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Directional wave effects on large offshore structures of arbitrary shape Sinha, Sanjay 1985

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DIRECTIONAL WAVE EFFECTS ON LARGE OFFSHORE OF ARBITRARY SHAPE STRUCTURES by SANJAY SINHA B.Sc.(Engg), Ranchi U n i v e r s i t y , Ranchi, I n d i a , 1981 M.Tech., Indian I n s t i t u t e Of Technology, Madras, I n d i a , 1983 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES Department Of C i v i l E ngineering We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA MAY 1985 © SANJAY SINHA, 1985 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department O f C i v i l Engineering The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ^/xdjlr ^rr^L^. D E - 6 <"*/fm ABSTRACT A numerical method i s d e s c r i b e d to study d i r e c t i o n a l wave e f f e c t s on l a r g e o f f s h o r e s t r u c t u r e s of a r b i t r a r y shape, based on an e x t e n s i o n of l i n e a r d i f f r a c t i o n wave theory f o r r e g u l a r waves. A computer program has been developed to compute l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude opera t o r s and hence the l o a d i n g and response s p e c t r a f o r both l o n g - and s h o r t - c r e s t e d random waves. Cosine powered d i r e c t i o n a l spreading f u n c t i o n s which are independent of frequency have been used to account f o r the sh o r t c r e s t e d n e s s of waves. Comparisions of the r e s u l t s f o r l o n g - and s h o r t - c r e s t e d seas show that there i s a s i g n i f i c a n t r e d u c t i o n i n the l o a d i n g , and hence i n the response, due to s h o r t c r e s t e d n e s s of waves. The p r o b a b i l i s t i c p r o p e r t i e s of the components of the l o a d i n g and response are d e s c r i b e d . Since the sea s u r f a c e i s assumed to f o l l o w a Gaussian d i s t r i b u t i o n , these are a l s o random Gaussian v a r i a b l e s . In s h o r t - c r e s t e d waves, the l o a d i n g and response components occur both i n - l i n e and t r a n s v e r s e to the p r i n c i p a l wave d i r e c t i o n . Thus the maximum h o r i z o n t a l l o a d i n g and response may occur i n an a r b i t r a r y h o r i z o n t a l d i r e c t i o n . An a n a l y t i c a l method i s developed to d e s c r i b e a l s o the p r o b a b i l i s t i c p r o p e r t i e s of the maxima of the components and the maxima of t h e i r h o r i z o n t a l r e s u l t a n t s . In the present study, r e s u l t s are d e s c r i b e d f o r a f r e e l y f l o a t i n g box. Comparisons are made with p u b l i s h e d i i r e s u l t s and are found to be q u i t e f a v o u r a b l e . Table of Contents ABSTRACT i i LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS i X NOMENCLATURE X 1. INTRODUCTION 1 1 . 1 GENERAL 1 1.2 LITERATURE REVIEW 2 1 .3 SCOPE OF THE PRESENT STUDY 6 1.4 DESCRIPTION OF METHOD 7 2. LINEAR DIFFRACTION THEORY 11 2.1 INTRODUCTION 11 2.2 DESCRIPTION OF RANDOM WAVES 11 2 . 3 LONG-CRESTED REGULAR WAVES 13 2.3.1 WAVE FORCES 17 2.3.2 BODY MOTIONS 19 2.3.3 NUMERICAL INTEGRATION 22 2.4 LONG-CRESTED RANDOM WAVES 24 2.5 SHORT-CRESTED RANDOM WAVES 26 3. PROBABILISTIC PROPERTIES OF LOADING AND RESPONSE 30 3.1 INTRODUCTION 30 3.2 PROBABILISTIC PROPERTIES OF COMPONENTS 31 3.2.1 FREQUENCY OF UPCROSSING OF COMPONENTS 33 3.3 PROBABILISTIC PROPERTIES OF THE HORIZONTAL RESULTANT OF COMPONENTS 34 3.3.1 A SPECIAL CASE 38 3.4 PROBABILISTIC PROPERTIES OF RESULTANT MAXIMA 39 i v 3.5 PROBABILITY DISTRIBUTION OF EXTREMES OF RESULTANT MAXIMA 45 3.6 HUNTINGTON AND GILBERT'S APPROACH 47 4. RESULTS AND DISCUSSION 49 4.1 LOADING AND RESPONSE SPECTRA 49 4.2 P R O B A B I L I S T I C PROPERTIES OF LOADING AND RESPONSE 54 4.2.1 LOADING AND RESPONSE COMPONENTS 55 4.2.2 HORIZONTAL RESULTANTS OF LOADING AND RESPONSE 56 4.3 A DESIGN PROCEDURE ..58 4.3.1 A WORKED EXAMPLE 59 5. CONCLUSIONS 61 REFERENCES 63 APPENDIX I . GREEN'S FUNCTION 67 APPENDIX I I . PROBABILITY DENSITY FUNCTION OF RESULTANT ...71 V LIST OF TABLES Table page 2.1 Normalizing f a c t o r of d i r e c t i o n a l s p r e a d i n g 73 f u n c t i o n . 4.1 Load r e d u c t i o n f a c t o r R^. i n s h o r t - c r e s t e d 74 seas f o r d i f f e r e n t s v a l u e s . 4.2 Response r e d u c t i o n f a c t o r R^j i n 75 s h o r t - c r e s t e d seas f o r d i f f e r e n t s v a l u e s . 4.3 Expected val u e s of the extremes of the 76 maxima of l o a d i n g . 4.4 Expected values of the extremes of the 77 maxima of response. v i LIST OF FIGURES FIGURE page 2.1 Sketch of a u n i - d i r e c t i o n a l wave spectrum. • 78 2.2 Sketch of a d i r e c t i o n a l wave spectrum. 78 2.3 D e f i n i t i o n sketch of a f l o a t i n g body showing 79 component motions. 2.4 D e f i n i t i o n sketch of the i n c i d e n t wave 79 d i r e c t i o n . 2.5 D i r e c t i o n a l spreading f u n c t i o n f o r d i f f e r e n t 80 values of s. 4.1 D e f i n i t i o n sketch of a f l o a t i n g box. 80 4.2 Loading t r a n s f e r f u n c t i o n s i n lo n g - and 81 s h o r t - c r e s t e d seas. 4.3 Response amplitude o p e r a t o r s i n l o n g - and 84 s h o r t - c r e s t e d seas. 4.4 U n i - d i r e c t i o n a l wave spectrum used i n 87 computations. 4.5 Loading s p e c t r a i n long- and s h o r t - c r e s t e d 88 seas. 4.6 Response s p e c t r a i n lo n g - and s h o r t - c r e s t e d 91 seas. 4.7 Expected val u e s of the extreme of the maxima 94 of l o a d i n g and response i n l o n g - and s h o r t - c r e s t e d seas. 4.8 Expected v a l u e s of the extreme of the maxima 95 of l o a d i n g and response components i n s h o r t - c r e s t e d seas. v i i 4.9 Expected values of the extremes of the maxima 96 of h o r i z o n t a l r e s u l t a n t s i n s h o r t - c r e s t e d seas. 4.10 Frequency of u p c r o s s i n g of components of 97 l o a d i n g and response i n l o n g - and s h o r t - c r e s t e d seas. 4.11 Frequency of u p c r o s s i n g of h o r i z o n t a l 97 r e s u l t a n t s i n s h o r t - c r e s t e d seas. 4.12 Comparision of present method with t h a t of 98 Huntington and G i l b e r t (1979) f o r a s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r . v i i i ACKNOWLEDGEMENTS In p r e s e n t i n g t h i s t h e s i s , I wish to express my s i n c e r e g r a t i t u d e to Dr M. de St. Q. Isaacson f o r h i s i n v a l u a b l e guidance and suggestions d u r i n g the e n t i r e course of the t h e s i s work. I a l s o thank a l l those who have d i r e c t l y or i n d i r e c t l y helped me i n completing t h i s t h e s i s s u c c e s s f u l l y . F i n a l l y , f i n a n c i a l support i n the form of a resea r c h a s s i s t a n t s h i p from the N a t u r a l Sciences and En g i n e e r i n g Research C o u n c i l , Canada, i s very much a p p r e c i a t e d . i x NOMENCLATURE c h a r a c t e r i s t i c body l e n g t h , matrix c o e f f i c i e n t s , see equation (2.26), matrix c o e f f i c i e n t s , see equation (2.31), h y d r o s t a t i c s t i f f n e s s matrix c o e f f i c i e n t s , n o r m a l i z i n g f a c t o r , water depth, d i r e c t i o n a l spreading f u n c t i o n , expected v a l u e , source s t r e n g t h d i s t r i b u t i o n f u n c t i o n , zero u p c r o s s i n g frequency, frequency of maxima, maximum value of f or f , x z frequency of u p c r o s s i n g of component, frequency of u p c r o s s i n g of h o r i z o n t a l r e s u l t a n t , f o r c e or moment components, g r a v i t a t i o n a l c o n s t a n t , Green's f u n c t i o n , i n c i d e n t wave h e i g h t , s i g n i f i c a n t wave he i g h t , moment or product of i n e r t i a , m o d i f i e d B e s s e l f u n c t i o n of order zero, Jacobian of t r a n s f o r m a t i o n , wave number, body mass, mass matrix c o e f f i c i e n t s , n-th s p e c t r a l moment of x component, n-th s p e c t r a l moment of y component, number of maxima, number of f a c e t s , hydrodynamic pressure, 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 , p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n , 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 of extreme of maxima, p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n of extreme of maxima, radius of g y r a t i o n , r e d u c t i o n f a c t o r , see equation (2.51), spreading index, see equation (2.44), waterplane area, e q u i l b r i u m body s u r f a c e , wave spectrum, waterplane moments, waterplane moments, i n - l i n e l o a d i n g or response spectrum, t r a n s v e r s e l o a d i n g or response spectrum, c r o s s l o a d i n g or response spectrum, l o a d i n g or response spectrum, time, peak p e r i o d of wave spectrum, t r a n s f e r f u n c t i o n or response amplitude operator, x i X random G a u s s i a n v a r i a b l e i n x d i r e c t i o n , ( x , y , z ) c o o r d i n a t e a x e s , X m maxima o f x, V extreme o f maxima o f x, y random G a u s s i a n v a r i a b l e i n y d i r e c t i o n , z r e s u l t a n t o f x and y components, Z B z c o o r d i n a t e of c e n t r e o f buoyancy, Z G z c o o r d i n a t e of c e n t r e o f g r a v i t y , zm maxima o f z, zm ext r e m e of maxima o f z, 1 v e c t o r of p o i n t ( £ , T } , $ ) . <k a m p l i t u d e o f body m o t i o n , added mass c o e f f i c i e n t s , X j k damping c o e f f i c i e n t s , K r o n e c k e r d e l t a , CcJ wave a n g u l a r f r e q u e n c y , v e l o c i t y p o t e n t i a l , P w a ter d e n s i t y , 0 i n c i d e n t wave d i r e c t i o n , 0o p r i n c i p a l wave d i r e c t i o n , [u] c o r r e l a t i o n m a t r i x , X c o r r e l a t i o n c o e f f i c i e n t between x and y, x i i 1. INTRODUCTION 1 . 1 GENERAL Consid e r a b l e a t t e n t i o n has been p a i d to the development of o f f s h o r e regions i n recent years to comply with world-wide demand f o r energy. The huge reserves of o i l beneath the seabed can alone account f o r 20 percent of the world's t o t a l need. The h o s t i l e environmental c o n d i t i o n s and l a r g e r water depths of 200 - 400 meters have l e d to the development of a v a r i e t y of s t r u c t u r a l concepts. Concrete g r a v i t y p l a t f o r m l o c a t e d i n water depths of about 200 meters i n North Sea provide one such example. The more e f f i c i e n t design of such o f f s h o r e s t r u c t u r e s i s of paramount importance, both from economic as w e l l as from s a f e t y v i e w p o i n t s . Environmental l o a d i n g i s the c r i t i c a l component i n design and wave l o a d i n g i s o f t e n one of the most severe of such l o a d s . O f f s h o r e s t r u c t u r e s which c o n t a i n l a r g e components which modify the i n c i d e n t wave f i e l d are c o n s i d e r e d to l i e i n the d i f f r a c t i o n regime of wave l o a d i n g (e.g. Isaacson, 1979), and the computation of wave l o a d i n g i s then g e n e r a l l y based on l i n e a r d i f f r a c t i o n wave theory (summarized i n Sarpkaya and Isaacson, 1981). The l i n e a r d i f f r a c t i o n wave theory assumes the waves to be l o n g - c r e s t e d ( u n i - d i r e c t i o n a l ) and s i n u s o i d a l . However, r e a l seas are both random and s h o r t - c r e s t e d ( m u l t i - d i r e c t i o n a l with wave energy propagating s i m u l t a n e o u s l y over a range of 1 2 d i r e c t i o n s ) . T h i s d i r e c t i o n a l spreading of waves g e n e r a l l y tends to decrease the wave loads i n comparison to those p r e d i c t e d f o r u n i - d i r e c t i o n a l waves. T h i s r e d u c t i o n i n design loads, obtained by i n c o r p o r a t i n g the e f f e c t s of d i r e c t i o n a l spreading, appears to be a very s i g n i f i c a n t improvement in design procedures. Thus i t i s e s s e n t i a l to develop new design procedures which not only account f o r the randomness of waves but a l s o f o r the d i r e c t i o n a l spreading of waves. 1 .2 LITERATURE REVIEW The major . r e s e a r c h e f f o r t i n t o wave l o a d i n g on l a r g e o f f s h o r e s t r u c t u r e s has c o n s i d e r e d l o n g - c r e s t e d r e g u l a r or random waves. A thorough summary of t h i s i s given by Sarpkaya and Isaacson (1981). On the other hand very l i t t l e a t t e n t i o n has been given to the e f f e c t of the d i r e c t i o n a l spreading of the waves. T h i s l a c k of a t t e n t i o n i s p a r t l y due to the absence of r e l i a b l e measured d i r e c t i o n a l wave s p e c t r a . However, more r e c e n t l y s e v e r a l attempts have been made to i n c o r p o r a t e the e f f e c t s of d i r e c t i o n a l spreading on l o a d i n g and response p r e d i c t i o n s . On the b a s i s of comparisons of measured p a r t i c l e v e l o c i t i e s i n a storm with p r e d i c t i o n s assuming i n turn u n i -and m u l t i - d i r e c t i o n a l waves, F o r r i s t a l l et. al. (1978) have emphasized the importance of a c c o u n t i n g f o r d i r e c t i o n a l spreading of waves. Lovaas (1984) has a l s o emphasized the importance of wave d i r e c t i o n a l i t y on the l o a d i n g and 3 r e s p o n s e o f o f f s h o r e s t r u c t u r e s . When t a k i n g d i r e c t i o n a l s p r e a d i n g i n t o a c c o u n t , t h e fo r m of t h e d i r e c t i o n a l wave s p e c t r u m a d o p t e d i s a p r i m a r y c o n s i d e r a t i o n . Borgman (1969) p r o v i d e d d i f f e r e n t e x p r e s s i o n s f o r d i r e c t i o n a l s p e c t r a f o r use i n o f f s h o r e d e s i g n . He d e s c r i b e d a method f o r e s t i m a t i n g d i r e c t i o n a l wave s p e c t r a f r o m an a r r a y o f wave r e c o r d s , b a s e d on a F o u r i e r s e r i e s a n a l y s i s o f c r o s s - s p e c t r u m d a t a . He c h e c k e d t h e method a g a i n s t a s e t o f computer s i m u l a t e d wave r e c o r d s h a v i n g a known t h e o r e t i c a l d i r e c t i o n a l wave s p e c t r u m and i s shown t o be q u i t e f a v o u r a b l e . Borgman ( 1 9 7 2 ) , a n d more r e c e n t l y O c h i (1982) have r e v i e w e d methods o f e v a l u a t i n g d i r e c t i o n a l wave s p e c t r a f r o m m easured d a t a and methods o f a n a l y s i s o f t h e d i r e c t i