On Froude-Cauchy similitude

ON FR OUDE - CA UCHY SIMILITUDE B. Le Mehaute

Associate Director

National Engineering Science Company Pasadena, California

ABSTRACT

It is common engineering practice in a hydraulic model study involving both gravity waves and a solid structure to measure the hydro-dynamic forces on the model of the structure and then calculate the res ult-ing internal structural stresses. Because of the large variety of ava il-able elastic material, and the latest development in solid state physics, it is now feasible to measure directly these structural stresses in a scale model study.

It is shown how the similitude of elastic for ce s in structure s subjected to wave action can be made compatible with the Froude s imili-tude valid for hydraulic motion. Several examples are presented to illustrate the method. These include the study of the elastic response of an ice floe, the motion of an underwater membrane-type oil storage tank, the behavior of the Mohole riser, and the motion of a Texas Tower type of structure under wave action. Results obtained in the NESCO wave tank are also presented.

INTRODUCTION

The rules of similitude can be obtained by three different approaches: dimensional analysis, inspectional analysis, or the most general method which consists of deducing the conditions of similitude from the Navier-Stokes equation and the equations of elasticity.

In hydraulic textbooks, similitude is usually presented as a natural consequence of dimensional analysis. Then the similitude of Froude, Reynolds, Mach, Weber, Cauchy, etc. are presented. But no scale model has ever been built according to an equality of Webe l' numbe 1', or even of Reynolds number. Practically, the Reynolds similitude does not exist in scale model technology, but the similitude of head loss, a function of the Reynolds number, is sometime s adjusted.

Because of the inherent inadequacies of dimensional analysis, the engineer will often rather deduce the laws of similitude by "inspectional analysis" (Le Mehaute, 1962). Knowledge and understanding of the phe-nomena under study is necessary for deducing the rules of similitude, for neglecting phenomena of secondary importance, and for deciding the relative importance of scale effects. Inspectional analysis is also required for the interpretation of scale model results. The problem needs to be mathematically formulated under a differential form, but does not need to be integrated. The law of similitude can be deduced from the law of

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I + G + P + F + E + C = 0

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f o r c e . O n the s c a l e m o d e l ,

I ' + G ' + P ' + F ' + E ' + C ' = 0

u r e

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T H E R E S P O N S E O F A F L O A T I N G E L A S T I C S H E E T T O W A V E A C T I O N

^ c a t a s t r o p h i c Ice f l o e s i n the A r c t i c have o c c a s i o n a l l y b e e n s p l i t b > ^^-^ns o f a i r f i s s u r e s . T h e s e f i s s u r e s s e e m t o be c a u s e d b y r a p i d v a r i a t 3 - s w e l l t e m p e r a t u r e , s u d d e n changes i n b a r o m e t r i c p r e s s u r e , o r s e a r m o s t l y g e n e r a t e d b y d i s t a n t s t o r m s . T h e f a c t t h a t t h e s e c r a c k s a p p »

d u r i n g t h e w i n t e r s e e m s to s u p p o r t the i d e a t h a t d i s t a n t s t o r m s a r e s o m e -t i m e s r e s p o n s i b l e f o r -the f i s s u r e s . The p u r p o s e o f t h i s s t u d y was t o o b t a i n t h e o r e t i c a l a n d e x p e r i i ^ ^ ' ^ * ^ ^ i n f o r m a t i o n o n the d y n a m i c i n t e r a c t i o n b e t w e e n f l o a t i n g i c e s h e e t s a n d g r a v i t y w a v e s . A l t h o u g h s p e c i f i c a l l y d e s i g n e d f o r t h e s t u d y o f i c e fl-O*;' i t i s e v i d e n t t h a t b o t h t h e o r y and e x p e r i m e n t s w i l l a p p l y t o the s t u d y ° ^ a n y k i n d o f e l a s t i c m a t e r i a l . F r o m the t h e o r e t i c a l v i e w p o i n t , the p r o b l e m c o n s i s t s o f f i n d i n g the t h r e e v e l o c i t y p o t e n t i a l s c o r r e s p o n d i n g t o the i n c i d e n t and r e f l e c t s w a v e m o t i o n (^j , the m o t i o n u n d e r the i c e f l o e <j)2 and the t r a n s m i t * ® m o t i o n (t>3 ( v <j'i = o, i = l , 2, 3)' and t o m a t c h the se v e l o c i t y p o t e n t i a l s a l o n g a v e r t i c a l a t tlie j u n c t i o n s of the n e i g h b o r i n g r e g i o n s b y e q u a t i n g , p r e s s u r e and h o r i z o n t a l v e l o c i t y ( H e n d r i c k s o n , W e b b , and Q u i g l e y , ' (see F i g u r e l ) . T h e f r e e s u r f a c e c o n d i t i o n f o r (^-^ and <\>^ i s , of c o u r s e , t h ® C a u c h y - P o i s s o n c o n d i t i o n : (FR., + g ' t ) = 0 . F o r <\> „ , t h i s c o n d i t i o n has t o be r e p l a c e d b y p<}>^ + p g w = - p ( x , t )

