DIRECTIONAL WAVE EFFECTS ON LARGE OFFSHORE STRUCTURES OF ARBITRARY SHAPE 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., I n d i a n I n s t i t u t e Of T e c h n o l o g y , M a d r a s , I n d i a , A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS MASTER FOR THE DEGREE OF OF APPLIED SCIENCE in FACULTY OF GRADUATE STUDIES D e p a r t m e n t Of C i v i l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH MAY © COLUMBIA 1985 SANJAY SINHA, 1985 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by t h e head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . It is understood t h a t copying o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department O f C i v i l Engineering The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1 Y 3 Date DE-6 <"*/fm ^/xdjlr ^rr^L^. ABSTRACT A n u m e r i c a l method wave effects on large s h a p e , b a s e d on an theory for i s described offshore extension regular waves. developed t o compute loading amplitude operators spectra for both Cosine powered independent and and in loading assumed random loading probabilistic Gaussian and transverse the and response random functions show that for there and the is a i n the o f t h e components of the waves. Since the sea s u r f a c e a Gaussian d i s t r i b u t i o n , response which a r e hence of In waves. of t h e r e s u l t s f o r loading, properties variables. loading horizontal the loading these short-crested components occur both is are also waves, the in-line and t o t h e p r i n c i p a l wave d i r e c t i o n . Thus t h e maximum horizontal describe follow response spreading and r e s p o n s e a r e d e s c r i b e d . to has been and short-crested seas wave functions the r e s p o n s e , due t o s h o r t c r e s t e d n e s s The program waves. C o m p a r i s i o n s reduction of a r b i t r a r y have been u s e d t o a c c o u n t short-crested significant directional diffraction computer transfer directional of linear hence long- of f r e q u e n c y shortcrestedness long- and study structures of A to and r e s p o n s e may occur d i r e c t i o n . An a n a l y t i c a l method a l s o the p r o b a b i l i s t i c components and the is properties maxima of i n an a r b i t r a r y developed to of t h e maxima o f their horizontal resultants. In freely the present floating study, box. results Comparisons i i are described a r e made w i t h for a published results and are found t o be quite favourable. Table of Contents ABSTRACT i L I S T OF TABLES i v L I S T OF FIGURES v i ACKNOWLEDGEMENTS i i iX NOMENCLATURE X 1. 1 2. INTRODUCTION 1 . 1 GENERAL 1 1.2 LITERATURE REVIEW 2 1 .3 SCOPE OF THE PRESENT STUDY 6 1.4 DESCRIPTION OF METHOD 7 LINEAR DIFFRACTION THEORY 11 2.1 INTRODUCTION 11 2.2 DESCRIPTION OF RANDOM WAVES 11 2 . 3 LONG-CRESTED REGULAR WAVES 3. 13 2.3.1 WAVE FORCES 17 2.3.2 BODY MOTIONS 19 2.3.3 NUMERICAL 22 INTEGRATION 2.4 LONG-CRESTED RANDOM WAVES 24 2.5 SHORT-CRESTED RANDOM WAVES 26 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 3.3 PROBABILISTIC PROPERTIES RESULTANT OF COMPONENTS OF THE 33 HORIZONTAL 3.3.1 A SPECIAL CASE 34 38 3.4 PROBABILISTIC PROPERTIES OF RESULTANT MAXIMA i v 39 3.5 P R O B A B I L I T Y DISTRIBUTION R E S U L T A N T MAXIMA 3.6 HUNTINGTON AND 4. OF EXTREMES OF 45 G I L B E R T ' S APPROACH 47 R E S U L T S AND D I S C U S S I O N 4.1 L O A D I N G AND RESPONSE 4.2 P R O B A B I L I S T I C RESPONSE 4.2.1 49 SPECTRA PROPERTIES OF LOADING AND 54 L O A D I N G AND R E S P O N S E 4.2.2 H O R I Z O N T A L RESPONSE 49 COMPONENTS RESULTANTS OF LOADING 55 AND 56 4.3 A D E S I G N PROCEDURE 4.3.1 5. ..58 A WORKED E X A M P L E CONCLUSIONS 61 REFERENCES 63 A P P E N D I X I . GREEN'S APPENDIX I I . 59 FUNCTION 67 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 R E S U L T A N T V ...71 L I S T OF TABLES Table 2.1 page Normalizing f a c t o r of d i r e c t i o n a l spreading 73 function. 4.1 4.2 Load reduction seas for different Response r e d u c t i o n short-crested 4.3 f a c t o r R^. in short-crested s values. f a c t o r R^j i n seas f o r d i f f e r e n t Expected values 74 75 s values. of the extremes of t h e 76 maxima o f l o a d i n g . 4.4 Expected values of t h e e x t r e m e s o f t h e maxima o f r e s p o n s e . vi 77 L I S T OF FIGURES FIGURE page 2.1 S k e t c h o f a u n i - d i r e c t i o n a l wave s p e c t r u m . 2.2 S k e t c h o f a d i r e c t i o n a l wave s p e c t r u m . 78 2.3 D e f i n i t i o n sketch 79 component 2.4 of a f l o a t i n g body • showing 78 motions. D e f i n i t i o n sketch o f t h e i n c i d e n t wave 79 direction. 2.5 D i r e c t i o n a l spreading values D e f i n i t i o n sketch 4.2 Loading transfer short-crested of a f l o a t i n g functions 80 box. i n long- 80 and 81 seas. Response a m p l i t u d e short-crested 4.4 for different o f s. 4.1 4.3 function operators i n long- and 84 seas. U n i - d i r e c t i o n a l wave s p e c t r u m u s e d in 87 computations. 4.5 Loading spectra 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 long- and s h o r t - c r e s t e d 91 seas. 4.7 Expected values of loading and r e s p o n s e short-crested 4.8 o f t h e e x t r e m e o f t h e maxima loading i n l o n g - and seas. Expected values of o f t h e e x t r e m e o f t h e maxima and r e s p o n s e c o m p o n e n t s i n short-crested 94 seas. vii 95 4.9 E x p e c t e d v a l u e s of of horizontal the e x t r e m e s of resultants in the maxima 96 short-crested seas. 4.10 F r e q u e n c y of loading and upcrossing response short-crested 4.11 F r e q u e n c y of resultants 4.12 Comparision of components of in long- 97 and seas. upcrossing of in short-crested horizontal seas. of present method w i t h t h a t H u n t i n g t o n and Gilbert (1979) f o r surface-piercing circular v i i i 97 a cylinder. of 98 ACKNOWLEDGEMENTS In p r e s e n t i n g t h i s thesis, gratitude t o Dr M. de S t . Q. guidance and suggestions t h e s i s work. I a l s o indirectly during thank a l l t h o s e assistantship financial from support the C o u n c i l , Canada, to express Isaacson h e l p e d me i n c o m p l e t i n g Finally, Research I wish for h i s invaluable the e n t i r e who this course have thesis i n t h e form Natural my s i n c e r e Sciences of directly or successfully. a research and E n g i n e e r i n g i s v e r y much a p p r e c i a t e d . ix of the NOMENCLATURE characteristic body length, matrix coefficients, see e q u a t i o n (2.26), matrix coefficients, see e q u a t i o n (2.31), hydrostatic s t i f f n e s s matrix normalizing water factor, depth, directional expected source zero coefficients, spreading function, value, strength distribution upcrossing frequency function, frequency, o f maxima, maximum v a l u e of f or f , x z frequency of upcrossing 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 o r moment gravitational of component, components, constant, Green's f u n c t i o n , incident wave significant height, wave height, moment o r p r o d u c t of inertia, modified Bessel f u n c t i o n of o r d e r Jacobian of t r a n s f o r m a t i o n , wave number, body mass, mass m a t r i x coefficients, zero, n - t h s p e c t r a l moment o f x component, n - t h s p e c t r a l moment o f y component, number o f maxima, number o f f a c e t s , hydrodynamic pressure, probability density function, probability distribution function, probability density function of extreme of maxima, probability of d i s t r i b u t i o n function o f extreme maxima, radius of g y r a t i o n , reduction f a c t o r , see e q u a t i o n spreading i n d e x , see e q u a t i o n waterplane area, equilbrium body wave (2.44), surface, spectrum, waterplane moments, waterplane moments, in-line loading transverse cross (2.51), loading loading loading or response spectrum, or response or response or response spectrum, spectrum, spectrum, time, peak p e r i o d transfer o f wave function operator, xi spectrum, or response amplitude X random (x,y,z) coordinate X maxima m Gaussian extreme y random z resultant B Z G z m z m 1 <k jk in y direction, axes, o f maxima Gaussian o f x, variable of x and y components, of c e n t r e of buoyancy, z coordinate of c e n t r e of gravity, maxima o f z, extreme vector o f maxima of point amplitude mass damping wave o f z, ( £ , T } , $ ) . o f body motion, coefficients, coefficients, Kronecker CcJ direction, z coordinate added X in x o f x, V Z variable delta, angular velocity frequency, potential, P water density, 0 incident 0o principal [u] correlation X correlation wave wave direction, direction, matrix, coefficient xi i between x and y, 1. INTRODUCTION 1 . 1 GENERAL Considerable attention of offshore regions world-wide demand beneath the seabed world's total larger water development gravity in design for can recent hostile of of variety Sea 200 located The - 400 offshore importance, both from comply huge with r e s e r v e s of o i l f o r 20 p e r c e n t meters have of s t r u c t u r a l p r o v i d e one such to development of the environmental conditions i n water of to the years alone account depths a been p a i d energy. need. The platform North in has depths led to and the concepts. Concrete o f about 200 meters s u c h example. The more efficient structures of paramount economic as is well as from safety viewpoints. Environmental d e s i g n and such loading wave l o a d i n g loads. is the i s often Offshore one which considered to l i e i n the d i f f r a c t i o n is Isaacson, then modify 1979), generally the and based which incident contain regime o f wave of wave diffraction i n Sarpkaya diffraction wave t h e o r y assumes t h e waves t o be both energy random and Isaacson, sinusoidal. short-crested propagating 1 over linear long-crested real a loading loading The (multi-directional simultaneously are wave t h e o r y 1981). However, of large field (summarized ( u n i - d i r e c t i o n a l ) and and linear severe wave the computation on component i n o f t h e most structures components (e.g. critical seas are w i t h wave range of 2 directions). tends This to decrease predicted design for loads, directional directional t h e wave obtained improvement in design develop design by in of waves g e n e r a l l y comparison waves. This incorporating appears to procedures. procedures randomness of waves but of loads uni-directional spreading, new spreading be Thus w h i c h not a l s o f o r the in effects of a very is those reduction the it to significant essential only account directional to f o r the spreading waves. 1 .2 LITERATURE REVIEW The major . r e s e a r c h offshore random s t r u c t u r e s has waves. S a r p k a y a and attention A been spreading of to absence the into considered thorough Isaacson has effort given long-crested summary (1981). On the t o the wave l o a d i n g on of regular this other effect is the to loading reliable i n c o r p o r a t e the and On measured directional and basis i n a storm multi-directional emphasized the spreading of importance effects of d i r e c t i o n a l have due wave been spreading on response p r e d i c t i o n s . the velocities little t h e waves. T h i s l a c k o f a t t e n t i o n i s p a r t l y of by directional s p e c t r a . However, more r e c e n t l y s e v e r a l a t t e m p t s made or given hand v e r y of large of of comparisons wave measured with p r e d i c t i o n s assuming waves, F o r r i s t a l l importance waves. of of Lovaas et. accounting (1984) has directionality on particle in turn uni- al. (1978) have for directional a l s o emphasized the loading the and 3 response of offshore When form taking Borgman directional described from a against known of wave for records, cross-spectrum a s e t of computer quite (1972), methods measured data properties various i s use offshore in on a He simulated wave expressions spectra Fourier series t h e method records spectrum He wave checked wave primary design. directional based a the having a a n d i s shown t o favourable. Borgman reviewed adopted account, different data. directional into provided for estimating o f wave theoretical spreading spectrum (1969) spectra method an a r r a y analysis be directional of the d i r e c t i o n a l consideration. for structures. and more of evaluating and o f wave methods recently directional of a n a l y s i s energy. Pinkster models o f wave numerical Ochi wave (1982) have spectra from of the d i r e c t i o n a l (1984) has reviewed elevation the in directional seas. The above studies multi-directional waves considered effects & Thompson wave and linear calculation on a on large of loading random multi-directional their Several offshore seas, and theoretical waves w i t h found directional cylinder. They waves t o f o r both a