o n a l p r o p e r t i e s of wave e n e r g y . P i n k s t e r (1984) has r e v i e w e d t h e v a r i o u s n u m e r i c a l m o d e l s o f wave e l e v a t i o n i n d i r e c t i o n a l s e a s . The above s t u d i e s t r e a t t h e p r o p e r t i e s o f m u l t i - d i r e c t i o n a l waves i n i s o l a t i o n . S e v e r a l a u t h o r s have c o n s i d e r e d t h e i r e f f e c t s on o f f s h o r e s t r u c t u r e s . H u n t i n g t o n & Thompson ( 1 9 7 6 ) , and H u n t i n g t o n (1979) s t u d i e d d i r e c t i o n a l wave e f f e c t s on a l a r g e v e r t i c a l c i r c u l a r c y l i n d e r . They e x t e n d e d l i n e a r wave d i f f r a c t i o n t h e o r y f o r r e g u l a r waves t o t h e c a l c u l a t i o n o f l o a d i n g t r a n s f e r f u n c t i o n s f o r b o t h l o n g -and s h o r t - c r e s t e d random s e a s , and f o u n d a f a v o u r a b l e c o m p a r i s o n between t h e i r t h e o r e t i c a l p r e d i c t i o n s f o r u n i -and m u l t i - d i r e c t i o n a l waves w i t h e x p e r i m e n t a l r e s u l t s . T h e s e 4 authors s t u d i e s i n d i c a t e that the r e d u c t i o n i n l o a d i n g due to d i r e c t i o n a l spreading of waves i s q u i t e s i g n i f i c a n t . Dean (1977) s t u d i e d d i r e c t i o n a l wave e f f e c t s on wave l o a d i n g using a " h y b r i d " method which combines l i n e a r and n o n l i n e a r wave t h e o r i e s . Dean (1977) extended t h i s method to account f o r the case i n which p r i n c i p a l wave d i r e c t i o n v a r i e s with the wave frequency, as may occur near the c e n t r e of a h u r r i c a n e . When t h i s e f f e c t i s taken i n t o account, a f u r t h e r f o r c e r e d u c t i o n i s p r e d i c t e d . Hackley (1979) simulated n u m e r i c a l l y the wave kinematics i n a random d i r e c t i o n a l sea as a f u n c t i o n of time on the b a s i s a l i n e a r Gaussian model and thereby c a l c u l a t e d the t o t a l f o r c e on a p i l e using the Morison equation. He concluded that the p r e d i c t i o n of the extreme peak f o r c e s obtained from d e t e r m i n i s t i c t h e o r i e s tends to be somewhat c o n s e r v a t i v e . Shinozuka et. al. (1979) a l s o s t u d i e d the e f f e c t s of s h o r t c r e s t e d n e s s of waves i n the time domain and a l s o p r e d i c t e d a r e d u c t i o n i n the l o a d i n g and response of o f f s h o r e s t r u c t u r e s . B a t t j e s (1982) s t u d i e d the e f f e c t s of s h o r t c r e s t e d n e s s on wave loads on long h o r i z o n t a l s t r u c t u r e s . Borgman and Y f a n t i s (1981) d e r i v e d e x p r e s s i o n s f o r the sp e c t r a and c r o s s - s p e c t r a of the h o r i z o n t a l components of the f o r c e s on the jacke t s t r u c t u r e s i n d i r e c t i o n a l seas based on the Morison equation and a s p e c i f i e d d i r e c t i o n a l wave spectrum. 5 Lambrakos (1982) i n v e s t i g a t e d the wave loads and movements of a p i p e l i n e l y i n g on the sea f l o o r . The d i r e c t i o n a l seas were represented by a double F o u r i e r s e r i e s , i n v o l v i n g 51 f r e q u e n c i e s and 21 d i r e c t i o n s . He found that the wave-induced loads and response are s t r o n g f u n c t i o n s of the p r i n c i p a l wave d i r e c t i o n , but w i t h i n reasonable l i m i t s not s t r o n g l y dependent on the degree of wave spreading. Teigen (1983) compared the r e s u l t s of p h y s i c a l model t e s t s and computer simulated r e s u l t s f o r a t e n s i o n l e g p l a t f o r m i n l o n g - and s h o r t - c r e s t e d seas. He found a c o n s i d e r a b l e r e d u c t i o n i n the t o t a l energy f o r the main response modes i n s h o r t - c r e s t e d seas compared to those based on l o n g - c r e s t e d seas. However, the extreme values of the responses (maxima or minima) were not found to be s i g n i f i c a n t l y a f f e c t e d by the d i r e c t i o n a l i t y of the waves. D a l l i n g a et.al. (1984) s t u d i e d the e f f e c t s of d i e c t i o n a l spreading of waves on the t r a n s p o r t of jackup p l a t f o r m s on a barge and concluded that the e f f e c t s of wave d i r e c t i o n a l i t y i n i r r e g u l a r waves i s s m a l l . T i c k e l l and Elwany (1979) have extended the Morison equation to analyse the h o r i z o n t a l wave-induced f o r c e s on v e r t i c a l s t r u c t u r a l members i n s h o r t - c r e s t e d seas. They compared p r o b a b i l i s t i c models with numerical s i m u l a t i o n s and with p r e l i m i n a r y experimental d a t a . Huntington and G i l b e r t (1979) p r e d i c t e d the expected values of the extremes of i n - l i n e and t r a n s v e r s e f o r c e and 6 moment components on a v e r t i c a l c y l i n d e r i n s h o r t - c r e s t e d s e a s . T h ey t h e n d e v e l o p e d a method by w h i c h t h e e x p e c t e d v a l u e o f t h e r e s u l t a n t o f e x t r e m e s o f f o r c e o r moment components i n s h o r t - c r e s t e d s e a s may be c a l c u l a t e d . T h i s can be o b t a i n e d i n terms o f t h e s p e c t r a and c r o s s - s p e c t r a of t h e i n - l i n e and t r a n s v e r s e f o r c e o r moment components. They r e p o r t e d v e r y good agreement between t h e o r e t i c a l p r e d i c t i o n s and e x p e r i m e n t a l r e s u l t s . 1.3 SCOPE OF THE PRESENT STUDY The p r i m a r y aim of t h e p r e s e n t work i s t o s t u d y d i r e c t i o n a l wave e f f e c t s on t h e l o a d i n g and r e s p o n s e o f l a r g e o f f s h o r e s t r u c t u r e s o f a r b i t r a r y s h a p e , and t o e x t e n d t h i s u s i n g s t a t i s t i c a l methods t o p r e d i c t t h e extreme l o a d s and r e s p o n s e s t h a t a r e l i k e l y t o o c c u r d u r i n g t h e o p e r a t i n g l i f e o f a s t r u c t u r e . T h e s e p r e d i c t i o n s a r e t h e n compared t o t h o s e b a s e d on t h e a s s u m p t i o n o f l o n g - c r e s t e d s e a s . The method u s e d i s q u i t e g e n e r a l and c a n be a p p l i e d t o any l a r g e o f f s h o r e s t r u c t u r e o f g e n e r a l shape t o compute t h e l o a d i n g t r a n s f e r f u n c t i o n s and t h e r e s p o n s e a m p l i t u d e o p e r a t o r s i n l o n g - a n d s h o r t - c r e s t e d random s e a s . The l o a d i n g t r a n s f e r f u n c t i o n s and t h e r e s p o n s e a m p l i t u d e o p e r a t o r s a r e o f g r e a t s i g n i f i c a n c e i n d e s i g n b e c a u s e . t h e y r e l a t e i n c i d e n t wave s p e c t r a t o t h e s p e c t r a o f t h e l o a d s a n d r e s p o n s e s o f i n t e r e s t . The p r o b a b i l i s t i c p r o p e r t i e s o f t h e components of t h e l o a d i n g and r e s p o n s e have been o b t a i n e d u s i n g a method 7 d e s c r i b e d by Clough and Penzien (1975). A new a n a l y t i c a l method has been developed to compute the extremes of maxima of the h o r i z o n t a l r e s u l t a n t s of the i n - l i n e and t r a n s v e r s e l o a d i n g and response components. The method i s q u i t e general and can be a p p l i e d i n a l l s i t u a t i o n s r e q u i r i n g the s t a t i s t i c a l p r o p e r t i e s of the amplitude of the r e s u l t a n t of two ortnogonal components with Gaussian d i s t r i b u t i o n s and zero means. T h i s i s s i g n i f i c a n t i n design s i n c e the designer can set l i m i t s on l o a d i n g and response of o f f s h o r e s t r u c t u r e s that are l i k e l y to occur d u r i n g t h e i r o p e r a t i n g l i f e . The r e s u l t s are compared to those of Huntington and G i l b e r t (1979). The t h e o r e t i c a l development i s d e s c r i b e d i n Chapters 2 and 3. R e s u l t s and d i s c u s s i o n are" presented i n Chapter 4. Con c l u s i o n s are presented in Chapter 5. A b r i e f d e s c r i p t i o n of the Green's f u n c t i o n used i s given i n Appendix I. The d e t a i l e d d e r i v a t i o n of the p r o b a b i l i t y theory used i n Chapter 4 i s given i n Appendix I I . 1.4 DESCRIPTION OF METHOD The method b a s i c a l l y comprises of three p a r t s . The f i r s t p a r t i n v o l v e s the s o l u t i o n of the wave d i f f r a c t i o n problem f o r r e g u l a r u n i - d i r e c t i o n a l waves using a three-dimensional wave source method. The second p a r t extends the above method to s h o r t - c r e s t e d random waves. F i n a l l y , the s t a t i s t i c s of the extremes of the h o r i z o n t a l loads and responses are developed u s i n g p r o b a b i l i t y theory. 8 The s o l u t i o n of the f i r s t p a r t has been d e s c r i b e d by s e v e r a l authors, i n c l u d i n g G a r r i s o n and Chow (1972), F a l t i n s e n and Michelsen (1974), Hogben and Standing (1974), G a r r i s o n (1978) and Isaacson (1985). The method n e g l e c t s v i s c o u s e f f e c t s and assumes the wave height to be s u f f i c i e n t l y small f o r n o n l i n e a r terms in v a r i o u s equations to be n e g l e c t e d . The f l u i d motion i s d e s c r i b e d by a v e l o c i t y p o t e n t i a l which i s a combination of components a s s o c i a t e d with the i n c i d e n t waves, s c a t t e r e d waves, and f o r c e d waves due to each mode of motion of the body. The v e l o c i t y p o t e n t i a l s a t i s f i e s the L a p l a c e equation, the seabed boundary c o n d i t i o n and l i n e a r i z e d boundary c o n d i t i o n s at the body s u r f a c e , f r e e s u r f a c e and f a r f i e l d . The unknown v e l o c i t y p o t e n t i a l components a s s o c i a t e d with the scattered" and f o r c e d waves are d e s c r i b e d by d i s t r i b u t i o n s of p o i n t wave sources over the immersed e q u i l b r i u m s u r f a c e of the body. The i n t e g r a l equations o b t a i n e d i n t h i s manner i n v o l v e a known t h r e e - d i m e n s i n a l Green's f u n c t i o n . These i n t e g r a l equations are s o l v e d by d i s c r e t i z i n g the body i n t o plane q u a d r i l a t e r a l elements, with the source strengths assumed t o be constant over each element. T h i s transforms the i n t e g r a l equations i n t o s e t s of a l g e b r a i c equations with the source s t r e n g t h s at the c e n t r e of each element i n i t i a l l y unknown. The accuracy of the s o l u t i o n depends on the d i s c r e t i z a t i o n of the s u r f a c e and improves with i n c r e a s i n g number of elements. Once the v e l o c i t y p o t e n t i a l s are known, the 9 r e q u i r e d hydrodynamic c o e f f i c i e n t s are determined by a p p l y i n g the l i n e a r i z e d B e r n o u l l i equation t o o b t a i n the pre s s u r e components and i n t e g r a t i n g these over the s t r u c t u r e s u r f a c e . The above approach i s then extended to long- and s h o r t - c r e s t e d random waves. In the case of l o n g - c r e s t e d random waves, the wave energy i s taken to be concentrated along the p r i n c i p a l wave d i r e c t i o n only, and the theory f o r r e g u l a r waves i s a p p l i e d to a s e r i e s of component wave t r a i n s of u n i t amplitude and spanning a range of f r e q u e n c i e s i n order to o b t a i n the l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude o p e r a t o r s as f u n c t i o n s of frequency. In the case of s h o r t - c r e s t e d waves, the sea s t a t e i s taken to comprise of a double s e r i e s of component wave t r a i n s of d i f f e r e n t f r e q u e n c i e s and d i r e c t i o n s . Thus the theory f o r r e g u l a r waves i s a p p l i e d to these components i n order to o b t a i n the l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude o p e r a t o r s as f u n c t i o n s of frequency and d i r e c t i o n . The s p e c t r a and c r o s s - s p e c t r a of the l o a d i n g and response components are then obtained by m u l t i p l y i n g the l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude operators with a s p e c i f i e d i n c i d e n t wave spectrum. In the case of s h o r t - c r e s t e d waves, d i r e c t i o n a l wave spectrum i s taken as a s p e c i f i e d u n i - d i r e c t i o n a l wave spectrum m u l t i p l i e d by a s p e c f i e d d i r e c t i o n a l spreading f u n c t i o n which i s independent of frequency. 10 F i n a l l y , the p r o b a b i l i s t i c p r o p e r t i e s of the components of the l o a d i n g and response and of t h e i r h o r i z o n t a l r e s u l t a n t s are d e s c r i b e d . The components of the l o a d i n g and response are d e s c r i b e d by Gaussian d i s t r i b u t i o n s with zero means. T h i s i s because the water su r f a c e e l e v a t i o n a l s o has a Gaussian d i s t r i b u t i o n with zero mean and the a n a l y s i s i s l i n e a r . The expected values of the extremes of maxima of the i n - l i n e and t r a n s v e r s e l o a d i n g and response components and of t h e i r h o r i z o n t a l r e s u l t a n t s f o r a s p e c i f i e d sea s t a t e are then obtained. T h i s procedure r e q u i r e s the moments of the sp e c t r a and c r o s s - s p e c t r a of components of the l o a d i n g and response as the primary input parameters. 2. LINEAR DIFFRACTION THEORY 2.1 INTRODUCTION In the present chapter a general numerical method i s de s c r i b e d to study d i r e c t i o n a l wave e f f e c t s on la r g e o f f s h o r e s t r u c t u r e s of a r b i t r a r y shape. The method extends the three-dimensional wave source method, d e s c r i b e d f o r example by G a r r i s o n and Chow (1972), Hogben and Standing (1974), G a r r i s o n (1978), F a l t i n s e n & Michelsen (1974) and Isaacson (1985), which p r o v i d e s wave loads and responses i n lo n g - c r e s t e d r e g u l a r waves, to both l o n g - and s h o r t - c r e s t e d random waves. The method i s a p p l i c a b l e to any l a r g e o f f s h o r e s t r u c t u r e s of general shape. The d e s c r i p t i o n of random waves i s given i n Se c t i o n 2.2. The l i n e a r d i f f r a c t i o n theory f o r r e g u l a r waves i s d e s c r i b e d i n S e c t i o n 2.3. The exte n s i o n of l i n e a r d i f f r a c t i o n theory to both l o n g - and s h o r t - c r e s t e d waves i s de s c r i b e d i n Secti o n s 2.4 and 2.5 r e s p e c t i v e l y . 