w h e r e w i s the p l a t e d e f l e c t i o n and p i s the p r e s s u r e e x e r t e d b y t t i ® p l a t e o n t h e w a t e r . The e q u a t i o n g o v e r n i n g the m o t i o n o f the p l a t e

D 9 x •T + P h 4 ^s 9 t ' P ( x , t ) w h e r e D = E h " 12 (1 - v ' )

i s the f l e x u r a l r i g i d i t y of the e l a s t i c p l a t e , E the e l a s t i c m o d u l u s c> ^ the p l a t e , h the p l a t e t h i c l m e s s , v the P o i s s o n ' s r a t i o o f p l a t e m-S-'^^ pg t h e d e n s i t y o f t h e i c e , and p the f l u i d d e n s i t y . T h e p l a t e f i b e r s t r e s s i s g i v e n b y r i a l , E h 2 ( 1 - v^) d w A c c o r d i n g to t h e F r o u d e s i m i l i t u d e , the s c a l e f o r the f o r c i n g f u n c t i p ( x , t ) i s the l i n e a r s c a l e \ . C o n s e q u e n t l y the s c a l e f o r

m u s t a l s o be \ , s c a l e f o r D i s and P s ^ dx " dt" M o r e s p e c i f i c a l l y , the s c a l e f o r a4 O W _{so the} T h i s c a n be a c h i e v e d b y v a r y i n g E , h , o r v . T h e s i m p l e s t m e t h o d c o n s i s t s , o f c o u r s e , o f c h o o s i n g the s c a l e f o r h t o be l i n e a r , k e e p i n g the s a m e v a l u e f o r v , i n w h i c h case one f i n d s Ej^^^^g-j^ ~ - A f i e l d •'^ ^• I t i s t h e n e a s i l y v e r i f i e d t h a t t h e s c a l e f o r f i b e r s t r e s s i s V. C o n s i d e r i n g n o w the t e r m Pg h { 9 ^ w / 9 t ^ ) , i t i s s e e n t h a t i t i s s u f f i c i e n t t h a t p = p f o r a c h i e v i n g a c o m p l e t e F r o u d e - C a u c h y . , ^ m o d e l A f i e l d s i m i l i t u d e . C o n s i d e r i n g t h a t the e l a s t i c i t y m o d u l u s f o r i c e i s a r o u n d l O ^ l b / i n ^ , i t w a s f o u n d t h a t a sheet o f p o l y e t h y l e n e w i t h a r e l a t i v e d e n s i t y of 0. 92, a m o d u l u s o f e l a s t i c i t y o f 2x10^^ I b / i n ^ , and a P o i s s o n ' s r a t i o o f 0. 43 w i l l f u l f i l l the s i m i l i t u d e c o n d i t i o n a t a s c a l e of \ = 1/50. E x p e r i m e n t s w e r e p e r f o r m e d i n the N E S C O w a v e t a n k w i t h a s h e e t o f p o l y e t h y l e n e 16 f e e t l o n g , f o u r f e e t w i d e a n d t w o i n c h e s t h i c k , r e p r e s e n t i n g a t w o - d i m e n s i o n a l i c e f l o e 800 f e e t l o n g and 8. 3 f e e t t h i c k ( L e M é h a u t é and S z c z e s n y , 1965) (see F i g u r e 2). T h i s s h e e t w a s m o n i -t o r e d b y s -t r a i n g a g e s , p r e s s u r e -t r a n s d u c e r s and d i s p l a c e m e n -t p i c k - u p d e v i c e s and s u b j e c t e d t o w a v e s w i t h p e r i o d s f r o m one s e c o n d t o f o u r s e c o n d s c o r r e s p o n d i n g to o c e a n w a v e s w i t h p e r i o d s r a n g i n g f r o m s e v e n to 28 s e c o n d s . T h e f i g u r e s i n c l u d e d show the i n s t a l l a t i o n ( F i g u r e 2) a n d s o m e e x a m p l e s o f t h e e x p e r i m e n t a l r e s u l t s ( F i g u r e 3 ) . I t w a s c o n -c l u d e d t h a t the p r o b a b i l i t y of i -c e -c r a -c k s due to w a v e a -c t i o n a l o n e i s v e r y r e m o t e . U N D E R W A T E R O I L S T O R A G E T A N K As o i l d r i l l i n g s t r u c t u r e s a r e i n s t a l l e d f a r t h e r a n d f a r t h e r o f f -s h o r e i t m a y b e c o m e u n e c o n o m i c a l t o p i p e l i n e c r u d e o i l f r o m w e l l -s f o r s t o r a g e i n t a n k s o n s h o r e . H e n c e u n d e r w a t e r o i l s t o r a g e t a n k s a n d t a n k e r s a r e n o w c o n s i d e r e d as a n a l t e r n a t i v e e n g i n e e r i n g s o l u t i o n . A m e m b r a n e - t y p e o i l s t o r a g e t a n k a n c h o r e d at the b o t t o m of the o c e a n at d e p t h s o f 100 t o 200 f e e t i s c o n s i d e r e d . A t s u c h d e p t h s t h e w a v e m o t i o n i s c o n s i d e r a b l y r e d u c e d , b u t s t i l l n o t n e g l i g i b l e . A s i d e of m a n y o t h e r a d v a n t a g e s , a f l e x i b l e r u b b e r t y p e s t r u c t u r e w i l l r e d u c e the w a v e f o r c e s c o n s i d e r a b l y . H o w e v e r , s o m e r e s o n a n c e e f f e c t s m u s t be e l i m i n a t e d . Such an a n c h o r e d bag w i l l l o o k l i k e a p i l l o w (see F i g u r e 4 ) .