predictions experimental have Huntington f o r regular functions of authors studied circular theory transfer properties structures. (1979) vertical diffraction between the in isolation. and Huntington wave short-crested comparison and (1976), effects extended the their treat long- favourable for results. uni- These 4 authors studies i n d i c a t e that to d i r e c t i o n a l spreading (1977) s t u d i e d using method wave t h e o r i e s . Dean for the the case force reduction the total force concluded that a linear may occur Dean on near simulated wave and loading nonlinear to account the centre into account, a with of a further using reduction on calculated equation. al. tends (1979) to be also wave i n the l o a d i n g (1982) s t u d i e d loads on He forces somewhat studied o f waves i n t h e t i m e domain structures. Battjes shortcrestedness Morison wave o f time o f t h e e x t r e m e peak theories et. of s h o r t c r e s t e d n e s s a the prediction Shinozuka predicted the G a u s s i a n model and t h e r e b y on a p i l e the numerically d i r e c t i o n a l sea as a f u n c t i o n from d e t e r m i n i s t i c conservative. offshore significant. w h i c h combines l i n e a r e f f e c t i s taken i n a random the b a s i s also as (1979) on effects due i s predicted. Hackley obtained effects loading i n w h i c h 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 When t h i s kinematics wave in (1977) e x t e n d e d t h i s method wave f r e q u e n c y , hurricane. of waves i s q u i t e directional a "hybrid" the r e d u c t i o n the and and r e s p o n s e of the long effects of horizontal structures. Borgman spectra the and Y f a n t i s and c r o s s - s p e c t r a forces on the (1981) d e r i v e d of the jacket b a s e d on t h e M o r i s o n e q u a t i o n wave spectrum. horizontal structures and expressions a f o r the components of i n d i r e c t i o n a l seas specified directional 5 Lambrakos movements of directional series, that a seas the investigated pipeline were i n v o l v i n g 51 lying of reasonable the frequencies not the wave the sea by a and floor. The double Fourier and wave found response are strong direction, but within s t r o n g l y d e p e n d e n t on t h e d e g r e e o f spreading. Teigen (1983) compared t e s t s and computer platform in the r e s u l t s simulated long- considerable and reduction results for short-crested i n the total of p h y s i c a l a energy on l o n g - c r e s t e d responses seas. However, (maxima significantly or the minima) were et.al. (1984) diectional spreading platforms on a b a r g e and c o n c l u d e d directionality Tickell equation compared in irregular and Elwany to analyse vertical o f waves on probabilistic of in models w i t h experimental H u n t i n g t o n and G i l b e r t values (1979) members of t h e the based the to be waves. effects transport that main of found a of of jackup the e f f e c t s o f wave i s small. the h o r i z o n t a l structural preliminary waves the the to those not studied leg found values a f f e c t e d by t h e d i r e c t i o n a l i t y Dallinga He for extreme model tension seas. r e s p o n s e modes i n s h o r t - c r e s t e d s e a s compared with loads a n d 21 d i r e c t i o n s . He loads principal limits on represented wave-induced functions wave (1982) have e x t e n d e d t h e M o r i s o n wave-induced short-crested numerical forces seas. simulations on They and data. (1979) the extremes of i n - l i n e predicted the expected and t r a n s v e r s e f o r c e and 6 moment components on a vertical seas. They then developed value of the resultant components be in-line and i n terms of very good experimental by seas may force agreement or in short-crested which extremes of the spectra and transverse reported a method in short-crested obtained cylinder of the expected force o r moment be c a l c u l a t e d . This can and c r o s s - s p e c t r a moment between of the components. theoretical They predictions results. 1.3 S C O P E OF T H E P R E S E N T STUDY The primary directional wave large offshore this using and aim structures that of a structure. those based offshore methods These to predict t o occur of significance i n design spectra the i s random amplitude because . they spectra during the a r e then of compared t o the relate loads seas. t o any large loading in transfer are of incident and The operators loading operators loads operating the amplitude The extend t h e extreme t o compute seas. study and t o a n d c a n be a p p l i e d shape to and response of long-crested and the response and the response to loading predictions general and short-crested work of a r b i t r a r y shape, of general functions functions the assumption i s quite structure transfer long- on t h e present on are likely life used the effects statistical responses method of great wave responses of interest. The loading probabilistic and response properties have been of t h e components obtained using a of the method 7 described method of has the C l o u g h and been d e v e l o p e d and can be applied A t o compute t h e the all analytical e x t r e m e s of maxima in-line and method transverse is quite situations general requiring o r t n o g o n a l components w i t h Gaussian distributions in design since set limits structures life. The Gilbert and is significant that on loading are likely r e s u l t s are response to occur during compared t o t h o s e the of of and designer offshore their operating Huntington t h e o r e t i c a l development is described 3. Results are" p r e s e n t e d are and 4 i s given part problem and basically the Finally, the and the 2 i n Chapter 4. A brief i s given description i n Appendix probability I. The used in parts. The theory II. above wave responses are of of three the wave uni-directional source method statistics comprises solution regular three-dimensional extends the 5. in Chapters METHOD involves for of used i n Appendix DESCRIPTION OF method in Chapter function derivation The discussion presented Green's detailed Chapter resultant of The the loads the (1979). Conclusions first and of the amplitude can of new the means. T h i s 1.4 in properties zero of (1975). r e s p o n s e components. The statistical two Penzien h o r i z o n t a l r e s u l t a n t s of loading and by of to the method. waves The developed using of using second short-crested extremes diffraction a part random waves. the horizontal probability theory. 8 The solution several authors, Faltinsen Garrison and be with which the been d e s c r i b e d and (1985). assumes for nonlinear fluid The the motion (1972), Standing (1974), method of motion satisfies the surface, free of in various the linearized surface by waves wave s o u r c e s over are the associated The equation, far described by waves velocity the field. seabed The the the unknown scattered" distributions equilbrium be velocity boundary c o n d i t i o n s at and immersed a forced body. p o t e n t i a l components a s s o c i a t e d w i t h forced to equations components Laplace neglects height i s described of by Chow wave terms i s a combination b o u n d a r y c o n d i t i o n and and has Garrison Isaacson The mode potential velocity part i n c i d e n t waves, s c a t t e r e d waves, and to each body first ( 1 9 7 4 ) , Hogben and and small neglected. potential due Michelsen effects sufficiently the including (1978) and viscous to of of surface point of the body. The a integral known are quadrilateral constant equations strengths The of solved discretizing the into at involve f u n c t i o n . These integral the body of centre the of into strengths transforms s e t s of a l g e b r a i c e q u a t i o n s the Once i n t h i s manner source over each element. T h i s surface elements. by Green's elements, with accuracy the obtained three-dimensinal equations be equations each element assumed the with and improves with the velocity potentials the to integral the initially s o l u t i o n d e p e n d s on plane source unknown. discretization increasing are number known, of the 9 required hydrodynamic applying the l i n e a r i z e d coefficients Bernoulli are determined equation p r e s s u r e components and i n t e g r a t i n g to these over obtain by the the structure surface. The above short-crested random along the waves of u n i t order response the trains of applied amplitude obtain to the of a different double to of waves and components a r e t h e n loading transfer with a s p e c i f i e d of f r e q u e n c i e s transfer f u n c t i o n s and waves, frequency. the sea s t a t e i s of component and d i r e c t i o n s . to these transfer short-crested obtained wave waves, d i r e c t i o n a l uni-directional specfied directional frequency. of wave in response direction. and multiplying the In wave s p e c t r u m spectrum Thus t h e loading amplitude spectrum. spreading function and the by wave components f u n c t i o n s and f u n c t i o n s and r e s p o n s e incident wave a range cross-spectra response for o f component o p e r a t o r s as f u n c t i o n s of frequency spectra long-crested t o be c o n c e n t r a t e d series series i s applied l o n g - and o n l y , and t h e t h e o r y a frequencies to o b t a i n the loading of i s taken of s h o r t - c r e s t e d order specified case loading for regular The the and s p a n n i n g theory amplitude extended o p e r a t o r s as f u n c t i o n s of case t o comprise In wave d i r e c t i o n is amplitude In taken to then wave e n e r g y the p r i n c i p a l trains is random waves. waves, regular in approach the operators case i s taken multiplied which of as a by a i s independent 10 Finally, of the the p r o b a b i l i s t i c loading and response resultants are described. response are means. T h i s a Gaussian linear. properties described their by G a u s s i a n d i s t r i b u t i o n s w i t h zero with surface zero elevation also o f t h e e x t r e m e s o f maxima o f t h e and of horizontal resultants for a specified then obtained. spectra This loading procedure and c r o s s - s p e c t r a response as the primary has mean and t h e a n a l y s i s i s in-line their horizontal and The e x p e c t e d v a l u e s transverse of The components o f t h e l o a d i n g i s because the water distribution and o f t h e components and r e s p o n s e components a n d requires t h e moments o f t h e o f components o f t h e input parameters. sea s t a t e a r e loading and 2. LINEAR 2.1 DIFFRACTION THEORY INTRODUCTION In the present described offshore the to chapter study a general directional numerical wave example by G a r r i s o n wave source a n d Chow Garrison Isaacson (1985), which p r o v i d e s long-crested (1978), regular structures of general Hogben wave l o a d s t o both long- linear described in diffraction described Section 2.2 DESCRIPTION The long- wave One energy principal is the (1974) a n d in and s h o r t - c r e s t e d offshore in for regular extension Section waves i s of linear a n d s h o r t - c r e s t e d waves i s OF RANDOM WAVES in wave energy to describing terms o f l o n g - c r e s t e d i s assumed to direction f r e q u e n c y . The o t h e r which Standing 2.4 a n d 2.5 r e s p e c t i v e l y . T h e r e a r e two a p p r o a c h e s waves. extends and responses i s given theory 2.3. t o both i n Sections large shape. diffraction theory and i s a p p l i c a b l e t o any l a r g e The d e s c r i p t i o n o f random waves The is described for F a l t i n s e n & Michelsen waves, random waves. The method on method method, (1972), (1974), 2.2. effects 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 . The three-dimensional method be and concentrated to i s i n terms of i s assumed frequency and d i r e c t i o n . 