2.2 DESCRIPTION OF RANDOM WAVES There are two approaches to d e s c r i b i n g random ocean waves. One i s i n terms of l o n g - c r e s t e d seas i n which the wave energy i s assumed t o be conce n t r a t e d along the p r i n c i p a l wave d i r e c t i o n and to be d i s t r i b u t e d over frequency. The other i s i n terms of s h o r t - c r e s t e d seas i n which the energy i s assumed to be d i s t r i b u t e d over both frequency and d i r e c t i o n . 11 1 2 Long-crested seas are g e n e r a l l y d e s c r i b e d by u n i - d i r e c t i o n a l wave spectrum, as i n d i c a t e d i n F i g u r e 2.1. A v a r i e t y of u n i - d i r e c t i o n a l wave s p e c t r a are i n use and have been summarized, f o r example, by Sarpkaya and Isaacson (1981 ) . In the r e a l ocean environment, waves propagate over a range of d i r e c t i o n s , and the s u p e r p o s i t i o n of such waves gives r i s e to a s h o r t - c r e s t e d waves, i n which the f r e e s u r f a c e has a two-dimensional r a t h e r than a one-dimensional s p a t i a l v a r i a t i o n . S h o r t - c r e s t e d seas are d e s c r i b e d by a d i r e c t i o n a l wave spectrum which depends both on the frequency and on the d i r e c t i o n of wave propagation, as sketched i n F i g u r e 2.2. S h o r t - c r e s t e d random seas are g e n e r a l l y d e s c r i b e d by a u n i - d i r e c t i o n a l wave spectrum m u l t i p l i e d by a d i r e c t i o n a l spreading f u n c t i o n . The sea s t a t e i s g e n e r a l l y assumed to be a s t a t i o n a r y , ergodic random pr o c e s s , with wave e l e v a t i o n d e s c r i b e d by a Gaussian d i s t r i b u t i o n . In the long term the wave c o n d i t i o n s are assumed to correspond to a s e r i e s of such sea s t a t e s , each i n i t s e l f c o n s i d e r e d s t a t i o n a r y with d i f f e r e n t s i g n i f i c a n t wave h e i g h t , peak p e r i o d and the p r i n c i p a l wave d i r e c t i o n . The long term v a r i a t i o n of sea s t a t e s and i t s e f f e c t on o f f s h o r e s t r u c t u r e s i s a d i s t i n c t problem not co n s i d e r e d i n t h i s study. 1 3 2.3 LONG-CRESTED REGULAR WAVES The f o l l o w i n g development f o r l o n g - c r e s t e d r e g u l a r waves i s based l a r g e l y on that given by Isaacson (1985). The theory i s based on the assumptions that the f l u i d i s in c o m p r e s s i b l e , i n v i s c i d and the flow i s i r r o t a t i o n a l . The wave height i s assumed to be s m a l l . The i r r o t a t i o n a l i t y c o n d i t i o n enables the f l u i d motion to be d e s c r i b e d by a v e l o c i t y p o t e n t i a l . The s m a l l wave amplitude assumption enables l i n e a r i z e d boundary c o n d i t i o n s to be a p p l i e d at the body s u r f a c e , f r e e s u r f a c e , and f a r f i e l d . A r e g u l a r small amplitude wave t r a i n of h e i g h t H and angular frequency co propagates i n water of con s t a n t depth d past a l a r g e , a r b i t r a r y , f r e e l y f l o a t i n g body as i n d i c a t e d in F i g u r e 2.3. Let Oxyz form a right-handed C a r t e s i a n c o o r d i n a t e system, with the wave t r a i n moving a t an angle 6 with the x a x i s as i n d i c a t e d i n F i g u r e 2.4 and z measured upward from the s t i l l water l e v e l . The body o s c i l l a t e s with s i x degrees of freedom, namely surge, sway, heave, r o l l , p i t c h , and yaw. Surge, sway and heave are t r a s l a t i o n a l motions p a r a l l e l to the x, y, z axes r e s p e c t i v e l y , and r o l l , p i t c h and yaw are r o t a t i o n a l motions about the same th r e e axes. Due to l i n e a r i z a t i o n , the problem can be t r e a t e d as a s u p e r p o s i t i o n of seven problems: the f l u i d motion produced by each of the s i x component motions of the body as w e l l as the i n t e r a c t i o n of a r e g u l a r wave t r a i n with the r e s t r a i n e d body. 1 4 Each component motion i s harmonic with a frequency equal to that of the i n c i d e n t waves CJ and can thus be rep r e s e n t e d as: a k = $ k exp ( - i u t ) (2.1) where i s a displacement f o r k = 1,2,3 and a r o t a t i o n f o r k = 4,5,6, $ k i s complex amplitude of each component motion and t i s time. Since only a steady s t a t e s o l u t i o n i s being sought, the time dependence occurs as exp(-icot) and hence does not enter the boundary v a l u e problem e x p l i c i t l y . The flow p o t e n t i a l i s made up of components a s s o c i a t e d with the i n c i d e n t waves ( s u b s c r i p t o ) , the s c a t t e r e d waves ( s u b s c r i p t 7), each of these being p r o p o r t i o n a l t o i n c i d e n t wave h e i g h t , and f o r c e d waves due to each mode of motion ( s u b s c r i p t 1,...,6), and each p r o p o r t i o n a l to the corr e s p o n d i n g motion amplitude. Thus the v e l o c i t y p o t e n t i a l 0 may be w r i t t e n as: - i i )H ^ <t> = [ ^(<t>o+<t>i) + £ -iwS^,, ] exp (- ic j t ) (2.2) z k=1 K K where 4>^ , k = 0,1,...,7 i s g e n e r a l l y complex. Each of the p o t e n t i a l s must s a t i s f y the Laplace equ a t i o n , and the boundary c o n d i t i o n s at the seabed and f r e e s u r f a c e . The l i n e a r i z e d boundary c o n d i t i o n on the body s u r f a c e i s given by: 15 n D (2.3) where n denotes-distance i n the d i r e c t i o n of u n i t v e c t o r normal to e q u i l b r i u m body s u r f a c e and d i r e c t e d outward from the body s u r f a c e . V n i s the v e l o c i t y of the s u r f a c e i t s e l f i n the d i r e c t i o n of n and i s given by: V = Z - icj$. n, exp(-icjt) n k=1 K K (2.4) i n which n. = n , 1 x' n 2 = n y , n 3 = n z ' n. = y n - z n , 4 1 z y' n c. = z n - x n , 5 x z' = x n - y n . (2.5) where n , n and n are the d i r e c t i o n c o s i n e s of the normal v e c t o r n i n the x, y and z d i r e c t i o n s . S u b s t i t u t i n g equations (2.2), (2.4) and (2.5) i n t o equation (2.3), we o b t a i n : 90 9n 0 UV7 30^ 6 30:3n 3n = Z -icj$.n. exp(-icot) k=1 K K (2.6) S e p a r a t i n g out the terms corre s p o n d i n g to the d i f f r a c t i o n problem (k=0,7) and each component of the r a d i a t i o n problem (k=1,...,6), equation (2.6) can be decomposed i n t o the 1 6 f o l l o w i n g f o r m : 30, 3n 30 ( 9n f o r k = 1 , . . . , 6 f o r k = 7 ( 2 . 7 ) T h e i n c i d e n t p o t e n t i a l 0 O i s k n o w n a n d i s g i v e n a s : 0 0 = ^ f s i n h k d ^ e x p [ i k ( x c o s 0 + y s i n f l ) ] ( 2 . 8 ) w h e r e k i s t h e w a v e n u m b e r . A c c o r d i n g t o p o t e n t i a l t h e o r y , t h e u n k n o w n p o t e n t i a l s 0^ m a y e a c h b e r e p r e s e n t e d a s d u e t o a d i s t r i b u t i o n o f p o i n t w a v e s o u r c e s o v e r t h e e q u i l b r i u m b o d y s u r f a c e S ^ . T h u s t h e p o t e n t i a l 0 ^ ( x ) a t a g e n e r a l p o i n t x = ( x , y , z ) m a y b e e x p r e s s e d a s : 0 k ( x > = SS f k ( £ } G ( £ ' i } d s ( 2 - 9 ) b w h e r e r e p r e s e n t s a s o u r c e s t r e n g t h d i s t r i b u t i o n f u n c t i o n , £ i s t h e p o i n t ( £ , T J , $ ) o n t h e b o d y s u r f a c e o v e r w h i c h t h e i n t e g r a t i o n i s p e r f o r m e d , a n d G ( x , £ ) i g t h e G r e e n ' s f u n c t i o n f o r t h e g e n e r a l p o i n t x d u e t o a s o u r c e o f u n i t s t r e n g t h a t £ . T h e G r e e n ' s f u n c t i o n i s c h o s e n s o t h a t i t s a t i s f i e s t h e L a p l a c e e q u a t i o n , t h e s e a b o u n d a r y c o n d i t i o n , t h e l i n e a r i z e d f r e e s u r f a c e b o u n d a r y c o n d i t i o n s a n d t h e r a d i a t i o n c o n d i t i o n . A l t e r n a t i v e e x p r e s s i o n s f o r i t 17 are g i v e n , i n Appendix I. I t remains f o r the source s t r e n g t h f u n c t i o n s f^ to be chosen so that the body s u r f a c e boundary c o n d i t i o n given by equation (2.7) i s s a t i s f i e d . Equation (2.7) together with equation (2.9) reduce to a set of s u r f a c e i n t e g r a l equations f o r f ^ : D f o r k = 1 ,..., 7 where x l i e s on the body s u r f a c e a t the p o i n t where boundary c o n d i t i o n i s a p p l i e d , n i s measured from x, and the i n t e g r a t i o n i s c a r r i e d out over £. From equations (2.8) and (2.9), the r i g h t hand s i d e of equation (2.10) i s given as: f 2n k f o r k =1,...,6 (2.11) sinh(kd) [ n z s i n h ( k ( z + d ) ) + i ( n cos0+n s i n 0 ) c o s h ( k ( z + d ) ) ] e x p [ i k ( x c o s 0 + y s i n f l ) ] x y f o r k=7 b k -The i n t e g r a l equations (2.10) are s o l v e d n u m e r i c a l l y f o r the source strengths f k and t h i s s o l u t i o n i s d e s c r i b e d i n s e c t i o n 2.3.3. 2.3.1 WAVE FORCES Once a l l the p o t e n t i a l s <p^ are known, the hydrodynamic pressure p may be obtained by the 18 l i n e a r i z e d B e r n o u l l i equation: (2.12) The components of wave f o r c e and moment a c t i n g on the body are thus given by: F. = -icjpj"- <j> n . dS f o r j = 1,...f6 (2.13) J • b 3 where F ^ F^, denote the f o r c e components i n the x, y, z d i r e c t i o n s and F^, Fg, Fg denote the moment components about the x, y, z axes. Equation (2.13) may be decomposed i n t o components a s s o c i a t e d with the p o t e n t i a l s ( 0 o + 0 7 ) and components a s s o c i a t e d with each of the f o r c e d p o t e n t i a l s 0 6. The f i r s t such component i s termed the e x c i t i n g f o r c e . The components a s s o c i a t e d with the f o r c e d p o t e n t i a l may f u r t h e r be decomposed i n t o components i n phase with the a c c e l e r a t i o n and i n phase with the v e l o c i t y of the corresponding body motion. Thus the t o t a l f l u i d force can be expressed as: (e) 6 F j = [F\ e'+ I (" 2Mj k + iwX ;. k)$ k]exp(-iwt) (2.14) f o r j = 1 ,...,6 The c o e f f i c i e n t s M^J and X^j are frequency dependent, are taken as r e a l and are termed the added masses and damping c o e f f i c i e n t s r e s p e c t i v e l y . S u b s t i t u t i n g 19 equations (2.2) and (2.14) i n t o equation (2.13) and c o l l e c t i n g terms f o r the e x c i t i n g f o r c e , added masses and damping c o e f f i c i e n t s , We o b t a i n the f o l l o w i n g e x p r e s s i o n s : F^ e )= pw2H Sc U o+07)n.dS (2.15) u.k = -pRe{ I j k } (2.16) X j k = -pcjlm{ I j k } (2.17) i n which Re{ } and Im{ } denote the r e a l and imaginary p a r t s and I j k i s given as: J j k = ' s b V j d s { 2 - 1 8 ) 2.3.2 BODY MOTIONS The equation of motion of the body may be w r i t t e n i n the f o l l o w i n g form: ^ [ V ( m j k + *jk> - i w X j k + c j k ] * k = F j * ( 2* 1 9 ) f o r j = 1 ,..., 6 where i s the mass matrix and C j k i s the h y d r o s t a t i c 20 s t i f f n e s s matrix. The mass matrix i s given as: m m 0 0 0 mz G 0 0 m 0 -mzG 0 0 0 0 m 0 0 0 0 -mz G 0 h -I xy -I mz G 0 0 X 5 -I 0 0 0 -I yz I zx yz 6_J (2.20) i n which m i s the mass of the body. z Q i s the z c o o r d i n a t e of c e n t r e of g r a v i t y of the body. I 4 , 1^, I g are the body's moment of i n e r t i a about the x, y, z axes r e s p e c t i v e l y and may be expressed as: 1 4 = m(r* + z*) 1 5 = m(rj + zg) 1 6 = m(rj) (2.21) i n which r x , r ^ and r z are the r a d i i of g y r a t i o n about the x, y, z axes r e s p e c t i v e l y . I „ , I„_ and I are the xy x body's product of i n e r t i a which are given as: xy yz zx ; v p b xydv J v p b yzdv / v p f a zxdv ( 2 . 2 2 ) where i s the d e n s i t y of the body and v i s the volume 2 1 of the body. The h y d r o s t a t i c s t i f f n e s s matrix i s given as: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C 3 3 C 3 4 C 3 5 0 0 0 C 3 4 C 4 4 0 0 0 0 C 3 5 0 C 5 5 0 0 0 0 0 0 0 ( 2 . 2 3 ) i n which C 3 3 = " 9 s ' C 3 4 = ~ P 9 S 2 f ' 4 4 = p g S 2 2 + mg(z B - zr) = pgS,! + mg(z n - z r ) c 3 5 = -pgst ] ( 2 . 2 4 ) i n which z„ i s the z c o o r d i n a t e of c e n t r e of buoyancy. z G i s the z c o o r d i n a t e of c e n t r e of g r a v i t y of body. S i s the waterplane area and S 1 r S 2, S n , S 2 2 are the waterplane moments: S = /dS, S, = /xdS, S 2 = JydS S n = Jx 2dS , S 2 2 = /y 2dS ( 2 . 2 5 ) The above i n t e g r a l s are taken over waterplane. 22 2.3.3 NUMERICAL INTEGRATION The submerged s u r f a c e i s d i s c r e t i z e d i n t o a f i n i t e number of f a c e t s N. The i n t e g r a l equation (2.10) i s then a p p l i e d at each f a c e t c e n t r e and we o b t a i n a set of l i n e a r a l g e b r a i c equations: L A . . f ^ k ) = b | k ) f o r i=1,...,N, k=1,...,7 (2.26) j = 1 -1 'J 1 (k) ( k) Here f . and b: denote f,,(x_.) and b. (x. ) r e s p e c t i v e l y and X j i s the value of x at the ce n t r e of j - t h f a c e t . The c o e f f i c i e n t s A^j are given as: A i j • " 5 i j + h /AS..H ( * i ' i ) d S ( 2 - 2 7 ) i n which i s the Kronecker d e l t a f u n c t i o n and ASj i s the area of the j t h f a c e t . When i * j , the in t e g r a n d i n equation (2.27) can be taken to be constant over the area of the f a c e t ASj, and thus A^j i s approximated as: A, * S i 8G ^ 27T ^ ( x . , 1 . ) (2.28) When i = j , 5 ^ = 1 accounts f o r the i n f l u e n c e of i - t h f a c e t on i t s own ce n t r e and we get: A . . = -1 (2.29) By the s o l u t i o n of equation (2.29), the source s t r e n g t h 23 may be evaluated at each f a c e t c e n t r e . Once the source strengths f ^ are known, the p o t e n t i a l s may be obtained by e v a l u a t i n g equation (2.9): H>\k) = Z B. . f ^ k ) (2.30) j = 1 J 3 (k) where <j>^ denotes the value at ^ ( x ^ ) . The c o e f f i c i e n t s B^j are given by: B i i = 4? *AS G(*i>V*S <2'31> J j Again when i * j , the i n t e g r a n d i s taken as constant over ASj and thus B^j i s approximated as: AS . B = — 1 G(x-,!•) (2.32) When i = j , a s i n g u l a r i t y occurs i n equation (2.31), and the corresponding e x p r e s s i o n f o r B ^ may be given as: /AST AS. B h = A + — - GiL (2.33) 47T 47T where excludes any c o n t r i b u t i o n from the s i n g u l a r term 1/R and A i s given as: A = — — / A C 1 dS (2.34) v/AST A S i R The above developement and e x p r e s s i o n s f o r d i f f e r e n t 24 f a c e t shapes are given, f o r example, by Hogben and Standing ( 1 9 7 4 ) . In p a r t i c u l a r , A = 2/7F f o r a c i r c u l a r f a c e t and A = 3 . 