F r o m a t h e o r e t i c a l v i e w p o i n t , the p r o b l e m c o n s i s t s o f d e t e r m i n i n g the m o t i o n o f t w o l i q u i d s o f d i f f e r e n t d e n s i t i e s s e p a r a t e d b y a m e m b r a n e . I t i s a c l a s s i c a l p r o b l e m i n h y d r o d y n a m i c s . I n t h i s p a r t i c u l a r c a s e , the

w, MAXIMUM A B S O L U T E D E F L E C T I O N , in TRAILING E D G E •o 0 20 O 8 8 n \c o o c 8 8 S 8 8 8 n in 8 8 o 8 n r\f\ • -8 -6 -4 -2 0 2 4 6 8 C O O R D I N A T E D I S T A N C E O F S H E E T , ff L E A D I N G E D G E S , . M A X I M U M A B S O L U T E BENDING S T R E S S , l b s / i n ^ TRAILING E D G E O 3 0 p ^ 0 0 0 1 -C n n o o c c 5 VJ .u — n rr O C ) 0 O O U.D n A 0 " c c 3 3 \J.H O O O 0 -8 -6 -4 -2 0 2 4 6 8 C O O R D I N A T E D I S T A N C E O F S H E E T , ft L E A D I N G E D G E FIGURE 3

p r o b l e m can be a p p r o x i m a t e d as f o l l o w s . Since the b a g i s f i x e d lilce a p i l l o w o n the b o t t o m , we w i l l a p p l y the a s s u m p t i o n of the l i n e a r l o n g w a v e t h e o r y t o the o i l m o t i o n . I f i s the a v e r a g e o i l d e p t h and r\ the v a r i a t i o n of o i l l e v e l a r o u n d , the e q u a t i o n s g o v e r n i n g the o i l m o t i o n m a y be w r i t t e n n + D ( u + V ) = 0 't o ^ x y ' p u^ = - p t X p V - - p ' P t w h e r e p * = P H + Pw^*) + ^ p ( t ) • H e r e p i s the h y d r o s t a t i c p r e s s u r e : t l P H = - P o g ^ + P w g ^ - ^Pw - Po) g ( ° o + ^)

w h e r e p ^ and p^ a r e the sea w a t e r and c r u d e o i l d e n s i t i e s r e s p e c t i v e l y a n d d the w a t e r d e p t h . P v / { t ) i s the p r e s s u r e v a r i a t i o n due t o t h e w a t e r w a v e . I t w i l l be a s s u m e d t h a t t h i s w a v e r e m a i n s u n d i s t u r b e d b y the b a g l o c a t e d at the b o t t o m , i . e . ,

c o s h k D , ^ ~ H C O S ( o - t - k x ) „ I f P = P g - r r - r ^ c o s h (o-t - k x ) = p g ' , . -, = B cos (crt - k d j '^w ^ w ^ 2 c o s h k d ' ' '^w 2 c o s h l c d