11 seas be random ocean i n which t h e along distributed short-crested seas t o be d i s t r i b u t e d o v e r the over in both 1 2 Long-crested uni-directional variety been seas are generally described wave s p e c t r u m , as i n d i c a t e d i n F i g u r e of u n i - d i r e c t i o n a l summarized, for by 2.1. A wave s p e c t r a a r e i n use and have example, by Sarpkaya and Isaacson (1981 ) . In the r e a l range of gives rise surface spatial o c e a n e n v i r o n m e n t , waves p r o p a g a t e directions, to a and has a t w o - d i m e n s i o n a l directional frequency and on sketched in spectrum the Figure 2.2. multiplied by a d i r e c t i o n a l sea s t a t e random by assumed process, each in itself significant direction. effect on considered to a of spreading with correspond considered The l o n g are described depends wave both a on the propagation, as random seas wave are spectrum function. stationary, t e r m t h e wave by a conditions to a s e r i e s of such sea s t a t e s , stationary with different peak p e r i o d and t h e p r i n c i p a l structures study. free by wave e l e v a t i o n d e s c r i b e d term v a r i a t i o n offshore the than a one-dimensional uni-directional In t h e l o n g wave h e i g h t , in this which which i s g e n e r a l l y assumed t o be a Gaussian d i s t r i b u t i o n . are seas in Short-crested described ergodic rather direction generally The waves, Short-crested wave a t h e s u p e r p o s i t i o n o f s u c h waves short-crested variation. over of is sea states a distinct wave and i t s problem not 1 3 2.3 LONG-CRESTED The following based is largely based on t h a t on wave h e i g h t to be the i s assumed be that and t h e f l o w i s condition a at regular small The the theory fluid is irrotational. enables the f l u i d velocity potential. enables l i n e a r i z e d applied waves i s The t o be s m a l l . by assumption regular by I s a a c s o n ( 1 9 8 5 ) . assumptions inviscid described for long-crested given irrotationality amplitude to WAVES development incompressible, The REGULAR t h e body surface, The s m a l l boundary free motion wave conditions surface, and f a r field. A angular past in f r e q u e n c y co p r o p a g a t e s i n w a t e r a large, Figure Let system, freely Oxyz d e g r e e s of freedom, and motions parallel pitch and yaw. yaw level. namely Surge, and are rotational of body. interaction indicated Cartesian and z measured sway, heave heave, respectively, about the f l u i d of a r e g u l a r with roll, are traslational and roll, t h e same three t h e p r o b l e m c a n be t r e a t e d by e a c h o f t h e s i x component m o t i o n s the 2.4 motions seven problems: d moving a t an a n g l e 6 surge, sway H and The body o s c i l l a t e s t o t h e x, y , z a x e s a x e s . Due t o l i n e a r i z a t i o n , superposition i n Figure water body a s right-handed w i t h t h e wave t r a i n from t h e s t i l l pitch, a of height of constant depth floating form t h e x a x i s as i n d i c a t e d upward six arbitrary, 2.3. coordinate with a m p l i t u d e wave t r a i n motion as produced o f t h e body a s w e l l wave t r a i n a as with the r e s t r a i n e d 1 4 E a c h component m o t i o n equal to that represented k = the incident = $ k k 4,5,6, $ i k frequency CJ a n d c a n t h u s be waves sought, the flow only a steady of each state for component solution is time dependence o c c u r s as exp(-icot) and t h e boundary v a l u e problem explicitly. p o t e n t i a l i s made up o f components t h e i n c i d e n t waves (subscript (2.1) complex a m p l i t u d e s h e n c e does n o t e n t e r with a f o r k = 1,2,3 and a r o t a t i o n motion and t i s time. Since The with exp(-iut) i s a displacement being harmonic as: a where of is ( s u b s c r i p t o ) , the 7 ) , each of these b e i n g wave h e i g h t , and f o r c e d (subscript 1,...,6), and scattered proportional waves due t o each each associated waves to incident mode of proportional c o r r e s p o n d i n g motion a m p l i t u d e . Thus t h e v e l o c i t y motion to the potential 0 may be w r i t t e n a s : - i i )H <t> = [ ^(<t>o <t>i) + z ^ + £ -iwS^,,] k=1 K K exp(-icjt) where 4>^ , k = 0,1,...,7 i s g e n e r a l l y Each equation, surface. surface of the potentials must The linearized i s g i v e n by: boundary complex. satisfy and t h e boundary c o n d i t i o n s (2.2) the Laplace a t t h e seabed and f r e e condition on the body 15 n where n d e n o t e s - d i s t a n c e normal to itself body surface. V i n the d i r e c t i o n V = Z k=1 n in i n the d i r e c t i o n equilbrium f r o m t h e body (2.3) D surface i s the n of n and of u n i t vector and d i r e c t e d velocity i s given of outward the surface by: - icj$. n, e x p ( - i c j t ) K (2.4) K which n. = n , 1 x' n n = 2 3 = n. = y n - z n , 4 z y' n . = z n - x n , 5 x z' = x n - y n . 1 n , n z' where n , n vector n i n t h e x, y and z Substituting (2.3), we and n are the d i r e c t i o n equations cosines of the normal directions. (2.2), (2.4) and 6 = Z -icj$.n. k=1 exp(-icot) (2.5) into equation obtain: 90 0 UV7 30: 30^ 9n 3n 3n Separating problem (2.5) c y K out the terms c o r r e s p o n d i n g (k=0,7) and e a c h component (k=1,...,6), e q u a t i o n (2.6) K to the diffraction of the r a d i a t i o n (2.6) c a n be decomposed into problem the 16 f o l l o w i n g form: f o r k=1 , .. ., 6 30, (2.7) 3n 30 ( for k=7 9n The 0 i n c i d e n t 0 ^ f s i n h k d ^ = where p o t e n t i a l k i s the 0^ may wave each o v e r p o t e n t i a l 0^(x) e x p r e s s e d a s : 0 (x> at S = where w h i c h £ the G r e e n ' s i s k ( s a t i s f i e s the the r a d i a t i o n due g e n e r a l £ } G ( p o i n t f o r at + i s g i v e n a s : (2.8) y s i n f l ) ] the £. to e q u i l b r i u m i n t e g r a t i o n s t r e n g t h the the f u n c t i o n c o n d i t i o n , and f b a n d t h e o r y , as £ ' i r e p r e s e n t s f u n c t i o n , i t the a S k u n i t p o t e n t i a l r e p r e s e n t e d s o u r c e s known number. to be i s O e x p [ i k ( x c o s 0 wave A c c o r d i n g 0 The a } d c o n d i t i o n . p o t e n t i a l s d i s t r i b u t i o n body s u r f a c e x = of p o i n t S^. Thus the ( x , y , z ) may be ( s o u r c e on s t r e n g t h body s u r f a c e p e r f o r m e d , a n d G ( x , £ ) p o i n t x f u n c t i o n e q u a t i o n , f r e e due s u r f a c e A l t e r n a t i v e t h e i s 2 - 9 ) d i s t r i b u t i o n t h e G r e e n ' s l i n e a r i z e d unknown s g e n e r a l L a p l a c e a p o i n t (£,TJ,$) i s t h e to a o v e r i g the s o u r c e of chosen so t h a t sea b o u n d a r y b o u n d a r y c o n d i t i o n s e x p r e s s i o n s f o r i t 17 are g i v e n , i n Appendix I . I t remains f o r the source functions condition (2.7) f ^ t o be c h o s e n so t h a t t h e body g i v e n by e q u a t i o n together surface with integral (2.7) equation equations is (2.9) strength s u r f a c e boundary satisfied. Equation reduce a s e t of to for f^: D for where x lies condition is integration (2.9), on t h e body applied, i s carried the r i g h t k = 1 ,..., 7 s u r f a c e a t t h e p o i n t where boundary n and is out over measured from x, £ . From e q u a t i o n s hand s i d e of e q u a t i o n the (2.8) and (2.10) i s g i v e n a s : f 2n b k f o r k =1,...,6 k (2.11) sinh(kd) [n sinh(k(z d)) + z + i ( n cos0+n s i n 0 ) c o s h ( k ( z + d ) ) ] e x p [ i k ( x c o s 0 x y for The integral equations source strengths section 2.3.3. 2.3.1 f k + ysinfl)] k=7 (2.10) a r e s o l v e d n u m e r i c a l l y f o r t h e and this solution is described in WAVE FORCES Once a l l the hydrodynamic potentials p r e s s u r e p may <p^ are known, be o b t a i n e d by t h e the 18 linearized Bernoulli equation: (2.12) The components of wave f o r c e and moment a c t i n g on t h e body a r e t h u s g i v e n b y : F. = -icjpj"- <j> n . dS J • b f o r j = 1,... 6 (2.13) f 3 F^, where F ^ y, z directions components be denote the f o r c e potentials the (0 + o into Fg, Fg components associated potentials with decomposed the into acceleration forced and in phase (2.13) may with the with each such force. The components in with may f u r t h e r be phase the with the v e l o c i t y of the total fluid force be e x p r e s s e d a s : (e) 6 = [F\ '+ I (" Mj e 2 k + iwX . )$ ]exp(-iwt) ; k k for The c o e f f i c i e n t s M ^ J a n d X ^ j are t h e moment first c o r r e s p o n d i n g body m o t i o n . T h u s t h e Fj x, The potential components the 0 . 6 i s termed t h e e x c i t i n g associated can denote associated 0 7 ) a n d components forced component F^, in a b o u t t h e x, y , z a x e s . E q u a t i o n decomposed of and components taken damping as are frequency (2.14) j = 1 ,...,6 dependent, r e a l a n d a r e t e r m e d t h e a d d e d masses and coefficients respectively. Substituting 19 equations (2.2) a n d collecting and terms damping (2.14) into equation (2.13) f o r t h e e x c i t i n g f o r c e , added coefficients, We obtain the and masses following expressions: F^ u. X in e ) pw H Sc = = -pRe{ I k j k which jk = (2.15) o j k } = -pcjlm{ I j k (2.16) } (2.17) Re{ } a n d Im{ } d e n o t e t h e r e a l p a r t s and I j J U +07)n.dS 2 k and imaginary i s given a s : 's V j d s { 2 b - 1 8 ) 2.3.2 BODY MOTIONS The in ^ equation o f m o t i o n o f t h e body may be w r i t t e n the f o l l o w i n g form: [ V ( m j k + *jk> - i w X jk + c jk *k ] for where i s t h e mass m a t r i x = j j= * * F and C j k ( 2 1 9 ) 1 ,..., 6 i s the hydrostatic 20 stiffness m m a t r i x . The mass m a t r i x m 0 0 0 0 m 0 -mz 0 0 m 0 0 -m z 0 h mz G 0 in 0 0 0 0 which m i s coordinate are G the mz G 0 0 0 -I -I xy -I I yz mass a n d may 4 = m(r* + z*) 1 5 = m(rj + zg) 1 6 = m(rj) r of the of g r a v i t y the x, y , z a x e s r e s p e c t i v e l y . xy yz zx where ; v , r ^ and r p z of i n e r t i a b J v p b / v p fa yz 6_J body. z Q o f t h e body. is I 4 the , 1^, I a b o u t t h e x, y , z z g axes (2.21) which body's p r o d u c t (2.20) zx be e x p r e s s e d a s : in x -I 5 t h e body's moment o f i n e r t i a 1 0 G 0 X of c e n t r e respectively i s given as: are the r a d i i of g y r a t i o n I „ , I„_ a n d I xy x which a r e given a s : about are the xydv (2.22) yzdv zxdv i s the density o f t h e body and v i s t h e volume 21 of the body. The h y d r o s t a t i c s t i f f n e s s 0 0 0 0 0 0 0 0 0 0 0 0 0 3 3 C 3 4 C 3 4 C 4 4 C 3 5 0 0 0 0 5 5 C ( 2 . 2 3 ) 0 0 0 0 0 0 3 5 C as: 0 0 which "9 ' s 3 3 = = pgS '44 C 2 2 = pgS,! in z C 0 0 C i s given 0 0 in matrix which 3 4 ~ = P is the B - z) + mg(z n - z ) waterplane t ] ( 2 . 2 4 ) r r of c e n t r e of of c e n t r e of g r a v i t y of area and S, = /xdS, = Jx dS 2 The above = -pgs 3 5 S 1 r S , 2 S , n S buoyancy. body. 2 2 S are the moments: S = /dS, n c 2 f z„ i s t h e z c o o r d i n a t e waterplane S S + mg(z i s the z c o o r d i n a t e G 9 , S 2 2 integrals = S 2 = JydS /y dS 2 are taken ( 2 . 2 5 ) over waterplane. 22 2.3.3 NUMERICAL The INTEGRATION submerged surface is discretized finite number o f f a c e t s N. The i n t e g r a l is then a p p l i e d a t each f a c e t of linear algebraic L A..f^ j= 1 - 'J and (2.10) c e n t r e and we o b t a i n a s e t k ) f o r i=1,...,N, k=1,...,7 (2.26) 1 (k) Here equation a equations: = b| k ) 1 into ( k) f. and b: d e n o t e f,,(x_.) and b. (x. ) r e s p e c t i v e l y X j i s the value of x at the c e n t r e of j - t h facet. The c o e f f i c i e n t s A ^ j a r e g i v e n a s : A i j in • " 5 i j which the area equation area + ^ A.. * i ' i ) d S ( 2 - (2.27) can *Si be taken A S j , and t h u s 2 7 ) f u n c t i o n and ASj i s o f t h e j t h f a c e t . When i * j , t h e i n t e g r a n d t o be c o n s t a n t in over the A ^ j i s approximated a s : 8G ^(x.,1.) (2.28) 27T When i = j , 5 ^ facet ( i s the Kronecker d e l t a of the facet A, By /AS..H h = 1 accounts f o r the i n f l u e n c e of i - t h on i t s own c e n t r e and we g e t : = -1 the s o l u t i o n (2.29) of equation (2.29), the source strength 23 may be source evaluated strengths obtained at f ^ are each known, = Z B. . f ^ j= 1 k) J k are given B i i Again over (2.30) *AS G j (*i>V* and thus B^j the c o r r e s p o n d i n g /AST The coefficients <' > 2 integrand i s taken i s approximated as constant (2.32) occurs expression in equation for B ^ may be (2.31), given and A A = — — / v/AST A above and as: (2.33) 47T excludes 1/R 31 as: iL where The ^(x^). AS. A + — - G = 47T term at S When i = j , a s i n g u l a r i t y h be (2.9): AS . = — 1 G(x-,!•) B B may ) when i * j , the ASj potentials the by: 4? = J c e n t r e . Once 3 (k) where <j>^ d e n o t e s t h e v a l u e B^j the by e v a l u a t i n g e q u a t i o n H>\ facet contribution i s given A C S any i 1 from the singular as: dS (2.34) R developement and expressions for different 24 facet shapes (1974). Standing facet and rectangular A are = — given, f o r example, In p a r t i c u l a r , A = A = 2/7F for a 3.525 by Hogben and for a square circular facet. For a facet: {ln[b + /b 1] + + 2 b ln[l (2.35) + •b b where b i s t h e a s p e c t Finally, ratio of t h e f a c e t . (2.15) equations and ( 2 . 