Inundation Effect and Quartic Approximation of Morison-Type Wave Loading
Transcript of Inundation Effect and Quartic Approximation of Morison-Type Wave Loading
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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
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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
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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
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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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