A p i s the p r e s s u r e d i f f e r e n c e b e t w e e n the t w o s i d e s of the m e m b r a n e . I f T i s the t e n s i o n i n the m e m b r a n e ,

m

A p = T (r^ + ^ ) = ^ = S i i ^ m ^ ' x x ' y y ' R R

w h e r e { r | x x + 'Hyy) i s the t o t a l c u r v a t u r e l / R , e the m e m b r a n e t h i c k -n e s s , a-nd N t h e s t r e s s p e r u -n i t a r e a . I -n the case of a b a g i -n the shape of a c y l i n d e r ,

n = 0 a n d ri = ^ , 'xx 'yy ^

a f t e r s o m e s t r a i g h t f o r w a r d o p e r a t i o n s i n w h i c h TI i s e l i m i n a t e d and i s s u p p o s e d t o be s m a l l , one o b t a i n s "^tt " "^xx " ^ ' ' " ^ - ^^^) w h e r e L P g J L P O « ' Po . I t i s r e c o g n i z e d t h a t Cp i s the w a v e c e l e r i t y i n a n e l a s t i c p i p e and C i s the w a v e c e l e r i t y o f a d e n s i t y w a v e . ^

T h i s e q u a t i o n has a n u m b e r of s o l u t i o n s . I n the case o f a n i n f i n i t e l y l o n g b a g , a p r o g r e s s i v e w a v e w i l l be g e n e r a t e d w i t h i n the b a g and r e s o n a n c e w i l l o c c u r w h e n the s u r f a c e w a v e c e l e r i t y e q u a l s the w a v e c e l e r i t y C g w i t h i n the b a g . I n the case of a b a g of f i n i t e l e n g t h , a s t a n d i n g w a v e m o t i o n i s s u p e r i m p o s e d t o the p r o g r e s s i v e w a v e m o t i o n w h i c h s t i l l e x i s t s b e c a u s e the b o u n d a r i e s a t the t w o ends of the tanlc a r e p a r t l y m o v a b l e . T h e r e s o n a n c e c o n d i t i o n f o r t h e s t a n d i n g w a v e o c c u r s w h e n T = 2 i / n C g w h e r e i i s the l e n g t h of the b a g and n i s an i n t e g e r .

A m o r e c o m p l e t e t h r e e - d i m e n s i o n a l t h e o r y i n w h i c h u and v a r e e l i m i n a t e d i n s t e a d of rj w i l l g i v e :

Pw - Po „ 2 ^ T D„ 4 D„ 2

w h e r e

9 x a x 9 y 9 y

and p^ i s the p r e s s u r e o n the o u t s i d e of the m e m b r a n e .

Some i n t e r e s t i n g s o l u t i o n s can be d e d u c e d f r o m t h i s e q u a t i o n once the b o u n d a r y c o n d i t i o n s a r e s p e c i f i e d . The s i m i l i t u d e c o n d i t i o n m a y a l s o be d e d u c e d f r o m t h i s e q u a t i o n and w i l l g i v e the s a m e r e s u l t s as t h e m o r e a p p r o x i m a t e t h e o r y a b o v e .

and t h e e f f e c t of n o n - l i n e a r i t y due to the l a w o f t e n s i o n - d i s p l a c e m e n t of the m e m b r a n e have a l s o b e e n i n v e s t i g a t e d t h o r o u g h l y ( L e M é h a u t é , 1964). T h e m o s t s i g n i f i c a n t r e s u l t s a r e p r e s e n t e d on the f o l l o w i n g s c h e m a t i c g r a p h s ( F i g u r e 5 ) . T h e m a x i m u m s t r e s s has b e e n c a l c u l a t e d t h e o r e t i c a l l y i n the case of r e s o n a n t and n o n - r e s o n a n t m o t i o n . T h e s e t h e o r e t i c a l d e v e l o p m e n t s , a l t h o u g h i n t e r e s t i n g , a r e not the p u r p o s e o f t h i s p a p e r , w h i c h i s the d e t e r m i n a t i o n of the l a w of s i m i l i t u d e .