1 8 ) c a n be w r i t t e n i n d i s c r e t i z e d form: F^ J = - IpgHk Z • j 1 e ) N 'jk 2.4 u ( 0 ) + * ! readily ) n..AS. J 1 (2.36) 1 (k) * i n j i A S i ( above a p p r o a c h be extended for long-crested to long-crested a p p r o a c h o f Bendat a n d P i e r s o l wave results, obtained at together with the spectra and cross-spectra incident components. The r e q u i r e d cross-spectra denotes components of ) 2 ' 3 7 ) LONG-CRESTED RANDOM WAVES The S^j 7 = regular wave uses s e r i e s o f wave spectrum of v a r i o u s loading derive the and response a r e d e n o t e d by S ^ j ( w ) f o r i , j = 1 , . . . , 6 . Here of response frequencies, and cross-spectrum and to the regular spectra the loading may random waves u s i n g t h e (1966). T h i s a waves the i - t h and j - t h o f l o a d o r r e s p o n s e a n d S^. d e n o t e s t h e s p e c t r u m t h e i - t h component of load or response. These s p e c t r a and 25 cross-spectra as are r e l a t e d to the incident wave s p e c t r u m S(w) follows: S^fu) where T^((o,d) amplitude ratio | T \ (CJ,0) | | T J ( C J , 0 ) | S (CJ) = for a specified loading calculated the i s a loading transfer operator of or response for a sinusoidal angle (2.38) of incident function direction per unit wave t r a i n 8 or a response and i s the wave amplitude a t frequency wave d i r e c t i o n measured u. from 8 is the x axis. From are equation (2.15) the transfer given as: T.(w,0) = - po) /- (0 +0 )n.dS 2 o 7 for and f u n c t i o n s f o r the loads from operators equations (2.15) a n d ( 2 . 1 9 ) , f o r the motions -pcj / 2 Tj(u,0) = j = 1,...,6 s the response of f l o a t i n g (0 + o amplitude body a r e g i v e n a s : 0?)njdS -g k (2.39) (2.40) Z [i 2 W (m j k + / z j k )-i X W : J k + for c j k ] j = 1 ,..., 6 26 2.5 SHORT-CRESTED RANDOM In of short-crested WAVES s e a s , t h e waves p r o p a g a t e d i r e c t i o n s and may be c o n s i d e r e d a s long-crested directions. S(co,8) random trains The d i r e c t i o n a l and can wave s p e c t r u m Sico) S(CJ,0) where D(co,6) wave wave a o v e r a range superposition of propagating in different spectrum be e x p r e s s e d i n t e r m s is of a denoted by uni-directional as: = S(OJ) D(CJ,0) (2.41) i s the d i r e c t i o n a l spreading function such that 7T S Equation D(CJ,<9) 66 (2.42) e n s u r e s t h a t directional in wave spectrum the energy assumed in directional a frequencies, that wave so t h a t i n the spectrum. spectrum is the d i r e c t i o n a l the It i s o f wave same energy at a l l s p r e a d i n g f u n c t i o n can of frequency: <* Did) A common form of based wave the angular d i s t r i b u t i o n taken as independent D(u,0) contained i s equal t o the energy c o n t a i n e d the corresponding u n i - d i r e c t i o n a l often be (2.42) = 1 (2.43) directional on a c o s i n e powered spreading distribution: function Did) i s 27 C(s)cos Die) 2 s |6>-0 I < (0-0 ) o = (2.44) 0-0 I o where 0 O */2 O i s the distribution to ensure principal i s centred. that equation C(s) direction C(s) > w/2 about w h i c h i s a normalizing (2.42) i s s a t i s f i e d the angular factor and chosen i s given = by: (2.45a) 2s J cos 0 17 d0 b ~1t The normalizing range of values alternative be written of i n t e r m s of loading F or long-crested directions = ( C(s) s expression C(s) The factor and f o r the in Table 2.1 for i s sketched in Figure normalizing factor gamma f u n c t i o n r the 2.5. C(s) + spectra S^J(CJ) waves given An may (2.45b) 1 random a as: | r(s+l) s response 0 are is listed due propagating to uncorrelated in different as: it Sj.fu) = / | T ( c j , 0 ) | |T . ( C J , 0 ) | S(co.e) i -it for where T J ( C J , 0 ) response and are the amplitude (2.40) d0 (2.46) 3 loading operators, respectively. If i , j= 1 ,. .. , 6 transfer functions defined in equations the directional or the (2.39) spreading 28 function D(o>,0) i s t a k e n a s i n d e p e n d e n t equation ( 2 . 4 6 ) c a n be w r i t t e n a s : S -(o}) = { J |T. i Since is | | T \ ( u , 0) | D ( 0 ) } t h e i n t e g r a t i o n over convenient function to 6 define i s now independent a directionally directin 0 i t S(CJ), averaged d0 } 2 i; of between the form: = |T .(w) | case 8, of transfer 1 / (2.48) 2 TT i n t h e common the (2.47) i , j = 1 , . . ., 6 enables the r e l a t i o n s h i p S.j(u) In S(CJ then as: T^^(CJ) — written frequency, for T..(w) = { ] | T . ( u , 0 ) | |T.(o),0) |D(0) This of spectra . to • S(u) long-crested the corresponding be (2.49) waves propagating transfer function i n the is given as: T (a;) = {Ti(ure0)T^(u,eQ)} ]/2 The rms i ; j response an values i n long- integration (2.50) of t h e components and s h o r t - c r e s t e d over r e s p e c t i v e l y . The rms a) of values s e a s may be o b t a i n e d equations of o f the l o a d i n g and the (2.38) and components by (2.47) of the 29 l o a d i n g and compared seas. We response to may factor R^j values. Thus oo the I = corresponding r a t i o of these two be long-crested reduction rms TT |T. (w,0) | | T . ( w , e ) | ° ~* D(0)d0}s(cj)dw • (2.51) °° l i | |T.(a>,0 ) | S(w)do) for i , j = 0 o 1 reduction component factor, loads uni-directional Since the assumed t o be in-line and then, and spreading long-crested uncorrelated, transverse directional may be responses waves i n o r d e r when d i r e c t i o n a l the in then characteristics the i\T (u 6 if values may as o rms rms seas response j This short-crested thus d e f i n e a l o a d i n g or a ;{; R?. in a p p l i e d d i r e c t l y to predicted to obtain function 1,..., 6 by assuming coresponding i s taken into values account. random waves f r o m d i r e c t i o n s l o a d i n g and to the principal wave s p e c t r u m response are are induced wave d i r e c t i o n i s symmetric. the even 3. 3.1 PROBABILISTIC PROPERTIES OF LOADING AND INTRODUCTION The spectra and described i n the interest to cross-spectra previous the in a certain unlikely structures. be limit from using of he will on t h e l o a d i n g the enable and statistical which of the themselves spectra methods w h i c h a r e chapter first describes the both and s h o r t - c r e s t e d long- loads wave direction. A method and described turn probabilistic and r e s p o n s e f o r In in-line short-crested and The overall which v a r i e s randomly i s described properties components a s w e l l both direction. i s thus a vector probabilistic random s e a s . and r e s p o n s e s o c c u r the p r i n c i p a l response in life the o f t h e components o f t h e l o a d i n g and designer response and response can of to occur the the operating properties direct needs t o know t h e properties to little chapter. The seas, and r e s p o n s e and r e s p o n s e w h i c h a r e l i k e l y The maximum l o a d i n g predicted this Rather, t o be e x c e e d e d d u r i n g cross-spectra in are storm c o n d i t i o n . T h i s s e t the design are of the l o a d i n g chapter designer. maxima o f t h e l o a d i n g to RESPONSE here loading to provide described the of the h o r i z o n t a l r e s u l t a n t of the a s o f t h e maxima o f t h i s r e s u l t a n t . These in a specified properties or i n magnitude a r e u s e d t o p r e d i c t t h e maximum l o a d i n g occuring transverse storm duration. The o f t h e components o f t h e l o a d i n g in Section 3.2, w h i l e 30 those of and r e s p o n s e probabilistic and r e s p o n s e a r e their horizontal 31 resultants and described in related the Sections probabilistic horizontal of 3.3 properties resultants approach summarized maxima and of are of these 3.4 resultants are respectively. The t h e e x t r e m e s o f maxima o f t h e described g i v e n by H u n t i n g t o n in Section and G i l b e r t 3.5. A (1979) i s i n S e c t i o n 3.6. 3.2 PROBABILISTIC PROPERTIES OF COMPONENTS The component direction, Gaussian water denoted surface "i s parameter spectral density Rayleigh many i s because also distribution, x to the possess a theory i s a narrow-band m o f x. The spectral On e spectrum, the the tail Detailed of level x so t h a t as above entailed in the d i s t r i b u t i o n t h e c a s e s when x p o s s e s s e s a x narrow-band width which probability reduces o t h e r h a n d , a s e —> becomes G a u s s i a n to the high frequency of and f o u r t h 0, o f x. F o r o f t h e component maxima below t h e mean treatments maxima depends on t h e z e r o t h , s e c o n d distribution. density the w h i c h d e p e n d s on t h e s p e c t r a l e of t h e s p e c t r u m o f t h e maxima x probability the to possess a w i t h z e r o mean, a n d a l i n e a r moments o f t h e s p e c t r u m to in t h e l o a d s and r e s p o n s e s . e itself corresponds response assumed have a d i s t r i b u t i o n width parameter the w i t h a z e r o mean. T h i s S i n c e x has a Gaussian themselves or simply as x i s taken elevation distribution to predict loading here distribution Gaussian used of the there to a 1, t h e are as i t , corresponding a wide spectrum. o f t h e maxima f o r spectrum and an 32 arbitrary by spectrum a r e g i v e n Cartwright The which and and L o n g u e t - H i g g i n s single x » m largest 13 to occur a The specified x m o f t h e maxima x m m T, is independent is distribution i n terms o f t h e d i s t r i b u t i o n expression and respectively. duration corresponding resulting Penzien, o r extreme v a l u e 3 in be d e f i n e d The (1956) (1952) i n a number M o f maxima c o n s i d e r e d occurs requirement. may by L o n g u e t - H i g g i n s primary function P(x ) m of given a the as maxima (Clough and 1975): P(TJ) = exp[- M exp(-i? /2)] (3.1) 2 where rj i s d e f i n e d a s x //m^, a n d m m 0 i s the variance of x defined as: m The 0 = J S o (GJ) dcj x probability differentiating Using expected (Clough equation of x number m frequency (3.1) w i t h may p(rj) c a n be o b t a i n e d be by r e s p e c t t o rj. developed and is f o r the given as 1975): = (21n(M)) of function ( 3 . 1 ) , an a p p r o x i m a t e e x p r e s s i o n and P e n z i e n , Efx/v/mu) The density equation value (3.2) x l / 2 + . 5772 (21n ( M ) ) " maxima M may of zero u p c r o s s i n g be o b t a i n e d f 0 1/ 2 (3.3) by m u l t i p l y i n g t h e by t h e s p e c i f i e d duration T, 33 M = f 0 T . One i m p o r t a n t and (3.3) no l o n g e r but now itself the is p o i n t t o be n o t e d depend i n v o l v e the s p e c t r a l on t h e f r e q u e n c y it will be r e a l i z e d number o f maxima i n t h e c a s e smaller than same peak that s i n c e the frequency now be r e l a t i v e l y 3.2.1 defined x in and moments of spectrum. duration the spectrum is shape w i t h t h e of zero upcrossing small. mean f r e q u e n c y of u p c r o s s i n g a t any l e v e l = / xp(x,x)di o denotes (3.4) a derivative random with k are uncorrelated, their function 1 m m 0 0 variables, respect joint t o time. probability and a Since x density i s given as: p(x,x) = where m x is a s ( P a p o u l i s , 1965) : w h i c h x and x a r e G a u s s i a n dot which FREQUENCY OF UPCROSSING OF COMPONENTS The f narrow-band spectral e, e q u a t i o n (3.3) f o r a given f o r an a r b i t r a r y frequency, spectral of narrow-band that of a parameter upcrossing, (1982) e m p h a s i z e s , e x a c t l y t h e same a s f o r t h e c a s e However, width of zero d e p e n d s on t h e z e r o t h and s e c o n d s p e c t r u m o f x. As O c h i will i s t h a t e q u a t i o n s (3.1) exp{-l( i i + §i)} 0 (3.5) 2 2 i s t h e v a r i a n c e o f x, m 2 i s t h e second spectral 34 moment o f x which and m i s a l s o equal are defined 2 t o v a r i a n c e o f k, and m 0 by: CO m = ; w S ( a > ) dco o xx Substituting obtain f As x equation (3.5) following expression = a 2¥ (^) l / 2 obtained t. - value f , w h i c h was 0 l^) ' 1 PROBABILISTIC equation (3.4), x = of we for f : x exp(-xV2m ) by s e t t i n g £ into (3.7a) 0 specific upcrossing 3.3 (3.'6) n n f , the frequency r e q u i r e d i n the of zero foregoing, is 0. (3.7b) 2 PROPERTIES OF THE HORIZONTAL RESULTANT OF COMPONENTS Let x and y be two G a u s s i a n means, a c t i n g i n o r t h o g o n a l plane. Their p(x,y)= joint random v a r i a b l e s w i t h directions probability — exp{/27rm m ( 1-X ) distribution ( 1 2 0 0 in x 2(1-X ) ? ( X x y /m m 0 + 0 horizontal i s given as: m 2 2 the JLl)} m 0 zero 35 -°° < x, y < -1 where and X m are m 0 m 0 and 0 defined 0 variances i n terms of the of spectra x and of y respectively x and y as: (3.9a) = J S o (3.9b) (oj)dco y the y correlation = c //m m 2 0 itself coefficient between x and y and 0 given (3.9c) o of the cross-spectrum of x and y and (3.9d) x Let z be the + / x + y . The 2 2 resultant variable p(z,0) z resultant of may z the probability be and 8 random v a r i a b l e s density developed 8 i s introduced of is as: oo = ; S (co) du o ^ 0 is follows: i s variance c the (3.8) = / S (w)do> o g i v e n as c are 0 X < +1 xx is X m < +00 and is the function as follows: joint obtained x and p(z) An i n t e r m s of p r o b a b i l i t y density a l o n e may a o b t a i n e d