5 2 5 f o r a square f a c e t . For a re c t a n g u l a r f a c e t : A = — {ln[b + / b 2 + 1] + b l n [ l + ( 2 . 3 5 ) •b b where b i s the aspect r a t i o of the f a c e t . F i n a l l y , equations ( 2 . 1 5 ) and ( 2 . 1 8 ) can be w r i t t e n i n d i s c r e t i z e d form: F ^ e ) = - IpgHk Z u ( 0 ) + * ! 7 ) ) n..AS. ( 2 . 3 6 ) J • j = 1 J 1 1 N (k) 'jk * i n j i A S i ( 2 ' 3 7 ) 2 . 4 LONG-CRESTED RANDOM WAVES The above approach for l o n g - c r e s t e d r e g u l a r waves may r e a d i l y be extended to l o n g - c r e s t e d random waves using the approach of Bendat and P i e r s o l (1966). T h i s uses the regular wave r e s u l t s , obtained at a s e r i e s of wave f r e q u e n c i e s , together with the i n c i d e n t wave spectrum t o d e r i v e the s p e c t r a and c r o s s - s p e c t r a of v a r i o u s l o a d i n g and response components. The r e q u i r e d l o a d i n g and response s p e c t r a and c r o s s - s p e c t r a are denoted by S^j(w) f o r i , j = 1 , . . . , 6 . Here S^j denotes the cross-spectrum of the i - t h and j - t h components of loa d or response and S^. denotes the spectrum of the i - t h component of loa d or response. These s p e c t r a and 25 c r o s s - s p e c t r a are r e l a t e d to the i n c i d e n t wave spectrum S(w) as f o l l o w s : S ^ f u ) = | T \ (CJ,0) | | T J ( C J , 0 ) | S ( C J ) (2.38) where T^((o,d) i s a l o a d i n g t r a n s f e r f u n c t i o n or a response amplitude operator f o r a s p e c i f i e d d i r e c t i o n 8 and i s the r a t i o of l o a d i n g or response per u n i t wave amplitude c a l c u l a t e d f o r a s i n u s o i d a l wave t r a i n at frequency u. 8 i s the angle of i n c i d e n t wave d i r e c t i o n measured from the x a x i s . From equation (2.15) the t r a n s f e r f u n c t i o n s f o r the loads are given as: po) 2/- ( 0 o + 0 7 ) n . d S f o r j = 1,...,6 (2.39) and from equations (2.15) and (2.19), the response amplitude ope r a t o r s f o r the motions of f l o a t i n g body are given as: - p c j 2 / s ( 0 o + 0 ? ) n j d S Tj(u,0) = -g (2.40) k Z i [ - W 2 ( m j k + / z j k ) - i W X : J k + c j k ] f o r j = 1 ,..., 6 T.(w,0) = -26 2.5 SHORT-CRESTED RANDOM WAVES In s h o r t - c r e s t e d seas, the waves propagate over a range of d i r e c t i o n s and may be con s i d e r e d as a s u p e r p o s i t i o n of lo n g - c r e s t e d random wave t r a i n s p ropagating i n d i f f e r e n t d i r e c t i o n s . The d i r e c t i o n a l wave spectrum i s denoted by S(co,8) and can be expressed i n terms of a u n i - d i r e c t i o n a l wave spectrum Sico) as: where D(co,6) i s the d i r e c t i o n a l spreading f u n c t i o n such that Equation (2.42) ensures that the energy c o n t a i n e d i n the d i r e c t i o n a l wave spectrum i s equal to the energy contained in the corresponding u n i - d i r e c t i o n a l wave spectrum. I t i s of t e n assumed that the angular d i s t r i b u t i o n of wave energy i n a d i r e c t i o n a l wave spectrum i s the same at a l l fre q u e n c i e s , so that the d i r e c t i o n a l spreading f u n c t i o n can be taken as independent of frequency: S ( C J , 0 ) = S ( O J ) D ( C J , 0 ) (2.41) 7T S D(CJ,<9) 66 = 1 (2.42) D(u,0) <* Did) (2.43) A common form of d i r e c t i o n a l s preading f u n c t i o n Did) i s based on a cosine powered d i s t r i b u t i o n : 27 Die) = C ( s ) c o s 2 s ( 0 - 0 o ) |6>-0O I < */2 0-0oI > w/2 (2.44) where 0 O i s the p r i n c i p a l d i r e c t i o n about which the angular d i s t r i b u t i o n i s c e n t r e d . C(s) i s a n o r m a l i z i n g f a c t o r chosen to ensure that equation (2.42) i s s a t i s f i e d and i s given by: C(s) = (2.45a) 1 7 2s J cos b0 d0 ~1t The n o r m a l i z i n g f a c t o r C(s) i s l i s t e d i n T a b l e 2.1 f o r a range of v a l u e s of s and i s sketched i n F i g u r e 2.5. An a l t e r n a t i v e e x p r e s s i o n f o r the n o r m a l i z i n g f a c t o r C(s) may be w r i t t e n i n terms of the gamma f u n c t i o n r a s : C(s) = F ( s + 1 | (2.45b) r ( s + l ) The l o a d i n g or response s p e c t r a S ^ J ( C J ) due t o u n c o r r e l a t e d l o n g - c r e s t e d random waves propagating i n d i f f e r e n t d i r e c t i o n s 0 are given as: it S j . f u ) = / |T i(cj,0) | |T .(C J,0) | S(co.e) d0 (2.46) -it  3 f o r i , j = 1 ,. .. , 6 where T J ( C J , 0 ) are the l o a d i n g t r a n s f e r f u n c t i o n s or the response amplitude o p e r a t o r s , d e f i n e d i n equations (2.39) and (2.40) r e s p e c t i v e l y . I f the d i r e c t i o n a l s p r e a d i n g 28 f u n c t i o n D(o>,0) i s taken as independent of frequency, then equation ( 2 . 4 6 ) can be w r i t t e n as: Si-(o}) = { J |T. | | T \ ( u , 0) |D(0)} S ( C J ( 2 . 4 7 ) f o r i , j = 1 , . . ., 6 Since the i n t e g r a t i o n over 6 i s now independent of S ( C J ) , i t i s convenient to d e f i n e a d i r e c t i o n a l l y averaged t r a n s f e r f u n c t i o n T^^(CJ) as: T..(w) = { ] |T.(u,0) | |T.(o),0) |D(0) d0 } 1 / 2 ( 2 . 4 8 ) — TT T h i s enables the r e l a t i o n s h i p between the s p e c t r a to be w r i t t e n i n the common form: . • S . j ( u ) = |Ti;.(w) | 2 S ( u ) ( 2 . 4 9 ) In the case of l o n g - c r e s t e d waves propagating i n the d i r e c t i n 80, the corresponding t r a n s f e r f u n c t i o n i s giv e n as: T i ; j(a;) = {Ti(ure0)T^(u,eQ)} ]/2 ( 2 . 5 0 ) The rms value s of the components of the l o a d i n g and response i n l o n g - and s h o r t - c r e s t e d seas may be obtained by an i n t e g r a t i o n over a) of equations ( 2 . 3 8 ) and ( 2 . 4 7 ) r e s p e c t i v e l y . The rms values of the components of the 29 l o a d i n g and response i n s h o r t - c r e s t e d seas may then be compared to the corresponding rms v a l u e s i n l o n g - c r e s t e d seas. We may thus d e f i n e a l o a d i n g or a response r e d u c t i o n f a c t o r R^j as the r a t i o of these two c h a r a c t e r i s t i c s rms v a l u e s . Thus oo TT ; { ; |T. (w,0) | |T.(w ,e) | D(0)d0}s(cj)dw R?. = ° ~* • (2.51) I j °° i\Ti(ul60 | |T.(a>,0o) | S(w)do) o -1 f o r i , j = 1 , . . . , 6 T h i s r e d u c t i o n f a c t o r , then, may be a p p l i e d d i r e c t l y to the rms component loads and responses p r e d i c t e d by assuming u n i - d i r e c t i o n a l waves i n order to o b t a i n coresponding v a l u e s when d i r e c t i o n a l spreading f u n c t i o n i s taken i n t o account. Since the l o n g - c r e s t e d random waves from d i r e c t i o n s are assumed to be u n c o r r e l a t e d , l o a d i n g and response are induced i n - l i n e and t r a n s v e r s e to the p r i n c i p a l wave d i r e c t i o n even i f the d i r e c t i o n a l wave spectrum i s symmetric. 3. PROBABILISTIC PROPERTIES OF LOADING AND RESPONSE 3.1 INTRODUCTION The s p e c t r a and c r o s s - s p e c t r a of the l o a d i n g and response d e s c r i b e d i n the pr e v i o u s chapter are of l i t t l e d i r e c t i n t e r e s t to the de s i g n e r . Rather, he needs to know the maxima of the l o a d i n g and response which are l i k e l y to occur in a c e r t a i n storm c o n d i t i o n . T h i s w i l l enable the designer to s et the design l i m i t on the l o a d i n g and response which are u n l i k e l y to be exceeded d u r i n g the o p e r a t i n g l i f e of the s t r u c t u r e s . The maximum l o a d i n g and response can themselves be p r e d i c t e d from the p r o p e r t i e s of the s p e c t r a and c r o s s - s p e c t r a using s t a t i s t i c a l methods which are d e s c r i b e d in t h i s c h a p t e r . The chapter f i r s t d e s c r i b e s the p r o b a b i l i s t i c p r o p e r t i e s of the components of the l o a d i n g and response f o r both l o n g - and s h o r t - c r e s t e d random seas. In s h o r t - c r e s t e d seas, loads and responses occur both i n - l i n e and t r a n s v e r s e to the p r i n c i p a l wave d i r e c t i o n . The o v e r a l l l o a d i n g or response i s thus a ve c t o r which v a r i e s randomly i n magnitude and d i r e c t i o n . A method i s d e s c r i b e d here to provide the p r o b a b i l i s t i c p r o p e r t i e s of the h o r i z o n t a l r e s u l t a n t of the components as w e l l as of the maxima of t h i s r e s u l t a n t . These in t u r n are used to p r e d i c t the maximum l o a d i n g and response o c c u r i n g i n a s p e c i f i e d storm d u r a t i o n . The p r o b a b i l i s t i c p r o p e r t i e s of the components of the l o a d i n g and response are d e s c r i b e d i n S e c t i o n 3.2, while those of t h e i r h o r i z o n t a l 30 31 r e s u l t a n t s and of the maxima of these r e s u l t a n t s are d e s c r i b e d i n Sect i o n s 3.3 and 3.4 r e s p e c t i v e l y . The p r o b a b i l i s t i c p r o p e r t i e s of the extremes of maxima of the h o r i z o n t a l r e s u l t a n t s are d e s c r i b e d i n S e c t i o n 3.5. A r e l a t e d approach given by Huntington and G i l b e r t (1979) i s summarized i n S e c t i o n 3.6. 3.2 PROBABILISTIC PROPERTIES OF COMPONENTS The component of the l o a d i n g or the response i n the x d i r e c t i o n , denoted here simply as x i s taken to possess a Gaussian d i s t r i b u t i o n with a zero mean. T h i s i s because the water s u r f a c e e l e v a t i o n " i s assumed a l s o to possess a Gaussian d i s t r i b u t i o n with zero mean, and a l i n e a r theory i s used to p r e d i c t the loads and responses. Since x has a Gaussian d i s t r i b u t i o n , the maxima of x themselves have a d i s t r i b u t i o n which depends on the s p e c t r a l width parameter e of the spectrum of x. The s p e c t r a l width parameter e i t s e l f depends on the z e r o t h , second and f o u r t h s p e c t r a l moments of the spectrum of x. For e 0, which corresponds to a narrow-band spectrum, the p r o b a b i l i t y d e n s i t y of the maxima x m of the component x reduces to a Rayl e i g h d i s t r i b u t i o n . On the other hand, as e —> 1, the p r o b a b i l i t y d e n s i t y becomes Gaussian so t h a t there are as many maxima below the mean l e v e l as above i t , corresponding to the high frequency t a i l e n t a i l e d i n a wide spectrum. D e t a i l e d treatments of the d i s t r i b u t i o n of the maxima f o r the cases when x possesses a narrow-band spectrum and an 32 a r b i t r a r y spectrum are given by Longuet-Higgins (1952) and by Car t w r i g h t and Longuet-Higgins (1956) r e s p e c t i v e l y . The s i n g l e l a r g e s t or extreme value x m of the maxima x 13 3 m m which occurs i n a number M of maxima c o n s i d e r e d independent and to occur i n a s p e c i f i e d d u r a t i o n T, i s a primary requirement. The co r r e s p o n d i n g d i s t r i b u t i o n f u n c t i o n P ( x m ) may be d e f i n e d i n terms of the d i s t r i b u t i o n of the maxima x m» The r e s u l t i n g e x p r e s s i o n i s given as (Clough and Penzien, 1975): P ( T J ) = exp[- M exp ( - i? 2/2)] (3.1) where rj i s d e f i n e d as xm//m^, and m0 i s the v a r i a n c e of x d e f i n e d as: m0 = J S (GJ) dcj (3.2) o x x The 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 p(rj) can be obtained by d i f f e r e n t i a t i n g equation (3.1) with r e s p e c t to rj. Using equation (3.1), an approximate exp r e s s i o n f o r the expected value of x m may be developed and i s given as (Clough and Penzien, 1975): Efx/v/mu) = ( 2 1 n ( M ) ) l / 2 + . 5772 (21n ( M ) ) " 1 / 2 (3.3) The number of maxima M may be obtained by m u l t i p l y i n g the frequency of zero u p c r o s s i n g f 0 by the s p e c i f i e d d u r a t i o n T, 33 M = f 0 T . One important p o i n t to be noted i s that equations (3.1) and (3.3) no longer i n v o l v e the s p e c t r a l width parameter e, but now depend on the frequency of zero u p c r o s s i n g , which i t s e l f depends on the zero t h and second s p e c t r a l moments of the spectrum of x. As Ochi (1982) emphasizes, equation (3.3) i s e x a c t l y the same as f o r the case of narrow-band spectrum. However, i t w i l l be r e a l i z e d that f o r a giv e n d u r a t i o n the number of maxima i n the case of a narrow-band spectrum i s small e r than that f o r an a r b i t r a r y s p e c t r a l shape with the same peak frequency, s i n c e the frequency of zero u p c r o s s i n g w i l l now be r e l a t i v e l y s m a l l . 3.2.1 FREQUENCY OF UPCROSSING OF COMPONENTS The mean frequency of u p c r o s s i n g at any l e v e l x i s d e f i n e d as ( P a p o u l i s , 1965) : f = / x p ( x , x ) d i (3.4) x o i n which x and x are Gaussian random v a r i a b l e s , and a dot denotes a d e r i v a t i v e with r e s p e c t t o time. Since x and k are u n c o r r e l 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 i s given as: p(x,x) = 1 exp{-l( i i + §i)} (3.5) m0m2 0 2 where m0 i s the v a r i a n c e of x, m2 i s the second s p e c t r a l 34 moment of x which i s a l s o equal to v a r i a n c e of k, and m0 and m2 are d e f i n e d by: CO mn = ; w nS x x(a>) dco (3.'6) o S u b s t i t u t i n g equation (3.5) i n t o equation (3.4), we o b t a i n f o l l o w i n g e x p r e s s i o n f o r f x : f x = 2¥ (^) l / 2exp(-xV2m 0) (3.7a) As a s p e c i f i c value of f , the frequency of z e r o u p c r o s s i n g f 0 , which was r e q u i r e d i n the foregoing, i s obtained by s e t t i n g x = 0. t. - £ l ^ ) 1 ' 2 (3.7b) 3.3 PROBABILISTIC PROPERTIES OF THE HORIZONTAL RESULTANT OF  COMPONENTS Let x and y be two Gaussian random v a r i a b l e s with z e r o means, a c t i n g i n orthogonal d i r e c t i o n s i n the h o r i z o n t a l plane. T h e i r j o i n t p r o b a b i l i t y d i s t r i b u t i o n i s given as: p(x,y)= 1 — exp{- ( x ? /27rm0m0( 1-X 2) 2(1-X 2) m( 2 X x y + JLl)} /m0m0 m0 35 -°° < x, y < + 0 0 -1 < X < +1 ( 3 . 8 ) where m0 and m0 are the v a r i a n c e s of x and y r e s p e c t i v e l y and are d e f i n e d i n terms of the s p e c t r a of x and y as: ( 3 . 