frequency response functions based on equivalent statisticalcubicization of the drag force. They observed that the structural

response spectra obtained exhibited a significant resonance

phenomenon near the frequency of 3cop, where cop is the peak frequency

of the wave spectrum. The other type of nonlinear effect attributable to

wave forces can be related to the variable submerged height of the

structure near the mean water level, or the so-called inundat ion effect

(Tickell, et al . , 198 5; Tung, 1996; Liaw, 2000). This nonl inear

inundation effect produces even-order super-harmonic force

components that can cause the wave force spectra to have peaks a

which are oRen close to the fundamental structural frequency with

range between 0.25 and 0.12 H z (Gudmestad, 1988; Kjeoy, et al . ,Liaw, 2000). This can obviously lead to significant nonlinea r str

responses near 2cop. Further, depending on the frequency ratio

structure and wave, the nonlinear effect of inundation can, in

cases, be even more significant than that of the drag force distr

along the height of the structure (Liaw, 2000).

In a previous paper (Liaw, 2000) a quadratic model for wave

was developed: the drag force itsel f was approximated by a line

and the inundat ion force was correspondingly represented

quadratic term. Wave force spectrum based on such a quadratic

can properly estimate the nonl inea r wave forces by includi

superharmonic components with frequencies up to 2COp, and is ad

for analysing structures with fundamental natural frequencies

near 2O~p. However, for structures with frequencies near 3o)p, a

approximation for the drag force is necessary; the corresp

approximation for the inundation force should then be quartic.

paper, Taylor series expansion is first employed to obta

approximate expression for the inundation effect that is shown

function composed of the wave elevation and wave kinematics

mean water level; the force is modelled as a polynomial of terms

fourth-order. Secondly, the least squares approximation method i

to obtain the polynomial coefficients that are the weights of di

terms. JONSWAP wave spectrum is used in the numerical evalu

of the polynomial coefficients as well as the join t probability d

function of wave elevation and water particle velocity at the mean

level.

APPROXIMATION OF INUNDATION EFFECT A

CONCENTRATED LOADThe empirical Morison wave force, f, per unit length of a v

cylinder (Fig. 1) is given as:

f = Cu A ~ + CvAolu[u

where C u and C o are the inertia and drag coefficients respectiv

= x p D 2 / 4 a nd A v = p D / 2 ; u , the water particle velocity normal

structural member; p, mass density of water; D, diameter of cy

2 9 5



T o b e t t e r d e m o n s t r a t e t h e n o n l i n e a r ef f e c t o f i n u n d a t i o n , a s i n g l e

C M a n d C O , a r e a s s u m e d t o b e c o n s t a n t s a l o n g t h e h e i g h t o f t h e

c y l i n d e r . I f t h e m e t h o d o f m o d e - s u p e r p o s i t i o n i s a p p l i e d t o s o l v e f o r t h e

s t ruc tu ra l re sponses , the s t ruc tu ra l moda l fo rce F i s ob ta ined by

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

a l o n g t h e s u b m e r g e d h e i g h t o f t h e s t r u c t u r e , i .e .,

F = [ O ( z ) f ( z ) d z (2 )- d

E n d F i x e d t o D e c k

V M W L ( z - O )

C y l i n d e r

D

S e a B o t t o m ( z = - d ) I f

i i i

F i g u r e l. W a v e f o r c e d u e t o i n t e r m i t t e n t w a v e s

In o rder to inc lu de the va r iab le su r face e ffect , the m oda l fo rce shou ld be

i n t e g ra t e d f r o m t h e s e a f l o o r ( z = - d ) u p t o t h e i n s t a n t a n e o u s f r e e

su r face o f the wave , q . I t can be wr i t ten as :0 t/

F : f O ( z ) f ( z ) d z + f O ( z ) f ( z ) d z (3 )- d 0

A p p a r e n t l y , t h e f i rs t t e r m o n t h e r i g h t h a n d s i d e o f E q . ( 3 ) i s t h e

c o m m o n m o d a l w a v e f o r c e, w h i c h i s a n o n l i n e a r fu n c t i o n o f th e w a v e

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

represen ts the I n u n d a t i o n E f f e c t ( IE ) , w h i c h i s o f o n e h i g h e r o r d e r t h a n

t h e f ir st . I n t e r m s o f T a y l o r s e r i e s e x p a n s i o n , t h e s e c o n d t e r m c a n b e

e x p a n d e d w i t h r e s p e c t to t h e v a r i a b l e a t M e a n - W a t e r - L e v e l (M W L w i t h

z = 0), i .e . ,

F uE ) = _ ~ ( O ) f ( O ) . r l + ( ~ ' ( O ) f ( O ) + O ( O ) f ' ( O ) ) . l r l 2 (4 )