W i t h o u t any f u r t h e r d e v e l o p m e n t , i t i s seen t h a t the s i m U - i t u d e r e q u i r e m e n t s a r e : w h i c h g i v e s D \ and w h e r e s u b s c r i p t s m a n d f r e f e r t o m o d e l and f i e l d r e s p e c t i v e l y . T h i s c o n d i t i o n m a y be f u l f i l l e d i f the s c a l e m o d e l t a n k i s f i l l e d w i t h k e r o s e n e , a l c o h o l , o r g a s o l i n e , i n o r d e r to c o r r e c t f o r t h e d e n s i t y o f f r e s h w a t e r i n s t e a d o f sea w a t e r . F o r e l a s t i c i t y . 'e E • = \^ " e E " = \^ p R . O m L'^o . f w h i c h g i v e s I . e . , i n p r a c t i c e , e E ^ ^2 e j : P„ P.. e E I = e E l

o r m o r e g e n e r a l l y , i n the case w h e r e the l a w of t e n s i o n - d i s p l a . c e m e n t i s n o t l i n e a r , one s h o u l d have T x e j ^ ^ = X'^ T x e | £ . F o r the p a r t i c u l a r m a t e r i a l u n d e r c o n s i d e r a t i o n (a F i r e s t o n e p r o d u c t 1/4 i n c h t h i c k ) i t w a s f o u n d t h a t the s i m i l i t u d e c o n d i t i o n f o r e l a s t i c i t y w i l l be r e s p e c 1 ; e d p r o -v i d e d one uses a -v e r y t h i n r u b b e r ( l / 6 4 i n c h ) f o r the s c a l e m o d e l a t a s c a l e of 1/30.

DISPLACEMENT IN THE TANK

f o r the m o t i o n w i t h i n t h e tanl^ w i l l a l s o be s a t i s f i e d , m a k i n g the F r o u d e -C a u c h y — Re y n o l d s s i m i l i t u d e c o m p a t i b l e . I n d e e d , i t i s e a s i l y v e r i f i e d t h a t X V 3 / 2 s i n c e the k i n e m a t i c v i s c o s i t y f o r o i l i s 10~ . 5 X 10" f o r g a s o l i n e . 1. 6 X 10 f o r a l c o h o l , and M O H O L E RISER A N D T E X A S T O W E R

The a n a l y s i s of the d y n a m i c s of the M o h o l e r i s e r has b e e n c a r r i e d out i n g r e a t d e t a i l ( G a l e f , et a l , 1965). A s c a l e m o d e l s t u d y o f the r i s e r and f l o a t i n g p l a t f o r m has b e e n c o n s i d e r e d f o r m e a s u r i n g the f o r c e s o n the r i s e r n e a r the p l a t f o r m w h e r e t h e y a r e the g r e a t e s t . The d i a g r a m s i r i F i g u r e 6 a r e s e l f e x p l a n a t o r y .

T h e m o t i o n of the p l a t f o r m u n d e r w a v e a c t i o n obeys the F r o u d e s i m i l i t u d e p r o v i d e d m a s s and v o l u m e a r e d i s t r i b u t e d a c c o r d i n g l y . I t i s e v i d e n t t h a t the m o t i o n of the deep p a r t of t h e r i s e r w i l l n o t be i n t o o good s i m i l i t u d e due t o s c a l e e f f e c t s . B u t the c o u p l i n g f o r c e b e t w e e n the r i s e r and the p l a t f o r m , w h e r e a c t u a l l y the l a r g e s t f o r c e s a r e e x e r t e d , can be s t u d i e d i n s i m i l i t u d e of e l a s t i c i t y . A l t h o u g h t h e s e r u l e s of s i m i l i t u d e c a n be d e d u c e d f r o m the g e n e r a l e q u a t i o n s as e s t a b l i s h e d i n the p r e -v i o u s l y m e n t i o n e d r e p o r t , -v/e w i l l s i m p l y c o n s i d e r the e q u a t i o n f o r t h e m o m e n t at the t o p of the r i s e r , i . e . , at the c o n n e c t i o n b e t w e e n the r i s e r and t h e p l a t f o r m . T h i s m o m e n t i s M e ^f^~Êl + H F R w h e r e 0 R I H F s the a n g l e of d e v i a t i o n of the r i s e r s the t e n s i o n s t h e i n e r t i a

s the monaent due to the h y d r o d y n a m i c f o r c e s ( d r a g a n d i n e r t i a l f o r c e s ) .