9 a ) ( 3 . 9 b ) X i s the c o r r e l a t i o n c o e f f i c i e n t between x and y and i s given as f o l l o w s : X 2 = c0//m0mo ( 3 . 9 c ) c 0 i s v a r i a n c e of the cross-spectrum of x and y and i s i t s e l f given as: oo c 0 = ; S x (co) du ( 3 . 9 d ) o ^ Let z be the r e s u l t a n t of the random v a r i a b l e s x and y: z = + /x 2+y 2. The 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 p(z) of the r e s u l t a n t z may be developed as f o l l o w s : An a u x i l l i a r y v a r i a b l e 8 i s introduced and the j o i n t p r o b a b i l i t y d e n s i t y p ( z , 0 ) of z and 8 i s o b t a i n e d i n terms of p(x,y) by a t r a n s f o r m a t i o n procedure. The p r o b a b i l i t y d e n s i t y p(z) of z alone may then be obtained by a s u i t a b l e i n t e g r a t i o n of t h i s m0 = / S x x(w)do> o m0 = J S (oj)dco o y y 36 j o i n t p r o b a b i l i t y d e n s i t y p ( z , 0 ) . F i n a l l y , the p r o b a b i l i t y d e n s i t y p(z) may be i n t e g r a t e d to o b t a i n the p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n P ( z ) . T h i s procedure i s d e s c r i b e d , f o r example, by Bury (1975). In the present case, the a u x i l l i a r y v a r i a b l e 0 i s d e f i n e d as 0 = tan 1 ( y / x ) so t h a t the t r a n s f o r m a t i o n from (z,0) to (x,y) i s : X = ZCOS0 y = z s i n 0 The 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 p(z,0) may be expressed i n terms of 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 p ( x , y ) , given by equation (3.8), as f o l l o w s : where | J | i s the' determinant of the Jacobian of the t r a n s f o r m a t i o n . In view of the t r a n s f o r m a t i o n given by equation (3.10), | J | = z i n the present case. The p r o b a b i l i t y d e n s i t y , p ( z ) may be obtained by i n t e g r a t i n g t h i s j o i n t p r o b a b i l i t y d e n s i t y with r e s p e c t to 0: (3.10) p(z,0) = | J | p(x,y) (3.11) p(z) = J p(z,0) d0 (3.12) By v i r t u e of equations (3.10) and (3.11) t h i s may be w r i t t e n 37 as: p(z) = / z p(ZCOS0,zsinfl) 66 (3.13) Applying equation (3.8) to equation (3.13) and then c a r r y i n g out the i n t e g r a t i o n with respect to 8, over the range 0 to 2ff, an exp r e s s i o n f o r the 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 p(z) fo r the r e s u l t a n t z may be developed as f o l l o w s : r z 2 m0+m0 exp{- } p(z) = v/moirioO-X 2) l o t -2(1-X 2) 2m0m0 A} 2(1-X 2) f o r z>0 (3.14) f o r z<0 in which A = /(m 0 - m 0 ) 2 + 4X 2m 0m 0/2m 0m 0 (3.15) and I 0 i s the m o d i f i e d B e s s e l ' s f u n c t i o n of order zero. T h i s d e r i v a t i o n i s c a r r i e d out i n Appendix I I . The p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n P(z) of the r e s u l t a n t z may be obtained by i n t e g r a t i n g the p r o b a b i l i t y d e n s i t y p(z) as: z P(z) = ; p(z)dz (3.16) o S u b s t i t u t i n g equation (3.14) i n t o equation (3.16), we get: 38 P(z) = J I e x p ( - K 2 z : 2 ) l 0 ( K 3 z 2 ) d z f o r z>0 (3.17) where K,, K 2 and K 3 are c o n s t a n t s which are d e f i n e d as; K, = / m 0m 0(l-X 2) (3.18a) K 2 = (m 0 + m 0)/4(1-X 2)m 0m 0 (3.18b) K 3 = A/2(1-X 2) (3.18c) Upon c a r r y i n g out the i n t e g r a t i o n i n equation (3.17) with r e s p e c t to z, and u s i n g a i n t e g r a l equation f o r I 0 , we o b t a i n the f l l o w i n g e x p r e s s i o n : P(z) = 1 . 7r exp{-(K 2+K 3cos0)z 2} 1 - J 30 27TK, (K 2+K 3cos0) f o r z^O (3.19) f o r z<0 3.3.1 A SPECIAL CASE When the standa r d d e v i a t i o n s i n two component d i r e c t i o n s are equal and a l s o the two components are u n c o r r e l a t e d , then equations (3.14) and (3.19) reduce 39 t o : p(z) = 2m2, z / % exp(- ) 2m2, for z>0 (3.20) fo r z<0 P(z) =• 1 - exp(- ) 2m? f o r z>0 f o r z<0 (3.21) where m0 i s the v a r i a n c e of e i t h e r component. As expected, these equations correspond to the R a y l e i g h d i s t r i b u t i o n . 3.4 PROBABILISTIC PROPERTIES OF RESULTANT MAXIMA The p r o b a b i l i t y d e n s i t y of the r e s u l t a n t z was developed i n S e c t i o n 3.3, but i t i s the d i s t r i b u t i o n of the maxima z m which i s of more p r a c t i c a l i n t e r e s t , s i n c e t h i s w i l l be used i n t u r n to develop the d i s t r i b u t i o n of extremes of z . We m now c o n s i d e r the d e r i v a t i o n of the p r o b a b i l i t y d i s t r i b u t i o n of' z^. I t i s convenient, i n i t i a l l y , to d e f i n e v a r i o u s c h a r a c t e r i s t i c f r e q u e n c i e s of the random v a r i a b l e s z ( t ) . f denotes number of maxima per u n i t time r e g a r d l e s s of t h e i r magnitude. Thus 1/f c i s the average p e r i o d between the s u c c e s s i v e maxima r e g a r d l e s s of magnitude, f denotes the number of u p c r o s s i n g s per u n i t time with respect t o a 40 c e r t a i n t h r e s h o l d l e v e l z. T h i s may be expressed i n terms of 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 of z and z, where a dot denotes a d e r i v a t i v e with respect to time, as: f z = J z p(z,z) dz (3.22) — oo S p e c i f i c v a l u e s of f z which are encountered are as f o l l o w s . f 0 i s the zero u p c r o s s i n g frequency, which i s the number of zero u p c r o s s i n g s per u n i t time and i s the v a l u e of f at z = 0. F i n a l l y , f i s the maximum value of f z w i t h r e s p e c t to z. We now c o n s i d e r the p r o b a b i l i t y d i s t r i b u t i o n of z^. A maximum z m of the random process z ( t ) occurs when z = 0, and z < 0 s i m u l t a n e o u s l y . Thus, the i n f o r m a t i o n about the d i s t r i b u t i o n of maxima can be obtained by c o n s i d e r i n g the 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 of z, z and z, and the 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 p ( z m ) may be d e f i n e d i n terms of t h i s as ( L i n , 1976): 1 * ° p ( z m ) = " — | i / 2 p(z,z=0,z) dz (3.23) f -oo C Since the 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 of z, z and z i s extremely d i f f i c u l t to o b t a i n , an approximation to equation (3.24) i s a p p l i e d : 1 d ( f z ) p(z ) = - _ L ?_ (3.24) m f dz c T h i s approximation i s very good i f the component s p e c t r a are 41 narrow-banded. In the case of x and y p o s s e s s i n g wide band s p e c t r a , t h i s approximation leads to an overestimate of p ( z m ) . However, the use of equation (3.24) s i m p l i f i e s the problem to that of f i n d i n g the 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 of z, z only. The procedure used to o b t a i n P ( z m ) on the b a s i s of equation (3.24) proceeds along the l i n e s to that used to o b t a i n the d i s t r i b u t i o n of z as d e s c r i b e d i n s e c t i o n 3.3. The j o i n t d i s t r i b u t i o n of p(x,y,x,y) i s f i r s t o btained. The a u x i l l i a r y v a r i a b l e s 6 and 6 are then introduced and j o i n t p r o b a b i l i t y d e n s i t y p(z,z,0,0) i s o b t a i n e d i n terms of p(x,y,x,y) by a t r a n s f o r m a t i o n procedure. The j o i n t p r o b a b i l i t y d e n s i t y p(z,z) i s then obtained by s u i t a b l e i n t e g r a t i o n . T h i s j o i n t p r o b a b i l i t y d e n s i t y i s then a p p l i e d to an approximate formula, d e r i v e d from equation (3.24), f o r the p r o b a b i l i t y d e n s i t y p(z ) of the maxima z . r J u r m m A complete expr e s s i o n f o r the j o i n t p r o b a b i l i t y d e n s i t y p(x,y,x,y) i s somewhat i n t r a c t a b l e and an assumption made i n order to develop a s u i t a b l e r e s u l t i s that the cross-spectrum S i s taken as z e r o . T h i s i s true f o r an xy axisymmetric s t r u c t u r e and i n any case i s g e n e r a l l y a resonable approximation. Since x, y, i , y are Gaussian random v a r i a b l e s with zero means, 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 i s given as: p(x,y,x,y)= — ! — e x p { - l [ X ] T M 1 [ X ] } (3.25) 42 T where [X] i s the transpose of the v e c t o r c o n t a i n i n g the random v a r i a b l e s and i s : [X] T= {x,y,x,y} (3.26) [u] i s a c o r r e l a t i o n matrix with components given as: Oo = Re {/ S, .(CJ) dcj} f o r i=x,y and j=ic,y (3.27) o -1 A i s the determinant of the c o r r e l a t i o n matrix n. The v a r i o u s s p e c t r a o c c u r i n g i n equation (3.27) are r e l a t e d as f o l l o w s : S . • = C J 2 S , S . . = C J 2 S (3.28) xx xx' yy yy Thus, the c o r r e l a t i o n matrix M may be expressed i n terms of known s p e c t r a l moments of x and y d e f i n e d as f o l l o w s : m = J u> n S x ( C J ) dw, m' = / wn S (w) dco (3.29) o o y y The c o r r e l a t i o n matrix i s e v e n t u a l l y given i n terms of these 43 moments as: in] = m0 0 0 0 0 m0 0 0 0 0 m2 0 0 0 0 m2 (3.30) As a r e s u l t of the assumption t h a t = 0 , the non-diagonal i n t e r a c t i o n c o e f f i c i e n t s i n t h i s matrix are a l l zero and the c o r r e l a t i o n matrix reduces to a d i a g o n a l matrix. The a u x i l l i a r y v a r i a b l e s 6 and 6 are now i n t r o d u c e d where 0 = tan 1 ( y / x ) as b e f o r e . The t r a n s f o r m a t i o n between the C a r t e s i a n v a r i a b l e s x, y, k, y and the p o l a r v a r i a b l e s z, z, 0, 0* i s : x = ZCOS0 y = z s i n 0 x = zcos0 - z0*sin0 y = z s i n 0 - z0cos0 (3.31) (3.32) The j o i n t p r o b a b i l i t y d e n s i t y p(z,0,z,0*) may be obtained i n terms of p(x,y,x,y) by the t r a n s f o r m a t i o n : p(z,0,z,0) = | J | p(x,y,x,y) (3.33) 44 where | J | i s the determinant of the Jacobian of the tr a n s f o r m a t i o n . On the b a s i s of equations (3.31) and (3.32) |J | = z 2 i n the present case. The 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 of z and z which i s now r e q u i r e d may be obtained by i n t e g r a t i n g the j o i n t p r o b a b i l i t y d e n s i t y p(z,z,0,0) over the complete ranges of 6 and 8: 2TT 0 0 . p(z,z) = J { J | J | p(x,y,x,y)dd}d0 (3.34) o -°° Combining equations (3.22) and (3.33) we o b t a i n : oo 2TT 0 0 f = J / / z z 2 p(x,y,x,y) dz dd d$ (3.35) — 00 O — OD S u b s t i t u t i n g equation (3.25) f o r the j o i n t p r o b a b i l i t y d e n s i t y and i n t e g r a t i n g f i r s t with respect to 6 then z we ob t a i n : 2TT f = J k A(0)exp{-(cos 20/2m o + s i n 2 0 / 2 m o ) z 2 } dd (3.36) z o where k = z / ( 2 7 r) 3 / / 2/m 0m 0 (3.37a) Aid) = / m 2 s i n 2 0 + m 2cos 20 (3.37b) S u b s t i t u t i n g equation (3.36) i n t o equation (3.24) we f i n a l l y o b tain the f o l l o w i n g e x p r e s s i o n f o r the 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 of maxima: 45 p(z_) = — J . kA(6>) [z 2B(0)-1 ] exp{-B(0)z 2/2} dd (3.38) III r-f c 0 where B(0) = cos26>/m0 + s i n 2 0 / m o (3.39) It should be mentioned here that the assumption u n d e r l y i n g equation (3.24) i m p l i e s that df /dz i s to be e v a l u a t e d only f o r a l e v e l z greater than t h a t of the expected value of the random process z ( t ) . The l a t t e r may be e v a l u a t e d from p ( z ) , given by equation (3.14). T h i s assumption allows the 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 p ( z m ) , given by equation (3.38), to be evaluated only f o r the values g r e a t e r than the expected values of the process z ( t ) . F i n a l l y , the e x p r e s s i o n f o r the p r o b a b i l i t y d i s t r i b u t i o n of z m may be o b t a i n e d by i n t e g r a t i n g the p r o b a b i l i t y d e n s i t y given by e q u a t i o n (3.24) and i s thereby evaluated u s i n g f o l l o w i n g r e l a t i o n : f P ( z ) = 1 - j2 (3.40) m t c 3.5 PROBABILITY DISTRIBUTION OF EXTREMES OF RESULTANT MAXIMA The p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n P , ( z m ) of the s i n g l e l a r g e s t or extreme amplitude z m of the maxima z m that occurs i n a sample of M independent maxima i s given as (Longuet-Higgins, 1952, Benjamin and C o r n e l l , 1970): P , ( z J = P M U J (3.41) 1 m m 4 6 where P( ) corresponds to the d i s t r i b u t i o n of z^, given here by equation ( 3 . 4 0 ) . The corresponding 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 p, may then be obtained from: d P M ( z ) / - \ m P' ( zm ) = — d zm = M P M _ 1 ( z ) p ( z J ( 3 . 4 2 ) m r m S u b s t i t u t i n g from equations ( 3 . 3 8 ) and ( 3 . 4 0 ) , the 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 p , ( z m ) f o r the extreme of the maxima may thus be e v a l u a t e d . Now that the 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 P i ( z m i s known, v a r i o u s c h a r a c t e r i s t i c v a l u e s of z may be obtained. m •* • In p a r t i c u l a r , the expected value of extreme of maxima of r e s u l t a n t z may be obtained using f o l l o w i n g r e l a t i o n : E(z ) = / z m p , ( z ) d z m ( 3 . 4 3 ) m * m r m m z Since the expre s s i o n obtained f o r the 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 p(z ) was v a l i d only f o r l e v e l s of z g r e a t e r than the expected value E ( z ) , the lower l i m i t of i n t e g r a t i o n should a l s o be g r e a t e r than the expected v a l u e of the random process z ( t ) . The c a l c u l a t i o n of E ^ m ) r e q u i r e s a numerical i n t e g r a t i o n i n c o r p o r a t i n g the r e s u l t s o b t a i n e d here. E ^ m ) depends on both v/m7 and the number of maxima M. The l a t t e r i s g iven by s p e c i f i e d d u r a t i o n T m u l t i p l i e d by f m , the 47 maximum value of f with respect t o z: M (3.44) Hence the r e q u i r e d computations i n c l u d e the determination of with z, as given by equation (3.38). 