by suitable z = of the auxilliary probability t r a n s f o r m a t i o n p r o c e d u r e . The t h e n be y: density p ( x , y ) by p(z) integration of of a z this 36 joint probability density p(z) may distribution e x a m p l e , by In (z,0) X as to Finally, integrated obtain to function P(z). This Bury the defined be density p(z,0). the the procedure probability probability i s described, for (1975). present 0 = tan 1 case, the ( y / x ) so that auxilliary the variable 0 is transformation from (x,y) i s : = ZCOS0 (3.10) y The = zsin0 joint expressed p(x,y), probability i n t e r m s of given p(z,0) = where by is transformation. p(z) By equation probability (3.8), as p(z,0) may density the' determinant view (3.10), |J| density,p(z) probability = J p(z,0) v i r t u e of of function follows: of the the Jacobian transformation = z in may be obtained density with the respect present to by of given case. the by The integrating this 0: d0 equations be (3.11) In probability joint joint function | J | p(x,y) |J| equation density (3.12) (3.10) and (3.11) t h i s may be written 37 as: p ( z ) = / z p ( Z C O S 0 , z s i n f l ) 66 Applying equation (3.8) t o e q u a t i o n out the i n t e g r a t i o n 2ff, an e x p r e s s i o n for (3.13) the r e s u l t a n t with (3.13) a n d t h e n r e s p e c t t o 8, o v e r f o r the p r o b a b i l i t y z may be d e v e l o p e d r z v/moirioO-X ) exp{- 2 the range 0 to density function p(z) as follows: 2 m +m 0 2m m 0 0 2(1-X ) 2 lot- carrying 0 A} } f o r z>0 2(1-X ) 2 p(z) = (3.14) f o r z<0 in which A = /(m - m ) 0 and I 0 + 4X m m /2m m The is carried probability resultant (3.15) 2 0 0 i s the modified Bessel's derivation density 2 0 0 0 function of order zero. This out i n Appendix I I . distribution function z may be o b t a i n e d by i n t e g r a t i n g P(z) of the the probability p(z) a s : P(z) z = ; p(z)dz o Substituting equation (3.16) (3.14) i n t o e q u a t i o n (3.16), we g e t : 38 P(z) where K,, = J K 2 I exp(-K z and K 2 : )l (K z )dz 3 (3.17) a r e c o n s t a n t s w h i c h a r e d e f i n e d as; 3 K, = / m m (l-X ) K 2 = (m 0 K 3 = (3.18a) 2 0 0 + m )/4(1-X )m m (3.18b) 2 0 0 0 A/2(1-X ) (3.18c) 2 Upon c a r r y i n g out t h e i n t e g r a t i o n i n respect obtain f o r z>0 2 0 2 to z, and using a equation integral (3.17) equation with f o r I , we 0 the f l l o w i n g e x p r e s s i o n : . 7r e x p { - ( K + K c o s 0 ) z } 2 1 1 - 27TK, 2 J = for 30 (K +K cos0) 2 P(z) 3 (3.19) for 3.3.1 z^O 3 A SPECIAL When the CASE standard d i r e c t i o n s are equal uncorrelated, z<0 then deviations and a l s o equations the in two (3.14) and two component components (3.19) r e d u c e are 39 to: z / exp(- 2m , p(z) % ) for 2 = (3.20) 1 - exp(P(z) ) for z<0 for z>0 2m? =• (3.21) for where z>0 2m, 2 m is 0 expected, the these variance equations of either z<0 component. correspond As t o the Rayleigh distribution. 3.4 PROBABILISTIC PROPERTIES OF RESULTANT The probability Section which in 3.3, d e n s i t y o f t h e r e s u l t a n t z was d e v e l o p e d but it i s the d i s t r i b u t i o n i s o f more p r a c t i c a l turn to MAXIMA interest, since develop the d i s t r i b u t i o n o f t h e maxima z this the d e r i v a t i o n of the p r o b a b i l i t y of' is It characteristic denotes number m a g n i t u d e . Thus successive convenient, frequencies initially, to distribution define maxima is c the average regardless number o f u p c r o s s i n g s various o f t h e random v a r i a b l e s z ( t ) . o f maxima p e r u n i t t i m e r e g a r d l e s s 1/f m be u s e d will o f e x t r e m e s o f z . We m now c o n s i d e r z^. in per unit period of magnitude, time with of t h e i r between f f the denotes the respect to a 40 certain joint threshold level probability v a l u e s of i s the 0 f Finally, We f now maximum z z < 0 of m the as maxima equation p(z m time and Thus, can which as follows. i s the number of i s the v a l u e f z with of z ( t ) occurs function p(z ) m of by may be at z = z^. when z = 0, information obtained f r e s p e c t to distribution the be are about z. A and the c o n s i d e r i n g the d e n s i t y f u n c t i o n of z, z a n d density z, and the d e f i n e d i n terms ( L i n , 1976): 1 * ° | i/ -oo C joint extremely This of = " — f ) S i n c e the is unit frequency, random p r o c e s s probability m z, where a dot as: which are encountered z simultaneously. this p ( z i n terms of (3.22) c o n s i d e r the p r o b a b i l i t y probability of r e s p e c t to time, i s t h e maximum v a l u e of distribution joint with zero upcrossing zero u p c r o s s i n g s per 0. expressed = J z p ( z , z ) dz — oo z Specific f be d e n s i t y f u n c t i o n of z and denotes a d e r i v a t i v e f z. T h i s may 2 p ( z , z = 0 , z ) dz probability difficult to (3.23) d e n s i t y f u n c t i o n of obtain, an z, z and approximation z to (3.24) i s a p p l i e d : 1 ) = - _L f c d approximation ( z ?_ dz f ) i s v e r y good (3.24) i f t h e component s p e c t r a are 41 n a r r o w - b a n d e d . In spectra, p(z ). this problem to function of The The that z, z use the leads used variables probability density by to This approximate of p(z m r expression i s somewhat order develop cross-spectrum band overestimate of simplifies S m ) on to described 6 are the is derived ) of introduced suitable is taken in obtained joint joint terms The by of joint suitable i s then applied (3.24), for probability density an assumption result as and maxima z . m i n t r a c t a b l e and a to The from equation f o r the used obtained. procedure. the of 3.3. obtained then basis in section is first then the that probability density formula, p(x,y,x,y) to z as is z lines p(x,y,x,y) p(z,z) u A complete wide probability density P( transformation joint J the p(z,z,0,0) p r o b a b i l i t y density r joint obtain of 6 and a density integration. an (3.24) f i n d i n g the distribution probability to equation distribution p(x,y,x,y) y possessing only. auxilliary the x and (3.24) p r o c e e d s a l o n g joint t o an of of of procedure equation case approximation However, t h e m obtain the zero. is This made i n that i s true the for an generally a xy axisymmetric resonable Since structure and in any case is approximation. x, zero means, given as: y, i , y are their p(x,y,x,y)= — ! — joint Gaussian random probability exp{-l[X]T M 1 [X]} variables density with function is (3.25) 42 T where [X] random v a r i a b l e s and i s : [X] = T [u] is the transpose of the v e c t o r c o n t a i n i n g the {x,y,x,y} (3.26) i s a correlation matrix w i t h components given as: Oo = Re {/ S, .(CJ) dcj} f o r i=x,y and j=ic,y o - (3.27) 1 A i s the various determinant spectra of the correlation occuring i n equation matrix n. (3.27) a r e r e l a t e d The as follows: S. • xx = CJ2S , S. . xx' Thus, t h e c o r r e l a t i o n known m = yy The c o r r e l a t i o n x dw, matrix (3.28) yy matrix s p e c t r a l moments = J u> n S ( C J ) o CJ2S M may be e x p r e s s e d o f x and y d e f i n e d a s m' = / w o n S of follows: (w) dco y i n terms (3.29) y i s eventually given i n terms of these 43 moments a s : m in] = 0 0 0 0 m 0 0 0 0 m 0 0 0 0 0 0 2 m (3.30) 2 As a r e s u l t of the assumption interaction coefficients correlation auxilliary tan 1 (y/x) Cartesian matrix as in this reduces variables before. 6 that to =0, matrix a v a r i a b l e s x, y, k, non-diagonal are a l l zero diagonal a n d 6 a r e now The the matrix. introduced transformation and t h e The where between 0 = the y a n d t h e p o l a r v a r i a b l e s z, z, 0, 0* i s : x = ZCOS0 (3.31) y = zsin0 x = z c o s 0 - z0*sin0 (3.32) y = z s i n 0 - z0cos0 The joint probability d e n s i t y p(z,0,z,0*) may be o b t a i n e d i n t e r m s o f p ( x , y , x , y ) by t h e 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| is transformation. |J| = z determinant On t h e b a s i s i n the present 2 The is the joint now Jacobian of (3.31) a n d the (3.32) case. may density the of e q u a t i o n s probability density required probability of function be o b t a i n e d of z and z which by i n t e g r a t i n g p(z,z,0,0) over the joint the complete ranges of 6 and 8: 2TT p(z,z) = J o Combining equations oo f = J (3.34) (3.22) a n d (3.33) we 2TT 0 0 / / z z O — 00 obtain: p ( x , y , x , y ) dz dd d$ 2 (3.35) — OD Substituting density . 0 0 { J | J | p(x,y,x,y)dd}d0 -°° equation (3.25) and i n t e g r a t i n g first for with the joint respect to 6 probability then z we obtain: 2TT f z = J o kA(0)exp{-(cos 0/2m 2 + sin 0/2m )z } 2 o 2 o dd (3.36) where k = z/(27r) Aid) = /m sin 0 2 2 Substituting obtain function 3 / / 2 /m m 0 + m cos 0 equation the following o f maxima: (3.37a) 0 (3.37b) 2 2 (3.36) i n t o e q u a t i o n expression (3.24) we finally f o r the p r o b a b i l i t y density 45 p(z_) J . kA(6>) [ z B ( 0 ) - 1 ] e x p { - B ( 0 ) z / 2 } dd = — 2 r- III c f (3.38) 2 0 where B(0) It = cos 6>/m + sin 0/m 2 should equation that the assumption (3.24) i m p l i e s t h a t d f /dz z g r e a t e r than random p r o c e s s by density t o be expected z ( t ) . The equation probability (3.38), o be m e n t i o n e d h e r e for a l e v e l given (3.39) 2 0 t h a t of latter (3.14). evaluated the expected may be This function evaluated only i s t o be assumption given m f o r the only v a l u e of evaluated p(z ), underlying from the p(z), allows by the equation v a l u e s g r e a t e r than the v a l u e s of t h e p r o c e s s z ( t ) . Finally, the z expression distribution of probability d e n s i t y g i v e n by m may be for obtained equation the by probability integrating (3.24) and the i s thereby evaluated using following r e l a t i o n : P(z) m 3.5 = f 1 - j2 t c (3.40) PROBABILITY DISTRIBUTION OF The single occurs probability largest in 1 a = P distribution or extreme a m p l i t u d e sample (Longuet-Higgins, P,(zJ m EXTREMES OF M U J m 1952, of M function z m P , ( z ) of the m o f t h e maxima z independent Benjamin and RESULTANT MAXIMA m that maxima i s g i v e n Cornell, as 1970): (3.41) 46 where P( ) c o r r e s p o n d s by equation function The ( 3 . 4 0 ) . p, may to the d i s t r i b u t i o n then corresponding be o b t a i n e d o f z^, g i v e n probability here density from: dP (z) /- \ m P' m = — m M ( z ) d z = M P Substituting probability maxima may Now M _ 1 ( z ) p ( z J m m (3.42) r from equations density thus be that function p,(z ) m the particular, resultant E(z Since m ) probability / z z * m m r should dz m value E ( z ) , is o f e x t r e m e o f maxima o f following relation: (3.43) for the probability only for levels the lower than of z limit the expected greater of value density than integration o f t h e random z (t). The calculation integration of E ^ m ) requires i n c o r p o r a t i n g the r e s u l t s depends on is by s p e c i f i e d given m obtained. m obtained a l s o be g r e a t e r process o f z may be m •* • value using function P i ( z m p ( z ) was v a l i d expected f o r t h e extreme of the density values expected p,(z) the expression function the the z may be o b t a i n e d = the ( 3 . 4 0 ) , evaluated. known, v a r i o u s c h a r a c t e r i s t i c In and ( 3 . 3 8 ) both v/m7 obtained a numerical here. E and t h e number o f maxima M. The duration T multiplied by f ^ m ) latter m , the 47 maximum v a l u e o f f with r e s p e c t t o z: M (3.44) Hence t h e r e q u i r e d c o m p u t a t i o n s This is obtained w i t h z, a s g i v e n i n c l u d e t h e d e t e r m i n a t i o n of n u m e r i c a l l y from by e q u a t i o n the v a r i a t i o n of f (3.38). 