T h e s e c o n d t e r m o n t h e r i g h t - h a n d - s i d e o f E q . ( 4 ) i s u s u a l l y a h i g h e r -

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

of the cy l inder , e .g . ~ P ( z ) = c o s ( ~ z / 2 d ) f o r t h e l e g s o f j a c k - u p

p la t fo rms , i s a ssumed . Thus

--- O(O )f (O) • 7 /+ O (O )f ' (0 ) . 2 r /2 (5 )( l e )

I

I t c a n a l s o b e e a s i l y s h o w n t h a t t h e f i rs t p a r t o f t h e r i g h t - h a n d - s i d e o f

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

r e l a t iv e l y h i g h e r o r d e r a n d h a s a m u c h l o w e r m a g n i t u d e t h a n t h e f i rs t ;

t h e r a t io b e t w e e n t h e m i s k r/ , w i t h k b e i n g t h e w a v e n u m b e r . T h u s , t h e

c o n c e n t r a t e d f o rc e r e p r e s e n t i n g t h e i n u n d a t i o n e f f e ct c a n b e w r i t te n a s :

F u E ) = f ( O ) r l = F l r e ) + F ( S ) (6 )w h e r e

F t ' E ) = f ~ ( O ) r l = C i A, f i (0 )q (7 )

a n d

FD E , : UD (O)T] ---- C o A ~ u ( O ) [ u ( O ) l r l ( 8 )

F/:E) i s th e i n e r t i a p a r t a n d F o ~ ) t h e d r a g p a r t o f t h e t o t a l i n u n d a t i o n

IE) 1E)fo rce F :zEj , of which the co r res pon d ing mod a l fo rce i s F ( = ¢P(0)F .

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

c y l i n d e r i s c o n s i d e r e d . F u r t h e r m o r e , t h e h y d r o d y n a m i c c o e f f i c

c o n c e n t r a t e d f o r c e a c t i n g a t M W L o f t h e s t r u c t u r e . M o r e o v e r

F / E ) a n d F ~ ~ ) a r e e v e n - o r d e r f u n c t i o n s o f t h e w a v e e l e v a t i o n

cause s ign i f ican t s t ruc tu ra l re sponse s a t even o rder f requ enc ies

2o)p an d 4o)p.

Q U A R T I C A P P R O X I M A T I O N O F I N U N D A T I O N D

F ORC E

A s e x p r e s se d i n E q . ( 8 ) , t h e i n u n d a t i o n d r a g f o rc e , F ~ S ) ,

p r o d u c t o f 1 / a n d t h e d i s t r i b u t e d d r a g f o r c e a t M W L , w h i c h i s a S

L a w S y s t e m , l u ] u , w i t h t h e s i g n o f w a t e r p a r t i c le v e l o c i t y u a t

Cons ide r ing f i r s t ly the d is t r ibu ted d rag fo rce i t se l f , i t c

a p p r o x i m a t e d ( B o r g m a n , 1 9 6 9 ) u s i n g t h e l e a s t s q u a re s m e t h o d , i .e

] u [u = c r ~ 8 u = a l u (L ine ar approx . )

o r

[ u l u _ = . [ o ' , ~ ] u + [ ~ f / 3 c r , ] u 3 = a ~ s u + a 3 s u 3 ( C u b i c a p p r o x . )

w h e r e t r i s t h e s t a n d a r d d e v i a t i o n o f u . H o w e v e r , t h e

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

f o r c e b e c a u s e o f t h e i n h e r e n t c o r re l a t i o n b e tw e e n r / a n d u (

s tochas t ic ana lys is o f s t ruc tu ra l re sponses to a un id i rec t io na l , s ta t

z e r o - m e a n l i n e a r G a u s s i a n w a v e t r a i n o f w a v e e l e v a t i o n r / , t h e

A i r y w a v e t h e o r y i s c o n s i d e r e d , i . e .,

on ?u = r ( z ) o ) r l , - f f~ = r ( z ) c o O t

a n d

o ) 2 = k g t a n h k d , r ( z ) = c o s h k ( z + d ) , ( - d < z < 0 ) s i n h k d

As ind ica ted in Eq . (11) , the re i s a l inea r t rans fo rm proc ess be tw

a n d u . I n th e f o l l o w i n g d e r i v a t i o n , u w i l l b e u n d e r s t o o d a s u

brev i ty . I f po lynom ia l approx im at ion o f F J F~) up to four th o

cons ide red , we have :

R = u [ u [ r ~ v = a 2 4 r l u + a 4 4 ~ u 3 (Quar t ic approx . )

I t s h o u l d b e n o t e d t h a t t h i s q u a r t i c a p p r o x i m a t i o n n e c e s s a r i l y c o

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

t h a n f o u r . H e r e, w e d e f i n e Q = E l ( R - v ) 2] a s t h e m e a n s q u a r e

b e t w e e n R a n d v . B y s e t t i n g t h e d e r i v a t i v e o f Q w i t h r e s p e c t

co r resp ond in g coef f ic ien t equa l to ze ro , i . e . ,

I c3Q = 0

0 a 2 4

O Q = 0

0 a 4 4

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

a r e o b t a i n e d :

I E[r12u2]a24 + E[r12u '~ la44 = E[[u lr l2u 2] = 2E [r/2u 3 .. .