3.6 HUNTINGTON AND GILBERT'S APPROACH Huntington and G i l b e r t (1979) have pre s e n t e d an a l t e r n a t i v e method by which the extreme v a l u e s of maxima of the r e s u l t a n t of two orthogonal components may be estimated. They s t a t e d the problem as f o l l o w s : Given a vect o r Gaussian p r o c e s s ( x ( t ) , y ( t ) ) with s p e c t r a and cross-spectrum S (co) , S (co) , S (co), what i s xx yy xy the l e v e l of magnitude z = /x 2+y 2 which with high p r o b a b i l i t y p w i l l not be exceeded d u r i n g time T ? They s o l v e d t h i s problem by assuming that f o r larg e z, the upcrossings are independent e v e n t s . Then the upcrossings occur with a Poisson d i s t r i b u t i o n determined by the mean frequency of u p c r o s s i n g f . Hence the p r o b a b i l i t y of no event i n time T i s given by: Thus the problem reduces to t h a t of f i n d i n g the mean Th i s i s obtained n u m e r i c a l l y from the v a r i a t i o n of f p(z) = e x p ( - f z T ) (3.45) frequency of u p c r o s s i n g f They p r o v i d e a lengthy 48 expres s i o n f o r f , analogous but d i f f e r e n t to equation (3.36), which depends only on the s p e c t r a l moments of x and y. T h e i r approach d i f f e r s s i g n i f i c a n t l y to the one developed here. In the pr e v i o u s approach, the d i s t r i b u t i o n of the extreme of any number of independent maxima has been obtained v i a the d i s t r i b u t i o n of maxima of random process z ( t ) . Thus, the approach used here i s able to d e s c r i b e i n turn the d i s t r i b u t i o n of the random process z ( t ) , the d i s t r i b u t i o n of the maxima z . the d i s t r i b u t i o n of the m extremes of the maxima z^, and f i n a l l y the expected value of z m« Huntington and G i l b e r t ' s (1979) approach i n v o l v e s only the p r o b a b i l i t y d i s t r i b u t i o n of the extremes z . 4. RESULTS AND DISCUSSION 4.1 LOADING AND RESPONSE SPECTRA In the present chapter, r e s u l t s are d e s c r i b e d f o r a f r e e l y f l o a t i n g box i n d i c a t e d i n F i g u r e 4.1 of s i z e L x B x D = 90 m x 90 m x 40 m where L i s the box l e n g t h , B i s the beam, and D i s the d r a f t . The r a d i i of g y r a t i o n about the x, y, z axes are 33.04 m, 32.09 m, 32.92 m r e s p e c t i v e l y and the (x,y,z) c o o r d i n a t e s of the ce n t r e of g r a v i t y of the box are (0,0,10.62m). T h i s example i s the same one as that c o n s i d e r e d by F a l t i n s e n and Michelsen (1974) and i s chosen in present study i n order t o pr o v i d e s u i t a b l e comparisons with t h e i r r e s u l t s . R e s u l t s have been ob t a i n e d f o r deep water wave c o n d i t i o n s u s i n g 48 f a c e t s to represent the box s u r f a c e . The numerical method used i s q u i t e g e n e r a l and can be used to compute l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude operators (RAO'S) f o r l a r g e o f f s h o r e s t r u c t u r e s of general shape both i n long- and s h o r t - c r e s t e d seas. The l o a d i n g t r a n s f e r f u n c t i o n s and the response amplitude operators i n l o n g - c r e s t e d seas have been compared with those p u b l i s h e d by F a l t i n s e n and Michelsen (1974). The d i r e c t i o n a l l y averaged l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude operators (see e q u a t i o n , 2.48) i n s h o r t - c r e s t e d seas have been compared t o those f o r l o n g - c r e s t e d seas. F i g u r e s 4.2 and 4.3 present the l o a d i n g t r a n s f e r f u n c t i o n s and the RAO'S i n l o n g - and s h o r t - c r e s t e d seas, 49 50 d e f i n e d i n equations (2.38) and (2.39). In the case of s h o r t - c r e s t e d seas, the l o a d i n g t r a n s f e r f u n c t i o n s and the RAO'S have been o b t a i n e d u s i n g d i r e c t i o n a l s preading f u n c t i o n s given by equation (2.44) with the spreading index s t a k i n g on values 1, 2,..., 10 so that the i n f l u e n c e of d i r e c t i o n a l spreading of waves may be assessed. However, i n the f i g u r e s r e s u l t s f o r s h o r t - c r e s t e d seas are presented only f o r three spreading f u n c t i o n s (s=1, 2, 3) i n order to show the t r e n d of the r e s u l t s . In a l l cases, the p r i n c i p a l wave d i r e c t i o n i s along the x a x i s ( 6 0 = 0*). One i n t e r e s t i n g f e a t u r e of the f i g u r e s i s that the sway, r o l l and yaw components of l o a d i n g and response are absent i n the case of symmetrical l o a d i n g i n l o n g - c r e s t e d seas, whereas they are a l l present i n the case of symmetrical l o a d i n g i n s h o r t - c r e s t e d seas. T h i s supports the statement made in S e c t i o n 2.5 that the l o a d i n g and response occur both i n - l i n e and t r a n s v e r s e to the p r i n c i p a l wave d i r e c t i o n , even i f the d i r e c t i o n a l wave spectrum i s symmetric. For i n c r e a s i n g v a l u e s of s, which corresponds to a more -c o n c e n t r a t e d d i r e c t i o n a l wave spectrum, the r e s u l t s f o r the i n - l i n e components of l o a d i n g and response (surge and p i t c h ) in s h o r t - c r e s t e d seas approach those f o r l o n g - c r e s t e d seas. This corresponds to i n c r e a s i n g energy being c o n c e n t r a t e d i n these i n - l i n e components with i n c r e a s i n g s, as expected. On the other hand, the energy a s s o c i a t e d with the t r a n s v e r s e components, sway and r o l l , i n s h o r t - c r e s t e d seas decreases 51 with i n c r e a s i n g s, again as expected. These o b s e r v a t i o n s can be e a s i l y drawn from the Figures 4.2 and 4.3. Comparisons between the l o a d i n g t r a n s f e r f u n c t i o n s T ^ J ( C J ) i n long- and s h o r t - c r e s t e d seas, which may be c a r r i e d out with reference to F i g u r e 4.2, show t h a t the e x c i t i n g f o r c e i n sway and the e x c i t i n g moment i n p i t c h i n s h o r t - c r e s t e d seas with a c o s 2 0 angular d i s t r i b u t i o n of wave energy are about 88% of the c o r r e s p o n d i n g values f o r l o n g - c r e s t e d seas with the same t o t a l energy. The e x c i t i n g f o r c e i n sway and the e x c i t i n g moment in r o l l i n s h o r t - c r e s t e d seas with the same angular d i s t r i b u t i o n of wave energy are about 52% of the c o r r e s p o n d i n g i n - l i n e components (surge and p i t c h r e s p e c t i v e l y ) i n l o n g - c r e s t e d seas with the same t o t a l energy or 59% of these i n - l i n e components i n s h o r t - c r e s t e d seas. These r e s u l t s compare well with those of Huntington & Thompson (1976) and Dean (1977) which were obtained f o r the case of a s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r . On the other hand, there i s no e f f e c t on heave f o r c e s due to wave d i r e c t i o n a l i t y (see F i g u r e 4.2c). T h i s can be e x p l a i n e d as f o l l o w s : For any s t r u c t u r e heaving i n s t i l l water, the r a d i a t e d waves can be r e l a t e d to damping c o e f f i c i e n t s . By the Haskind r e l a t i o n s , these damping c o e f f i c i e n t s can i n t u r n be r e l a t e d to the heave f o r c e due to an i n c i d e n t wave t r a i n . Hence t h i s f o r c e can be expressed s o l e y i n terms of heave r a d i a t i o n problem and thus i s independent of i n c i d e n t wave d i r e c t i o n . 52 The yaw moment (see Fi g u r e 4.2e) i n s h o r t - c r e s t e d seas i s about 1.2% of the value of the p i t c h moment i n the l o n g - c r e s t e d seas with the same t o t a l energy. However, the yaw moment remains same with d i f f e r e n t d i r e c t i o n a l spreading f u n c t i o n s . T h i s may again be e x p l a i n e d by a s i m i l a r argument as t h a t given f o r heave f o r c e s . The r e d u c t i o n i n the lo a d i n g s due to other d i r e c t i o n a l s preading f u n c t i o n s are summarized i n Table 4.1. The d e f i n i t i o n s of the l o a d i n g r e d u c t i o n f a c t o r s i n Table 4.1 are a c c o r d i n g to equation (2.51). However, i n the case of the t r a n s v e r s e (sway and r o l l ) components i n s h o r t - c r e s t e d seas, the l o a d i n g r e d u c t i o n f a c t o r s are d e f i n e d with r e s p e c t to the i n - l i n e (surge and p i t c h ) components i n l o n g - c r e s t e d seas; and the l o a d i n g r e d u c t i o n f a c t o r f o r yaw i s d e f i n e d with respect to the p i t c h l o a d i n g i n l o n g - c r e s t e d seas. Comparisons between the response amplitude o p e r a t o r s T ^ J ( C J ) i n long- and s h o r t - c r e s t e d seas, which may be c a r r i e d out with r e f e r e n c e to Fi g u r e 4.3, are s i m i l a r to those f o r loads d i s c u s s e d a l r e a d y . The surge and p i t c h amplitudes i n short c r e s t e d - s e a s are about 87% and 85% r e s p e c t i v e l y of the correspo n d i n g values i n l o n g - c r e s t e d seas with the same t o t a l energy. The sway and r o l l amplitudes i n s h o r t - c r e s t e d seas are about 52% and 57% r e s p e c t i v e l y of the i n - l i n e (surge and p i t c h ) components i n l o n g - c r e s t e d seas, or about 60% and 67% r e s p e c t i v e l y of the i n - l i n e components i n s h o r t - c r e s t e d seas. 53 There i s no e f f e c t of wave d i r e c t i o n a l i t y on the heave amplitude (see F i g u r e 4.3c). The yaw amplitude (see F i g u r e 4.3e) i s about 2.7% of the p i t c h amplitude i n l o n g - c r e s t e d seas. I t remains same with d i f f e r e n t values of spre a d i n g index s. The absence of wave d i r e c t i o n a l e f f e c t s on the heave and yaw amplitudes i s as expected. T h i s i s because the v a r y i n g parameters such as the added mass c o e f f i c i e n t s , the damping c o e f f i c i e n t s and the e x c i t i n g l o a d i n g s i n heave and yaw, which i n f l u e n c e the responses, are independent of i n c i d e n t wave d i r e c t i o n . The r e d u c t i o n i n the responses due to other s p r e a d i n g f u n c t i o n s are summarized i n Table 4.2. The d e f i n i t i o n s of response r e d u c t i o n f a c t o r s are s i m i l a r to those of the corresponding l o a d i n g r e d u c t i o n f a c t o r s . The l o a d and response s p e c t r a due to a. s p e c i f i e d i n c i d e n t wave c o n d i t i o n s are now presented. In the present study the example u n i - d i r e c t i o n a l wave spectrum used i s the ISSC spectrum sketched i n F i g u r e 4.4, with s i g n i f i c a n t wave height H s = 1 5 m and peak p e r i o d T p = 1 5 sec. The ISSC spectrum i s given as: S ( C J ) = a exp(-pya> 9 )/ws (4.1) where a = 488 H|/T p, 0 = 1948/T p. F i g u r e s 4.5 and 4.6 present the l o a d i n g and response s p e c t r a . These have been o b t a i n e d by m u l t i p l y i n g the wave 54 s p e c t r a l d e n s i t y by the corresponding l o a d i n g t r a n s f e r f u n c t i o n s (equation 2.39) and the response amplitude operators (equation 2.40) at a l l f r e q u e n c i e s . These f i g u r e s i n d i c a t e that the r e l a t i o n s h i p between the s p e c t r a f o r long-and s h o r t - c r e s t e d seas are s i m i l a r to those f o r the loadin g t r a n s f e r f u n c t i o n s and response amplitude operators presented i n F i g u r e s 4.2 and 4.3 and are d i s c u s s e d a l r e a d y . 4.2 PROBABILISTIC PROPERTIES OF LOADING AND RESPONSE Once the l o a d i n g and response s p e c t r a have been obtained, t h e i r p r o p e r t i e s may then be used to p r e d i c t the extreme of maxima of the l o a d i n g and response that are l i k e l y to. occur during a s p e c i f i e d sea s t a t e . In s h o r t - c r e s t e d seas, l o a d i n g and response occur both i n - l i n e (surge and p i t c h ) and tr a n s v e r s e (sway and r o l l ) to the p r i n c i p a l wave d i r e c t i o n . Thus, the extreme of maxima of the r e s u l t a n t of i n - l i n e and tr a n s v e r s e components of the l o a d i n g and response occur i n a random d i r e c t i o n with random magnitude. T h i s random q u a n t i t y c o n t a i n s i n f o r m a t i o n about chances of maxima o c c u r i n g i n the two components d i r e c t i o n s i m u l t a n e o u s l y . In the present s e c t i o n , the r e s u l t s are given f o r the p r o b a b i l i s t i c p r o p e r t i e s of the l o a d i n g and response components and for the h o r i z o n t a l r e s u l t a n t of i n - l i n e and tr a n s v e r s e components of the l o a d i n g and response. 55 4.2.1 LOADING AND RESPONSE COMPONENTS For the case of l o n g - c r e s t e d seas, f i g u r e 4.7 presents the expected value E(x m)//m^ of the extreme of maxima of components of the l o a d i n g and response, normalized with respect to the corresponding standard d e v i a t i o n /m^ of x, as a f u n c t i o n of the number of maxima M. T h i s corresponds simply to equation (3.3) and a p p l i e s e q u a l l y to a l l components of l o a d i n g and response. F i g u r e 4.8 p r e s e n t s the corresponding r e s u l t s f o r s h o r t - c r e s t e d seas and shows expected values E(x m)/t/m^ of the extreme of maxima as f u n c t i o n s of the number of maxima M, f o r d i f f e r e n t v a l u e s of the spreading index s. F i g u r e s 4.8(a) and 4.8(b) provide the r e s u l t s f o r i n - l i n e (surge and p i t c h ) and t r a n s v e r s e (sway and r o l l ) components r e s p e c t i v e l y . To provide s u i t a b l e comparisons between the r e s u l t s f o r l o n g - and s h o r t - c r e s t e d seas, the expected v a l u e s i n both F i g u r e s 4.8(a) and 4.8(b) have been normalized with respect to the standard d e v i a t i o n v/m0 of the i n - l i n e (surge and p i t c h ) component i n l o n g - c r e s t e d seas. As d i s c u s s e d i n S e c t i o n 4.1, the heave and yaw l o a d i n g and response are independent of the d i r e c t i o n a l spreading of waves. Thus the v a r i o u s s p e c t r a l moments of the heave and yaw l o a d i n g and response s p e c t r a are the same i n both l o n g - and s h o r t - c r e s t e d seas. Hence t h e i r s t a t i s t i c a l p r o p e r t i e s are u n a f f e c t e d due to spre a d i n g 56 of waves in s h o r t - c r e s t e d seas. Comparisons between r e s u l t s f o r long- and s h o r t - c r e s t e d seas have shown a s i m i l a r r e l a t i o n f o r the expected v a l u e s of the extreme of the maxima of i n - l i n e and t r a n s v e r s e components of the l o a d i n g and response as d i s c u s s e d i n the context of l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude o p e r a t o r s i n S e c t i o n 4.1. Huntington and G i l b e r t (1979) have a l s o p r e d i c t e d a s i m i l a r r e l a t i o n f o r the expected v a l u e s of the extreme of maxima of the i n - l i n e and t r a n s v e r s e components of l o a d i n g f o r a s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r . 