3.6 HUNTINGTON AND GILBERT'S APPROACH Huntington method and G i l b e r t by resultant which the (1979) have p r e s e n t e d extreme o f two o r t h o g o n a l values of components an alternative maxima may be of the estimated. They s t a t e d t h e p r o b l e m a s f o l l o w s : Given spectra the a vector and level cross-spectrum of probability Gaussian magnitude p will S z process (x(t),y(t)) (co) , S (co) , S xx yy xy = /x +y 2 which 2 n o t be e x c e e d e d d u r i n g They s o l v e d t h i s p r o b l e m by a s s u m i n g the u p c r o s s i n g s a r e independent occur with a Poisson d i s t r i b u t i o n frequency event events. of i n time upcrossing frequency with high T ? that f o r large z, Then t h e u p c r o s s i n g s determined Hence (co), what i s by the the probability mean of no T i s g i v e n by: p(z) Thus t h e f . time with problem of = exp(-f T) (3.45) z reduces upcrossing to f that of They finding provide the a mean lengthy 48 expression for (3.36), which f , analogous depends o n l y but different on t h e s p e c t r a l to equation moments of x and the one y. Their approach developed of here. t h e extreme obtained via z(t). Thus, turn the significantly In t h e p r e v i o u s a p p r o a c h , the distribution the d i s t r i b u t i o n distribution of the used here i s able of the random maxima z . m the o f t h e maxima z ^ , and f i n a l l y to describe process distribution in z(t), of the the the expected v a l u e of Huntington and G i l b e r t ' s (1979) a p p r o a c h the probability of the extremes distribution been o f maxima o f random p r o c e s s z « m to o f any number o f i n d e p e n d e n t maxima has the approach distribution extremes differs involves z . only 4. RESULTS AND 4.1 LOADING In AND RESPONSE the present freely floating DISCUSSION SPECTRA chapter, results are box i n d i c a t e d i n F i g u r e described 4.1 y, z a x e s a r e 33.04 m, (x,y,z) coordinates (0,0,10.62m). considered in present with conditions compute using loading the same t h e x, one as that (1974) and i s c h o s e n provide obtained i s quite transfer in long- functions suitable for and general functions offshore comparisons 2.48) and 4.2 and c a n in the 4.3 RAO'S wave seas. with used to amplitude of general The loading operators in those published by directionally averaged and r e s p o n s e a m p l i t u d e operators short-crested for long-crested and be and r e s p o n s e short-crested ( 1 9 7 4 ) . The transfer functions water t h e box s u r f a c e . The and t h e r e s p o n s e a m p l i t u d e and M i c h e l s e n t o those deep structures s e a s have been compared Figures functions about o f g r a v i t y o f t h e box a r e 48 f a c e t s t o r e p r e s e n t equation, compared i s the 32.92 m r e s p e c t i v e l y and t h e is to (RAO'S) f o r l a r g e long-crested Faltinsen been method used both transfer (see have loading operators shape i n order B results. Results numerical example of g y r a t i o n F a l t i n s e n and M i c h e l s e n study their 32.09 m, of the c e n t r e This by The r a d i i a of s i z e L x B x D = 90 m x 90 m x 40 m where L i s t h e box l e n g t h , beam, and D i s t h e d r a f t . for have been seas. present i n long- 49 seas the loading transfer and s h o r t - c r e s t e d seas, 50 defined i n equations short-crested RAO'S have functions values for show t h e results three One for interesting roll and absent i n the case whereas symmetrical (2.44) w i t h 10 the that be x axis (6 symmetrical they are of f u n c t i o n s and the all spreading spreading the index influence of a s s e s s e d . However, i n seas = 0 the components o f of the are 2, 3) In a l l c a s e s , f e a t u r e of yaw case the f u n c t i o n s (s=1, presented in order the to principal 0*). figures is l o a d i n g and loading in response are long-crested loading i n s h o r t - c r e s t e d seas. This supports the occur in-line direction, and even if t h a t the transverse the in the of made i n S e c t i o n 2.5 present that case statement both so short-crested results. i s along In directional o f waves may spreading sway, using 2,..., t r e n d of t h e wave d i r e c t i o n seas, 1, (2.39). loading transfer equation spreading figures only the and obtained g i v e n by directional the seas, been s t a k i n g on (2.38) the l o a d i n g and to the directional response principal wave wave spectrum is symmetric. For increasing -concentrated in-line in directional of s, w h i c h c o r r e s p o n d s wave s p e c t r u m , components of l o a d i n g and corresponds these in-line other components, to increasing components w i t h hand, the energy sway and roll, the response s h o r t - c r e s t e d seas approach those This the values t o a more results for ( s u r g e and for long-crested energy the pitch) seas. being concentrated increasing s, as e x p e c t e d . a s s o c i a t e d w i t h the in short-crested seas in On transverse decreases 51 with be increasing easily s, a g a i n drawn from t h e Comparisons in long- T^J(CJ) out with force and sway short-crested are exciting the 88% sway and seas w i t h wave are energy and same due the to other wave water, seas. These & Thompson the case hand, t h e r e i s no pitch The wave for in roll in distribution in 59% of long-crested these (1976) and a of in-line in-line r e s u l t s compare of in exciting corresponding energy or as directionality f o l l o w s : For the radiated coefficients. coefficients soley the exciting values moment angular of carried Dean well (1977) surface-piercing cylinder. explained t o an for energy. respectively) total of H u n t i n g t o n w h i c h were o b t a i n e d On pitch in corresponding same 52% be d i s t r i b u t i o n of exciting the functions the moment same t o t a l in short-crested those the the about (surge the of the short-crested circular that 2 in with show seas w i t h a c o s 0 a n g u l a r force components transfer s e a s , w h i c h may and can 4.3. loading 4.2, about with and to Figure seas w i t h seas the 4.2 short-crested long-crested components expected. These o b s e r v a t i o n s Figures between reference in energy as By can the any independent terms of of can Haskind in turn heave be Figure structure waves i n c i d e n t wave t r a i n . in (see e f f e c t on be 4.2c). This heaving related relations, related to Hence t h i s heave the i n c i d e n t wave d i r e c t i o n . to can still damping heave f o r c e problem be be damping these f o r c e can radiation in forces due expressed and thus i s 52 The is yaw moment about 1.2% long-crested yaw (see Figure of the seas with value moment r e m a i n s same w i t h that given The f o r heave reduction definitions of the l o a d i n g the transverse seas, to the i n - l i n e with and the respect out discussed short crested-seas in and 52% and short-crested in However, i n the case of 4.1 short-crested with in respect long-crested f a c t o r f o r yaw i s d e f i n e d i n long-crested seas, seas. operators w h i c h may be c a r r i e d amplitudes in 87% a n d 85% r e s p e c t i v e l y o f t h e long-crested 57% respectively seas. in Table The 4.3, a r e s i m i l a r t o t h o s e f o r seas amplitudes respectively and p i t c h ) components 67% 4.1. factors The s u r g e a n d p i t c h a r e about directional response amplitude e n e r g y . The sway a n d r o l l (surge 60% the to Figure already. values spreading Table factors are defined reduction between reference seas a r e about in a n d p i t c h ) components loading loads total (2.51). t o the p i t c h loading corresponding due t o o t h e r reduction i n l o n g - and s h o r t - c r e s t e d with the by a s i m i l a r argument summarized reduction Comparisons T^J(CJ) are equation (surge i n the However, directional (sway a n d r o l l ) components the loading seas; different i n the loadings to energy. seas forces. functions according t h e p i t c h moment be e x p l a i n e d spreading are of t h e same t o t a l f u n c t i o n s . T h i s may a g a i n as 4.2e) i n s h o r t - c r e s t e d of with in-line same in short-crested of in long-crested the the the seas, in-line o r about components in 53 There amplitude 4.3e) i s no e f f e c t (see F i g u r e 4.3c). is seas. o f wave d i r e c t i o n a l i t y about index s. The The yaw a m p l i t u d e 2.7% o f t h e p i t c h I t r e m a i n s same w i t h absence on t h e heave amplitude different of (see Figure in long-crested values of wave d i r e c t i o n a l spreading effects on t h e heave and yaw a m p l i t u d e s i s as expected. varying a s t h e a d d e d mass c o e f f i c i e n t s , t h e parameters such damping c o e f f i c i e n t s yaw, which incident influence wave The and t h e e x c i t i n g the reduction corresponding study factors and The sketched height 15m = s of spectra definitions to due those to of of the In a. s p e c i f i e d the wave s p e c t r u m i n F i g u r e 4.4, w i t h and spreading factors. wave c o n d i t i o n s a r e now p r e s e n t e d . H and independent due t o o t h e r similar t h e example u n i - d i r e c t i o n a l spectrum are 4.2. are response ISSC s p e c t r u m peak p e r i o d T p present used i s the significant wave = 1 5 s e c . The ISSC i s given as: S(CJ) = a exp(-pya> )/w 9 where a = 488 H|/T , 0 = p Figures i n Table loading reduction load incident responses, i n the responses f u n c t i o n s a r e summarized The l o a d i n g s i n heave direction. reduction response T h i s i s because the 4.5 and 4.6 (4.1) s 1948/T . p present s p e c t r a . T h e s e have been o b t a i n e d the by loading and multiplying response the wave 54 spectral density functions and (equation that transfer i n Figures the loading and a specified transverse and amplitude These figures the s p e c t r a f o r long- f o r the loading amplitude operators 4.2 and 4.3 and a r e d i s c u s s e d AND have already. RESPONSE been obtained, t h e n be u s e d t o p r e d i c t t h e extreme of and r e s p o n s e t h a t are l i k e l y s e a s t a t e . In s h o r t - c r e s t e d occur transfer response response and r e s p o n s e s p e c t r a p r o p e r t i e s may response the PROPERTIES OF LOADING maxima o f t h e l o a d i n g during and seas a r e s i m i l a r t o those 4.2 PROBABILISTIC their 2.39) loading 2.40) a t a l l f r e q u e n c i e s . functions presented corresponding t h e r e l a t i o n s h i p between short-crested Once the (equation operators indicate by both in-line (sway and r o l l ) (surge seas, and t o the p r i n c i p a l to. occur loading pitch) wave direction. Thus, t h e e x t r e m e o f maxima o f t h e r e s u l t a n t o f i n - l i n e transverse components random d i r e c t i o n w i t h contains information of the l o a d i n g about two components direction section, the results properties of the l o a d i n g the horizontal components and r e s p o n s e o c c u r random m a g n i t u d e . T h i s are given for and r e s p o n s e resultant of the l o a d i n g random of In the in a i n the present probabilistic components in-line and r e s p o n s e . the and quantity c h a n c e s o f maxima o c c u r i n g simultaneously. and and and for transverse 55 4.2.1 LOADING AND For the presents maxima RESPONSE COMPONENTS case of the expected of long-crested figure 4.7 v a l u e E ( x ) / / m ^ o f t h e extreme of m components of the loading corresponding normalized with respect to the deviation /m^ of a x, as maxima M. T h i s c o r r e s p o n d s applies equally seas, to and response, standard f u n c t i o n o f t h e number o f simply to equation a l l components of (3.3) and loading and response. Figure 4.8 short-crested of the Figures between results for values E(x )/t/m^ m and v a l u e s of t h e s p r e a d i n g 4.8(b) provide the ( s u r g e and p i t c h ) and t r a n s v e r s e components respectively. the r e s u l t s expected been deviation in and shows e x p e c t e d for different 4.8(a) in-line have seas the corresponding extreme o f maxima a s f u n c t i o n s o f t h e number o f maxima M, the presents for long- values 0 and i n both normalized v/m To p r o v i d e with of the i n - l i n e index results (sway and suitable s. for roll) comparisons short-crested seas, F i g u r e s 4.8(a) and 4.8(b) respect to the standard ( s u r g e and p i t c h ) component long-crested seas. As discussed in l o a d i n g and r e s p o n s e spreading in 4.1, a r e independent t h e heave and of the both statistical l o a d i n g and r e s p o n s e spectra moments o f are l o n g - and s h o r t - c r e s t e d s e a s . Hence p r o p e r t i e s a r e u n a f f e c t e d due yaw directional o f waves. Thus t h e v a r i o u s s p e c t r a l t h e heave and yaw same Section to the their spreading 56 of waves i n s h o r t - c r e s t e d s e a s . Comparisons short-crested expected and in Huntington amplitude and relation maxima loading of Gilbert horizontal presents now (1979) f o r the expected components function loads f o r the s e a s . The expected normalized with in-line circular values Section 4.1. extreme of cylinder. AND RESPONSE p r o p e r t i e s of motions, of i n - l i n e case the Figure 4.9 o f t h e extreme o f m number s = and t r a n s v e r s e 1, again as maxima M i n s h o r t - c r e s t e d in Figure 4.9 have been respect t o the standard d e v i a t i o n V m ^ i n s h o r t - c r e s t e d seas. Figure e q u a l l y to the r e s u l t a n t t o surge as also predicted a v a l u e s E(z )//STQ" component corresponding in-line functions v a l u e s of the and the r e s u l t a n t the and and t r a n s v e r s e components o f f o r the p a r t i c u l a r of have to the p r o b a b i l i s t i c the expected of in RESULTANTS OF LOADING resultant maxima applies operators for a surface-piercing Turning