I n e r t i a l f o r c e s due t o w a t e r w i l l be i n r e l a t i v e l y g o o d s i m i l i t u d e p r o v i d e d the e x t e r n a l r i s e r d i a m e t e r s a r e r e l a t e d b y t h e g e o m e t r i c s c a l e . F o r p r a c t i c a l r e a s o n s , the c r o s s s e c t i o n o f t h e r i s e r , w h i c h i s an a n n u l u s , has t o be r e p r o d u c e d as a f u l l d i s k i n the m o d e l (see n e x t p a g e ) . The i n e r t i a l f o r c e due t o the m a s s of the r i s e r i t s e l f w i l l be i n s i m i l i t u d e p r o v i d e d

the d e n s i t y of t h e m o d e l e q u a l s the a v e r a g e d e n s i t y o f the r i s e r , i n c l u d i n g the s t e e l a n n u l u s and the w a t e r i n s i d e . T h u s

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w h e r e p i s the d e n s i t y and A the c r o s s s e c t i o n . S u b s c r i p t m r e f e r s to t h e m o d e l , w to the f l u i d w i t h i n the r i s e r , and R t o the s t e e l a n n u l u s of the r i s e r . I t i s p o i n t e d out t h a t a r e l a t i v e l y l a r g e e r r o r i n p has

o n l y a s e c o n d a r y e f f e c t on the r i s e r b e h a v i o r . ^

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The d r a g f o r c e w i l l n o t be i n good s i m i l i t u d e , b u t t h i s e f f e c t i s a l s o n e g l i g i b l e n e a r the p l a t f o r m . The C a u c h y s i m i l i t u d e i s c o m p a t i b l e w i t h the F r o u d e s i m i l i t u d e p r o v i d e d : ( S u b s c r i p t m r e f e r s to m o d e l , R to r i s e r . ) M n m

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T E l l T E I | ^ T h e s c a l e m o d e l c h a r a c t e r i s t i c s f o r a f u l l d i s k a r e : , A i T = TTR ^ E m m ü m A i w h e r e —^ i s the r e l a t i v e e l o n g a t i o n , and m . r 4 4 w h i l e f o r the a n n u l u s r i s e r t h e y a r e T S 2 . R ^ A R ^ E - ^ and

I ^ . R , ^ A R ^ I n s e r t i n g th.ese e x p r e s s i o n s w i t h i n the s i m i l i t u d e c o n d i t i o n g i v e s , w i t h ( A i / i ) = ( A i ^ A ) : I t i s i n t e r e s t i n g to m e n t i o n t h a t the s i m i l i t u d e c o n d i t i o n f o r t e n s i o n r e -q u i r i n g ( T ^ / T j ^ ) = \ ^ g i v e s : T h e f i r s t c o n d i t i o n f o r E^^ p r e v a i l s i n s o f a r as the r e a c t i o n of the p l a t f o r m u p o n the r i s e r i s c o n c e r n e d . T h i s c o n d i t i o n i s a c t u a l l y a l s o a good a p p r o x i m a t i o n f o r r e p r o d u c i n g the m o t i o n of the r i s e r i n s i m i l i t u d e . I n d e e d , i n t h i s case i t i s seen t h a t i n e r t i a l f o r c e s , b e n d i n g f o r c e s , and t e n s i o n a r e a l m o s t i n s i m i l i t u d e f r o m the p l a t f o r m to the b o t t o m . I t has b e e n f o u n d t h a t a w i r e of p o l y t e t r a f l u o r e t h y l e n e , r e a d i l y a v a i l a b l e o n the m a r k e t , f u l f i l l s a l l the s i m i l i t u d e r e q u i r e m e n t s at a s c a l e of a p p r o x i m a t e l y 1/200. I n d e e d , i t s r e l a t i v e d e n s i t y i s 2. 2 and i t s m o d u l u s o f e l a s t i c i t y i s 0. 58 X 10^ p s i (the s c a l e w i l l be d e t e r m i n e d e x a c t l y a f t e r t h e s e v a r i a b l e have b e e n m e a s u r e d o n a s a m p l e ) .

A t s u c h a s c a l e , the w a t e r d e p t h at the w e l l l o c a t i o n u n d e r the w a v e t a n k w i l l have t o be 90 f e e t . I f one m i g h t c o m p r o m i s e b y s t o p p i n g the d e p t h on the f l o o r of the b u i l d i n g , i t w o u l d be e q u i v a l e n t t o a p p r o x i -m a t e l y 1000 f e e t i n the o c e a n . A n o t h e r s o l u t i o n w i l l c o n s i s t of d i s t o r t i n g the r i s e r v e r t i c a l l y , i . e . , b y m a k i n g ~ ^^j^ x \ , and A i A A i _ m _ R i m s u c h t h a t A i T h e n , = ^ R 2 ^ ^ I f t h e d e p t h o f t h e s c a l e m o d e l i s s i x f e e t , f o r e x a m p l e , A w i l l be e q u a l to 15.