4.2.2 HORIZONTAL RESULTANTS OF LOADING AND RESPONSE Turning now to the p r o b a b i l i s t i c p r o p e r t i e s of the h o r i z o n t a l r e s u l t a n t loads and motions, F i g u r e 4.9 presents the expected values E(zm)//STQ" of the extreme of the maxima of the r e s u l t a n t of i n - l i n e and t r a n s v e r s e components f o r the p a r t i c u l a r case s = 1, again as f u n c t i o n of the number of maxima M i n s h o r t - c r e s t e d seas. The expected v a l u e s i n F i g u r e 4.9 have been normalized with respect to the standar d d e v i a t i o nV m ^ of the i n - l i n e component i n s h o r t - c r e s t e d seas. F i g u r e 4.9 a p p l i e s e q u a l l y to the r e s u l t a n t of loads and responses corresponding t o surge and sway and to p i t c h and r o l l i n t u r n . The number of maxima M d i s c u s s e d i n the context of the v a r i o u s expected v a l u e s of extremes may be obtained 57 by equation (3.44), M = f m r , where T i s the s p e c i f i e d d u r a t i o n . In the case of the components, f i s a maximum at the zero l e v e l so that f = f 0 . On the other hand, i n m the case of the h o r i z o n t a l r e s u l t a n t s , f must be m obtained n u m e r i c a l l y from the v a r i a t i o n of f with z as z given in equation (3.36). F i g u r e 4.10 presents the frequency of u p c r o s s i n g f / f 0 of components of the l o a d i n g and response and i s a p p l i c a b l e to both long- and s h o r t - c r e s t e d seas. T h i s i s because the frequency of upcrossing f i s normalized with r e s p e c t to the frequency of zero u p c r o s s i n g f 0 which i s d i f f e r e n t f o r each component of "the l o a d i n g and response i n long- and s h o r t - c r e s t e d seas. F i g u r e 4.11 presents the frequency of upcrossing f / f 0 of the r e s u l t a n t of the i n - l i n e (surge or p i t c h ) and t r a n s v e r s e (sway or r o l l ) components of the l o a d i n g and response i n s h o r t - c r e s t e d seas f o r the p a r t i c u l a r case s = 1. The frequency of u p c r o s s i n g f i n t h i s case i s normalized Z with r e s p e c t to the frequency of zero u p c r o s s i n g f 0 of the i n - l i n e component (surge or p i t c h ) of l o a d i n g and response i n s h o r t - c r e s t e d seas. F i g u r e 4.11 a p p l i e s e q u a l l y to the r e s u l t a n t of loads and responses corresponding to surge and sway and to p i t c h and r o l l i n t u r n . Comparisons of the fr e q u e n c i e s of u p c r o s s i n g of the components ( F i g u r e 4.10) and of the h o r i z o n t a l r e s u l t a n t s i n s h o r t - c r e s t e d seas (Figure 4.11) show t h a t 58 f i s a maximum at the zero l e v e l , f = f 0 , and decays x m u e x p o n e n t i a l l y as z i n c r e a s e s . On the other hand, f , i s z zero at the zero l e v e l and i n c r e a s e s to the maximum f m before i t decays e x p o n e n t i a l l y . In f a c t f o r s = 1, we have f m / f o = 1.58 and t h i s occurs at z/Vnvo" = 0.78. F i n a l l y , F i g u r e s 4.12(a) and 4.12(b) present a comparison of the present method with that of Huntington and G i l b e r t (1979) f o r a s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r i n s h o r t - c r e s t e d seas. The v a r i a n c e of the i n - l i n e component has been taken to be three times that of the t r a n s v e r s e component as s t a t e d i n the example given by Huntington and G i l b e r t (1979). F u r t h e r , the i n - l i n e and t r a n s v e r s e components are u n c o r r e l a t e d . The comparison i s q u i t e favourable and goes some way towards e s t a b l i s h i n g the v a l i d i t y of the present method. 4.3 A DESIGN PROCEDURE F i n a l l y , i n order to i l l u s t r a t e the p r a c t i c a l a p p l i c a b i l i t y of the present method, a design procedure i s presented here to c a l c u l a t e design loads and responses both i n long- and s h o r t - c r e s t e d seas i n a s p e c i f i e d sea s t a t e . The design procedure takes the f o l l o w i n g s t e p s : 1. L i n e a r d i f f r a c t i o n theory i s used to c a l c u l a t e the l o a d i n g t r a n s f e r f u n c t i o n s and response amplitude o p e r a t o r s i n both l o n g - as w e l l as s h o r t - c r e s t e d seas corresponding to a s p e c i f i e d d i r e c t i o n a l spreading f u n c t i o n . 59 amplitude o p e r a t o r s to o b t a i n the cor r e s p o n d i n g l o a d i n g and response s p e c t r a and c r o s s - s p e c t r a . S u i t a b l e i n t e g r a t i o n of the v a r i o u s l o a d i n g and response s p e c t r a and c r o s s - s p e c t r a are c a r r i e d out to o b t a i n the r e q u i r e d moments, i n c l u d i n g the z e r o t h , f i r s t and second moments. These s p e c t r a l moments are then used to estimate the expected v a l u e s and the frequency of u p c r o s s i n g of the v a r i o u s l o a d i n g and response components and t h e i r h o r i z o t a l r e s u l t a n t s u sing the theory presented i n the chapter 3. The number of maxima M i s obtained by m u l t i p l y i n g the maximum frequency of u p c r o s s i n g f with the s p e c i f i e d storm d u r a t i o n T. The v a r i o u s expected values of extremes of maxima of the components of l o a d i n g and response and t h e i r h o r i z o n t a l r e s u l t a n t i n a s p e c i f i e d - storm d u r a t i o n ( i . e . corresponding to the number of maxima obtained i n step 5), are obtained using equations (3.3) and (3.36). 4.3.1 A WORKED EXAMPLE R e s u l t s are presented below f o r the f r e e l y f l o a t i n g box d e s c r i b e d i n S e c t i o n 4.1 f o r both l o n g - and s h o r t - c r e s t e d seas. The r e s u l t s f o r s h o r t - c r e s t e d seas are given f o r the p a r t i c u l a r case of a c o s i n e squared d i r e c t i o n a l spreading f u n c t i o n (s=1). The s p e c i f i e d storm d u r a t i o n i s 12 hours and the sea s t a t e i s 60 d i r e c t i o n a l spreading f u n c t i o n ( S=1). The s p e c i f i e d storm d u r a t i o n i s 12 hours and the sea s t a t e i s d e s c r i b e d by the ISSC spectrum with H g = 15 m and T p = 15 sec, sketched i n F i g u r e 4.4. The s p e c i f i e d wave c o n d i t i o n i s thus as f o l l o w s : Storm d u r a t i o n =12 hours U n i - d i r e c t i o n a l wave spectrum = ISSC S i g n i f i c a n t wave h e i g h t , H g = 15 m Peak p e r i o d , T p = 1 5 sec Spreading index, s =1 Step 1 was c a r r i e d out to gi v e the r e s u l t s shown i n F i g u r e s 4.2 and 4.3. Step 2 was c a r r i e d out us i n g the s p e c i f i e d d i r e c t i o n a l wave spectrum, to gi v e the r e s u l t s shown i n F i g u r e s 4.5 and 4.6. Step 3 was c a r r i e d out to ob t a i n the v a r i o u s s p e c t r a l moments of the l o a d i n g and response s p e c t r a . Step 5 was then c a r r i e d out, us i n g F i g u r e s 4.10 and 4.11, to o b t a i n number of maxima i n 12 hours d u r a t i o n . Step 6 was c a r r i e d out u s i n g F i g u r e s 4.7 4.10 to o b t a i n the extreme l o a d i n g and response c o n d i t i o n under 12 hours storm d u r a t i o n . The r e s u l t s are given i n Ta b l e s 4.3 and 4.4 f o r the lo a d i n g and response r e s p e c t i v e l y . These t a b l e s p r o v i d e the expected v a l u e s of the extreme of the maxima of the components and t h e i r h o r i z o n t a l r e s u l t a n t s . 5. CONCLUSIONS 1 . L i n e a r d i f f r a c t i o n wave theory has been extended to estimate the l o a d i n g and response of l a r g e o f f s h o r e s t r u c t u r e s of a r b i t r a r y shape i n both l o n g - and s h o r t - c r e s t e d random seas. The r e s u l t s from t h i s procedure are the s p e c t r a and c r o s s - s p e c t r a of the components of l o a d i n g and response. 2. The i n - l i n e components (surge and p i t c h ) of the lo a d i n g and response i n s h o r t - c r e s t e d seas are l e s s than the corresponding values based on the assumption of a l o n g - c r e s t e d sea. The r e d u c t i o n i n these i n - l i n e components of the l o a d i n g and response depends on the degree of spreading of waves, with a g r e a t e r r e d u c t i o n corresponding to a g r e a t e r degree of sprea d i n g . The re d u c t i o n i n these i n - l i n e components of the l o a d i n g and response i s expected t o be q u i t e s i g n i f i c a n t i n desi g n . 3. The t r a n s v e r s e components (sway and r o l l ) of the lo a d i n g and response, p r e d i c t e d to be zero i n l o n g - c r e s t e d seas, are q u i t e s i g n i f i c a n t i n s h o r t - c r e s t e d seas. The magnitidues of these t r a n s v e r s e components are greater f o r a g r e a t e r degree of spreading of waves. 4. For d e c r e a s i n g spreading of waves, the r e s u l t s f o r s h o r t - c r e s t e d seas approach those f o r l o n g - c r e s t e d seas. 5. There i s no e f f e c t of wave d i r e c t i o n a l i t y on the heave or yaw l o a d i n g and response. 6. The expected values of the extreme of maxima of i n - l i n e components (surge and p i t c h ) of l o a d i n g and response i n 61 62 a s p e c i f i e d d u r a t i o n are l e s s than the corresponding values based on the assumption of l o n g - c r e s t e d seas. 7. The expected values of the extreme of maxima of t r a n s v e r s e components (sway and r o l l ) of l o a d i n g and response in a s p e c i f i e d d u r a t i o n , p r e d i c t e d to be zero i n l o n g - c r e s t e d seas, are q u i t e s i g n i f i c a n t i n s h o r t - c r e s t e d seas. 8. The s t a t i s t i c a l p r o p e r t i e s of the heave and yaw l o a d i n g and response components are u n a f f e c t e d by the d i r e c t i o n a l spreading of waves i n s h o r t - c r e s t e d seas. 9. An a n a l y t i c a l method has been developed by which the p r o b a b i l i s t i c p r o p e r t i e s of the h o r i z o n t a l l o a d i n g or response r e s u l t a n t of the i n - l i n e (surge or p i t c h ) and t r a n s v e r s e (sway or r o l l ) components of the l o a d i n g and response i n s h o r t - c r e s t e d seas can be d e s c r i b e d . The method i s general and can be used i n a l l s i t u a t i o n s with two orthogonal components which are Gaussian with zero means and which have d i f f e r e n t standard d e v i a t i o n s . The method r e q u i r e s the f i r s t and second moments of s p e c t r a as primary input parameters. 10. Experiments are r e q u i r e d to measure the l o a d i n g and response of l a r g e or s l e n d e r ( f i x e d or f l o a t i n g ) o f f s h o r e s t r u c t u r e s i n d i r e c t i o n a l seas with or without u n d e r l y i n g c u r r e n t s . On the other hand, t h i s w i l l serve as a t o o l f o r v e r i f y i n g the p r e d i c t i o n s based on v a r i o u s a n a l y t i c a l or numerical procedures. REFERENCES 1. Abramowitz, M. and Stegun, I.A., 1964, Handbook of mathematical functions, N a t i o n a l Bureau of Standard Mathematics S e r i e s , Dover p u b l i c a t i o n s I n c . , New York. 2. B a t t j e s , A. J . , 1982, E f f e c t s of s h o r t c r e s t e d n e s s on wave loads on long s t r u c t u r e s , Applied Ocean Research, V o l . 4, No. 3, pp. 165-172. 3. Bendat, J.S. and P i e r s o l , A.G., 1966, Measurement and analysis of random data, J . Wiley & Sons, New York. 4. Benjamin, J.R. and C o r n e l l , A.C., 1970, Probability, s t a t i s t i c s and decision for c i v i l engineers, McGraw H i l l , New York. 5. Borgman, L.E., 1969, D i r e c t i o n a l s p e c t r a models for design use, Proc. Offshore Technology Conference, Houston, Paper No. OTC 1069, V o l . 1, pp. 721-746. 6. Borgman, L.E., 1972, S t a t i s t i c a l models f o r ocean waves and wave f o r c e s , Advances in Hydroscience, V o l . 8, pp. 139-181 . 7. Borgman, L.E. and Y f a n t i s , V., 1981, D i r e c t i o n a l s p e c t r a l d e n s i t y of f o r c e s on f i x e d platforms, Confer ence on Directional Wave Spectra Applications, Berkley, pp. 315-332. 8. Bury, K.V., 1975, S t a t i s t i c a l models in applied science, J . Wiley & Sons, New York. 9. C a r t w r i g h t , D.E. and Longuet-Higgins, M.S., 1956, The s t a t i s t i c a l d i s t r i b u t i o n of the maxima of a random f u n c t i o n , Proc. Roy. Soc. , Ser. A, V o l . 237, pp. 212-232. 10. Clough, W.R. and Penzien, J . , 1975, Dynamics of structures, McGraw H i l l , New York. 11. D a l l i n g a , R.P., A a l b e r s , A.B., Vegt v e r den, J.W.W., 1984, Design aspects f o r t r a n s p o r t of Jackup platforms 63 64 on barge, Proc. Offshore Technology Conference, Houston, Paper No. OTC 4733, pp. 195-202. 12. Dean, R.G., 1977, Hyb r i d method of computing wave l o a d i n g , Proc. Offshore Technol ogy Confer e nee, Houston, Paper No. OTC 3029, pp. 483-492. 13. F a l t i n s e n , O.M. and Mic h e l s e n , F.C., 1974, Motions of l a r g e s t r u c t u r e s i n waves at zero Froude number, Proc. Int. Symp. on the Dynami cs of Marine Vehicles and Structures in Waves, U n i v e r s i t y C o l l e g e , London, pp. 97-140. 14. F o r i s t a l l , G.Z., Ward, E.G., Cardone, V . J . and Borgman, L.E. , 1978, D i r e c t i o n a l s p e c t r a and kinematics of su r f a c e g r a v i t y waves i n T r o p i c a l storm D e l i a , /. Phys. Oceanography, V o l . 8, pp. 808-909. 15. G a r r i s o n , C.J. and Chow, P.Y., 1972, Wave f o r c e s on submerged bodies, /. Waterways, Harbours and Coastal Eng. Div., ASCE, V o l . 98, No. WW3, pp. 375-392. 16. G a r r i s o n , C.J., 1978, Hydrodynamic l o a d i n g on l a r g e o f f s h o r e s t r u c t u r e s : Three-dimensional source d i s t r i b u t i o n methods, In Numerical Methods in Offshore Engineering, eds., O.C. Z i e n k i e w i c z , R.W. Lewis, and K.G. Stagg, J . Wiley & Sons, England, pp. 97-140. 17. Hackley, M.B., 1979, Wave fo r c e s i m u l a t i o n s t u d i e s i n d i r e c t i o n a l seas, Proc. Conf. Behaviour of Offshore St ruct ures, BOSS'79, London, V o l . I, pp. 187-219. 