the relation of l o a d i n g t r a n s f e r the i n - l i n e 4.2.2 HORIZONTAL the long- o f t h e l o a d i n g and r e s p o n s e the context response similar for s e a s have shown a s i m i l a r t r a n s v e r s e components and results v a l u e s o f t h e e x t r e m e o f t h e maxima o f discussed of between o f l o a d s and of 4.9 responses a n d sway a n d t o p i t c h and r o l l i n turn. The number o f maxima M d i s c u s s e d i n t h e c o n t e x t the v a r i o u s expected of v a l u e s o f e x t r e m e s may be o b t a i n e d 57 by equation duration. at T i s the of t h e components, so t h a t in equation (3.36). f specified i s a maximum f = f . On t h e o t h e r hand, i n m of the h o r i z o n t a l resultants, f must be m n u m e r i c a l l y from t h e v a r i a t i o n o f f w i t h z a s z obtained Figure 0 m level the case f / f M = f r , where In the case the zero given (3.44), 4.10 presents of components applicable to both respect which is different the frequency of u p c r o s s i n g o f t h e l o a d i n g and r e s p o n s e and l o n g - and s h o r t - c r e s t e d s e a s . because the frequency with 0 to of the upcrossing frequency f of z e r o f o r e a c h component response in long- presents the resultant of the i n - l i n e is This i s normalized upcrossing of upcrossing f / f 0 0 4.11 of the ( s u r g e 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 t h e l o a d i n g a n d r e s p o n s e short-crested seas the p a r t i c u l a r frequency f o f "the l o a d i n g a n d and s h o r t - c r e s t e d s e a s . F i g u r e frequency is for of u p c r o s s i n g f in this in case s = 1. The is normalized case Z with the respect in-line response equally t o the frequency component in (surge or p i t c h ) short-crested to the corresponding of z e r o u p c r o s s i n g f resultant to surge seas. of of Figure loads loading 4.11 and and sway and t o p i t c h 0 of and applies responses and r o l l in turn. C o m p a r i s o n s o f t h e f r e q u e n c i e s of u p c r o s s i n g o f t h e components resultants (Figure 4.10) and i n s h o r t - c r e s t e d seas of the ( F i g u r e 4.11) horizontal show t h a t 58 f x is a maximum a t t h e z e r o exponentially zero at before in-line (1979) , and d e c a y s for a and by H u n t i n g t o n seas. component and = 4.12(b) 1, we present that of The a Huntington surface-piercing has been t a k e n transverse fors a t z/Vnvo" = 0.78. method w i t h short-crested component in-line occurs 4.12(a) of t h e p r e s e n t in the given level Figures Gilbert cylinder 0 hand, f , i s z a n d i n c r e a s e s t o t h e maximum f m = 1.58 and t h i s comparison of u i t d e c a y s e x p o n e n t i a l l y . In f a c t Finally, and f = f m a s z i n c r e a s e s . On t h e o t h e r the zero fm/fo have level, circular v a r i a n c e of t h e t o be t h r e e times that a s s t a t e d i n t h e example Gilbert (1979). Further, the and t r a n s v e r s e components a r e u n c o r r e l a t e d . The comparison i s q u i t e f a v o u r a b l e a n d goes some way t o w a r d s establishing the v a l i d i t y of the present method. 4.3 A DESIGN PROCEDURE Finally, i n order of the present to calculate short-crested procedure 1. method, a d e s i g n design seas takes Linear to illustrate operators in a specified sea applicability i s presented both here i n l o n g - and state. The design the f o l l o w i n g s t e p s : transfer in corresponding function. procedure l o a d s and responses diffraction loading the p r a c t i c a l theory functions both to is used and to c a l c u l a t e the response amplitude l o n g - as w e l l as s h o r t - c r e s t e d seas a specified directional spreading 59 amplitude and operators response Suitable spectra t o o b t a i n the c o r r e s p o n d i n g loading s p e c t r a and c r o s s - s p e c t r a . integration o f t h e v a r i o u s l o a d i n g and and c r o s s - s p e c t r a a r e c a r r i e d r e q u i r e d moments, including response out t o o b t a i n the z e r o t h , f i r s t and the second moments. These spectral moments a r e t h e n expected values various loading horizotal chapter The and t h e f r e q u e n c y and response resultants estimate the of u p c r o s s i n g of the components using the theory number o f maxima M i s o b t a i n e d storm to and their presented i n the 3. maximum The used frequency of by upcrossing f multiplying with the the s p e c i f i e d d u r a t i o n T. various expected components resultant v a l u e s o f e x t r e m e s o f maxima o f t h e o f l o a d i n g and r e s p o n s e in a corresponding specified- to 5), are obtained and t h e i r storm horizontal duration (i.e. t h e number o f maxima o b t a i n e d in step using equations (3.3) and (3.36). 4.3.1 A WORKED EXAMPLE Results are presented box described short-crested are given directional storm in below Section 4.1 s e a s . The r e s u l t s for the p a r t i c u l a r spreading duration is 12 function hours f o r the f r e e l y for both floating long- for short-crested case of a cosine (s=1). and the and seas squared The specified sea state is 60 directional storm duration described 15 spreading by is 12 hours and the t h e ISSC s p e c t r u m w i t h sec, sketched The (S=1). function in Figure specified wave c o n d i t i o n Spreading Step 1 Figures s carried out to give wave s p e c t r u m , to condition loading the then S t e p 6 was c a r r i e d obtain under results out to give the 12 h o u r s extreme storm are given and response expected values components a n d t h e i r shown i n using the the r e s u l t s out to s p e c t r a l moments o f t h e l o a d i n g a n d 4.10 a n d 4.11, t o o b t a i n 4.10 The the r e s u l t s 4.5 a n d 4.6. S t e p 3 was c a r r i e d various hours d u r a t i o n . hours =1 r e s p o n s e s p e c t r a . S t e p 5 was Figures = = 1 5 sec directional the p = 15 m g p shown i n F i g u r e s is = ISSC 4.2 a n d 4.3. S t e p 2 was c a r r i e d specified obtain H state i s thus as f o l l o w s : spectrum wave h e i g h t , index, was wave sea = 15 m and T =12 Uni-directional Peak p e r i o d , T g specified 4.4. Storm d u r a t i o n Significant H The carried out, using number o f maxima i n 12 out u s i n g loading Figures 4.7 and response duration. i n Tables 4.3 a n d 4.4 r e s p e c t i v e l y . These t a b l e s f o r the provide o f t h e e x t r e m e o f t h e maxima o f t h e horizontal resultants. 5. 1. Linear diffraction estimate of short-crested are components of l o a d i n g in values long-crested sea. degree to a i n these quite for 5. cross-spectra a greater (surge and p i t c h ) based The on o f waves, greater the with degree these in these depends a greater of a in-line on the reduction spreading. The o f t h e l o a d i n g and significant t o be z e r o in i n design. seas approach i n long-crested short-crested transverse short-crested of components The are greater o f waves. waves, those seas. seas, the results for long-crested o f wave d i r e c t i o n a l i t y for seas. on t h e heave and r e s p o n s e . The e x p e c t e d v a l u e s components than the (sway a n d r o l l ) o f t h e l o a d i n g degree o f spreading no e f f e c t loading assumption of i n - l i n e components spreading is this of the of the seas a r e l e s s reduction For decreasing There from and and r e s p o n s e . components of or yaw l o a d i n g 6. and significant magnitidues long- spectra response, p r e d i c t e d are both results i s e x p e c t e d t o be q u i t e The t r a n s v e r s e and in offshore The spreading corresponding response large of the l o a d i n g and response of reduction of shape short-crested corresponding been e x t e n d e d t o seas. The i n - l i n e components components 4. the has response arbitrary random response theory and procedure and 3. wave the loading structures 2. CONCLUSIONS (surge o f t h e e x t r e m e o f maxima o f and p i t c h ) 61 of loading in-line and response i n 62 a specified values 7. The b a s e d on response values components in long-crested 9. statistical and response An response method two as 10. of as a t o o l analytical of seas. maxima of loading and t o be zero significant in the yaw unaffected been d e v e l o p e d of the the roll) and heave and are has of the in-line (surge components of can seas be used can first and seas. the loading or or p i t c h ) and loading and described. in a l l situations Gaussian with standard the which the be loading by by horizontal deviations. s e c o n d moments of The with zero The spectra parameters. required large or to measure slender structures in directional underlying of predicted components w h i c h a r e input Experiments are offshore of short-crested requires response roll) quite w h i c h have d i f f e r e n t primary corresponding waves i n s h o r t - c r e s t e d properties orthogonal method of method i s general means and extreme are properties (sway or in the long-crested duration, seas, resultant transverse the components analytical response of of (sway and spreading probabilistic than seas. 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D a l l i n g a , R.P., Aalbers, A.B., V e g t v e r d e n , J.W.W., 1984, D e s i g n a s p e c t s f o r t r a n s p o r t o f Jackup platforms 63 64 on b a r g e , Proc. Offshore Technology Paper No. OTC 4733, p p . 195-202. Conference, Houston, 12. Dean, R.G., 1977, H y b r i d method of computing wave loading, Proc. Offshore Technol ogy Confer e nee, H o u s t o n , Paper No. OTC 3029, p p . 483-492. 13. F a l t i n s e n , O.M. and M i c h e l s e n , F.C., 1974, M o t i o n s o f large structures i n waves a t z e r o F r o u d e number, Proc. Int. Symp. on the Dynami cs of Marine Vehicles and Structures in Waves, University C o l l e g e , London, p p . 97-140. 14. F o r i s t a l l , G.Z., Ward, E.G., C a r d o n e , V . J . and Borgman, L.E. , 1978, D i r e c t i o n a l spectra and k i n e m a t i c s o f s u r f a c e g r a v i t y waves i n T r o p i c a l s t o r m D e l i a , /. Phys. Oceanography, V o l . 8, p p . 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 b o d i e s , /. 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, H y d r o d y n a m i c loading on l a r g e offshore structures: Three-dimensional source distribution methods, I n Numerical Methods in Offshore Engineering, e d s . , O.C. Zienkiewicz, R.W. Lewis, and K.G. S t a g g , J . W i l e y & Sons, E n g l a n d , pp. 97-140. 17. H a c k l e y , M.B., 1979, Wave f o 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 s e a s , Proc. Conf. Behaviour of Offshore St ruct ures, BOSS'79, London, V o l . I , p p . 187-219. 18. Hogben, N. a n d S t a n d i n g , R.G., 1974, Wave l o a d s on l a r g e b o d i e s , 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, p p . 258-277. 19. H u n t i n g t o n , S.W. and Thompson, D.M., 1976, F o r c e s on large cylinder 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, H o u s t o n , Paper No. OTC 2539, V o l . I I , pp. 169-183. 20. H u n t i n g t o n , 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 in short c r e s t e d seas. I n Mechanics of Wave Induced Forces on Cyli nders, e d . T. L . Shaw, P i t m a n , London, PP. 65 636-649. 21. H u n t i n g t o n , S.W. and G i l b e r t , G., 1979, E x t r e m e f o r c e s in short crested seas, Proc. Offshore Technology Conference, Houston, Paper No. OTC 3595, V o l . I l l , pp. 2075-2084. 22. I s a a c s o n , M. de S t . Q., 1979, Wave i n d u c e d f o r c e s i n t h e diffraction r e g i m e , 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 arbitrary shape, Coastal/Ocean Engineering Report, Department of C i v i l Engineering, U n i v e r s i t y of B r i t i s h C o l u m b i a , V a n c o u v e r , C a n a d a . 24. Lambrakos, K.F., 1982, M a r i n e p i p e l i n e d y n a m i c response to waves from directional wave spectra, Ocean Engineering, V o l . 9, No. 4, pp. 385-405. 25. L i n , Y.K., dynamics, Florida. 26. Longuet-Higgins, M.S., 1952, On the statistical distribution of the h e i g h t s of s e a waves, J. Marine Research, V o l . 11, pp. 245-266. '1976, Probabilistic Robert E. K r i e g e r theory of structural P u b l i s h i n g Company, M a l b a r , 27. L o v a a s , J.H., 1984, Hydrodynamic l o a d s and response of marine structures, Proc. Symp. on Description and Modelling of Directional Seas, Paper No. D-1, Technical 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 p r e d i c t i o n o f random seas, Advances V o l . 13, pp. 217-375. 29. 