= E [ r / u ] .> ,E t r l ' u 4 ] a 2 4 + E [ r l 2 u 6 ] a , , E [ l u [ rl E u ] = 2 : 5

I t i s n o t e d t h a t+ < ~ + ~

E [ r f ' u " L , , = ~ p f ' u " p ( q , u ) d u d q

296



C o n s i d e r n o w t h e t w o r a n d o m v a r i a b l e s r / a n d u , w h i c h a re

c o r r el a t e d, t h e i r j o i n t 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 ( J P D F ) c a n b e

expressed as :

exp - 1 2 ~_~_ 2 r/ u u 2

p ( ~ , u ) ( 1 7 )

2 x t r . tr . 1 C - p ,~

whe re p~ , i s the co r re la t ion coef f ic ien t o f r / and u , i .e .

P ~ = g [ ( ~ - z ( ~ ) X u - e ( u ) ) ]

O'rtO"

Note tha t E [ r /] = E[u] = 0 . T h e r e f o r e ,

(18)

E[(r/- E(r/)Xu E(u))]= E[r/u]= ~H(co)S. co)dco (19 )

whe re H(co) i s the f requenc y t rans fe r func t ion f rom r / to u , i .e .

H(co) = mr(0 ) = co co th kd (20)

I t i s obv ious tha t the po ly nom ia l coef f ic ien ts , a24 and a44 in Eq . (15) ,

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

f u n c t i o n o f th e w a v e s p e c t r u m , S ~ ( c o ) . I f J O N S W A P w a v e s p e c t r u m i s

cons ide red , i . e .,

/ - / s c o p2 4 - 1 7 ~S , ( c o) = e x p [ ~ ] y q ( 2 1)

161oco' ( co l e% )4whe re H~. i s the s ign i f ican t wave he igh t ; cot ,, the p eak wav e f req uency ; 3' ,

t h e p e a k e n h a n c e m e n t f a c to r . V a l u e s o f / o a n d q a r e :

I o = 0 . 2 / 0 - 0 . 2 8 7 L n ( r ) ) ( 2 2 )

q = e xp [- (co / cop - l ) 2 / 2 o ' ~ ( 2 3 )

w i t h ( 7 = 0 . 0 7 f o r co / co , < 1 o r t r = 0 . 0 9 f o r co / co , > 1 . T o ev a l u at e

t h e c o r r e l a ti o n c o e f fi c i e n t, w e c a n l e t x = r / / t r , a n d y = u / o . tha t

leads to t r . = t ry = 1 . i. e . .

p ( r h u ) = p ( x , y ) ( 2 4 )t r . t r .

w h e r e th e JP D F o f x a n d y i s :

e x p l . . . . .- _1 7 ~, ( x , - 2 p ~ r x y + y 2 ) }

p ( x , y ) = [ 2 ( i - p ~ , ) i 25 )

a n d

SH(co)S"(c°)dc° ( 2 6 )

P'Y = P ~ = ~ ~ SH 2 CO)S.(CO)dco

F r o m E q . ( 1 2 ), w e a l s o h a v e

( ~o )' = ( k d ) t a n h ( k d ) . ~ _ I ) ' 1z x ) s ( 2 7 )

whe re co = and s=d/gTp2; t h e l a t te r i s a r e l a t i v e m e a s u r e o f t h e w a t e r

d e p t h . G i v e n s a n d c o , k d c a n b e o b t a i n e d f r o m E q . ( 2 7 ) . E q . ( 2 6 )c a n t h e n b e a p p l i e d t o e v a l u a t e t h e c o r r e l a t i o n c o e ff i c i en t f o r a g i v e n

p e a k e n h a n c e m e n t f a c to r 3 '.

F i g u r e 2 s h o w s t h e r e l a t i o n s h i p b e t w e e n t h e c o r r e l a t i o n c o e f fi c i e n t,

P~u , and the re la t ive wa te r dep th , s , fo r th ree d i f fe ren t va lues o f peak

e n h a n c e m e n t fa c t o r 3' o f t h e w a v e s p e c t ru m . I t c a n b e o b s e r v e d t h a t a l l

va lues o f , o~, a re ve ry c lose to 1 .0 , wh ic h ind ica tes the fac t tha t r / and

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

theory .

1 . 0 0 .

Pn u

0 . 9 5 .