18. Hogben, N. and Standing, R.G., 1974, Wave loads on l a r g e bodies, Proc. Int. Symp. on Dynami cs of Marine Ve hi cles and St ruct ur es in Waves, U n i v e r s i t y C o l l e g e , London, pp. 258-277. 19. Huntington, S.W. and Thompson, D.M., 1976, Forces on l a r g e c y l i n d e r i n m u l t i d i r e c t i o n a l random waves, Proc. Offshore Technology Conf er ence, Houston, Paper No. OTC 2539, V o l . I I , pp. 169-183. 20. Huntington, S.W., 1979, Wave l o a d i n g on l a r g e c y l i n d e r s i n short c r e s t e d seas. In Mechanics of Wave Induced Forces on Cyli nders, ed. T. L. Shaw, Pitman, London, PP. 65 636-649. 21. Huntington, S.W. and G i l b e r t , G., 1979, Extreme f o r c e s i n short c r e s t e d seas, Proc. Offshore Technology Conference, Houston, Paper No. OTC 3595, V o l . I l l , pp. 2075-2084. 22. Isaacson, M. de St. Q., 1979, Wave induced f o r c e s i n the d i f f r a c t i o n regime, In Mechanics of Wave Induced Forces on Circular Cylinders, ed. T. L. Shaw, Pitman, London, pp. 68-89. 23. Isaacson, M. de St. Q., 1985, Wave e f f e c t s on l a r g e o f f s h o r e s t r u c t u r e s of a r b i t r a r y shape, Coastal/Ocean Engineering Report, Department of C i v i l E n g i n e e r i n g , U n i v e r s i t y of B r i t i s h Columbia, Vancouver, Canada. 24. Lambrakos, K.F., 1982, Marine p i p e l i n e dynamic response to waves from d i r e c t i o n a l wave s p e c t r a , Ocean Engineering, V o l . 9, No. 4, pp. 385-405. 25. L i n , Y.K., '1976, Probabilistic theory of structural dynamics, Robert E. K r i e g e r P u b l i s h i n g Company, Malbar, F l o r i d a . 26. Longuet-Higgins, M.S., 1952, On the s t a t i s t i c a l d i s t r i b u t i o n of the heights of sea waves, J. Marine Research, V o l . 11, pp. 245-266. 27. Lovaas, J.H., 1984, Hydrodynamic loads and response of marine s t r u c t u r e s , Proc. Symp. on Description and Modelling of Directional Seas, Paper No. D-1, T e c h n i c a l U n i v e r s i t y , Copenhagen, Denmark. 28. O c h i , M.K., 1982, S t o c h a s t i c a n a l y s i s and p r o b a b i l i s t i c p r e d i c t i o n of random seas, Advances in Hydroscienee, V o l . 13, pp. 217-375. 29. P a p o u l i s , A., 1965, Probability, random variables, stochastic processes, McGraw H i l l , New York. 30. P i n k s t e r , J.A., 1984, Numerical m o d e l l i n g of d i r e c t i o n a l seas, Proc. Symp. on Description and Modelling of Directional Seas, Paper No. C-1, T e c h n i c a l U n i v e r s i t y , 66 Copenhagen, Denmark. 31. Sarpkaya, T. and Isaacson, M., 1981, Mechanics of wave forces on offshore structures, Van Nostrand Reinhold, New York. 32. Shinozuka, M., Fang, S.L.S. and N i s h i t a n i , A., 1979, Time-domain s t r u c t u r a l response i n a short c r e s t e d sea. Journal of Energy Resources Technology, T r a n s a c t i o n s of ASME, V o l . 101, pp. 270-275. 33. Teigen, P:S., 1983, The response of TLP i n sho r t c r e s t e d waves, Proc. Offshore Technology Conference, Houston, Paper No. OTC 4642, V o l . I I I . 34. T i c k e l l , R.G., and Elwany, M.H.S., 1979, A p r o b a b i l i s t i c d e s c r i p t i o n of a member i n short c r e s t e d seas, In Mechanics of Wave Induced Forces on Cylinders, ed. T.L. Shaw, Pitman, London, pp. 561-576. APPENDIX I. GREEN'S FUNCTION The Green's f u n c t i o n f o r the three-d i m e n s i o n a l case can e i t h e r be expressed i n terms of an i n t e g r a l or an i n f i n i t e s e r i e s . The i n t e g r a l expreesion i s 1 . 1 G(x,£) = g,+ 2 PV /F,F 2 J0(nr)6n - i C 0 c o s h [ k ( $ + d ) ] c o s h [ k ( z + d ) ] J 0 ( k r ) (AI.1) where PV denotes the p r i n c i p a l value of i n t e g r a l . The i n f i n i t e s e r i e s e xpression i s G(x,£) = -iC 0cosh[k(S+d) ] c o s h [ k ( z + d ) ] H 0 ( 1 ) ( k r ) + 4 Z C c o s [ M _ ( S + d ) ] c o s [ u ( z+ d ) ] K 0 ( u r) (AI.2) . 4 11 i l l 11 i l l m= l where r = [ ( x - £ ) 2 + ( y - T ? ) 2 ] l / 2 R = [ r 2 + ( z - $ ) 2 ] l / 2 = [ r 2 + ( z + 2 d + $ ) 2 ] l / 2 P t - ^ MtanhMd-v 67 68 F 2 = C3cp{-Md)cosh^( S+d) )cosh(Ai(z+d) ) c o s h ( ^ d ) r _ 2 T T U 2 - K 2 ) *~o -(k2-v2)d+i> r _ m *~m " u2 + v2 (u^+v2)a+v v °= k t a n h ( k d ) = 4^ a n d M m a r e t h e r e a l p o s i t i v e r o o t s o f t h e e q u a t i o n j u m t a n ( n m d ) + ^  = 0 (A I . 3 ) t a k e n i n a s c e n d i n g o r d e r . J 0 i s t h e B e s s e l ' s f u n c t i o n s o f f i r s t k i n d , H 0 ^ 1 ^ i s t h e H a n k e l f u n c t i o n o f t h e f i r s t k i n d , K 0 i s t h e m o d i f i e d B e s s e l ' s f u n c t i o n o f f i r s t k i n d , a l l t h e s e b e i n g o f o r d e r z e r o . 3G The g r a d i e n t a t x i s a l s o r e q u i r e d a nd i s g i v e n a s : I<2'r> • H v If »t <*'•«> where n x ( x - $ ) + n y ( y - r ? ) n r = ( A I . 5 ) 69 7\C Thus the two expr e s s i o n s f o r corresponding to equations (AI.1) and (AI.2) are given as: [n ( x - $ ) + n (y -Tj)+n (z-$)] 9G x f_ 9 n R3 [ n x ( x - U + n y (y-T/)+n 2(z+2d+t) ] R I 3 + 2/MF, [ F 3 J 0 (*ir)n - F 2 J , (nr)n_]dn o r z i k C 0 c o s h [ k ( t + d ) ] [ s i n h ( k ( z + d ) ) J 0 ( k r ) n + c o s h ( k ( z + d ) ) J , ( k r ) n r ] (AI.6) and !§• - i k c o s h ( k ( t + d ) ) [ c o s h ( k ( z + d ) ) ] H ( 1 } ( k r ) n r - s i n h ( k ( z + d ) ) H 0 ( 1 J n ] z 4 Z M mC mcos(/i m($+d)) [cos(ju m(z+d)) K , ( M m r ) n r m= 1 + s i n ( M m ( z + d ) ) K 0 ( M m r ) n z ] r e s p e c t i v e l y (AI.7) where sinh(/i(z+d)) F 3 = exp(-Md) ( A I . 8 ) cosh(Md) Where kr i s l a r g e , the s e r i e s form converges r a p i d l y because 70 of the behaviour of K 0 ( j x r ) . When kr tends to z e r o , the B e s s e l ' s f u n c t i o n K 0 ( M C ) tends to i n f i n i t y and the s e r i e s form can not be used. In t h i s case, i n t e g r a l form i s more convenient. A d e t a i l e d treatment of above mentioned Green's f u n c t i o n i s given by Isaacson (1985). APPENDIX I I . PROBABILITY DENSITY FUNCTION OF RESULTANT Equation (3.13) i n chapter 3 i s given as: p(z) = j z p( z c o s 0 , z s i n 0 ) d0 (AII.1) S u b s t i t u t i n g equation (3.8) i n t o equation (AII.1) and i n t e g r a t i n g with r e s p e c t to 8 from 0 to 2TT, we get: z 2 * p(z) = fexp[ - z 2 ( m o c o s 2 0 + m o s i n 2 0 - 2X y/m om osin0cos0) 27TK, O / 2 ( 1 - X 2 ) m o m o ] d 0 ( A l l . 2 ) where K, = /m 0m 0(1-X 2) Ap p l y i n g t r i g o n o m e t r i c a l i d e n t i t i e s , ( A l l . 2 ) can be expressed as: z 2TT p(z) = j ^ - e x p ( - K 2 z 2 ) / e x p [ - z 2 ( K c c o s 2 0 - K g s i n 2 0 ) ] d 0 ( A I I . 3 ) where K 2 = (m 0+m 0)/4(1-X 2)m 0m 0 K = (m 0-m 0)/4(l-X 2)m 0m 0 c K_ = 2X/m 0m 0/4(1-X 2)m 0m 0 71 72 Equ a t i o n ( A l l . 3 ) can be w r i t t e n as: z 2 7 r p ( z ) = — — e x p ( - K 2 z 2 ) /exp{-z 2A cos(20+0) }d0 ( A l l . 4 ) 27TK, o where A = /K 2 + K 2 ' ( A l l . 5 ) and 0 = tan 1 ( K /K ) ( A l l . 6 ) S u b s t i t u t i n g X = 0+0 i n t o equation ( A l l . 4 ) , we get: 0+47T p(z) = j^- e x p ( - K 2 z 2 ) Jfexp(-z 2A cosX)dX ( A l l . 7 ) 1 0 Equ a t i o n ( A l l . 7 ) can be w r i t t e n as: z ' 4TT p(z) = r r - e x p ( - K 2 z 2 ) -jr- fexp{-z2h cosX) dX ( A l l . 8 ) 1 4 7 r o Us i n g i d e n t i t y (9.6.16) from Abramowitz and Stegun (1964), eq u a t i o n ( A l l . 7 ) can f i n a l l y be w r i t t e n as: p(z) = ^ e x p ( - K 2 z 2 ) I 0 ( z 2 A ) ( A l l . 9 ) 1 where I 0 i s the m o d i f i e d B e s s e l ' s f u n c t i o n of order ze r o . 73 s C(s) 1 2/rr 2 8 / 3 7 T 3 16 / 5 7 T 4 1 28/35* 5 256/63* 6 1 024/231 TT 7 2048 / 4 2 9 7 T 8 32768/6435* 9 65536/12155TT 10 2621 44/461 89ir T a b l e 2.1 N o r m a l i z i n g f a c t o r of d i r e c t i o n a l s p r e a d i n g f u n c t i o n s 74 s SURGE SWAY HEAVE ROLL PITCH YAW 1 0.880 0.516 1 .0 0.515 0.888 0.016 2 0.924 0.426 1.0 0.427 0.930 0.014 3 0.945 0.372 1.0 • 0.374 0.949 0.013 4 0.956 0.333 1.0 0.337 0.961 0.012 5 0.964 0.307 1.0 0.310 0.967 0.012 6 0.969 0.288 1.0 0.291 0.972 0.012 7 0.971 0.272 1.0 0.276 0.974 0.012 8 0.972 0.260 1.0 0.265 0.975 0.012 9 0.972 0.251 1.0 0.255 0.975 0.012 10 0.972 0.242 1.0 0.246 0.974 0.012 TABLE 4.1 L o a d i n g r e d u c t i o n f a c t o r R^ .. i n s h o r t - c r e s t e d s e a s f o r d i f f e r e n t s v a l u e s . 75 s SURGE SWAY HEAVE ROLL PITCH YAW 1 0.874 0.520 1 .0 0.567 0.847 0.040 2 0.919 0.427 1 .0 0.456 0.898 0.030 3 0.941 0.371 1.0 0.391 0.923 0.030 4 0.957 0.333 1.0 0.348 0.938 0.029 5 0.961 0.305 1.0 0.317 0.947 0.028 6 0.966 0.285 1 .0 0.295 0.954 0.028 7 0.969 0.271 1 .0 0.279 0.957 0.028 8 0.970 0.258 1 .0 0.265 0.959 0.027 9 0.970 0.249 1 .0 0.255 0.960 0.027 10 0.970 0.241 1 .0 0.246 0.960 0.026 Table 4.2 Motion r e d u c t i o n f a c t o r R. ^  i n s h o r t - c r e s t e d seas f o r d i f f e r e n t s v a l u e s . EXTREME LOADING LONG-CRESTED SHORT-CRESTED Surge Sway Re s u l t a n t (surge and sway) Heave R o l l P i t c h R e s u l t a n t ( r o l l and p i t c h ) Yaw 630143 kN 282966 kN 7411021 kN-m 554522 kN 327674 kN 570506 kN 282966 kN 3807432 kN-m 6549120 kN-m 6772205 kN-m 615837 kN-m Table 4.3 Expected v a l u e s of the extremes of the maxima l o a d i n g . EXTREME RESPONSE LONG-CRESTED SHORT-CRESTED Surge Sway Re s u l t a n t (surge and sway) Heave R o l l P i t c h R e s u l t a n t ( r o l l and p i t c h ) yaw 7.13m 12.90 m 0.766 rad 6.26 m 3.70 m 6.44 m 12.90 m 0.433 rad 0.646 rad 0.670 rad 0.12 rad Table 4.4 Expected v a l u e s of the extremes of the maxima response. f / f 0 F i g u r e 2.1 Sketch of a u n i - d i r e c t i o n a l wave spectrum. F i g u r e 2.2 Sketch of a d i r e c t i o n a l wave spectrum. 79 i 1 _' |t,(heove) v* / y V / £5( pitch) ^ C,Csur Q.) ^—X W//////P/////A ^ « — F i g u r e 2.3 D e f i n i t i o n s ketch of a f l o a t i n g body showing component motions. F i g u r e 2.4 D e f i n i t i o n s ketch of the i n c i d e n t wave d i r e c t i o n . e - 0 O ( deg . ) F i g u r e 2.5 D i r e c t i o n a l spreading f u n c t i o n f o r d i f f e r e n t s v a l u e s . .L. n i i n u n n n a t m n / n n n F i g u r e 4 . 1 D e f i n i t i o n s ketch of a f l o a t i n g box 81 0.2 0.4 0.6 u> rad/sec —i 0.8 o 3.5 3H 2.5 H rsi EH 2 1.5 0.2 (b) EXCITING FORCE IN SWAY S = 1 — s = 2 — s = 3 0.4 0.6 iii rad/sec — i 0.8 F i g u r e 4.2 Loading t r a n s f e r f u n c t i o n s i n long- and s h o r t - c r e s t e d seas. 82 o 5 4.5 E 3.5 H 2 •* 3 "3 " r o 2.5 ro EH 2H 1.5 1 0.5 0.2 «> 0.40 o 0.35 e i E-0.30 H 0.25 0.20 H 0.15 ^ 0.10 0.2 ( c ) EXCITING FORCE IN HEAVE ( a l l s ) i 1 r — 0.4 0.6 oi r a d / s e c — i 0.8 (d) EXCITING MOMENT I N ROLL S = 1 S = 2 » s — 3 0.4 0.6 u> r a d / s e c • 0.8 F i g u r e 4.2 ( c o n t . ) L o a d i n g t r a n s f e r f u n c t i o n s i n l o n g - a n d s h o r t - c r e s t e d s e a s . 83 o 0.8 x 0.7 E I 2 0.6 ^ 0.5-m in E-> 0.4 0.3 0.2 9 15-| o -— 14-X -13-12-11-6 10-a i 9-2 8-7-3 6-v o 5-4-3-2-1-0 -0.2 (e) EXCITING MOMENT IN PITCH l o n g - c r e s t e d s = 1 — i — 0.4 0.6 0.8 w rad/sec (f ) EXCITING MOMENT IN YAW ( a l l s) 0.4 0.6 u> rad/sec — i 0.8 F i g u r e 4.2 (cont.) Loading t r a n s f e r f u n c t i o n s i n l o n g - and s h o r t - c r e s t e d seas. 84 O.H • , . , , , 0.2 0.4 0.6 0.8 u) rad/sec F i g u r e 4.3 Response amplitude o p e r a t o r s i n long-s h o r t - c r e s t e d seas. and 8 5 (c) HEAVE . ( a l l s) u rad/sec (d) ROLL Figure 4.3 (cont.) Response amplitude o p e r a t o r s i n lo n g - and s h o r t - c r e s t e d seas. 86 co I 8 .O - 1 o x 7.0-j 6.0 g 5.0 H (0 4.0-3 3.0H in in 2.0 -\ L O H 0.0 0.2 (e) PITCH -long-crested -s = 1 s = 2 s = 3 0.4 0.6 OJ rad/sec — i 0.8 -* 0.9 n I o X 0.8 0.7 O 0.6 \ u -r- o.4-| 3 E-i 0.3 H 0.2 H o .H 0.2 ( f ) YAW ( a l l s) — I • 1— 0.4 0.6 CJ rad/sec 0.8 Fig u r e 4 . 3 (cont.) Response amplitude o p e r a t o r s in long- and s h o r t - c r e s t e d seas. 87 co rad/sec F i g u r e 4 . 4 U n i - d i r e c t i o n a l wave spectrum used i n computation. 88 oH 1 1 • i > 1 0.2 0.4 0.6 0.8 co rad/sec o 55-1 co rad/sec F i g u r e 4 . 5 Loading s p e c t r a i n l o n g - and s h o r t - c r e s t e d seas. F i g u r e 4.5 (cont.) Loading s p e c t r a i n l o n g - and s h o r t - c r e s t e d seas. 9 0 o X (0 u 25 n 20 A u 15 a; V) i E I z 10 5 ID 3 0.2 (e) EXCITING MOMENT IN PITCH l o n g - c r e s t e d s = 1 S = 2 s = 3 0.4 0.6 CJ rad/sec 0.8 o o X 40 35 m 30 VJ \ U cu 25 tn I £ I z 20 ^ 15 H 10 5H 0.2 ( f ) EXCITING MOMENT IN YAW ( a l l s) i 0.6 CJ rad/sec "o.8 Figure 4.5 (cont.) Loading s p e c t r a i n long- and s h o r t - c r e s t e d seas. 91 0-| , , , — , , , 0.2 0.4 0.6 0.8 co rad/sec Figure 4.6 Response s p e c t r a i n l o n g - and s h o r t - c r e s t e d seas. 92 o.oo-l . 1 . 1 . 1 0.2 0.4 0.6 0.8 w rad/sec Figure 4.6 (cont.) Response s p e c t r a i n long- and s h o r t - c r e s t e d seas. 9 3 0.45-1 0.40 0.35 o <u 0.30 (A I <o 0.25 3 0.20 H in in co. 0.15 H 0.10 0.05 H 0.00 (e) PITCH •long-crested s = 1 s = 2 s = 3 0.2 0.4 0.6 co rad/sec 0.8 1.75 H 1.50 H o 0) (/) I m u ~. 1-25 3 CO H 0.75 H 0.50 0.2 ( f ) YAW ( a l l S) — i > 1 — 0.4 0.6 co rad/sec 0.8 F i g u r e 4.6 (cont.) Response s p e c t r a i n long- and s h o r t - c r e s t e d seas. 94 Figure 4.7 Expected values of the extreme of the maxima of l o a d i n g and response i n l o n g - c r e s t e d seas. 5.2 4.8 H 4.4 H 95 e 3.6 H 3.2 H 2.8 2.4 H IN-LINE COMPONENT 2 4 3-i 2.8-2.6-2.4-2.2-2-\ 1.8-1.6-W 1.4-1.2-1-0.8-l o g 1 Q ( M ) TRANSVERSE COMPONENT 2 - r-4 l o g 1 ( J ( M ) — S = 1 - s = 2 - s = 3 5 5 6 F i g u r e 4.8 Expected values of the extreme of the maxima of loa d i n g and response i n s h o r t - c r e s t e d seas. 96 F i g u r e 4.9 E x p e c t e d v a l u e s of the extreme of the maxima of h o r i z o n t a l r e s u l t a n t s i n s h o r t - c r e s t e d seas. 97 Figure 4.10 Frequency of u p c r o s s i n g of the components of lo a d i n g and response i n long- and s h o r t - c r e s t e d seas. 1.6 H N Figure 4.11 Frequency of u p c r o s s i n g of h o r i z o n t a l r e s u l t a n t s of l o a d i n g and response i n s h o r t - c r e s t e d seas. 9 8 0.8 0.4-o e <N 3-(a) FREQUENCY OF UPCROSSING Present method Huntington and G i l b e r t - r -4 5 (b) EXPECTED VALUE Present method Huntington and G i l b e r t o i 2 3 4 5 6 i o g l 0 ( M ) Figure 4.12 Comparison of present method with that of Huntington and G i l b e r t (1979) f o r a s u r f a c e - p i e r c i n g c i r c u l a r c y l i n d e r . 

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