30. Papoulis, stochastic A., 1965, Probability, processes, McGraw H i l l , New and in probabilistic Hydroscienee, random York. variables, P i n k s t e r , J.A., 1984, N u m e r i c a l m o d e l l i n g o f 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, Technical University, 66 Copenhagen, Denmark. 31. S a r p k a y a , T. and I s a a c s o n , M., forces on offshore structures, New Y o r k . 1981, Mechanics of wave Van N o s t r a n d R e i n h o l d , 32. S h i n o z u k a , M., F a n g , S.L.S. and Nishitani, 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. T e i g e n , P:S., 1983, The r e s p o n s e o f TLP i n s h o r t c r e s t e d waves, Proc. Offshore Technology Conference, Houston, P a p e r No. OTC 4642, V o l . I I I . 34. T i c k e l l , R.G., and E l w a n y , 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 in short crested seas, In Mechanics of Wave Induced Forces on Cylinders, e d . T.L. Shaw, P i t m a n , L o n d o n , pp. 561-576. APPENDIX I . GREEN'S FUNCTION The Green's function either be expressed series. The i n t e g r a l G(x,£) = where PV d e n o t e s infinite i n t e r m s o f an i n t e g r a l 0 i C cosh[k($+d)]cosh[k(z+d)]J (kr) 0 0 0 value of i n t e g r a l . ]cosh[k(z+d)]H ( 1 ) 0 (kr) + 4 Z C cos[M_(S+d)]cos[u (z+d)]K (u r) . 4 1 11 ill 111 ill m= l 0 where = [ (x-£) + 2 R = = P t ( y - T ? ) [r +(z-$) ] 2 2 2 ] l / 2 l / 2 [r +(z+2d+$) ] 2 (AI.1) expression i s G(x,£) = -iC cosh[k(S+d) r infinite J (nr)6n 2 the p r i n c i p a l series o r an expreesion i s 1. 1 g,+ 2 PV / F , F - The f o r the three-dimensional case can 2 l / 2 ^ MtanhMd-v 67 (AI.2) 68 F {-Md)cosh^( S+d) ) c o s h ( A i ( z + d ) = C3cp 2 ) cosh(^d) _ - r *~o 2TTU2-K2) (k -v )d+i> 2 2 u +v m 2 _ *~m " r 2 (u^+v )a+v 2 v °= k t a n h ( k d ) and M m = ^4 a r e t h e r e a l p o s i t i v e r o o t s o f t h e e q u a t i o n ju tan(n d) +^ = 0 m taken i nascending J the (AI.3) m order. i s the Bessel's 0 Hankel Bessel's function function functions of thef i r s t of first of f i r s t kind, kind, K 0 i s a l lthese kind, the being H ^ ^ 1 0 i s modified of order zero. The gradient 3G a t x i s also I< '> • H v If » 2r t required a n d i sg i v e n a s : <*•'«> where n (x-$) n (y-r?) + x n r = y (AI.5) 69 T h u s t h e two e x p r e s s i o n s ( A I . 1 ) and (AI.2) a r e g i v e n [n 9G 9 7\C for corresponding + equations as: (y-Tj)+n (x-$) n to (z-$)] f_ x R3 n [n (x-U+n x y (y-T/)+n (z+2d+t) ] 2 R + 2/MF, [ F J o 3 (*ir)n 0 I 3 -F J, (nr)n_]dn 2 r z ikC cosh[k(t+d)][sinh(k(z+d))J (kr)n 0 0 + cosh(k(z+d))J,(kr)n ] (AI.6) r and !§• - cosh(k(z+d))]H( ikcosh(k(t+d))[ - sinh(k(z+d))H ( 1 J 0 1 } (kr)n n r ] z 4 Z M C cos(/i ($+d)) [cos(ju (z+d)) m= 1 m m m m + sin(M (z d))K (M r)n ] + m 0 m z K,( M m r)n r respectively (AI.7) where sinh(/i(z+d)) F 3 = exp(-Md) (AI.8) cosh(Md) Where k r i s l a r g e , t h e s e r i e s f o r m c o n v e r g e s r a p i d l y b e c a u s e 70 of the b e h a v i o u r of K ( j x r ) . Bessel's form c a n function not convenient. function When 0 be K0(MC) used. treatment Isaacson tends to i n f i n i t y In t h i s c a s e , A detailed i s g i v e n by tends kr integral to zero, and the form o f above m e n t i o n e d (1985). the series is more Green's APPENDIX Equation p(z) I I . PROBABILITY DENSITY FUNCTION OF RESULTANT (3.13) i n c h a p t e r = j z p(zcos0,zsin0) Substituting integrating p(z) with z = equation 2 27TK, 3 i s given as: d0 (AII.1) (3.8) i n t o r e s p e c t t o 8 from * fexp[ -z (m cos 0 2 equation and 0 t o 2TT, we g e t : + m sin 0 2 2 o (AII.1) o - 2X y / m m s i n 0 c o s 0 ) o o O /2(1-X )m m ]d0 2 o o (All.2) where K, = / m m ( 1 - X ) 2 0 Applying 0 trigonometrical identities, (All.2) can be expressed as: z = j ^ - p(z) 2TT exp(-K z ) /exp[-z (K cos20 2 2 2 c where K = (m +m )/4(1-X )m m 0 = (m -m )/4(l-X )m m 0 K_ = 2 X / m m / 4 ( 1 - X ) m m 0 K 2 2 c 0 0 0 2 0 0 0 2 0 0 0 71 - K sin20)]d0(AII.3) g 72 Equation p(z) (All.3) z = — — 27TK, c a n be w r i t t e n as: 2 7 r exp(-K z ) /exp{-z A 2 c o s ( 2 0 + 0 ) }d0 2 2 (All.4) o where A = /K 2 + K ' 2 (All.5) and 0 = t a n ( K /K 1 Substituting ) (All.6) X = 0+0 into equation (All.4), we get: 0+47T p(z) = j^- exp(-K z ) p(z) 0 (All.7) c a n be w r i t t e n z ' = rr- exp(-K z ) -jr- 2 2 1 Using p(z) where I = 0 (All.7) ^ (9.6.16) can exp(-K z ) 2 i s the m o d i f i e d 4TT fexp{-z h c o s X ) dX from Abramowitz finally 2 1 as: 2 o 4 7 r identity equation (All.7) 2 2 1 Equation Jfexp(-z A cosX)dX 2 be w r i t t e n and S t e g u n ( 1 9 6 4 ) , as: I (z A) (All.9) 2 0 Bessel's (All.8) function of o r d e r zero. 73 s C(s) 1 2/rr 2 8/37T 3 16/57T 4 1 28/35* 5 256/63* 6 1 024/231 TT 7 2048/4297T 8 32768/6435* 9 65536/12155TT 2621 44/461 89ir 10 Table 2.1 Normalizing factor of functions directional spreading 74 s SURGE SWAY 1 0.880 0.516 2 0.924 3 HEAVE ROLL PITCH YAW 1 .0 0.515 0.888 0.016 0.426 1.0 0.427 0.930 0.014 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 o r f a c t o r d i f f e r e n t s R^.. i n v a l u e s . s h o r t - c r e s t e d s e a s 75 s SURGE SWAY 1 0.874 0.520 2 0.919 3 ROLL PITCH YAW 1 .0 0.567 0.847 0.040 0.427 1 .0 0.456 0.898 0.030 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 M o t i o n reduction for HEAVE f a c t o r R. ^ i n s h o r t - c r e s t e d different s values. seas EXTREME LOADING LONG-CRESTED SHORT-CRESTED Surge 630143 kN 554522 kN Sway 327674 kN Resultant 570506 kN (surge and sway) Heave 282966 kN Roll 7411021 Pitch kN-m Resultant (roll 3807432 kN-m 6549120 kN-m 6772205 kN-m and p i t c h ) 615837 Yaw Table 282966 kN 4.3 Expected values of the extremes loading. kN-m o f t h e maxima EXTREME RESPONSE LONG-CRESTED SHORT-CRESTED Surge 7.13m 6.26 m Sway 3.70 m Resultant 6.44 m (surge and sway) Heave 12.90 m 12.90 Roll 0.433 r a d Pitch 0.766 r a d Resultant (roll 0.646 r a d 0.670 r a d and p i t c h ) yaw Table m 0.12 r a d 4.4 Expected values of the extremes response. o f t h e maxima f/f Figure 0 2.1 S k e t c h o f a u n i - d i r e c t i o n a l Figure 2.2 S k e t c h o f a d i r e c t i o n a l wave wave spectrum. spectrum. 79 i |t,(heove) V / 1 v* Figure 2.3 D e f i n i t i o n Figure 2.4 Definition y £ ( pitch) 5 _' ^ ^—X / W//////P/////A C,Csur .) Q ^ sketch of a floating component m o t i o n s . sketch of the i n c i d e n t « — body wave showing direction. e - 0 Figure 2.5 Directional O (deg.) spreading function values. for different .L. n Figure i i n u n n n 4.1 Definition a t m sketch n /n of a n n floating box s 81 0.2 0.4 —i 0.6 0.8 u> r a d / s e c o (b) EXCITING FORCE IN SWAY 3.5 — — S = 1 s = 2 s = 3 3H 2.5 H rsi EH 2 1.5 —i 0.4 0.2 Figure 4.2 Loading iii 0.8 0.6 rad/sec transfer functions s h o r t - c r e s t e d seas. in long- and 82 5 o (c) EXCITING FORCE I N HEAVE 4.5 (all E 3.5 H 2 •* 3 s) "3 "ro ro 2.5 EH 2H 1.5 1 0.5 0.2 i 0.4 — i 0.8 r— 0.6 1 oi r a d / s e c «> o 0.40 (d) E X C I T I N G MOMENT I N R O L L 0.35 » e i s S = 1 S = 2 — 3 0.30 H 0.25 E- 0.20 H 0.15 ^ 0.10 0.2 0.4 0.6 u> r a d / s e c Figure 4.2 0.8 • (cont.) Loading transfer functions short-crested seas. i n long- and 83 0.8 o (e) EXCITING MOMENT IN PITCH x long-crested s = 1 0.7 0.6 E I 2 ^ 0.5- m in E-> 0.4 0.3 0.2 9 o — X —i— 0.4 0.6 w 0.8 rad/sec 15-| - ( f ) EXCITING MOMENT IN YAW 14- - 13- ( a l l s) 12116 10- ai 9- 2 87- 3 6vo 5432100.2 Figure 0.4 u> r a d / s e c 0.6 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 short-crested seas. —i 0.8 long- and 84 O.H • 0.2 , . 0.4 u) Figure 4.3 Response , , , 0.6 0.8 rad/sec amplitude operators s h o r t - c r e s t e d seas. in long- and 85 ( c ) HEAVE . (all u rad/sec (d) Figure 4.3 s) ROLL ( c o n t . ) Response a m p l i t u d e o p e r a t o r s short-crested seas. i n l o n g - and 86 co I 8.O-1 o (e) x PITCH 7.0-j -long-crested -s = 1 s = 2 s = 3 6.0 g 5.0 H (0 4.0- 3 3.0H in in 2.0 -\ LO H 0.0 -* I 0.2 0.9 —i 0.4 OJ r a d / s e c n o X 0.8 0.6 ( f ) YAW 0.8 (all s) 0.7 O 0.6 \ u -r- 3 E-i o.4-| 0.3 H 0.2 H o.H 0.2 Figure 4.3 • —I 0.4 CJ 1— 0.6 0.8 rad/sec ( c o n t . ) Response a m p l i t u d e o p e r a t o r s short-crested seas. i n l o n g - and 87 co r a d / s e c Figure 4.4 Uni-directional wave computation. spectrum used in 88 oH 0.2 1 • 1 0.4 i 0.6 > 1 0.8 co r a d / s e c o 55-1 co r a d / s e c Figure 4 . 5 Loading spectra in long- and short-crested seas. Figure 4.5 (cont.) Loading short-crested spectra seas. in long- and 90 25 n (e) EXCITING MOMENT IN PITCH o long-crested s = 1 X 20 A S = 2 s = 3 (0 u u 15 a; V) i E I z 10 5 ID 3 0.4 0.2 o o X m 0.8 0.6 CJ r a d / s e c ( f ) EXCITING MOMENT IN YAW 40 ( a l l s) 35 30 VJ \ U cu 25 tn I £ I 20 ^ z 15 H 10 5H 0.2 Figure 4.5 CJ (cont.) rad/sec i 0.6 Loading spectra s h o r t - c r e s t e d seas. "o.8 in long- and 91 0-| 0.2 , , 0.4 , —, 0.6 , , 0.8 co r a d / s e c Figure 4.6 Response s p e c t r a in long- and short-crested seas. 92 o.oo-l . 0.2 1 . 0.4 w Figure 4.6 rad/sec (cont.) Response short-crested 1 0.6 spectra seas. . 1 0.8 in long- and 93 0.45-1 (e) PITCH 0.40 •long-crested s = 1 s = 2 s = 3 0.35 o <u (A I <o 0.30 0.25 3 0.20 H co. 0.15 H 0.10 in in 0.05 H 0.00 0.2 0.4 co r a d / s e c 0.6 0.8 (f) YAW (all 1.75 H S) o (/) 1.50 H 0) I m u ~. 3 1-25 CO H 0.75 H 0.50 —i 0.2 > 0.4 1 — 0.6 0.8 co r a d / s e c Figure 4.6 (cont.) Response spectra s h o r t - c r e s t e d seas. in long- and 94 Figure 4.7 E x p e c t e d v a l u e s of the e x t r e m e o f t h e maxima l o a d i n g and r e s p o n s e i n l o n g - c r e s t e d s e a s . of 95 5.2 IN-LINE COMPONENT 4.8 H 4.4 H e H 3.6 3.2 H 2.8 1 — S = - s -s = 2 = 3 2.4 H 2 5 4 log (M) 1 Q 3-i 2.8- TRANSVERSE COMPONENT 2.62.42.22\ W 1.81.61.41.21- 0.8- -r 4 - 2 log Figure 1 ( J 5 6 (M) 4 . 8 E x p e c t e d v a l u e s o f t h e e x t r e m e o f t h e maxima l o a d i n g and r e s p o n s e i n s h o r t - c r e s t e d s e a s . of 96 F i g u r e 4.9 E x p e c t e d v a l u e s o f t h e e x t r e m e o f t h e maxima 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. of 97 F i g u r e 4.10 loading 1.6 Frequency of u p c r o s s i n g of the components of and r e s p o n s e i n l o n g - and s h o r t - c r e s t e d s e a s . H N Figure 4.11 F r e q u e n c y o f l o a d i n g and 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 response i n s h o r t - c r e s t e d seas. 98 (a) FREQUENCY OF UPCROSSING 0.8 P r e s e n t method H u n t i n g t o n and 0.4- -r Gilbert - 5 4 (b) EXPECTED VALUE o e <N P r e s e n t method H u n t i n g t o n and 3- o i 2 3 4 5 Gilbert 6 iog ( M ) l 0 Figure 4.12 Comparison of present method with t h a t of Huntington and Gilbert (1979) for a surface-piercing circular cylinder.
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Directional wave effects on large offshore structures of arbitrary shape Sinha, Sanjay 1985
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Title | Directional wave effects on large offshore structures of arbitrary shape |
Creator |
Sinha, Sanjay |
Publisher | University of British Columbia |
Date Issued | 1985 |
Description | A numerical method is described to study directional wave effects on large offshore structures of arbitrary shape, based on an extension of linear diffraction wave theory for regular waves. A computer program has been developed to compute loading transfer functions and response amplitude operators and hence the loading and response spectra for both long- and short-crested random waves. Cosine powered directional spreading functions which are independent of frequency have been used to account for the shortcrestedness of waves. Comparisions of the results for long- and short-crested seas show that there is a significant reduction in the loading, and hence in the response, due to shortcrestedness of waves. The probabilistic properties of the components of the loading and response are described. Since the sea surface is assumed to follow a Gaussian distribution, these are also random Gaussian variables. In short-crested waves, the loading and response components occur both in-line and transverse to the principal wave direction. Thus the maximum horizontal loading and response may occur in an arbitrary horizontal direction. An analytical method is developed to describe also the probabilistic properties of the maxima of the components and the maxima of their horizontal resultants. In the present study, results are described for a freely floating box. Comparisons are made with published results and are found to be quite favourable. |
Subject |
Ocean wave power Offshore structures |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-05-28 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0062942 |
URI | http://hdl.handle.net/2429/25139 |
Degree |
Master of Applied Science - MASc |
Program |
Civil Engineering |
Affiliation |
Applied Science, Faculty of Civil Engineering, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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