S i m i l a r r u l e s of s i m i l i t u d e c a n be deduced as e a s i l y f o r s t u d y i n g t h e r e s p o n s e of a T e x a s T o w e r t y p e of s t r u c t u r e t o w a v e a c t i o n . The l i m i t a t i o n o f t h i s k i n d o f s t u d y i s , o f c o u r s e , d e t e r m i n e d b y the r e l a t i v e i m p o r t a n c e o f t h e d r a g f o r c e , w h i c h has t o be s m a l l b y c o m p a r i s o n w i t h i n e r t i a l f o r c e . F o r t h i s r e a s o n , t h i s k i n d o f s t u d y w i l l be s u i t a b l e f o r r e l a t i v e l y l a r g e p i l e s . C O N C L U S I O N I t has b e e n s e e n t h r o u g h a m i n i m u m o f i n s p e c t i o n a l a n a l y s i s how i t i s p o s s i b l e t o m a k e the C a u c h y s i m i l i t u d e c o m p a t i b l e w i t h the F r o u d e s i m i l i t u d e . T h e f e w e x a m p l e s w h i c h have b e e n t r e a t e d a r e o n l y r e p r e -s e n t a t i v e c a -s e -s f r o m m a n y p r o b l e m -s . A w h o l e f a m i l y of p r o b l e m -s o f h y d r o e l a s t i c i t y can be s u c c e s s i v e l y s t u d i e d b y a p p l i c a t i o n o f the F r o u d e -C a u c h y s i m i l i t u d e : h y d r o d y n a m i c i m p a c t , w a v e s l a m m i n g , w a v e b r e a k i n g on a v e r t i c a l w a l l , h y d r o e l a s t i c v i b r a t i o n s , e t c . E a c h p r o b l e m needs to be a n a l y z e d i n d e p e n d e n t l y i n o r d e r t o a s s e s s the v a l u e and the l i m i t of a s c a l e m o d e l s t u d y . I n m o s t cases the s c a l e w i l l have t o be c h o s e n as a f u n c t i o n o f the e l a s t i c p r o p e r t i e s of the m a t e r i a l . H o w e v e r , the l a r g e v a r i e t y o f p l a s t i c o r r u b b e r t y p e m a t e r i a l s o f f e r s a w i d e r a n g e of p o s s i b l e s c a l e s .

The d e v e l o p m e n t of s o l i d s t a t e p h y s i c s now p e r m i t s the m o n i t o r -ing o f s c a l e m o d e l s t r u c t u r e s b y h i g h l y s e n s i t i v e s t r a i n gages and p r e s s u r e t r a n s d u c e r s . I t has b e e n seen t h a t e v e n p o l y e t h y l e n e can be e q u i p p e d w i t h s t r a i n gages w i t h o u t f e a r of r e d u n d a n c y : o n l y t w o s t r a i n gages out of Z4 w e r e l o s t d u r i n g the t e s t s .

The o l d m e t h o d c o n s i s t s of m e a s u r i n g the p r e s s u r e f o r c e s and c a l c u l a t i n g the s t r u c t u r a l f o r c e s . V e r y o f t e n , the b e s t m e t h o d i n s c a l e m o d e l t e c h n o l o g y s h o u l d be the d i r e c t m e a s u r e m e n t o f the s t r e s s w i t h i n h y d r a u l i c s t r u c t u r e s b y a p p l i c a t i o n o f the F r o u d e - C a u c h y s i m i l i t u d e . A C K N O W L E D G E M E N T S T h e i c e f l o e s t u d y was s p o n s o r e d b y the U . S . N a v a l C i v i l E n g i n e e r i n g L a b o r a t o r y u n d e r c o n t r a c t N B y - 3 2 2 5 6 u n d e r the d i r e c t i o n of M r . J . D y k i n s . T h e s t u d y of a n u n d e r w a t e r m e m b r a n e w a s c a r r i e d out f o r G l o b a l M a r i n e E x p l o r a t i o n C o m p a n y and F i r e s t o n e T i r e and R u b b e r C o m p a n y u n d e r the g u i d a n c e of M r . G . L o c l w o o d . T h e M o h o l e s t u d y w a s c a r r i e d out as a f e a s i b i l i t y s t u d y f o r B r o w n and R o o t u n d e r the d i r e c t i o n o f M r . A . G a l e f as N E S C O p r o j e c t m a n a g e r .