0.90 .q

~ ~ 0.9"-- - - ' - - "- - -" - ' - - 0 .9

11

- - y = 1 . 0

- - - y = 3 . 3

. . . . y=5.0

0.85I E - 3 o . 8 1 . . . . . . . o ' . ] . . . . . . . . ;

s = d / g T p 2

F i g u r e 2 . C o r r e l a t i o n c o e f f i c i en t s o f r / a n d u

=

y = l . 0

= .

11 Io11

F i g u r e 3a . J P D F o f r / a n d u w i t h T = I . 0

2

1

~ = o

-I

-2

-3.3

y=5 .0

= .

n Icr

F i g u r e 3 b . J P D F o f I / a n d u w i t h y = 5 . 0

T h e c o n t o u r p l o t s o f J P D F o f r / a n d u f o r t w o d i f f e r e n

o f p c o r r e s p o n d i n g r e s p e c t i v e ly t o y = l . 0 a rt d 5 . 0 i n t h e d e e p - w

a r e g i v e n i n F i g u r e 3 . I t i s o b v i o u s t h a t t h e c o r r e l a t i o n i s a f u n

t h e w a t e r d e p t h ( a s i n d i c a t e d b y s ) a n d t h e s h a r p n e s s o f t

s p e c t r u m . H o w e v e r , e v e n f o r t h e d e e p - w a t e r ( l a r g e s ) a n d b r

case (3' = 1 fo r P ie rso n-M osko wi tz spec t rum ) , the JPD F is s

2 9 7



narrowly distributed with P,u = 0.929. One also notices that

p O T, u ) = p ( - r l , - u ) , p ( - q , u ) = p ( q , - u ) a nd

~ T m u " p ( q ' u ) d u d r l = c r: ° ' : S ~ m y " p ( x ' y ) d y d x (27)- ¢ o 0 - ~ 0

The polynomial coefficients a2+ and a44 can be numerically evaluated

from Eq. (15) for a given set values ofs and 7 for JONSWAP spectrum.

1 . 3 4 -

1 . 3 3 •

1 . 3 2 -

1 . 3 1 -

1 3 0 ,0 . 8 1 :

4 1 3

~ /50 . 8 0

a 2 4

a 1 3 1 . 3 1 7 3

. . . . . . . . . . . . • 1 . 3 1 5 6" i' = 1 . 0 1 . 3128

- - - 7 = 3.3.. . . 7=5.0

0,8089

, - 0 . 8 0 7 6~ o " ~ . - " . . . . . . . . . . 0 . 80 6 9

. / f / + ° a 4 4

. . . . . . . . i . . . . . . . . , . . . . . . . . i

I E - 3 0 . 0 1 0 .1 1

d/gTp2

Figure 4. Quartic coefficients of approximation

Figure 4 shows the variation of the quartic coefficients, a24 and a44,

versus Y and s. The values of a24 and a44 are presented as ratios to the

cubic approximation coefficients, a13 and a33, for the distributed drag

force as given in Eq.(10). Apparently, both ratios are not very sensitive

to either s or ~,; they can be regarded as two constants, 4/3 and 4/5.

Consequently, the quartic approximation coefficients for the inundation

drag force can be expressed as:

4a13_ 4(,fZ 2o. /

and

(28a)

a 4 4 ~ 5 5 L V 7 /" 3 a - n )

Therefore, the complete expression for the approximate modal

inundation force in Eq. (6) is

F (m) m qb(0)f(0)r/

= , I , ( 0 ) c . A , + , I , ( 0 ) C o A o ( a 2 , ,l u ( 0 ) + a 4 4 r ] u S ( 0 ) )

a t

(29)

CONCLUSIONS

The inundation effect of varying wave surface on structures can be

simplified as concentrated inundation forces acting at the mean water

level (MWL). Corresponding to the common practice of approximating

the drag force by a cubic polynomial, the inundation drag force can be

approximated by a quartic polynomial, which involves only even orderterms. The polynomial coefficients depend upon the correlation

coefficient of wave elevation and water particle velocity at MWL and

can be obtained using the least squares method. Based on the

assumptions of linear wave theory, the join t probability density function

(JPDF) and the correlation coefficient of the wave elevation and water

particle velocity at the mean water level are evaluated. It is shown that,

for wave conditions specified by JONSWAP-type wave spectrum, the

JPDF and the correlation coefficient are functions of two parameters:the water depth parameter s=d/gTp 2 and the peak enhancement factor of

the wave spectrum 7. The numerical results reveal that the

polynomial coefficients are not very sensitive to the two para

Simple closed-form expressions of the quartic coefficients

inunda tion drag force can be obtained, and they can be related

cubic coefficients for the distributed drag force. Consequently, a

analytical relationship between the wave elevation and the total

wave force, including the inundation effect, is established, wh

serve as the basis for developing the nonlinear frequency

transfer function of wave forces.

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