T h e w r i t e r w o u l d l i k e to a c k n o w l e d g e the m e m b e r s of the s t a f f w h o p e r f o r m e d o n t h e v a r i o u s p r o j e c t s , and i n p a r t i c u l a r , D r s . J . H e n d r i c k s o n and R . K o h , M e s s r s . A . G a l e f and A . K i n a n , and M i s s L . W e b b .

R E F E R E N C E S

G a l e f , A , , e t a l (1965). D y n a m i c S t r e s s A n a l y s i s o f the M o h o l e R i s e r S y s t e m : N E S C O R e p o r t S N - 1 8 3 .

H e n d r i c k s o n , J . , W e b b , L . , and Q u i g l e y , R . (1962). Study of N a t u r a l F o r c e s A c t i n g o n F l o a t i n g I c e F i e l d s : U . S . N a v a l C i v i l E n g i n e e r i n g L a b o r a t o r y R e p o r t . L e M é h a u t é , B . (1962). T h e o r y , E x p e r i m e n t s , A P h i l o s o p h y of H y d r a u l i c s J o u r n a l of t h e H y d r a u l i c s D i v i s i o n , P r o c e e d i n g s o f t h e A m e r i c a n S o c i e t y o f C i v i l E n g i n e e r s , V o l . 88, N o . H Y 1. L e M é h a u t é , B . (1964). Study of a M e m b r a n e T y p e U n d e r w a t e r O i l S t o r a g e T a n k : N E S C O R e p o r t SN 2 2 1 . L e M é h a u t é , B . , and S z c z e s n y , C. (1965). E x p e r i m e n t a l I n v e s t i g a t i o n of I c e F l o e u n d e r Wave A c t i o n : U . S . N a v a l C i v i l E n g i n e e r i n g L a b o r a t o r y R e p o r t .

A m A P P E N D I X I N O T A T I O N c r o s s s e c t i o n of the m o d e l of the r i s e r A ^ c r o s s s e c t i o n o f t h e m a t e r i a l o f t h e r i s e r

•A^ c r o s s s e c t i o n of the f l u i d w i t h i n the r i s e r

B a m p l i t u d e C-g w a v e c e l e r i t y w i t h i n the o i l s t o r a g e t a n k D f l e x u r a l r i g i d i t y D^ a v e r a g e o i l d e p t h e t h i c k n e s s of the m e m b r a n e E e l a s t i c i t y m o d u l u s g g r a v i t y a c c e l e r a t i o n h t h i c k n e s s o f t h e p l a t e H w a v e h e i g h t H F m o m e n t of the h y d r o d y n a m i c f o r c e I i n e r t i a of the r i s e r k i l e n g t h of the o i l s t o r a g e t a n k L w a v e l e n g t h M m o m e n t of the r i s e r o n the p l a t f o r m n i n t e g e r p p r e s s u r e Pf^ h y d r o s t a t i c p r e s s u r e p^ p r e s s u r e o u t s i d e the m e m b r a n e p ^ p r e s s u r e due t o w a t e r w a v e s R r a d i u s of c u r v a t u r e of the m e m b r a n e 2TT L

S f i b e r s t r e s s X t t i m e v a r i a b l e T w a v e p e r i o d T t e n s i o n of the m e m b r a n e m T t e n s i o n of the r i s e r u h o r i z o n t a l c o m p o n e n t of v e l o c i t y i n the OX d i r e c t i o n V h o r i z o n t a l c o m p o n e n t of v e l o c i t y i n t h e OY d i r e c t i o n w p l a t e d e f l e c t i o n X h o r i z o n t a l c o o r d i n a t e y h o r i z o n t a l c o o r d i n a t e z v e r t i c a l c o o r d i n a t e

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•n m r v a r i a t i o n of o i l l e v e l a r o u n d D o angle of d e v i a t i o n of the r i s e r l i n e a r s c a l e P o i s s o n ' s r a t i o isiinematic v i s c o s i t y f o r the p r o t o t y p e k i n e m a t i c v i s c o s i t y o n the scale m o d e l p d e n s i t y of w a t e r p d e n s i t y of the m o d e l of the r i s e r '^m p d e n s i t y of the m a t e r i a l of the r i s e r

d e n s i t y of sea w a t e r ; a l s o d e n s i t y of the f l u i d w i t h i n the r i s e r

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